UNIVERSITY OF TORONTOFaculty of Arts and ScienceDECEMBER EXAMINATIONS 2010STA 302 H1F / STA 1001 HFDuration - 3 hoursAids Allowed: CalculatorLAST NAME: FIRST NAME:STUDENT NUMBER:• There are 19 pages including this page.• The last page is a table of formulae that may be useful. For all questions you can assumethat the results on the formula page are known unless the question states otherwise.• Pages 14 through 18 contain output from SAS that you will need to answer Question 5.• Total marks: 851 2ab 2cd 3 4 5a5b 5c 5d(i-iii) 5d(iv-vi) 6 7, 81Continued1. (10 marks) Beside each description, write the letter of the term from the list belowthat provides the best match.(I) What to include when the effect of X1on Y is different for different valuesof X2.(II) The proportion of variation explained by the regression line.(III) An observed response minus its estimated mean according to some model.(IV) A measure of how influential a particular observation is.(V) A method for estimating regression coefficients.(VI) A test comparing a model of interest to the model with only an intercept.(VII) A statistic used to identify problems of multicollinearity.(VIII) A statistic for comparing models with different sets of explanatory vari-ables.(IX) Another name for the estimate of σ2in regression analysis.(X) A measure of how unusual the x-values are for a particular observation.(A) Analysis of Variance(B) Analysis of Variance F -test(C) R-squared(D) Adjusted R-squared(E) t-test(F) Residual(G) Standardized residual(H) Fitted value(I) Interaction(J) Indicator(K) Explanatory variable(L) Response variable(M) Outlier(N) Least Squares(O) Correlation(P) Degrees of freedom(Q) Cook’s Distance(R) Leverage(S) Variance Inflation Factor(T) Residual mean square(U) Mean square of regression(V) Residual sum of squares(W) Regression sum of squares(X) Total sum of squares(Y) Variance(Z) Extra sum of squares2Continued2. Suppose that we believe that a response variable Y is related to a non-random ex-planatory variable x by the model Yi= βxi+ ei, i = 1, . . . , n. That is, we believethat it is appropriate to use a model that goes through the origin. Assume that thefollowing conditions hold:• The errors e1, . . . , enhave expectation 0.• The errors have common variance σ2.• The errors are uncorrelated.(a) (3 marks) Show that the least squares estimator of β isˆβ =nXi=1xiYi,nXi=1x2i(b) (3 marks) Assuming that the model is correct, show thatˆβ is an unbiased esti-mator of β.3Continued(Question 2 continued.)(c) (2 marks) Find Var(ˆβ).(d) (2 marks) Suppose that the model Yi= βxi+ eiis correct, but the modelYi= β0+ β1xi+ eiis used. Show that Var(ˆβ1) ≥ Var(ˆβ).4Continued3. A multiple linear regression model with dependent variable Y and 3 explanatoryvariables was fit to 15 observations. The residual sum of squares was found to be 22.0and it was also found that(X0X)−1=0.5 0.3 0.2 0.60.3 6.0 0.5 0.40.2 0.5 0.2 0.70.6 0.4 0.7 3.0(a) (1 mark) What degrees of freedom would be used when finding a confidenceinterval for β1?(b) (1 mark) What is the estimate of the error variance?(c) (1 mark) What is the estimated variance of the estimator of β2?5Continued4. Consider the multiple regression modelY = Xβ + e, e ∼ N(0, σ2I)(a) (3 marks) Show thatˆe = (I −H)e.(b) (1 mark) Why is E(ee0) = Var(e)?(c) (4 marks) Show that I − H is idempotent and symmetric.(d) (3 marks) Show that Var(ˆe|X) = σ2(I − H).6Continued5. The data considered in this question are the same data considered in Assignment 1,taken from a 2007 Wall Street Journal article on the decline of U.S. house prices. Thedata are indicators of the real-estate market in 28 U.S. cities. The variables consideredin this question are:Response variable:• PriceChange – The percent change in average price of a home from one year ago.Explanatory variables:• LoansOverdue – The percentage of mortgage loans that are 30 days or more overdue.• InventoryChange – The percent change in housing inventory from one year ago. Apositive value indicates that more houses are on the market.• EmployOutlook – A character variable that classifies the projected growth in thenumber of jobs as one of Strong, Average, or Weak. (An observation that had anemployment outlook of Very Weak in the original data has been re-classified as Weak.)• iEmployOutIsWeak – An indicator variable that is 1 if EmployOutlook is Weak and0 otherwise.• iEmployOutIsAverage – An indicator variable that is 1 if EmployOutlook is Averageand 0 otherwise.• iEmpWeak LoansOD – The product of iEmployOutIsWeak and LoansOverdue.• iEmpAvg LoansOD – The product of iEmployOutIsAverage and LoansOverdue.On pages 14 through 18 there is SAS output for the analysis of these data. Thequestions below relate to the SAS output.(a) ANALYSIS 1 (page 14) was carried out using only observations having EmployOutlookeither Strong or Weak. (That is, cities with Average employment outlook wereremoved from the data for this analysis only.) The questions in part (a) relateto ANALYSIS 1.i. (2 marks) What is the estimated difference in the mean of percent changein average price of a home between cities with Strong and cities with Weakemployment outlook?ii. (2 marks) Can you conclude that there is a difference in the mean of percentchange in average price of a home between cities with Strong and cities withWeak employment outlook? Justify your answer.7Continued(Question 5 continued.)(b) ANALYSIS 2 (page 15) was carried out using all of the available data. It is asimple linear regression using LoansOverdue as the explanatory variable. Thequestions in part (b) relate to ANALYSIS 2.i. (4 marks) Four numbers in the SAS output have been replaced by letters.What are they?(A) =(B) =(C) =(D) =ii. (2 marks) R2is only 22%. As a consequence, can we conclude that there isnot a linear relationship between PriceChange and LoansOverdue? Explain.iii. (5 marks) On page 15 you are given a plot of the standardized residualsversus the predicted values and a normal quantile plot of the standardizedresiduals for this analysis. What are you looking for in each plot and whatdo you conclude?8Continued(Question 5 continued.)(c) ANALYSIS 3 (page 16) was carried out on all of the available data. It is amultiple regression using LoansOverdue and InventoryChange as explanatoryvariables. The questions in part (c) relate to ANALYSIS 3.i. (1 mark) Write down the model that is being fit. Do not