Chengyuan Yin School of mathematics Econometrics Chengyuan Yin School of mathematics

10. Prediction in the Classical Regression Model Econometrics 10. Prediction in the Classical Regression Model

Forecasting Objective: Forecast Distinction: Ex post vs. Ex ante forecasting Ex post: RHS data are observed Ex ante: RHS data must be forecasted Prediction vs. model validation. Within sample prediction “Hold out sample”

Prediction Intervals Given x0 predict y0. Two cases: Estimate E[y|x0] = x0; Predict y0 = x0 + 0 Obvious predictor, b’x0 + estimate of 0. Forecast 0 as 0, but allow for variance. Alternative: When we predict y0 with bx0, what is the 'forecast error?' Est.y0 - y0 = bx0 - x0 - 0, so the variance of the forecast error is x0Var[b - ]x0 + 2 How do we estimate this? Form a confidence interval. Two cases: If x0 is a vector of constants, the variance is just x0 Var[b] x0. Form confidence interval as usual. If x0 had to be estimated, then we use a random variable. What is the variance of the product? (Ouch!) One possibility: Use bootstrapping.

Forecast Variance Variance of the forecast error is 2 + x0’ Var[b]x0 = 2 + 2[x0’ (X’X)-1x0] If the model contains a constant term, this is In terms squares and cross products of deviations from means. Interpretation: Forecast variance is smallest in the middle of our “experience” and increases as we move outside it.

Butterfly Effect 5.1 in the 6th edition

Salkever’s Algebraic Trick Salkever’s method of computing the forecasts and forecast variances Multiple regression of produces the least squares coefficient vector followed by the predictions. Residuals are 0 for the predictions, so s2( * )-1 gives the covariance matrix for the coefficient estimates and the variances for the forecasts. (Very clever, useful for understanding. Not actually used in modern software.)

Dummy Variable for One Observation A dummy variable that isolates a single observation. What does this do? Define d to be the dummy variable in question. Z = all other regressors. X = [Z,d] Multiple regression of y on X. We know that X'e = 0 where e = the column vector of residuals. That means d'e = 0, which says that ej = 0 for that particular residual. Fairly important result. Important to know.

Oaxaca Decomposition Two groups, two regression models: (Two time periods, men vs. women, two countries, etc.) y1 = X11 + 1 and y2 = X22 + 2 Consider mean values, y1* = E[y1|mean x1] = x1* 1 y2* = E[y2|mean x2] = x2* 2 Now, explain why y1* is diferent from y2*. (I.e., departing from y2, why is y1 different?) (Could reverse the roles of 1 and 2.) y1* - y2* = x1* 1 - x2* 2 = x1*(1 - 2) + (x1* - x2*) 2 (change in model) (change in conditions)

The Oaxaca Decomposition