Suppose you\'ve been hired by the President\'s Office at the UO to help reduce d

Question

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 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_i = beta_1 + beta_2 F_i + beta_3 S_i + beta_4 A_i + u_i where: D_i is the damage in dollars to the ith dorm F_i is the number of freshmen living in the ith dorm S_i is the total number of students living in the ith dorm A_i 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: (a) Conduct separate t tests of the following two null hypotheses: H_0: beta_2 = 0 and H_0: beta_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) Using the F test statistic, derive this statistic in terms of R^2 instead of in terms of RSS. (c) Suppose you estimate the model excluding the two variables in part (a). You obtain an R^2 of 0.40 for this regression. Use the given information to conduct an F test of the joint null hypothesis: H_0: beta_2 = beta_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) Based on these results, discuss the potential evidence for multicollinearity 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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