The OLS Model and Categorical Variables

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The OLS Model and Categorical Variables

• ECON 222, lecture notes 2
• Petra Todd

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Discrete regressors, categorical variables

• Suppose we analyze factors that determine
  workers earnings
• Have data on
• Wages, education, labor market experience,
  gender, union status, race/ethnicity

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OLS estimators

• Chooses ß coefficients to minimize the sum of
  squared residuals

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Consider special case of regression only on a
constant term
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Interpretation of Model Coefficients

• Suppose model estimation yields
• Yi30,000-5,000 femalei
• (0.006) (0.03)
• Where p-values are given in parentheses
• What is the average earnings for males? Females?
• Is the difference between males and females
  statistically significant from zero at a 5 level?

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Matrix representation of the OLS Regression
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• If one of the columns of x is a constant term,
  get that residuals sum to zero

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Properties of the OLS estimator for Nonstochastic
regressor case

• Recall the definition of the expectation of a
  random variable

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Show that the OLS estimator is unbiased
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Variance of a random variable
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Variance of the OLS estimator
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Distribution of OLS estimator

• Assume error terms are normally distributed
  eN(0,s2I)
• Note that A e N(0,As2A)

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• Observe that the term
• Is getting large with N (as you get more terms in
  the sum).

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• To do hypothesis testing, we need a statistic for
  which the variance is stable (not shrinking to
  zero).
• Standardize by square-root of N

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Testing Restrictions on the OLS Model
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Digression on Quadratic Forms
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Testing nonlinear restrictions the Delta Method
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Deriving Partial Regression Coefficients
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Show
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We will need the following properties
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