Univariate Linear Prediction Models

When building a forecasting model for an economic variable, the variable’s ownpast time series is often the first thing that comes to mind. Using only this predictor has given rise to a surprisingly successful class of time-series specifications known as autoregressive moving average (ARMA) models. Dating back to the seminal work by Box and Jenkins (1970), these models have become the workhorse of the forecast profession and an extensive literature covers estimation and forecasting with ARMA models. Indeed, ARMA models form the backbone of many commercial forecasts and remain the centerpiece of many applied forecasting courses.

The popularity and success of ARMA models can be explained by several phenomena. First and foremost, these models pose a very minimalist demand on the forecaster’s information set. The only information required by these models is the history of the variable of interest, $Z_t=\{Y_1,Y_2,...,Y_{t-1},Y_t\}$ . This information can be used to construct an ARMA forecast even in situations where the forecaster has no real idea of a model for Y and hence does not know which other variables to obtain data on. For example, an inventory manager will often have information on past inventories, but may not have information, let alone a fully specified model, to suggest why there are fluctuations in the demand for a product. Even if a sales model were available, it might be too expensive to collect the data required for estimation of the model’s parameters.

Second, because of their minimal information requirements, ARMA forecasts make for an excellent “baseline” against which the forecasts from more complicated methods can be compared. By excluding other variables, ARMA forecasts show how useful the past of a time series is for predicting its future. The value added by using more complicated forecasting methods or including additional predictor variables can then be measured by the extent to which such forecasts reduce the average loss relative to the loss associated with the best ARMA model.

Third, the use of ARMA models is underpinned by theoretical arguments. Specifically, the Wold representation theorem shows that all covariance stationary processes can be represented by a (possibly infinite-order) moving average representation. ARMA models form an exhaustive set for such processes and have certain optimality properties among linear projections of a variable on its own past and past shocks to the series. This result does not say that ARMA models are optimal in a global sense. For example, it may be better to use nonlinear transformations of past values of the series in the prediction model or to condition on a wider information set. However, if attention is restricted to linear functions of past values of the series, then it is natural to start with ARMA prediction models.

Fourth, despite—or, maybe because of—their simplicity, ARMA models have proved surprisingly difficult to beat in empirical work. Provided that the lag order of ARMA models is sensibly determined, they rarely produce extreme forecasts and are good at capturing persistence in economic variables through low-order autoregressive or moving average components.

Full expositions of the material covered in this chapter are available in many textbooks, including Brockwell and Davis (1996) at the more technical end and Diebold (2007) at a level that is easier to access. The purpose of this chapter is to provide a brief survey of an area that is part of classical work in forecasting and thus provides foundations for material covered in subsequent chapters, most notably chapter 8 on univariate nonlinear forecasting models and chapter 9 on multivariate forecasting with vector autoregressions.

The chapter proceeds as follows. Section 7.1 offers a brief review of the basic properties of ARMA models. Section 7.2 discusses parameter estimation and lag length selection for these models. Section 7.3 introduces forecasting schemes for ARMA models. Section 7.4 covers extensions to deterministic and seasonal components. Section 7.5 introduces exponential smoothing and unobserved components models and section 7.6 concludes.

7.1 ARMA MODELS AS APPROXIMATIONS

We first introduce some basic concepts from time-series analysis required for understanding the forecasting models. Next, we introduce the basic autoregressive (AR) and moving average (MA) models.