US real GNP, 1875–1929 and 1950 –1993

4. US real GNP, 1875–1929 and 1950 –1993

It is possible to inquire into the plausibility of TS between 1875 and 1993 while discounting the experience of 1930 –1949. This would still require the existence of a fixed trend line over the whole period, and presumably the same second moment structure around that line before 1930 and after 1949. To investigate this possibility, we established a TS regime by fitting to the 1875–1929 period TS models from the ARMAp,q class given below. ~1 2 f 1 L 2 . . . 2 f p L p ~ y t 2 a 2 bt 5 ~ 1 2 u 1 L 2 . . . 2 u q L q e t 3 As before p,q were selected through SBC, which yielded a first-order autoregression for all four series GNP-BG, GNP-R, GNP-BGPC, GNP-RPC. 18 We then used these fitted models to generate forecasts from base years 1950, 1951, . . . , 1992. This yields 43 one-year-ahead forecasts, 42 two-year-ahead forecasts, and so on, that can be compared with the observed true values of the series. To give a basis for comparison, we also fitted DS models of the form 1 to the 1875–1929 data. In all cases, the random walk model was selected by SBC. We used these fitted models to generate forecasts in the later period. The ratios of the forecast mean squared errors MSE are shown in Table 2. Clearly the TS models are seriously inferior based on this criterion, particularly for the nonper capita series. Although the high ratios, MSETSMSEDS, for the raw data constitute very strong evidence against TS, they should not be interpreted as strong evidence in favor of DS. The TS assumption of a fixed trend line over all time is a very strong one. When this assumption is false, forecasts will not readily adapt to new generating structures as will forecasts from a random walk. Of course, it could be argued that the relatively poor performance of the TS forecasts could be due in part to sampling error in the estimated parameters, particularly those of the trend function, given that we are extrapolating so far ahead. To allow for this possibility we again carried out parametric bootstrap tests of the TS specifications. Specifically, we generated 2000 series of 119 observations from the TS models fitted to the 1875–1929 data. We then fitted random walk models to the first 55 observations of each sample correspond- ing to 1875–1929 and computed forecasts of the last 44 observations corresponding to 1950 –1993, precisely in the manner that generated the entries of Table 2. For the one-step Table 2 Ratios of mean squared errors for trend-stationary to difference-stationary models for h-years-ahead forecasts, 1950 –1993, based on models fitted to 1875–1929 data. h GNP-R GNP-RPC GNP-BG GNP-BGPC 1 8.912 1.676 13.67 1.701 0.004 0.185 0.001 0.154 2 11.25 1.732 16.48 1.683 3 12.15 1.723 17.03 1.606 4 13.14 1.844 17.83 1.661 5 20.68 2.318 22.70 1.828 Note: Figures in brackets are parametric bootstrap test p-values of the adequacy of the trend-stationary specification. 93 P. Newbold et al. Journal of Economics and Business 53 2001 85–102 ahead forecasts, we report in that table the portion of times that the DS forecasts outperform the TS forecasts by more than was found for the actual data sets i.e. the proportion of times that the simulated ratio of mean squared errors exceeded the actual entry. These are then p-values of a test of the TS specification. While it is possible to get these large values for the ratio, MSETSMSEDS, with a true TS series it is unlikely. How unlikely? We find from Table 2 that TS can be rejected at very low significance levels for the nonper capita series. For the per capita series, the p-values are 0.185 for the Romer data and 0.154 for the Balke-Gordon data. The results thus show that if the true model is TS, then the ratio MSE TSMSE DS for the simulated series will exceed the actual value only 0.4 of the time for GNP-R, 18.5 of the time for GNP-RPC, 0.1 of the time for GNP-BC and 15.4 of the time for GNP-BGPC. Given that the competitor is just a simple random walk, this represents moderately strong evidence against TS. The forecasting performance of the TS models remains surprisingly poor in a TS world, when allowing for sampling error in the parameter estimates. Further insight can be obtained through a more detailed examination of the one-year- ahead forecasts. Figs. 3 and 4 show logarithms of RGNP together with one-year-ahead forecasts generated from DS and TS models fitted to the Balke-Gordon data, 1875–1929. The pictures for models fitted to the Romer data are similar. Fig. 3, for the nonper capita series clearly illustrates the difficulty of tying forecasts to a fixed trend. The forecasts from the TS models are consistently too high, particularly in the later years. As shown in Table 2, and illustrated in Fig. 4, forecasts of per-capita GNP from a TS model are somewhat less poor. Now, forecasts from the TS model are too low for much of the period. It is certainly Fig. 3. GNP-BC: Actual and forecast values, 94 P. Newbold et al. Journal of Economics and Business 53 2001 85–102 clear that the one-year-ahead forecast errors are not zero-mean white noise, as would be the case if the same TS model held in 1875–1929 and 1950 –1993. The source of the results in Table 2 can also be seen through graphs of the complete series. Figs. 5 and 6 show the two Balke-Gordon series. We have drawn on these graphs the trend lines obtained from fitting a TS model to the 1875–1929 data, and extrapolated those lines. Substantial deviations from the extrapolated trend lines in 1950 –1993 are apparent, partic- ularly for the nonper capita series. Further examination of the one-year-ahead forecast errors is useful. For the forecast errors generated from DS models, we fitted processes of the form 3. In fact, p,q 5 0,0 was selected by SBC in all cases. Table 3 shows t-ratios associated with the estimated intercept and slope parameters. Although the slope parameter estimates are moderately far from zero for the nonper capita series, there is not very strong evidence against the contention that these forecast errors are zero-mean white noise—that is, that the difference-stationary model fitted to 1875–1929 data has generated optimal forecasts in 1950 –1993. The TS forecasts are based on first order autoregressions. If, in fact, the 1950 –1993 data are generated by a random walk, it can be shown that the one-year-ahead forecast errors e t should then follow an ARIMA 0,1,1 process De t 5 b 1 ~ 1 2 uLe t 4 Table 4 shows parameter estimates from fitting this model to the forecast errors. Although the moving average parameter estimates are large, corresponding to the large autoregressive Fig. 4. GNP-BGBC: Actual and forecast values. 95 P. Newbold et al. Journal of Economics and Business 53 2001 85–102 parameter estimates used to generate the forecasts, they are still far from one. Moreover, the residual autocorrelations from the fitted models 4 do not suggest serious misspecification. In fact, fitting models of the form 1, with e t in place of y t , to the forecast errors, the ARIMA0,1,1 model was selected by SBC for both nonper capita series. This model was a close second choice for the two per capita series. However, the first choice in both cases was an ARIMA1,1,1 model, with estimated moving average parameter one. This should not be taken as strong evidence of overdifferencing, as maximum likelihood estimation is very likely to produce such an outcome in sample sizes as small as this. Our analysis of forecasts of the years 1950 –1993, generated by models fitted to the years 1875–1929 suggests that, even discounting the experience of the intervening years, station- arity around a single trend over the whole period 1875–1993 is implausible. Indeed there is little evidence against a random walk in the post-World War II years. As Diebold and Senhadji 1996 suggest, distinction between TS and DS processes can be important for forecasting purposes. We have seen that tying forecasts to an inappropriate fixed trend can generate seriously suboptimal outcomes.

5. Conclusions