Question

6. Two researchers are investigating the effects of time spent studying on the examination marks earned by students on a certain course. For a sample of 100 students, they have the examination mark, M, total hours spent studying, H, hours on primary study, P, and hours spent on revision. R. By definition, H = P+ R. The sarnple means of H. P. and R are 100 hours, 95 hours, and 5 hours, respectively. The sample correlation coefficients are 0.98 for H and P, and 0.10 for H and R, and -0.11 for P and R. The standard deviations of the distributions of H, P, and R are 10.1, 10.1 and 2.1, respectivel:y Researchers A decides to regress M on P and R and fit the following regression(standard errors in parentheses) M 45.60.15P 0.21R R2 0.243 (1) (2.8) (0.03) (0.14) Researchers B decides to regress M on H, P and R. However, the regression application refuses to fit the regression with all three ex planatory variables. Instead, it drops R and the regression output is R2-0.243 M-45.6 + 0.21H+-0.05P (2.8) (0.14) (2) (0.14) (a) Researcher A says that her specification has better explanatory power than that of Researcher B because the coefficient of her main variable, P, has a high t statistic. Explain whether this assertion is correct (b) She says that the insignificant coefficient of R in 1) is to be expected because the students, on average, spent much less time on revision than on primary study. Explain whether this assertion is correct (c) Researcher B says that, assuming that his specification is in fact correct, not being able to include R in the regression has given rise to omitted variable bias, and this is responsible for the negative coefficient of P. Explain whether this assertion is correct. d) A commentator, drawing attention to the high correlation be- tween H and P, says that the real reason for the implausible neg- ative coefficient of P obtained by Researcher B is multicollinear- ity. Explain whether this assertion is correct. Is the negative coefficient of P implausible?

Answer 1

a) Regression model of researcher A has a better explanatory power because of following conditions compared to researcher B

-The independent variable P in the first model is significant at 1% and 5%.but in the case of second case none of the independent variables are significant at 1%,5% and 10%(P values are insignificant)

-In the second case, There exists multicollinearity ie there is strong colinearity among the predictors. It is given that the sample correlation coefficient among predictors H and P is .98 which shows a high positive correlation.

-When multicollinearity exists among predictors in a model there is chances for the t ratios of the coefficients to be statistically insignificant and R square which will measure the overall measure of goodness of fit to be very high.

Here the t ratios are insignificant for all coefficients including constant as a result of the presence of multicollinearity. In both models, the r square is the same but in the second model, the r square can be high due to multicollinearity. So regression model by researcher a has better r square which is free from multicollinearity.

b) The assertion cannot be said correctly because the model has certain limitations. The first model shows that the R variable has insignificant t ratio. This coefficient has insignificant t ratios because they do not have an impact on the dependent variable or cannot be proved statistically. here both models have very low r square which indicates the overall measure of goodness of fit is very low.

There is a negative weak correlation between the hours of revision and hours on the primary study

c)The assertion of the researcher b is wrong because he has chosen the wrong combination of predictors. He could have chosen the predictors H and R which would have avoided the multicollinearity problem that exists in the present model.

d)It is true what the commentator has said. There exist multicollinearity problem between the H and P and this led to the negative coefficient of P.Existence of multicollinearity have led to the insignificance of the t ratios in the second model