Chapter 2 Regression-based Forecasting

One way to compute the conditional expectation is the linear regression model. Here, our information set contains data on all relevant explanatory variables available at the time of forecast, i.e,

\[\begin{equation} \Omega_t={X_{1t}, X_{2t},...X_{Kt}} \end{equation}\]

Hence, we get the following equality:

\[\begin{equation} E(y_t|\Omega_t)=E(y_{t}|X_{1t}, X_{2t}, X_{3t},...,X_{Kt}) \end{equation}\]

The right hand side of the above equation is the multiple regression model of the form: \[\begin{equation} y_{t}=\beta_0+\beta_1 X_{1t}+\beta_2 X_{2t}+..+\beta_K X_{Kt}+\epsilon_t \end{equation}\]

We can easily estimate the above model using Ordinary Least Squares (OLS) and compute the predicted value of \(y\): \[\begin{equation} \widehat{y}_t = \widehat{\beta_0} +\widehat{\beta_1} X_{1t} +\widehat{\beta_2} X_{2t}+...+ \widehat{\beta_k} X_{Kt} \end{equation}\]

The above equation can be used to compute the optimal forecast. Suppose, we are interested in computed the \(h\) period ahead forecast for \(y\). Then, using the above equation we get: \[\begin{equation} \widehat{y}_{t+h} = \widehat{\beta_0} +\widehat{\beta_1} X_{1t+h} +\widehat{\beta_2} X_{2t+h}+...+ \widehat{\beta_k} X_{Kt+h} \end{equation}\]

2.1 Scenario Analysis and Conditional Forecasts

One way to use a regression model to produce forecasts is called scenario analysis where we produce a different forecast for the dependent variable under each possible scenario about the future values of the independent variables. For example, what will be the forecast for inflation if the Federal Reserve Bank raises the interest rate? Would our forecast differ depending on the size of the increase in the interest rate?

2.2 Unconditional Forecasts

An alternative is to separately forecast each independent variable and then compute the forecast for the dependent variable. Yet another alternative is to use lagged variables as independent variables. Depending on the number of lags, we can forecast that much ahead into future (see Distributed Lag Section for details).

2.3 Some practical issues

  1. To forecast the dependent variable we first need to compute a forecast for the independent variable. Errors in this step induce errors later.

  2. Spurious regression: It is quite possible to find a strong linear relationship between two completely unrelated variables over time if they share a common time trend.

  3. Model Uncertainty: We do not know the true functional form for the regression model and hence our estimated model is only a proxy for the true model.

  4. Parameter Uncertainty: This kind of forecast uses regression coefficients that are computed using a fixed sample. Over time with new data, there will be changes in these coefficients.