25 Solved Questions on Business Statistics II - Final Exam | IDS 371, Exams of Business Statistics

Download 25 Solved Questions on Business Statistics II - Final Exam | IDS 371 and more Exams Business Statistics in PDF only on Docsity! IDS 371 - Business Statistics II SOLUTIONS TO FINAL EXAM OF FRIDAY, 21-JULY-2006 There were wenty-five questions, four points each. Multiple-choice and short-answer. Open book: open books and notes. REGRESSION AND CORRELATION IN FINANCIAL INVESTMENTS ANALYSIS Recall that a stock’s “beta” is computed by simple linear regression. To describe “beta” and tell how it is computed, answer the following questions. 1. What is Y? ___ ANSWER: The rate of return (ROR) of the particular asset (stock or mutual fund) of interest. 2. What is X? ________________________________________________________ ANSWER: The ROR of the market as a whole (e.g., the ROR of the S&P500). 3. What is n ? __________________________________________________ ANSWER: The sample size, which in this application is typically 60 months (five years of monthly data). 4. What does it mean if beta is less than 1 ? ANSWER: The change in ROR of the particular asset is less than that of the market as a whole, so this asset is considered less risky than the market as a whole. 5. Variance of a portfolio of two stocks. Suppose Var(X) = 49 and Var(Y) = 9. For what values of the correlation of X and Y is the Var( .3X + .7Y ) less than Var(Y) ? SOLUTION: Var(.3X + .7Y) = Var(aX + bY) = a2 Var(X) + b2 Var(Y) + 2 a b SD(X) SD(Y) Corr(X,Y) = .32 (49) + .72 (9) + 2 (.3)(.7) (7) (3)  = .09 (49) + .492 9 + 2 (.3)(.7) (7) (3)  = 8.82 + 8.82  < 9. This gives  < +.0204 . IDS 371 / Summer ’06 / Sclove / Solutions to Final Exam p. 2 MULTIPLE REGRESSION. THE "MARKET MODEL": "BETA" IN BULL AND BEAR MARKETS Monthly rates of return (%) of a mutual fund, the Cheetah Fund, were regressed on those of the S&P500. The fitted equation was predicted value of Y = 0.2 + 0.600 X + 0.800 X L (*) where Y = rate of return of Cheetah Fund X = rate of return of S&P500 and L = 1 if Bull market, 0 if Bear market. 6. What is equation (*) in a Bear Market ? ANSWER: L = 0, so pred.val. of Y = 0.2 + 0.6 X 7. What is equation (*) in a Bull Market ? ANSWER: L = 1, so pred.val. of Y = 0.2 + 0.6 X + 0.8 X (1) = 0.2 + 1.4 X . 7. What is the estimate of the Cheetah Fund's "beta" in a Bear market? ANSWER: “beta” = coefficient of X = 0.6 8. What is the estimate of the Cheetah Fund's "beta" in a Bull market ? ANSWER: “beta” = coefficient of X = 1.4 10. What is the predicted value of Y if X = 0 ? ANSWER: pred.val. of Y if X = 0 is the constant, 0.2 IDS 371 / Summer ’06 / Sclove / Solutions to Final Exam p. 5 TWO-WAY ANALYSIS OF VARIANCE WITH INTERACTION Consider the analysis-of-variance table below, for a 32 factorial experiment with two observations per cell. No. of Factor Type Levels Temp fixed 3 Press fixed 3 Analysis of Variance for Yield Source DF SS MS F P ---------------------------------------------- Temp 2 0.30111 0.15056 8.47 .009 Press 2 0.76778 0.38389 21.59 .000 Temp*Press 4 x = ? y = ? z = ? .470 Error 9 0.16000 0.01778 ---------------------------------------------- Total 17 1.29778 MSTot = ? ---------------------------------------------- 18. The value of MSTot, the variance of all 18 observations, is not shown. Compute it. SOLUTION: Each variance, or "MS", is equal to the corresponding SS/DF. The variance is SS(Total)/DF(Total) = 1.29778/17 = 0.07634 ________________________________ 19. What is the value of SS(Temp)? ___ 0.30111 ____________________ 20. What is the value of F for Temp ? ____ 8.47 _________________________ 21. What is the P-value for Interaction ? __ .470 _____________________________ 22. Do the effects of Temp depend upon the levels of Press? Why, or why not? ANSWER: The p-value for Temp*Press is high; the amount of interaction is not statistically significant. We conclude that the effects of Temp do not depend upon the levels of Pressure. IDS 371 / Summer ’06 / Sclove / Solutions to Final Exam p. 6 TIME SERIES ANALYSIS 23. Prediction. What is the moving-average predictor for yt based on the four most recent observations ? ANSWER: pred.val. of yt = ( yt-1 + yt-2 + yt-3 + yt-4) / 4 . 24. Prediction. What is the sum-of-digits predictor for yt based on the three most recent observations ? ANSWER: The sum-of-digits weighted moving average of three observations weights the threee most recent observations with 3, 2, and 1: pred.val. of yt = ( 3yt-1 + 2yt-2 + 1yt-3 ) / (3+2+1) = (3/6)yt-1 + (2/6) yt-2 + (1/6)yt-3 = (1/2)yt-1 + (1/3) yt-2 + (1/6)yt-3 . 25. Autoregression. Suggest autoregressive prediction expressions for quarterly time series data { yt }, as follows: Example: First-order (that is, lag four): predicted value of yt = b0 + b1 yt-4 ___________________________________________________________________________ Second-order: ANSWER: predicted value of yt = b0 + b1 yt-4 + b2 yt-8 Lags one and four ANSWER: predicted value of yt = b0 + b1 yt-1 + b2 yt-4 ____________________________________________________________ lags one, lag four, and lag five ANSWER: predicted value of yt = b0 + b1 yt-1 + b2 yt-4 + b3 yt-5 Created 2006: July 21 last edited 2006: July 22