Heterogeneity and dynamics in individual wages and labour market histories

UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de Fundamentos da An´alise Econ´omica HETEROGENEITY AND DYNAMICS IN INDIVIDUAL WAGES AND LABOUR MARKET HISTORIES by Laura Hospido Quintana Memoria presentada para optar al grado de Doctor por la Universidad de Santiago de Compostela Junio de 2007 —————— Memoria presentada para optar ´o grao de Doutor pola Universidade de Santiago de Compostela Xu˜no de 2007 HETEROGENEITY AND DYNAMICS IN INDIVIDUAL WAGES AND LABOUR MARKET HISTORIES Esta memoria foi presentada o d´ıa23deNovembrode2007noSal´on de Graos da FacultadedeEcon´omicas da Universidade de Santiago de Compostela, ante o Tribunal composto por: Dr. Alberto Meixide Vecino (Presidente). Departamento de Fundamentos da An´alise Econ´omica. Facultade de CC. Econ´omicas, Universidade de Santiago de Compostela. Dr. Wenceslao Gonz´alez Manteiga (Secretario).Departamento de Estat´ıstica eInvestigaci´on Operativa. Facultade de Matem´aticas, Universidade de Santiago de Compostela. Dr. Costas (Konstantinos) Meghir. Departamento de Econom´ıa, University College London. Dr. Jean-Marc Robin. EUREQua - Equipe de Recherche en Economie Quantitative de Paris 1, Universit´edeParis1-Panth´eon Sorbonne. Dr. Enrique Sentana Iv´a˜nez. Profesor de Econometr´ıa de Series Temporais no Centro de Estudios Monetarios y Financieros (CEMFI), Madrid. Calificaci´on: Sobresaliente Cum Laude. To my family Starvation is the characteristic of some people not having enough food to eat. It is not the characteristic of there being not enough food to eat. AMARTYA SEN La vejez empieza cuando se pierde la curiosidad. JOS´ E SARAMAGO A sociedade non pode en xustiza prohibir o exercicio honrado das s´uas facultades ´a metade do x´enero humano. CONCEPCI´ ON ARENAL Acknowledgements / Agradecimientos / Agradecementos El proceso de elaboraci´on de esta Tesis Doctoral ha contado con el apoyo de muchas personas. Estas l´ıneas son para todas ellas. En primer lugar quiero dar las gracias a Manuel Arellano por su excelente labor de supervisi´on, su paciencia y ayuda constantes. Me gustar´ıa expresarle toda mi gratitud por la generosidad y entusiasmo con los que comparte su conocimiento de la econom´ıa en general y la econometr´ıa en particular. A ´el le debo mucho de lo aprendido en estos a˜nos de intenso trabajo y mi inter´es por esta rigurosa manera de medir los datos para tratar de entender el mundo en el que vivimos. En deuxi`eme lieu je tiens `a remercier St´ephane Bonhomme pour ses commentaires et ses conseils pr´ecieux sur pratiquement tous les contenus de la th`ese ainsi que pour son appui infatigable tout au long de ma recherche. Tambi´en quiero expresar mi agradecimiento por sus sugerencias a Samuel Bentolila, Jinyong Hahn, Pedro Mira y Enrique Sentana. My stay at the Economics Department of University College London during the Fall of 2006 was a fantastic experience. I wish to thank Costas Meghir, Nicola Pavoni and Marcos Vera for comments and encouragement. Special thanks are due to Richard Blundell for his suggestions, help and time. En el Centro de Estudios Monetarios y Financieros (CEMFI) he disfrutado de un entorno de investigaci´on y humano muy estimulante. Debo agradecer al CEMFI como instituci´on y a Rafael Repullo, su director, el haberme ofrecido estas excelentes condiciones de trabajo y haberme facilitado la asistencia a cursos y conferencias. Mi reconocimiento va tambi´en para todos los profesores y el personal del CEMFI. Desexo agradecer ´o Departamento de Fundamentos da An´alise Econ´omica da Universidade de Santiago de Compostela ´a mi˜na admisi´on no seu programa de doutoramento. Agradezo a colaboraci´on de Juan Jos´e Ares, Roberto Bande, Melchor Fern´andez, Alvii xiv LIST OF TABLES B.2 AR(1) with fixed effects. Properties of ˆα(T= 16) . . . . . . . . . . . . . . 149 B.3 AR(1) with multiple fixed effects. Properties of ˆαfor α= 0.5 . . . . . . . . 149 B.4 AR(1)-EARCH(1) with fixed effects. Properties of ˆα, ˆ βfor α= 0.5, β = 0.5 150 B.5 AR(1)-EARCH(1) with multiple fixed effects. Properties of ˆα, ˆ βfor α= 0.5 (T= 16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.6 My sample vs. Meghir and Pistaferri (2004) . . . . . . . . . . . . . . . . . 150 B.7 Sample 1. Distribution of observations by year . . . . . . . . . . . . . . . . 151 B.8 Sample 1. Distribution of observations by education . . . . . . . . . . . . . 151 B.9 Sample 1. Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . 152 B.10 Sample 1. αand βestimates . . . . . . . . . . . . . . . . . . . . . . . . . . 152 B.11 Correlations with observed variables . . . . . . . . . . . . . . . . . . . . . . 153 B.12 Meghir and Windmeijer (1999). αand βestimates . . . . . . . . . . . . . 154 B.13 Main Descriptive Statistics of the following Distributions . . . . . . . . . . 154 B.14 Mean elasticities with respect to yit−1at different quantiles . . . . . . . . . 154 B.15 Mean marginal effects with respect to past shocks at different quantiles . . 155 B.16 Sample 2. Distribution of observations by year . . . . . . . . . . . . . . . . 156 B.17 Sample 2. Distribution of observations by education . . . . . . . . . . . . . 156 B.18 Sample 2. Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . 157 B.19 Sample 2. αand βestimates . . . . . . . . . . . . . . . . . . . . . . . . . . 157 B.20 Attrition. αand βestimates . . . . . . . . . . . . . . . . . . . . . . . . . . 157 B.21 Consumption Growth Equation . . . . . . . . . . . . . . . . . . . . . . . . 158 C.1 Distribution of observations by year . . . . . . . . . . . . . . . . . . . . . . 161 C.2 Distribution of individuals by number of observations . . . . . . . . . . . . 161 C.3 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 C.4 Distribution of Individuals over Jobs by Birth Cohort (percent) . . . . . . 162 C.5 Sample Correlations across Individuals . . . . . . . . . . . . . . . . . . . . 162 LIST OF TABLES xv C.6 Sample Wage Annual Growth . . . . . . . . . . . . . . . . . . . . . . . . . 163 C.7 Logwages on number of jobs . . . . . . . . . . . . . . . . . . . . . . . . . . 163 C.8 Autorregresive Model of Earnings . . . . . . . . . . . . . . . . . . . . . . . 164 C.9 Autorregresive Model of Earnings with Job Changes . . . . . . . . . . . . . 164 C.10 Wage Variance Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 D.1 Design 1: Static Probit for different values of T. . . . . . . . . . . . . . . 169 D.2 Design 2 with δ0= 1: Static Probit for different values of T. . . . . . . . 169 D.3 Design 2 with δ0= 0.5: Static Probit for different values of T. . . . . . . . 170 D.4 Design 3: Static Probit for different values of T. . . . . . . . . . . . . . . 170 D.5 Design 4: Dynamic Probit for different values of T. . . . . . . . . . . . . . 171 D.6 Design 5: Dynamic Probit for different values of T. . . . . . . . . . . . . . 172 D.7 Design 6: Static Probit for different values of T. . . . . . . . . . . . . . . 172 D.8 Design 7: Dynamic Probit for different values of T. . . . . . . . . . . . . . 173 List of Figures E.1 Mean hourly wage by gender (pta 1992) . . . . . . . . . . . . . . . . . . . 176 E.2 Sample proportions by gender . . . . . . . . . . . . . . . . . . . . . . . . . 177 E.3 The mean of log wages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 E.4 The variance of log wages . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 E.5 Distribution of Residuals in First Differences . . . . . . . . . . . . . . . . . 180 E.6 Distribution of Standarized Residuals in First Differences . . . . . . . . . . 181 E.7 Kernel densities of logwages and simulated logwages . . . . . . . . . . . . . 182 E.8 Kernel density of individual means . . . . . . . . . . . . . . . . . . . . . . 183 E.9 Kernel density of individual logvariances . . . . . . . . . . . . . . . . . . . 184 E.10 Mean Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 E.11 Mean Marginal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 E.12 Probability of Job Change . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 E.13 Probability of Job Change by Exit Reason . . . . . . . . . . . . . . . . . . 188 xvii Introduction This doctoral thesis considers new models and estimation methods for the analysis of the wage distribution and the labour market histories, from a dynamic perspective. In this analysis I use panel data, that is, repeated observations over time for the same individuals. It is a well-known fact that individual wages evolve over time. In the data, we observe different patterns due to the cyclical aggregate conditions of the economy. We also find heterogeneous wage profiles across groups of individuals according to different observed characteristics: gender, age, education, and many others. Lastly, even among quite homogeneous groups, there exists heterogeneity at the individual level (e.g. ability), unobserved to the econometrician, but which would have an impact over the evolution of earnings along the professional careers of workers. Another well-established fact is that individuals move between different labour market states - they are alternately employed, unemployed or out of the labour force - and, conditioning on working, they transit between different jobs. The way how workers build their own work histories also differs across individuals and over time. Therefore, the starting point of this thesis is the idea that differences in individual labour market histories may help to better understand differences on individual earnings dynamics1. For instance, in the case of gender differentials, we would expect that gender differences in work histories would help to explain a substantial part of the male-female wage gap. In fact, several arguments in the literature have connected job mobility with the 1Throughout, I use the terms earnings and wages indistinctly. 1 2Introduction existence and persistence of the wage gap over time. It has been argued that if women job mobility is more restricted due to variables like husband’s residence and children’s care, then wage gains predicted by search and job-matching models (Burdett, 1978; Jovanovic, 1979) will be smaller (Keith and Williams, 1995). Similar arguments could be extended to heterogeneous individuals in other dimensions, either observed (for age, Topel and Ward (1992) documented a sizeable impact of mobility in earnings of young males) or unobserved, and even, they could be extended to heterogeneity at the individual-job specific level (Postel-Vinay and Robin (2002) stressed the relevance of match effects in a model with within and between jobs wage dynamics). Specifically, this thesis deals with the consideration of different levels of heterogeneity, both observed (chapter 1) and unobserved (chapters 2 and 3), individual (chapter 2) and job-specific (chapter 3), in empirical models for the dynamics of the distribution of earnings and labour market trajectories of workers along their careers. Chapter 4 represents a technical contribution, useful in several economic applications. The first chapter studies gender differences in the wage growth and job mobility of young workers using data from the Spanish section of the European Community Household Panel (1994-2001). First, I build an experience measure that - as opposed to the conventional potential experience variable - considers the existence of discontinuities in the professional career of workers and, second, I analyse job mobility patterns for males and females, separately. From the comparison between the proposed experience measure - accumulated experience - and the one used normally - potential experience - it turns out that wage returns to experience are higher with the more accurate measure and that difference is greater for women than men. This result suggests the existence of a gender wage penalty to interruptions. Regarding job changes, the findings indicate that turnover rates are similar for men and women among young workers. Differences come from the side of some characteristics that are relevant for one of the two groups and not for the Introduction 3 other, specially in case of promotion or in transitions to non-employment. For men, holding a position with responsibility or having a family it turns out to be important when changing job. On the contrary, for women it is relevant the type of journey or the size of the firm. Finally, in addition to the gender penalty to interruptions, I also find that early-career wage growth is greater for men than for women, and this is specially true in years when job changes occur. Similar results have been documented for data from U.S. (Light and Ureta, 1992; Loprest, 1992), Italy (Del Bono and Vuri, 2006) and Finland (Napari, 2007). The second chapter, main body of the thesis, contributes to the earnings dynamics literature modelling not only the unobserved individual heterogeneity and time series properties of the conditional mean of earnings given its past (as in Lillard and Willis, 1978; MaCurdy, 1982; Abowd and Card, 1989, among others), but also allowing for richer sources of heterogeneity and dynamics in the conditional variance (Meghir and Pistaferri, 2004). In particular, I propose a dynamic panel data model with individual effects both in the mean and in a conditional ARCH type variance function. The second contribution consists on shedding some light on how the volatilities of individual wages behave in a period of increasing aggregate inequality as it has happened in the last three decades in the U.S. (Juhn, Murphy, and Pierce, 1993). From a methodological point of view, this chapter applies and extends new estimation methods based on corrected likelihood functions. The use of this newly developed biascorrected likelihood approach makes it possible to reduce the estimation bias to a term of order 1/T2in a fixed-Tcontext. The small sample performance of bias corrected estimators is investigated in a Monte Carlo simulation study. The simulation results show that the bias of the maximum likelihood estimator is substantially corrected for designs that are broadly calibrated to the Panel Study of Income Dynamics. The empirical analysis is conducted on data drawn from the 1968-1993 PSID. I find 4Introduction that it is important to account for individual unobserved heterogeneity and dynamics in the variance, and that the latter is driven by job mobility. I also find that the model explains the non-normality observed in logwage data. In the last part of the empirical analysis, I look at the model’s implications for consumption growth, in a simple precautionary savings framework (Browning and Lusardi, 1996). The main result is that an increase in individual risk induces a reduction in current consumption, and this effect is more important for the less educated people, slightly significant for the graduate and insignificant for the college educated. This result goes in line with the idea that there are more insurance possibilities for these latter (Blundell, Pistaferri y Preston, 2005). Directly connected with chapter 2, the third chapter develops a model that explicitly considers job changes in the dynamics of wages and in the heterogeneity pattern. I propose an error components model designed to more thoroughly describe the impact of job mobility on the dynamics and heterogeneity of individual wages than previous references. In particular, the specification proposed has two different parameters to capture dynamics within jobs and across jobs, and the unobserved heterogeneity shows a richer pattern, as well, composed of both individual and job-specific effects. The potential endogeneity of job mobility in relation to earnings is circumvented using an instrument variable estimation method that controls for those unobserved heterogeneity components. In the empirical application, I use data on work histories drawn from the PSID, which allows the distinction between voluntary and involuntary job-to-job changes. With respect to the main results, I find that - once we control for individual and job-specific effects - the dynamics within jobs is almost zero, whereas across jobs is significant but small. For the dynamics, the distinction between voluntary and involuntary transitions turns out to be irrelevant. However, that distinction matters in the case of the components of the cross-sectional variance. The estimated variance of the job-specific effects represents around one third of the variance for the individual fixed effects. If I consider a subsample Introduction 5 that only includes involuntary job changes, the estimated variance of the heterogeneity across jobs increases up to one half. Finally, the fourth chapter represents a technical contribution related to the computational calculation in practice of bias corrections of the type presented in the second chapter. Chapter 4 considers estimation of non-linear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the Incidental Parameters problem (Neyman and Scott, 1948), that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. The chapter shows how to use an iterated algorithm which simplifies estimation in a nonlinear model with multiple fixed effects and also discusses the application of this computational simplification to bias corrected concentrated likelihoods. Some Monte Carlo experiments illustrate the results. 12 Chapter 1 1.3 Accumulated Experience 1.3.1 Building the Experience Variable The experience measure called potential experience, POTEXit, defined for a given individual iand time tas ageit - years of schoolingi6, implies an unlikely assumption. The assumption implicit under this measure is that individuals work continuously since they finish their studies. This implies, for instance, that two individuals with the same years of education begin to work at the same time and do not suffer any interruption from that moment. In practice, potential experience may approximate quite wrong the capacities acquired by different individuals throughout their professional career. This problem is specially worrisome in the case of female earnings, since women seem more willing to interrupt their careers due to family matters, like care giving activities, both to children and elderly parents. But discontinuities can be also common among young male workers, due to periods of job-shopping or fixed-term contract endings (the second motive is specially relevant among young workers in Spain nowadays10). As an illustration, I calculate the fraction of time that individuals of a subsample of the ECHP spend working in five years11. Table A.2 shows the proportion of males and females that work at least a given number of months over that period. We can see that being continuously employed is not so common, and it is even less likely for women. In fact, the proportion of individuals that during this period work more than the 90 per cent of the time is only 38 per cent for men and 22 per cent for women. The measure of experience proposed as an alternative, accumulated experience, ACCEXit, is built as the sum of a set of variables that measure the fraction of time 10According to data from the Encuesta de poblaci´on activa (EPA), in the last trimester of 2006 the rate of temporality was 33.8 per cent. For the interval of age from 25 to 29 years it rises above 44 per cent. The number is even greater in the youngest segment (less than 25). 11The subsample consists on individuals that has finalized their studies and that are observed from 25 to 29 years. In total, there are 2,184 observations in the sample. 1.3. Accumulated Experience 13 (number of months in a year) that an individual ihas spent working in the last year, Xi(t−1), two years ago, Xi(t−2), three years ago, Xi(t−3),... until the beginning of her professional career, Xi1. That is, ACCEXit =Xi(t−1) +Xi(t−2) +Xi(t−3) +...+Xi1= t−1 X s=1 Xi(t−s).(1.1) With this measure we can easily take into account the existence of interruptions on individual labour market histories. 1.3.2 Accumulated Experience and Potential Experience In order to establish a comparison between the two measures of experience, I estimate by OLS wage equations as proposed by Mincer (1974), separately for men and women12. The dependent variable, yit,is the logarithm of real gross hourly wage. In addition to experience measures, I include a set of explanatory variables common in all the specifications (some have temporal variation, Wit,and other are constant at the individual level, Zi): individual characteristics as birth cohort, dummy variables that indicate if the person is married or if there are children in the household, and educational level; characteristics related to the job position as tenure, type of employer, type of contract, part-time, firm size and type of occupation; and the labour market situation by means of time and region dummies13. Only the birth year dummies are not commonly included in empirical wage equations. I include them because there proves to be a marked decline in wages for 12In the empirical application, a Chow test rejected at the 5% level the null hypothesis of equality of coefficients for men and women. 13A detailed explanation of these variables is offered in Appendix 1.B. 14 Chapter 1 successive birth cohorts. Formally, yit =α0+α1ACCEXit +α2(ACCEXit)2+α′ 3Wit +α′ 4Zi+uit,(1.2) yit =δ0+δ1POTEXit +δ2(POTEXit)2+δ′ 3Wit +δ′ 4Zi+vit,(1.3) where the error terms, uit and vit,are assumed to be white noises. As a first approximation to the relationship between the two measures of experience, if we consider a regression of the potential measure over the accumulated one, it turns out that accumulated experience explains 57 per cent of the variation in potential experience for men and 39 per cent in the case of women. If - in addition - I include the age at which individuals start working, the R2rises to 77 per cent for males and 67 per cent for females. These results indicate that women in the sample delay their entrance to the market and suffer more interruptions throughout their careers. Tables A.3 and A.4 show the estimates for the specification with the accumulated experience and the potential experience measures, respectively. Looking at the two first columns of both tables (specification I), we can see that the sign of the coefficients seems the correct. As we would expect, variables related to human capital, accumulated or potential experience and - mainly - educational level, have positive and significant effect on wages14. Also being older or working as a civil servant, part time, in bigger firms, in positions with responsibility or as a manager or graduate, has positive effect on wages. With the potential experience measure, having higher values of tenure has positive and significant effect for both men and women. Moreover, the presence of children at home has positive effect for males, whereas it is not significantly different from zero for females. The same happens, but with opposite sign, in the case of temporary contract. With the potential experience measure, being 14In the case of experience, I am referring to the joint effect of the linear and the quadratic term. 1.3. Accumulated Experience 15 married is positively related with wages in the case of women. Next, I consider additional variables that try to capture the importance of career interruptions (specification II, columns 3 and 4). In Table A.3, I include a variable that measures the difference between potential and accumulated experience. This difference would be positive due to two reasons: (a) if the individual does not start working just after finishing studies, and (b) if the individual interrupts her career. In order to isolate the second effect, I also include a dummy variable of late incorporation to the job market. The main result is that this difference has significant negative effect for women whereas for men it is not significantly different from zero. In a similar way, in Table A.4, I introduce a variable called interruptions that is equal to 1 if individual iat year thas worked less than 12 months (12 months is the amount assumed by the potential experience measure). Again, we can observe that the coefficient is not significant for males whereas for females has a negative and significant effect. Another interesting feature is the comparison of wage returns to experience between the two experience variables. If these returns are different, the use of one measure or the other would have implications (specially for gender comparisons). The effect over wages of a marginal increase in experience is equal to the partial derivative with respect to experience (equations (1.2) and (1.3), respectively). In particular, α2+ 2α3∗ACCEXit,(1.4) δ2+ 2δ3∗POTEXit.(1.5) Table A.5 shows the effects obtained from the previous estimates. Beginning with specification I, the main result is that with the accumulated experience measure returns to experience are higher and the gender differential decreases (at least for low levels 16 Chapter 1 of experience15). Additionally, the difference with respect to the potential experience measure is greater for women than for men. With regard to specification II, now returns to experience continue to be bigger with the accumulated measure although the distance with respect to the potential one seems to be less. In short, using a measure of experience more accurate than the potential one has consequences. As we have seen, with the proposed measure returns to experience are larger than with the potential experience, and this is specially true for women. I use the accumulated experience measure in the rest of the work. 1.3.3 Checking Endogeneity Since the experience variable proposed considers individual heterogeneity in the accumulation of experience, it might arise an endogeneity problem due to a correlation between this measure and unobservable wage determinants. In such a case, OLS estimates would be inconsistent. Next, I take advantage of the panel structure panel to assess this possibility. I assume that the random error in (2), uit, can be decomposed into a fixed individual component, ηi, and a random component, ǫit, both with zero mean and constant variance. Additionally, I assume - as beforethat the transitory error term, ǫit, is uncorrelated with all the explanatory variables. With regard to the individual component, ηi,a first approximation would be a fixed effects approach. Under this approach, individual heterogeneity could be arbitrarily correlated with the regressors. However, this methodology is very demanding for the sample considered here, since the time variation in the first differences of the explanatory variables is not very large. A second approximation, that represents an intermediate solution between OLS and fixed effects, consists on considering that the η′ iswould be correlated 15In the sample, many of the observed labour market histories do not last more than three or four years. Due to the lack of observations, estimates for far away horizons are based mainly on extrapolations. 1.3. Accumulated Experience 17 with some of the explanatory variables (accumulated experience and some elements of Wit or Zi) but uncorrelated with the rest. This is the efficient instrumental variables method proposed by Hausman and Taylor (1981). A disadvantage of this method is that we have to impose which are the variables that are correlated with the individual effects and which are not. Only if the assumption is correct, the estimator would be consistent. Formally, yit =α0+α1ACCEXit +α2(ACCEXit)2+α′ 31Wit1+α′ 32Wit2+α′ 41Zi1+α′ 42Zi2+ηi+ǫit, (1.6) where the η′ isare correlated with ACCEXit,(ACCEXit)2, Wit2and Zi2, but uncorrelated with Wit1and Zi1.The method takes the variables that are uncorrelated with the η′ isas instruments for the variables that do are correlated. Instruments are: (a) each endogenous variable with time variation (ACCEXit,(ACCEXit)2, Wit2) in deviations from individual means, (b) each exogenous variable with time variation (Wit1) both in deviations from individual means and individual means, and (c) each exogenous variable without time variation (Zi1). Therefore we get identification if we have enough exogenous variables with time variation to use as instruments for the endogenous variables that do not change. Here, Wit2includes marital status, tenure, type of employer, type of contract, part time, firm size and occupation, and Zi2includes education.As exogenous variables, as Booth et al. (2002), I consider that Zi1contains birth cohort and Wit1,regional unemployment rate16 and children. Now (Table A.6), estimated coefficients for accumulated experience are slightly higher than in the previous section (more for women than for men). Those variables still have an effect significantly different from zero, like education, part time or firm size, maintain their positive relation with wages. Nevertheless, now imprecision is greater. This causes that 16Given the reduced regional mobility in the sample, this variable has larger time variation than region dummies. Notice that in the ECHP regional distribution is at NUTS1 level, that is not exactly the same as Autonomous Communities distribution (see variable definition in Appendix 1.B). 18 Chapter 1 variables as type of contract or occupation become insignificant. Given the limited time variation in the sample as well as the fact of having to assume the scheme of correlations between the individual effects and the regressors, I take these estimations with caution17. 1.3.4 Job Changes and Wage Growth Job mobility is closely related to wage dynamics. I have already mentioned that the first years of the professional career concentrate a big amount of the wage growth that individuals accumulate throughout their life and mobility plays an important role on this pattern. In addition, it has been stated that among American and Italian workers this early wage growth is greater for men than for women. Time evolution for gender gross wage gap in the sample appears in Figure E.1. Although, at labour market entry the gender wage gap is hardly perceivable, in a few years this gap become noticeable. In two years, wage growth for men is 15.94 per cent and for women 14.28 per cent. In four years, we have an accumulated growth of 26.09 per cent for males and only a 17.68 for females, whereas in six years those numbers are 44.66 and 29.28, respectively. To analyse to what extent job changes affect wages, I include variables that indicate job changes in the wage equations18. Consider equation (1.3): yit =δ0+δ1POTEXit + second order terms and other variables + vit, 17In fact, in a specification where children variable is not exogenous any more, estimates for experience coefficients are even closer to the ones obtained with OLS. However, imprecision is even greater in this case. 18It would be more appropriate to consider a joint model for wages and job changes since, if there exists correlation between the unobservable determinants of wages and those of job mobility, we would have a sample selection problem. A model with self-selection is out of the scope of this work and constitutes an interesting point for future research. 1.3. Accumulated Experience 19 where POTEXit =AGEit −Y EARS OF SCHOOLINGi−6 =t+ (AGEi0−Y EARS OF SCHOOLINGi−6) = t+ci, with AGEi0denoting age at which individuals enter the sample and cian individual specific constant. If we omit second order terms and other variables (or considering yit as the part of the logwage unexplained by them), we could rewrite (3) as yit =δ0+δ1t+δ1ci+vit, and, in terms of growth rates, ∆yit =δ1+ ∆vit ⇒ˆ δ1=∆y. In other words, we can interpret the estimated coefficient ˆ δ1,corresponding to the potential experience variable, as the mean wage growth. If we add interactions of potential experience and job changes, we would obtain estimates of the mean wage growth with job change. Table A.7 shows that the mean wage growth with job change is 0.044 for men (linear term and interaction term jointly significant at 99 per cent) and only 0.008 for women (jointly significant at 90 per cent). Without job change, mean wage growth is 0.029 and 0.015, respectively. According to these estimates, the early-career wage growth of males is favoured by job mobility, but the same does not happen in the case of females. Next section will address whether there are gender differences in the determinants of job mobility. Those differences may be causing that males and females wages do not grow at the same rate. 20 Chapter 1 1.4 Job mobility In this section, I analyse whether there are gender differences on the mobility patterns of young workers, the probabilities of each type of job change, or the factors that affect these movements. For undertaking this task, I consider all the transitions from the first job of each individual and I make distinctions with respect to the type of change (promotion, layoff and quit) and with respect to the destiny of the change (job to job or job to nonemployment, that is, unemployed or out of the labour force). I assume that an individual experiments at most one job change per year, because the reason for changing is only available for one transition each year19. 1.4.1 Definitions TO STAY: transition job to job, without change of employee nor duties. PROMOTION: transition job to job, without change of employee but in a better position. LAYOFF: transition job to job or job to nonemployment, if the reason for changing is forced by the employer, end of the fixed-term contract or by business closing. QUIT: transition job to job or job to nonemployment, due to other reason (better position, getting married, studies, military service, illness or own inability, taking care of children or older people, ...). 19Notice that this is a quite restrictive assumption, since the temporality rate among the individuals in the sample is more than 40 per cent. In fact, if we counted the cases for which two or more job changes occur in a year, those changes represent around a 30 per cent of the total transitions. In any case, this is a limitation imposed by the own nature of the information available. 1.4. Job mobility 21 1.4.2 Estimation Results I consider a multinomial logit to model the transitions across jobs and from employment to nonemployment20. I am interested in how ceteris paribus changes in the elements of a set of variables affect the probabilities of each type of change. For j= 0,1,2,3,4,5, I define the probabilities P(y=j|x),where now y={to stay, promotion, change job to job through layoff, change job to job through quit, change job to nonemployment through layoff, change job to nonemployment through quit}, and xare personal and job characteristics that have influence on the probability of changing. The multinomial logit assumes a logistic form for those probabilities. I estimate the model by maximum likelihood21. In the sample there are 1470 transitions, 736 for men and 734 for women. The 55 per cent of the transitions in the case of men and the 58 per cent for women imply job changes. In fact, gender differences do not come by the side of the number of job changes they suffer. In Figure E.2 we can see that gender differences arise if we distinguish by type. Transitions job to job through layoff are higher for women and transitions job to nonemployment through quit are higher for men, although they are small in absolute terms. Next, I consider the estimation of the multinomial logit model, separately for men and women. In a first specification I include as explanatory variables age, family (married 20In the empirical analysis of job mobility, discrete choice models and continuous duration models have been used. Both methodologies constitute alternative ways of modelling the same underlying process. Duration models consider the probability that a given job ends in a certain time interval conditioned on having lasted until then. Discrete choice models consider a sequence of successes or failures that are observed in each time interval, understanding by success the job change and failure, to stay in the same position. Royalty (1998) points as a main advantage of the continuous duration models the fact that the results do not depend on the considered time interval (Heckman and Singer, 1984), problem that can arise with the discrete duration and discrete choice models, in which we need to choose a given point in time when the decision takes place. Nevertheless, a model as the multinomial logit, equivalent to a discrete duration model with constant hazard rates, may have a simpler interpretation in terms of how the variables affect the probabilities of each event. For this reason, and also because many variables in the data are measure annually, I use here a discrete framework with annual intervals. 21For a description of the multinomial logit see Wooldridge (2001). 28 Chapter 1 Educational level: dummies defined for the highest degree obtained by the individual (primary education, graduate, college). Tenure: it is constructed from the answers that individuals give when they are asked in which year they began to work with the present employer. It is obtain as the difference between the current year and the year the individual begins to work with the present employer. It is used as a continuous variable, or as dummies of less than 1 year, from 1 to 2 years, and more than 2 years of tenure. Personal characteristics: age (continuous variable), sex (two dummies), marital status (married as opposed to another situation), presence of children at home (dummy variable), family (married and/or with children). Economic Centres: in the ECHP regional division is at NUTS1 level. - Northwest: Galicia, Asturias and Cantabria. - Northeast: Basque Country, Navarra, Rioja, and Aragon. - Madrid. - Centre: Castilla - Leon, Castilla la Mancha, Extremadura. - East: Catalonia, Valencia, The Balearics Islands. - South: Andalusia, Murcia, Ceuta and Melilla. - The Canary Islands. Centres is a dummy variable that takes the value 1 if an individual lives in Madrid, Catalonia or Basque Country, regions with a higher economic activity. Type of contract: temporary or permanent work. Type of journey: part-time or full-time work. Degree of responsibility: dummy variable whether a position involves supervision duties. Firm size: from 1 to 4 employees, from 5 to 49 employees, and 50 or more employees. Occupation: I have grouped the variable occupation in four categories. Since the 1.B. Definition of Variables 29 results can be sensible to this grouping, the groups are establish based on similar requirements on qualification and responsibility. The four categories are: MANAGERS and PROFESSIONALS: Directors of the Public Administrations, Professions associated to college degrees in the fields of pure and natural sciences, health and education, Professionals of the Law, Social sciences and humanities, Technical experts on pure and natural sciences, health and education and Professionals of support in financial, commercial operations and in the administrative management. CLERICAL and SERVICES: Clerical employees and workers of catering and personal services, protection and security, and sales workers AGRICULTURE and MANUFACTURE: Qualified workers in agriculture and fishing, qualified workers on construction, extractive industries, food, drinks and tobacco, wood and textile industry, qualified craftsmen and workers in the metallurgy, operators and fitters of industrial machinery, and transport. UNSKILLED: Non-qualified services and commerce workers, farming and fishing labourers, labourers of mining industry, manufacturing construction, industries and transport. Transitions: categorical variable that takes six values, one for each transition (according to the definitions included in section 1.4.1): staying, promotion, change job to job through layoff, change job to nonemployment through layoff, change job to job through quit, and change job to nonemployment through quit. Time effects: eight dummies, one for each year. Chapter 2 Modelling Heterogeneity and Dynamics in the Volatility of Individual Wages 2.1 Introduction Estimates of individual earnings processes are useful for a variety of purposes, which include testing between different models of the determinants of earnings distributions, building predictive earnings distributions, or calibrating consumption and saving models. While several papers have focused on modelling the heterogeneity and time series properties of the conditional mean of earnings given its past (Lillard and Willis, 1978; MaCurdy, 1982; Abowd and Card, 1982, among others), the modelling of the conditional variance has been mostly neglected. However, in many applications it is important to understand the behavior of higher order moments of the process. This would be the case if we consider an individual trying to forecast her future earnings, in order to guide savings or other decisions. As the individual faces various sorts of uncertainty, we shall be interested in forecasting not only the level of earnings but also its variance. The properties of the variance will be important for describing wage profiles over time and for better understanding what drives fluctuations in them. A richer specification can 31 32 Chapter 2 contribute also to modelling choices in models that use the earnings process as an input. In fact, recent studies stress the relevance of considering a variance that varies with time and across individuals (Meghir and Windmeijer, 1999; Chamberlain and Hirano, 1999; Meghir and Pistaferri, 2004; Albarr´an, 2004; Alvarez and Arellano, 2004). There are also many papers that study the increase in the cross-sectional variance of earnings since the 70’s until today (Juhn, Murphy, and Pierce, 1993, and many others). This growth in the aggregate variance is associated with an increase in inequality. Much less is known about the behaviour of the conditional variance given observed and unobserved individual characteristics. In this chapter, I propose a likelihood-based panel data model for the heterogeneity and dynamics of the conditional mean and the conditional variance of individual wages. In particular, I build a dynamic panel data model with linear individual effects in the mean and multiplicative individual effects in the conditional ARCH type variance function. Therefore, with this model, we can say to what extent the time evolution of the variance is determined by permanent individual heterogeneity or by state dependence effects. This distinction would be crucial, for instance, in the case of precautionary savings as the consumer would behave differently if she knows that the risk she suffers is permanently higher, than if it is only due to a period of higher volatility. It is well known that failure to control for individual unobserved heterogeneity can lead to misleading conclusions. This problem is particularly severe when the unobserved heterogeneity is correlated with explanatory variables. Such a situation arises naturally in a dynamic context. Here, I adopt a fixed effects perspective leaving the distribution for the unobserved heterogeneity completely unrestricted and treating each effect as one different parameter to be estimated. There is an extensive literature on how to estimate linear panel data models with fixed effects (see Chamberlain, 1984, and Arellano and Honor´e, 2001, for references), but 2.1. Introduction 33 there are no general solutions for non-linear cases. If the number of individuals Ngoes to infinity while the number of time periods Tis held fixed, estimation of non-linear models with fixed effects by maximum likelihood suffers from the so-called Incidental Parameters Problem (Neyman and Scott, 1948). This problem arises because the unobserved individual characteristics are replaced by inconsistent sample estimates, which biases estimates of model parameters. In particular, the bias of the maximum likelihood estimator is of order 1/T. The number of periods available for many panel data sets is such that it is not less natural to talk of time-series finite sample bias than of fixed-Tinconsistency or underidentification. In this light, an alternative reaction to the fact that micro panels are short is to ask for approximately unbiased estimators as opposed to estimators with no bias at all. This approach has the potential of overcoming some of the fixed-Tidentification difficulties and the advantage of generality. Methods of estimation of nonlinear fixed effects panel data models with reduced bias properties have been recently developed (see Arellano and Hahn, 2006a, for a review). There are automatic methods based on simulation (Hahn and Newey, 2004), bias correction based on orthogonalization (Cox and Reid, 1987; Lancaster, 2002) and their extensions (Woutersen, 2002; Arellano, 2003), analytical bias correction of estimators (Hahn and Newey, 2004; Hahn and Kuersteiner, 2004), bias correction of the moment equation (Carro, 2006; Fern´andez-Val, 2005) and bias corrections for the concentrated likelihood (DiCiccio and Stern, 1993; Severini, 1998a; Pace and Salvan, 2005). Following this perspective, I build a modified likelihood function for estimation and inference. Using a bias-corrected concentrated likelihood makes it possible to reduce the estimation bias to a term of order 1/T2, without increasing its asymptotic variance.This is very encouraging since the goal is not necessarily to find a consistent estimator for fixed T, but one with a good finite sample performance and a reasonable asymptotic approximation for the samples used in empirical studies. 34 Chapter 2 The contributions of the chapter are twofold. First, I develop several versions of the modified likelihood based on DiCiccio and Stern (1993), Severini (1998a), Pace and Salvan (2005), and Arellano and Hahn (2006b) adapted to a dynamic conditional variance model. Second, I show how this approach works in practice for a specific empirical setting. The small sample performance of bias corrected estimators is investigated in a Monte Carlo study. The simulation results show that the bias of the maximum likelihood estimator is substantially corrected for samples designs that are broadly calibrated to the one used in the empirical application. The empirical analysis is conducted on data drawn from the 1968-1993 Panel Study of Income Dynamics (PSID). These models and data are interesting because we do not know much how the volatilities of individual wages behave in a period of increasing aggregate inequality. I find that it is important to account for individual unobserved heterogeneity and dynamics in the variance, and that the latter is driven by job mobility. I also find that the model explains the non-normality observed in logwage data. In a similar sample for male earnings, Meghir and Pistafferi (2004) find strong evidence of state dependence effects as well as evidence of unobserved heterogeneity in the variances1. They also propose an autoregressive conditional heteroskedasticity panel data model of earnings dynamics, but they separate into a permanent component and a transitory component of earnings shocks. This can be appropriate in models where the author makes assumptions about the nature of the different shocks that affect the income process. Nevertheless, a model with a permanent component I(1) imposes a unit root, i.e., a value for the autoregressive coefficient in the mean equal to one, whereas recent evidence suggests a value for this coefficient around 0.4−0.5 (Alvarez and Arellano, 2004). I use 1Also Lin (2005), using a subsample of the dataset considered by Meghir and Pistaferri (2004), finds statistically significant evidence of ARCH effects in earnings dynamics. He considers an ARCH-fixed effects estimator in a “quasi-lineal” setting. Here we consider a different econometric framework, which let us handle models with multiple effects and estimators without being constrained to the availability of differencing schemes. 2.1. Introduction 35 a single-shock, multiple effects model instead2. This parsimonious specification would be useful for describing and estimating wage distributions (Chamberlain and Hirano, 1999). Meghir and Pistaferri recover orthogonality conditions for the estimation. Their method depends critically on the linear specification for the variance. But even in this case, they recognize that they cannot do fixed-Tconsistent GMM estimation because they have weak instruments. So, they implement a WG-GMM estimator which is only consistent when T→ ∞.What is specially worried about this is that they have a bias of order 1/T as opposed to my estimator which has a bias of order 1/T2.This difference is very important, as we will see in the simulations with respect to the MLE which also has a bias of order 1/T. Even worse, because the WG-GMM estimator use arbitrary moment conditions and thus it is less efficient than MLE. I choose an exponential specification that implies a conditional variance always nonnegative regardless of the parameter values and in addition it has a steady-state distribution (Nelson, 1992). What is interesting is that the estimation method does not depend on the particular specification. It could also use without major changes a quadratic specification as the one of Meghir and Pistaferri. Two limitations of the model are the following: (i) so far there is not adjustment for measurement error; and (ii) there is not explicit treatment of job changes. It is known that measurement error is important for PSID wages and that part of the wages variance may be due to job mobility, so these issues need to be addressed in further work. The rest of the chapter is organised as follows. Section 2.2 presents the model and the likelihood function. Section 2.3 reviews the alternative approaches for correcting the likelihood adapted to this particular setting. Section 2.4 shows some simulations to study the finite sample performance of the bias corrections for the concentrated likelihood. In Section 2.5, I present the empirical application on individual wages and in Section 2.6 the implications of the model for consumption growth. Section 2.7 concludes. 2Meghir and Windmeijer (1999) and Albarr´an (2004) use single-shock models as well but they do not have an application to data. 36 Chapter 2 2.2 The Model and the Likelihood Function 2.2.1 The Model I consider the following model of standardized logwages where iand tindex individuals and time, respectively:3 yit =αyit−1+ηi+eit =αyit−1+ηi+h1/2 it ǫit; (i= 1, ..., N;t= 1, ..., T) with Eyit|yt−1 i,Θi=αyit−1+ηi, and hit =V ar yit|yt−1 i,Θi=Ee2 it|yt−1 i,Θi = exp (ψi+β[|ǫit−1|−E(|ǫit−1|)]) =h(ǫit−1, ψi). In these expressions, {yi0, ..., yiT }N i=1 are the observed data, Θi= (ηi, ψi)′are the individual unobserved fixed effects, eit is an ARCH process, and {ǫit}is an i.i.d. sequence with zero mean and unit variance4. The log formulation implies that hit is always nonnegative, regardless of the parameter values (Nelson, 1992). Finally, I denote the vector of common parameters as Γ = (α, β)′. For the conditional mean, I consider an autoregressive specification where the parameter αmeasures the persistence on the level of wages to shocks, ηidescribe permanent 3In the sequel, for any random variable (or vector of variables) Z,zit denotes observation for individual iat period t, and zt i={zi0, ..., zit}, i.e. the set of observations for individual ifrom the first period to period t. 4In the empirical analysis, I approximate the absolute value function by means of a differentiable function. 2.2. The Model and the Likelihood Function 37 unobserved heterogeneity and eit reflects shocks that individuals receive every period. Departing for the classical AR(1) process, I permit that the variances, given past observations, change over time and across individuals. This particular ARCH type specification allows me to capture two patterns of wage volatility. The first one is individual heterogeneity, ψi: wage volatilities of different individuals can vary differently. For instance, there can be different variances of wages between civil servants and workers of a sales department and also between workers of sales departments in big and small firms. The second one is dynamics, β, reflecting that periods of high volatility in wages tend to be consecutive and vice versa. This feature would be noticeable not only for sellers, but also for funds managers or, in general, for workers that receive bonuses. 2.2.2 The Likelihood Function Under the assumption that ǫit ∼N(0,1),that is, ǫit|yt−1 i,Θi∼N(0,1) then, conditional on the past, the model is normal heteroscedastic yit|yt−1 i,Θi∼N(αyit−1+ηi, hit), and the individual likelihood, conditioned on initial observations, and fixed effects, is f(yi1, ..., yiT |yi0,Θi0) = T Y t=1 f(yit|yit−1,Θi0,Γ0). The log-likelihood for one observation, ℓit, differs from the linear model with normal errors through the time-dependence of the conditional variance. For any individual iand t > 1, we can write ln f(yit|yit−1,Θi,Γ) = ℓit (Γ,Θi)∝ −1 2ln (h(ǫit−1, ψi)) −1 2 (yit −αyit−1−ηi)2 h(ǫit−1, ψi). 44 Chapter 2 2.3.3 Trace Based Approach for Pseudo Likelihoods Since ΥiΓ,b Θi(Γ)= 0,a trimmed version of Υi(Γ) might work. That is, b Υi(Γ) = Ω0+ r X l=1 (Ωl+ Ω′ l), Ωl=1 T−l T X t=l+1 1−l r+ 1∂ℓit Γ,b Θi(Γ) ∂Θ· ∂ℓit−lΓ,b Θi(Γ) ∂Θ′. In principle rcould be chosen as a suitable function of Tto ensure bias reduction but, given that in practice Twill be small and that the procedure is known to fail for values of rat both ends of the admissible range (r= 0 and r=T−1), in practice rwill be chosen equal to 2 or 3. 2.4 Monte Carlo Evidence The practical importance of these bias corrections depends on how much bias is removed for the relatively small Tthat is often relevant in econometric applications. In this section, I provide some simple versions of the model showing that these corrections can remove a large part of the bias even with small T. 2.4.1 The linear dynamic panel model with fixed effects Consistent estimates of αfor fixed Tare available in the AR(1) case. I consider this model first to compare the bias correcting estimators described above with the one proposed by Lancaster (2002). 2.4. Monte Carlo Evidence 45 The model design is yit =αyit−1+ηi+ǫit,(t= 1, ..., T;i= 1, ..., N) ǫit ∼N(0,1), ηi∼N(0,1), yi0∼Nηi (1 −α),1 (1 −α2). The data are generated for T= 8 and 16, N= 500 and 1000,and for α= 0.5,and 0.8.I have simulated samples for different samples sizes because I expect the modified MLE to improve much more with Tthan with N. And I have also simulated samples for different values of αbecause the larger the αthe greater the serial correlation of yit, thus I expect that the estimator performs worse. Here the MLE of αis ˆα≡arg max α 1 N N X i=1 "1 T T X t=1 ℓit (α, ˆηi(α))#=PN i=1 PT t=1 ˜yit ˜yit−1 PN i=1 PT t=1 ˜y2 it−1 , where ˆηi(α)≡arg max η 1 T T X t=1 ℓit (α, η) = ¯yi−α¯yi(−1), and ¯yi=1 T T P t=1 yit, ¯yi(−1) =1 T T P t=1 yit−1, ˜yit =yit −¯yi,˜yit−1=yit−1−¯yi(−1).I also consider several bias-correcting estimators of αthat are obtained by maximization of a modified concentrated log likelihood like eα≡arg max α 1 N N X i=1 ℓmi (α, bηi(α)) . - Determinant Based Approach Using Expected Quantities: in this case, b Hi(α) = −1 T T X t=1 ∂2ℓit (α, bηi(α)) ∂η2= 1, 46 Chapter 2 ¯ Υi(α, ηi;α0, ηi0) = TE0"∂ℓi(α, η) ∂ηi−E∂ℓi(α, η) ∂ηi2yi0# =TV ar0∂ℓi(α, η) ∂ηiyi0=TV ar [¯vi|yi0]. where ¯vi=1 T T P t=1 ∂ℓit(α,η) ∂η ,6and as it is shown in Appendix 2.B ¯ Υi(α, η;α0, η0) = 1 + T(α0−α)2ωT(α0) + 2T(α0−α)ψT(α0), with ωT(α0) = 1 T2h1 + (1 + α0)2+1 + α0+α2 02+...+1 + α0+...+αT−2 02i, ψT(α0) = 1 T21 + α0+...+αT−2 0+1 + α0+...+αT−3 0+...+ 1. Thus ¯ Υi(α, bηi(α) ; ˆα, ˆηi) = 1 + T(ˆα−α)2ωT(ˆα) + 2T(ˆα−α)ψT(ˆα). It follows that in this case ℓmi (α, ˆηi(α) ; ˆα, ˆηi) = −1 2T T X t=1 (yit −αyit−1−ˆηi(α))2−1 2Tln ¯ Υi(α, ˆηi(α) ; ˆα, ˆηi). - Determinant Based Approach Using a Parametric Bootstrap Estimate of V ar [ˆηi(α)]: now ℓmi (α, ˆηi(α)) = −1 2T T X t=1 (yit −αyit−1−ˆηi(α))2−1 2ln d V ar [ˆηi(α)] , where d V ar [ˆηi(α)] = 1 M M X m=1 [ˆηm i(α)−ˆηi(α)]2, and mindexes the simulated samples by parametric bootstrap. 6In what follows I omit the argument in ℓit for notational simplicity. 2.4. Monte Carlo Evidence 47 - Trace Based Approach with Trimming: this approach uses a trimmed version of Υi(α),that is, b Υi(α) = Ω0+ 2 r X l=1 Ωl, where Ωl=1 T−l T X t=l+1 1−l r+ 1∂ℓit ∂ηi·∂ℓit−l ∂ηi , for rsmall. So, ℓmi (α, ˆηi(α)) = −1 2T T X t=1 (yit −αyit−1−ˆηi(α))2−1 2Tb H−1 i(α)b Υi(α). - Following Lancaster (2002), I consider the Approximate Conditional Likelihood: ℓmi (α, ˆηi(α)) = −1 2T T X t=1 (yit −αyit−1−ˆηi(α))2+bT(α) T, where bT(α) = 1 T"T−1 X t=1 T−t tαt#. Before presenting the results I want to mention that I use Individual Block-Bootstrap, that is, fixed-Tlarge-Nnon parametric bootstrap for calculating the standard errors of the estimates. The assumption of independence across individual allows me to draw complete time series for each individual to capture the time series dependence, that is, I draw yi= (yi1, ..., yiT )′Stimes to obtain the simulated data ny(s) i, y(s) i(−1)oS s=1 .For each sample I obtain the corresponding estimates of α0,ˆα(1), ..., ˆα(S),and the empirical distribution as an approximation of the distribution of ˆα.7 Table B.1 reports estimates, based on 300 Monte Carlo runs, for T= 8 and N= 500. I find some differences in the performance between these four types of bias corrections. 7Notice that, opposite to the block bootstrap procedure used in time-series literature (Hall and Horowitz, 1996; Horowitz, 2003), here I do not need to choose any bandwidth. 48 Chapter 2 I have also found that iterating bias correction, in the case of the first two corrections, improves a bit the estimation but for brevity I do not report here these results. An example of that is included in the next subsection. We see in the table that the fixed effects MLE is downward biased by around 35-40 percent in both cases. Bias corrections, except the one proposed by Lancaster (2002) that is consistent for fixed T, all perform better when α= 0.5. In this latter case, the corrections reduce the bias for at least a half. In addition, we can see that the mean of the standard errors estimated by individual block-bootstrap is a good approximation to the Monte Carlo standard deviation. Table B.2 presents estimates for T= 16 and N= 500 8. We can see that for α= 0.5, the MLE has still an important bias, but the modified MLEs are closer to the true value. As before, corrections perform worse when α= 0.8. 2.4.2 The linear dynamic panel model with multiple fixed effects One of the advantages of the bias-correcting estimators with respect to the estimator proposed by Lancaster is their generality. With only a slight modification of the previous expressions it is possible to deal with a more complex model, as an AR(1) model with fixed effects in the conditional mean, ηi,and in the conditional variance, σ2 i. Now the model design is yit =αyit−1+ηi+eit =αyit−1+ηi+σiǫit,(t= 1, ..., T;i= 1, ..., N) ǫit ∼N(0,1), ηi∼N(0,1), ψi= log σ2 i∼N(−3.0,0.8) , yi0∼Nηi (1 −α),σ2 i (1 −α2). The data are generated for T= 8 and 16, N= 500,and for α= 0.5.I denote as 8I do not report here the results for N= 1000, because increasing the number of individuals from N= 500 to N= 1000 has little effect on the magnitude of the estimated bias (much less effect that increasing T). 2.4. Monte Carlo Evidence 49 Θi= (ηi, σ2 i)′the vector of fixed effects. The MLE of αis ˆα≡arg max α 1 N N X i=1 "1 T T X t=1 ℓit α, b Θi(α)# = arg max α 1 N N X i=1 "−1 2ln ˆσ2 i(α)−1 2T T X t=1 (yit −αyit−1−ˆηi(α))2 ˆσ2 i(α)#, where b Θi(α) =   ˆηi(α) ˆσ2 i(α)  =   ¯yi−α¯yi(−1) 1 T T P t=1 (yit −αyit−1−(¯yi−α¯xi))2  , and ¯yi=1 T T P t=1 yit, ¯yi(−1) =1 T T P t=1 yit−1, ˜yit =yit −¯yi,˜yit−1=yit−1−¯yi(−1).Again, I consider several bias-correcting estimators of αthat are obtained by maximization of a modified concentrated log likelihood like eα≡arg max α 1 N N X i=1 ℓmi α, b Θi(α). - Determinant Based Approach Using Expected Quantities: now Hi(α) = −1 T T X t=1    ∂2ℓit ∂η2 ∂2ℓit ∂η∂σ2 ∂2ℓit ∂σ2∂η ∂2ℓit ∂(σ2)2    =1 T T X t=1    1 σ2 i (yit−αyit−1−ηi) σ4 i (yit−αyit−1−ηi) σ4 i(yit−αyit−1−ηi)2 σ6 i−1 2σ4 i   , and ¯ Υi(α, Θi;α0,Θi0) =TE0∂ℓi(α, Θi) ∂Θi−E∂ℓi(α, Θi) ∂Θi∂ℓi(α, Θi) ∂Θ′ i−E∂ℓi(α, Θi) ∂Θ′ iyi0. 50 Chapter 2 Now, I obtain ¯ Υiα, b Θi(α) ; ˆα, b Θias a mean of {Υm i(α)}M m=1 calculated in data simulated as nQT t=1 fyit|yi0,ˆα, b ΘioN i=1. That is, ¯ Υiα, b Θi(α) ; ˆα, b Θi=1 M M X m=1 Υm i(α), where Υm i(α) = 1 T T X t=1 T X s=1 ("∂ℓit ∂Θi− 1 T T X r=1 ∂ℓir ∂Θi!#·"∂ℓis ∂Θ′ i− 1 T T X r=1 ∂ℓir ∂Θ′ i!#), and ∂ℓit ∂Θi =   ∂ℓit ∂η ∂ℓit ∂σ2   =   (yit−αyit−1−ηi) σ2 i (yit−αyit−1−ηi)2−σ2 i 2σ4 i   . This leads to ℓmi α, b Θi(α) ; ˆα, b Θ=1 T T X t=1 ℓit α, b Θi(α)+1 2Tln det b Hi(α) −1 2Tln det ¯ Υiα, b Θi(α) ; ˆα, b Θi. - Determinant Based Approach Using a Bootstrap Estimate of V ar hb Θi(α)i: this approach is based on using the bootstrap estimate d V ar hb Θi(α)i=1 M M X m=1 hb Θm i(α)−b Θi(α)ihb Θm i(α)−b Θi(α)i′, which leads to ℓmi α, b Θi(α)=1 T T X t=1 ℓit α, b Θi(α)−1 2ln det b Hi(α)d V ar hb Θi(α)i. - Trace Based Approach with Trimming: this approach uses a trimmed version of Υi(α), 2.4. Monte Carlo Evidence 51 that is, b Υi(α) = Ω0+ r X l=1 (Ωl+ Ω′ l), where Ωl=1 T−l T X t=l+1 1−l r+ 1∂ℓit ∂Θi·∂ℓit−l ∂Θ′ i , for rsmall. So, ℓmi α, b Θi(α)=1 T T X t=1 ℓit α, b Θi(α)−1 2Tb H−1 i(α)b Υi(α). Table B.3 reports estimates for T= 8 and 16, and N= 500.We see in the table that the fixed effects MLE is downward biased in both cases. Here we can see that iterating bias correction improves substantially the estimation. In fact, bias corrections reduce the bias for at least a half and this bias practically disappears when I iterate the corrections. 2.4.3 The AR(1)-EARCH(1) panel model with fixed effects Now the model design is yit =αyit−1+eit =αyit−1+h1/2 it ǫit,(t= 1, ..., T;i= 1, ..., N) hit = exp ψi+βqǫ2 it−1+ Λ −p2/π=h(ǫit−1, ψi), ǫit ∼N(0,1), ψi∼N(−3.0,0.8) . where Λ is a small positive number used to approximate the absolute value function by means of a rotated hyperbola, and p2/π is an approximation for Epǫ2 it−1+ Λgiven that ǫit−1∼N(0,1).The process is started at yi0= 0, then 700 time periods are generated before the sample is generated. I denote as Γ = (α, β).The data are generated for T= 8 and 16, N= 1000, α = 0.5,and β= 0.5.For each sample I have estimated Γ by maximum likelihood and, at the moment, by the trimming modified maximum likelihood. 52 Chapter 2 The MLE of Γ is b Γ≡arg max Γ 1 N N X i=1 "1 T T X t=1 ℓit Γ,b ψi(Γ)#, where b ψi(Γ) ≡arg max ψ 1 T T X t=1 ℓit (Γ, ψ). Since here I can not get a explicit expression of the fixed effects estimators as functions of αand β, I do a double maximization, strictly speaking Nmaximizations inside the one for Γ. I use a Quasi-Newton’s Method algorithm to maximize the log likelihood function with respect to Γ, and in each step b ψi(Γ) is computed such that, for this given value of Γ,the individual log likelihood is maximized with respect to ψ. The MMLE is e Γ = arg max Γ 1 N N X i=1 ℓmi Γ,b ψi(Γ) = arg max Γ 1 N N X i=1 "1 T T X t=1 ℓit Γ,b ψi(Γ)−ˆ bi(Γ) T#, where ˆ bi(Γ) = 1 2hb H−1 i(Γ) b Υi(Γ)i, for b Hi(Γ) = −1 T T X t=1 ∂2ℓit ∂ψ2, and a trimmed version of Υi(Γ) with rsmall b Υi(Γ) = Ω0+ 2 r X l=1 Ωl, 2.4. Monte Carlo Evidence 53 Ωl=1 T−l T X t=l+1 1−l r+ 1∂ℓit ∂ψi·∂ℓit−l ∂ψi . In this case I calculate numerical first and second derivatives. Table B.4 reports estimates for T= 8 and 16,and N= 1000.In this case ˆαis not biased, and with the trimming correction I correct an otherwise seriously biased MLE of β. 2.4.4 The AR(1)-EARCH(1) panel model with multiple fixed effects Here the model design is yit =αyit−1+ηi+eit =αyit−1+ηi+h1/2 it ǫit,(t= 1, ..., T;i= 1, ..., N) hit = exp ψi+βqǫ2 it−1+ Λ −p2/π=h(ǫit−1, ψi), ǫit ∼N(0,1); ηi∼N(0,1) ; ψi∼N(−3.0,0.8) . The process is started at yi0= 0, then 700 time periods are generated before the sample is generated. I denote as Γ = (α, β).The data are generated for T= 16, N= 1000, α0= 0.5,and β0= 0.5.For each sample I have estimated Γ by maximum likelihood and, at the moment, by the trimming modified maximum likelihood. The MLE of Γ is b Γ≡arg max Γ 1 N N X i=1 "1 T T X t=1 ℓit Γ,b Θi(Γ)#, where b Θi(Γ) ≡arg max Θ 1 T T X t=1 ℓit (Γ,Θ) , 60 Chapter 2 σit =hit (ψi, ǫit−1)1/2= exp ψi 2+β 2qǫ2 it−1+ Λ −p2/π = exp   ψi 2+β 2  v u u t yit−1−αyit−2−ηi hit−1(ψi, ǫit−2)1/2!2 + Λ −p2/π    , and ∂µit ∂yit−1 =α, ∂σit ∂yit−1 =σit ×β 2×1 ǫ2 it−1+ Λ1/2× yit−1−αyit−2−ηi hit−1(ψi, ǫit−2)1/2!×∂ǫit−1 ∂yit−1 =σit ×β 2×ǫit−1 ǫ2 it−1+ Λ1/2×1 hit−1(ψi, ǫit−2)1/2. Thus I can calculate a mean elasticity at different parts of the wage distribution as ετ(log wit−1) = 1 NT N X i=1 T X t=1 ∂log Qτ(wit) ∂log wit−1. The first column in Table B.14 shows that those elasticities increase with the quantiles. That is, there are different elasticities below and above the median, where the mean elasticity is just equal to the corrected estimate of alpha, ˆαT.In Table B.14 and in Figure E.10, we can see that this pattern is very different for individuals with low (second column) or high (third column) values of the estimated fixed effects in the variance. Impulse-response function: functions of ǫit−sNow, Qτ(log w) = µ+qτσ. 2.5. Estimation Results 61 and in the conditional case, regarding µand σas functions of ǫit−1, ∂Qτ(log wit) ∂ǫit−1 =∂µ ∂ǫit−1 +qτ ∂σ ∂ǫit−1 . In particular, for the model considered here µit =αyit−1+ηi=ααyit−2+ηi+hit−1(ψi, ǫit−2)1/2ǫit−1+ηi σit = exp ψi 2+β 2qǫ2 it−1+ Λ −p2/π, and ∂µit ∂ǫit−1 =αhit−1(ψi, ǫit−2)1/2, ∂σit ∂yit−1 =σit ×β 2 ǫit−1 ǫ2 it−1+ Λ1/2!. Thus I can calculate a mean marginal effect at different parts of the logwage distribution as b E∂Qτ(log wit) ∂ǫit−1=1 NT N X i=1 T X t=1 ∂Qτ(log wit) ∂ǫit−1. Notice that ∂Qτ(log wit) ∂ǫit−1 =∂log Qτ(wit) ∂log wit−1×hhit−1(ψi, ǫit−2)1/2i. Now, for ∂Qτ(log wit) ∂ǫit−2 =∂µ ∂ǫit−2 +qτ ∂σ ∂ǫit−2 . 62 Chapter 2 In particular, for the model considered here and µit =α2αyit−3+ηi+hit−2(ψi, ǫit−3)1/2ǫit−2+ (1 + α)ηi+αhit−1(ψi, ǫit−2)1/2ǫit−1 =α3yit−3+1 + α+α2ηi+αhit−1(ψi, ǫit−2)1/2ǫit−1+α2hit−2(ψi, ǫit−3)1/2ǫit−2, σit = exp ψi 2+β 2qǫ2 it−1+ Λ −p2/π = exp   ψi 2+β 2  v u u t yit−1−αyit−2−ηi hit−1(ψi, ǫit−2)1/2!2 + Λ −p2/π    , and ∂µit ∂ǫit−2 =α2hit−2(ψi, ǫit−3)1/2+1 2αβhit−1(ψi, ǫit−2)1/2ǫit−1ǫit−2 pǫ2 it−2+ Λ, ∂σit ∂ǫit−2 =σit β 2 ǫit−1 ǫ2 it−1+ Λ1/2!×σit−1 β 2 ǫit−2 ǫ2 it−2+ Λ1/2!. The first panel in Table B.15 shows the mean marginal effects with respect to ǫit−1over different quantiles of the logwage distribution and the second panel, the case with respect to ǫit−2.In Figure E.11 we can see that past shocks seem to have effect over logwages even two periods apart. 2.5.3 Job changes It is important taking into account that in a model where individual heterogeneity is treated as fixed effects we abstract for job changes. A specification like this yit =αyit−1+ηi+eit, works worse if there are many job changes in the sample because ηiis fixed. In order to evaluate this concern, I consider a sample where individuals in different jobs are treated 2.6. Implications for Consumption Growth 63 as different individuals. That is, for each individual yit =αyit−1+ηi1+eit; individual iin job 1, yit =αyit−1+ηi2+eit; individual iin job 2. I use data on 1,346 and 17,485 observations. I do the same sample selection as before. Sample composition by year and by education, and demographic characteristics are presented in Tables B.16-B.18. Results are reported in Table B.19. We can see that the significant ARCH effects in the variance disappears as soon as we consider a sample without job changes. 2.5.4 Attrition A final issue is the extent to which attrition from the PSID has biased the results. In this chapter, I assume that attrition is all accounted for by the permanent characteristics in the individual fixed effects. To provide some evidence for this I compare the estimates in my sample to those obtained using only individuals who are 16 or more years in the sample (921 individuals). This kind of selection mimics attrition bias since it eliminates individuals observed for a shorter time period. The estimates based on this sample are included in Table B.20. The main conclusion is that the corrected estimates are not very different to those reported in Table B.10. 2.6 Implications for Consumption Growth Given the results above I provide now an example that illustrates the effects that individual risk can have in explaining precautionary saving, that is, additional saving that results from the knowledge that the future is uncertain. Here, I follow most of the literature and I consider that additional saving is achieved by consuming less. 64 Chapter 2 Over the last 30 years there has been a well-documented increase in cross-sectional income inequality in the US, and some authors have suggested that households are now exposed to more earnings instability than they were (Gottschalk and Moffitt, 1994). This figure suggests that precautionary saving motives associated with an increase in income risk could have become more important. In the presence of complete insurance, either formal or informal, it should only be the component of risk that is common to all individuals in an economy that affects consumption. Banks, Blundell, and Brugiavini (2001) find that it is not the common component of risk, but instead the cohort-specific risks which dominate consumption growth. Their results corroborate the notion that if income uncertainty has been growing over the recent past then the failure of insurance between agents makes the precautionary motive for saving an increasingly important self-insurance mechanism. They use series of repeated cross sections of British households data, but they can not consider individual-specific risk due to the lack of panel data. Here, I evaluate the independent role of individual wage risk in consumption growth. 2.6.1 Consumption Model Let us consider the following intertemporal consumption model13 (Browning and Lusardi, 1996), where individuals choose consumption so as to maximize an intertemporal utility function subject to the intertemporal budget constraint: max {Ct+k}T−t k=0 Et T−t X k=0 h(1 + δ)−kU(Ct+k, Dt+k)i s.t. At+1+k= (1 + rt+k)·(At+k+Yt+k−Ct+k) AT+1 ≥0 (k= 0, ..., T −t) 13I omit the individual index for simplicity. 2.6. Implications for Consumption Growth 65 where, for each period s,Csis consumption, Yslabour income or earnings, rsreal interest rate, Asfinancial wealth (at the beginning of the period), δsubjective intertemporal rate, and Dsdemographic characteristics. I assume the date of death is known and there are not explicit liquidity constraints. The optimal intertemporal allocation of consumption verifies the Euler equation, that is, Et1 + rt 1 + δ·UC(Ct+1, Dt+1) UC(Ct, Dt)= 1 where Uc(·) denotes the first derivative of the utility function with respect to consumption. I assume a CRRA utility function: U(Ct, Dt) = 1 1−ρexp (ϕ′Dt)·C1−ρ t where ρ > 0 is the relative risk aversion coefficient. So, 1 + rt 1 + δ·exp (ϕ′∆Dt+1)·Ct+1 Ct−ρ = 1 + ξt+1, where Etξt+1= 0.Taking logs and using the usual approximation for logs I obtain the linearized Euler equation: ∆ ln Ct+1 =1 ρln (rt−δ) + 1 ρϕ′∆Dt+1 +1 2ρV artξt+1+vt+1. The first term on the RHS of the equation takes into account the intertemporal substitution effect: an increase in rt,opportunity cost of current consumption, implies a higher growth of future consumption. The second term considers how different stages of the life cycle are reflected on the consumption profile, by changes in circumstances implicit in demographic variables. Finally, the third term on the RHS of the equation captures precautionary saving. A rise in the expected variance of earnings innovations represents 66 Chapter 2 an increase in earnings risk and should depress period tconsumption hence increasing the growth of consumption between tand t+ 1.In other words, a positive parameter implies that risk induces a delay in spending and current consumption is therefore reduced. Notice that V artξt+1reflects uncertainty regarding future realizations of any uninsurable variable relevant for consumption. Thus, it is not sufficient to enter the wage risk term alone. A scaling term is required by which “poorer” individuals are more responsive to changes in earnings risk, πt=Yt−1 Ct−12.In consequence, ∆ ln Ct+1 =1 ρln (rt−δ) + 1 ρϕ′∆Dt+1 +γπtσ2 t+1 +vt+1 where σ2 t+1 is a measure of the conditional variance of the wage shock. 2.6.2 Estimation and results I use food consumption data from the PSID (1974-1987). In my sample14, I estimate by OLS15 the following empirical equation: ∆ ln Cit+1 =δt+β′∆Dit+1 +γπitσ2 it+1 +vit+1, where σ2 it+1 is replaced by ˆσ2 it+1 =hit+1 ˆǫit;ˆ Γ,ˆ Θi,initial conditions. Looking at the estimate for the γparameter in Table B.21, column 2, I obtain a 14The sample includes 1,191 individuals and 15,192 observations. 15It would be interesting to follow the same approach as before considering a complete likelihood function: Lnow =Lbefore +X i,t −1 2ln σ2 v−1 2σ2 vh∆ ln ˆ Ct+1 −γπtσ2 t+1i2 where ∆ ln ˆ Ct+1 is obtained from first stage regressions of ∆ ln Ct+1 on δtand ∆Dt+1. 2.7. Conclusions 67 significant and positive effect of this term on the consumption growth. As stated above, an increase in individual risk induces a reduction in current consumption and, therefore, an increase in the growth of consumption between tand t+ 1. Regarding the interactions with education (columns 3 and 4), we can see that this positive effect is more important for the less educated people, slightly significant for the graduate and insignificant for the college educated. This result goes in line with the idea that there are more insurance possibilities for these latter. 2.7 Conclusions In this chapter I propose a model for the conditional mean and the conditional variance of individual wages. It is a non linear dynamic panel data model with multiple individual fixed effects. For estimating the parameters of the model I assume a distribution for the shocks and apply bias corrections to the concentrated likelihood. This corrects the bias of the estimated parameters from O(T−1) to O(T−2), so the estimator has a good finite sample performance and a reasonable asymptotic approximation for moderate T. In fact, Monte Carlo results show that the bias of the MLE is substantially corrected for samples designs that are broadly calibrated to the PSID dataset. The main advantage of this approach is its generality. As we have seen, the method is generally applicable to take into account dynamics and multiple fixed effects. Another advantage is that the fixed effects are estimated as part of the estimation process. The empirical analysis is conducted on data drawn from the 1968-1993 PSID dataset. In line with previous literature, I find a corrected estimate for the autoregressive coefficient in the mean around 0.5 (Alvarez and Arellano, 2004), and positive ARCH effects for the variance (Meghir and Pistafferri, 2004). Job changes are driving this dynamics in the variance. I also find important fixed differences across individuals in the variance. In addition, it turns out that this located-scaled model explains the non-normality observed 68 Chapter 2 in logwage data. I then illustrate some implications that ARCH effects may have in the field of savings. Finally there are three issues, at least, that require further research: measurement error in PSID wages, a more comprehensive model that include job changes, and the comparison with female workers in terms of wage profiles. Appendix of Chapter 2 2.A Bias of the Concentrated Likelihood Following Arellano and Hahn (2006a, 2006b), let us obtain the expression for the First Order Bias of the Concentrated Likelihood at an arbitrary value of the common parameter Γ. Let ℓi(Γ,Θi) = PT t=1 ℓit (Γ,Θi)/T where ℓit (Γ,Θi) = ln f(yit|yit−1,Γ,Θi) denotes the log likelihood of one observation. Let Θi(Γ) = arg max Θi plimT→∞ℓi(Γ,Θi), and b Θi(Γ) = arg max Θi ℓi(Γ,Θi), so that under regularity conditions Θi(Γ0) = Θi0. Following Severini (2000) and Pace and Salvan (2005), the concentrated likelihood for unit i ˆ ℓi(Γ) = ℓiΓ,b Θi(Γ), can be regarded as an estimate of the unfeasible concentrated log likelihood ¯ ℓi(Γ) = ℓiΓ,Θi(Γ). 2.A. Bias of the Concentrated Likelihood 69 Now, define uit (Γ,Θi) = ∂ℓit (Γ,Θi) ∂Γ, vit (Γ,Θi) = ∂ℓit (Γ,Θi) ∂Θi , ui(Γ,Θi) = 1 T T X t=1 uit (Γ,Θi), vi(Γ,Θi) = 1 T T X t=1 vit (Γ,Θi), Hi(Γ) = −lim T→∞ E"∂viΓ,Θi(Γ) ∂Θ′ i#. When Θi0is a vector of fixed effects, the Nagar expansion for b Θi(Γ) −Θi(Γ) takes the form b Θi(Γ) −Θi(Γ) = H−1 i(Γ) viΓ,Θi(Γ)+1 TBi(Γ) + Op1 T3/2,(A.1) where Bi(Γ) = H−1 i(Γ) Ξi(Γ) vec H−1 i(Γ) +1 2E ∂ ∂Θ′vec∂viΓ,Θi(Γ) ∂Θ′!′H−1 i(Γ) ⊗H−1 i(Γ)vec (Υi(Γ))#, and Υi(Γ) = Υi(Γ; Γ0,Θi0) = lim T→∞ TE hviΓ,Θi(Γ)viΓ,Θi(Γ)′i, Ξi(Γ) = Ξi(Γ; Γ0,Θi0) = lim T→∞ TE "∂viΓ,Θi(Γ) ∂Θ′⊗viΓ,Θi(Γ)′#. Next, expanding ℓiΓ,b Θi(Γ)around Θi(Γ) for fixed Γ, ℓiΓ,b Θi(Γ)−ℓiΓ,Θi(Γ) =∂ℓiΓ,Θi(Γ) ∂Θ′b Θi(Γ) −Θi(Γ) +1 2b Θi(Γ) −Θi(Γ)′∂2ℓiΓ,Θi(Γ) ∂Θ∂Θ′b Θi(Γ) −Θi(Γ)+Op1 T3/2 76 Chapter 3 heterogeneity and time series properties of individual wage processes (Lillard and Willis, 1978; MaCurdy, 1982; Abowd and Card (1989), among others), but many have ignored job mobility and the distinction between dynamics within and between jobs. In the second chapter of this thesis, I consider a model for the heterogeneity and dynamics of the conditional mean and the conditional variance of individual wages. In the empirical analysis of that chapter - conducted on data drawn from the 1968-1993 Panel Study of Income Dynamics (PSID) - I find that it is important to account for individual unobserved heterogeneity and dynamics also in the conditional variance, and that the latter is driven by job mobility. In line with those results, this chapter develops a model that explicitly considers job changes in the dynamics of wages and in the heterogeneity pattern. In particular, the specification proposed has two different parameters to capture dynamics within jobs and across jobs, and the unobserved heterogeneity shows a richer pattern, as well, composed of both individual and job-specific effects. As pointed out by Low et al. (2007), it is important to distinguish between movements in earnings that reflect choice and those which reflect uncertainty. Those authors address this issue by allowing for endogenous labour supply and job mobility which implies that a proportion of earnings fluctuations, usually interpreted as risk, are in fact attributed to choice. Here, the potential endogeneity of job mobility in relation to earnings is circumvented using an instrument variable estimation method that controls for individual and job-specific unobserved heterogeneity. This match effect will change across jobs but it will remain constant within a position3. Differently to Lillard (1999), Abowd and Kang (2002) and Low et al. (2007), I adopt a fixed effects perspective leaving the distribution for the unobserved heterogeneity components completely unrestricted and treating each effect as one different parameter to be estimated. In the empirical application, I use data on work histories drawn from the PSID, which 3The importance of match effects in explaining wages has been stressed by Topel and Ward (1992), Abowd, Kramarz and Margolis (1999), Postel-Vinay and Robin (2002) and Bonhomme and Jolivet (2006). 3.2. The Data 77 allows the distinction between voluntary and involuntary job-to-job changes. In the data, once we control for individual and job-specific effects, the dynamics within jobs is almost zero, whereas across jobs is significant but small. For the dynamics, the distinction between voluntary and involuntary transitions turns out to be irrelevant. However, that distinction matters in the case of the components of the cross-sectional variance. The estimated variance of the job-specific effects represents around one third of the variance for the individual fixed effects. If I consider a subsample that only includes involuntary job changes, the estimated variance of the heterogeneity across jobs increases up to one half. This chapter contributes to the literature by more thoroughly describing the impact of job mobility on the dynamics and heterogeneity of individual wages than previous references. First, the model permits that job changes may be correlated with individual and job specific characteristics. Second, it is agnostic regarding the distribution of these individual and job effects. Third, it can be estimated with no need to explicitly model the job mobility process. Finally, the model also allows calculating different components of variance within and between jobs. The rest of the chapter is organised as follows. Section 3.2 describes the data. Section 3.3 presents the model. Section 3.4 explains the estimation strategy and section 3.5 shows the estimation results. Finally, section 3.6 concludes with a future research agenda. 3.2 The Data The data come from the PSID for the period 1968-1993. The PSID began in 1968 by interviewing over 5,000 families. Of these, about 3,000 families were representative of the US population as a whole (the core sample), and about 2,000 were low-income families (the Census Bureaus SEO sample). Thereafter, these same families have subsequently been interviewed every year, as have any new families formed from the original group of 78 Chapter 3 families4. The survey contains abundant information on individual characteristics, income and labour market status. The data set should follow individuals over a sufficiently long period of time to observe preand postjob changes earnings histories. 3.2.1 Sample Construction In the empirical analysis, I use the core sample. I restrict my study to heads of households since survey questions on the PSID regarding employment history are only asked to household heads5and, only from 1979, also to wives. In addition, I select males aged 25 to 55 - to focus the analysis during the working life - with no missing records on race, education or region of residence. I drop those with top coded wages, the self-employed, those with less than 8 years of usable data on earnings and those with missing records on the question reason of change. Finally, I have an unbalanced panel that contains 2,013 individuals and 27,845 observations from 1968 to 19926. Step-by-step details on sample selection are reported in Appendix 3.A, and sample composition by year, individuals by number of observations and demographic characteristics are presented in Tables C.1-C.3. 3.2.2 Job Changes Definition I determine that a job change takes place if current tenure of the worker is less than a year and if there is information available regarding the type of job change. The type of change is defined by the answer to the question, “What happened to the job you had before - did the company go out of business, were you laid off, promoted, or what?”. That question 4A family member who moves out of a PSID family is eligible for interviewing as a separate family unit if he or she is a sample member and he or she is 18 years old or older and living in a different, independent household. 5A household head is defined as the adult of the family. When there is more than one adult in the family, the PSID assigns the primary male adult as the household head. 6Since time reference for wage records is the previous year in every survey wave, I use information only until 1992. 3.2. The Data 79 was only asked to individuals who report being with their present employer for less than twelve months (otherwise the question is skipped and coded as not applicable), so this make me feel confident regarding the variable tenure7. As pointed out by Polsky (1999) from 1984-88, this question was asked of all respondents who reported that their current job started after January of the previous year. To correct for this possible inconsistency, no job change is reported for those with current tenure greater than one year. From the answers to the question regarding reason of change, I define a job change as an involuntary job separation or job loss in case of business or plant closing or due to being laid off or fired; and as quit, in case of voluntary change. The sample only includes job-to-job changes, because monthly calendar information (that would provide information regarding spells of unemployment with durations of less than a year) is not available in the PSID prior 1984. 3.2.3 Descriptive Analysis of the Raw Data The descriptive analysis will emphasize a number of salient facts about job mobility and the relationship between this and earnings dynamics. Job mobility Among the 2,013 sample individuals, there 699 individuals (around 35 percent) who never change job, whereas the remaining individuals change at least once (on average they have 3.40 different jobs). As pointed out by Topel and Ward (1992), the most prominent and widely documented facts about job mobility are that average rates of job changing decline with age or experience and, specially, with current job tenure. These facts are consistent with the 7Because the PSID did not collect information on specific employers, the identification of job changes in this data set has been quite controversial. Many of the difficulties related to measuring job tenure in the PSID were evaluated by Brown and Light (1992). The tenure question also switched from being coded in intervals prior to 1976 to being measured in months, and from asking about “position” tenure to “employer” tenure. In any case, these difficulties diminish here since I am not interested in the exact value of the variable but if it is less or more than one year. 80 Chapter 3 predictions of job-matching and search models8(Johnson, 1978; Burdett, 1978; Jovanovic, 1979). Figure E.12 shows those patterns in the sample. Regarding “vintage effects”, it is less clear if people entering the labour market more recently have patterns of labour mobility different from those of earlier cohorts. Table C.4 presents the distribution of jobs by birth cohort. The 1921-1941 cohort contains a larger proportion of individuals who only have one job than individuals born between 1941-1960. Although sample selection may be relevant, since workers are more exposed to job changes as they grow older and more recent entrants are less likely to be observed in higher-order jobs, the results in the table suggest an increase in job instability for the most recent cohort in the sample. With respect to the job-exit reason, if we look (Figure E.13) at average rates of job changing by cohorts we find that younger cohorts of workers are more likely to be laid off from their jobs than older cohorts but the difference is bigger in case of quit. More striking is the comparison across skill groups. For all groups the main reason for leaving job is quitting, but the difference with respect to layoff is more important for graduate and - specially - for college people than for dropouts. Job mobility and earnings dynamics In order to get a first impression of the impact that job changes have over the evolution of earnings (and as a check of the definitions above), I calculate the cross-sectional sample correlations for consecutive logwage observations on years when no-change, a job loss or a job quit has happened. I deflate nominal annual earnings by the GNP Personal Consumption Expenditure Deflator (base 1992). Table C.5 summarizes those calculations. As we would expect, when a job change occurs the correlation diminish, and that reduction is bigger in case of loss than in case 8In a matching model, job mobility is the consequence of a voluntary change to a better position where the worker is more productive and receives a higher pay. Search models are based in the existence of imperfect information. In these models, jobs are experience goods. As time goes by, the firm acquires more information and it can adjust the salary better. Under this approach, job mobility is the result of a “poor” matching looking for a better chance. 3.3. The Model 81 of a voluntary change. Table C.6 displays average annual wage growth for workers within jobs and between jobs by type of exit. Within job average annual wage growth is lower than between job average wage growth in case of voluntary transitions. In case of job loss I obtain a drop on real wages. I find the same qualitative pattern among different demographic groups. As pointed out by Dustmann and Meghir (2005), the fact that within job average annual wage growth is lower than between job average wage growth does not imply that, on average, job quitters have higher wages than stayers. As they did, I regress log wages on dummies for the number of jobs workers have held up to then, also including age and year dummies. Estimates for the first seven jobs, reported in the first column of Table C.7, indicate that workers with more jobs have lower wages. Once I include individual fixed effects in the regression (column 4), the number of jobs is positively related with wages. In fact, if I exclude from the sample movers who transit only through job loss (columns 2 and 5), I obtain a positive relationship between number of jobs and wages. On the contrary, if I exclude those who change voluntary (columns 3 and 6), I obtain that workers with more jobs have lower wages even after including individual fixed effects. 3.3 The Model In this section I propose an empirical model to study the dynamics of individual earnings over time, within a job and over the career of a worker in one or more different jobs. 3.3.1 Basic Specification Building on the autoregressive model developed in Lillard and Willis (1978), for a worker ithat is observed for Tiperiods always at the same job, I consider the following standard 82 Chapter 3 specification yit =αyit−1+vit =αyit−1+ηi+ǫit; (t= 1, .., Ti),|α|<1, where yit is the log earnings of an individual iin period t, the parameter αmeasures the persistence on the level of those earnings to shocks, ηiis an unobserved time invariant individual component, like ability, and ǫit is a purely transitory person-period component, that is, Eǫit|yt−1 i, ηi= 0, where yt−1 i= (yi1, ..., yit−1)′.9I abstract from additive aggregate effects by regarding yit as a deviation from a time effect10. Given the model and the initial condition, yi1,the wage profile of an individual iwho always stays at the same job would evolve as yi2=αyi1+vi2=αyi1+ηi+ǫi2 yi3=αyi2+vi3=αyi2+ηi+ǫi3 . . . that is, her wage today would be αtimes her wage yesterday (where the parameter α measures the persistence on the level of wages to shocks) plus a random term, vit, due to 9In the sequel, for any random variable (or vector of variables) Z,zit denotes observation for individual iat period t, and zt i={zi1, ..., zit}, i.e. the set of observations for individual ifrom the first period to period t. 10As is usual in the earnings dynamics literature, the variable yit - strictly speaking - represents log earnings residuals from first stage regressions on some observed variables -apart from year dummies (that capture the aggregate conditions of the economy) - as age, race and other individual characteristics. So we would keep in mind the following structure: wit =xitβ+uit uit =γi+υit υit =αυit−1+ǫit where wit is the log annual wages of an individual iin period t, xit is a vector of exogenous variables, and uit is a random error with two components , an unobserved individual heterogeneity component and an autoregressive component. The connection with the specification proposed above would be yit = ˆuit and ηi= (1 −α)γi. 3.3. The Model 83 an unobserved time invariant individual component, ηi,like ability, and a purely transitory person-period component ǫit. On the contrary, for a worker hthat changes job between t= 3 and t= 4,I would consider yh2=αyh1+ηh+ǫh2 yh3=αyh2+ηh+ǫh3 yh4=α∗yh3+η∗ h+ǫh4      job change ⇒hends job at t=3 and starts a new one at t=4 yh5=αyh4+η∗ h+ǫh5 . . . This specification departs from the standard one in two main features related to job mobility: 1. The dynamics captured by the autoregressive parameters is different in years when workers change job, α∗=α+β, than within the same job, α. 2. The unobserved individual heterogeneity have a job-specific matching component. In other words, I consider individual and job specific fixed effects, ηi(t)=µi+φij; that is, within the same job we would have ηi(t)=ηi(t−1), but between jobs ηi(t)= µi+φ∗ ij 6=ηi(t−1) =µi+φij.As mentioned before, I adopt a fixed effects perspective leaving the distribution for the unobserved heterogeneity completely unrestricted both within jobs as well as between jobs. To sum up, the general formulation of the model is the following yit =αyit−1+βdit−1yit−1+vit =αyit−1+βdit−1yit−1+ηi(t)+ǫit =αyit−1+βdit−1yit−1+µi+φij +ǫit; (i= 1, ...N;t= 2, .., Ti),(3.1) 84 Chapter 3 where dit is an indicator of worker iending current job at time t.11 Given the model, within job, the transitory shocks will be uncorrelated with lagged earnings, but not with present or future earnings. Similarly, I do not need to assume the strict exogeneity of the job changes, in the sense of being uncorrelated to past, present, and future time-varying shocks. Apart from possibly being correlated with the unobserved heterogeneity components, I will consider that job changes may be predetermined, that is, they might be correlated with errors at certain periods but not at others. In particular, we could think on dit as a function of past errors, dit =f(ǫit−1, ǫit−2, ǫit−3, ...),and unobserved heterogeneity components - that is, the individual’s work history - but as being uncorrelated to present and future shocks. Specifically, I am imposing that Eǫit|yt−1 i, dt i= 0.(3.2) Although it would be preferable to also allow for correlation between dit and ǫit,that would lead us to consider selection models which is out of the scope of this thesis. Even so, this specification has several advantages. First, it permits the estimation of a model in which job changes can be correlated with individual and job specific characteristics. Second, I do not need to do any assumption regarding the distribution of these individual and job effects. Third, I do not need either to explicitly model the job mobility process. The model also allows to calculate different components of variance within and between jobs. Moreover, note that neither time series nor conditional heteroskedasticity are assumed. That is, the unconditional variances of the errors, denoted as Eǫ2 it=σ2 t, 11I should formally have a jsubscript on wages but since it does not add clarity I have dropped it. 3.4. Identification and Estimation method 85 are allowed to change with tand to differ from the conditional variances Eǫ2 it|yt−1 i, ηi. As before, we could consider unobserved heterogeneity components in those conditional variances, both at the individual and job-specific level. 3.3.2 Specification by Type of Exit In the empirical analysis I will also consider an extended specification that reflects different dynamics across individuals and time according to the type of job change yit =αyit−1+βldloss it−1yit−1+βqdquit it−1yit−1+µi+φij +ǫit; (i= 1, ...N;t= 2, .., Ti),(3.3) where dloss it is a dummy variable equal to one if worker iat time tends current job due to an involuntary job separation or job loss; and dquit it equal one if worker iat time tends current job because she has decided to moved to a new job. I consider the kind of individual and stochastic effects which preserve the same properties as the basic specification. 3.4 Identification and Estimation method In this section I discuss the conditions under which I achieve parameter identification. In the model, wages are observed conditional on individuals working; within-job wages, which identifies the parameter αand the individual component ηi, are only observed if the individual does not change job; between-job wage growth, which helps identify heterogeneity across jobs, φij, and differences on dynamics on years of change, β, is observed only for job movers. Further, participation and mobility decisions can be all 92 Chapter 3 1. Drop members of the Latino sample (10,022 individuals) = Sample (42,983 individuals). 2. Keep only those who are continuously heads of their households = Sample (16,038 individuals). 3. Keep only males aged 25 to 55 over the period = Sample (8,190 individuals). 4. Drop those with a spell of self-employment = Sample (6,303 individuals). 5. Drop those with missing race, education and region of residence records = Sample (6,047 individuals). 6. Drop those with top-coded earnings records and those with missing earnings = Sample (5,479 individuals). 7. Drop those with outlying earnings records, that is, a change in log earnings greater than 5 or less than -3 = Sample (5,384 individuals). 8. Drop those with missing records on reason of job change question and those with noncontinuous data = Sample (5,345 individuals). 9. Keep only those who are in the sample for 8 years or more = FINAL SAMPLE: Males, 1968-1992 (2,013 individuals and 27,845 observations). Chapter 4 Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note1 4.1 Introduction In a typical nonlinear micropanel data model with fixed effects there are hundreds or thousands of individual coefficients to estimate together with a relatively small number of common parameters. A well known computational simplification in the linear model is to obtain first the maximum likelihood (ML) estimates of the common parameters from a regression on the data in deviations from individual means, and secondly retrieve ML estimates of the effects from averaged residuals one by one. A similar computational simplification is available for Newton-Raphson and related algorithms for nonlinear fixed effects models, which exploits the block-diagonal structure of the Hessian. This simplification has been discussed in Hall (1978), Chamberlain (1980), and Greene (2004) for nonlinear models with a scalar fixed effect. The first purpose of this work is to show how to use an iterated algorithm of this type in a nonlinear model with multiple fixed effects. As first noted by Neyman and Scott (1948), when the time series dimension Tis small 1This chapter is part of a joint work with Manuel Arellano. 93 94 Chapter 4 relative to the cross-sectional dimension n, ML estimates of the common parameters can be severely biased, specially in dynamic models. This Incidental Parameters problem arises because the unobserved individual characteristics are replaced by noisy estimates, which bias estimates of model parameters. In particular, the bias of the MLE is of order 1/T. In some special cases it is possible to obtain fixed Tlarge nconsistent estimators of certain common parameters, but these situations are more the exception than the rule. Alternatively, a number of additional approaches have been proposed to obtain approximately unbiased estimators as opposed to estimators with no bias at all2. One of these approaches consists of estimation from a bias corrected objective function relative to some target criterion3. In this chapter we discuss the application of computationally efficient algorithms to modified concentrated likelihoods of this type to obtain estimators without bias to order 1/T in nonlinear panel models with multiple fixed effects. The chapter is organized as follows. Section 4.2 introduces the model and notation. Section 4.3 explains how the iterated algorithm works. Section 4.4 discusses its application to bias corrected concentrated likelihoods. Section 4.5 presents some simulation results. Finally, Section 4.6 concludes. Detailed derivations are given in the Appendix. 4.2 Model and Notation Let us consider the following model for the joint density of Trandom vectors conditioned on initial observations, strictly exogenous variables, and fixed effects: f(yi1, ..., yiT |yi0, xi1, ..., xiT , αi0) = T Y t=1 fyit |yi(t−1), xit, αi0, θ0 2See Arellano and Hahn, 2006a, for a review of this literature on bias-adjusted estimation methods for nonlinear panel data models with fixed effects. 3See Pace and Salvan (2005) for adjustments of this type for a generic concentrated likelihood with independent observations, Arellano and Hahn (2006a) for static nonlinear panel models and Arellano and Hahn (2006b) and the second chapter of this thesis for the dynamic case. 4.3. Efficient Newton-Raphson iteration 95 where θ0is a vector of common parameters and αi0is a vector of fixed effects. We observe the random sample {yi0, ..., yiT , xi0, ..., xiT }n i=1 and we denote α0= (α′ 10, ..., α′ n0)′ and δ0= (θ′ 0, α′ 0)′. Let the log likelihood of one observation be ℓit (θ, αi) = ln fyit |yi(t−1), xit, αi, θ and let ℓi(θ, αi) = PT t=1 ℓit (θ, αi). 4.3 Efficient Newton-Raphson iteration Let us consider the estimator   bθ bα  = arg max θ,α n X i=1 ℓi(θ, αi) and let first and second derivatives be denoted by dθi =∂ℓi(θ, αi) ∂θ , dαi =∂ℓi(θ, αi) ∂αi Hθθi =∂2ℓi(θ, αi) ∂θ∂θ′, Hααi =∂2ℓi(θ, αi) ∂αi∂α′ i , Hθαi =∂2ℓi(θ, αi) ∂θ∂α′ i . The Kth step of the iteration of a computationally efficient algorithm for obtaining bθand bαtakes the form θ[K]−θ[K−1] =−"n X i=1 Hθθi −HθαiH−1 ααiHαθi#−1n X i=1 dθi −HθαiH−1 ααidαi(4.1) αi[K]−αi[K−1] =−H−1 ααi dαi +Hαθi θ[K]−θ[K−1],(i= 1, ..., n) (4.2) where all derivatives are evaluated at θ[K−1] and αi[K−1]. This result can be easily proved using partitioned inverse formulae (a detailed deriva- 96 Chapter 4 tion is in the Appendix 4.A). It is a standard result in nonlinear estimation of models with many group effects.4 4.4 Adjusted Concentrated Likelihood When Tis short we may be interested to consider an estimator that maximizes a bias corrected concentrated likelihood of the type reviewed in Arellano and Hahn (2006a): bθc= arg max θ n X i=1 [ℓi(θ, bαi(θ)) + βi(θ, bαi(θ))] where bαi(θ) = arg max αℓi(θ, α) and βi(θ, αi) is an adjustment term. As long as the adjustment term depends on α, the iterated algorithm discussed above cannot be directly used for estimating bθc. Note that   bθc bαc   = arg max θ,α n X i=1 [ℓi(θ, αi) + βi(θ, bαi(θ))] where bαc=bαbθc. Thus, if we use the analysis of covariance algorithm discussed in the previous section we still need to calculate bαi(θ) for given values of θ. 4An alternative Gauss-Newton algorithm which leads to a regression-based iteration is discussed in Appendix 4.B. 4.5. Monte Carlo Study 97 An Alternative, Computationally Effective Estimator Alternatively, we can consider an estimator of the form   eθ eα  = arg max θ,α n X i=1 [ℓi(θ, αi) + βi(θ, αi)] for which the iterated algorithm can be used. This is equivalent to: eθ= arg max θ n X i=1 [ℓi(θ, eαi(θ)) + βi(θ, eαi(θ))] where eαi(θ) = arg max α[ℓi(θ, α) + βi(θ, αi)] . The statistic eαi(θ) can be regarded as a Bayesian estimator that uses eβi(θ,αi)as the prior distribution of αifor a given value of θ. Thus, under general conditions, eαi(θ) will be asymptotically equivalent to bαi(θ), and eθwill have similar (bias reducing) properties as bθ(see Severini, 1998, section 4, for a discussion on the use of adjusted concentrated likelihoods using alternative estimates of nuisance parameters). It appears that eθis not only computationally convenient, but it may also exhibit improved finite sample properties in certain situations due to the replacement of bαi(θ) by eαi(θ). 4.5 Monte Carlo Study In this section Monte Carlo simulations are used to evaluate the performance of the efficient algorithm in different sample sizes, and its application to bias corrected concentrated likelihoods of nonlinear models. We consider four examples in this section, but keeping the simulation design as consistent as possible across the models: the static probit with 98 Chapter 4 scalar fixed effects, the dynamic probit with scalar fixed effects, the static probit with multiple fixed effects and the dynamic probit with multiple fixed effects. Thus, yit =1[wit +ǫit >0] where ǫit ∼N(0,1) .We compare estimates of common parameters estimated by ML and bias-corrected ML5. 4.5.1 Probit designs with Scalar fixed effects We consider five different data-generating processes: three for a static probit and two more for a dynamic probit. Static Probit •Design 1 (Bester and Hansen, 2005): w(1) it =ηi0+β0xit with ηi0∼N(xi0,1) , xit =1 2xit−1+uit, uit ∼N(0,1) , xi0∼N(0,1) ,and β0= 1. Models were fit with T={8,12}and N= 100.Each model was fit 1,000 times with random draws for ǫit.The conditioning data, xit,and ηi0were held constant. •Design 2 (Greene, 2004): w(2) it =ηi0+β0xit +δ0dit 5Other studies, that consider nonlinear designs with scalar fixed effects (Bester and Hansen, 2005; Carro, 2006; and Fern´andez-Val, 2005), show that the bias in the ML estimator is similar in magnitude for the logit and the probit models and that bias corrections also perform similarly. Here, we focus on probit designs and extend the analysis to consider multiple fixed effects. 4.5. Monte Carlo Study 99 with ηi0=√T¯xi+ai, ai∼N(0,1) , xit ∼N(0,1) , hit ∼N(0,1) , dit =1[xit +hit >0] , β0= 1 and δ0={1,0.5}. •Design 3: w(3) it =ηi0+β0xit +δ0dit with ηi0= 0,∀i, xit ∼N(0,1) , hit ∼N(0,1) , dit =1[xit +hit >0] , β0= 1 and δ0= 0.5. For designs 2 and 3 models were fit with T={6,8,10,12}and N= 1,000.Each model was fit 100 times with random draws for ǫit.The conditioning data, xit, dit and ηi0 were held constant. Adjusted Concentrated Likelihood We have that Pr (yit = 1|wit) = Φ (wit) = Φit where Φ is the normal cdf. For design 1 Pr (yit = 1|ηi, xi) = Φ (ηi+βxit), whereas for designs 2 and 3 Pr (yit = 1|ηi, xi, di) = Φ (ηi+βxit +δdit). Let’s consider αi=ηiand θ=βfor design 1, and θ= (β, δ) for designs 2 and 3. Let the log-likelihood of one observation be ℓit (θ, αi) = yit ln Φit + (1 −yit) ln (1 −Φit), 100 Chapter 4 and let ℓi(θ, αi) = PT t=1 ℓit (θ, αi). We can obtain the MLE, bθ, as the argument that maximizes the log-likelihood function   bθ bα  = arg max θ,α n X i=1 ℓi(θ, αi). Or, equivalently, bθ, is the estimator that maximizes the concentrated log-likelihood function: bθ= arg max θ n X i=1 [ℓi(θ, bαi(θ))] where bαi(θ) = arg max αℓi(θ, α). The corrected concentrated MLE, bθc,is the argument that maximizes bθc= arg max θ n X i=1 [ℓi(θ, bαi(θ)) + βi(θ, bαi(θ))] where βi(θ, αi) is an adjustment term. The corrected computationally efficient MLE, eθ, is an estimator of the form   eθ eα  = arg max θ,α n X i=1 [ℓi(θ, αi) + βi(θ, αi)] or equivalently, eθ= arg max θ n X i=1 [ℓi(θ, eαi(θ)) + βi(θ, eαi(θ))] where eαi(θ) = arg max α[ℓi(θ, α) + βi(θ, αi)] . Following Arellano and Hahn (2006a), for a static model with scalar fixed effects, the form 4.5. Monte Carlo Study 101 of the adjustment term will be: βi(θ, αi) = −1 2 −1 T T X t=1 ∂2ℓit (θ, αi) ∂α2 i!−11 T T X t=1 ∂ℓit (θ, αi) ∂αi2 . Table D.1 lists the means of the empirical sampling distribution for the ML and biascorrected ML estimators (BC-C for the corrected concentrated MLE, bθc, and BC-E for the corrected computationally efficient MLE, eθ) in Design 1. The most relevant feature is the upward bias in the ML estimates of β. For each choice of T, the bias corrected estimates perform better both in terms of the bias and the precision. The results for the ML and the BC-C are consistent with the ones in Bester and Hansen (2005). Moreover, the BC-E is slightly better, in addition to the improvement in terms of computational time. Tables D.2 and D.3 display the means of the empirical sampling distribution for the ML and bias-corrected ML estimators in Design 2 with δ0= 1 and δ0= 0.5, respectively. This design was proposed by Greene (2004) in order to examine the small sample bias of the fixed effects MLE. For all values of T, the BC estimates offer substantial improvements over ML (again, both in the bias and in the SD). As we would expect, all the estimates improve quickly with the number of time periods. Also we observe that size distortions are bigger for δ0= 1 than for δ0= 0.5,but corrections perform well in any case. What it is important is that we obtain considerable bias reduction for a Tas small as 6 or 8. Table D.4 lists the means of the empirical sampling distribution for the ML and biascorrected ML estimators in Design 3. In this design, we simulate the data generating process with ηi0= 0,∀i, but we estimate as if we would have individual heterogeneity in the data. Even in this case the fixed effects MLE is severely biased. BC estimates offer a great improvement over ML. 108 Chapter 4 and now Υi(θ, αi) is obtained as ¯ Υiθ, αi;ˆ θ, bαi=1 M M X m=1 Υm i(θ, αi), where Υm i(θ, αi) = 1 T T X t=1 T X s=1 "∂ℓit ∂αi− 1 T T X r=1 ∂ℓir ∂αi!#·"∂ℓit ∂α′ i− 1 T T X r=1 ∂ℓir ∂α′ i!#. Again, BC-C-Trimming, bθ1c,and BC-C-Expectation, bθ2c,maximize bθjc = arg max θ n X i=1 ℓi(θ, bαi(θ)) + βji (θ, bαi(θ)),with (j= 1,2) . And BC-E-Trimming, eθ1, and BC-E-Expectation, eθ2,would be estimators of the form   e θj eαj   = arg max θ,α n X i=1 ℓi(θ, αi) + βji (θ, αi) or equivalently, eθj= arg max θ n X i=1 ℓi(θ, eαi(θ)) + βji (θ, eαi(θ)),with (j= 1,2) . Table D.8 lists the means of the empirical sampling distribution for the ML and bias-corrected ML estimators in Design 7. Adding more fixed effects clearly increases complexity since the model is more demanding in terms of T. This is reflected in the fact that size distortions are bigger than for the scalar case (Table D.6). In any case, we obtain improved estimates applying the corrections based on trimming (again slightly better for the BC-E than BC-C). Regarding the correction based on expected terms, further research is needed in terms of the number of simulated samples required to obtain 4.6. Conclusions 109 the expectations since the results are very sensitive to the value of this parameter. In the scalar case with values of this parameter Maround 200-300 was enough for obtaining negligible differences across designs. 4.6 Conclusions In this chapter, we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminates the estimates of the common parameters due to the nonlinearity of the problem. We show how to use an iterated algorithm which simplifies estimation in a nonlinear model with multiple fixed effects and we also discuss its application to bias corrected concentrated likelihoods. Simulations show that the estimator proposed is not only computationally convenient but it is also as good as others in a variety of probit designs. Different adjustments of the likelihood function result in bias corrected estimators that perform comparably to other bias corrections proposed in the literature. We can think in many microeconometric applications that use nonlinear panel data models. The results of the chapter suggest that bias corrected estimates will be very useful in relevant empirical settings given the sample sizes of the panels more often used by researchers and, moreover, because they allow us to introduce more individual heterogeneity to address endogeneity concerns in a robust way. 110 Chapter 4 Appendix of Chapter 4 4.A Newton-Raphson iteration The Kth step of the Newton-Raphson iteration takes the form δK=δK−1−∂2L(δK−1) ∂δ∂δ′−1∂L (δK−1) ∂δ , or for shortness ∆δ=−∂2L ∂δ∂δ′−1∂L ∂δ where L(δ) = Pn i=1 ℓi(θ, αi) and ∂L ∂δ =          ∂L ∂θ ∂L ∂α1 . . . ∂L ∂αn          = T X t=1          Pn i=1 ∂ℓit(θ,αi) ∂θ ∂ℓ1t(θ,α1) ∂α1 . . . ∂ℓnt(θ,αn) ∂αn          =  dθ dα    ∂2L ∂δ∂δ′= T X t=1          Pn i=1 ∂2ℓit(θ,αi) ∂θ∂θ′ ∂2ℓ1t(θ,α1) ∂θ∂α′ 1... ∂2ℓnt(θ,αn) ∂θ∂α′ n ∂2ℓ1t(θ,α1) ∂α1∂θ′ ∂2ℓ1t(θ,α1) ∂α1∂α′ 10 . . .. . ..... . . ∂2ℓnt(θ,αn) ∂αn∂θ′0... ∂2ℓnt(θ,αn) ∂αn∂α′ n          =  Hθθ Hθα H′ θα Hαα    and dα=      dα1 . . . dαn      , Hαα =      Hαα10 . . ..... . . 0. . . Hααn      , Hθα =Hθα1. . . Hθαn  4.A. Newton-Raphson iteration 111 so that dθ=Pn i=1 dθi and Hθθ =Pn i=1 Hθθi, and HθαH−1 αα =Hθα1H−1 αα1. . . HθαnH−1 ααn  HθαH−1 αα Hαθ = n X i=1 HθαiH−1 ααiHαθi Letting   Hθθ Hθα H′ θα Hαα    −1 =  Hθθ Hθα Hθα′Hαα    where Hθθ =Hθθ −HθαH−1 αα Hαθ−1 Hθα =−HθθHθαH−1 αα Hαα =H−1 αα +H−1 αα HαθHθθHθαH−1 αα the partitioned formula gives:   ∆θ ∆α  =−  Hθθ Hθα Hθα′Hαα     dθ dα    or ∆θ=−Hθθdθ+Hθαdα ∆α=−Hθα′dθ+Hααdα. 112 Chapter 4 We have Hθθ ="n X i=1 Hθθi −HθαiH−1 ααiHαθi#−1 Hθαdα=−HθθHθαH−1 αα dα=−Hθθ n X i=1 HθαiH−1 ααidαi and −∆θ=Hθθdθ+Hθαdα=Hθθdθ−Hθθ n X i=1 HθαiH−1 ααidαi =Hθθ dθ− n X i=1 HθαiH−1 ααidαi! so that ∆θ=−"n X i=1 Hθθi −HθαiH−1 ααiHαθi#−1n X i=1 dθi −HθαiH−1 ααidαi. Similarly, we have7 ∆α=−H−1 αα (dα+Hαθ∆θ) so that ∆αi=−H−1 ααi (dαi +Hαθi∆θ),(i= 1, ..., n) 7Note that −∆α=Hθα′dθ+Hααdα =−H−1 αα HαθHθθdθ+H−1 αα +H−1 αα HαθHθθHθαH−1 αα dα =H−1 αα dα+HαθHθθHθαH−1 αα dα−HαθHθθdθ =H−1 αα dα−HαθHθαdα−HαθHθθdθ =H−1 αα dα−Hαθ Hθαdα+Hθθdθ =H−1 αα (dα+Hαθ∆θ). 4.B. A regression-based iteration 113 4.B A regression-based iteration Alternatively, we may consider a Gauss-Newton approach after enforcing block diagonality. The motivation is the same as in Berndt, Hall, Hall, and Hausman (1974) in that the nonzero components of the Hessian are approximated by outer product terms. The advantages of this procedure are that it only requires first derivatives and that it leads to a regression-based iteration. Let us introduce the notation dθit =∂ℓit (θ, αi) ∂θ , dαit =∂ℓit (θ, αi) ∂αi Ψθθi = T X t=1 dθitd′ θit,Ψααi = T X t=1 dαitd′ αit,Ψθαi = T X t=1 dθitd′ αit. The Kth step of the iteration of the Gauss–Newton algorithm for obtaining bθand bαtakes the form θ[K]−θ[K−1] =−"n X i=1 Ψθθi −ΨθαiΨ−1 ααiΨαθi#−1n X i=1 dθi −ΨθαiΨ−1 ααidαi αi[K]−αi[K−1] =−Ψ−1 ααi dαi + Ψαθi θ[K]−θ[K−1],(i= 1, ..., n) where all derivatives are evaluated at θ[K−1] and αi[K−1]. Thus, θ[K]−θ[K−1] =− n X i=1 T X t=1 e dθit e d′ θit!−1n X i=1 T X t=1 e dθit where e dθit =dθit −e Πidαit and e Πi= ΨθαiΨ−1 ααi, 114 Chapter 4 so that the e dθit are the residuals of individual-specific regressions of dθit on dαit. Next, θ[K]−θ[K−1] can be calculated as a pooled regression of minus one on e dθit. Finally, αi[K]−αi[K−1] can be obtained as a regression of −1 + d′ θit θ[K]−θ[K−1]on dαit: αi[K]−αi[K−1] =− T X t=1 dαitd′ αit!−1T X t=1 dαit 1 + d′ θit θ[K]−θ[K−1],(i= 1, ..., n). General Conclusions This doctoral thesis considers new models and estimation methods for the analysis of the wage distribution and the labour market histories, from a dynamic perspective. The first chapter studies gender differences in wage growth and job mobility in the initial stages of workers’ careers. Similar studies has been conducted on data from US, Italy and Finland, and in chapter 1, I extend the analysis to data from the Spanish section of the European Community Household Panel (1994-2001). First, I propose an experience measure that - as opposed to the conventional potential experience variable - considers the existence of discontinuities in the professional career of workers. Secondly, I analyse gender differences in job mobility patterns among young workers. From the comparison between the proposed experience measure - accumulated experience - and the one used normally - potential experience - it turns out that wage returns to experience are higher with the more accurate measure and that difference is greater for women than men. This result suggests the existence of a gender wage penalty to interruptions. Regarding job changes, the findings indicate that turnover rates are similar for men and women among young workers. Differences come from the side of some characteristics that are relevant for one of the two groups and not for the other, specially in case of promotion or in transitions to non-employment. For men, holding a position with responsibility or having a family it turns out to be important when changing job. On the contrary, for women it is relevant the type of journey or the size of the firm. Finally, in addition to the gender penalty to interruptions, I also find that early-career wage growth is greater for men than 115 116 General Conclusions for women, and this is specially true in years when job changes occur. In the second chapter, I consider a model for the heterogeneity and dynamics of the conditional mean and the conditional variance of individual wages. In particular, I propose a dynamic panel data model with individual effects both in the mean and in a conditional ARCH type variance function. I posit a distribution for earning shocks and build a modified likelihood function for estimation and inference in a fixed-Tcontext. Using a newly developed bias-corrected likelihood approach it is possible to reduce the estimation bias to a term of order 1/T2. The small sample performance of bias corrected estimators is investigated in a Monte Carlo simulation study. The simulation results show that the bias of the maximum likelihood estimator is substantially corrected for designs that are broadly calibrated to the Panel Study of Income Dynamics (PSID). The empirical analysis is conducted on data drawn from the 1968-1993 PSID. Focusing on US data is interesting because we do not know much how the volatilities of individual wages behave in a period of increasing aggregate inequality. In the data, I find that it is important to account for individual unobserved heterogeneity and dynamics in the variance, and that the latter is driven by job mobility. I also find that the model explains the nonnormality observed in logwage data. Finally, the chapter includes an illustration of some implications that ARCH effects would have on consumption growth. The main conclusion is that an increase in individual risk induces a reduction on current consumption, and this effect is more important for the less educated people, slightly significant for the graduate and insignificant for the college educated. This result goes in line with the idea that there are more insurance possibilities for these latter. The third chapter presents a model that explicitly considers job changes in the dynamics of wages and in the heterogeneity pattern. I propose a specification with two different parameters to capture dynamics within jobs and across jobs, and where the unobserved heterogeneity shows a richer pattern, as well, composed of both individual and job-specific General Conclusions 117 effects. The potential endogeneity of job mobility in relation to earnings is circumvented using an instrument variable estimation method that controls for those unobserved heterogeneity components. In the empirical application, I use data on work histories drawn from the PSID. Regarding results, once we control for individual and job-specific effects, the dynamics within jobs is almost zero, whereas across jobs is significant but small. For the dynamics, the distinction between voluntary and involuntary transitions turns out to be irrelevant. However, that distinction matters in the case of the components of the cross-sectional variance. The estimated variance of the job-specific effects represents around one third of the variance for the individual fixed effects. If I consider a subsample that only includes involuntary job changes, the estimated variance of the heterogeneity across jobs increases up to one half. A natural next step in my research agenda would be the comparison of the results from chapters 2 and 3 with the corresponding to European countries. Another interesting extension would be the consideration of the labour market participation decision and, thus, the inclusion of women and transitions job-to-nonemployment and nonemploymentto-job into the analysis. The fourth chapter is mainly a methodological contribution, related to the computational calculation in practice of bias corrections of the type presented in chapter 2. In particular, chapter 4 considers estimation of nonlinear panel data models that include multiple individual fixed effects. 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Appendix A Tables of Chapter 1 127 Table A.1: Descriptive Statistics Observations Males Females individual-year Mean SD Mean SD Number of individuals 543 577 Number of observations 1537 1726 Age (last observation) 23.12 3.82 25.09 4.23 Children 0.02 0.13 0.09 0.28 Married 0.06 0.23 0.21 0.41 Hours per week 39.86 7.79 37.29 10.20 Hourly Wage (pta. 1992) 633.45 287.53 601.94 343.58 Accumulated Experience 1.08 1.28 1.04 1.23 Potential Experience 3.86 2.73 4.82 4.18 Tenure 0.75 1.16 0.74 1.08 Years of schooling 12.85 3.45 13.64 3.54 Primary Education 0.26 0.45 0.19 0.39 Graduate Education 0.35 0.47 0.28 0.45 College Education 0.39 0.49 0.53 0.50 Part-time 0.12 0.33 0.25 0.44 Temporary 0.44 0.49 0.40 0.49 Managers and Professionals 0.22 0.42 0.28 0.45 Clerical and Services 0.17 0.38 0.46 0.50 Agriculture and Manufacture 0.37 0.48 0.09 0.29 Unskilled workers 0.24 0.42 0.17 0.37 More than 50 wage-earners 0.31 0.46 0.31 0.46 Between 5 and 49 wage-earners 0.51 0.50 0.41 0.49 Between 1 and 4 wage-earners 0.18 0.39 0.28 0.45 SD: Standard deviation. Table A.2: Proportion of males/females that work more than a fraction of time # months 6 18 30 42 54 Males 90.65 84.17 71.22 62.59 38.13 Females 81.17 63.64 46.10 33.70 22.08 Note: fraction of time measured as the number of months employed in the last 5 years. 130 Appendix A Table A.3: OLS Regressions by gender. Accumulated Experience Dependent variable: logwage rate [I] Males [I] Females [II] Males [II] Females Accumulated Experience 0.061*** 0.059*** 0.057*** 0.057*** [0.022] [0.022] [0.022] [0.022] (Accumulated Experience)2-0.005 -0.007 -0.005 -0.008* [0.004] [0.005] [0.004] [0.005] Difference Potential - 0.005 -0.008* Acccumulated Experience [0.009] [0.004] Delayed Entrance to Market 0.034 0.019 [0.046] [0.026] Birth year: 1955-1973 0.177*** 0.157*** 0.088* 0.182*** [0.044] [0.039] [0.046] [0.035] Birth year: 1974-1977 0.080** 0.048 0.065* 0.063** [0.033] [0.035] [0.034] [0.028] Married -0.041 0.046 -0.033 0.051* [0.053] [0.037] [0.052] [0.030] Children 0.195** -0.011 0.163 0.005 [0.097] [0.046] [0.100] [0.043] Graduate Education 0.077** 0.045* 0.082** 0.026 [0.034] [0.027] [0.033] [0.029] College Education 0.170*** 0.141*** 0.193*** 0.151*** [0.034] [0.028] [0.037] [0.031] Tenure: <1 year -0.039 -0.045 -0.045 -0.044 [0.037] [0.032] [0.037] [0.032] Tenure: 1 - 2 years -0.010 -0.033 -0.016 -0.032 [0.031] [0.030] [0.031] [0.030] Civil servant 0.211*** 0.208*** 0.217*** 0.204*** [0.052] [0.051] [0.052] [0.038] Temporary -0.062*** -0.005 -0.059*** -0.007 [0.025] [0.027] [0.025] [0.024] Part-time 0.118** 0.103*** 0.115** 0.103*** [0.050] [0.029] [0.050] [0.027] >50 wage-earners 0.233*** 0.205*** 0.230*** 0.204*** [0.041] [0.031] [0.041] [0.026] 5 - 49 wage-earners 0.126*** 0.146*** 0.124*** 0.146*** [0.036] [0.027] [0.036] [0.024] Managers and Professionals 0.153*** 0.305*** 0.161*** 0.293*** [0.045] [0.044] [0.045] [0.039] Clerical and Services -0.045 0.030 -0.046 0.023 [0.034] [0.033] [0.033] [0.030] Agriculture and Manufacture 0.025 0.034 0.028 0.029 [0.027] [0.039] [0.027] [0.035] Constant 5.987*** 5.873*** 5.967*** 5.897*** [0.070] [0.087] [0.070] [0.080] Observations 973 1040 973 1040 R20.43 0.47 0.43 0.47 Note: ***, **, * significant at 99%, 95%, 90% level, respectively. Robust standard errors in brackets. Year and region dummies included. Omitted group: Birth year>1977, primary education, more than 2 years of tenure, between 1 and 4 wage-earners, unskilled. 131 Table A.4: OLS Regressions by gender. Potential Experience Dependent variable: logwage rate [I] Males [I] Females [II] Males [II] Females Potential Experience 0.033*** 0.011* 0.033*** 0.012* [0.011] [0.006] [0.011] [0.006] (Potential Experience)2-0.002* -0.001*** -0.002* -0.001*** [0.001] [0.000] [0.001] [0.000] Interruptions 0.003 -0.053** [0.021] [0.023] Birth year: 1955-1973 0.093** 0.187*** 0.093** 0.184*** [0.047] [0.034] [0.047] [0.034] Birth year: 1974-1977 0.063* 0.068** 0.063* 0.065** [0.034] [0.027] [0.034] [0.027] Married -0.024 0.057* -0.023 0.058** [0.054] [0.059] [0.054] [0.029] Children 0.202*** 0.010 0.201*** 0.013 [0.101] [0.043] [0.101] [0.043] Graduate Education 0.085** 0.029 0.085** 0.027 [0.033] [0.029] [0.034] [0.028] College Education 0.199*** 0.120*** 0.200*** 0.121*** [0.037] [0.031] [0.037] [0.031] Tenure: <1 year -0.081** -0.087*** -0.083** -0.053 [0.031] [0.030] [0.035] [0.032] Tenure: 1 - 2 years -0.042 -0.058** -0.044 -0.031 [0.030] [0.029] [0.031] [0.031] Civil servant 0.213*** 0.197*** 0.213*** 0.198*** [0.052] [0.038] [0.052] [0.038] Temporary -0.058** -0.011 -0.058** -0.008 [0.025] [0.024] [0.025] [0.024] Part-time 0.109** 0.094*** 0.109** 0.097*** [0.051] [0.027] [0.051] [0.027] >50 wage-earners 0.234*** 0.204*** 0.234*** 0.201*** [0.032] [0.026] [0.041] [0.026] 5 - 49 wage-earners 0.127*** 0.145*** 0.127*** 0.143*** [0.030] [0.024] [0.037] [0.024] Managers and Professionals 0.163*** 0.297*** 0.163*** 0.297*** [0.045] [0.039] [0.045] [0.039] Clerical and Services -0.044 0.024 -0.044 0.023 [0.034] [0.030] [0.034] [0.030] Agriculture and Manufacture 0.036 0.040 0.036 0.041 [0.027] [0.035] [0.027] [0.035] Constant 5.957*** 6.178*** 5.957*** 5.926*** [0.075] [0.083] [0.075] [0.080] Observations 973 1040 973 1.040 R20.43 0.47 0.43 0.47 Note: ***, **, * significant at 99%, 95%, 90% level, respectively. Robust standard errors in brackets. Year and region dummies included. Omitted group: Birth year>1977, primary education, more than 2 years of tenure, between 1 and 4 wage-earners, unskilled. 132 Appendix A Table A.5: Effect on the log-wage of a marginal change in the experience Years of Accumulated Experience Potential Experience [I] [II] [I] [II] Males Females Males Females Males Females Males Females x=0 0.061*** 0.059*** 0.057*** 0.057*** 0.033*** 0.011*0.033*** 0.012* [0.006] [0.008] [0.010] [0.008] [0.003] [0.085] [0.003] [0.060] x=1 0.051*** 0.045*** 0.047*** 0.041*** 0.029*** 0.009*0.029*** 0.010* [0.002] [0.003] [0.003] [0.003] [0.002] [0.100] [0.002] [0.082] x=2 0.041*** 0.031*** 0.037*** 0.025*** 0.025*** 0.007 0.025*** 0.008 [0.002] [0.007] [0.002] [0.008] [0.001] [0.162] [0.001] [0.119] x=3 0.031** 0.017 0.027** 0.009 0.021*** 0.005 0.021*** 0.006 [0.040] [0.272] [0.034] [0.383] [0.001] [0.238] [0.001] [0.181] x=4 0.021 0.003 0.017 -0.007 0.017*** 0.003 0.017*** 0.004 [0.366] [0.943] [0.311] [0.833] [0.001] [0.363] [0.001] [0.286] Mean 0.049*** 0.042*** 0.045*** 0.037*** 0.017*** 0.002 0.017*** 0.003 [0.002] [0.002] [0.002] [0.003] [0.001] [0.440] [0.001] [0.353] Note: ***, **, * significant at 99%, 95%, 90% level, respectively. p-values in brackets. Mean: average experience value in the sample. 133 Table A.6: GLS/IV Regressions by gender Dependent variable: logwage rate Males Females Accumulated Experience 0.068*** 0.072*** [0.026] [0.026] Experience squared -0.006 -0.010*** [0.003] [0.004] Birth year: 1955-1973 0.090 0.216** [0.148] [0.093] Birth year: 1974-1977 0.046 0.078 [0.104] [0.080] Married 0.037 0.058* [0.060] [0.035] Children 0.415*** -0.039 [0.099] [0.059] College Education 0.405** 0.292* [0.203] [0.158] Tenure: <1 year -0.028 -0.021 [0.035] [0.030] Tenure: 1 - 2 years 0.020 -0.018 [0.029] [0.026] Civil servant 0.060 0.089** [0.052] [0.045] Temporary 0.009 -0.006 [0.023] [0.021] Part-time 0.108** 0.149*** [0.051] [0.028] >50 wage-earners 0.114*** 0.095*** [0.035] [0.034] 5 - 49 wage-earners 0.074** 0.072*** [0.032] [0.028] Managers and Professionals -0.029 -0.016 [0.055] [0.053] Clerical and Services -0.042 -0.039 [0.042] [0.045] Agriculture and Manufacture 0.036 0.014 [0.031] [0.048] Regional unemployment rate -0.005 -0.003 [0.004] [0.004] Constant 5,967*** 5,856*** [0.127] [0.136] Observations 973 1040 Number of individuals 425 459 Note: ***, **, * significant at 99%, 95%, 90% level, respectively. Robust standard errors in brackets. Year dummies included. Omitted group: Birth year>1977, non college education, more than 2 year of tenure, between 1 and 4 wage-earners, unskilled.