4.2 PLUG-IN ESTIMATORS

4.2 PLUG-IN ESTIMATORS

The plug-in approach is commonly used in classical forecasting. Any estimator of $\theta$ or $\beta$ requires the use of some form of objective function, which may or may not relate to the loss function for the forecasting problem. Here we break down plugin estimators into three groups—those estimated using the correct loss function to construct parameter estimates; those that use a different loss function to estimate the parameters; and those that use the correct loss function but with restrictions imposed. First we present a simple application of a plug-in estimator.

4.2.1 Plug-In Estimators Based on the Forecaster’s Loss Function

Given a loss function and an assumed joint density for the data, $(Z,Y)$ , we can sometimes construct a forecast model, $f(z,\beta )$ , that captures the relevant feature of the predictive density. For example, a linear model might arise under MSE loss if the joint density of the data is normal. Typically, however, the joint density for the data is unknown and so both the form of the function, $f(z,\beta )$ , and $\beta$ are unknown. In practice, we must therefore use an approximation to the correct form of the forecast model $f(z,\beta )$ and choose some estimator for $\beta .$ .

In many situations where $f(z,\beta )$ takes the form of a parametric model, loss-based or plug-in methods will result in the same or very similar estimators. By this we mean that $|{\hat{\beta }}_{\text{textsc}t,\mathrm{pI}}-{\hat{\beta }}_T|{\to }^{p}0$ , where ${\hat{\mathbf{\beta }}}_{\text{textsc}\text{textsc}T,\text{textsc}ph}$ is the plug-in estimator. This would hold, for example, under MSE loss and a model for the conditional expectation with the plugin estimator based on OLS. This follows directly as the OLS estimator minimizes the sum of squared residuals, $\begin{array}{r}{\sum }_t(y_{t+1}-f(z_t,\beta ){)}^2\end{array}$ , and so the objective functions are the same.

Aside from squared error loss, forecast models that use estimates from least squares problems may not have this property. The biggest concern with plug-in estimators is therefore that the loss function used to estimate the model parameters may not line up with the forecaster’s loss function. We examine this issue in the next subsection.

A second concern is related to data heterogeneity which could mean that ${\hat{\beta }}_{T,\mathrm{PI}}{}^P$ ${\beta }^{\ast \ast }$ rather than ${\hat{\mathbf{\beta }}}_{T,\mathrm{PI}}\to P{\beta }^{\ast }$ . In practice this means that the forecast model is good “on average” rather than being good at predicting next period’s realization conditional on the current value of $Z.$

Example 4.2.1 (Mean-variance investor’s use of plug-in estimators). Consider the portfolio choice of an investor with initial wealth, $W_T=1$ . The investor chooses portfolio weights, ${\omega }_T=f(Z_T)$ , which gives rise to future wealth, $W_{T+1}={\omega }_T^{\prime }{y}_{T+1:}$ where ${\gamma }_{T+1}$ is a vector of gross returns on a set of risky assets. Given the investor’s information at time $T,Z_T,$ , the distribution of future asset returns is $y_{T+1}\sim ({\mu }_T,{\Sigma }_T)$ , so $E\left[W_{T+1}|Z_T\right]={\omega }_T^{\prime }{\mu }_T,\mathrm{Var}(W_{T+1}|Z_T)={\omega }_T^{\prime }{\Sigma }_T{\omega }_T$ . Suppose the investor has

mean–variance preferences

$$U(W_{T+1})=E[W_{T+1}|Z_T]-\frac{a}{2}\mathrm{Var}(W_{T+1}|Z_T),\text{\hspace{1em}(4.6)}$$

where $a>0$ captures the investor’s risk aversion. Taking conditional expectations of (4.6), we have

$$E[U(W_{T+1})|Z_T]={\omega }_T^{\prime }{\mu }_T-\frac{a}{2}{\omega }_T^{\prime }{\Sigma }_T{\omega }_T.$$

Maximizing expected utility subject to the constraint ${\omega }^{\prime }(z_T)\iota =1$ , we get the optimal portfolio weights, (see Aït-Sahalia and Brandt (2001)),

$${\omega }_T=\frac{{\Sigma }_T^{-1}{\mu }_T}{a}+{\Sigma }_T^{-1}\iota \left(\frac{a-{\iota }^{\prime }{\Sigma }_T^{-1}{\mu }_T}{a{\iota }^{\prime }{\Sigma }_T^{-1}\iota }\right).\text{\hspace{1em}(4.7)}$$

A two-step plug-in approach could be to first obtain estimates of ${\mu }_T$ and ${\Sigma }_T,e.g.$ $\begin{array}{r}{\hat{\mu }}_T=T^{-1}{\sum }_{t=1}^T{y}_t\end{array}$ , and $\begin{array}{r}{\hat{\Sigma }}_T=(T-1{)}^{-1}{\sum }_{t=1}^T(y_t-{\hat{\mu }}_T)(y_t-{\hat{\mu }}_T{)}^{\prime }\end{array}$ . In a second step, these estimates can be substituted into (4.7) to get the plug-in estimate of the portfolio weights:

$${\omega }_T=\frac{{\hat{\Sigma }}_T^{-1}{\hat{\mu }}_T}{a}+{\hat{\Sigma }}_T^{-1}\iota \left(\frac{a-{\iota }^{\prime }{\hat{\Sigma }}_T^{-1}{\hat{\mu }}_T^{\prime }}{a{\iota }^{\prime }{\hat{\Sigma }}_T^{-1}\iota }\right).\text{\hspace{1em}(4.8)}$$

Notice that ${\omega }_T$ is a highly nonlinear function of ${\hat{\mu }}_T,{\hat{\Sigma }}_T$ and so the sampling distribution of the estimated parameters may well lead to undesirable sampling properties of the associated expected utility function.