Age Structure Effects and Consumption in Norway, 1968(3) - 1998(4)

Erlandsen, Solveig K. Working Paper Age Structure Effects and Consumption in Norway, 1968(3) - 1998(4) Working Paper, No. 2003/1 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Erlandsen, Solveig K. (2003) : Age Structure Effects and Consumption in Norway, 1968(3) - 1998(4), Working Paper, No. 2003/1, ISBN 82-7553-208-6, Norges Bank, Oslo, https://hdl.handle.net/11250/2498627 This Version is available at: https://hdl.handle.net/10419/209815 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no ANO 2003/1 Oslo February 6, 2003 Working Paper Research Department Age structure effects and consumption in Norway, 1968(3) – 1998(4) by Solveig Erlandsen ISSN 0801-2504 ISBN 82-7553-208-6 Working papers from Norges Bank can be ordered by e-mail: [email protected] or from Norges Bank, Subscription service, P.O.Box. 1179 Sentrum N-0107 Oslo, Norway. Tel. +47 22 31 63 83, Fax. +47 22 41 31 05 Working papers from 1999 onwards are available as pdf-files on the bank’s web site: www.norges-bank.no, under "Published". Norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. Working papers fra Norges Bank kan bestilles over e-post: [email protected] eller ved henvendelse til: Norges Bank, Abonnementsservice Postboks 1179 Sentrum 0107 Oslo Telefon 22 31 63 83, Telefaks 22 41 31 05 Fra 1999 og senere er publikasjonene tilgjengelige som pdf-filer på www.norges-bank.no, under "Publisert". Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Age structure effects and consumption in Norway, 1968(3)—1998(4)∗ Solveig Erlandsen Abstract In this paper the effects of a changing age distribution on aggregate consumption are analysed. This is done by estimating a Norwegian consumption function which controls for age structure effects. The model is estimated on quarterly time series data from 1968(3) to 1998(4). The results show that changes in the age composition affect aggregate consumption significantly, giving support to the predictions of the Life Cycle Hypothesis that young adults and old persons have a higher average propensity to consume than the middle-aged. The consumption model encompasses a model which does not control for age composition effects. Keywords: Demography, consumption, time series models. JEL classification: C22, E21, J1. ∗This paper is a revised version of my Master dissertation at the University of Oslo. Comments and suggestions from my supervisor, Ragnar Nymoen, are gratefully acknowledged. I would also like to thank Farooq Akram, Gunnar Bårdsen, Ida Wolden Bache, Øyvind Eitrheim, Michael Riis Jacobsen, Terje Skjerpen, Ingvild Svendsen and Fredrik Wulfsberg for valuable comments on econometric issues and computer software. Comments from seminar participants at two seminars in Norges Bank are also gratefully acknowledged. The work on this dissertation has been done during my student internship in Research Department, Norges Bank. In addition to financial support, Norges Bank has also provided excellent working conditions. The views expressed are those of the author and should not be interpreted as reflecting those of Norges Bank. 1 Introduction Economic theory predicts that the age composition of the population affects macroeconomic variables. The Life Cycle Hypothesis (LCH),1for instance, suggests that an individual’s consumption and saving decisions are dependent upon the individual’s age. This follows from the consumption smoothing assumption of the theory, which implies that an individual borrows when young, saves as middle-aged, and dissaves when old. And, since aggregate consumption is the sum of all the individuals’ consumption, the age composition of the population is thus expected to influence aggregate consumption.2 In this paper I investigate empirically whether changes in the age structure of the population affect Norwegian aggregate consumption. Like many other Western countries, Norway experienced low birth rates in the 1930s and a baby boom in the 25 years following World War II. As a result, the age composition of the Norwegian population have exhibited large variations over the last decades, making it possible to identify potential age structure effects. The effects of a changing age distribution on consumption are investigated by introducing an age structure variable in a consumption function. The age structure variable is a modified version of the variable suggested by McMillan and Baesel (1990). It is constructed as the number of persons in age groups which are expected to have low propensity to consume to the number of persons in age groups which are expected to have high propensity to consume. Hence, the variable can be considered as a life cycle measure. To estimate the consumption model the Johansen method is applied to investigate whether a single-equation method can estimate the consumption model efficiently. Then, as it turns out that it can, the consumption function is modelled and estimated by a single-equation method. All the estimations are based on quarterly time series data from 1968(3) to 1998(4). Despite the theoretical prediction that changes in the age composition of the population can affect consumption, demographic changes are not often taken into account in empirical consumption studies. On Norwegian data, Frøiland (1999) is one of few studies that includes a demographic variable in an empirical consumption function. By applying quarterly data for the period 1967(3)-1997(3), he finds significant short-run effects of changes in the age structure on aggregate non-housing consumption. Other studies have investigated the consumption behaviour of different socio-economic groups by using Norwegian time series data. For instance, Magnussen (1994) finds that there are considerably differences between the income elasticities of pensioners and workers, while Brubakk (1994) reports that the properties of an empirical consumption model for non-durables and services improve when the income growth of wageearners and self-employed workers enter as regressors. Brubakk’s (1994) study is not directly related to age composition effects. Its starting point is nevertheless similar to the one in this paper, namely that since consumers are heterogeneous, changes in the composition of the different groups of consumers may affect aggregate consumption. The number of empirical studies which takes into account age structure effects on aggregate consumption is also relatively small on international data.3Among these is Berg (1996), 1Modigliani and Brumberg (1954, 1979) and Ando and Modigliani (1963). 2This argument must, however, not always hold, as changes in the age structure of the population also can effect other equilibrating mechanisms in the economy. As noted by Easterlin (1987), a large cohort size can, for instance, reduce the real wage for the members of the cohort, as increased competition for jobs generates a downward pressure on real wages. 3The impact of changes in the age composition of the population on other macroeconomic variables than consumption have been reported in several studies. McMillan and Baesel (1990) investigate for instance age composition effects on GDP, inflation, unemployment, and real interest rates on U.S. data, while Malmberg and Lindh (2000) examine age composition effects on the Swedish saving rate, investment rate, current account, 1 who investigates the relationship between the population age distribution and consumption on Swedish annual data from 1950 to 1995. He concludes that different age shares of the population have significant effects on the demand for ’pure consumption’,4and reports that the estimated coefficients of the different age shares variables are consistent with the predictions oftheLCH.FairandDominguez(1991)studytheeffects of a changing age composition on U.S. consumption data. Using quarterly data from 1954 to 1988 they find strong support for the hypothesis that the age structure of the population affects aggregate consumption. The pattern of the age coefficients Fair and Dominguez (op. cit.) report is in line with the predictions of the LCH.5Heien (1972), who estimates a multiperiod consumption function on annual U.S. data from 1948 to 1965, also reports a significant relationship between the age structure of the population and consumption of non-durables and services. However, his results contradicts those of Fair and Dominguez (1991).6Not all studies find significant age composition effects. Denton and Spencer (1976) is an example of this. By using both Canadian times series data and cross-section data for 21 OECD countries, Denton and Spencer (op. cit.) find that neither the average household size nor the age distribution of the population influence consumption significantly.7 The remainder of the paper is organized as follows. The theoretical background for introducing age structure effects in an aggregate consumption function is presented in Section 2. In Section 3 the empirical methodology is described, while data are presented in Section 4. A Norwegian consumption model which controls for age structure effects is then modelled and estimated in Section 5. Section 6 concludes. 2 Theoretical background Several theories predict that demographic factors affect aggregate consumption. The Life Cycle Hypothesis (LCH), first developed by Modigliani and Brumberg (1954, 1979) and later extended by Ando and Modigliani (1963), is by far the most influential one of these, and the focus here will be on this theory. The LCH is presented below. 2.1 The Life Cycle Hypothesis The LCH shares many common features with Friedman’s (1957) Permanent Income Hypothesis (PIH). The key idea of both theories is that an individual uses saving and borrowing to smooth the path of consumption over the life cycle. There is, however, an important difference between the two theories, namely that the LCH by explicitly recognizing the finite lives of individuals can, as its name implies, deal with the changes in income and consumption ”needs” that usually occur over an individual’s life cycle. In general one can also say that the focus in the LCH and PIH differs; the LCH emphasises the role of age and asset accumulation as determinants of consumption, while the PIH is more concerned with the measurement links GDP and inflation. 4See Berg (op. cit.) for a definition of the pure consumption concept. 5Their results are robust and unambigious for service consumption, nondurable consumption and housing investment, while the results for durable expenditures are mixed. 6More specifically, Heien (op. cit.) finds that consumption expenditures rise as the median age of the population above 24 years of age rises. 7The differences in the findings of these studies may be due to the way the age composition effects are modelled in the consumption functions. This issue is discussed in more detail in Section 3.2. 2 between current and permanent income.8The LCH as well as the PIH are based on utility maximizing consumers who face an intertemporal choice between consumption and saving. By assuming the existence of perfect capital markets the consumer choice theory provides a rationale for a decoupling of the time-pattern of earnings and assets from the desired pattern of consumption. This microeconomic basis of the theories is briefly presented below,9before the special characteristics of the LCH is outlined. Consider a utility maximizing individual who lives for L periods, and who receives utility only from present and future consumption.10 For simplicity the individual’s subjective discount rate is set to zero. The individual’s utility function is assumed to be separable, and hence it can be expressed as (2.1) U= L X t=1 u(ct),u 0(·)>0,u 00(·)<0, where u(·)denotes the instantaneous utility function, i.e. the felicity function, and ctis consumption in period t. Assume that the individual earns income, y,intheLperiodsof her life, and that a0denotes the individual’s initial wealth. The individual takes both income and wealth as given. Capital markets are assumed to be perfect, and the interest rate is for simplicity set to zero. The individual’s intertemporal budget constraint is thus (2.2) L X t=1 ct=a0+ L X t=1 yt. Given the utility function and the intertemporal budget constraint, the Lagrangian for the individual’s maximization problem is (2.3) L= L X t=1 u(ct)+λÃa0+ L X t=1 yt— L X t=1 ct!, and, hence, the first-order condition for ctis (2.4) u0(ct)=λ. Twopointsareworthnotinginequation(2.4). First, that the Lagrange-multiplier, λ,is constant over time, and second, that the marginal utility of consumption is a function only of current consumption. Thus, since the marginal utility of consumption is constant over time, this implies that also consumption must be constant, i.e. c1=c2=... = cL. Given this, equation (2.2) can be rewritten to (2.5) ct=1 LÃa0+ L X τ=1 yτ!for all t, where the term in the parentheses is the individual’s total lifetime resources. Equation (2.5) states thus that consumption in any period tis not dependent on the income earned in that 8The two theories can thus be seen as complements rather than substitutes to each other. 9This presentation is in large based on Romer (1996). 10The individual is assumed not to receive or leave any bequest. 3 period, but on total lifetime resources. Furthermore, it shows that the individual divide consumption equally between the periods. Based on this microeconomic theory, the LCH emphasizes the role of age and wealth as determinants of consumption. The following example of the basic, or ”stripped down” version of the LCH, given in Modigliani and Brumberg (1954), can illustrate this. Consider an individual who lives for 50 years, L = 50, of which she earns income in the first 40 years and isretiredinthelast10years.Theincomeprofile is flat over the earning span, and the income earned each year is denoted by y. The individual has no initial wealth, she leaves no bequest, and interest rates are set to zero. Under the assumption that consumption is constant over the life cycle, the individual will then consume 1 L=1 50 of her total lifetime income each year. Consequently, the individual will accumulate wealth over the earning span. At the time of retirement wealth is at its peak, consisting of 20 per cent of total lifetime income. Thereafter, during the retirement years the wealth gradually decreases as the individual consumes out of it, reaching zero at the end of her life. Two important features of the LCH can be illustrated by this example. First, that the marginal propensity to consume (MPC) out of income depends on the individual’s age. If an unexpected and temporary additional income unit, ∆y,isearned in the first year the individual lives, consumption in this year (and in the following years) will increase by 1 50 of ∆y. If, on the other hand, the temporary income shock occurs when the individual is at age 35, the MPC will be 1 16 . Second, that wealth has an important role in determining consumption. In this version of the LCH both the income stream and the consumption level are assumed to be constant, and known with certainty, over the earning and life spans, respectively. As demonstrated by Modigliani and Brumberg (1979) this assumption as well as other simplifying assumptions of the ”stripped down” version can be replaced by more realistic ones. As pointed out by, among others, Modigliani (1986), labour income is far from being constant over the earning span in average. Rather it tends to increase with experience, reaching its peak either some time before or at the time of retirement. Thereafter it falls as a consequence of retirement. Also consumption varies usually with age. Different ”needs” over the life cycle, for instance because of changes in the family size, is a major reason for this. The life cycle of the family size appears to have a hump-shaped profile, peaking somewhat earlier than income. Rødseth (1992) notes that the concept of consumption smoothing in the LCH hence could be interpreted as consumption relative to ”needs” rather than to the level of consumption. Allowing income to vary over the life cycle may imply that young workers save less than the middle-aged. This is due to that young workers have lower average income than the middleaged, at the same time as their ”needs” are larger because of ”home building” and relatively large family sizes. For the retired persons, both versions of the model predict that they are to dissave. So far the implications of the LCH for the consumption behaviour of an individual, or a household, have been described. Based on the derivation in Ando and Modigliani (1963), a way to aggregate the consumption function over the population is now presented. Given that the utility function of an individual is homogeneous with respect to consumption at different points in time, the consumption function for an individual at age T can be written as a proportional function of the individual’s lifetime resources11 (2.6) cT t=kT tvT t, where cstill denotes consumption, kis the proportionality factor, vis the present value of 11Given that capital markets still are assumed to be perfect. 4 The age share groups are highly correlated with each other, cf. the correlation matrix for thegroupsinTable2. Thetableshowsthattheagegroups0-19and50-66arepositively correlated with each other and negatively correlated with the age groups 20-49 and 67-. 5 Modelling consumption including an age structure variable 5.1 Long-run consumption functions including age share variables As a starting point for the modelling procedure, it is first investigated whether the different age share groups presented in Section 4.2 have significant effects on consumption. The purpose of doing this is twofold. First, to get an early indication of the fruitfulness of modelling age structure effects in the consumption function. And, second, the age share coefficients can add information about how to compose the ”best” age structure variable. This issue will be discussed in more detail in the next section. In a similar manner to what Berg (1996) does, the (logs of) the four age share groups are included separately in four different static long-run consumption regressions. The reason for including the age share groups separately in the regressions is to avoid multicollinearity problems.18 The consumption models are estimated by OLS, and the estimation period is from 1968(3) to 1998(4). The estimation results are presented in Table 3.19,20 The (logs of the) age share variables are denoted by share 0-19, share 20-49, share 50-66 and share 67-, and, as shown in the table, all the age share coefficients are significant on a 1% level. Furthermore, the signs of the coefficients of the latter three variables are in accordance with the predictions of the LCH. The positive coefficient signs of share 20-49 and share 67and the negative coefficient sign of share 50-66 imply that the young workers and old persons have higher aggregate average propensity to consume (APC) than the middle-aged persons. The age share coefficient of children and youths is negative. Theory says little about the effect on consumption of changes in the share of children and youths in the population. However, as children and youths are financially dependent upon their parents in average, it can be argued that aggregate consumption is to decline when the share of these groups increase. The signs of the coefficients on all the age share groups are consistent with Berg’s (1996) findings on Swedish data. The income and wealth elasticities are not very sensitive to the choice of age share variables. As shown in the table, the estimated income elasticities vary between 0.53 and 0.58, while the wealth elasticity is estimated to be around 0.23 when the different age share variables are included in the regressions. Compared to the elasticities reported by Brodin and Nymoen (1992), cf. equation (3.1), the income and wealth elasticities are neither much affected by the introduction of the new variables in the long-run consumption functions. The short-run dynamics are not modelled in the regressions reported in Table 3. The residuals of the model are therefore expected to be autocorrelated. The p-values of the fifth-order residual autocorrelation test statistics, denoted by AR1-5 in the table,21 reveal that this is the case. As mentioned above, one of the purposes of this section is to investigate whether the age 18As reported in Table 2 the age share variables are highly collinear. 19The dummy variables are not reported in the table. 20EViews Version 4.0 is used to compute the coefficients and the standard errors, while PcGive Version 9.3 has computed the AR 1-5 and ADF test statistics. 21See Hendry and Doornik (1996) for details about the test. 11 Table 3: Long-run consumption functions; dependent variable: c (1) (2) (3) (4) Constant 1.99 (0.24) 2.22 (0.29) 1.34 (0.14) 2.50 (0.43) y0.53 (0.04) 0.57 (0.03) 0.58 (0.03) 0.55 (0.04) w0.22 (0.02) 0.23 (0.02) 0.24 (0.02) 0.23 (0.02) share 0-19 −0.37 (0.08) share 20-49 0.31 (0.07) share 50-66 −0.17 (0.04) share 670.23 (0.07) AR 1-5 0.00** 0.00** 0.00** 0.00** ADF -4.92** -4.92** -4.86** -4.57* Seasonal dummy variables (not reported) were also included in the regressions. Newey and West (1987)HAC standard errors in parenthesis. * and ** indicate 5% and 1% significance levels, respectively. AR (1-5) denotes residual autorcorrelation test and ADF denotes the augmented Dickey-Fuller test on residuals. structure of the population affects aggregate consumption rather than to develop a congruent consumption model. Thus, instead of removing the residual autocorrelation by modelling the short-run dynamics, Newey-West heteroscedasticity and autocorrelation consistent (HAC) standard errors22 are applied to obtain reliable t-values. The standard errors are given in parentheses in Table 3. ADF denotes the augmented Dickey-Fuller test statistics for unit roots on the residuals in Table 3. The null hypothesis of unit roots in the residuals is rejected on a 1% level for the three first regression models and on a 5% level for the last model.23 According to the Engle and Granger (1987) test for cointegration, these results suggest that at least one cointegrating relationship is present in each of the models. 5.2 The age structure variable Based on the estimation results in the previous section, an age structure variable is to be included in a dynamic consumption model. By grouping the population in two groups, those with low APC and those with high APC, or, in other words, the age share groups with negative and positive coefficient signs, respectively, in Table 3, the following measure is obtained age =log·Population (0 −19 + 50 −66) Population (20 −49 + 67−)¸. 22Newey and West (1987). 23The critical values are —4.79 and —4.19 at 1% and 5% levels, respectively. These values are taken from MacKinnon (1991). 12 Figure 3: Age structure variable, age 1970 1975 1980 1985 1990 1995 2000 0 .70 0 .75 0 .80 0 .85 0 .90 0 .95 1.00 1.05 AGE When such a variable is constructed and used in regressions two points are worth noting. First, that the sum of the freely estimated partial effects of the different age groups not necessarily are the same as the effect of the composite age measure. Nevertheless, the measure is expected to have a negative effect on consumption since the age groups with low APC are in the numerator. Second, that the measure imposes restrictions on the coefficients of the different age groups relative to each other.24 Figure 3 plots the age structure variable over the period 1968(3)-1998(4). It shows that the number of childeren, youths and middle-aged persons relative to the rest of the population was decreasing until the beginning of the 1990s, and that it has been rising since then. 5.3 Multivariate cointegration analysis25 Efficient estimation of a dynamic consumption model in a single-equation approach relies on the model containing only one cointegration vector and that the right-hand side variables are weakly exogenous. In this section, it is investigated whether these criteria are satisfied for the consumption model by applying the Johansen multivariate system method.26 24By applying a χ2-distributed likelihood ratio (LR) test in preliminary multivariate estimations it has been tested whether a symmetry restriction on the coefficients of the two age groups share (20-49 + 67-) and share (0-19 + 50-66) is valid. It turns out that it is. Although these two variables are not exactly the same as those in age, the test indicates that the symmetry restriction on the coefficients of the variables in the numerator and denumerator in age also is valid. 25The estimations in Section 5.3 are carried out in PcFiml Version 9.3. 26See Johansen (1988). 13 Table 4: System diagnostic tests for fifth-order VAR Vector AR 1-5 F(45, 244) 1.185 [0.210] Vector normality χ2(6) 7.788 [0.254] Vector heteroscedasticity F(204, 357) 0.663 [0.999] 5.3.1 Formulating the VAR The starting point for the Johansen approach is the formulation of the unrestricted VAR. The approach requires that the VAR is correctly specified, and accordingly, that the residuals of the model are white noise. The specification of the VAR and the results of the vector error mis-specification tests are shown below. As is common when dealing with quarterly data, five lags on each of the potentially endogenous variables of the system, namely on ct,ytand wt, are initially included. The demographic variable agetis also expected to influence the long-run equilibrium of the model. It is, however, assumed to be weakly exogenous to the system,27 and hence it enters the VAR as a non-modelled variable. To allow for linear trends in the levels of the data, a constant term is included unrestrictedly in the system. In addition to the constant term, three centered seasonal dummy variables and two other dummy variables, named VAT and AUDI,enter the VAR unrestrictedly. The reason for using centered seasonal dummies is that, since they sum to zero over time, they do not influence the asymptotic distributions of the reduced rank test statistics. VAT is a dummy for the introduction of the VAT in 1970, while the AUDI variable consists of the two variables ∆STOP, which is an income policy dummy for the wage and price freeze in 1978,28 and the change in inflation, ∆∆4cpi.29 Since the VAR includes both dummies and a non-modelled stochastic variable (age), the available critical values of the trace-statistic are probably somewhat removed from the (unknown) true critical values. However, the inclusion of age which comes close to being I(1) over the sample, suggest that we follow the recommandations of Harbo et al. (1998) and include a linear trend component, restricted to lie in the cointegration space, in the VAR. Table 4 reports the system mis-specification tests for the VAR over the sample period 1968(3)-1998(4). As the names of the tests imply, the tests checks for vector error autocorrelation, non-normality and heteroscedasticity.30 None of the tests indicate vector error mis-specifications, and hence this fifth-order VAR is the basis for the cointegration analysis. 5.3.2 Testing for cointegration Having specified the VAR, the next step is to determine the cointegration rank of the system. The results of the Johansen test is presented in Table 5. In the table, λtrace denotes the observed trace statistic, while λtrace,adj. is its small sample adjusted counterpart.31 Since the 27The assumption that the age structure of the population is weakly exogenous to the system does not necessarily hold, as economic factors, such as income, wealth and consumption, can influence the birth and death rates of the population. These effects are however expected to be small, and hence they are neglected here. 28See Appendix A for details on the dummy variables. 29More specifically, AUDI=∆STOP —∆∆4cpi. This symmetry restriction on the coefficients is imposed in Eitrheim et al. (2002), and to simplify the modelling procedure I do the same. 30See Doornik and Hendry (1997) for details of the tests. 31The adjusted test statistic is based on Reimers (1992). 14 Table 5: Johansen tests for cointegration Eigenvalues λtrace λiH0H1λtrace λtrace,adj. 95% 0.1914 r = 0 r ≥1 46.33 40.64 49.6 0.1045 r ≤1r≥2 20.42 17.91 30.5 0.0553 r ≤2r≥36.95 6.10 15.2 Figure 4: Unrestricted cointegration vectors 1970 1975 1980 1985 1990 1995 2000 2.1 2.2 2.3 vector1 1970 1975 1980 1985 1990 1995 2000 5.9 6 6.1 vector2 1970 1975 1980 1985 1990 1995 2000 -9.5 9 .25 -9 vector3 model is conditioned upon the age variable, the asymptotic critical values at 5% significance level, denoted by 95% in the table, are taken from Harbo et al. (1998). These are, however, only indicative since dummy variables enter the system. Strictly interpreted the results of the trace test indicate that there are no cointegrating vectors in the system. Being aware of, however, that the critical values only are indicative, it can be argued that there is a cointegrating vector in the system. Figure 4, which plots the unrestricted cointegrating vectors, supports this argument. The first vector seems clearly to be stationary, while it is more difficult to judge whether the last two vectors are stationary or not. As the argument of the existence of a cointegrating vector also is supported by economic theory, I proceed assuming that there is a cointegrating vector in the system. Brodin and Nymoen (1992) and Eitrheim et al. (2002), who estimate a Brodin and Nymoen (1992) type of consumption function on Norwegian data for the period 1968(3)-1998(4), also find little formal support for cointegration. Given the stability of their long run relationships over a decade, one may suspect that the formal test has little power. The linear trend component was included in the VAR for inference purposes in the trace test. By imposing the restriction that there is a cointegrating vector, a χ2-distributed LR test 15 Table 6: Normalized eigenvectors, β cywage 1.000 -0.609 -0.190 0.150 -5.099 1.000 2.984 0.488 -8.208 7.012 1.000 -4.280 Table 7: Testing the significance of each of the variables in the cointegrating vector cywage χ2(1) 16.078 16.945 10.877 11.205 p-value 0.000** 0.000** 0.001** 0.001** can be used to test for the significance of the trend variable in the system. The p-value of the test statistic is 0.55, and hence the null hypothesis that the coefficient of the trend is equal to zero is accepted. The trend is therefore omitted from the I(0)-system in the following analysis. In Table 6 the estimated eigenvectors β’, when the trend variable is dropped from the VAR, are presented. The β-matrix is reported in normalized form, with one of the elements in each row of β’ set equal to 1. As the table reports, normalizing the estimated cointegrating vector on consumption gives the following long-run relationship (5.1) ct=0.609yt+0.190wt−0.150aget. The equation shows that the coefficient of the age structure variable has, as expected, a negative sign. Hence, when the number of children, youths and persons between 50 and 66 years of age increases with one percent relative to the rest of the population, consumption is expected to decrease by 0.15 percent in the long-run, given that yand ware constant. Comparing the estimated elasticities with those of Eitrheim et al. (2002), who report longrun income and wealth elasticities on Norwegian data to be 0.65 and 0.23, respectively, the introduction of the age variable seems to reduce the impact of both income and wealth on consumption. The LR tests for the significance of each of the variables in the cointegration vector show that all the variables are significant on a 1% level, cf. Table 7. Table 8 reports the estimated adjustment, or loading, coefficients α. The estimated adjustment coefficient for the consumption equation is —0.655, showing that consumption adjusts relatively fast to changes in the underlying equilibrium relationship. Table 8: Loading factors, α c -0.655 -0.002 -0.001 y -0.108 -0.004 -0.003 w -0.084 -0.026 0.000 16 A test for weak exogeneity of ytand wtcorresponds to a test of the significance of their respective rows in Table 8. By transforming the system to an I(0) system, a LR test can be used to test for weak exogeneity. Table 9 reports the results of this test, and it shows that the joint test of ytand wtbeing weakly exogenous to the system is accepted, with a p-value at 0.82. Table 9: Joint test for weak exogeneity (y, w) χ2(2) 0.400 p-value 0.818 Having found a cointegrating vector and that ytand wtare weakly exogenous in the system, the consumption function can be estimated efficiently by a single-equation approach. 5.4 Single-equation approach In this section, the consumption function is estimated as an equilibrium correction model (EqCM) with the long-run relationship found in the system analysis being imposed on it. A general-to-specific modelling strategy is used as a guideline for reducing the model from the general unrestricted model (GUM) to the final model.32 The GUM is formulated such that it is consistent with the VAR system outlined in Section 5.3, and hence the first-difference of each of the variables ct,ytand wtenter the model with four lags. The long-run, or equilibrium correction, relationship in equation (5.1) is imposed on the GUM. To simplify the modelling procedure, the same lag lengths as in Eitrheim et al. (2002) is used in the EqCM term, and hence the term enters the GUM in the following way (5.2) EqCMt=ct−0.609yt−4−0.190wt+0.150aget. The GUM is estimated by OLS, and the estimation period is from 1968(3) to 1998(4). The estimated coefficients of the GUM are reported in Table 10 with standard errors in parentheses. The table shows that the null hypothesis of all the diagnostic tests are accepted on a 5% significance level, indicating no residual mis-specification in the GUM. To investigate the parameter constancy of the model plots of the recursive residuals can provide valuable information. Plots of 1-step ahead residuals, which is obtained by a estimation of the GUM by recursive least squares, are shown in the first window of Figure 5. The bands in the figure depict the ±2standard errors. The plot indicates an outlier in 1988, which can be due to the wage freeze of that year.33 The plot shows also that some of the other observations in the 1980s are near the border of the band, leading to a slight increase in the ±2standard error band in the 1980s. The band is, however, very stable in the 1990s, indicating a well-specified model for this period. The second plot of the figure, which shows the points of the 1-step Chow tests, confirms this picture. The break-point Chow test is depicted in the third plot. 32All the estimations in Section 5.4 is done in PcGive Version 9.3. 33Including a dummy variable that takes account of the wage freeze can remove the outlier. However, preliminary estimations show that the income coefficient is very sensitive to the inclusion of such a variable. Hence, since this can indicate that the period of the wage freeze contains valuable information about the consumption-income relationship, the dummy is left out. 17 Table 10: OLS estimates of general unrestricted model (GUM) ∆ct=1.159 (0.241) —0.091 (0.114) ∆ct−1+0.051 (0.103) ∆ct−2+0.048 (0.097) ∆ct−3 +0.427 (0.075) ∆ct−4+0.324 (0.062) ∆yt+0.233 (0.072) ∆yt−1+0.248 (0.076) ∆yt−2 +0.273 (0.080) ∆yt−3+0.190 (0.086) ∆yt−4+0.102 (0.064) ∆wt+0.062 (0.065) ∆wt−1 +0.151 (0.067) ∆wt−2+0.012 (0.063) ∆wt−3—0.090 (0.062) ∆wt−4—0.612 (0.127)EqCMt−1 —0.057 (0.011)CS1t—0.062 (0.012)CS2t—0.039 (0.011)CS3t+0.069 (0.012)VAT t +0.185 (0.066)(∆STOPt—∆∆4cpit) R2=0.973 σ=1.41% T=122 Model diagnostics Test Test statistic p-value AR 1—5 F(5, 96) =1.436 0.218 ARCH 4 F(4, 93) = 0.735 0.571 Normality χ2(2) = 1.732 0.421 Heteroscedasticity F(37, 63) = 0.913 0.612 RESET F(1, 100) = 0.128 0.721 InthisploteachofthepointsarethevalueoftheChow-testforthatdateagainstthefinal period. Accordingly, since the model is not well-specified for the 1980s but for the 1990s, the pointvaluesarenotstableoverthesampleperiod,inparticularforthevaluesinthe1980s. Likewise for the forecasting Chow-test depicted in the fourth plot. This test increases the forecasting horizon from the initial sample in the recursive least squares estimation to the complete sample period, and hence the forecasting failures of the model in the 1980s affect all the following points. In all the Chow-tests the points are scaled by their 1% critical value in the figure. This recursive analysis of the residuals of the GUM indicates that there may be more to model in the 1980s. Since the model, however, seems well-specified for the 1990s the GUM is kept unchanged. Several of the right-hand side variables in the GUM are insignificant on a 1% significance level. This suggests that the GUM can be simplified. By following the steps of the PcGets algorithm suggested by Hendry and Krolzig (1999) and Hendry (2000) it is investigated whether a reduced congruent version of the GUM can be obtained. By the use of ”pre-search” reduction tests, multiple search paths and encompassing tests of the terminal models of the different search paths against their union, it turns out that a reduced version of the GUM is congruent. Accordingly, this model is chosen as the final model and is presented in Table 11. The number of explanatory factors is reduced from 21 in the GUM to 14 in the final model, and all the variables of the final model are significant on a 1% level. The estimated equation standard error is 1.43%, slightly higher than the 1.41% of the GUM. It is, however, lower than the equation standard error of the consumption model in Eitrheim et al. (2002) at 1.53%. The coefficient of EqCMt−1at —0.69 is somewhat higher in absolute value than 18 Figure 5: Recursive residuals of the GUM 1980 1985 1990 1995 2000 -.02 0 .02 Res1Step 1980 1985 1990 1995 2000 .25 .5 .75 1 1.25 1% 1up CHOWs 1980 1985 1990 1995 2000 .25 .5 .75 1 1.25 1% Ndn CHOWs 1980 1985 1990 1995 2000 .5 .75 1 1.25 1.5 1% Nup CHOWs the long-run adjustment coefficient in the GUM (—0.61), but only slightly higher than the one found in the system analysis (—0.66). It is thus still indicating a fast adjustment towards equilibrium. Figure 6 plots the actual and fitted values of the model (for readability these are reported for ∆4ct) and of the residuals. The plot of the residuals does not indicate any residual mis-specification. This is supported by the diagnostic tests, which shows that the null hypothesis of no mis-specification of all the tests are accepted on a 5% level. The test statistic for fifth-order autocorrelation has though become significant on a 10% level, partly due to the exclusion of several of the lagged first-differences of ct. Recursive plots of the estimates with ±2standard errors for the variables of the model are given in Figure 6. The plots show that most of the parameters are constant over the sample period, in particular in the 1990s. The exceptions are the parameters for ∆ct−4and ∆yt−3, which seem to be trending slightly upwards. As distinct from the model of Eitrheim et al. (2002), the coefficient of the EqCM term in the final model is constant over the sample period. In the model of Eitrheim et al. (op. cit.) this parameter is declining in the 1980s before it stabilizes. With the exception of the age structure variable, the consumption model of Eitrheim et al. (2002) contains the samevariablesasthefinal model, and the two models are estimated on the same data set. The introduction of the age structure variable in the model is thus having a stabilizing effect on the long-run relationship. Recursive residuals of the model together with residual Chow-tests are plotted in Figure 8. The figure shows that the outliers in the 1980s have become somewhat more significant than in the GUM. 19 Figure 6: Fitted values (∆4ct) and residuals of the final model. 1970 1975 1980 1985 1990 1995 2000 0 .05 0 .00 0 .05 0 .10 D4c Fitted 1970 1975 1980 1985 1990 1995 2000 -2 -1 0 1 2 3 Residuals (Dc) 1980 1990 2000 1 1.5 Constant 1980 1990 2000 0 .25 .5 dc_4 1980 1990 2000 0 .2 .4 dy 1980 1990 2000 0 .25 dy_1 1980 1990 2000 .2 .4 dy_2 1980 1990 2000 .2 .4 .6 dy_3 1980 1990 2000 .25 .5 .75 dy_4 1980 1990 2000 -.25 0 .25 .5 dw_2 1980 1990 2000 -.75 -.5 EqCM Figure 7: Recursive estimates of final model 20 27 WORKING PAPERS (ANO) FROM NORGES BANK 20022003 Working Papers were previously issued as Arbeidsnotater from Norges Bank, see Norges Bank’s website http://www.norges-bank.no 2002/1 Bache, Ida Wolden Empirical Modelling of Norwegian Import Prices Research Department 2002, 44p 2002/2 Bårdsen, Gunnar og Ragnar Nymoen Rente og inflasjon Forskningsavdelingen 2002, 24s 2002/3 Rakkestad, Ketil Johan Estimering av indikatorer for volatilitet Avdeling for Verdipapirer og internasjonal finans Norges Bank 33s 2002/4 Akram, Qaisar Farooq PPP in the medium run despite oil shocks: The case of Norway Research Department 2002, 34p 2002/5 Bårdsen, Gunnar, Eilev S. Jansen og Ragnar Nymoen Testing the New Keynesian Phillips curve Research Department 2002, 38p 2002/6 Lindquist, Kjersti-Gro The Effect of New Technology in Payment Services on Banks’Intermediation Research Department 2002, 28p 2002/7 Sparrman, Victoria Kan pengepolitikken påvirke koordineringsgraden i lønnsdannelsen? En empirisk analyse. Forskningsavdelingen 2002, 44s 2002/8 Holden, Steinar The costs of price stability - downward nominal wage rigidity in Europe Research Department 2002, 43p 2002/9 Leitemo, Kai and Ingunn Lønning Simple Monetary Policymaking without the Output Gap Research Department 2002, 29p 2002/10 Leitemo, Kai Inflation Targeting Rules: History-Dependent or Forward-Looking? Research Department 2002, 12p 2002/11 Claussen, Carl Andreas Persistent inefficient redistribution International Department 2002, 19p 28 2002/12 Næs, Randi and Johannes A. Skjeltorp Equity Trading by Institutional Investors: Evidence on Order Submission Strategies Research Department 2002, 51p 2002/13 Syrdal, Stig Arild A Study of Implied Risk-Neutral Density Functions in the Norwegian Option Market Securities Markets and International Finance Department 2002, 104p 2002/14 Holden, Steinar and John C. Driscoll A Note on Inflation Persistence Research Department 2002, 12p 2002/15 Driscoll, John C. and Steinar Holden Coordination, Fair Treatment and Inflation Persistence Research Department 2002, 40p 2003/1 Erlandsen, Solveig Age structure effects and consumption in Norway, 1968(3) – 1998(4) Research Department 2003, 27p Solveig Erlandsen: Age structure effects and consumption in Norway, 1968(3) – 1998(4) Working Paper 2003/1 KEYWORDS: Demography Consumption Time series models - 13233