Growth Cycles in XXth Century European Industrial Productivity: Unbiased Variance Estimation in a Time-varying Parameter Model (FEP Working Paper Nº 144, Faculdade de Economia da Universidade do Porto, Maio 2004)

Faculdade de Economia do Porto - R. Dr. Roberto Frias - 4200-464 - Porto - Portugal Tel . (351) 225 571 100 - Fax. (351) 225 505 050 - http://www.fep.up.pt WORKING PAPERS Growth Cycles in XXth Century European Industrial Productivity: Unbiased Variance Estimation in a Time-varying Parameter Model Álvaro Aguiar Manuel M. F. Martins Investigação - Trabalhos em curso - nº 144, Maio 2004 FACULDADE DE ECONOMIA UNIVERSIDADE DO PORTO www.fep.up.pt CEMPRE – Centros de Estudos Macroeconómicos e Previsão Growth Cycles in XXth Century European Industrial Productivity: Unbiased Variance Estimation in a Time-varying Parameter Model Álvaro Aguiar and Manuel M. F. Martins CEMPRE*, Faculdade de Economia, Universidade do Porto [email protected] mm[email protected] Abstract This note applies the median unbiased estimation of coefficient variance, proposed by Stock and Watson (1998), to the extraction of the time-varying trend growth rate of industrial productivity in fifteen European countries, over most of the XXth Century, by means of an unobservable components univariate decomposition. In addition to the description of the procedure, this illustration is particularly useful in explaining why the method is especially appropriate for comparison of trends growth rates extracted from time series with diverse degrees of variability. Keywords: unobservable components model; industrial productivity; growth cycles; Europe. JEL codes: C22, N64. Resumo Esta nota aplica a estimação mediana não enviesada da variância, sugerida por Stock e Watson (1998), à extracção da taxa de crescimento tendencial, variável ao longo do tempo, da produtividade industrial em 15 países europeus, durante grande parte do século XX, através de uma decomposição univariada de componentes não observáveis. Para além de descrever o procedimento, esta ilustração é particularmente útil para explicar porque é que o método é especialmente apropriado para comparar taxas de crescimento tendenciais extraídas de séries temporais com graus de variabilidade diversos. Palavras-chave: modelos de componentes não observáveis;produtividade industrial; ciclos de crescimento; Europa. Classificação JEL: C22, N64. * CEMPRE - Centro de Estudos Macroeconómicos e Previsão - is supported by the Fundação para a Ciência e a Tecnologia, Portugal, through the Programa Operacional Ciência, Tecnologia e Inovação (POCTI) of the Quadro Comunitário de Apoio III. 1 1. Introduction This note describes the econometric procedure suggested by Stock and Watson (1998) for unbiased estimation of coefficient variance in a time-varying parameter model, and illustrates it with the estimation of time-varying trend growth rates of industrial productivity throughout the XXth century, for Portugal and other fourteen European countries. Identification of the trend path of a time series is often an important empirical issue in macroeconomic analysis. One of the valuable methods currently available for such identification consists of specifying an unobservable components model – typically assuming that trend and cycle follow, respectively, a random walk and a stationary auto-regressive process, with uncorrelated innovations – and carrying out estimation by maximum likelihood using the Kalman filter to compute the likelihood function. In non-stationary time-series such as, for example, (the log of) real output or productivity, it is useful to allow the trend to include a drift which itself follows a random walk, so that the estimate of this time-varying drift is directly interpretable as a time-varying trend growth rate - see Harvey (1989). However, if the variation of the trend growth rate is small – which seems plausible for most economic time-series – the maximum likelihood estimator of the variance of its changes is biased towards zero, because a large amount of probability piles-up at zero in the density function. Stock and Watson (1998) have suggested a solution to this “pile-up problem”, designing a procedure for median unbiased estimation of the variance of the changes in the trend growth rate, within an unobservable components model. The intuition behind their method is, in essence, that the magnitude of time-variation in the trend growth rate can be uncovered from the statistics of tests for a break in the regression of the actual growth rate on a constant throughout the sample period. By means of Monte Carlo integration, they have computed a look-up table, where a scale parameter for the variance can be appropriately selected, from the results of various stability test statistics – including the maximum of a sequential GLS Chow test that we use in this illustration. The unobservable components trend-cycle model, extended with Stock and Watson’s (1998) procedure, has been used by Roberts (2001) to study the trends of US hours worked and productivity throughout the last four decades of the XXth century. French (2001) has also used 2 this framework to estimate the trend of US total factor productivity in 1960-1999, comparing its performance to some alternative models. In this note, we apply the procedure to the estimation of growth cycles in XXth century European industrial productivity, as a background to the comparative analysis of Portuguese industrial productivity growth pursued in Aguiar and Martins (2004). In that paper we use univariate time-varying trend growth rates to analyze the path of Portuguese industrial productivity in international perspective, as well as the diverse intra-industry patterns. The unobservable components trend-cycle model with a stochastic trend drift turns out to be quite appropriate to analyze long-term industrial productivity, since the XXth Century has been characterized, in the industrialized countries, by secular growth in productivity levels, with phases of considerable and persistent acceleration/deceleration of productivity. By applying Stock and Watson’s (1998) procedure, we ensure that the variability of each particular estimated trend series mirrors the variability that can be extracted from the respective underlying time-series of actual productivity growth. In doing so, we include the comparison of variances in the comparisons of trends in productivity growth rates between countries or industries. We believe that, from an economic point of view, this method improves on others that blindly impose a unique smoothing intensity. This note briefly describes, in section 2, the unobservable components model and the econometric procedure allowing for its estimation according to Stock and Watson (1998). In section 3, the method is illustrated with an account of the estimation of the trend growth rate of Portuguese industrial productivity 1910-2000, and with the report of the main results of similar estimations for fourteen other European countries. Section 4 briefly concludes, and an appendix describes the data and its sources. 2. The unobservable components model and unbiased estimation In this section we describe the trend-cycle unobservable components model used as a background to Aguiar and Martins (2004), and the procedure suggested by Stock and Watson (1998) for unbiased estimation of the variance of changes in the trend growth rate. The time-varying parameter model The model decomposes the level of log industrial productivity ( ) into the sum of a nonstationary stochastic trend ( ) and a stationary stochastic cycle ( ), t y trend t ycycle t y 3        += =− ++= += − − −− )4( )3( )2( )1( 1 1 11 cycle t cycle t cycle t g ttt trend tt trend t trend t cycle t trend tt yy gg gyy yyy ερ ε ε . Specifically, the trend is assumed to follow a random walk process with a stochastic drift (equation 2), which itself follows a random walk process (equation 3), while the transitory component is modelled as a stationary auto-regressive process (equation 4). To allow for estimation, the model is written in state-space format,    ++= ++= − α εαα εα tttt y tttt cTr dzy 1 , where is the observed time-series, , t y0=== y ttt dc ε [ ] 011 = z, and the state vector - comprising the unobservable components - is           = t cycle t trend t t g y y α ; and the transition matrix governing the unobservable components dynamics is           = 100 00 101 ρ Tr . The vector of stochastic innovations of the system,           = g t cycle t trend t t ε ε ε ε α , is assumed to follow a multivariate normal probability distribution, , with ( QN t,0~ α ε )           = 2 2 2 00 00 00 g cycle trend Q ε ε ε σ σ σ , which incorporates the independence between trend and cycle’s innovations - a standard identifying assumption of the unobservable components model. 4 The model has three time-varying parameters - the unobservable variables in α - and four hyper-parameters - g t cycle t trend t εε ε σ σ σ ρ and ,,, . Once adequate initial conditions are established for both sets of parameters and for the filter variance of the unobservable components, the Kalman filter recursive equations can be iterated, and the parameters of the system can be estimated through maximisation of its likelihood function - Harvey, 1989. However, as remarked in the introduction, maximum likelihood fails to generate unbiased estimates of . While the other hyper-parameters are unbiased, the maximum likelihood estimate of is typically biased towards zero, and thus towards non-significance. As a result, the unrestricted maximum likelihood estimation of the model produces smooth trends and cycles of the level, that are large in amplitude and very persistent - as is characteristic of unobservable components models with orthogonal trend-cycle –, while the trend growth rate displays no apparent time-variation, irrespective of the actual fluctuations in the rate of change of the time-series. Hence, the usefulness of Stock and Watson’s (1998) addition to the method. 2 g ε σ 2 g ε σ The procedure for unbiased estimation of parameter variance Stock and Watson’s (1998) procedure starts off by regressing the first differences of (here, the log of industrial productivity) on a constant. Typically, in empirical macroeconomic analysis, feasible generalised least squares (GLS) is needed, in place of OLS, as the residuals from OLS regressions have a non-white-noise structure that can be well described by a finite order auto-regressive process. t y The regression of the growth rate of the observed series on a constant is run over the entire sample (T observations) and sequentially over the sub-samples obtained by splitting the full sample at Ts, for all 0.15 T ≤ Ts ≤ 0.85 T (a standard 15 percent trimming). In the process, the sequential GLS Chow statistics, FT(s), testing for breaks at dates Ts, are extracted. Letting SSRt1,t2 denote the sum of squared residuals from the GLS regression over observations t1 ≤ t ≤ t2, each Chow statistic is () TTsTs TTsTsT T SSRSSR kT k SSRSSRSSR sF ,1,1 ,1,1,1 )( )( + + + − − − =. 5 The higher FT statistic - which is Quandt's likelihood ratio statistic QLR - is retained, and Stock and Watson's table 3 (1998, page 354) is used to obtain the corresponding median-unbiased estimator of a scale coefficient λ . )(sup sFTT = Then, the Kalman filter can be run: hyper-parameters ρ and cycle t ε σ are freely estimated by maximum likelihood; the variance of the innovation to the trend growth rate is restricted as 2 2 2 trendg T εε σ λ σ       =, and the variance of the innovation to the trend level is normalized as a function of the variance of the innovation to the cycle and of its auto-regressive parameter, as 2 2 21 cycletrend εε σ ρ σ         =; so that compliance with Stock and Watson’s (1998) specification - implicit in their Monte Carlo results - is attained. After convergence, the fixed-interval Kalman smoother is run (Harvey, 1989, page 154), in order to generate estimates of the time-varying parameters – trend level, cycle level, and trend growth rate, - that are, in each period, conditional on the whole sample information. In particular, the smoothed estimates of the time-varying trend growth rate are the crucial ones for the analysis of industrial productivity growth cycles in Aguiar and Martins (2004). 3. Growth Cycles of Industrial Productivity in XXth Century Europe In this section we apply the model and econometric procedure just described to the estimation of time-varying trend growth rates of industrial productivity in fifteen European countries throughout most of the XXth century, beginning with a more detailed report of the estimation process for the case of Portugal. The data - described in the appendix - are annual time-series of labor productivity in the industry of Portugal, Belgium, Finland, France, Germany, Italy, Norway, Spain, Sweden, UK, Austria, Denmark, Greece, Ireland and Netherlands. The sample period is 1910-2000, except for the last five countries, for which the sample is 1927-2000. Productivity has been computed as the ratio of industrial production to industrial labor force. The interpretation of productivity trends during the I and II World War periods is inappropriate, because, with the only exception of Portugal, all productivity series have been 6 interpolated over those periods, as their very sharp and erratic variations precluded a reasonable estimation of the trend. Figure 1, measuring on the left hand side scale the log level of Portuguese industrial productivity between 1910 and 2000, clearly shows that it has trended upwards during most of the XXth century. The graph also affords some preliminary indication that Portuguese industrial productivity growth has gone through some persistent shifts throughout the century, thus suggesting the relevance of using a time-varying growth rate model. For example, the pace of productivity growth has been larger between the early 1950s and the mid 1970s than during the two previous decades; between the mid-70s and the mid-80s there has been no noticeable rise in productivity; and in the final 15 years of the century Portuguese industrial productivity seems to have augmented at a pace similar to the one observed between the mid50s and the mid-60s. Figure 1 also shows - measured on the right hand side scale - the annual growth of Portuguese industrial productivity, ie the first differences of the series measured on the left hand side axis. The (typical) high volatility of the series of annual growth rates hampers any precise definition of the amplitude and length of the apparent growth cycles. Next, in order to filter the noise, we extract the trend growth rate of Portuguese industrial productivity using the procedure set up by Stock and Watson (1998). The results of the sequential GLS Chow FT test for testing the stability of a regression of Portuguese industrial productivity on a constant are depicted in figure 2. Inclusion of one lagged observation proved to be enough to achieve non-correlated residuals, a common feature of annual productivity data. The maximum value of the FT statistic turns out to be 15.14736 and is located at 1974 - table 1. This value, according to Stock and Watson’s (1998) look-up table (Table 3, page 354), yields a coefficient λ = 11.87096, which, divided by the number of observations T, results in 0.1319. Hence, the model described above, in section 2, is applied to the log of Portuguese industrial productivity, imposing the restrictions 222 1319.0 trendg εε σσ = 2 2 21 cycletrend εε σ ρ σ         =, and estimating the hyper-parameters ρ and cycle t ε σ , together with the time-varying parameters (trend level, cycle level, and trend growth rate), by maximum likelihood with the Kalman filter. 7 As regards the starting values of the state variables, the iterations have been initiated setting to zero the stationary variable (the cycle level) and setting the non-stationary variables (trend level and trend growth rate) equal to their observed counterparts – ie actual log productivity and actual growth of productivity at the sample onset, respectively. The filter variances have been initiated at for all the parameters, corresponding to 95 percent confidence bands as wide as about 10 percentage points. 2 05.0 Focusing on the relevant results for the analysis, figure 3 shows the actual annual growth, the secular average growth and the estimate of the time-varying trend growth rate of Portuguese industrial productivity 1911-2000. The unbiased estimates of the trend growth rate vary from a minimum of 1.6 percent in the late 30s to a maximum of 4.5 percent in the mid-60s, and are about 3.7 percent at the end of the century. The deviations between the trend growth rate and the average secular rate establish the growth cycles that are presented in table 2. Following a first half of the century in which trend growth has been below the average secular growth, from 1951 on Portugal experienced a positive growth cycle of industrial productivity, which has only been discontinued during the erratic years between the 1974 revolution and the advent of political stabilisation, economic reform, and accession to the European Community in the mid-80s. We now turn to a very brief description of the main results regarding the other fourteen European countries considered in Aguiar and Martins (2004), in a comparative perspective. Table 1 reports the statistics, their date of occurrence, and the implied values for )(sup sFT λ and T/ λ . The table shows the wide range of variability in the series of productivity annual growth across these countries and, thus, highlights the relevance of using a specific and unbiased estimate of the variances of each country’s changes in trend growth rate. The fifteen charts in figures 4-7 assemble the annual growth, average secular growth, and estimates of time-varying trend growth rates of industrial productivity of each country, with a uniform scale. Following a standard taxonomy, figure 4 describes the G4 countries (France, Germany, Italy and UK, the European members of the G7), figure 5 comprises the small Northern countries (Austria, Belgium, and Netherlands), figure 6 depicts the Scandinavian countries (Denmark, Finland, Norway and Sweden), and figure 7 the countries that are subject to European Union policies towards economic convergence with the richer members, labelled Cohesion countries (Greece, Ireland, Spain and Portugal). 8 Figure 3. Growth, Average Growth, and Trend Growth of Portuguese Industrial Productivity -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 15 Figure 4. Growth, Average Growth, and Trend Growth of Industrial Productivity - G4 European Countries -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 FRANCE -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 GERMANY -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 UNITED KINGDOM -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 ITAL Y 16 Figure 5. Growth, Average Growth, and Trend Growth of Industrial Productivity - Small Northern European Countries -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 BELGIUM -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1928 1930 1932 1934 1936 1938 1940 1942 1944 1946 1948 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 AUSTRIA -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1928 1930 1932 1934 1936 1938 1940 1942 1944 1946 1948 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 NETHERLANDS 17 Figure 6. Growth, Average Growth, and Trend Growth of Industrial Productivity - Scandinavian European Countries -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1928 1930 1932 1934 1936 1938 1940 1942 1944 1946 1948 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 DENMARK -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 FINLAND -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 NORWAY -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 SWEDEN 18 Figure 7. Growth, Average Growth, and Trend Growth of Industrial Productivity - Choesion European Countries -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1928 1930 1932 1934 1936 1938 1940 1942 1944 1946 1948 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 IRELAND -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1928 1930 1932 1934 1936 1938 1940 1942 1944 1946 1948 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 GREECE -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 SPAIN -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1911 1914 1917 1920 1923 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 PORTUGAL 19 Table 1. Sup FT Statistic, λ, and λ/T - Fifteen European Countries QLR statistic Country Sample Sup FT Portugal 1910-2000 15.14736 1974 11.87096 0.13190 Germany 1910-2000 11.54345 1926 9.74805 0.10831 Italy 1910-2000 10.45865 1947 8.86577 0.09851 France 1910-2000 8.84317 1930 7.79346 0.08659 United Kingdom 1910-2000 10.36092 1950 8.80061 0.09778 Netherlands 1927-2000 12.38808 1986 10.43523 0.14295 Belgium 1910-2000 25.17260 1937 17.07763 0.18975 Austria 1927-2000 6.71246 1950 6.03227 0.08263 Denmark 1927-2000 18.70736 1983 13.69272 0.18757 Sweden 1910-2000 24.70231 1933 16.86204 0.18736 Norway 1910-2000 28.88790 1930 18.27865 0.20310 Finland 1910-2000 6.60118 1950 5.91861 0.06576 Spain 1910-2000 16.38453 1959 12.56819 0.13965 Ireland 1927-2000 33.35979 1979 20.01673 0.27420 Greece 1927-2000 16.15063 1949 12.42582 0.17022 Industrial productivity timeseries Date of ocurrence of Sup FT λλ/Τ Table 2. XXth Century Industrial Growth-Cycles in Fifteen European Countries Average Secular Growth Country Below Above Below Above Portugal 2.83% 1911-1950 1951-1973 1974-1984 1985-2000 Germany 2.90% 1911-1932 1933-2000 Italy 2.78% 1911-1946 1947-2000 France 2.45% 1911-1945 1946-2000 United Kingdom 2.39% 1911-1944 1945-2000 Netherlands 2.72% 1928-1949 1950-1982 1983-2000 Belgium 2.36% 1911-1945 1946-2000 Austria 3.77% 1928-1950 1951-2000 Denmark 2.79% 1928-1951 1952-2000 Sweden 2.53% 1911-1931 1932-2000 Norway 3.08% 1911-1949 1950-2000 Finland 3.33% 1911-1946 1947-2000 Spain 1.80% 1911-1945 1946-1988 1989-2000 Ireland 4.45% 1928-1980 1981-2000 Greece 3.42% 1928-1946 1947-1977 1978-2000 Trend Growth Rate versus Average Secular Growth 20 Working papers mais recentes Nº 143 Jorge M. S. Valente and Rui A. F. S. Alves, Beam search algorithms for the early/tardy scheduling problem with release dates, April 2004 Nº 142 Jorge M. S. Valente and Rui A. F. S. 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