Question: Describe how to obtain nonlinear least squares

> Two-way random effects model. We modify the random effects model by the addition of a time-specific disturbance. Thus, where Write out the full disturbance covariance matrix for a data set with n = 2 and T = 2.

> 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 / . At every observation, the individual- and time- specific dummy variables sum to

> What are the probability limits of (1/n) LM, where LM is defined in (11-42) under the null hypothesis that 2u = 0 and under the alternative that 2u ≠ 0?

> The National Institute of Standards and Technology (NIST) has created a Web site that contains a variety of estimation problems, with data sets, designed to test the accuracy of computer programs. (The URL is http://www.itl.nist.gov/div898/strd/.) One of

> Unbalanced design for random effects. Suppose that the random effects model of Section 11.5 is to be estimated with a panel in which the groups have different numbers of observations. Let Ti be the number of observations in group i. a. Show that the pool

> Suppose that the fixed effects model is formulated with an overall constant term and n - 1 dummy variables (dropping, say, the last one). Investigate the effect that this supposition has on the set of dummy variable coefficients and on the least squares

> 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 b. Estimate the fixed effects model of (11-11), and then test the

> Obtain the reduced form for the model in Exercise 8 under each of the assumptions made in parts a and in parts b(1) and b(9).

> Consider the following two-equation model: 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:

> For the model Assume that yi2 + yi3 = 1 at every observation. Prove that the sample covariance matrix of the least squares residuals from the three equations will be singular, thereby precluding computation of the FGLS estimator. How could you proceed i

> Consider the system The disturbances are freely correlated. Prove that GLS applied to the system leads to the OLS estimates of 1 and 2 but to a mixture of the least squares slopes in the regressions of y1 and y2 on x

> Consider the two-equation system 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 t

> Prove that in the model generalized least squares is equivalent to equation-by-equation ordinary least squares if X1 = X2. The general case is considered in Exercise 14.

> The model satisfies all the assumptions of the seemingly unrelated regressions 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 ï&c

> For the model in Application 1, test the hypothesis that = 0 using a Wald test and a Lagrange multiplier test. Note that the restricted model is the cobb–Douglas log linear model. The LM test statistic is shown in (7-22). To carry out the test, you wil

> Consider estimation of the following two-equation model: A sample of 50 observations produces the following moment matrix: a. Write the explicit formula for the GLS estimator of [1, 2]. What is the asymptotic covaria

> Prove the general result in point 2 in Section 10.2.2, if the X matrices in (10-1) are identical, then full GLS is equation-by-equation OLS. Hints: If all the Xm matrices are identical, then the inverse matrix in (10-10) is /. Also, Use these results

> Prove that an under identified equation cannot be estimated by 2SLS.

> Prove that

> For the model show that there are two restrictions on the reduced form coefficients. Describe a procedure for estimating the model while incorporating the restrictions.

> The following model is specified: 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 parame

> A sample of 100 observations produces the following sample data: The underlying seemingly unrelated regressions model is a. Compute the OLS estimate of , and estimate the sampling variance of this estimator. b. Compute the FGLS estimate

> Suppose that the regression model is yi =  + i, where a. Given a sample of observations on yi and xi, what is the most efficient estimator of m? What is its variance? b. What is the OLS estimator of , an

> Suppose that the regression model is y =  + , where has a zero mean, constant variance, and equal correlation, , across observations. Then if i ≠ j. Prove that the least squares estim

> Suppose that y has the pdf And For this model, prove that GLS and MLE are the same, even though this distribution involves the same parameters in the conditional mean function and the disturbance variance.

> 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 F7.2, estimate a, k, ï&

> In the generalized regression model, suppose that / is known. a. What is the covariance matrix of the OLS and GLS estimators of ? b. What is the covariance matrix of the OLS residual vector e = y - Xb? c. What is the covariance matrix

> In the generalized regression model, if the K columns of X are characteristic vectors of
, then ordinary least squares and generalized least squares are identical. (The result is actually a bit broader; X may be any linear combination of exactly K char

> Finally, suppose that / must be estimated, but that assumptions (9-22) and (9-23) are met by the estimator. What changes are required in the development of the previous problem?

> Now suppose that the disturbances are not normally distributed, although / is still known. Show that the limiting distribution of the previous statistic is (1/J) times a chi-squared variable with J degrees of freedom. (Hint: The denominator converges t

> This and the next two exercises are based on the test statistic usually used to test a set of J linear restrictions in the generalized regression model, where  is the GLS estimator. Show that if / is known, if the disturbances are nor

> The following table presents a hypothetical panel of data: a. Estimate the group wise heteroscedastic model of section 9.7.2. Include an estimate of the asymptotic variance of the slope estimator. Use a two-step procedure, basing the FGLS estimator at

> The model satisfies the group wise heteroscedastic regression model of section 9.7.2 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 &

> Suppose that in the group wise heteroscedasticity model of section 9.7.2, Xi is the same for all i. What is the generalized least squares estimator of ? How would you compute the estimator if it were necessary to estimate 2i?

> 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 s for each

> For the model in Exercise 9, suppose that is normally distributed, with mean zero and variance 2[1 + (x)2]. Show that 2 and 2 can be consistently estimated by a regression of the least squares residuals on a constant and x2. Is this estimator efficie

> To continue the analysis in Application 5, consider a nonparametric regression of G/Pop on the price. Using the nonparametric estimation method in Section 7.5, fit the nonparametric estimator using a range of bandwidth values to explore the effect of ban

> For the model in Exercise 9, what is the probability limit of Note that s2 is the least squares estimator of the residual variance. It is also n times the conventional estimator of the variance of the OLS estimator, How does this equation compare with

> What is the covariance matrix, of the GLS estimator and the difference between it and the OLs estimator, The result plays a pivotal role in the development of specification tests in Hausman (1978).

> Prove that in the control function estimator in (8-16), you can use the predictions, z'p, instead of the residuals to obtain the same results apart from the sign on the control function itself, which will be reversed.

> Prove that the control function approach in (8-16) produces the same estimates as 2SLS.

> This is easy to show. In the expression for , if the kth Colum n in X is one of the columns in Z, say the lth, then the kth column in (Z'Z)-1Z'X will be the lth column of an L * L identity matrix. This result means that the kth column in = Z (Z'Z)-1Z

> Consider the linear model, Let z be an exogenous, relevant instrumental variable for this model. Assume, as well, that z is binary—it takes only values 1 and 0. Show the algebraic forms of the LS estimator and the IV estimator for both

> At the end of section 8.7, it is suggested that the OLs estimator could have a smaller mean squared error than the 2sLs estimator. Using (8-4), the results of Exercise 1, and Theorem 8.1, show that the result will be true if How can you verify that thi

> Derive the results in (8-32a) and (8-32b) for the measurement error model. Note the hint in Footnote 4 in section 8.5.1 that suggests you use result (A-66) when you need to invert

> For the measurement error model in (8-26) and (8-27), prove that when only x is measured with error, the squared correlation between y and x is less than that between y* and x*. (Note the assumption that y* = y.) Does the same hold true if y* is also mea

> In the discussion of the instrumental variable estimator, we showed that the least squares estimator, bLs, is biased and inconsistent. Nonetheless, bLs does estimate something—plim Derive the asymptotic covariance matrix of bLs and sh

> In Application 1 in chapter 3 and Application 1 in chapter 5, we examined Koop and Tobias’s data on wages, education, ability, and so on. We continue the analysis here. (The source, location and configuration of the data are given in th

> Verify the following differential equation, which applies to the Box–cox transformation: Show that the limiting sequence for  = 0 is These results can be used to great advantage in deriving the actual second derivatives

> Dummy variable for one observation. Suppose the data set consists of n observations, (yn, Xn) and an additional observation, The full data set contains a dummy variable, d, that equals zero save for one (the last) observation. Then, the full data set is

> Reverse regression continued. suppose that the model in Exercise 3 is extended to y = x* + d + , x = x* + u. For convenience, we drop the constant term. Assume that x*, e, and u are independent normall

> Estimate the parameters of the model in Example 10.4 using two-stage least squares. Obtain the residuals from the two equations. Do these residuals appear to be white noise series? Based on your findings, what do you conclude about the specification of t

> Carry out an ADF test for a unit root in the rate of inflation using the subset of the data in Appendix Table F5.2 since 1974.1. (This is the first quarter after the oil shock of 1973.)

> Using the macroeconomic data in Appendix Table F5.2, estimate by least squares the parameters of the model where ct is the log of real consumption and yt is the log of real disposable income. a. Use the Breusch and Pagan LM test to examine the residuals

> (This exercise requires appropriate computer software. The computations required can be done with RATS, EViews, Stata, LIMDEP, and a variety of other software using only preprogrammed procedures.) Quarterly data on the consumer price index for 1950.1 to

> Data for fitting an improved Phillips curve model can be obtained from many sources, including the Bureau of Economic Analysis’s (BEA) own Web site, www. economagic.com, and so on. Obtain the necessary data and expand the model of Example 20.3. Does addi

> The data used to fit the expectations augmented Phillips curve in Example 20.3 are given in Appendix Table F5.2. Using these data, reestimate the model given in the example. Carry out a formal test for first-order autocorrelation using the LM statistic.

> The application in chapter 3 used 15 of the 17,919 observations in Koop and Tobias’s (2004) study of the relationship between wages and education, ability, and family characteristics. (see Appendix Table F3.2.) We will use the full data set for this exer

> Stochastic Frontier Model. section 10.3.1 presents estimates of a cobb–Douglas cost function using Nerlove’s 1955 data on the U.s. electric power industry. christensen and Greene’s 1976 update of this study used 1970 data for this industry. The christens

> Continuing the analysis of the previous application, note that these data conform precisely to the description of corner solutions in section 19.3.4. The dependent variable is not censored in the fashion usually assumed for a tobit model. To investigate

> The Mroz (1975) data used in Example 19.10 (see Appendix Table F5.1) also describe a setting in which the tobit model has been frequently applied. The sample contains 753 observations on labor market outcomes for married women, including the following va

> We examined Ray Fair’s famous analysis (Journal of Political Economy, 1978) of a Psychology Today survey on extramarital affairs in Example 18.18 using a Poisson regression model. Although the dependent variable used in that study was a count, Fair (1978

> Appendix Table F18.3 contains data on ship accidents reported in McCullagh and Nelder (1983). The data set contains 40 observations on the number of incidents of wave damage for oceangoing ships. Regressors include aggregate months of service, and three

> The GSOEP data are an unbalanced panel, with 7,293 groups. Continue your analysis in Application 3 by fitting the Poisson model with fixed and with random effects and compare your results. (Recall, like the linear model, the Poisson fixed effects model m

> Several applications in the preceding chapters using the German health care data have examined the variable DocVis, the reported number of visits to the doctor. The data are described in Appendix Table F7.1. A second count variable in that data set that

> Continuing the analysis of the first application, we now consider the self-reported rating, v1. This is a natural candidate for an ordered choice model, because the simple five-item coding is a censored version of what would be a continuous scale on some

> Appendix Table F17.2 provides Fair’s (1978) Redbook Magazine survey on extramarital affairs. The variables in the data set are as follows: and three other variables that are not used. The sample contains a survey of 6,366 married women

> Appendix Table F17.2 provides Fair’s (1978) Redbook survey on extramarital affairs. The data are described in Application 1 at the end of chapter 18 and in Appendix F. The variables in the data set are as follows: and three other variab

> Data on U.S. gasoline consumption for the years 1953 to 2004 are given in Table F2.2. Note the consumption data appear as total expenditure. To obtain the per capita quantity variable, divide GASEXP by GASP times Pop. The other variables do not need tran

> Consider a model for the mix of male and female children in families. Let Ki denote the family size (number of children), Ki = 1………. Let Fi denote the number of female children, Fi = 0â&#1

> A regression model that describes income as a function of experience is /

> Does the Wald statistic reject the null hypothesis too often? construct a Monte carlo study of the behavior of the Wald statistic for testing the hypothesis that g equals zero in the model of Section 15.5.1. Recall that the Wald statistic is the square o

> (This application will require an optimizer. Maximization of a user-supplied function is provided by commands in Stata, R, SAS, EViews or NLOGIT.) Use the following pseudo-code to generate a random sample of 1,000 observations on y from a mixed normals p

> The geometric distribution used in Examples 14.13, 14.17, 14.18, and 14.22 would not be the typical choice for modeling a count such as DocVis. The Poisson model suggested at the beginning of section 14.11.1 would be the more natural choice (at least at

> Binary Choice. This application will be based on the health care data analyzed in Example 14.13 and several others. Details on obtaining the data are given in Appendix F Table 7.1. We consider analysis of a dependent variable, y, that ta

> The data in Appendix Table F6.1 are an unbalanced panel on 25 U.S. airlines in the pre-deregulation days of the 1970s and 1980s. The group sizes range from 2 to 15. Data in the file are the following variables. (Variable names contained in the data file

> The data in Appendix Table F10.4 were used by Grunfeld (1958) and dozens of researchers since, including Zellner (1962, 1963) and Zellner and Huang (1962), to study different estimators for panel data and linear regression systems. [See Kleiber and Zeile

> Several applications in this and previous chapters have examined the returns to education in panel data sets. Specifically, we applied Hausman and Taylor’s approach in Examples 11.17 and 11.18. Example 11.18 used Cornwell and Rupert&aci

> Using the estimated health outcomes model in Example 10.8, determine the expected values of ln Income and Health Satisfaction for a person with the following characteristics: Female = 1, Working = 1, Public = 1, AddOn = 0, Education = 14, Married = 1, HH

> Christensen and Greene (1976) estimated a “generalized Cobb–Douglas” cost function for electricity generation of the form ln C = a + b ln Q + g[1 (ln Q)2] + dk ln Pk + dl ln Pl + df ln Pf + e. Pk, Pl, and Pf indicate unit prices of capital, labor, and fu

> Using the cost function estimates in Example 10.2, we obtained an estimate of the efficient scale, / We can use the delta method in Section 4.5.4 to compute an asymptotic standard error for the estimator of Q* and a confidence interval. The estimators of

> Reverse regression. A method of analyzing statistical data to detect discrimination in the workplace is to fit the regression y = a + x' + d + , (1) where y is the wage rate and d is a dummy variable indicating either membership (d = 1) or nonmember s

> Partial Frisch and Waugh. In the least squares regression of y on a constant and X, to compute the regression coefficients on X, we can first transform y to deviations from the mean y and, likewise, transform each column of X to deviations from the respe

> Change in the sum of squares. suppose that b is the least squares coefficient vector in the regression of y on X and that c is any other K *1 vector. Prove that the difference in the two sums of squared residuals is (y - Xc)=(y - Xc) - (y - Xb)=(y - Xb)

> The two-variable regression. For the regression model y = a + bx + e, a. show that the least squares normal equations imply iei = 0 and ixiei = 0. b. show that the solution for the constant term is a = y - bx. c. show that the solution for b is b = 3

> The data in Appendix Table f5.2 may be used to estimate a small macroeconomic model. Use these data to estimate the model in Example 10.5. Estimate the parameters of the two equations by two-stage and three-stage least squares.

> Reverse regression continued. This and the next exercise continue the analysis of Exercise 2. In Exercise 2, interest centered on a particular dummy variable in which the regressors were accurately measured. Here we consider the case in which the crucial

> A regression model with K = 16 independent variables is fit using a panel of seven years of data. The sums of squares for the seven separate regressions and the pooled regression are shown below. The model with the pooled data allows a separate constant

> Prove that under the hypothesis that RB = q, the estimator where J is the number of restrictions, is unbiased for s2.

> Use the test statistic defined in Exercise 7 to test the hypothesis in Exercise 1.

> An alternative way to test the hypothesis RB - q = 0 is to use a Wald test of the hypothesis that L* = 0, where L* is defined in (5-23). Prove that Note that the fraction in brackets is the ratio of two estimators of s2. By virtue of (5-28) and the prece

> Prove the result that the R2 associated with a restricted least squares estimator is never larger than that associated with the unrestricted least squares estimator. conclude that imposing restrictions never improves the fit of the regression.

> Prove the result that the restricted least squares estimator never has a larger covariance matrix than the unrestricted least squares estimator.

> The expression for the restricted coefficient vector in (5-23) 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 s2(X'X)-1 - s2(X'X)-1R'[R(X'

> The regression model to be analyzed is y = X1B1 + X2B2 + E , where X1 and X2 have K1 and K2 columns, respectively. The restriction is B2 = 0. a. Using (5-23), prove that the restricted estimator is simply [b1*, 0], where b1* is the least squares coeffici

> 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.

> The Grunfeld investment data in Appendix Table 10.4 constitute a classic data set that has been used for decades to develop and demonstrate estimators for seemingly unrelated regressions.28 Although somewhat dated at this juncture, they remain an ideal a

> Show that the model of the alternative hypothesis in Example 5.7 can be written As such, it does appear that H0 is a restriction on H1. However, because there are an infinite number of constraints, this does not reduce the test to a standard test of rest

> The log likelihood function for the linear regression model with normally distributed disturbances is shown in (14-39) in section 14.9.1. show that at the maximum likelihood estimators of b for B and e'e/n for s2, the log likelihood is an increasing func

> Suppose the true regression model is given by (4-7). The result in (4-9) shows that if pX.z is nonzero and g is nonzero, then regression of y on X alone produces a biased and inconsistent estimator of B. suppose the objective is to forecast y, not to est