20.2 UNIVARIATE FORECASTING MODELS

20.2 UNIVARIATE FORECASTING MODELS

Define $d_t$ to be deterministic if $\dot{}{d}_t$ is perfectly predictable into the future. For example, $d_t$ can be a constant, a time trend, or a sine wave. When the trending behavior of a series is caused by deterministic terms, model estimation and many of the aspects of forecast evaluation follow the discussions in earlier chapters. Because $d_t$ is perfectly predictable (if there are no nondeterministic terms), we can consider the forecasting model $y_t=f_t(z_t,\beta )+{\epsilon }_t$ to estimate $\beta$ under MSE loss. For any horizon, forecasts can be constructed from ${\hat{\mathbf{f}}}_{t+h|t}=f_t({\mathbf{z}}_{t+h},\hat{\mathbf{\beta }})$ since $z_{t+h}$ is known at time t. Hence, no issues arise in relation to whether direct or iterated forecasts should be used. Estimates for $\hat{\mathbf{\beta }}$ are consistent and asymptotically normal if the model is sufficiently smooth, so model selection issues are similar to those covered in earlier chapters.

Consider the following linear model that may have a unit root:

$$\begin{array}{rl}& y_t={\varphi }^{\prime }{d}_t+u_t,t=1,\dots ,T,\\ & (1-\rho L)u_t=v_t,t=2,\dots ,T,\\ & u_1=\xi .\end{array}$$

Here $d_t$ consists of strictly exogenous deterministic terms and $\beta =({\varphi }^{\prime },\rho {)}^{\prime }$ . We consider models with either a constant or a constant and a time trend, so $d_t=1$ or $d_t=(1,t{)}^{\prime }$ and so $\varphi$ is either univariate or $\varphi =({\varphi }_1,{\varphi }_2{)}^{\prime };\xi$ is the initial condition. We can allow additional serial correlation through $v_t=c(L){\epsilon }_t$ , where ${\epsilon }_t$ is a zeromean white noise term with variance ${\sigma }_{\epsilon }^2$ . The lag polynomial describing the dynamic behavior of ${\gamma }_t$ has been factored so that $\rho =1-\gamma /T$ corresponds to the largest root of the polynomial, and we assume that c(L ) is one summable, i.e., ${\sum }_{i=0}^{\infty }|c_i|i<\infty$ We could alternatively assume that $v_t$ is globally covariance stationary and that $\begin{array}{r}{T}^{-1/2}{\sum }_{t=1}^T{v}_t{\to }^{d}N(0,{\omega }^2)\end{array}$ , where ${\omega }^2$ is the spectral density of $v_t$ at frequency ${}^{0,}$ divided by $2\pi$ . In either case, an autoregressive model for $y_t-{\varphi }^{\prime }{d}_t$ is regarded as an approximation to the stationary dynamics of the model.

When ${\gamma }_t$ is known to have a unit root, many results are equivalent to those obtained in previous chapters with a unit root imposed. Specifically, in this case the predicted variable can be differenced and modeling can proceed with the differenced data. If we use the iterated forecasting method of section 7.3, we can predict $\Delta y_{t+1},\dots ,\Delta y_{t+h}$ and use the sum $\begin{array}{r}{y}_t+{\sum }_{\tau =1}^h\Delta y_{t+\tau }\end{array}$ as our forecast for the level of $y_{t+h}$ . The direct forecasting method regresses $y_{t+h}-y_t$ on variables known at time t, which may include lags of the change in ${\gamma }_t$ , as well as other terms that are not trending apart from deterministic terms, to produce a forecast $f_h(z_t,\beta )$ . The forecast of the level of $y_{t+h}$ is then obtained as $f_{t+h|t}=y_t+f_h(z_t,\beta )$ . Similar results apply under loss functions other than MSE.

If we are unsure whether ${\gamma }_t$ contains a unit root, issues related to estimation error similar to those discussed in earlier chapters arise. Magnitudes of estimation errors are different in this setting, however, because parameter estimates are no longer asymptotically normally distributed and typically depend on how close the root is to unity. Forecasters have the choice between (i) ignoring the possibility that $\rho \ne 1$ and so estimating the autoregression in differences; (ii) estimating $\rho$ from an autoregression in levels; or (iii) combining the two approaches with a pre-test for a unit root. Comparison of the risk function in (i) and (ii) is straightforward—for $\rho$ close enough to 1, a smaller risk is obtained by imposing a value of 1 rather than estimating the parameter. $\mathrm{As}\rho$ moves further away from 1, the error from imposing the incorrect value of the largest root will increase and becomes larger than the loss due to estimation of the parameter. As in the stationary case, the risk function that arises from imposing $\rho =1$ when it is not true is unbounded as $\rho$ diverges from 1, whereas the risk from estimating $\rho$ is bounded for all values of $\rho$ .

By recursive substitution, the model in (20.2) with $c(L)=1$ yields

$$\begin{array}{l}{y}_{T+h}-y_T={\epsilon }_{T+h}+\rho {\epsilon }_{T+h-1}+\cdot \cdot \cdot +{\rho }^{h-1}{\epsilon }_{T+1}+({\rho }^h-1)(y_T-{\varphi }^{\prime }{d}_T)\\ +{\varphi }^{\prime }(d_{T+h}-d_T)\\ =\sum _{i=1}^h{\rho }^{h-i}{\epsilon }_{T+i}+({\rho }^h-1)(y_T-{\varphi }^{\prime }{d}_T)+{\varphi }^{\prime }(d_{T+h}-d_T),\end{array}\text{\hspace{1em}(20.3)}$$

and so the h-step-ahead forecast (with known parameters is) $f_{T+h|T}=y_T+({\rho }^h-1)(y_T-{\varphi }^{\prime }{d}_T)+{\varphi }^{\prime }(d_{T+h}-d_T)$ . We next compare each of the three strategies discussed above. If we impose a unit root on the model and $d_t=1$ , then the forecast becomes simply ${\gamma }_T$ . If there is also a time trend, then the forecast is $y_T+{\hat{\varphi }}_{2}h$ , where ${\hat{\varphi }}_2$ is an estimator of ${\varphi }_2$ . Under the second strategy that estimates the parameters $(\rho ,\varphi )$ , the forecasts are $y_T+({\hat{\rho }}^h-1)(y_T-{\hat{\varphi }}^{\prime }{d}_T)+{\hat{\varphi }}^{\prime }(d_{T+h}-d_T)$ . Different estimators can be used for $(\rho ,\varphi )$ in this case. There is no obvious “optimal” estimator for the autoregressive parameters and many estimators have been suggested. Typically, however, either OLS or median unbiased estimators are employed.3 In addition, least squares can be used to estimate $\varphi .$ , although Canjels and Watson (1997) and Ng and Vogelsang (2002) have also suggested GLS estimators. Alternatively, we can consider estimating $(\rho ,\varphi )$ directly by simply regressing ${\gamma }_t$ on $(y_{t-1},d_t)$ . The final strategy that has been considered in the literature is a hybrid of the first two approaches, namely selecting the relevant forecast using estimation or imposing a unit root after a pre-test for a unit root.

To analyze the behavior of the forecast errors, we employ large sample approximations for which $\rho =1+\gamma /T$ , although we suppress the dependence of $\rho$ on $T$ as is typical in this literature. For the initial condition let

$$\xi =\alpha \omega (2\gamma {)}^{-1/2}{T}^{1/2},$$

where ${\omega }^2=c(1{)}^2{\sigma }_{\epsilon }^2$ so the initial condition is asymptotically of the same order as the stochastic part of the model when $\alpha \ne 0$ . Setting $\alpha =1$ corresponds to drawing the initial condition from its unconditional distribution when $\rho <1$ . Under these conditions we have

$$\begin{array}{r}{T}^{-1/2}(u_{[Ts]})\Rightarrow \omega M(s)=\{\begin{array}{ll}\omega W(s)& \mathrm{for}\gamma =0,\\ \omega \alpha e^{-\gamma s}(2\gamma {)}^{-1/2}+\omega {\int }_0^s{e}^{-\gamma (s-\lambda )}dW(\lambda )& \mathrm{for}\gamma \ne 0,\end{array}\end{array}\text{\hspace{1em}(20.4)}$$

where $W(\cdot )$ is a standard univariate Brownian motion and (20.4) defines $M(s)$ . See Elliott (2006) for further details.