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 =(XTX)–1XTy for the coefficients vector β in the model y = XTβ + ε was unbiased with variance-covariance matrix given by Var(b|X) = σ2(XTX)–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(XTX)–1. That is, b|X ~ N(β2(XTX)–1). This appendix presents a simple technique for simulating the large sample properties of the least squares estimator.

Consider the simple linear regression model yi = 4 + 10xi + εi with one dependent and one explanatory variable. For simulation purposes, we will assume that xi ~ N(10,25) and εi ~ 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 xi and εi, plugging these values into the regression equation to get the corresponding value of the dependent variable, yi. The simulation ends by running a regression of yi versus xi 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 ...

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