1 Economics 435 Spring 1997 D. Maki MIDTERM EXAMINATION Instructions: There are three parts to this examination, each with two questions. Answer one question from each part (a total of three questions). Questions are equally weighted. Total time allowed = three hours. Part I. Answer one of the following two questions. 1. The following (edited) SHAZAM output arises from 12 monthly observations on sales(q) and price(p) of hamburgers at a mythical fast food outlet. |_stat q p NAME N MEAN ST. DEV VARIANCE Q 12 924.17 174.67 30509. P 12 1.1975 0.10627 0.11293E-01 |_ols q p R-SQUARE = 0.9212 R-SQUARE ADJUSTED = 0.9133 VARIANCE OF THE ESTIMATE-SIGMA**2 = 2643.7 LOG OF THE LIKELIHOOD FUNCTION = -63.2130 VARIABLE ESTIMATED STANDARD T-RATIO NAME COEFFICIENT ERROR P -1577.6 145.9 -10.81 CONSTANT 2813.3 175.3 16.05 |_genr lq=log(q) |_genr lp=log(p) |_ols lq lp/loglog R-SQUARE = 0.8809 R-SQUARE ADJUSTED = 0.8690 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.46437E-02 LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -65.4516 VARIABLE ESTIMATED STANDARD T-RATIO NAME COEFFICIENT ERROR 10 DF LP -1.9273 0.2241 -8.600 CONSTANT 7.1528 0.4417E-01 161.9 a. Do you think the LINLIN or LOGLOG equation is most appropriate? Why? Is there any strong evidence in favour of either form? Explain. b. Using the LINLIN equation, construct a 95% confidence interval for the predicted mean of q given p = 1. (Tabled t(.025,10) = 2.228). c. Using the LOGLOG equation, test the hypothesis that the elasticity of demand is -1. 2 2. The following table gives information on the price of a Big Mac in 18 countries at two time points, 1992 and 1994, measured in Canadian dollars. Country Price-92 Price-94 Country Price-92 Price-94 Argentina 4.99 3.90 | Germany 3.73 4.36 Australia 2.39 2.75 | Hong Kong 1.65 1.78 Belgium 4.30 4.36 | Italy 3.85 4.10 Brazil 2.19 3.90 | Japan 5.23 3.45 Britain 3.68 3.84 | Mexico 3.35 2.83 Canada 2.86 2.86 | Russia 2.30 2.38 Chile 3.17 2.85 | Switzerland 5.50 6.49 China 1.43 1.63 | Taiwan 3.26 3.25 France 4.40 4.55 | Thailand 2.64 2.64 Someone regressed the 1994 price on the 1992 price with the (edited) printout: |_stat p94 p92 NAME N MEAN ST. DEV VARIANCE P94 18 3.4400 1.1533 1.3300 P92 18 3.3844 1.1898 1.4155 |_ols p94 p92 R-SQUARE = 0.6424 R-SQUARE ADJUSTED = 0.6201 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.50529 VARIABLE ESTIMATED STANDARD T-RATIO NAME COEFFICIENT ERROR 16 DF P92 0.77693 0.1449 5.362 CONSTANT 0.81051 0.5183 1.564 Assume a naive version of purchasing power parity theory predicts that over time, prices of a homogeneous commodity should approach equality across countries, when expressed in units of a common currency. a. Without restricting yourself to regression analysis, list a couple of ways you could test whether the above data support this version of PPP. b. Do the regression results reported provide any way to test this version of PPP? If so, perform the test(s). c. If both variables were expressed in U.S. dollars instead of Canadian dollars, using the exchange rate $1 U.S. = $1.39 Canadian, would any of the results in the regression equation change? Explain. Part II. Answer one of the following two questions. 3. Given three variables, Y, X2 and X3: a. If the population model is Y=B1+B2*X2+B3*X3+u and we regress Y=b1+b2*X2+v, do we know whether the absolute value of b2 is greater or less than the magnitude of the OLS estimator of B2 which would obtain if the multiple regression model were estimated instead (call this B2hat)? What about the magnitudes of the estimated variances of b2 versus B2hat? Is b2 biased? If so, what is the expression for the bias? Explain your answers. 3 b. Repeat part a if it is known that X2 and X3 are uncorrelated in the population, but not necessarily uncorrelated in the sample. c. Repeat part a if it known that X2 and X3 are uncorrelated in both the population and the sample. 4. Assume the Capital Asset Pricing Model (CAPM) states that the return on a given company's stock (Y) is a linear function of the average market return on all stocks (X2), while Arbitrage Pricing Theory (APT) claims some other variable (X3) is also important as a determinant of Y. Someone claims to test APT by obtaining a time series of data on the three variables for some company, and then performing the following two-step estimation: Step I: Regress X2=a1+a2*X3+w Step II: Regress Y=B1+B2*w+B3*X3+u and then conclude that APT is better than CAPM if B3 is statistically significant in the Step II regression. a. Do you think this is a good way to test APT? Explain. b. If instead you estimated Y=b1+b2*X2+b3*X3+v, how would b1 compare with B1? How would b2 compare with B2? How would the standard errors of these estimators compare? Explain. Part III. Answer one of the following two questions. 5. How would you estimate the parameters of the following models? Are there any that are not linear in the parameters? What would these functions look like if they were graphed in X-Y space? (Assume all observations on X, Y; and all parameters, are positive). a. XY = B1+B2*X+u b. Y = 1/[1+exp(B1+B2*X+u)] c. Y = exp[B1-B2*(1/X)+u] d. Y = B1+exp(B2*X2+u) e. Y2 = B1+B2*X+u 6. Answer all of the (unrelated) parts of this question. a. If X1, X2 and X3 are mutually uncorrelated and have the same standard deviation, show that the correlation between (X1+X2) and (X2+X3) equals one-half. b. Consider the model: (1/Y)=B1+B2*(1/X)+u. Assume X and Y are restricted to positive values. What does this model look like in Y-X space? How would you estimate the parameters? Can you think of any example where this model might be appropriate? Explain. c. In a simple regression situation, explain how a Chow test could be used to test for nonlinearity. How else could you test for this? d. In the population model: Y = B1*X2B2*X3B3*u, if estimation is done by converting to logarithms, what are the CLR assumptions?