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| m61036 | To extend Zellner and Revankar’s model in a fashion similar to theirs, we can use the Box–Cox transformation for the dependent variable as well. Use the method of Example 17.6 (with θ = λ) to repeat the study of the preceding two exercises. How do your results change? |
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| m61062 | Two samples of 50 observations each produce the following moment matrices. (In each case, X is a constant and one variable.)
a. Compute the least squares regression coefficients and the residual variances s2 for each data set. Compute the R2 for each regression.
b. Compute the OLS estimate of the coefficient vector assuming that the coefficients and disturbance variance are the same in the two regressions. Also compute the estimate of the asymptotic covariance matrix of the estimate.
c. Test the hypothesis that the variances in the two regressions are the same without assuming that the coefficients are the same in the two regressions.
d. Compute the two-step FGLS estimator of the coefficients in the regressions, assuming that the constant and slope are the same in both regressions. Compute the estimate of the covariance matrix and compare it with the result of partb. |
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| m61066 | A two-way fixed effects model, suppose that the fixed effects model is modified to include a time-specific dummy variable as well as an individual-specific variable. Then yit = αi + γt + β xit + εit. At every observation, the individual- and timespecific dummy variables sum to 1, so there are some redundant coefficients. The discussion in Section 13.3.3 shows that one way to remove the redundancy is to include an overall constant and drop one of the time specific and one of the timedummy variables. The model is, thus, yit = μ + (αi − α1) + (γt − γ1) + β xit + εit. (Note that the respective time- or individual-specific variable is zero when t or i equals one.) Ordinary least squares estimates of β are then obtained by regression of yit − yi −yt + y on xit − xi.−xt + x. Then (αi − α1) and (γt − γ1) are estimated using the expressions in (13-17) while m = y – b x. Using the following data, estimate the full set of coefficients for the least squares dummy variable model:
Test the hypotheses that (1) the “period” effects are all zero, (2) the “group” effects are all zero, and (3) both period and group effects are zero. Use an F test in each case. |
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| m61067 | Two-way random effects model, we modify the random effects model by the addition of a time specific disturbance. Thus, yit = α + β xit + εit + ui + vt, where
Write out the full covariance matrix for a data set with n = 2 and T = 2. |
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| m61068 | Use the data in Section 13.9.7 (the Grunfeld data) to fit the random and fixed effect models. There are five firms and 20 years of data for each. Use the F, LM, and/or Hausman statistics to determine which model, the fixed or random effects model, is preferable for these data. |
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| m61149 | Use MacKinnon, White, and Davidson’s PE test to determine whether a linear or loglinear production model is more appropriate for the data in Appendix Table F6.1. (The test is described in Section 9.4.3 and Example 9.8.) |
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| m61192 | Use the data in Section 13.9.7 (the Grunfeld data) to fit the random and fixed effect models. There are five firms and 20 years of data for each. Use the F, LM, and/or Hausman statistics to determine which model, the fixed or random effects model, is preferable for these data. |
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| m61227 | Use the Lagrange multiplier test to test the hypothesis in Exercise 1. |
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| m61281 | Using only the nonlimit observations, repeat Exercise 2 in the context of the truncated regression model, estimate μ and σ by using the method of moment’s estimator outlined in Example 22.2. Compare your results with those in the previous exercises. |
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| m61284 | Using the Box–Cox transformation, we may specify an alternative to the Cobb–Douglas model as Using Zellner and Revankar’s data in Appendix Table F9.2, estimate α, βk, βl, and λ by using the scanning method suggested in Section 9.3.2. (Do not forget to scale Y, K, and L by the number of establishments.) Use (9-16), (9-12), and (9-13) to compute the appropriate asymptotic standard errors for your estimates, compute the two output elasticities, ∂ lnY/∂ ln K and ∂ lnY/∂ ln L, at the sample means of K and L. |
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| m61286 | Using the data and model of Example 2.3, carry out a test of the hypothesis that the three aggregate price indices are not significant determinants of the demand for gasoline. |
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| m61288 | Using the data in Exercise 3, use the Oberhofer–Kmenta method to compute the maximum likelihood estimate of the common coefficient vector |
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| m61289 | Using the data of Exercise 5, reestimate the parameters using a two-step FGLS estimator. Try the estimator used in Example 11.4. |
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| m61290 | Using the gasoline market data in Appendix Table F2.2, use the partially linear regression method in Section 16.3.3 to fit an equation of the form ln(G/Pop) = β1 ln(Income) + β2 lnPnew cars + β3 ln Pused cars + g(lnPgasoline) + ε |
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| m61292 | Using the macroeconomic data in Appendix Table F5.1, estimate by least squares the parameters of the model ct = β0 + β1yt + β2ct−1 + β3ct−2 + εt, where ct is the log of real consumption and yt is the log of real disposable income.
a. Use the Breusch and Pagan test to examine the residuals for autocorrelation.
b. Is the estimated equation stable? What is the characteristic equation for the autoregressive part of this model? What are the roots of the characteristic equation, using your estimated parameters?
c. What is your implied estimate of the short-run (impact) multiplier for change in yt on ct? Compute the estimated long-run multiplier. |
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| m61293 | Using the matrices of sums of squares and cross products immediately preceding Section 3.2.3, compute the coefficients in the multiple regression of real investment on a constant, real GNP and the interest rate. Compute R2. |
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| m61297 | Using the results in Example 18.7, estimate the asymptotic covariance matrix of the method of moments estimators of P and λ based on m 1 and m 2 [Note: You will need to use the data in Example C.1 to estimate V.] |
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| m61298 | Using the results in Exercise 1, test the hypothesis that the slope on x1 is 0 by running the restricted regression and comparing the two sums of squared deviations |
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| m61313 | Verify the following differential equation, which applies to the Box–Cox transformation:
Show that the limiting sequence for λ = 0 is these results can be used to great advantage in deriving the actual second derivatives of the log-likelihood function for the Box–Cox model. |
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| m61317 | Verify the rank and order conditions for identification of the second and third behavioral equations in Klein’s Model I. |
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