12.5 CONCLUSION

Forecasting a binary outcome is interesting in its own right because the highly restrictive space of outcomes allows interesting simplifications of the loss functions.

Because the conditional mean is the probability of a particular outcome, and because in binary situations the probability of one outcome completely defines the distribution of the outcomes, estimating the conditional mean is equivalent to predictive density estimation. This contrasts with the usual case where the focus is on point forecasts which leads to estimating the conditional mean under MSE loss.

Under binary loss, the conditional mean estimate is a probability whereas outcomes are (typically) 0 or 1 and so the conditional mean is not a point forecast.

Loss functions relevant to the binary forecasting problem are very simple because there are only four outcomes, two of which are correct forecasts, two of which are not. Often, writing down a loss function therefore becomes a question of how to trade off between two possible mistakes. The simplicity of the loss function means that transforming a distributional (or conditional mean) forecast to a point forecast simplifies to whether the probability of an outcome is above a cutoff that is determined solely from the loss function.

Because of the simplicity of the loss function, it is possible to construct a one-to-one relationship between averages over loss functions for point forecasts and loss functions for distributional forecasts. This tight relationship allows a deep understanding of how scoring rules (loss functions for distributional forecasts) operate.

Volatility and Density Forecasting

The preceding chapters mostly focused on methods for generating point forecasts. Point forecasts were motivated as solutions to a problem of the form

$$\underset{f(z)}{\min }\int L(f(z),y{)}_{PY}(y|z)dy,\text{\hspace{1em}(13.1)}$$

and hence depend on both the loss function $L(f(z),y)$ and the density of the outcome conditional on available data, $p_Y(y|z)$ . Point forecasts are a feature of the predictive density, $p_Y(y|z)$ , and in some situations this kind of summary statistic is sufficient, most notably under squared error loss where only a location measure is required for the forecast.

This chapter presents the object of the forecasting exercise as that of providing a density forecast rather than a point forecast. A number of reasons have been suggested in the literature for why a forecaster might provide a predictive density, $p_Y(y|z)$ , rather than a point forecast. The primary reason is that the forecasts will be employed by a variety of users with different loss functions. Point forecasts obtained as the solution to (13.1) are loss function specific and so providing a point forecast or even a few summary features of the predictive density is insufficient for many of the prospective uses of the forecast.

Second, just as with point estimates, point forecasts convey no sense of the precision of the forecast. Consider an inflation forecast reported to a policy maker. If current inflation is running at 3% per annum and the predicted inflation rate is 4%, then the policy maker may be tempted to raise interest rates to keep inflation in check. However, if the density of the forecast is such that outcomes anywhere between 2% and 6% are likely, then the information in the point forecast is quite imprecise and the policy maker may be less likely to undertake decisive action. Policy makers’ inherent need for full distribution forecasts is recognized by the former Chair of the Federal Reserve, Alan Greenspan (2004): “A central bank needs to consider not only the most likely future path for the economy, but also the distribution of possible outcomes about that path.”

Similarly, a portfolio manager whose point forecast of stock returns over some holding period is 5% will want to know the degree of uncertainty surrounding both this forecast and possible future outcomes. The former will depend on the estimation error of the forecast model and it matters greatly whether the standard error of the mean forecast is 2% or 10%—in the latter case, estimation uncertainty essentially swamps the “signal” in the point forecast. This matters, particularly to risk-averse investors. Risk-averse investors’ portfolio selections will generally be more cautious if uncertainty about the future is perceived to be high, since this will increase the risk of suffering large losses.

A third reason for being interested in density forecasts arises when we are interested in multistep forecasting with nonlinear models. As explained in chapter 8, the full density matters whenever we iterate on a nonlinear prediction model since the nonlinear effects typically depend not just on the conditional mean, but also on where in the set of possible outcomes future values occur.

Density forecasts—forecasts of the likelihood of different outcomes—provide information on the uncertainty surrounding a forecast and hence can be used to address these issues. Alternatively, we could provide a summary measure of the spread of the distribution rather than the entire distribution. This leads to interval forecasts which are more informative than point forecasts but, except for special cases such as the Gaussian or other parsimoniously parameterized distributions, not as informative as density forecasts. Interval forecasts, like confidence intervals, give an idea of the uncertainty surrounding the outcome without requiring the complete information embedded in the density forecast.

The basic density forecasting problem involves a single outcome variable, $y_{t+1}$ and conditioning variables, $z_t$ . A density forecast characterizes the conditional distribution of $y_{t+1}$ , given $Z_t$

$$\begin{array}{r}\begin{array}{r}{p}_Y(y_{t+1}|Z_t)=P(y_{t+1}|Z_t).\end{array}\end{array}\text{\hspace{1em}(13.2)}$$

As written here, the conditional distribution or predictive density makes use of all relevant variables in the information set. In practice, a subset of conditioning variables is generally used. In the simplest case where only autoregressive dynamics is considered, this would comprise the past history of the variable itself.

A very popular approach to constructing density forecasts is to first model any conditional dynamics in the mean and volatility of the series and then apply flexible parametric or nonparametric methods to model the “normalized” residual, i.e., the demeaned series scaled by the conditional volatility. This conditional location–scale approach has given rise to a large family of volatility models which have proved successful in financial forecasting in particular, but also in recent macroeconomic forecasting studies.

This chapter considers some of the most popular families of volatility models and density specifications that have been used in academic studies and discusses how they can be used to construct density forecasts. We concentrate on parametric modeling of both the volatility and density, but also discuss alternatives involving semiparametric and nonparametric methods and their usage in constructing forecasts. We first cover a range of parametric volatility models in common use. As is typical of the arguments for and against parametric versus nonparametric estimates, the trade-off in density estimation is between obtaining better estimates for the parametric method when the parametric form is close to the true but unknown density versus the flexibility (and lower risk of misspecification) of the nonparametric methods.

Section 13.1 discusses the role of the loss function in constructing density forecasts. Section 13.2 goes over a range of parametric models for volatility forecasting. Section 13.3 provides a brief introduction to forecasting methods based on realized variance measures. Parametric, semiparametric, and nonparametric estimation of density models is covered in section 13.4. Section 13.5 turns to methods for generating interval and quantile forecasts. Multivariate density models are discussed in section 13.6, and copula models are covered in section 13.7. Section 13.8 concludes the chapter.