Econometric: chapter 3: Multiple Regression Analysis Estimation

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Problems 1 Using the data in GPA2 on 4,137 college students, the following equation was estimated by OLS: colgpa = 1.392 - .0135 hsperc + .00148 sat n = 4,137, R2 = .273, where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that, for example, hsperc = 5 means the top 5% of the class), and sat is the combined math and verbal scores on the student achievement test. (i) Why does it make sense for the coefficient on hsperc to be negative? (ii) What is the predicted college GPA when hsperc = 20 and sat = 1,050? (iii) Suppose that two high school graduates, A and B, graduated in the same percentile from high school, but Student A's SAT score was 140 points higher (about one standard deviation in the sam- ple). What is the predicted difference in college GPA for these two students? Is the difference large? (iv) Holding hsperc fixed, what difference in SAT scores leads to a predicted colgpa difference of 50, or one-half of a grade point? Comment on your answer.model. 8 Suppose that average worker productivity at manufacturing firms (avgprod) depends on two factors, average hours of training (avgtrain) and average worker ability (avgabil): avgprod = Bo + Biavgtrain + Byavgabil + u. Assume that this equation satisfies the Gauss-Markov assumptions. If grants have been given to firms whose workers have less than average ability, so that avgtrain and avgabil are negatively correlated, what is the likely bias in B, obtained from the simple regression of avgprod on avgtrain?Given regression model colgpa=1.392-0.0135 hsperc+0.00148sat To find answers as asked a) here coefficient of hsperc is -0.0135. It makes sense because hsperc tells about the percentile in the high school. The lesser value hsperc will take the higher rank student will be. (For eg-if hsperc=2% , this tells that student is in top 2% of class and hence he will be higher ranked). Now we can see this as the student who has higher rank in high school will get more gpa in college and hence the relationship between hsperc and colgpa is inverse relationship. B) put hsperc=20 and sat=1050, we get colgpa = 1.392 - 0.0135 x 20 + 0.00148 x 1,050 = 2.676 c) given hsperc is same for 2 students but sat score of A is 140 points greater than sat score of B Here variable hsperc is held constant, then as sat score is increased by 1 unit, colgpa will increase by 0.00148 units. So for 140 points increase, colgpa will increase by:- 0.00148 x 140 = 0.2072 Hence predicted difference will be 0.2072 grade point This difference is very small d) to find difference in sat score so that predicted difference in colgpa is 0.5 grade point dcolgpa = .00148 dsat. We need to find dsat when dcolgpa=0.5, We get dcolgpa 0.5 dsat = = 337.84 0.00148 0.00148 We see that of sat score of one student is 338 points more than other while they both have same hsperc, then the difference in colgpa will be very less (only 0.5) indicating that sat score is weak predictor of colgpa Explanation: dcolgpa represent change in colgpa and dsat represent change in sat scoreg b~ W N The Gauss-Markov assumptions are crucial in the context of linear regression. These assumptions are: . Linearity: The relationship between the dependent variable and the independent variables is linear. . Independence: Observations are independent of each other. . Homoscedasticity: Residuals have constant variance. . No perfect multicollinearity: Independent variables are not perfectly correlated. . No endogeneity: There is no correlation between the independent variables and the error term. . No omitted variables bias: All relevant variables are included in the model. Now, let's analyze the situation described in your question. The given regression model is: avgprod = 8 + B avgtrain + Bravgabil + u The key information is that there is a negative correlation between avgtrain and avgabil. This means that when avgtrain increases, avgabil tends to decrease and vice versa. In a simple regression where you regress avgprod on avgtrain (ignoring avgabil), the negative correlation between avgtrain and avgabil might lead to a violation of the assumption of no perfect multicollinearity. This is because avgtrain and avgabil are correlated, and this can affect the estimation of the coefficient 3;. When there is multicollinearity, the estimated coefficients may be biased. In this case, the bias in 3 is likely to be upward (positive). This means that the simple regression may overstate the true positive relationship between avgtrain and avgprod. In summary, the likely bias in B, obtained from the simple regression of avgprod on avgtrain is upward (positive), and this is due to the negative correlation between avgtrain and avgabil violating the assumption of no perfect multicollinearity