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Econometrics: OLS estimates

 Let X and Y be column vectors and the sample has the size n:


The vector beta contains both the coefficient of X as well as the coefficient for the intercept.


This is equivalent to writing this equation:

We will assume that the expected value of the error term is 0 (this can be done by construction), and that X is uncorrelated to epsilon. Assume the following is for population data.

Let's prove that statement:

Since we know that the covariance between X and epsilon is 0, it follows that the expected value of X times the error term must also be 0.


Since we do not have access to the actual expected value of the distribution, let's use the sample data instead:

  • The hat on the covariance signifies that this is a MM estimator.

Let's use the fact that the sample mean of epsilon is also equal to 0, then:

 

Using the estimation for the covariance and that the expected value of the error term equals 0:

Then we have:


To get this result, you must use the properties of the summation operator.


So far we have found the OLS estimates for beta-0 hat, beta-1 hat. To see why this is called the ordinary least square estimates, let epsilon hat be the residual and defined as follows:


We want to minimize this argument by finding the optimal values of beta-0 hat and beta-1 hat. Take the partial derivatives with respect to beta-0 hat and beta-1 hat and set the functions to 0 and solve for them. We'll get the same result for beta-0 hat and beta-1 hat. Thus, we have minimized the squared residuals.

We finally have our OLS regression line, which is an estimate of the population regression:

  • Note that the hats on the regression parameters signify that they are estimates.

Reference: Wooldridge, Jeffrey. Introductory Econometrics, 3rd edition. Thomson South-Western (2006). p 29-38.
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