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| m60784 | The data used to fit the expectations augmented Phillips curve in Example 12.3 are given in Table F5.1. Using these data, reestimate the model given in the example. Carry out a formal test for first order autocorrelation using the LM statistic. Then, reestimate the model using an AR(1) model for the disturbance process. Since the sample is large, the Prais–Winsten and Cochrane–Orcutt estimators should give essentially the same answer. Do they? After fitting the model, obtain the transformed residuals and examine them for first order autocorrelation. Does the AR(1) model appear to have adequately “fixed” the problem? |
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| m60803 | The expression for the restricted coefficient vector in (6-14) may be written in the form b* = [I − CR] b + w, where w does not involve b. What is C? Show that the covariance matrix of the restricted least squares estimator is σ2 (X X)−1 − σ2(X X)−1 R [R(X X)−1R ]−1 R(X X)−1 and that this matrix may be written as Var[b |X]{[Var(b |X)]−1 − R [Var(Rb) |X]−1R}Var[b |X]. |
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| m60836 | The following 20 observations are drawn from a censored normal distribution:
The applicable model is y*i = μ + εi, y + i = y*i if μ + εi > 0, 0 otherwise, εi ~ N [0, σ2]. Exercises 1 through 4 in this section are based on the preceding information. The OLS estimator of μ in the context of this To bit model is simply the sample mean. Compute the mean of all 20 observations. Would you expect this estimator to over or underestimate μ if we consider only the nonzero observations, then the truncated regression model applies. The sample mean of the non limit observations is the least squares estimator in this context. Compute it and then comment on whether this sample mean should be an overestimate or an underestimate of the true mean. |
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| m60837 | You might find it useful to read the early sections of Chapter 21 for this exercise.) The extramarital affairs data analyzed in Section 22.3.7 can be reinterpreted in the context of a binary choice model. The dependent variable in the analysis is a count of events. Using these data, first recode the dependent variable 0 for none and 1 for more than zero.Now, first using the binary probit estimator, fit a binary choice model using the same independent variables as in the example discussed in Section 22.3.7. Then using a semiparametric or nonparametric estimator, estimate the same binary choice model. A model for binary choice can be fit for at least two purposes, for estimation of interesting coefficients or for prediction of the dependent variable. Use your estimated models for these two purposes and compare the two models. |
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| m60838 | The following hypothetical data give the participation rates in a particular type of recycling program and the number of trucks purchased for collection by 10 towns in a small mid-Atlantic state:
The town of Eleven is contemplating initiating a recycling program but wishes to achieve a 95 percent rate of participation. Using a probit model for your analysis,
a. How many trucks would the town expect to have to purchase in order to achieve their goal?
b. If trucks cost $20,000 each, then is a goal of 90 percent reachable within a budget of $6.5 million? (That is, should they expect to reach the goal?)
c. According to your model, what is the marginal value of the 301st truck in terms of the increase in the percentageparticipation? |
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| m60839 | The following is a panel of data on investment (y) and profit (x) for n = 3 firms over T = 10 periods.
a. Pool the data and compute the least squares regression coefficients of the model yi t = α + βxit + εit.
b. Estimate the fixed effects model of (13-2), and then test the hypothesis that the constant term is the same for all three firms.
c. Estimate the random effects model of (13-18), and then carry out the Lagrange multiplier test of the hypothesis that the classical model without the common effect applies.
d. Carry out Hausman’s specification test for the random versus the fixed effect model. |
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| m60840 | The following model is specified: y1 = γ1y2 + β11x1 + ε1, y2 = γ2y1 + β22x2 + β32x3 + ε2. All variables are measured as deviations from their means. The sample of 25 observations produces the following matrix of sums of squares and cross products:
a. Estimate the two equations by OLS.
b. Estimate the parameters of the two equations by 2SLS. Also estimate the asymptotic covariance matrix of the 2SLS estimates.
c. Obtain the LIML estimates of the parameters of the first equation.
d. Estimate the two equations by 3SLS.
e. Estimate the reduced-form coefficient matrix by OLS and indirectly by using your structural estimates from Partb. |
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| m60841 | The following regression is obtained by ordinary least squares, using 21 observations. (Estimated asymptotic standard errors are shown in parentheses.) yt = 1.3 + 0.97yt−1 + 2.31xt , D − W = 1.21. (0.3) (0.18) (1.04) Test for the presence of autocorrelation in the disturbances. |
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| m60842 | The following sample moments for x = [1, x1, x2, x3] were computed from 100 observations produced using a random number generator: The true model underlying these data is y = x1 + x2 + x3 + ε.
a. Compute the simple correlations among the regressors.
b. Compute the ordinary least squares coefficients in the regression of y on a constant x1, x2, and x3.
c. Compute the ordinary least squares coefficients in the regression of y on a constant x1 and x2, on a constant x1 and x3, and on a constant x2 and x3.
d. Compute the variance inflation factor associated with each variable.
e. The regressors are obviously collinear. Which is the problem variable? |
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| m60843 | The following table presents a hypothetical panel of data:
a. Estimate the group wise heteroscedastic model of Section 11.7.2. Include an estimate of the asymptotic variance of the slope estimator. Use a two-step procedure, basing the FGLS estimator at the second step on residuals from the pooled least squares regression.
b. Carry out the Wald, Lagrange multiplier, and likelihood ratio tests of the hypothesis that the variances are all equal. For the likelihood ratio test, use the FGLS estimates.
c. Carry out a Lagrange multiplier test of the hypothesis that the disturbances are uncorrelated acrossindividuals. |
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| m60849 | The full model of Example 2.3 may be written in logarithmic terms as lnG/pop = α + βp ln Pg + βy lnY + γnc ln Pnc + γuc ln Puc + γpt ln Ppt + β year + δd ln Pd + δn ln Pn + δs ln Ps + ε. Consider the hypothesis that the microelasticities are a constant proportion of the elasticity with respect to their corresponding aggregate. Thus, for some positive θ (presumably between 0 and 1), γnc = θδd, γuc = θδd, γpt = θδs. The first two imply the simple linear restriction γnc = γuc. By taking ratios, the first (or second) and third imply the nonlinear restriction
a. Describe in detail how you would test the validity of the restriction.
b. Using the gasoline market data in Table F2.2 , test the restrictions separately and jointly. |
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| m60906 | The J test in Example 8.2 is carried out using over 50 years of data. It is optimistic to hope that the underlying structure of the economy did not change in 50 years. Does the result of the test carried out in Example 8.2 persist if it is based on data only from 1980 to 2000? Repeat the computation with this subset of the data. |
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| m60907 | The Lagrange multiplier test of the hypothesis Rβ − q = 0 is equivalent to aWald test of the hypothesis that λ = 0, where λ is defined in (6-14). Prove that χ2 = λ {Est.Var[λ]} −1 λ = (n − K) [e *e*/e /e – 1]. Note that the fraction in brackets is the ratio of two estimators of σ2. By virtue of (6-19) and the preceding discussion, we know that this ratio is greater than 1. Finally, prove that the Lagrange multiplier statistic is equivalent to JF, where J is the number of restrictions being tested and F is the conventional F statistic given in (6-6). |
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| m60930 | The model satisfies the Group Wise heteroscedastic regression model of Section 11.7.2. All variables have zero means. The following sample second-moment matrix is obtained from a sample of 20 observations:
a. Compute the two separate OLS estimates of β, their sampling variances the estimates of σ21 and σ22, and the R2’s in the two regressions.
b. Carry out the Lagrange multiplier test of the hypothesis that σ21 = σ22.
c. Compute the two-step FGLS estimate of β and an estimate of its sampling variance. Test the hypothesis that β equals 1.
d. Carry out the Wald test of equal disturbance variances.
e. Compute the maximum likelihood estimates of β, σ21, and σ22 by iterating the FGLS estimates to convergence.
f. Carry out a likelihood ratio test of equal disturbance variances.
g. Compute the two-step FGLS estimate of β, assuming that the model in (14-7) applies. (That is, allow for cross-sectional correlation.) Compare your results with those of part c. |
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| m60931 | The model y1 = β1x1 + ε1, y2 = β2x2 + ε2 satisfies all the assumptions of the classical multivariate regression model. All variables have zero means. The following sample second-moment matrix is obtained from a sample of 20 observations:
a. Compute the FGLS estimates of β1 and β2.
b. Test the hypothesis that β1 = β2.
c. Compute the maximum likelihood estimates of the model parameters.
d. Use the likelihood ratio test to test the hypothesis in part b. |
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| m60964 | The regression model to be analyzed is y = X1β1 + X2β2 + ε, where X1 and X2 have K1 and K2 columns, respectively. The restriction is β2 = 0. a. Using (6-14), prove that the restricted estimator is simply [b1*, 0], where b1* is the least squares coefficient vector in the regression of y on X1. b. Prove that if the restriction is β2 = β02 for a nonzero β02, then the restricted estimator of β1 is b1* = (X 1X1) −1 X 1 (y − X2β02). |
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| m61002 | The Two Variable Regression for the regression model y = α + β x + ε
(a) Show that the least squares normal equations imply Σiei = 0 and Σi xi ei = 0. (b) Show that the solution for the constant term is a = y – bx.
(c) Show that the solution for b is b = [Σni = 1(xi – x) (yi – y)]/[Σni=1(xi – x)2].
(d) Prove that these two values uniquely minimize the sum of squares by showing that the diagonal elements of the second derivatives matrix of the sum of squares with respect to the parameters are both positive and that the determinant is 4n[Σni=1 xi2) nx2] = 4n[Σni=1 (xi – x)2], which is positive unless all values of x are the same. |
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| m61025 | (This exercise requires appropriate computer software. The computations required can be done with RATS, EViews, Stata, TSP, LIMDEP, and a variety of other software using only preprogrammed procedures.) Quarterly data on the consumer price index for 1950.1 to 2000.4 are given in Appendix Table F5.1. Use these data to fit the model proposed by Engle and Kraft (1983). The model is πt = β0 + β1πt−1 + β2πt−2 + β3πt−3 + β4πt−4 + εt where πt = 100 ln [pt/pt−1] and pt is the price index.
a. Fit the model by ordinary least squares, then use the tests suggested in the text to see if ARCH effects appear to be present.
b. The authors fit an ARCH (8) model with declining weights, Fit this model. If the software does not allow constraints on the coefficients, you can still do this with a two-step least squares procedure, using the least squares residuals from the first step. What do you find?
c. Bollerslev (1986) recomputed this model as a GARCH (1, 1). Use the GARCH (1, 1) form and refit your model. |
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| m61031 | Three variables, N, D, and Y, all have zero means and unit variances. A fourth variable is C = N + D. In the regression of C on Y, the slope is 0.8. In the regression of C on N, the slope is 0.5. In the regression of Don Y, the slope is 0.4.What is the sum of squared residuals in the regression of C on D? There are 21 observations and all moments are computed using 1/(n − 1) as the divisor. |
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| m61035 | To continue the analysis in Question 4, consider a nonparametric regression of G/Pop on the price. Using the nonparametric estimation method in Section 16.4.2 fit the nonparametric estimator using a range of bandwidth values to explore the effect of bandwidth. |
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