5.4 Properties of the OLS Estimator

If assumptions (1) to (5) hold, then the OLS estimator \(\boldsymbol{\hat{\beta}}\) is:

1. A consistent estimator of \(\boldsymbol{{\beta}}\)
2. Asymptotically normally distributed
3. Having a variance of \(\mathbb{V}(\boldsymbol{\hat{\beta}}) = \sigma^2(\mathbf{X}'\mathbf{X})^{-1}\)

Notice that we did not assume normality of \(\epsilon_{i},y_{i}\) or \(x_{i}\).

Let’s discuss each of these properties in a little bit more detail.

5.4.1 1. Consistentcy of \(\boldsymbol{{\beta}}\)

\(\boldsymbol{\hat{\beta}}\) is the OLS estimator of \(\boldsymbol{{\beta}}\). A consistent estimator is one for which, as the sample size (\(n\)) increases, the estimate converges in probability to the value that the estimator is designed to estimate. This is often stated as \(plim(\boldsymbol{\hat{\beta}})=\boldsymbol{{\beta}}\). Stated differently, as the sample size grows, the OLS coefficients converge to the true coefficients.

5.4.2 2. Asymptotic Normality

Asymptotic normality is another property of the OLS estimator when all assumptions are met. “Asymptotic” refers to how an estimator behaves as the sample size tends to infinity. “Normality” refers to the normal distribution, so an estimator that is asymptotically normal will have an approximately normal distribution as the sample size gets larger.

5.4.3 Variance of \(\hat{\beta}\)

Having a variance of \(\mathbb{V}(\boldsymbol{\hat{\beta}}) = \sigma^2(\mathbf{X}'\mathbf{X})^{-1}\) is another property of the OLS estimator when the previously stated assumptions are met. This means, for example, we can estimate the standard errors from the main diagonal of \(\sigma^2(\mathbf{X}'\mathbf{X})^{-1}\) and perform significance testing based on this variance.