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free/or 0.5$ |
| m67641 | Consider the following joint PDF:
f(x, y) = 12 / 5 x(2 - x - y); 0 < x < 1; 0 < y < 1
= 0 otherwise
a. Find P(x > 0.5) and P(y < 0.5)
b. What is the conditional density of X given that Y = y, where 0 < y < 1? |
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| m67648 | Consider the following model:
Rt = A1 + A2Mt + A3Yt + u1t
Yt = B1 + B2Rt + u2t
where Y = income (measured by gross domestic product, GDP), R = interest rate (measured by 6-month Treasury bill rate, %), and M = money supply (measured by Ml). Assume that M is determined exogenously.
a. What economic rationale lies behind this model? (Hint: See any macroeconomics textbook.)
b. Are the preceding equations identified?
c. Using the data given in Table 11-2 (on the textbook s Web site), estimate the parameters of the identified equation(s). Justify the method(s) you use. |
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| m67649 | Consider the following model:
Y1t= A1 + A2Y2t + A3X1t + u1t
Y2t = B1 + B2Y1t + u2t
where the Y s are the endogenous variables, the X s the exogenous, and the u s the stochastic error terms. Based on this model, the following reduced form regressions are obtained
Y1t = 6 + 8X1t
Y2t = 4 + 12X1t
a. Which structural coefficients, if any, can be estimated from these reduced form equations?
b. How will our answer change if it is known a priori that 1. A2 = 0 and 2. A1 = 0? |
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| m67650 | Consider the following model:
Yi = B0 + B1Xi + B2D2i + B3D3 + ui
where Y = annual earnings of MBA graduates
X = years of service
D2 = 1 if Harvard MBA
= 0 if otherwise
D3 = l if Wharton MBA
= 0 if otherwise
a. What are the expected signs of the various coefficients?
b. How would you interpret B2 and B3?
c. If B2 > B3, what conclusion would you draw? |
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| m67651 | Consider the following model:
Yi = B1 + B2D2i + B3D3i + B4 (D2i D3i) + B5Xi + ui
where Y = the annual salary of a college teacher
X = years of teaching experience
D2 = 1 if male
= 0 if otherwise
D3 = 1 if white
= 0 if otherwise
a. The term (D2iD3i) represents the interaction effect. What does this expression mean?
b. What is the meaning of B4?
c. Find E(Yi | D2 = 1, D3 = 1, Xi) and interpret it. |
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| m67652 | Consider the following model:
Yt = B1 + B2X2t B3X3t + B4X4t + ut
Suppose the error term follows the AR(1) scheme in Eq. (10.6). How would you transform this model so that there is no autocorrelation in the trans-formed model? |
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| m67653 | Consider the following model:
Yt = B1 + B2Xt + B3Xt - 1 + B4Xt - 2 + B3Xt - 3 + ut
where Y = the consumption
X = the income
t = the time
This model states that consumption expenditure at time Ms a linear function of income not only at time t but also of income in three previous time periods. Such models are called distributed lag models and represent what are called dynamic models (i.e., models involving change over time).
a. Would you expect multicollinearity in such models and why?
b. If multicollinearity is suspected, how would you get rid of it? |
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| m67654 | Consider the following models:
Model I: Consumption i = B1 + B2 income i + ui
Model II: Consumption i = A1 + A2 wealth i + vi
a. How would you decide which of the models is the "true" model?
b. Suppose you regress consumption on both income and wealth. How would this help you decide between the two models? Show the necessary details. |
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| m67666 | Consider the following reformulation of the model given in Problem 11.18.
Rt = A1 + A2Mt + A3Yt + u1t
Yt = B1 + B2Rt + B31t + u2t
where in addition to the variables defined in the preceding problem, I stands for investment (measured by gross private domestic investment, GPDI). Assume that M and J are exogenous.
a. Which of the preceding equations is identified?
b. Using the data in Table 11-2 (on the textbook s Web site), estimate the parameters of the identified equation(s).
c. Comment on the difference in the results of this and the preceding problem. |
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| m67669 | Consider the following regression:
Ŷi = - 66.1058 + 0.0650 Xi r2 = 0.9460
se = (10.7509) ( ) n = 20
t = ( ) (18.73)
Fill in the missing numbers. Would you reject the hypothesis that true B2 is zero at a = 5%? Tell whether you are using a one-tailed or two-tailed test and why. |
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| m67670 | Consider the following regression model:
Ŷt = - 49.4664 + 0.88544X2t + 0.09253X3t; R2 = 0.9979; d = 0.8755
t = (- 2.2392) (70.2936) (2.6933)
where
Y = the personal consumption expenditure (1982 billions of dollars)
X2 = the personal disposable income (1982 billions of dollars) (PDI)
X3 = the Dow Jones Industrial Average Stock Index
The regression is based on U.S. data from 1961 to 1985.
a. Is there first-order autocorrelation in the residuals of this regression? How do you know?
b. Using the Durbin two-step procedure, the preceding regression was trans-formed per Eq. (10.15), yielding the following results:
Yt* = - 17.97 + 0.89X*2t + 0.09X*3t; R2 = 0.9816; d = 2.28
t = (30.72) (2.66)
Has the problem of autocorrelation been resolved? How do you know?
c. Comparing the original and transformed regressions, the t value of the PDI has dropped dramatically. What does this suggest?
d. Is the d value from the transformed regression of any value in determining the presence, or lack thereof, of autocorrelation in the transformed data? |
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| m67671 | Consider the following regression results:
Ŷt= - 0.17 + 5.26Xt 2 = 0.10, Durbin-Watson = 2.01
t = (- 1.73) (2.71)
where Y = the real return on the stock price index from January of the current year to January of the following year
X = the total dividends in the preceding year divided by the stock price index for July of the preceding year
t = time
On Durbin-Watson statistic, see Chapter 10. The time period covered by the study was 1926 to 1982. 2 stands for the adjusted coefficient of determination. The Durbin-Watson value is a measure of autocorrelation. Both measures are explained in subsequent chapters.
a. How would you interpret the preceding regression?
b. If the previous results are acceptable to you, does that mean the best investment strategy is to invest in the stock market when the dividend/price ratio is high? |
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| m67676 | Consider the following set of hypothetical data:
Suppose you want to do a multiple regression of Y on X2 and X3.
a. Can you estimate the parameters of this model? Why or why not?
b. If not, which parameter or combination of parameters can you estimate? |
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| m67679 | Consider the following two regressions based on the U.S. data for 1946 to 1975.26 (Standard errors are in parentheses.)
where C = aggregate private consumption expenditure
GNP = gross national product
D = national defense expenditure
t = time
The objective of Hanushek and Jackson s study was to find out the effect of defense expenditure on other expenditures in the economy.
a. What might be the reason(s) for transforming the first equation into the second equation?
b. If the objective of the transformation was to remove or reduce heteroscedasticity, what assumption has been made about the error variance?
c. If there was heteroscedasticity, have the authors succeeded in removing it? How can you tell?
d. Does the transformed regression have to be run through the origin? Why or why not?
e. Can you compare the R2 values of the two regressions? Why or why not? |
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| m67681 | Consider the following two-equation model:
Y1t = A1 + A2Y2t + A3X1t + ult
Y2t = B1 + B2Y1t + B3X2t + u2t
where the Y s are the endogenous variables, the X s the exogenous variables, and the u s the stochastic error terms.
a. Obtain the reduced form regressions.
b. Determine which of the equations is identified.
c. For the identified equation, which method of estimation would you use and why?
d. Suppose, a priori, it is known that A3 = 0. How would your answers to the preceding questions change? Why? |
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| m67689 | Consider the model
Yi = B1 + B2Xi + B3X2i + B4X3i + ui
where Y = the total cost of production and X = the output. Since X2 and X3 are functions of X, there is perfect collinearity. Do you agree? Why or why not? |
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| m67731 | Continue with Problem 3.12, but suppose we now regress X on Y.
a. Present the result of this regression and comment.
b. If you multiply the slope coefficients in the two regressions, what do you obtain? Is this result surprising to you?
c. The regression in Problem 3.12 may be called the direct regression. When would a reverse regression be appropriate?
d. Suppose the r2 value between X and Y is 1. Does it then make any difference if we regress Y on X or X on Y? |
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| m67732 | Continue with Problem 6.20. What would the regression equation be if you were to include interaction dummies for the three qualitative variables in the model? |
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| m67733 | Continue with Problem 8.14. Is there multicollinearity in the previous problem? How do you know?
Problem 8.14
You are given the annual data in Table 8-5 for the United States for the period 1971 to 1986. Consider the following aggregate demand function for passenger cars:
InYi = B1 + B2InX2t + B3InX3t + B4InX4t + B5InX5t + B6InX6t + ut
where In = the natural log
DEMAND FOR NEW PASSENGER CARS IN
THE UNITED STATES, 1971 TO 1986 |
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| m67734 | Continue with Problem C.14, but now assume that X and y are positively correlated with a correlation coefficient of 0.6.
Problem C.14
If X ~ N(10, 3) and Y ~ N(15, 8), and if X and Y are independent, what is the probability distribution of
a. X + Y
b. X - Y
c. 3X
d. 4X + 5y |
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