Chapter 5 - Properties of our Estimators

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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 least-squares (OLS) estimator
  • Develop the formula for the standard error of the OLS coefficient
  • Describe the consistency property of the OLS estimator


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 CR1-CR3, 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.