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Chapter 8 Exercises 85Chapter 8 Model Estimation8.1 Given the following table (Table 8.1 from the textbook):Table 8.1 — Hypothetical Demand for OrangesDay(i) Quantity (Yi) Cents (Xi) (Yi – Y) (Xi – X) (Xi – X)(Yi – Y) (Xi – X)2(Yi – Y)21 55 100 –45 30 –1,350 900 20252 70 90 –30 20 –600 400 9003 90 80 –10 10 –100 100 1004 100 70 0 0 0 0 05 90 70 –10 0 0 0 1006 105 70 5 0 0 0 257 80 70 –20 0 0 0 4008 110 65 10 –5 –50 25 1009 125 60 25 –10 –250 100 62510 115 60 15 –10 –150 100 22511 130 55 30 –15 –450 225 90012 130 50 30 –20 –600 400 900Sums: 1,200 840 0 0 –3,550 2,250 6,300Source: Jan Kmenta, Elements of Econometrics, second edition, Macmillan Publishing Company, 1986.(a) Plot the scatter plot of X and Y on the following figure.(b) Calculate the means of X and Y:11niiX Xn 11niiY Yn (c) Use the information in the table to calculate the covariance of X and Y:11( , ) ( )( )ni iiCov X Y X X Y Yn   86 Chapter 8 ExercisesXY10 20 30 4040506070901001101201308050 60 70 80 90 100(d) The sign of the covariance indicates how Y varies with X. In this case, when X tends to be above its mean, Y tends to be below its mean, and when X tends to be below its mean, Y tends to be above its mean. Thus, these variables are (directly, inversely) related.(e) Use the information in the table to calculate the variance of X and Y:211( ) ( )niiVar X X Xn  211( ) ( )niiVar Y Y Yn  (f) The least-squares estimate of  isCov(X ,Y)Var(X ) =Chapter 8 Exercises 87(g) The least-squares estimate of  is = Y – X = (h) Write the estimated least-squares equation:Y =  + X = (i) Verify that this equation passes through the point of the means by calculating Y at X = X:Y =  + X = (j) Plot this equation on the same graph as the scatter plot. Interpret the estimated coefficients.(k) Compute Yi for each observation and complete the appropriate column in the following extension of Table 8.1:Table 8.1 (continued)Day(i) Quantity (Yi) Cents (Xi)Yi(Yi – Yi) (Yi – Yi)2(Yi – Y) (Yi – Y)21 55 1002 70 903 90 804 100 705 90 706 105 707 80 708 110 659 125 6010 115 6011 130 5512 130 50Sums: 1,200 840(l) Calculate i = Yi – Yi and i2 for each observation, and complete the appropriate columns inthe table.(m) Calculate and interpret the standard error of regression:88 Chapter 8 ExercisesS = ei2n – k – 1(n) Calculate and interpret the standard error of the estimated slope coefficient:S S(Xi – X )2 = (o) Calculate and interpret R2 using��222( )( )( )( )iiY YVar YRVar YY Y (p) Calculate the adjusted R2 using the following formula; R2 is (smaller, larger) than R2:R2 = 1 – (1 – R2)(n – 1)(n – k – 1) = 8.2 The capital asset pricing model (CAPM) says thatj = c + j(m – c)where j is the expected return to any asset j, c is a risk-free return, and m is a measure of the return to the market as a whole. The term j(m – c) is called the risk-adjustment factor (or risk premium), the extra amount over the sure return c that is required to compensate for the risk of buying asset j. To implement the CAPM, we can estimate beta for any asset using historical data.The sure return c is subtracted from both sides of the equation, the empirically observed return rjfor asset j is substituted for j, and the empirically observed return rm is substituted for m. Themodel to be estimated isrjt – ct = j + j(rmt – ct) + twhere t is the periodic observation index, (rjt – ct) is the periodic rate of return for asset j above the risk-free rate ct in period t, j is an intercept added to avoid forcing the estimated equation through the origin, and (rmt – ct) is the market return above the risk-free rate. As before, t is a random error term. Empirical experience with this model has been fairly successful with researchers obtaining estimates of betas that are statistically significant.Suppose that you collect data on ten years of monthly returns rjt for two companies, Mobil (j = 1) and Motorola (j = 2). The market rate rmt used is a volume-adjusted composite monthly return based on all stock transactions from the New York Stock Exchange and theChapter 8 Exercises 89American Exchange over the same ten year-time span. The risk-free return ct is the return on 30-day U.S. Treasury bills. Ordinary least squares regression yields (t-ratios in parentheses):Mobil:Y1 = 0.0079 + 0.7815(rmt – ct)R12 = 0.3810(1.019) (5.974)Motorola:Y2 = 0.0063 + 1.3381(rmt – ct)R22 = 0.5536(0.672) (8.482)(a) What is the estimated beta of the asset for Mobil? _____________________. What is the estimated beta of the asset for Motorola? _____________________.(b) Comment on the magnitude and precision of each of the estimated betas. Is this about what you would expect? _______________________________________________________ _______________________________________________________________________.(c) Comment on the magnitude and precision of each of the estimated intercept terms. Is this about what you would expect? ______________________________________________ _______________________________________________________________________.(d) Calculate the standard error of the estimate for 1. _______________________________. Calculate the standard error of the estimate for 2. _______________________________.(e) How would you test a null hypothesis that each company’s risk is the same as the average risk over the entire market? That is, test that  = 1 against the alternative that  ≠ 1. Usingthe rule of thumb, are either of these estimated betas different from unity?(f) The estimated CAPM can be used to measure the proportion of the variation in the dependent variable explained by the variation in the independent variable. In the CAPM context, R2 is said to be a measure of the impact of market (systematic) risk, and 1 – R2 is ameasure of the impact of asset-specific (unsystematic) risk. For Mobil, what is the proportion of the total variation attributed to market risk, and what is the proportion of the total variation attributed to asset-specific risk? ____________. For Motorola, what is the proportion of the total variation attributed to market risk, and what is the proportion of the total variation attributed to asset-specific risk? _____________.90 Chapter 8 Exercises(g)


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