matrix-free Hypothesis Testing in the Regression Model Introduction Kurt Schmidheiny Unversität Basel Short Guides to Microeconometrics Fall 2018

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1 Short Guides to Microeconometrics Fall 2018 Kurt Schmidheiny Unversität Basel Hypothesis Testing in the Regression Model matrix-free Introduction This handout extends the handout on The Multiple Linear Regression model and refers to its definitions and assumptions in section 2. This handout describes in examples how typical hypothesis are tested. 1 Small sample, two-sided t-test 2 2 Small sample, one-sided t-test 3 3 Small sample, F -Test 4 4 Small sample, joint F -Test 5 5 Large sample, two-sided z -Test 6 6 Large sample, one-sided z -Test 7 7 Large sample, Wald-Test 8 8 Large sample, joint Wald-Test 9 Statistical Tables 11 Version: , 15:39

2 Hypothesis Testing in the Regression Model 2 1 Small sample, two-sided t-test term u i is normally distributed and homoskedastic (OLS3a, OLS4a). H 0 : β 1 = q = 0.06 vs. H A : β t = β 1 q ŝe[ β 1 ] = Step 5: Get critical value from t-table, e.g. = t crit = t 1 α/2,n K 1 = t 0.975,26 = t = < t crit = H 0 is not rejected The effect of an additional year of education on wage is not significantly different from 6% at the 5% significance level.

3 3 Short Guides to Microeconometrics 2 Small sample, one-sided t-test term u i is normally distributed and homoskedastic (OLS3a, OLS4a). H 0 : β 1 = q = 0.06 vs. H A : β 1 < 0.06 If ˆβ immediately go to step 6 and do not reject H 0. t = β 1 q ŝe[ β 1 ] = Step 5: Get critical value from t-table, e.g. = t crit = t α,n K 1 = t 0.05,26 = t 1 α,n K 1 = t 0.95,26 = t = < t crit = H 0 is rejected The effect of an additional year of education on wage is significantly smaller than 6% at the 5% significance level.

4 Hypothesis Testing in the Regression Model 4 3 Small sample, F -Test term u i is normally distributed and homoskedastic (OLS3a, OLS4a). H 0 : β 2 /(2β 3 ) = 25 β β 3 = 0, J = 1 H A : β β 3 0 F = 3.21 Step 5: Get critical value from F -table, e.g. F crit = F 1 α,j,n K 1 = F 0.95,1,26 = 4.23 F = 3.21 < F crit = 4.23 H 0 is not rejected The optimal work experience is not significantly different from 25.

5 5 Short Guides to Microeconometrics 4 Small sample, joint F -Test term u i is normally distributed and homoskedastic (OLS3a, OLS4a). H 0 : β 2 = 0 and β 3 = 0, J = 2 H A : β 2 0 and/or β 3 0 F = 5.78 Step 5: Get critical value from F -table, e.g. F crit = F 1 α,j,n K 1 = F 0.95,2,26 = 3.37 F = 5.78 > F crit = 3.37 H 0 is rejected Work experience has a significant effect on wages at the 5% level.

6 Hypothesis Testing in the Regression Model 6 5 Large sample, two-sided z -Test term u i may be heteroskedastic (OLS3d, OLS4b). H 0 : β 1 = q = 0.06 vs. H A : β z = β 1 q ŝe[ β 1 ] = = Note: ŝe[ β 1 ] can be heteroscedasticity- or cluster-robust. Step 5: Get critical value from standard normal table, e.g. z crit = z 1 α/2 = z = z = < z crit = H 0 is not rejected The effect of an additional year of education on wage is not significantly different from 6% at the 5% significance level.

7 7 Short Guides to Microeconometrics 6 Large sample, one-sided z -Test term u i may be heteroskedastic (OLS3d, OLS4b). H 0 : β 1 = q = 0.06 vs. H A : β 1 < 0.06 If ˆβ immediately go to step 6 and do not reject H 0. z = β 1 q ŝe[ β 1 ] = = Note: ŝe[ β 1 ] can be heteroscedasticity- or cluster-robust. Step 5: Get critical value from standard normal table, e.g. z crit = z α = z 0.05 = z 0.95 = z = > z crit = H 0 is not rejected The effect of an additional year of education on wage is not significantly smaller than 6% at the 5% significance level.

8 Hypothesis Testing in the Regression Model 8 7 Large sample, Wald-Test term u i may be heteroskedastic (OLS3d, OLS4b). H 0 : β 2 /(2β 3 ) = 25 β β 3 = 0, J = 1 H A : β 2 /(2β 3 ) 25 W = J F = 3.07 Note: can be based heteroscedasticity- or cluster-robust standard errors. Step 5: Get critical value from χ 2 -table, e.g. χ 2 crit = χ 2 1 α,j = χ ,1 = 3.84 W = 3.07 < χ 2 crit = 3.84 H 0 is not rejected The optimal work experience is not significantly different from 25.

9 9 Short Guides to Microeconometrics 8 Large sample, joint Wald-Test term u i may be heteroskedastic (OLS3d, OLS4b). H 0 : β 2 = 0 and β 3 = 0, J = 2 H A : β 2 0 and/or β 3 0 W = J F = Note: can be based heteroscedasticity- or cluster-robust standard errors. Step 5: Get critical value from χ 2 -table, e.g. χ 2 crit = χ 2 1 α,j = χ ,2 = 5.99 W = > χ 2 crit = 5.99 H 0 is rejected Work experience has a significant effect on wages at the 5% level.

10 Hypothesis Testing in the Regression Model 10

11 11 Short Guides to Microeconometrics Standard Normal Distribution (cdf) The table reports Φ(x) = P r(x x). x

12 Hypothesis Testing in the Regression Model 12 t-distribution (Percentiles) The table reports x given P = P r(x x) and the degrees of freedom ν. P ν

13 13 Short Guides to Microeconometrics Chi-squared Distribution (Percentiles) The table reports x given P = P r(x x) and the degrees of freedom ν. P ν

14 Hypothesis Testing in the Regression Model 14 F-Distribution (95-% Percentiles) The table reports x given P r(x x) = 0.95 and the degrees of freedom n 1 (nominator) and n 2 (denominator). n1 n