Applied Econometrics Using the SAS® System

Appendix C

SIMULATING THE LARGE SAMPLE PROPERTIES OF THE OLS ESTIMATORS

In Chapter 1 we saw that under the least squares assumptions, the estimator b =(X^TX)^–1X^T_y for the coefficients vector β in the model y = X^Tβ + ε was unbiased with variance-covariance matrix given by Var(b|X) = σ²(X^TX)^–1. Here, σ² = Var(ε|X). We also saw that if ε|X ~ N(0,σ²), then the asymptotic distribution of b|X is normal with mean β and variance-covariance σ²(X^TX)^–1. That is, b|X ~ N(β,σ²(X^TX)^–1). This appendix presents a simple technique for simulating the large sample properties of the least squares estimator.

Consider the simple linear regression model y_i = 4 + 10x_i + ε_i with one dependent and one explanatory variable. For simulation purposes, we will assume that x_i ~ N(10,25) and ε_i ~ N(0,2.25). Note that the random nature of the regressor is simply being used to generate values for the explanatory variable. A single simulation run for this model comprises generating n = 50 values of x_i and ε_i, plugging these values into the regression equation to get the corresponding value of the dependent variable, y_i. The simulation ends by running a regression of y_i versus x_i using the 50 simulated values. Proc Reg is used to estimate the values of β₁, β₂, and σ². This simulation is then repeated 10,000 times. Therefore, we end up having 10,000 estimates of the coefficients and of σ². Proc Means is then used to generate basic summary statistics for these 10,000 estimates. The output generated can be used to determine ...