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| m58696 | Consider estimation of the following two equation model: y1 = β1 + ε1, y2 = β2x + ε2. A sample of 50 observations produces the following moment matrix:
a. Write the explicit formula for the GLS estimator of [β1, β2].What is the asymptotic covariance matrix of the estimator?
b. Derive the OLS estimator and its sampling variance in this model.
c. Obtain the OLS estimates of β1 and β2, and estimate the sampling covariance matrix of the two estimates. Use n instead of (n − 1) as the divisor to compute the estimates of the disturbance variances
d. Compute the FGLS estimates of β1 and β2 and the estimated sampling covariance matrix.
e. Test the hypothesis that β2 = 1. |
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| m58698 | Consider GMM estimation of a regression model as shown at the beginning of Example 18.8. Let W1 be the optimal weighting matrix based on the moment equations. Let W2 be some other positive definite matrix. Compare the asymptotic covariance matrices of the two proposed estimators. Show conclusively that the asymptotic covariance matrix of the estimator based on W1 is not larger than that based on W2. |
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| m58700 | Consider, sampling from a multivariate normal distribution with mean vector μ = (μ1, μ2, . . . , μM) and covariance matrix σ2I. The log-likelihood function is Show that the maximum likelihood estimates of the parameters are Derive the second derivatives matrix and show that the asymptotic covariance matrix for the maximum likelihood estimators is Suppose that we wished to test the hypothesis that the means of the Mdistributions were all equal to a particular value μ0. Show that the Wald statistic would be where ¯y is the vector of sample means.br> |
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| m58706 | Consider the following two-equation model: y1 = γ1y2 + β11x1 + β21x2 + β31x3 + ε1, y2 = γ2y1 + β12x1 + β22x2 + β32x3 + ε2.
a. Verify that, as stated, neither equation is identified.
b. Establish whether or not the following restrictions are sufficient to identify (or partially identify) the model:
(1) β21 = β32 = 0,
(2) β12 = β22 = 0,
(3) γ1 = 0,
(4) γ1 = γ2 and β32 = 0,
(5) σ12 = 0 and β31 = 0,
(6) γ1 = 0 and σ12 = 0,
(7) β21 + β22 = 1,
(8) σ12 = 0, β21 = β22 = β31 = β32 = 0,
(9) σ12 = 0, β11 = β21 = β22 = β31 = β32 = 0. |
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| m58707 | Consider the multiple regression of y on K variablesXand an additional variable z. Prove that under the assumptions A1 through A6 of the classical regression model, the true variance of the least squares estimator of the slopes on X is larger when z is included in the regression than when it is not. Does the same hold for the sample estimate of this covariance matrix? Why or why not? Assume that X and z are nonstochastic and that the coefficient on z is nonzero. |
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| m58708 | Consider the probit model analyzed in Section 17.8. The model states that for given vector of independent variables, Prob[yi = 1 | xi ] = Ф [x iβ], Prob [yi = 0 | xi ] = 1 − Prob[yi = 1 | xi]. We have consideredmaximumlikelihood estimation of the parameters of this model at several points. Consider, instead, a GMM estimator based on the result that this suggests that we might base estimation on the orthogonality conditions construct a GMM estimator based on these results. Note that this is not the nonlinear least squares estimator. Explain—what would the orthogonality conditions be for nonlinear least squares estimation of this model? |
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| m58712 | Consider the simple regression yt = βxt + ε1 where E p[ε | x] = 0 and E [ε2 | x ] = σ2
(a) What is the minimum mean squared error linear estimator of β? Choose e to minimize Var [β] + [E(β – β)]2. The answer is a function of the unknown parameters].
(b) For the estimator in part a, show that ratio of the mean squared error of β to that of the ordinary least squares estimator b is Note that τ is the square of the population analog to the “t ratio” for testing the hypothesis that β = 0, which is given in (4-14). How do you interpret the behavior of this ratio as τ -> ∞? |
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| m58714 | Consider the system y1 = α1 + βx + ε1, y2 = α2 + ε2. The disturbances are freely correlated. Prove that GLS applied to the system leads to the OLS estimates of α1 and α2 but to a mixture of the least squares slopes in the regressions of y1 and y2 on x as the estimator of β. What is the mixture? To simplify the algebra, assume (with no loss of generality) that x = 0. |
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| m58715 | Consider the two-equation system y1 = β1x1 + ε1, y2 = β2x2 + β3x3 + ε2. Assume that the disturbance variances and covariance are known. Now suppose that the analyst of this model applies GLS but erroneously omits x3 from the second equation. What effect does this specification error have on the consistency of the estimator of β1? |
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| m58718 | Construct the Lagrange multiplier statistic for testing the hypothesis that all the slopes (but not the constant term) equal zero in the binomial log it model. Prove that the Lagrange multiplier statistic is nR2 in the regression of (yi = p) on the xs, where P is the sample proportion of 1s. |
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| m58720 | Continuing the analysis of Section 14.3.2, we find that a trans log cost function for one output and three factor inputs that does not impose constant returns to scale is
ln C = α + β1 ln p1 + β2 ln p2 + β3 ln p3 + δ11½ ln2 p1 + δ12 ln p1 ln p2 + δ13 ln p1 ln p3 + δ22 ½ ln2 p2 + δ23 ln p2 ln p3 + δ33 ½ ln2 p3 +γy1 ln Y ln p1 + γy2 ln Y ln p2 + γy3 ln Y ln p3 + βy ln Y + βyy½ ln2 Y + εc. The factor share equations are
S1 = β1 + δ11 ln p1 + δ12 ln p2 + δ13 ln p3 + γy1 ln Y + ε1,
S2 = β2 + δ12 ln p1 + δ22 ln p2 + δ23 ln p3 + γy2 ln Y + ε2,
S3 = β3 + δ13 ln p1 + δ23 ln p2 + δ33 ln p3 + γy3 ln Y + ε3.
a. The three factor shares must add identically to 1. What restrictions does this requirement place on the model parameters?
b. Show that the adding-up condition in (14-39) can be imposed directly on the model by specifying the Trans log model in (C/p3), (p1/p3), and (p2/p3) and dropping the third share equation. Notice that this reduces the number of free parameters in the model to 10.
c. Continuing Part b, the model as specified with the symmetry and equality restrictions has 15 parameters. By imposing the constraints, you reduce this number to 10 in the estimating equations. How would you obtain estimates of the parameters not estimated directly? The remaining parts of this exercise will r |
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| m58721 | Continuing to use the data in Exercise 1, consider once again only the nonzero observations. Suppose that the sampling mechanism is as follows: y* and another normally distributed random variable z has population correlation 0.7. The two variables, y* and z, are sampled jointly. When z is greater than zero, y is reported. When zis less than zero, both zand y are discarded. Exactly 35 draws were required to obtain the preceding sample. Estimate μ and σ. |
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| m58740 | Data for fitting an improved Phillips curve model can be obtained from many sources, including the Bureau of Economic Analysis’s (BEA) own website, Economagic. Com and so on, obtain the necessary data and expand the model of example 12.3, does adding additional explanatory variables to the model reduce the extreme pattern of the OLS residuals that appears in Figure 12.3? |
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| m58741 | Data on t = strike duration and x = unanticipated industrial production for a number of strikes in each of 9 years are given in Appendix Table F22.1. Use the Poisson regression model discussed in Section 21.9 to determine whether x is a significant determinant of the number of strikes in a given year. |
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| m58742 | Data on U.S. gasoline consumption for the years 1960 to 1995 are given in Table F2.2.
a. Compute the multiple regression of per capita consumption of gasoline, G/pop, on all the other explanatory variables, including the time trend, and report all results. Do the signs of the estimates agree with your expectations?
b. Test the hypothesis that at least in regard to demand for gasoline consumers do not differentiate between changes in the prices of new and used cars.
c. Estimate the own price elasticity of demand, the income elasticity, and the crossprice elasticity with respect to changes in the price of public transportation.
d. Reestimate the regression in logarithms so that the coefficients are direct estimates of the elasticities. (Do not use the log of the time trend.) How do your estimates compare with the results in the previous question? Which specification do you prefer?
e. Notice that the price indices for the automobile market are normalized to 1967, whereas the aggregate price indices are anchored at 1982. Does this discrepancy affect the results? How? If you were to renormalize the indices so that they were all 1.000 in 1982, then how would your results change? |
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| m58763 | Deleting an observation a common strategy for handling a case in which an observation is missing data for one or more variables is to fill those missing variables with 0s and add a variable to the model that takes the value 1 for that one observation and 0 for all other observations. Show that this ‘strategy’ is equivalent to discarding the observation as regards the computation of b but it does have an effect on R2. Consider the special case in which X contains only a constant and one variable. Show that replacing missing values of x with the mean of the complete observations has the same effect as adding the new variable. |
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| m58765 | Demand system estimation. Let Y denote total expenditure on consumer durables, nondurables, and services and Ed, En, and Es are the expenditures on the three categories. As defined, Y = Ed + En + Es. Now, consider the expenditure system Prove that if all equations are estimated by ordinary least squares, then the sum of the expenditure coefficients will be 1 and the four other column sums in the preceding model will bezero. |
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| m58768 | Derive the disturbance covariance matrix for the model What parameter is estimated by the regression of the OLS residuals on their lagged values? |
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| m58769 | Derive the log-likelihood function, first-order conditions for maximization, and information matrix for the model yi = x iβ + εi, εi ~ N [0, σ2(γ zi )2]. |
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| m58770 | Derive the log-likelihood function for the model in (13-18), assuming that εit and ui are normally distributed. |
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