The Multivariate Regression Model

Transcription

1 The Multivariate Regression Model Example Determinants of College GPA Sample of 4 Freshman Collect data on College GPA (4.0 scale) Look at importance of ACT Consider the following model CGPA ACT i 0 i i ACT 4 tests English/math/reading/science reasoning Composite scores from -36 Average score in 000 was Movement from to represents 7 percentage points in the distribution (56 th to 63th percentile) 3 College GPA Scatter Plot: ACT Score and College GPA ACT 4

2 College GPA Scatter Plot: ACT Score and College GPA. *run regression with one variable. reg college_gpa act Source SS df MS Number of obs = F(, 39) = 6. Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = college_gpa Coef. Std. Err. t P> t [95% Conf. Interval] act _cons Interpret the result: ACT 5 6 Is this an accurate estimate of (CGPA)/ (ACT)? ACT is but one measure of ability Noisy measure at best Are there other measures available? Consider another model (Think of this as the true model) CGPA ACT HSGPA i 0 i i i College GPA Scatter Plot: HS GPA and College GPA HS GPA 8

3 Scatter Plot: HS GPA and College GPA Scatter Plot: HS GPA and College GPA College GPA 3.0 College GPA HS GPA HS GPA 9 0 *run synthetic regression of hs_gpa on act reg hs_gpa act. * get correlations between key variables. corr college_gpa act hs_gpa (obs=4) colleg~a act hs_gpa college_gpa.0000 act hs_gpa Source SS df MS Number of obs = F(, 39) = 8.88 Model Prob > F = Residual R-squared = Adj R-squared = 0.3 Total Root MSE = hs_gpa Coef. Std. Err. t P> t [95% Conf. Interval] act _cons

4 , x (7) E[ ] ˆ x i 0 i i ˆ n i ( x x )( x x ) i i n i ( x x ) i we anticipate that 0 and we have shown that ˆ 0 E[ ˆ ] then E[ ] On average, the value we estimated in the False model will be greater than the one in the true model 3 4. * run multivariate regression. reg college_gpa act hs_gpa Source SS df MS Number of obs = F(, 38) = 4.78 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.3403 college_gpa Coef. Std. Err. t P> t [95% Conf. Interval] act hs_gpa _cons The coefficient on ACT in the false model was The coefficient in the True model is the coefficient falls by 77% Example : Class Size and Performance Data from 40 schools in CA Outcome is average on state test for reading and math in 6 th grade Average scores around 650 for state Key covariate: student/teacher ratio SCORE STR i 0 i i 5 6 4

5 Scatter Plot: Student Teacher Ratio vs. Average Test Scores Scatter Plot: Student Teacher Ratio vs. Average Test Scores Average Test Score Average Test Score Student-Teacher ratio Student-Teacher ratio 7 8. * run regression with one variable. reg average_score student_teacher Source SS df MS Number of obs = F(, 48) =.58 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = 8.58 average_sc~e Coef. Std. Err. t P> t [95% Conf. Interval] student_te~r _cons A one student increase in class size will reduce average Scores by.8 points Omitted variables Class size is but one covariate we could add Consider others that might be correlated with X that are omitted from model Example: % ESL These students tend to score lower on tests If they are also more or less likely to be in more crowded schools, then results could be biased Increasing a class size by 5 will reduce average scores By 5(.8)=.4 which is.4/654=.07 or by.7% 9 0 5

6 SCORE STR ESL i 0 i i i Think of this as the true model, E[ ] ˆ x x i 0 i i (7) ˆ n i ( x x )( x x ) i i n i ( x x ) i Scatter Plot: % ESL vs. Average Test Scores Scatter Plot: % ESL vs. Average Test Scores Average Test Score Average Test Score % ESL % ESL 3 4 6

7 Scatter Plot: Student Teacher Ratio vs. % ESL Scatter Plot: Student Teacher Ratio vs. % ESL % ESL Student-Teacher ratio 5 % ESL Student-Teacher ratio 6 averag~e studen~r esl_pct average_sc~e.0000 student_te~r esl_pct and we have shown that ˆ 0 E[ ] ˆ then E[ ] we anticipate that 0 On average, the value we estimated in the False model will be smaller than the one in the true model 7 8 7

8 Think of the prediction this way ˆ E[ ] ˆ β 0 ( ) ( )( ) 9 In the single variable model, the Student/teacher ratio is picking up two effects Larger class sizes reduce performance ESL students are more likely to be in more crowded schools, and they that tend to have lower scores Therefore, the model without ESL will estimate a too large of a negative number 30. * run multivariate regression. reg average_score student_teacher esl_pct Source SS df MS Number of obs = F(, 47) = 55.0 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = average_sc~e Coef. Std. Err. t P> t [95% Conf. Interval] student_te~r esl_pct _cons student increase in class size reduces test scores by 5(.) = 5.5 which is 5.5/654= or.8% -- half the Estimate impact as before A one percentage point increase in % ESL in school Will reduce average scores by.64 points 3. * demonstrate the partialing out. * nature of mv regressions.. * run a regression of STR on ESL. * output the residuals. reg student_teacher esl Source SS df MS Number of obs = F(, 48) = 5.5 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.8604 student_te~r Coef. Std. Err. t P> t [95% Conf. Interval] esl_pct _cons * output residuals. predict res_str, residual 3 8

9 . * run a regression of test scores. * on the student_teacher residuals. reg average_score res_str Source SS df MS Number of obs = F(, 48) = 4. Model Prob > F = 0.0 Residual R-squared = Adj R-squared = 0.00 Total Root MSE = average_sc~e Coef. Std. Err. t P> t [95% Conf. Interva res_str _cons Exact same number as before 33 9