As the U.S. economy goes into a downturn, we are going to be reminded that extrapolating trends is a hazardous enterprise. For instance, linear extrapolation of tax receipts (expressed as a share of GDP) is probably something that one should be wary of doing. And yet, as shown in some comments on previous posts (see here and here), there seems to be too much belief in what ocular regressions can tell one.
So here is post that, while it seems to be completely unrelated to issues of current interest, should be of concern to those who do forecasting. (Time series econometricians can skip this post, or you will be appalled at the things I’m going to say, in terms of the sloppiness of discourse. The technically inclined should read Jim Hamilton’s book, chapters 15-17.)
Let me return to the issue of Federal tax receipts (line 2, BEA GDP table 3.1) expressed as a share of GDP. Back in July 2006, I asserted that the increase in tax receipts was not so remarkable in the context of statistical uncertainty. A cursory glance at the data would seem to contradict my assertion. Indeed, the sky seemed to be the limit in July 2006.
Figure 1: Federal tax receipts to GDP ratio (blue), and linear trend estimated over 2003q1-06q1 period (red). Source: BEA NIPA release of 29 November 2007, Table 3.2, NBER and author’s calculations.
What troubled me was that the discourse was couched in terms of trends and period averages. I mentioned that estimated trends were sensitive to sample period. And here is an illustration of that fact. I’ve plotted below estimated (OLS) linear trends for several different superiods. One clear result is that the “trends” move around with the sample period.
Figure 2: Federal tax receipts to GDP ratio (blue), and linear trend estimated over 67q1-86q4 period (red), 77q1-96q4 (green), 87q1-06q4 (purple) and 67q1-06q4 (teal). Source: BEA NIPA release of 29 November 2007, Table 3.2, NBER and author’s calculations.
The sensitivity of estimated trends to sample periods is the quintessential feature of nonstationary time series, or more precisely integrated time series (a random walk is an example of an integrated time series). Technically, the OLS estimator of the trend coefficient does not converge to the population mean when dealing with an integrated series.
Now one might argue that the tax receipt to GDP ratio cannot literally be a nonstationary series; it’s bounded from below at zero and above at one. However, the question is whether over the sample period is bounded; or even if the series is stationary, but highly persistent, then one might still get this outcome.
What do formal tests indicate? A standard unit root test (ADF, using the Schwartz Bayesian information criterion for lag length, and allowing for constant and trend) fails to reject the null hypothesis. Using the Elliott-Rothenberg-Stock Dickey-Fuller test also fails to reject the unit root null. The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test which has a trend stationary null rejects at the 5% MSL (bandwidth for the kernel estimator is 10). In other words, the series is so persistent that it is better treated as an integrated process than as a trend stationary process (no surprise to people who work in the macroeconometrics). On the other hand the change (i.e., first difference) of the tax receipts/GDP ratio is stationary (or technically, rejects the unit root null, and fails to reject the trend stationary null). Hence, the Federal tax receipts to GDP ratio appears to be difference stationary.
So, reader, beware of fitting trends and extrapolating! What’s the recourse? When the series is difference stationary, then one can forecast by working with the first differenced series. I show the implications of doing so in figure 3. The blue line with boxes is the forecast using a linear trend; the red line with boxes is the forecast assuming the ratio “drifts” up over time (technically, here I’m modeling the ratio as a random walk with drift, although a more refined approach would probably model the series as a ARIMA(0,1,1)).
Figure 3: Federal tax receipts to GDP ratio (blue), and linear trend estimated over 87q1-07q3 period (blue) and plus/minus 2 standard error prediction intervals; and forecast from estimated drift (red) and plus/minus 2 standard error prediction intervals. Source: BEA NIPA release of 29 November 2007, Table 3.2, and author’s calculations.
I’ve included the ±2 standard error bands for the respective forecasts. One interesting aspect is that the prediction interval expands over time for the difference stationary forecast (in red). The prediction interval for forecast assuming trend stationarity does not expand over time. Of course, the test results suggest that the assumption of trend stationarity is not justified, so the prediction interval is included merely for illustrative purposes.
The key take-away from this last figure is that even when the forecast trends are very similar, the implied uncertainty regarding the forecasts values differs substantially. In the figure, the two forecasts just happen to parallel. However, the more appropriate forecast associated with the difference stationary assumption implies much greater uncertainty going out toward longer horizons. Assuming trend stationarity (when not appropriate) provides an false sense of certainty.
For more (albeit dated) cautionary tales, see “Beware of Econometricians Bearing Estimates,” JPAM (1991).