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| m67359 | Based on the data for the years 1962 to 1977 for the United States, Dale Bails and Larry Peppers obtained the following demand function for automobiles:
Ŷt = 5807 + 3.24Xt r2 = 0.22
se = (1.634)
where Y = retail sales of passenger cars (thousands) and X = the real disposable income (billions of 1972 dollars). These for b1 is not given.
a. Establish a 95% confidence interval for B2.
b. Test the hypothesis that this interval includes B2 = 0. If not, would you accept this null hypothesis?
c. Compute the t value under H0:B2 = 0. Is it statistically significant at the 5 percent level? Which t test do you use, one-tailed or two-tailed, and why? |
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| m67360 | Based on the following data, estimate the model:
(1/Yi) = B1 + B2Xi + ui
a. What is the interpretation of B2?
b. What is the rate of change of y with respect to X?
c. What is the elasticity of y with respect to X?
d. For the same data, run the regression
Yi = B1 + B2 (1/Xi) + ui
e. Can you compare the r2s of the two models? Why or why not?
f. How do you decide which is a better model? |
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| m67361 | Based on the GNP/money supply data given in Table 5-14 (found on the textbook s Web site), the following regression results were obtained (y = GNP, X = M2):
a. For each model, interpret the slope coefficient.
b. For each model, estimate the elasticity of the GNP with respect to money supply and interpret it.
c. Are all r2 values directly comparable? If not, which ones are?
d. Which model will you choose? What criteria did you consider in your choice?
e. According to the monetarists, there is a one-to-one relationship between the rate of changes in the money supply and the GDP. Do the preceding regressions support this view? How would you test this formally? |
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| m67363 | Based on the quarterly data for the U.K. for the period 1990-1Q to 1998-2Q, the following results were obtained by Asteriou and Hall. The dependent variable in these regressions is Log(IM) = logarithm of imports (t ratios in parentheses).
a. Interpret each equation.
b. In Model 1, which drops Log(PPI), the coefficient of Log(CPI) is positive and significant at about the 5% level. Does this make economic sense?
c. In Model 3, which drops Log(CPI), the coefficient of Log(PPI) is positive but statistically insignificant. Does this make economic sense?
d. Model 2 includes the logs of both price variables and their coefficients are individually statistically significant. However, the coefficient of Log(CPI) is positive and that of Log(PPI) is negative. How would you rationalize this result?
e. Do you think multicollinearity is the reason why some of these results are conflicting? Justify your answer.
f. If you were told that the correlation between PPI and CPI is 0.9819, would that suggest that there is a multicollinearity problem?
g. Of the three models given above, which would you choose and why? |
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| m67367 | Based on the U.K. data on annual percentage change in wages (Y) and the percent annual unemployment rate (X) for the years 1950 to 1966, the following regression results were obtained:
Ŷt = - 1.4282 + 8.7243(1/Xt)
se = (2.0675) (2.8478) r2= 0.3849
F(l, 15) = 9.39
a. What is the interpretation of 8.7243?
b. Test the hypothesis that the estimated slope coefficient is not different from zero. Which test will you use?
c. How would you use the F test to test the preceding hypothesis?
d. Given that = 4.8 percent and = 1.5 percent, what is the rate of change of Y at these mean values?
e. What is the elasticity of Y with respect to X at the mean values?
f. How would you test the hypothesis that the true r2 = 0? |
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| m67368 | Based on the U.S. data for 1965-IQ to 1983-IVQ (n = 76), James Doti and Esmael Adibi25 obtained the following regression to explain personal consumption expenditure (PCE) in the United States.
Ŷt = - 10.96 + 0.93X2t - 2.09X3t
t = (- 3.33) (249.06) (- 3.09)
R2 = 0.9996
F = 83,753.7
where Y = the PCE ($, in billions)
X2 = the disposable (i.e., after-tax) income ($, in billions)
X3 = the prime rate (%) charged by banks
a. What is the marginal propensity to consume (MPC)-the amount of additional consumption expenditure from an additional dollar s personal disposable income?
b. Is the MPC statistically different from 1? Show the appropriate testing procedure.
c. What is the rationale for the inclusion of the prime rate variable in the model? A priori, would you expect a negative sign for this variable?
d. Is b3 significantly different from zero?
e. Test the hypothesis that R2 = 0.
f. Compute the standard error of each coefficient. |
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| m67459 | By the conditional probability formula, we have
1. P(A | B) = P(AB) / P(B) and
2. P(B | A) = P(AB) / P(A) → P(AB) = P(B | A) P(A)
where → means "implies." If you substitute for P(AB) from the right-hand side of (2) into the numerator of (1), what do you get? How do you interpret this result? |
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| m67485 | Can you rewrite Eq. (2.22) to express X as a function of Y? How would you interpret the converted equation? |
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| m67519 | Check that all R2 values in Table 8-4 are statistically significant. |
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| m67553 | Comparing two r2s when dependent variables are different. Suppose you want to compare the r2 values of the growth model (5.19) with the linear trend model (5.23) of the consumer credit outstanding regressions given in the text. Proceed as follows:
a. Obtain In Yt, that is, the estimated log value of each observation from model (5.19).
b. Obtain the antilog values of the values obtained in (a).
c. Compute r2 between the values obtained in (b) and the actual y values using the definition of r2 given in Question 3.5.
d. This r2 value is comparable with the r2 value obtained from linear model (5.23).
Use the preceding steps to compare the r2 values of models (5.19) and (5.23). |
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| m67563 | Complete the following sentences:
a. In the double-log model the slope coefficient measures ...
b. In the lin-log model the slope coefficient measures ...
c. In the log-lin model the slope coefficient measures ...
d. Elasticity of Y with respect to X is defined as ...
e. Price elasticity is defined as ...
f. Demand is said to be elastic if the absolute value of the price elasticity is ..., but demand is said to be inelastic if it is ... |
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| m67567 | Complete the following table: |
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| m67584 | Compute the expected value and variance for the PDF given in Problem A.19.
Problem A.19
The PDF of a continuous random variable X is as follows:
f(X) = c(4x - 2x2) 0 ≤ x ≤ 2
= 0 otherwise |
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| m67588 | Compute the p values in the following cases:
a. t ≥ 1.72, d.f. = 24
b. Z ≥ 2.9
c. F ≥ 2.59, d.f. = 3 and 20, respectively
d. x2 ≥ = 19, d.f. = 30
If you cannot get an exact answer from the various probability distribution tables, try to obtain them from a program such as MINITAB or Excel. |
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| m67610 | Consider a random variable (r.v.) X ~ N (8, 16). State whether the following statements are true or false:
a. The probability of obtaining an X value of greater than 12 is about 0.16.
b. The probability of obtaining an X value between 12 and 14 is about 0.09.
c. The probability that an X value is more than 2.5 standard deviations from the mean value is 0.0062. |
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| m67614 | Consider an r.v. that follows the t distribution.
a. For 20 degrees of freedom (d.f.), what is the probability that the t value will be greater than 1.325?
b. What is the probability that the t value in C.5 (a) will be less than - 1.325?
c. What is the probability that a t value will be greater than or less than 1.325?
d. Is there a difference between the statement in C.5(c) and the statement, "What is the probability that the absolute value of t, | t|, will be greater than 1.325?" |
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| m67616 | Consider data on the weekly stock prices of Qualcomm, Inc., a digital wireless telecommunications designer and manufacturer, over the time period of 1995 to 2000. The complete data can be found in Table 5-16 on the textbook s Web site.
a. Create a scatter gram of the closing stock price over time. What kind of pattern is evident in the plot?
b. Estimate a linear model to predict the closing stock price based on time. Does this model seem to fit the data well?
c. Now estimate a squared model by using both time and time-squared. Is this a better fit than in part (b)?
d. Now attempt to fit a cubic or third-degree polynomial to the data as follows:
Yi = B0 + B1Xi + B2X2i + B3X3 + ui
where Y = stock price and X = time. Which model seems to be the best estimator for the stock prices? |
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| m67618 | Consider Figure 9-10, which plots the gross domestic product (GDP) growth, in percent, against the ratio of investment/GDP, in percent, for several countries for 1974 to 1985.28 The various countries are divided into three groups- those that experienced positive real (i.e., inflation-adjusted) interest rates, those that experienced moderately negative real interest rates, and those that experienced strongly negative interest rates.
a. Develop a suitable model to explain the percent GDP growth rate in relation to percent investment/GDP rate.
b. From Figure 9-10, do you see any evidence of heteroscedasticity in the data? How would you test its presence formally?
c. If heteroscedasticity is suspected, how would you transform your regression to eliminate it?
d. Suppose you were to extend your model to take into account the qualitative differences in the three groups of countries by representing them with
FIGURE 9-10
Real interest rates, investment, productivity, and growth in 33 developing countries from 1974 to 1985
dummy variables. Write the equation for this model. If you had the data and could estimate this expanded model, would you expect heteroscedasticity in the extended model? Why or why not? |
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| m67620 | Consider formulas (B.32) and (B.33). Let X stand for the rate of return on a security, say, IBM, and Y the rate of return on another security, say, General Foods. Let s2X = 16, s2Y = 9, and r = - 0.8. What is the variance of (X + Y) in this case? Is it greater than or smaller than var (X) + var (Y)? In this instance, is it better to invest equally in the two securities (i.e., diversify) than in either security exclusively? This problem is the essence of the portfolio theory of finance. |
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| m67625 | Consider the data given in Table 10-7 (on the textbook s Web site) relating to stock prices and GDP for the period 1980-2006.
a. Estimate the OLS regression
Yt = B1 + B2Xt + ut
b. Find out if there is first-order autocorrelation in the data on the basis of the d statistic.
c. If there is, use the d value to estimate the autocorrelation parameter ρ.
d. Using this estimate of p, transform the data per the generalized difference equation (10.14), and estimate this equation by OLS (1) by dropping the first observation and (2) by including the first observation.
e. Repeat part (d), but estimate ρ from the residuals as shown in Eq. (10.20). Using this estimate of p, estimate the generalized difference equation (10.14).
f. Use the first difference method to transform the model into Eq. (10.17) and estimate the transformed model.
g. Compare the results of regressions obtained in parts (d), (e), and (f). What conclusions can you draw? Is there autocorrelation in the transformed regressions? How do you know? |
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