The Parametric Forecasting Problem

The forecaster’s objective is to use data—outcomes of the random variable $Z-$ to predict the value of the random variable Y. Let $T$ be the date when the forecast is computed and define ${\mathcal{I}}_T$ as the sigma algebra generated by $\{Z_t{\}}_{t=1}^T,$ so ${\mathcal{I}}_T$ is a filtration $({\mathcal{T}}_{T-1}\subset {\mathcal{T}}_T)$ . Following conventional terminology in the forecasting literature, we refer to ${\mathcal{I}}_T$ as the information set at time T. We define $Z$ as $\{Z_t{\}}_{t=1}^T$ when the timing of the forecasting problem is clear and as $Z_T$ when we need to be clearer about the time subscript, and write objects as conditional on $Z$ or conditional on ${\mathcal{I}}_T$ as meaning the same thing. By $E\left[Y|z\right]$ we mean $E\left[Y|Z=z\right]$ or $E\left[Y|{\mathcal{I}}_T\right]$ . The random variable $Z$ typically includes past values of the predicted variable, as well as other possibly useful variables, $X_t,$ so often $\{Z_t{\}}_{t=1}^T=$ $\{Y_t,X_t{\}}_{t=1}^T$ . In some cases (such as ARIMA and VAR forecasts, which we discuss in more detail in chapters 7 and 9), the information set might include only past values of the predicted variable(s). The outcome, Y, may be a vector or could be univariate.

We assume throughout the analysis that a joint distribution for $\{Y,Z\}$ exists, which can be written as $p_Y(y|z,\theta )p_Z(z,\theta )$ , where the vector $\theta$ comprises all the parameters of the joint distribution. Through a small abuse of notation we refer to the true parameters as θ regardless of whether we are referring to the conditional distribution for Y given $Z$ or to the joint distribution. We refer to this joint distribution as the data-generating process or true model.

This chapter focuses on the construction of optimal point forecasts, i.e., finding the best possible $f(Z)$ for a general forecasting problem. There are many reasons for considering the optimality of a forecast. First, clearly we want the best possible forecast. Second, questions related to forecast optimality can be complicated and there will not be a single best forecasting approach even in the simplest of problems. Selecting a good forecasting approach often requires deeper thinking about the problem than merely stating a loss function and it need not be straightforward to determine a reasonable set of models for a realistic forecast problem. This point motivates the wide range of approaches considered in the empirical forecasting literature (and in part II of this book), with different approaches being tailored to different data-generating processes and forecast objectives. Claims of forecast optimality are typically limited to restrictive classes of models and so can be very weak—optimality often does not extend too far in the sense that there may be better forecast models available, given the same data.

We use “risk” as our measure of the performance of a forecasting method, as well as the best measure for distinguishing between methods. Using jargon from the statistics literature, the risk of a forecasting method is simply the expected loss when integrating over both Y and Z. This focus on risk is unlikely to be controversial. Many forecasting papers motivate their methods through Monte Carlo experiments that examine risk at a few, carefully chosen points in the space of the parameters and data-generating processes. Such risk evaluation exercises require us to think carefully about the model spaces and to avoid assuming that methods that work for some Monte Carlo studies will necessarily work more generally, especially for quite different data-generating processes. It also highlights the role of the loss function; results favoring a particular forecasting model for one loss function need not generalize to other loss functions.

Examining in a single chapter the forecasting problem in the context of decision theory requires us to cut many corners. Hence, this chapter is really an overview of some key points from decision theory and estimation theory adapted to the forecasting problem and using language from forecasting. We demonstrate results not by proofs with technical conditions but through relevant examples.

Forecast models may be parametric, semiparametric or nonparametric. Parametric models for the forecast are fully specified up to a finite-dimensional unknown vector, β. Nonparametric models typically have an infinite-dimensional set of unknown parameters. Semiparametric models fall in the middle. Throughout this chapter we assume that the true joint distribution of the data, or data-generating process, has a parametric representation with a joint density for the outcome variable and the conditioning information used by the forecaster. This is the setting that the title of the chapter refers to.

An alternative to examining point forecasts is to provide a predictive density. The construction of a predictive density arises naturally under the Bayesian approach. From the classical perspective, the estimation problem will be different from that of point forecasting. We will see below that point forecasts are features of the predictive distribution in the sense that knowing the predictive distribution allows the construction of the optimal point forecast. Hence, generating a predictive distribution is in a sense a harder problem since the estimation problem is to construct something more informative than the point estimate. For this reason it is often suggested that provision of a predictive distribution (known as density forecasting in the forecasting literature) is better than provision of a point forecast. However, when the predictive distribution must be estimated, additional concerns arise.

The chapter restricts our focus to forecasts with loss functions of the form L ( f, Y), i.e., we drop the dependence of L on Z, except through f . This reduces the notation and, as we saw in the previous chapter, most commonly used loss functions simplify in this manner.

The chapter proceeds as follows. Section 3.1 examines optimality of point forecasts under known model parameters. Section 3.2 considers classical (frequentist) approaches when parameters are estimated, again focusing on optimality issues. This is followed in section 3.3 by an examination of Bayesian methods for constructing point forecasts. Section 3.4 relates the two approaches and discusses density forecasts.

Section 3.5 compares the methods in the context of an empirical portfolio decision problem. Section 3.6 concludes.