**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 ^{T}X)^{–1}X^{T}_{y}** for the coefficients vector

**β**in the model

**y = X**+

^{T}β**ε**was unbiased with variance-covariance matrix given by

*Var*(

**b|X**) = σ

^{2}(

**X**)

^{T}X^{–1}. Here, σ

^{2}=

*Var*(

**ε|X**). We also saw that if

**ε|X**~

*N*(0,σ

^{2}), then the asymptotic distribution of

**b|X**is normal with mean

**β**and variance-covariance σ

^{2}(

**X**)

^{T}X^{–1}. That is,

**b|X**~

*N*(

**β**,σ

^{2}(

**X**)

^{T}X^{–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 + 10

*x*+ ε

_{i}_{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*and

_{i}*ε*, plugging these values into the regression equation to get the corresponding value of the dependent variable,

_{i}*y*. 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 β*

_{i}_{1}, β

_{2}, and σ

^{2}. This simulation is then repeated 10,000 times. Therefore, we end up having 10,000 estimates of the coefficients and of σ

^{2}. Proc Means is then used to generate basic summary statistics for these 10,000 estimates. The output generated can be used to determine ...