Lec4-2019.pdf

Lec4-2019.pdf - Introductory Econometrics Statistical...

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Introductory Econometrics Statistical Properties of the OLS Estimator, Interpretation of OLS Estimates and Effects of Rescaling Monash Econometrics and Business Statistics 2019 1 / 34

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Recap I The multiple regression model can be written in matrix form as y n × 1 = X n × ( k +1) β ( k +1) × 1 + u n × 1 I The OLS procedure finds a linear combination of X that is closest to the vector y , i.e. the length of its error vector (OLS residuals) is the shortest I This implies that the OLS residual vector is perpendicular to all columns of X , i.e., X 0 ˆ u = 0 which leads to the famous OLS formula b β = ( X 0 X ) - 1 X 0 y 2 / 34
I A consequence of orthogonality of residuals and columns of X is that n X i =1 ( y i - ¯ y ) 2 = n X i =1 y i - ¯ y ) 2 + n X i =1 ˆ u 2 i or SST = SSE + SSR I This leads to the definition of the coefficient of determination R 2 , which is a measure of goodness of fit R 2 = SSE / SST = 1 - SSR / SST 3 / 34

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I OLS formula gives us fitted values that are closest to the actual values in the sense that the length of the residual vector is the smallest possible. I But why should this be a good estimate of the unknown parameters of the conditional expectation function? I We ask if this formula produces the best information we can get from our data about the unknown parameters β 4 / 34
Lecture Outline I Interpretation of the OLS estimates in multiple regression (textbook reference 3-2a to 3-2e) I Properties of good estimators I Properties of the OLS estimator b β 1. OLS is unbiased: The expected value of b β (Textbook reference 3-3, and Appendix E-2) 2. The Variance of b β (Textbook reference 3-4, and Appendix E-2) 3. OLS is BLUE (Gauss-Markov Theorem): The Efficiency of b β (Textbook reference 3-5, and Appendix E-2) I Units of measurement: do the results qualitatively change if we change the units of measurement? (textbook reference 2-4a, 6-1 (exclude 6-1a)) 5 / 34

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Interpretation of OLS estimates Example: The causal effect of education on wage I Consider again the wage equation written as: wage = β 0 + β 1 educ + β 2 IQ + u where IQ is IQ score (in the population, it has a mean of 100 and sd = 15). I Primarily interested in β 1 , because we want to know the value that education adds to a person’s wage. I Without IQ in the equation, the coefficient of educ will show how strongly wage and educ are correlated, but both could be caused by a person’s ability. I By explicitly including IQ in the equation, we obtain a more persuasive estimate of the causal effect of education provided that IQ is a good proxy for intelligence. 6 / 34
Interpretation of OLS estimates Example: The causal effect of education on wage I If we only estimate a regression of wage on education, we cannot be sure if we are measuring the effect of education, or if education is acting as a proxy for smartness. This is important, because if the education system does not add any value other than separating smart people from not so smart, the society can achieve that much cheaper by national IQ tests!

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