Is Germany s GDP Trend-Stationary? A Measurement-With-Theory Approach 1

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1 Is Germany s GDP Trend-Stationary? A Measurement-With-Theory Approach 1 by Bernd Lucke University of Hamburg von-melle-park 5 D Hamburg Germany lucke@hermes1.econ.uni-hamburg.de January 2002 revised June 2002 Abstract: The time series properties of German GDP have been re-examined in recent research. Extending the sample to include GDP data from 1950 onwards, some researchers argued in favor of a trend-stationary rather than difference stationary representation of real log GDP. We show that this conclusion is based on an atheoretic trend model underlying the unit root tests. A simple linear trend model fails to take the post World-War II catch-up process properly into account. We use the Solow growth model to discriminate between transitional catch-up dynamics and long-run equilibrium growth. With the proper transformation of GDP data, we are able to use standard unit root tests and find that both ADF and KPSS tests suggest a difference stationary model. This evidence is supported by non-standard unit root tests which allow for polynomial trend representations. Keywords: Solow growth model, transistional dynamics, unit root tests JEL: C22, O41, O52 1 I thank Nicola Brandt for excellent research assistance and Carsten-Patrick Meier for making his GDP data available to me. The comments of two anonymous referees are gratefully acknowledged.

2 1. Introduction Much attention has been devoted to the stationarity properties of macroeconomic time series ever since Nelson and Plosser s (1982) seminal contribution argued that most US time series might better be represented by unit root rather than by trend stationary processes. This view has found widespread acceptance since, and countless papers have appeared which used unit root and cointegration techniques in applied macroeconomic models. As Rudebusch (1993) demonstrated, the matter is far more than a mere statistical issue, since difference stationary and trend stationary models estimated for, say, real GDP, imply very different economic dynamics at cyclical frequencies. For instance, impulse responses at the five-year horizon imply shock magnification by a factor of 1.6 under difference stationary specification while the same shock dies out rapidly (shrinkage factor of 0.15) under a trend stationary model estimated from the same set of data. Today, the dominant view in the profession today is still in favor of the unit root representation of GDP. But there have been numerous dissenting voices, e. g. Perron (1989), DeJong and Whiteman (1991), DeJong et al. (1992), Rudebusch (1993). In the case of (West) Germany, the unit root perspective has recently been challenged by Assenmacher (1998) and Meier (2001). Both authors use long annual time series for log real GDP (starting in 1950) and apply conventional unit root tests, in particular the Augmented Dickey-Fuller (ADF) Test. (Assenmacher uses an alternative with broken linear trend). Both conclude that the unit root hypothesis can be rejected in favor of the respective trend stationary alternative. While many authors have tested the unit root hypothesis for German GDP with quarterly data which are available since 1960, a distinctive feature of the Assenmacher-Meier (henceforth AM) approach is their usage of data starting as early as They can thus just rely on annual data, but this is nevertheless perfectly legitimate since it is well-known that the main determinant of the power of unit root tests is neither the lenght nor the sampling frequency of the series, but its span (Shiller and Perron (1985)). More problematic in AM s work is the specification of (broken) linear trends for the log of GDP. These seem inappropriate since a casual inspection of the German GDP series reveals that the level of GDP is well represented by a linear trend, but the logarithm (henceforth lngdp) seems to follow a strictly concave trend. This due to the fact that growth rates of German GDP have been very high after World War II and have settled to lower levels since. Assenmacher s work partially captures this feature by piecewise linear trends, but this approach is ad hoc. Therefore, this paper suggests to view German GDP growth in terms of the neoclassical growth model, where much of the post-world-war-ii growth would be interpreted as convergence to a steady-state growth path from an initial position characterized by severe war damages. It is hence the central idea of this paper that the exceptional German catch-up process must be properly taken into account if unit-root tests are applied to German data starting as early as The apparent problem inherent in unit root inference applied to the transitional dynamics of a Solovian growth process lies in the fact that it is difficult to distinguish convergence to a balanced growth path from (persistent or mean-reverting) fluctuations in the neighborhood of this balanced growth path. In other words: It could be that the balanced growth path is a unit root process, say, a random walk with drift. However, if Solovian transitional dynamics in the form of convergence to the balanced growth path is present (as we would expect for Germany), then the transitional dynamics might be mistaken for 2

3 evidence of mean-reverting fluctuations around the long-run path, i. e. unit-root tests might falsely reject the null. Moreover, under these circumstances, the estimated trend will generally be influenced both by the long-run growth path and by the (essentially monotone) convergence process. Hence the estimated trend has no economic interpretation and can, in particular, not be taken to indicate the long-run development of GDP. Conversely, if the true balanced growth path were surrounded by stationary fluctuations (i. e. if the unit root hypothesis were false), then Solovian convergence from below in the 1950s would lead to a linear trend whose estimated slope is upward biased. Hence, at the end of the sample the observed fluctuations would seem to deviate more and more from the estimated trend such that unit root tests would (falsely) tend to find persistence. The unit root hypothesis for German GDP can therefore be tested only under a correct specification of the underlying trend which, if the neoclassical growth model is accepted, involves transitional dynamics for a time series starting shortly after World War II. Thus the appropriate trend model is monotone and concave. In this paper, I will specify the Solow growth model with Cobb-Douglas production function in order to derive an analytic expression for the time path of GDP. (This specification results in Bernoulli s differential equation whose solution is well known). I will then test the unit root hypothesis for German GDP in an ADF-type test which allows for the correct trend specification. The sequel is organized as follows: Section II briefly reviews the time series properties of German GDP, in Section III I derive the correct trend path, and Section IV presents simple econometric exercises aimed at testing the unit root hypothesis. Section V reports the results of nonstandard unit root tests which allow for (atheoretic) polynomial trend representations and Section VI concludes. 3

4 II. Time Series Properties of German GDP Let us first have a look at both the level and the logarithm of (West) German real GDP from 1950 to 1998, cf. Figure 1. On a merely descriptive basis it is obvious that a linear trend is a rather good approximation for the level of GDP. Consequently, it is a poor approximation for its logarithm, which displays a concave pattern Figure GDP lngdp lngdp GDP The growth rates are given in Figure 2, both exact and approximated as the first difference of the logarithms. The difference, however, seems to be negligible, even for the growth rate of 12% in While the growth rates are quite volatile, a downward sloping trend for the first 25 years or so is clearly discernible. Over the whole sample period, this downward sloping trend might well converge against a positive constant as the neoclassical growth model would imply. 4

5 Figure growth rate of GDP exact approximate growth rate Applying a standard ADF-unit root test to lngdp gives the following result (t-statistics in parentheses): lngdp = lnGDP lngdp lngdp t + ε t t 1 t 1 t 2 t ( 3.95) ( 3.65) ( 1.94) ( 2.19) ( 2.15) (1) The t-statistic of 3.65 for the coefficient of the lagged level is significant at the 5% level of the Dickey-Fuller distribution and it seems as if this basically justifies AM s decision to reject the unit root hypothesis for lngdp. However, this result depends crucially on the very first observations of the sample period, i. e. on observations which are probably characterized by strong catch-up dynamics, or, in the terminology of the neoclassical growth model, by sizable steps towards convergence to a steady state growth path. The first observation in the dependent variable of (1) is the growth rate for 1953, since three lagged variables are used in the ADF regression. Successively shrinking the sample by starting in later years and running the same ADF-regression (1) for each reduced sample casts doubt on the validity of the inference suggested by (1), as most resulting test statistics are clearly insignificant. This is graphically displayed in Figure 3, where ADF t-statistics and appropriate 5% critical values are shown for the dates of the first dependent growth rate in the respective sample. For instance, a sample starting with the 1955 growth rate as dependent variable produces a significant ADF-statistic of 3.75, but almost all successive samples result in clearly insignificant ADF-statistics. (The only exception is the sample beginning with the

6 growth rate as dependent variable. Here we find a marginally significant ADF-statistic of as opposed to a 5% critical value of 3.56). Figure t-statistic ADF Critical value This behavior of the ADF-statistic across shrinking samples could, in principle, be due to a loss of power associated with fewer observations. But given the rapid decline in the size of the ADF-statistic with the loss of only three or four observations, this explanation seems unsatisfactory in particular since a more appealing reasoning (inappropriateness of the trend model) is readily at hand. One could try to defend the trend-stationary view of German GDP by using the result of West (1988), which states that the asymptotic distribution of the ADF-statistic of (1) is standard normal if the data generating process has a unit root and a non-zero trend. (More generally, it is required that the number of estimated parameters in (1) is exactly equal to the number of nonzero parameters of the data-generating process). Of course, a standard normal distribution for the ADF-t-statistic would mean that almost all statistics in Figure 3 are indeed significant at the 5% level. In order to check for this hypothesis, I follow Meier (2001) in calculating the F-test-statistic for the joint hypothesis that both the coefficient of the y t-1 -term and the linear trend in regressions of type (1) are zero 2. If the null is rejected, then either y t is trend stationary, or West s result applies and also suggests trend stationarity. Figure 4 shows the F-statistic across the same sample variation as in Figure 3. We see that the statistic is significant for samples starting prior to 1960, and generally insignificant for samples starting after 1959, the one exception being again Thus, again, the analysis suggests that rejection of the unit root null depends crucially on the early post-war years in 2 This statistic has a non-standard distribution tabulated by Dickey and Fuller (1981), the 5% critical value is 6.73 for 50 observations and 7.24 for 25 observations. 6

7 which the German economy was way off its balanced growth path. It is hence of prime importance for unit root inference to specify the correct trend model. The linear trend routinely included in ADF-regressions is certainly inappropriate to capture the kind of convergence dynamics which can be observed in these years. Figure F-Statistic Dickey-Fuller 5% critical value There is a simple reason, why the initial observations in the sample may unduely influence the results of ADF-type tests. Recall that ADF-tests are based on a regression where the growth rate of GDP is regressed on the lagged logarithm of GDP, i. e. lngdp = + β ln GDP ε t t 1 t Precisely the same type of regression, however, is used in tests for (conditional) β- convergence in the framework of the neoclassical growth model, cf. Baumol (1986) and the large literature which builds on his work. Perfect convergence is implied by a value of β=-1, since then lagged GDP cancels on both sides of the equation and current is independent of the starting value. Conversely, β=0 indicates that the growth rate of GDP is independent of the starting value and hence there is no convergence. Negative values, -1<β<0 imply ongoing convergence. In principle, the dynamics of the deterministic neoclassical groth model have nothing to do with the stationarity properties of time series models. In application to real world data, however, it is clear that convergence due to below-steady-state capital accumulation (catch-up processes) will bias the estimated coefficient of the lagged level in unit root tests downward, thereby increasing the likelihood of rejecting the null beyond the nominal, say, 5% level. This effect is particularly important if subsamples with strong catchup dynamics are included in the 7

8 analysis and it is hence suggestive to interpret Figures 3 and 4 as corroborating evidence. Consequently, unit root tests for data which are characterized by sizable catch-up dynamics are misspecified unless the convergence phenomenon can adequately be captured by, e. g., the deterministic time trend. III. Convergence to a Balanced Growth Path In order to find an analytic expression for the growth path of an economy which has lost a large part of its capital stock, let us consider the capital accumulation equation of the Solow growth model for the special case of Cobb-Douglas production. Under constant returns we can write this equation in intensive form as ( δ γ ) k & = sk n+ + k (2) where k=k/al is capital in efficiency units, is the capital share, labor L grows with constant rate n, exogenous factor productivity A grows with constant rate γand δ is the constant rate of depreciation. This equation is Bernoulli s differential equation, which can be transformed to linearity by substituting z=k 1-. The general solution in z as a function of time t is given by { ( )( δ γ ) } z = cexp 1 n+ + t + s n + δ+, (3) γ where c is an arbitrary constant. Output in intensive form is given by Y/AL=y=k, hence we have s ln y = ln cexp{ ( 1 )( n+ δ+ γ ) t} + 1 n + δ+ γ (4) For a given starting value y 0 at time t=0 we compute 1 0 c= y s n + δ+ γ. Further, we have s y*: = lim y = t n + δ+ γ 1 so that * c= y y, i. e. c<0 for y 0 <y*. 8

9 Using ln ln ln( ) ln ln ( ) ( γ ) Y = y+ AL = y+ AL + n+ t and imposing c<0 for Germany in the 1 2 ( ) s we have lny = ln( AL 0 0 ) + ( n+ γ) t+ s ln cexp{ ( 1 )( n+ δ+ γ ) t} + 1 n δ γ + + s = ln( AL 0 0 ) + ln( c) + ( n+ γ) t+ ln exp{ ( 1 )( n+ δ+ γ ) t} 1 1 c ( n + δ+ γ ) = : c1+ ct 2 + ln c3 exp{ ( 1 )( c2 + 1 δ ) t} = : c + ct+ S t (5) where c 1 is arbitrary and c 2, and c 3 are positive constants. Observe that c 3 >1 so that the logartithm in the last term is well defined. While standard unit root tests allow for a constant and a linear trend under the alternative, expression (5) shows that this routine specification neglects a term of the form S( t) : = /1 ( ) ln c3 exp{ ( 1 )( c2 + δ ) t} for economies whose GDP is characterized by Solovian transitional dynamics. However, this term converges to a constant when t approaches infinity, i. e. the specification of standard unit root tests is asymptotically correct. 9

10 IV. Empirical Implementation Equation (5) describes the time path of log output in a deterministic Solow-model converging to the steady state growth path. In a stochastic setting, (5) must be augmented with an error term u t : ( ) ln Y = c1+ ct 2 + S t + ut (5 ) The error term will typically be autocorrelated. Let us assume that u t is an AR(2) process, which happens to be an appropriate lag specification: u = ρ u + ρ u + ε (6) t 1 t 1 2 t 2 t where ε t is white noise. Now, lagging (5 ) once and multipliying with ρ 1, and lagging (5 ) twice and multiplying with ρ 2 gives two equations which can be subtracted from (5 ) to yield: ( ) ( ) ( ) ( 1) ( 2) lny ρ lny ρ lny = c 1 ρ ρ + c 1 ρ ρ t+ ρ c + 2ρ c t 1 t 1 2 t S t ρ S t ρ S t + ε 1 2 t (7) It would be tempting to estimate the parameters in (7) (e. g. by maximum likelihood or nonlinear least squares) and then construct a unit root test based on the estimated standard errors. This idea, however, cannot be implemented for at least four (related) reasons: First, expressions for finite sample standard errors under the unit root null are not available due to the nonstandard regression equation. Second, asymptotic standard errors under the unit root null are available from the Dickey-Fuller distribution (recall that S( ) is asymptotically constant), but these are inappropriate since they refer to a model without transitional dynamics. Moreover, standard formulae for asymptotic standard errors assume that the matrix of regressors is asymptotically nonsingular, but this condition is violated, since constant and S( )-terms in (7) are asymptotically perfectly collinear. Third, due to the asymptotic singularity, neither maximum likelihood nor nonlinear least squares estimates of (7) are consistent. Fourth, (7) is overparameterized and the parameters are not identified. In fact, parameters are not even identified in the much simpler equation (5 ). Approximate inference on the unit root hypothesis is possible, however, if some of the parameters in (7) are calibrated rather than estimated. Inference can then proceed as if the calibrated parameters were the true values and Monte Carlo simulations can be employed to check how sensitive the results are if calibration fails to specify the correct parameter values. Let us define the variable ( ) Z : = ln Y S t t t t (8) which is the logged output variable properly adjusted for transitional growth. Equation (7) is then equivalent to 10

11 ( 1 ) ( 2 ) ( 1 ) ( 1 ) Z = c ρ ρ + c ρ + ρ ρ ρ Z ρ Z + ρ ρ ct+ ε = : β + β Z + β Z + β t + ε t t 1 2 t t 0 1 t 1 2 t 1 3 t (9) a conventional ADF-test equation in the transformed variable Z t. In order to implement (9), the parameters in (8) must be fixed a priori. Fortunately, we roughly know their orders of magnitude: The rate of depreciation δ is approximately 0.05 with annual data. is the capital share, so 0.3 from the national accounts. The parameter c 2 (=n+γ) is the exogenous steady-state growth rate of real GDP, which will be around To find a value for c 3, observe that c ( + δ+ γ ) 1 1 s y* y * 1 = = = = c n c y* y y o 0 1 y *. Differentiating (4) with respect to time and setting t=0 we further find dln y dt t = y* y0 n + δ+ γ c( n + δ+ γ ) = = 1 s c + y0 n + δ+ γ ( ) 1 y * δ γ = 1 ( n + δ+ γ ) = y0 c3 1 ( n + + ) ( n+ + ) ( n + + ) δ γ δ γ c3 = 1+ = 1+ dln y dlny n γ dt dt t= 0 t= 0 Since the first observed growth rate in the sample is 0.09, I set c 3 =1+0.3*( )/( )=1.3. Let us momentarily treat these calibrated values as if they were the true values of the parameters, δ, c 2 and c 3. We can then proceed to test the unit root hypothesis with a standard ADF-test. Estimating equation (9) for samples starting in 1952 and subsequent years and running through 1998, we obtain t-statistics for the estimated β 1 -coefficient as displayed in Figure 5. 11

12 Figure ADF-statistic from (9) ADF-statistic from (1) 5% critical value Note first that the ADF-test based on (9) requires only one lag to render the residuals white noise unlike the model-free version in (1), where two lags were necessary. Observe further that the ADF-test based on (9) yields generally insignificant results again with the peculiar exception of the sample starting in In particular, using the largest possible sample ( ) does not produce a rejection of the unit root null. In fact, my conclusion is quite polar to AM s, since the general pattern seems to be that extending the sample into the 1950s implies smaller test statistics in absolute value and hence less evidence against the unit root hypothesis. Finally, note that the standard deviation of the test statistics based on (9) is 0.41, wile the standard deviation for the test statistics based on (1) is 0.56, thus it seems that the behavior of the ADF-test statistic for (9) is somewhat stabler (and hence possibly more reliable) than for (1). Thus, the conclusion seems to be that there is only very little evidence against the unit root hypothesis for German GDP, and this evidence is not robust against varying the sample size: The sample must start in 1968 if a significant test statistic is to be obtained. Not surprisingly, the evidence is even weeker if we take erroneous calibration of the parameters, δ, c 2 and c 3 into account, as I will briefly demonstrate. As an example, let us suppose that the parameter was falsely set at =0.3, while the true value is =0.4. (The analysis focuses on, since this parameter affects the definition of Ζ t through many different channels, some of which coincide with channels by which variations in other calibrated parameters would affect Z t ). Using the first two historical observation for 12

13 lngdp, we can use (7 ) to construct artificial lny series. To do this, we impose the unit root property (β=0) and calibrate =0.4, δ=0.05, c 2=0.02, c 3=1.4. Estimating (9) under the unit root hypothesis we obtain estimates for the constant, the coefficient of the lagged difference in (7 ) and the standard deviation of the error term, which happen to be 0.017, 0.308, and 0.020, respectively. With these settings, 50,000 artificial time series for lny are simulated and transformed in Z-series using a wrong value of =0.3, cf. equation (8). Estimating (9) with these data for succesively shrinking samples , ,..., gives rise to a total of 50,000 t-statistics on β 1 for each sample. From this information we obtain a fairly precise picture of the (appropriate finite sample) critical value for the ADF-test. Figure 6 illustrates the results. Not unexpectedly, the critical values are higher in absolute value for each sample, if calibration assigns incorrect values. The difference seems to be small, however. The empirical evidence for the unit root hypothesis is hence qualitatively the same as under the assumption of correct calibration: There is just one sample definition ( ) which produces a significant test statistic, for all other sample lengths (and in particular those with long spans) there is no sufficient evidence to reject the unit root null Figure % critical value, true calibration 5% critical value, wrong calibration Another issue merits attention. Work by Mankiw, Romer, and Weil (MRW) (1992) suggests that a value of 0.3 may be grossly misspecified, since capital should be interpreted as representing all accumulable inputs, in particular physical and human capital. While the MRW model is not nested in the above framework, their central idea would be well reflected by specifying a much higher value of, say 0.8. (This would imply that about 20% of total income goes to labor services which are not human capital intensive). While this idea might be suggestive at first sight, such a high value of is clearly inappropriate. This is easily seen by constructing the Z-variable in (9) for the two cases of 13

14 =0.3 and =0.8, denoted Z 3 and Z 8. (All other parameters are calibrated as set out above). As can be seen from Figure 7, Z 8 is not an appropriate data transformation, for if it were it would imply that the mean growth rate of GDP (corrected for transitional dynamics) is negative rather than positive! (Recall that (8) describes the long run equilibrium growth and that the constant in (8) is positive). Thus Z 8 involves an exaggerated correction for the post-world War II catch-up process. The reason for this is fairly simple: World War II caused large damages to physical capital in Germany, but the stock of human capital was probably much less affected, so the strong transitional dynamics in the 1950s and 1960s were basically due to a catch-up in physical capital only. Figure Z3 Z8 Since the unit root null cannot be rejected under appropriate trend extraction, one might want to test the alternative hypothesis of trend stationarity. The most widely used test in the literature which specifies a null of trend stationarity is due to Kwiatkowski et al. (KPSS) (1992). This test specifies the variable of interest to be a trend-stationary process plus a random walk with innovation variance σ 2. Under the null hypothesis σ 2 is zero, so that the random walk reduces to a constant. The KPSS test uses a semi-parametric estimation technique which requires the specification of a truncation parameter l. Table 1 gives results for the KPSS test for various values of l. I report test statistics both for Z 3 and Z 4, where Z 4 is Z as in (8) with =0.4 and c 3 =

15 Table 1 KPSS Test Statistics, sample l=2 l=4 l=8 l=12 l=16 Z *** 0.247*** 0.171** 0.146** 0.141* Z *** 0.245*** 0.171** 0.147** 0.142* asymptotic critical values: 1%: (***) 5%: (**) 10%: (*) There is apparently strong evidence against the hypothesis of trend stationarity from the KPSS test. All test statistics are significant at least at the 10% level of significance, some of them even at the 1% level. Interestingly, this is in line with results of Meier (2001), who also finds rejection of trend stationarity with the KPSS test and with the related test of Leybourne and McCabe (1994). However, Meier choses to discard these results in favor of specifying the test equation withour trend term. He then produces test statistics which are insignificant, but his model is clearly misspecified since it implies a long run growth rate of GDP which is zero a fact which seems to have escaped his attention. Finally, let us consider the KPSS test for the problematic sample , cf. Table 2. On the basis of these test results we are unable to reject the null for small truncation parameters, but do reject it at the 5% or 1% level for large values of l. This may indicate that the autocorrelation structure of the test equation s error term is particularly complicated for this sample definition, so that the appropriate dynamic specification requires long lags. In general, the only problem with high values of l is a loss of power (while small values of l may lead to inconsistent tests), but since we find significant statistics for l=12 and l=16 in Table 2, the power issue is irrelevant here and we may thus conclude that the null of trend stationarity is rejected even for the sample Table 2 KPSS Test Statistics, sample l=2 l=4 l=8 l=12 l=16 Z ** 0.340*** Z ** 0.309*** asymptotic critical values: 1%: (***) 5%: (**) 10%: (*) 15

16 V. Supplementary Tests Calibration is a hotly disputed issue in economics and particular since it does not have sound statistical foundations. Consequently, the above approach suffers from the drawback that it is impossible to derive the correct critical values and must instead rely on simulation. An alternative research strategy would be the use of unit root tests which allow for flexible trend models. Such tests have been developed by Park and Choi (1988), both for the unit root null and for the null of stationarity. Their approach allows the time series under investigation to fluctuate around arbitrary polynomial trends. While the correct trend model in (7) is not polynomial, a polynomial approximation may be quite good. For instance, estimating a second order autoregressive specification for lngdp with polynomial trend yields lngdp = lnGDP 0.431lnGDP t t t + ε 2 3 t t 1 t 2 t ( 4.03) ( 6.90) ( 2.96) ( 3.45) ( 3.27) ( 3.15) (10) Q(16) = 9.97 with apparently white noise residuals. (A fourth order trend term is not significant). Since Park and Choi derived the asymptotic distributions for their test, the issue boils down to either usage of the correct trend model with approximately correct critical values the approach of the preceding section or usage of an approximately correct trend model with true (asymptotic) critical values the approach of this section. Which of these two approaches is preferable, is largely a matter of taste, but it is certainly informative to pursue both of them. Let us first apply Park and Choi s J-test to lngdp. The test relies on a regression of the form k l i i t = µ i + µ i + t i= 0 i= k + 1, (11) ln GDP t t u where the true trend polynomial is of order k and the regression additionally provides for higher order terms up to order l. Let F(k,l) be the standard Wald-statistic for the null hypothesis µ k+ 1 =... = µ l and T be the number of observations. Under the unit root null, J ( kl,): = Fkl (,)/ T has a finite asymptotic distribution, while under the alternative of stationarity J (k,l) converges to zero. (Hence in order to reject the null, J (k,l) must be smaller than the 5% quantile of the asymptotic distribution). The J-test does not require to specify a certain number of lags. Instead, k and l must be chosen. From (10), k=3 seems a sensible choice, and Table 3 reports test results for various specifications of l. The unit root null cannot be rejected for any choice of l 3. 3 Qualitatively the same results obtain for k=2. In this case, however, J often assumes values even larger than the 95% quantile of the distribution, hence rejecting the null on the wrong side. This supports the view that k=2 is not the adequate polynomial trend specification. 16

17 Table 3 Park and Choi s J-Test Sample l J 5% critical value Park and Choi s G-test of the null of stationarity is also based on (11). The test statistic is 2 1 T 2 G kl, = F kl, σˆ ωˆ, where σˆ : = T uˆ is an estimate of the variance of u in (11) ( ) ( ) 2 2 t= 1 and ω ˆ2 is the estimate of the spectrum of u at frequency zero obtained from a lag window with v autocovariance terms. Under the null of stationarity, G(k,l) has an asymptotic chisquare distribution with l-k degrees of freedom. The results for lngdp for various choices of l and v are displayed in Table 4. t Table 4 P-values for Park and Choi s G-test Sample v l The general picture which emerges from Table 4 is as follows: For small values of v, i. e. spectral estimates with strong smoothing, the null hypothesis of stationarity can mostly not be rejected. As v increases, so does the evidence against the stationarity hypothesis. For v=12, i. e. a spectral window quite narrowly focused on frequency zero, all G-statistics are significant. Since already the KPSS tests indicated that lngdp s autocorrelation structure is quite protracted under the hypothesis of stationarity, small values for v may not be adequate. From this perspective the evidence against the the null may deem more reliable than the results which are compatible with the null. But even without reference to the KPSS tests, the results in Table 4 are not as ambiguous as they may seem. For in any case the data generating process seems to be near unit root, i. e. either exactly I(1) or stationary with a root close to one. Such near unit root processes have much spectral mass at frequency zero, in fact, the spectra of I(1) processes have poles at 17

18 frequency zero. Consequently, broad spectral windows as for instance implied by v=4 induce a strong downward bias in the estimate of the spectral density at frequency zero. Hence, small values of v result in bad spectral estimators for near unit root processes at frequency zero, and hence the G-test for small values of v cannot be considered reliable. But focussing on large values of v yields fairly clear evidence against the null of stationarity, and thus both of Park and Choi s tests suggest that lngdp is I(1). 18

19 VI. Conclusions It is the central idea of this paper that the exceptional German catch-up process must be properly taken into account if unit-root tests are applied to German data starting as early as The apparent problem inherent in unit root inference applied to the transitional dynamics of a Solovian growth process lies in the fact that it is difficult to distinguish convergence to a balanced growth path from (persistent or mean-reverting) fluctuations in the neighborhood of this balanced growth path. For the case of a Cobb-Douglas production function, it is possible to solve the Bernoulli differential equation which is at the heart of the Solow model and derive the appropriate correction for transitional growth. The unit root hypothesis for German GDP can therefore be tested under a correct specification of the underlying trend. This relies on parameter values which are difficult to estimate. Calibration, however, may be a useful device to proceed: First, the approximate magnitude of the parameters in such a parsimonius model may be infered from familiar macroeconomic stylized facts, and, second, Monte Carlao evidence suggests that erroneous calibration may not greatly influence power and level of the ADF unit root test. Using this strategy, I find no convincing evidence against the unit root null hypothesis. In particular, extending the span of the data over almost half a century results in test statistics which are clearly insufficient to reject the null. This is polar to some recent research which argued that inclusion of GDP data from the 1950s might provide the critical mass of evidence which allows for a rejection of the null. Clearly, the results presented in this paper suggest that such claims should be cautiously received. Quite contrary, testing the reverse null hypothesis of trend stationarity does result in significant statistics. Hence the ADF test and the KPSS test yield basically consistent results. The same is true for Park and Choi s J- and G-test: Again, the unit root null cannot be rejected, while is is difficult to argue in favor of the null of stationarity. Taken together, the unit root model seems to be the clearly preferable model for German GDP. 19

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