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Econ103_sping11_lec10c - ECON 103, Lecture 10: Multiple...

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Unformatted text preview: ECON 103, Lecture 10: Multiple Regression and Testing Maria Casanova May 3 (version 1) Maria Casanova Lecture 10 1. Introduction General Multiple Regression Model: Yi = β0 + β1 X1i + β2 X2i + ... + βk Xki + ui We have estimated the parameters β0 , β1 , ..., βk . Now, we will study whether the estimated coefficients are statistically significant. whether certain relationships between parameters hold. whether a group of parameters are jointly significant. Maria Casanova Lecture 10 1. Introduction Outline: Hypothesis tests for a single coefficient For example H0 : β2 = 0 or H0 : β3 = 1 Hypothesis tests regarding multiple coefficients For example H0 : β1 = β2 = β4 = 0 or H0 : β5 = 2β3 Example - the return to education Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Testing a hypothesis about a single coefficient is done in the same way as in the case of the simple regression. The CLT tells us that ˆ βj − βj ∼ N (0, 1) ˆ SE (βj ) ˆ In practice, we have to estimate SE (βj ), which in turn depends on σu , ˆ2 the estimated variance of the error term. We have n − k − 1 degrees of freedom to estimate this variance, thus an 2 unbiased estimator of σu is σu = ˆ2 1 n−k −1 Maria Casanova ui2 ˆ i Lecture 10 2. Hypothesis Tests for a Single Coefficient Testing a hypothesis about a single coefficient is done in the same way as in the case of the simple regression. The CLT tells us that ˆ βj − βj ∼ N (0, 1) ˆ SE (βj ) ˆ ˆ2 In practice, we have to estimate SE (βj ), which in turn depends on σu , the estimated variance of the error term. We have n − k − 1 degrees of freedom to estimate this variance, thus an 2 unbiased estimator of σu is σu = ˆ2 1 n−k −1 Maria Casanova ui2 ˆ i Lecture 10 2. Hypothesis Tests for a Single Coefficient Testing a hypothesis about a single coefficient is done in the same way as in the case of the simple regression. The CLT tells us that ˆ βj − βj ∼ N (0, 1) ˆ SE (βj ) ˆ ˆ2 In practice, we have to estimate SE (βj ), which in turn depends on σu , the estimated variance of the error term. We have n − k − 1 degrees of freedom to estimate this variance, thus an 2 unbiased estimator of σu is σu = ˆ2 1 n−k −1 Maria Casanova ui2 ˆ i Lecture 10 2. Hypothesis Tests for a Single Coefficient Testing a hypothesis about a single coefficient is done in the same way as in the case of the simple regression. The CLT tells us that ˆ βj − βj ∼ N (0, 1) ˆ SE (βj ) ˆ ˆ2 In practice, we have to estimate SE (βj ), which in turn depends on σu , the estimated variance of the error term. We have n − k − 1 degrees of freedom to estimate this variance, thus an 2 unbiased estimator of σu is σu = ˆ2 1 n−k −1 Maria Casanova ui2 ˆ i Lecture 10 2. Hypothesis Tests for a Single Coefficient As a consequence of this, for a small n, the distribution of the ˆ standardized βj is a t with n − k − 1 degrees of freedom: ˆ βj − βj ∼ tn−k −1 ˆ SE (βj ) The t distribution converges to a normal when n is large. ˆ βj − βj ∼ N (0, 1) ˆ SE (βj ) Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient As a consequence of this, for a small n, the distribution of the ˆ standardized βj is a t with n − k − 1 degrees of freedom: ˆ βj − βj ∼ tn−k −1 ˆ SE (βj ) The t distribution converges to a normal when n is large. ˆ βj − βj ∼ N (0, 1) ˆ SE (βj ) Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H0 : βj = 0) 2 Formulate the alternative hypothesis, either as two-sided (H1 : βj = 0) or as one-sided (H1 : βj < 0 or H1 : βj > 0) 3 Specify the level of significance α (e.g. α = 0.05) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α. 6 Decide whether you can or cannot reject the null hypothesis. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Test scores example: Test Scoresi = β0 + β1 STRi 1 + β2 PctLEi 2 + ui Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 = = −2.56 t act = 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 = = −2.56 t act = 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 = = −2.56 t act = 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 = = −2.56 t act = 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 = = −2.56 t act = 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 t act = = = −2.56 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 t act = = = −2.56 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 t act = = = −2.56 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient The estimated regression line is: Test Scoresi = 686.0 − 1.10 STRi − 0.65 PctLEi (8.73) (0.43) (0.03) We are interested in whether the STR affects test scores. 1 H 0 : β1 = 0 2 H 1 : β1 = 0 3 α = 5% 4 The actual value of the t -statistic is ˆ β1 − β1,0 −1.10 − 0 t act = = = −2.56 0.43 ˆ SE β1 5 The critical values are zα/2 = −1.96 and z1−α/2 = 1.96 6 We reject the null hypothesis. β1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient We can create a confidence interval for β1 , as before. For example, a 95% confidence interval for β1 is given by: ˆ ˆˆ ˆ β1 − 1.96 × SE (β1 ), β1 + 1.96 × SE (β1 ) In our example, this confidence interval (also reported by Stata) is: (−1.95, −0.25) We can also use the p -value to conduct the test. The p -value is 0.011 < 0.05 = α, therefore we reject the null hypothesis at the 5% level. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient We can create a confidence interval for β1 , as before. For example, a 95% confidence interval for β1 is given by: ˆ ˆˆ ˆ β1 − 1.96 × SE (β1 ), β1 + 1.96 × SE (β1 ) In our example, this confidence interval (also reported by Stata) is: (−1.95, −0.25) We can also use the p -value to conduct the test. The p -value is 0.011 < 0.05 = α, therefore we reject the null hypothesis at the 5% level. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient We can create a confidence interval for β1 , as before. For example, a 95% confidence interval for β1 is given by: ˆ ˆˆ ˆ β1 − 1.96 × SE (β1 ), β1 + 1.96 × SE (β1 ) In our example, this confidence interval (also reported by Stata) is: (−1.95, −0.25) We can also use the p -value to conduct the test. The p -value is 0.011 < 0.05 = α, therefore we reject the null hypothesis at the 5% level. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient We can create a confidence interval for β1 , as before. For example, a 95% confidence interval for β1 is given by: ˆ ˆˆ ˆ β1 − 1.96 × SE (β1 ), β1 + 1.96 × SE (β1 ) In our example, this confidence interval (also reported by Stata) is: (−1.95, −0.25) We can also use the p -value to conduct the test. The p -value is 0.011 < 0.05 = α, therefore we reject the null hypothesis at the 5% level. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient We can create a confidence interval for β1 , as before. For example, a 95% confidence interval for β1 is given by: ˆ ˆˆ ˆ β1 − 1.96 × SE (β1 ), β1 + 1.96 × SE (β1 ) In our example, this confidence interval (also reported by Stata) is: (−1.95, −0.25) We can also use the p -value to conduct the test. The p -value is 0.011 < 0.05 = α, therefore we reject the null hypothesis at the 5% level. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Another example with dummy variables. Remember that in this case we had to choose an excluded group. It turns out that in our LAwages data set there are 4 individuals who are recorded as both Hispanic and Black. Let’s drop these individuals. In Stata type drop if black==1 & hispanic==1 regress wage hispanic black, robust We obtain results that are very similar to those in the last lecture: Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) where ‘Other’ is the excluded group. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Let’s now choose ‘Hispanic’ as the excluded group: Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient With ‘Other’ as the excluded group we have Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) With ‘Hispanic’ as the excluded group the estimated regression is Wagei = 10.16 + 1.09 Blacki + 5.94 Otheri (0.83) (1.12) (1.12) These regressions are really the same. They both say that average wage for Others is 16.09, average wage for Blacks is 11.44, and average wages for Hispanics is 10.15. But because the variables are coded differently, the coefficients measure different aspects of the relationship, so the t -statistics on the coefficients test different things. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient With ‘Other’ as the excluded group we have Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) With ‘Hispanic’ as the excluded group the estimated regression is Wagei = 10.16 + 1.09 Blacki + 5.94 Otheri (0.83) (1.12) (1.12) These regressions are really the same. They both say that average wage for Others is 16.09, average wage for Blacks is 11.44, and average wages for Hispanics is 10.15. But because the variables are coded differently, the coefficients measure different aspects of the relationship, so the t -statistics on the coefficients test different things. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient With ‘Other’ as the excluded group we have Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) With ‘Hispanic’ as the excluded group the estimated regression is Wagei = 10.16 + 1.09 Blacki + 5.94 Otheri (0.83) (1.12) (1.12) These regressions are really the same. They both say that average wage for Others is 16.09, average wage for Blacks is 11.44, and average wages for Hispanics is 10.15. But because the variables are coded differently, the coefficients measure different aspects of the relationship, so the t -statistics on the coefficients test different things. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient With ‘Other’ as the excluded group we have Wagei = 16.09 − 5.94 Hispanici − 4.65 Blacki (0.75) (1.12) (1.03) With ‘Hispanic’ as the excluded group the estimated regression is Wagei = 10.16 + 1.09 Blacki + 5.94 Otheri (0.83) (1.12) (1.12) These regressions are really the same. They both say that average wage for Others is 16.09, average wage for Blacks is 11.44, and average wages for Hispanics is 10.15. But because the variables are coded differently, the coefficients measure different aspects of the relationship, so the t -statistics on the coefficients test different things. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient In the first regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Others (since ‘Other’ is the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 tests whether Blacks and Others have equal average wages. On the other hand, in the second regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Hispanics (since ‘Hispanic’ is now the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 now tests whether Blacks and Hispanics have equal average wages. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient In the first regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Others (since ‘Other’ is the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 tests whether Blacks and Others have equal average wages. On the other hand, in the second regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Hispanics (since ‘Hispanic’ is now the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 now tests whether Blacks and Hispanics have equal average wages. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient In the first regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Others (since ‘Other’ is the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 tests whether Blacks and Others have equal average wages. On the other hand, in the second regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Hispanics (since ‘Hispanic’ is now the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 now tests whether Blacks and Hispanics have equal average wages. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient In the first regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Others (since ‘Other’ is the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 tests whether Blacks and Others have equal average wages. On the other hand, in the second regression, the coefficient on the variable Blacki measures the difference between average wages for Blacks and average wages for Hispanics (since ‘Hispanic’ is now the excluded group). Hence the t -statistic for the hypothesis that βBlack = 0 now tests whether Blacks and Hispanics have equal average wages. Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Note that the first regression doesn’t directly test whether Blacks and Hispanics have equal average wages, and the second regression doesn’t directly test whether Blacks and Others have equal average wages. Therefore, you may want to choose which group to “exclude” based on what hypotheses you want to test. Or run both regressions, even though they are really the same thing. Or you can use an F -statistic (next slides). Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Note that the first regression doesn’t directly test whether Blacks and Hispanics have equal average wages, and the second regression doesn’t directly test whether Blacks and Others have equal average wages. Therefore, you may want to choose which group to “exclude” based on what hypotheses you want to test. Or run both regressions, even though they are really the same thing. Or you can use an F -statistic (next slides). Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Note that the first regression doesn’t directly test whether Blacks and Hispanics have equal average wages, and the second regression doesn’t directly test whether Blacks and Others have equal average wages. Therefore, you may want to choose which group to “exclude” based on what hypotheses you want to test. Or run both regressions, even though they are really the same thing. Or you can use an F -statistic (next slides). Maria Casanova Lecture 10 2. Hypothesis Tests for a Single Coefficient Note that the first regression doesn’t directly test whether Blacks and Hispanics have equal average wages, and the second regression doesn’t directly test whether Blacks and Others have equal average wages. Therefore, you may want to choose which group to “exclude” based on what hypotheses you want to test. Or run both regressions, even though they are really the same thing. Or you can use an F -statistic (next slides). Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients We might be interested in testing whether education and experience are jointly important for wages. whether the return to education is the same for men and women. The t -test procedure is valuable for testing statistical significance of an individual regression coefficient (or a linear combination of coefficients). However, the t-test procedure is not valid for testing joint hypotheses. In order to test a hypothesis involving multiple coefficients, we need to use the F -statistic. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients We might be interested in testing whether education and experience are jointly important for wages. whether the return to education is the same for men and women. The t -test procedure is valuable for testing statistical significance of an individual regression coefficient (or a linear combination of coefficients). However, the t-test procedure is not valid for testing joint hypotheses. In order to test a hypothesis involving multiple coefficients, we need to use the F -statistic. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients We might be interested in testing whether education and experience are jointly important for wages. whether the return to education is the same for men and women. The t -test procedure is valuable for testing statistical significance of an individual regression coefficient (or a linear combination of coefficients). However, the t-test procedure is not valid for testing joint hypotheses. In order to test a hypothesis involving multiple coefficients, we need to use the F -statistic. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients We might be interested in testing whether education and experience are jointly important for wages. whether the return to education is the same for men and women. The t -test procedure is valuable for testing statistical significance of an individual regression coefficient (or a linear combination of coefficients). However, the t-test procedure is not valid for testing joint hypotheses. In order to test a hypothesis involving multiple coefficients, we need to use the F -statistic. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Let’s consider the following example: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi , How do we interpret the coefficients? β0 is the intercept for males. β0 + β1 is the intercept for females. β2 is the return to education for males. β2 + β3 is the return to education for females. A joint hypothesis would be that wages of men and women are the same: H0 : β1 = 0 and β3 = 0 vs H1 : β1 = 0 and/or β3 = 0 How do we test this hypothesis? Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients This is the intuition behind the procedure to test joint hypotheses: We start by estimating the full (unconstrained ) model (as in the previous slide). We then estimate the constrained model, which incorporates the restrictions that are true under H0 (in this case, β1 = 0 and β3 = 0). The RSS for the constrained model is always larger than for the unconstrained model for 2 reasons: 1 The RSS always increases when variables are dropped from the model =⇒ This is an algebraic fact. 2 If X1 and X3 contribute to explain Y (i.e. β1 and/or β3 = 0), the RSS increases when we drop them from the model. The idea is to measure how much of the increase in the RSS is due to the restrictions. If a lot, we will reject the null hypothesis. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Formally, we follow these steps: 1 H0 : β1 = 0 and β3 = 0 2 H1 : β1 = 0 or β3 = 0. This means that at least one of the two coefficient is different from zero, or that “β1 and β3 are jointly significant ”. 3 α = 5%. 4 Estimate the unconstrained model: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi 5 Compute the RSS for the unconstrained model (RSSU ) Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Formally, we follow these steps: 1 H0 : β1 = 0 and β3 = 0 2 H1 : β1 = 0 or β3 = 0. This means that at least one of the two coefficient is different from zero, or that “β1 and β3 are jointly significant ”. 3 α = 5%. 4 Estimate the unconstrained model: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi 5 Compute the RSS for the unconstrained model (RSSU ) Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Formally, we follow these steps: 1 H0 : β1 = 0 and β3 = 0 2 H1 : β1 = 0 or β3 = 0. This means that at least one of the two coefficient is different from zero, or that “β1 and β3 are jointly significant ”. 3 α = 5%. 4 Estimate the unconstrained model: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi 5 Compute the RSS for the unconstrained model (RSSU ) Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Formally, we follow these steps: 1 H0 : β1 = 0 and β3 = 0 2 H1 : β1 = 0 or β3 = 0. This means that at least one of the two coefficient is different from zero, or that “β1 and β3 are jointly significant ”. 3 α = 5%. 4 Estimate the unconstrained model: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi 5 Compute the RSS for the unconstrained model (RSSU ) Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Formally, we follow these steps: 1 H0 : β1 = 0 and β3 = 0 2 H1 : β1 = 0 or β3 = 0. This means that at least one of the two coefficient is different from zero, or that “β1 and β3 are jointly significant ”. 3 α = 5%. 4 Estimate the unconstrained model: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi 5 Compute the RSS for the unconstrained model (RSSU ) Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Formally, we follow these steps: 1 H0 : β1 = 0 and β3 = 0 2 H1 : β1 = 0 or β3 = 0. This means that at least one of the two coefficient is different from zero, or that “β1 and β3 are jointly significant ”. 3 α = 5%. 4 Estimate the unconstrained model: Wagei = β0 + β1 Femalei + β2 Educi + β3 Fem Educi + εi 5 Compute the RSS for the unconstrained model (RSSU ) Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 6 Estimate the constrained model: Wagei = β0 + β2 Educi + εi 7 Compute the RSS for the constrained model (RSSC ). Note that RSSC is always larger than RSSU . 8 The F-statistic follows the F distribution with (m, n − k − 1) degrees of freedom: F= (RSSC −RSSU ) m RSSU (n−k −1) ∼ Fm,n−k −1 , where: m = number of linear restrictions k = number of regressors in the unconstrained model. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 6 Estimate the constrained model: Wagei = β0 + β2 Educi + εi 7 Compute the RSS for the constrained model (RSSC ). Note that RSSC is always larger than RSSU . 8 The F-statistic follows the F distribution with (m, n − k − 1) degrees of freedom: F= (RSSC −RSSU ) m RSSU (n−k −1) ∼ Fm,n−k −1 , where: m = number of linear restrictions k = number of regressors in the unconstrained model. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 6 Estimate the constrained model: Wagei = β0 + β2 Educi + εi 7 Compute the RSS for the constrained model (RSSC ). Note that RSSC is always larger than RSSU . 8 The F-statistic follows the F distribution with (m, n − k − 1) degrees of freedom: F= (RSSC −RSSU ) m RSSU (n−k −1) ∼ Fm,n−k −1 , where: m = number of linear restrictions k = number of regressors in the unconstrained model. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 6 Estimate the constrained model: Wagei = β0 + β2 Educi + εi 7 Compute the RSS for the constrained model (RSSC ). Note that RSSC is always larger than RSSU . 8 The F-statistic follows the F distribution with (m, n − k − 1) degrees of freedom: F= (RSSC −RSSU ) m RSSU (n−k −1) ∼ Fm,n−k −1 , where: m = number of linear restrictions k = number of regressors in the unconstrained model. Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 9 10 Compute the actual value of the F -statistic, F act . Find the critical value for the Fm,n−k −1 distribution from the tables α (let’s call it Fm,n−k −1 ). Remember: In the book you will find tables for the Fn1 /n2 distribution. In our notation, n1 = m and n2 = n − k − 1. You will find separate tables for different significance levels (1%, 5% and 10%). When n is large (typically n − k − 1 > 120), the F -statistic has a distribution Fm,∞ , which can also be found on the tables. 11 α Reject if F act > Fm,n−k −1 . Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 9 10 Compute the actual value of the F -statistic, F act . Find the critical value for the Fm,n−k −1 distribution from the tables α (let’s call it Fm,n−k −1 ). Remember: In the book you will find tables for the Fn1 /n2 distribution. In our notation, n1 = m and n2 = n − k − 1. You will find separate tables for different significance levels (1%, 5% and 10%). When n is large (typically n − k − 1 > 120), the F -statistic has a distribution Fm,∞ , which can also be found on the tables. 11 α Reject if F act > Fm,n−k −1 . Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 9 10 Compute the actual value of the F -statistic, F act . Find the critical value for the Fm,n−k −1 distribution from the tables α (let’s call it Fm,n−k −1 ). Remember: In the book you will find tables for the Fn1 /n2 distribution. In our notation, n1 = m and n2 = n − k − 1. You will find separate tables for different significance levels (1%, 5% and 10%). When n is large (typically n − k − 1 > 120), the F -statistic has a distribution Fm,∞ , which can also be found on the tables. 11 α Reject if F act > Fm,n−k −1 . Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 9 10 Compute the actual value of the F -statistic, F act . Find the critical value for the Fm,n−k −1 distribution from the tables α (let’s call it Fm,n−k −1 ). Remember: In the book you will find tables for the Fn1 /n2 distribution. In our notation, n1 = m and n2 = n − k − 1. You will find separate tables for different significance levels (1%, 5% and 10%). When n is large (typically n − k − 1 > 120), the F -statistic has a distribution Fm,∞ , which can also be found on the tables. 11 α Reject if F act > Fm,n−k −1 . Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 9 10 Compute the actual value of the F -statistic, F act . Find the critical value for the Fm,n−k −1 distribution from the tables α (let’s call it Fm,n−k −1 ). Remember: In the book you will find tables for the Fn1 /n2 distribution. In our notation, n1 = m and n2 = n − k − 1. You will find separate tables for different significance levels (1%, 5% and 10%). When n is large (typically n − k − 1 > 120), the F -statistic has a distribution Fm,∞ , which can also be found on the tables. 11 α Reject if F act > Fm,n−k −1 . Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients 9 10 Compute the actual value of the F -statistic, F act . Find the critical value for the Fm,n−k −1 distribution from the tables α (let’s call it Fm,n−k −1 ). Remember: In the book you will find tables for the Fn1 /n2 distribution. In our notation, n1 = m and n2 = n − k − 1. You will find separate tables for different significance levels (1%, 5% and 10%). When n is large (typically n − k − 1 > 120), the F -statistic has a distribution Fm,∞ , which can also be found on the tables. 11 α Reject if F act > Fm,n−k −1 . Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Notes about the F test: The constrained model always has fewer parameters than the unconstrained one. F act is always non-negative F act measures the relative increase in RSS when moving from the unconstrained model to the constrained model. It can be shown that: F= Maria Casanova 2 2 R U −R C m 2 (1−RU ) n −k −1 Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Notes about the F test: The constrained model always has fewer parameters than the unconstrained one. F act is always non-negative F act measures the relative increase in RSS when moving from the unconstrained model to the constrained model. It can be shown that: F= Maria Casanova 2 2 R U −R C m 2 (1−RU ) n −k −1 Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Notes about the F test: The constrained model always has fewer parameters than the unconstrained one. F act is always non-negative F act measures the relative increase in RSS when moving from the unconstrained model to the constrained model. It can be shown that: F= Maria Casanova 2 2 R U −R C m 2 (1−RU ) n −k −1 Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Notes about the F test: The constrained model always has fewer parameters than the unconstrained one. F act is always non-negative F act measures the relative increase in RSS when moving from the unconstrained model to the constrained model. It can be shown that: F= Maria Casanova 2 2 R U −R C m 2 (1−RU ) n −k −1 Lecture 10 -------------+-----------------------------Total |85266.962 65240 1.30697367 Adj R-squared = 0.1557 Root MSE = 1.0505 3. Hypothesis Tests Regarding Multiple Coefficients ----------------------------------------------------------------------lincome |Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+--------------------------------------------------------Let’s proceed with the example. We start by estimating the .19492 education |.1914982 .0017458 109.69 0.000 .1880764 unconstrained model: _cons |7.261253 .0234553 309.58 0.000 7.215281 7.307225 . gen fem_ed=female*education . reg wage female education fem_ed Source |SS df MS -------------+-----------------------------Model |17665.6042 3 5888.53474 Residual |67601.3577 65237 1.03624259 -------------+-----------------------------Total |85266.962 65240 1.30697367 Number of obs = 65241 F( 3, 65237) = 5682.58 Prob > F = 0.0000 R-squared = 0.2072 Adj R-squared = 0.2071 Root MSE = 1.018 ---------------------------------------------------------------------wage |Coef. Std. Err. t P>|t| [95% Conf. Interval] ------------+--------------------------------------------------------female | -.498354 .0458689 -10.86 0.000 -.588257 -.408451 education | .1937551 .0022496 86.13 0.000 .1893459 .1981643 fem_ed |-.0015693 .0034135 -0.46 0.646 -.0082598 .0051212 _cons | 7.481399 .0301938 247.78 0.000 7.422219 7.540579 . test female fem_ed ( 1) female = 0 Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients Next we estimate the constrained EXAMPLE STATA EXAMPLE: F-TEST model: . reg wage education Source |SS df MS -------------+-----------------------------Model |13276.8994 1 13276.8994 Residual |71990.0625 65239 1.10348201 -------------+-----------------------------Total |85266.962 65240 1.30697367 Number of obs = 65241 F( 1, 65239) =12031.82 Prob > F = 0.0000 R-squared = 0.1557 Adj R-squared = 0.1557 Root MSE = 1.0505 ---------------------------------------------------------------------Std. Err. t P>|t| [95% Conf. Interval] wage |Coef. ------------+--------------------------------------------------------education |.1914982 .0017458 109.69 0.000 .1880764 .19492 _cons |7.261253 .0234553 309.58 0.000 7.215281 7.307225 . gen fem_ed=female*education . reg lincome female education fem_ed Source |SS df MS -------------+-----------------------------Maria Casanova Lecture 10 Number of obs = 65241 F( 3, 65237) = 5682.58 3. Hypothesis Tests Regarding Multiple Coefficients The F -statistic is: F= (RSSC −RSSU ) m RSSU (n−k −1) = (71990−67601) 2 67601 (65241−3−1) = 2117 α The critical value F2,∞ , for α = 5%, is 3.00. Since 2117 > 3, we reject the null hypothesis. β1 and β3 are jointly significant (wages are not the same for men and women). Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients The F -statistic is: F= (RSSC −RSSU ) m RSSU (n−k −1) = (71990−67601) 2 67601 (65241−3−1) = 2117 α The critical value F2,∞ , for α = 5%, is 3.00. Since 2117 > 3, we reject the null hypothesis. β1 and β3 are jointly significant (wages are not the same for men and women). Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients The F -statistic is: F= (RSSC −RSSU ) m RSSU (n−k −1) = (71990−67601) 2 67601 (65241−3−1) = 2117 α The critical value F2,∞ , for α = 5%, is 3.00. Since 2117 > 3, we reject the null hypothesis. β1 and β3 are jointly significant (wages are not the same for men and women). Maria Casanova Lecture 10 3. Hypothesis Tests Regarding Multiple Coefficients The F -statistic is: F= (RSSC −RSSU ) m RSSU (n−k −1) = (71990−67601) 2 67601 (65241−3−1) = 2117 α The critical value F2,∞ , for α = 5%, is 3.00. Since 2117 > 3, we reject the null hypothesis. β1 and β3 are jointly significant (wages are not the same for men and women). Maria Casanova Lecture 10 . reg lincome female education fem_ed 3. Hypothesis Tests Regarding MS Multiple Coefficients Source |SS df Number of obs = 65241 -------------+-----------------------------F( 3, 65237) = 5682.58 Model |17665.6042 3 5888.53474 Prob > F = 0.0000 Residual |67601.3577 65237 1.03624259 R-squared = 0.2072 -------------+-----------------------------Adj R-squared = 0.2071 run the test directly 65240 1.30697367 either of the 1.018 Total |85266.962 in Stata, using Root MSE = following We can commands: ----------------------------------------------------------------------Std. Err. t P>|t| [95% Conf. Interval] -------------+--------------------------------------------------------female | -.498354 test female=fem ed=0 .0458689 -10.86 0.000 -.588257 -.408451 education | .1937551 .0022496 86.13 0.000 .1893459 .1981643 fem_ed |-.0015693 .0034135 -0.46 0.646 -.0082598 .0051212 These are_cons results: the | 7.481399 .0301938 247.78 0.000 7.422219 7.540579 lincome |Coef. test female fem ed . test female fem_ed ( 1) female = 0 ( 2) fem_ed = 0 F( 2, 65237) = 2117.60 Prob > F = 0.0000 . test female fem_ed education ( 1) female = 0 ( 2) fem_ed = 0 ( 3) education = 0 M F( 3, 65237) = 5682.58aria Casanova Lecture 10 . reg lincome female education fem_ed 3. Hypothesis Tests Regarding MS Multiple Coefficients Source |SS df Number of obs = 65241 -------------+-----------------------------F( 3, 65237) = 5682.58 Model |17665.6042 3 5888.53474 Prob > F = 0.0000 Residual |67601.3577 65237 1.03624259 R-squared = 0.2072 -------------+-----------------------------Adj R-squared = 0.2071 run the test directly 65240 1.30697367 either of the 1.018 Total |85266.962 in Stata, using Root MSE = following We can commands: ----------------------------------------------------------------------Std. Err. t P>|t| [95% Conf. Interval] -------------+--------------------------------------------------------female | -.498354 test female=fem ed=0 .0458689 -10.86 0.000 -.588257 -.408451 education | .1937551 .0022496 86.13 0.000 .1893459 .1981643 fem_ed |-.0015693 .0034135 -0.46 0.646 -.0082598 .0051212 These are_cons results: the | 7.481399 .0301938 247.78 0.000 7.422219 7.540579 lincome |Coef. test female fem ed . test female fem_ed ( 1) female = 0 ( 2) fem_ed = 0 F( 2, 65237) = 2117.60 Prob > F = 0.0000 . test female fem_ed education ( 1) female = 0 ( 2) fem_ed = 0 ( 3) education = 0 M F( 3, 65237) = 5682.58aria Casanova Lecture 10 . reg lincome female education fem_ed 3. Hypothesis Tests Regarding MS Multiple Coefficients Source |SS df Number of obs = 65241 -------------+-----------------------------F( 3, 65237) = 5682.58 Model |17665.6042 3 5888.53474 Prob > F = 0.0000 Residual |67601.3577 65237 1.03624259 R-squared = 0.2072 -------------+-----------------------------Adj R-squared = 0.2071 run the test directly 65240 1.30697367 either of the 1.018 Total |85266.962 in Stata, using Root MSE = following We can commands: ----------------------------------------------------------------------Std. Err. t P>|t| [95% Conf. Interval] -------------+--------------------------------------------------------female | -.498354 test female=fem ed=0 .0458689 -10.86 0.000 -.588257 -.408451 education | .1937551 .0022496 86.13 0.000 .1893459 .1981643 fem_ed |-.0015693 .0034135 -0.46 0.646 -.0082598 .0051212 These are_cons results: the | 7.481399 .0301938 247.78 0.000 7.422219 7.540579 lincome |Coef. test female fem ed . test female fem_ed ( 1) female = 0 ( 2) fem_ed = 0 F( 2, 65237) = 2117.60 Prob > F = 0.0000 . test female fem_ed education ( 1) female = 0 ( 2) fem_ed = 0 ( 3) education = 0 M F( 3, 65237) = 5682.58aria Casanova Lecture 10 . reg lincome female education fem_ed 3. Hypothesis Tests Regarding MS Multiple Coefficients Source |SS df Number of obs = 65241 -------------+-----------------------------F( 3, 65237) = 5682.58 Model |17665.6042 3 5888.53474 Prob > F = 0.0000 Residual |67601.3577 65237 1.03624259 R-squared = 0.2072 -------------+-----------------------------Adj R-squared = 0.2071 run the test directly 65240 1.30697367 either of the 1.018 Total |85266.962 in Stata, using Root MSE = following We can commands: ----------------------------------------------------------------------Std. Err. t P>|t| [95% Conf. Interval] -------------+--------------------------------------------------------female | -.498354 test female=fem ed=0 .0458689 -10.86 0.000 -.588257 -.408451 education | .1937551 .0022496 86.13 0.000 .1893459 .1981643 fem_ed |-.0015693 .0034135 -0.46 0.646 -.0082598 .0051212 These are_cons results: the | 7.481399 .0301938 247.78 0.000 7.422219 7.540579 lincome |Coef. test female fem ed . test female fem_ed ( 1) female = 0 ( 2) fem_ed = 0 F( 2, 65237) = 2117.60 Prob > F = 0.0000 . test female fem_ed education ( 1) female = 0 ( 2) fem_ed = 0 ( 3) education = 0 M F( 3, 65237) = 5682.58aria Casanova Lecture 10 4. Overall Regression F -Statistic A special type of F test is the one of the hypothesis that all slope coefficients equal to 0. In this case we test whether none of the X variables has any explanatory power (i.e. whether all β ’s except for the constant are equal to 0). Notice that for this particular test RSSC =TSS. To see why, notice that the constrained regression: Y = β0 + ε ˆ yields β0 = µY . The RSS will pick up any variation around the mean, i.e. the TSS. Maria Casanova Lecture 10 4. Overall Regression F -Statistic A special type of F test is the one of the hypothesis that all slope coefficients equal to 0. In this case we test whether none of the X variables has any explanatory power (i.e. whether all β ’s except for the constant are equal to 0). Notice that for this particular test RSSC =TSS. To see why, notice that the constrained regression: Y = β0 + ε ˆ yields β0 = µY . The RSS will pick up any variation around the mean, i.e. the TSS. Maria Casanova Lecture 10 4. Overall Regression F -Statistic A special type of F test is the one of the hypothesis that all slope coefficients equal to 0. In this case we test whether none of the X variables has any explanatory power (i.e. whether all β ’s except for the constant are equal to 0). Notice that for this particular test RSSC =TSS. To see why, notice that the constrained regression: Y = β0 + ε ˆ yields β0 = µY . The RSS will pick up any variation around the mean, i.e. the TSS. Maria Casanova Lecture 10 4. Overall Regression F -Statistic A special type of F test is the one of the hypothesis that all slope coefficients equal to 0. In this case we test whether none of the X variables has any explanatory power (i.e. whether all β ’s except for the constant are equal to 0). Notice that for this particular test RSSC =TSS. To see why, notice that the constrained regression: Y = β0 + ε ˆ yields β0 = µY . The RSS will pick up any variation around the mean, i.e. the TSS. Maria Casanova Lecture 10 4. Overall Regression F -Statistic In this case, the F statistic is: F= TSSC −RSSU m RSSU n−k −1 But TSSC = TSSU . Therefore: F= TSSU −RSSU m RSSU n −k −1 = ESSU m RSSU n−k −1 Notice that we only need one regression to compute the value of the F -statistic. This F-statistic is called the overall regression F-statistic. It is reported by Stata below the number of observations. Maria Casanova Lecture 10 4. Overall Regression F -Statistic In this case, the F statistic is: F= TSSC −RSSU m RSSU n−k −1 But TSSC = TSSU . Therefore: F= TSSU −RSSU m RSSU n −k −1 = ESSU m RSSU n−k −1 Notice that we only need one regression to compute the value of the F -statistic. This F-statistic is called the overall regression F-statistic. It is reported by Stata below the number of observations. Maria Casanova Lecture 10 4. Overall Regression F -Statistic In this case, the F statistic is: F= TSSC −RSSU m RSSU n−k −1 But TSSC = TSSU . Therefore: F= TSSU −RSSU m RSSU n −k −1 = ESSU m RSSU n−k −1 Notice that we only need one regression to compute the value of the F -statistic. This F-statistic is called the overall regression F-statistic. It is reported by Stata below the number of observations. Maria Casanova Lecture 10 4. Overall Regression F -Statistic In this case, the F statistic is: F= TSSC −RSSU m RSSU n−k −1 But TSSC = TSSU . Therefore: F= TSSU −RSSU m RSSU n −k −1 = ESSU m RSSU n−k −1 Notice that we only need one regression to compute the value of the F -statistic. This F-statistic is called the overall regression F-statistic. It is reported by Stata below the number of observations. Maria Casanova Lecture 10 4. Overall Regression F -Statistic In this case, the F statistic is: F= TSSC −RSSU m RSSU n−k −1 But TSSC = TSSU . Therefore: F= TSSU −RSSU m RSSU n −k −1 = ESSU m RSSU n−k −1 Notice that we only need one regression to compute the value of the F -statistic. This F-statistic is called the overall regression F-statistic. It is reported by Stata below the number of observations. Maria Casanova Lecture 10 ---------------------------------------------------------------------4. Overall -Regression F -Statistic P>|t| [95% Conf. Interval] lincome |Coef. Std. Err. t -------------+--------------------------------------------------------education |.1914982 .0017458 109.69 0.000 .1880764 .19492 _cons |7.261253 .0234553 309.58 0.000 7.215281 7.307225 . gen fem_ed=female*education . reg wage female education fem_ed Source |SS df MS -------------+-----------------------------Model |17665.6042 3 5888.53474 Residual |67601.3577 65237 1.03624259 -------------+-----------------------------Total |85266.962 65240 1.30697367 Number of obs = 65241 F( 3, 65237) = 5682.58 Prob > F = 0.0000 R-squared = 0.2072 Adj R-squared = 0.2071 Root MSE = 1.018 ---------------------------------------------------------------------wage |Coef. Std. Err. t P>|t| [95% Conf. Interval] ------------+--------------------------------------------------------female | -.498354 .0458689 -10.86 0.000 -.588257 -.408451 education | .1937551 .0022496 86.13 0.000 .1893459 .1981643 fem_ed |-.0015693 .0034135 -0.46 0.646 -.0082598 .0051212 _cons | 7.481399 .0301938 247.78 0.000 7.422219 7.540579 . test female fem_ed ( 1) female = 0 ( 2) fem_ed = 0 F( 2, 65237) = 2117.60aria Casanova M Lecture 10 ...
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