7.3.4 Forecasting Variables with Unit Roots

7.3.4 Forecasting Variables with Unit Roots

This chapter focuses on stationary ARMA processes, but we briefly explain how to proceed if not all of the roots of the autoregressive polynomial $\varphi (L)$ fall outside the unit circle. The most common case is when one or more of the roots equals unity, while the remaining roots fall outside the unit circle. Suppose that $d$ roots of $\varphi (L)$ lie on the unit circle for some integer d while the remaining ones lie outside the unit circle and consider the factorization $\varphi (L)=\tilde{\varphi }(L)(1-L{)}^d$ . Factoring the polynomial in this way, $\varphi (L)y_t=\theta (L){\epsilon }_t$ can be written as $\tilde{\varphi }(L){\tilde{y}}_t=\theta (L){\epsilon }_t.$ where ${\tilde{y}}_t=(1-$ $L{)}^d{y}_t$ is called the dth difference of ${\gamma }_t$ . By assumption, the roots of $\tilde{\varphi }(L)$ lie outside the unit circle so the differenced process, ${\tilde{y}}_t.$ , will be stationary and can be studied instead of ${\gamma }_t$ . This is known as differencing ${\gamma }_t$ to stationarity and is a commonly applied method. Processes with d $\ne 0$ need to be differenced to achieve stationarity and are called $\mathrm{ARIMA}(p,d,q)$ processes, where the ${}^{\mathfrak{e}}{\mathrm{T}}^{\mathfrak{p}}$ stands for “integrated,” $i.e.$ , the opposite of differencing. There are many practical difficulties involved with determining d as well as with handling the discreteness of the properties of the model as the roots move from being close to 1 to being equal to 1. These are addressed more generally in chapter 20.

The common practice of transforming variables by differencing them a suitable number of times to ensure that they are stationary is important for estimation and model selection. Forecasts of the differenced variables can then be transformed back to levels and compared to outcomes in levels, if desired. We briefly illustrate how this works.

Suppose we are interested in forecasting the future level of y, denoted $y_{t+h}.$ but that y may be integrated of first or second order, which we denote by I(1) or I(2), so that $\Delta y_t$ is stationary if y is I(1), while ${\Delta }^{2}y$ is stationary if y is I(2). When y is I(1), we model the first-differenced series and so predict $\Delta y_{t+h}=y_{t+h}-$ ${\gamma }_{t+h-1}$ . We denote this forecast by $\Delta f_{t+h|t}$ . When y is I(2), we model the seconddifferenced series and so predict ${\Delta }^2{y}_{t+h}=(y_{t+h}-y_{t+h-1})-(y_{t+h-1}-y_{t+h-2})=$ $y_{t+h}-2y_{t+h-1}+y_{t+h-2}$ . We denote this forecast by ${\Delta }^2{f}_{t+h|t}$

Forecasts of the level of $y_{t+h}$ can be constructed from the forecast of $y_{t+h},\Delta y_{t+h}$ and ${\Delta }^2{y}_{t+h}$ as follows:

$$f_{t+h\mid t}=\{\begin{array}{ll}{f}_{t+h\mid t}& \mathrm{if}{y}_t\mathrm{is}\mathrm{I}(0),\\ y_t+{\sum }_{i=1}^h\Delta f_{t+i\mid t}& \mathrm{if}{y}_t\mathrm{is}\mathrm{I}(1),\\ y_t+h\Delta y_t+{\sum }_{i=1}^h{\sum }_{j=1}^i{\Delta }^2{f}_{t+j\mid t}& \mathrm{if}{y}_t\mathrm{is}\mathrm{I}(2).\end{array}$$

7.4 DETERMINISTIC AND SEASONAL COMPONENTS

Many economic time series follow a seasonal pattern. For example, retail sales, employment numbers, and housing starts are linked to weather patterns and holidays as workers are temporarily laid off due to bad weather or hired because of busier seasons in retail and services. It is common to filter out the seasonal component and report economic activity numbers on a seasonally adjusted basis. However, the raw, unfiltered number can be of separate interest, in which case a good prediction model must account for seasonal variation.10