20.3.1 Empirical Example

20.3.1 Empirical Example

As an empirical illustration, we consider long-run forecasts of the quarterly inflation rate, i.e., the log-first difference in the CPI, the unemployment rate, and the interest rate. These time series are quite persistent as shown in previous chapters. We show results for forecast horizons h = {1, 2, 3, 4, 8, 12, 16, 20} quarters and compare forecasts from a random walk, random walk with drift, an AR model with lag length selected by the AIC, and forecasts from AR(1) and AR(4) models. The out-ofsample results reported in table 20.1 are generated for the period 1970–2014 using an expanding estimation window.

First consider the inflation rate. Here the results show that the autoregressive models generally perform better than the random walk model for horizons of 1 through 3 quarters, but that the random walk model is better at the longest 20 quarter horizon. The random walk model with drift produces very poor forecasts, especially at long horizons. For the unemployment rate the AR(4) model dominates at all horizons and the random walk model produces RMSE values that are generally a bit higher than those from the AR models. Conversely, the random walk model produces the lowest RMSE values for the interest rate series at both short and long horizons.

In all cases the random walk with drift model performs very poorly because the series investigated here are not trending deterministically in any particular direction. Whether the simple random walk specification or the autoregressive specification is best varies, however, across variables and across sample periods. Which model produces the best forecasts will depend on the true, but unknown, degree of persistence of the underlying series; this is consistent with our theoretical results.

20.4 FORECASTING WITH PERSISTENT REGRESSORS

A common forecasting situation involves a regressor that displays persistence — the regressor is reverting to its mean only very slowly, if at all—but appears to have predictive power based on tests for the exclusion of that regressor in a linear regression. Examples include using the forward premium to predict changes in exchange rates (Bilson, 1981), short-run interest rates, consumption, and stock prices for forecasting changes in income (Chen, 1991; Hall, 1978), or the dividend–price ratio or earnings–price ratio for forecasting future stock returns (Stambaugh, 1999; Valkanov, 2003). In many cases, variation in the outcome variable captured by such predictor variables tends to be modest, although often economically interesting, as in the case of a small, but highly persistent risk premium on a financial asset.

TABLE 20.1:
Root mean squared forecast errors (RMSE) for different forecasting models. The table reports RMSE values for the quarterly inflation, unemployment and interest rate series at different forecast horizons ranging from 1 through 20 quarters. The forecast evaluation period is 1970– 2014; parameters are estimated using a recursively expanding estimation window for the models with unknown parameters.

To capture this setting, consider the model

$$\begin{array}{c}{y}_t={\varphi }_y{d}_{yt}+\beta z_{t-1}+u_{yt},\\ z_t={\varphi }_z{d}_{zt}+w_{zt},\\ (1-\rho L)w_{zt}=u_{zt},\\ u_{z1}=\xi .\end{array}\text{\hspace{1em}(20.10)}$$

TABLE 20.2:

Size properties for one-sided test.

Note: Size is 5% for a one-sided upper-tail test.

The regressor, $z_t,$ follows the same process as ${\gamma }_t$ in (20.2). Assumptions on $u_{zt}$ are the same as in the earlier section. Regressions such as (20.10) typically do not allow $u_{yt}$ to be serially correlated. If $u_{yt}$ were serially correlated, we would want to model this correlation to improve predictability. Moreover, in empirical applications there is usually little or no serial correlation to be found. The terms $d_{zt}$ and $d_{yt}$ are assumed to be deterministic, with $d_{zt}\subseteq d_{yt}$

We further assume that the partial sums of the residuals $u_t=(u_{yt},u_{zt}{)}^{\prime }$ converge to a bivariate Brownian motion process with variance–covariance matrix, , given by10

$$\Sigma =\left(\begin{array}{cc}{\Sigma }_{11}& \delta {\Sigma }_{11}^{1/2}{\Sigma }_{22}^{1/2}\\ \delta {\Sigma }_{11}^{1/2}{\Sigma }_{22}^{1/2}& {\Sigma }_{22}\end{array}\right)$$

δ is the correlation between $u_{yt}$ and $u_{zt}$ . This is also the long-run correlation between these shocks when $u_{zt}$ is serially correlated but globally covariance stationary.

Monte Carlo simulation results have lead a number of authors to note that ttests of the hypothesis $H_0:\beta =0$ versus $H_1:\beta \ne 0$ are not well approximated by the standard normal distribution when $\rho$ is close to 1 and δ is far from 0. For $\delta >0$ , the test underrejects in the upper tail and overrejects in the lower tail; the opposite holds for $\delta <0;$ see Mankiw and Shapiro (1986), Elliott and Stock (1994), and Stambaugh (1999). Table 20.2 shows upper-tail rejections for tests with a nominal 5% level for $T=100$ and $\xi =0$ . The values for $(\delta ,\rho )$ are selected to be relevant choices for forecasting changes in stock returns with the dividend–price ratio (Stambaugh, 1999).

Econometric analysis has explained such findings either as a small sample bias problem that disappears asymptotically or by modeling $\rho$ as being local-to-unity and showing that there is always a region for $\rho$ in which such tests will overreject, regardless of the sample size. Under both approaches, we have a SUR model for $\delta \ne 0$ , and we can rotate $u_t$ so $u_{yt}={\Sigma }_{11}^{1/2}(\delta {\Sigma }_{22}^{-1/2}{u}_{zt}+(1-{\delta }^2{)}^{1/2}{u}_{y.zt})$ , where $u_{y.zt}$ is orthogonal to $u_{zt}$ . If there are no deterministic terms in the model, this allows us to write the OLS estimate of $\mu$ as

$$\begin{array}{l}(\hat{\beta }-\beta )=\frac{\sum z_{t-1}{u}_{yt}}{\sum z_{t-1}^2}\\ ={\Sigma }_{11}^{1/2}\left(\delta {\Sigma }_{22}^{-1/2}(\hat{\rho }-\rho )+(1-{\delta }^2{)}^{1/2}\frac{\sum z_{t-1}{u}_{y.zt}}{\sum z_{t-1}^2}\right),\end{array}$$

where $\hat{\rho }$ is the OLS estimate of $\rho$ obtained from an AR(1) regression on $z_t$ . The intuition behind the “small sample” approach becomes clear from. The second line has a sampling distribution centered on 0. The OLS coefficient is downward biased for $\rho >0;$ so if $\delta <0$ then $\hat{\mathbf{\beta }}$ will be upwards biased, as found in the simulations discussed above. The extent of the bias depends on the nature of the deterministic terms and is greater when constants are estimated in the regressions. Stambaugh (1999) presents analytical results for the size of the bias when $|\rho |<1$ Amihud and Hurvich (2004) suggest a correction for the bias, and base tests on their corrected estimator. Each of these methods requires that $|\rho |<1$ . The problem with this approach is that for roots “close” to 1, the assumptions underlying the correction fail and the correction does not solve the overrejection problem and means that one needs to know whether $\rho$ is close to 1. In general, if $\rho$ is far enough away from 1, the original size distortion problem is small as is the correction. Hence, such methods seem applicable only for “moderate” values of $\rho$ and it is difficult to tell the precise region where $\rho$ is both close enough to 1 that we need to correct the statistic, yet far enough away from 1 that the correction solves the overrejection problem.

Asymptotic approaches allow for $\rho$ smaller than 1 (but close to 1) as well as $\rho =1$ uniformly. Following this approach and employing the expression, Elliott and Stock (1994) use the local-to-unity representation ${\rho }_T=1-\gamma /T$ to show that when $\beta ={\beta }_0$ , t-statistics converge to the limiting distribution

$$t_{\beta ={\beta }_0}⟹\delta t^{\ast }+(1-{\delta }^2{)}^{1/2}{z}^{\ast },$$

where $z^{\ast }$ is a standard (mixed) normal random variable that is independent of $t^{\ast }{}_{;}$ and $t^{\ast }$ has a nonstandard distribution equivalent to the distribution of a t-test testing that $z_t$ has a coefficient ${\rho }_T$ when regressing $z_t$ on $z_{t-1}$ and lagged changes that correct for the serial correlation in $u_{2t}$ . For $\rho =1$ , this is the well-known Dickey–Fuller distribution for a t-statistic in a unit root test. This nonstandard distribution is a function of $\gamma ,$ the local-to-unity parameter. Thus, when δ is small or 0, there is little size distortion in testing whether or not $z_t$ is a good predictor of $y_{t+1};$ however, for values of δ further away from $0,$ we expect size distortions even asymptotically if $\rho$ is sufficiently close to 1.

Understanding whether or not this problem arises empirically is relatively straightforward since the extent of the problem depends on the parameters $(\delta ,\rho )$ and the sample size. We can estimate $\delta$ consistently from the data by using OLS on and estimating the spectral density matrix using standard methods outlined in Newey and West (1987) or Andrews (1991). Estimates of the correlation from this estimated matrix are consistent, though biased towards 0. No consistent estimator is available for the local-to-unity parameter, $\gamma .$ However, we can still understand empirically whether there is a problem because we can construct confidence intervals on $\rho$ that are informative about which values of $\gamma$ appear plausible. For any value of $\delta ,$ size distortions in conventional t-tests are largest when $\rho$ is near 1. For example, if $\delta =0.5$ , a nominal 10% equal-tailed test leads to lower-tail rejections of 11% and upper-tail rejections around 2%. When $\delta =0.9$ , the lower-tail rejections increase to 38% while the upper-tail rejections almost completely disappear—they are about one-tenth of 1%. As γ gets larger and $\rho$ decreases, these size distortions disappear.

Different approaches have been suggested for constructing confidence intervals on $\beta$ in (20.10). Cavanagh, Elliott, and Stock (1995) suggest a Bonferroni approach, using confidence intervals for $\rho$ along with the distribution in (20.13) to provide intervals for $\beta .$ Campbell and Yogo (2006) show gains over this method from a similar approach that is based on a rotated version of (20.10). Jansson and Moreira (2006) suggest a method that conditions on a sufficient statistic for $\rho ,$ which results in a test that maximizes power conditional on this statistic. Since this test is similar for all values of $\rho ,$ by construction it is unbiased. However, this property comes at the cost of reduced power—Jansson and Moriera show that for many alternatives the Campbell and Yogo (2006) approach has better power. Elliott (2011) shows that additional stationary covariates that help explain the simultaneity leading to a nonzero $\delta$ can be used to mitigate or remove size distortions. Elliott, Müller, and Watson (2015) develop a method for constructing hypothesis tests on $\beta$ that is optimal against a point in the alternative space and controls size for all values of δ. These latter two methods have the property that they do not necessarily require $\rho$ to be near 1.