5. Suppose you\'ve been hired by the President\'s Office at the UO to help reduc

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5. Suppose you've been hired by the President's Office at the UO to help reduce damage done to dorms by rowdy students. Your first step is to use regression analysis to study last Assignment 3 Due: Wednesday, July 19th quarter's damage to each dorm as a function of a the attributes of that dorm. You decide to work with the following multiple linear regression model: D, -B12F+B3Si+B4Aui where Di is the damage in dollars to the ith dorm Fi is the number of freshmen living in the ith dorm S is the total number of students living in the ith dorm Ai is the number of incidents involving alcohol in the ith dorm You collect a sample of dorms of size n - 14 and obtain the following estimation results Parameter Estimate Standard Error 2 3 4 R2-.82 120 608 1.35 54 50 280 1.01 20 (a) (9 points) Conduct separate t tests of the following two null hypotheses: H0 : 2- 0 and Ho : 3-0. Use a two-sided test and a 5% significance level for both tests Your answer should include the test statistics, the degrees of freedom, the critical value, and the result of the tests (b) (8 points) Using the F test statistic, derive this statistic in terms of R2 instead of in terms of RSS. (Hint: divide both sides by TSS) (c) (5 points) Suppose you estimate the model excluding the two variables in part (a) You obtain an R2 of 0.40 for this regression. Use the given information to conduct an F test of the joint null hypothesis: H0 : 2 = 3 = 0. Use the 5% significance level. Your answer should include the F-stat, the degrees of freedom, the critical value, and the result of the test (d) (4 points) Based on these results, discuss the potential evidence for multicollinear- ity in this regression

Explanation / Answer

2)

a) t-stat for coefficeint of b2 is 608/280 = 2.1714

for b3 , it is -1.35/1.01 = -1.3366

df = n-k-1 = 14- 4-1 = 9

t-critical for alpha = 0.05

is 2.262

since both |TS| < t-critical ,we fail to reject the null hypothesis and conclude that both variable are not individually significant

b) F = (R^2 / k )/(1 -R^2)/(n-k-1)

= (0.82/4)/((1-0.82)/9) = 10.25

c) r -reduced model

ur - unreduced model

F = (Rur^2 - Rr^2)/q) )/((1- Rur^2)/(n-k-1))

here Rr2 = 0.40 Rur2 = 0.82 , q = 2

hence

F = (0.82 -0.4)/ 2 /((1-0.82)/9)

=10.5

df1 = q = 2 ,df2 = n-k-1 = 9

critical value = 4.25

since TS > critical value , we reject the null and conclude that the overall model is significant

d)

Yes there is multicollinearity

What is happening is that the two variables Fi and Si are
highly correlated, and this multicollinearity makes it difficult to uncover the partial effect
of each variable; this is reflected in the individual t statistics. The F statistic tests whether
these variables are jointly significant, and multicollinearity between
Fi and Si is much less relevant for testing this hypothesis

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