LN+7+OLS+SE+and+F-test

LN+7+OLS+SE+and+F-test - Empirical Methods II (API-202A)...

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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 1 Lecture Notes 7 OLS Standard Errors and F-test I – OLS Standard Errors SE ( 1 ˆ ) Refresher: The need for hypothesis testing Regressions: estimate partial association in the population using a sample Any sample is subject to sampling variation Need hypothesis testing to test hypotheses about population coefficients – Is 1 ˆ statistically significant? Refresher: we need the standard deviation of our estimator 1 ˆ , also called the standard error of 1 ˆ , SE ( 1 ˆ ) , to test for the statistical significance of 1 ˆ . Hypothesis Test : 1) 0 : 1 0 H , 0 : 1 A H 2) ) ˆ ( 0 ˆ 1 1 SE t 3) Reject the null if |t|>1.96 (N>120) Refresher: the (robust) SE ( 1 ˆ ) is computed as (K=1):  2 2 2 2 1 ) ( ˆ ) ( ) ˆ ( X X X X SE i i i TODAY: Where does the formula for the SE ( 1 ˆ ) come from and what is its key determinant? -Key determinant: What makes our SE ( 1 ˆ ) smaller, and thus, our t-test bigger? -In other words, what makes our estimation more or less accurate (ie, with higher or lower variance).
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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 2 Where does the formula for the SE ( 1 ˆ ) comes from? A Big Picture View (the math involved is beyond the scope of this course) i) Recall from LN4 that we can write the formula that computes 1 ˆ in demeaned terms (K=1). 2 2 1 ) ( ) )( ( ) ( ) )( ( ˆ d i d i d i i i i X Y X X X Y Y X X ii) And that we can replace d i Y with the SEQ written also in demeaned terms, yielding: 2 1 1 ) ( ˆ d i i d i X X iii) With this formulation, we can start thinking about how to compute the SE ( 1 ˆ ): 2 2 1 1 1 ) ( ) ( ) ˆ ( ) ˆ ( d i i d i d i i d i X X Var X X Var Var SE iv) One can show that for large N (>120) the formula above can be estimated with our formula for SE ( 1 ˆ ) pasted below - the math involved is beyond the scope of this course. 1  2 2 2 2 1 ) ( ˆ ) ( ) ˆ ( X X X X SE i i i 1 See Stock and Watson page 180 or Wooldridge page 264 for more details.
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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 3 Which is the key determinant of SE ( 1 ˆ )? The key determinant of SE ( 1 ˆ ) is the variance of the error term: 2 ) ( i i Var The larger the error terms i are, the bigger 2 i is, the bigger SE( 1 ˆ ) is and the more imprecise our estimation is . PRACTICAL IMPLICATION : Adding relevant control variables not only allow us to reduce the risk of OVB, but also to reduce the size and variance of the errors, and as a result, obtain more accurate estimates of 1 ˆ . “…we are taking something out of the error term. ..” For example . Adding control variables to a randomized experiment (eg Tennessee STAR) o Define TREAT = 1 if person is in treatment group (smaller classes) 0 if person is in control group (regular classes) o Two alternative estimates: test scores = 0 ˆ + 1 ˆ TREAT + ˆ test scores = 0 ˆ + 1 ˆ TREAT + 2 ˆ
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LN+7+OLS+SE+and+F-test - Empirical Methods II (API-202A)...

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