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Lecture Slides: Powerpoint PDF
Learning Objectives

Demonstrate the concept of sampling error

Derive the Best, Linear and Unbiased properties of the ordinary leastsquares (OLS) estimator

Develop the formula for the standard error of the OLS coefficient

Describe the consistency property of the OLS estimator
Example
What We Learned

An unbiased estimator gets the right answer in an average sample.

Larger samples produce more accurate estimates (smaller standard error) than smaller samples.

Under assumptions CR1CR3, OLS is the best, linear unbiased estimator — it is BLUE.

We can use our sample data to estimate the accuracy of our sample coefficient as an estimate of the population coefficient.

Consistency means that the estimator will get the right answer if applied to the whole population
Math Error in the Book (pg 73)
We specify a population model in which the X variables are uncorrelated with the errors, i.e., \(cov[X_i, \varepsilon_i] = 0\). However, to prove unbiasedness of OLS, we need \(cov \left [ \frac{x_i}{\sum_{i=1}^n x^2_i}, \varepsilon_i \right ] = 0\). In many practical settings, the condition \(cov[X_i, \varepsilon_i] = 0\) implies \(cov \left [ \frac{x_i}{\sum_{i=1}^n x^2_i}, \varepsilon_i \right ] = 0\) and therefore that OLS is unbiased. In many other settings, the condition \(cov[X_i, \varepsilon_i] = 0\) implies the OLS is approximately unbiased. For example, if you have a large sample or if the model is correctly specified. Figure 5.2 in the book shows that OLS is unbiased in our schools example even with a sample size of 20. Nonetheless, \(cov[X_i, \varepsilon_i] = 0\) doesn't guarantee unbiasedness in every setting, especially if the model is misspecified. Click here for the technical details.