The exponential distribution has density / How would you obtain a random sample of observations from an exponential population?
> Suppose that the classical regression model applies but that the true value of the constant is zero. Compare the variance of the least squares slope estimator computed without a constant term with that of the estimator computed with an unnecessary consta
> Consider the simple regression yi = xi + i where E[/x] = 0 and E[2 / x] = 2 a. What is the minimum mean squared error linear estimator of ? [Hint: Le
> For the simple regression model yi = + i, i ( N[0, 2], prove that the sample mean is consistent and asymptotically normally distributed. Now consider the alternative estimator mn = w y , w = i = i . Note that // Prove that this is a c
> Let ei be the ith residual in the ordinary least squares regression of y on X in the classical regression model, and let ei be the corresponding true disturbance. Prove that plim(ei - ei) = 0.
> For the classical normal regression model y = Xβ + ( with no constant term and K regressors, what is plim /assuming that the true value of β is zero?
> Prove that / where b is the ordinary least squares estimator and k is a characteristic root of X′X.
> Suppose that you have two independent unbiased estimators of the same parameter /with different variances v1 and v2. What linear combination / is the minimum variance unbiased estimator of /
> Example 4.10 presents a regression model that is used to predict the auction prices of Monet paintings. The most expensive painting in the sample sold for $33.0135M (ln = 17.3124). The height and width of this painting were 35″ and 39.4″, respectively. U
> In Section 4.9.2, we consider regressing y on a set of principal components, rather than the original data. For simplicity, assume that X does not contain a constant term, and that the K variables are measured in deviations from the means and are standar
> Statewide aggregate production function. Continuing Example 10.1, data on output, the capital stocks, and employment are aggregated by summing the values for the individual states (before taking logarithms). The unemployment rate for each region, m, at t
> In (4-13), we find that when superfluous variables X2 are added to the regression of y on X1 the least squares coefficient estimator is an unbiased estimator of the true parameter vector, β = (β′1, 0′)′. Show that, in this long regression, e′e/(n - K1 -
> Consider a data set consisting of n observations, nc complete and nm incomplete, for which the dependent variable, yi, is missing. Data on the independent variables, xi, are complete for all n observations, Xc and Xm. We wish to use the data to estimate
> Is it possible to partition R2? The idea of “hierarchical partitioning” is to decompose R2 into the contributions made by each variable in the multiple regression. That is, if x1, , xK are entered into a regression one at a time, then ck is the increment
> In the December 1969 American Economic Review (pp. 886–896), Nathaniel Leff reports the following least squares regression results for a cross section study of the effect of age composition on savings in 74 countries in 1964: ln S/Y = 7.3439 + 0.1596 ln
> 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, GNP, and the interest rate. Compute R2.
> 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 D on Y, the slope is 0.4. What is the
> Regression without a constant. suppose that you estimate a multiple regression first with, then without, a constant. Whether the R2 is higher in the second case than the first will depend in part on how it is computed. using the (relatively) standard met
> Change in adjusted R2. Prove that the adjusted R2 in (3-30) rises (falls) when variable xk is deleted from the regression if the square of the t ratio on xk in the multiple regression is less (greater) than 1.
> 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 Ed = ad + bdY + g
> 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 an
> (We look ahead to our use of maximum likelihood to estimate the models discussed in this chapter in chapter 14.) In Example 9.3, we computed an iterated FGLS estimator using the airline data and the model The weights computed at each iteration were comp
> Adding an observation. A data set consists of n observations contained in Xn and yn. The least squares estimator based on these n observations is bn = (X= X )-1X= y . Another observation, xs and ys, becomes available. Prove that the least squares estimat
> Residual makers. What is the result of the matrix product M1M where M1 is defined in (3-19) and M is defined in (3-14)?
> Find the first two autocorrelations and partial autocorrelations for the MA(2) process
> It is commonly asserted that the Durbin–Watson statistic is only appropriate for testing for first-order autoregressive disturbances. The Durbin–Watson statistic estimates 2(1 - ) where r is the first-order autocorrelation of the residuals. What combina
> Derive the disturbance covariance matrix for the model What parameter is estimated by the regression of the OLS residuals on their lagged values?
> Does first differencing reduce autocorrelation? Consider the models where / Compare the autocorrelation of t in the original model with that of vt in / where
> Prove that the Hessian for the tobit model in (19-14) is negative definite after Olsen’s transformation is applied to the parameters.
> Derive the partial effects for the tobit model with heteroscedasticity that is described in section 19.3.5.b.
> 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 have population correlation 0.7. The two variables, y* a
> Using only the no limit observations, repeat Exercise 2 in the context of the truncated regression model. Estimate / by using the method of moments estimator outlined in Example 19.2. Compare your results with those in the previous exercises.
> This application is based on the following data set. a. compute the OLS regression of y on a constant, x1, and x2. Be sure to compute the conventional estimator of the asymptotic covariance matrix of the OLS estimator as well. b. compute the White esti
> We now consider the tobit model that applies to the full data set. a. Formulate the log likelihood for this very simple tobit model. b. Reformulate the log likelihood in terms of Then derive the necessary conditions for maximizing the log likelihood with
> The following 20 observations are drawn from a censored normal distribution: The applicable model is Exercises 1 through 4 in this section are based on the preceding information. The OLs estimator of  in the context of this tobit mo
> Consider estimation of a Poisson regression model for yi | xi. The data are truncated on the left—these are on-site observations at a recreation site, so zeros do not appear in the data set. The data are censored on the right—any response greater than 5
> For the zero-inflated Poisson (ZIP) model in section 18.4.8, we derived the conditional mean function, / a. For the same model, now obtain / Then, obtain Does the zero inflation produce overdispersion? (That is, is the ratio greater than one?) b. obtai
> We are interested in the ordered probit model. our data consist of 250 observations, of which the responses are using the preceding data, obtain maximum likelihood estimates of the unknown parameters of the model. (Hint: Consider the probabilities as t
> In the panel data models estimated in section 17.7, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint: Unlike our application in the linear mo
> Prove (17-26).
> A data set consists of n = n1 + n2 + n3 observations on y and x. For the first n1 observations, y = 1 and x = 1. For the next n2 observations, y = 0 and x = 1. For the last n3 observations, y = 0 and x = 0. Prove that neither (17-18) nor (17-20) has a so
> 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 recycli
> Construct the Lagrange multiplier statistic for testing the hypothesis that all the slopes (but not the constant term) equal zero in the binomial logit model. Prove that the Lagrange multiplier statistic is nR2 in the regression of (yi - p) on the xs, wh
> In Example 8.5, we have suggested a model of a labor market. From the “reduced form” equation given first, you can see the full set of variables that appears in the model—that is the “endogenous variables,” ln Wageit, and Wksit, and all other exogenous v
> Given the data set estimate a probit model and test the hypothesis that x is not influential in determining the probability that y equals one.
> Suppose that a linear probability model is to be fit to a set of observations on a dependent variable y that takes values zero and one, and a single regressor x that varies continuously across observations. Obtain the exact expressions for the least squa
> A binomial probability model is to be based on the following index function model: The only regressor, d, is a dummy variable. The data consist of 100 observations that have the following: Obtain the maximum likelihood estimators of and b, and estimate
> Suppose the distribution of yi |  is Poisson, We will obtain a sample of observations, yi, , yn. suppose our prior for ï¬ï€ is the inverted gamma, which will imply a. construct the likelihood function,
> Derive the first-order conditions for nonlinear least squares estimation of the parameters in (15-2). How would you estimate the asymptotic covariance matrix for your estimator of /
> The Weibull population has survival function How would you obtain a random sample of observations from a Weibull population? (The survival function equals one minus the cdf.)
> Consider sampling from a multivariate normal distribution with mean vector and covariance matrix 2I. The log-likelihood function is show that the maximum likelihood estimators of the parameters are / and Derive the second derivativ
> For random sampling from the classical regression model in (14-3), reparameterize the likelihood function in terms of / Find the maximum likelihood estimators of and δ and obtain the asymptotic covariance matrix of the estimators of these parameters.
> Show that the likelihood inequality in Theorem 14.3 holds for the Poisson distribution used in section 14.3 by showing that / is uniquely maximized at θ = θθ. (Hint: First show that the expectation is / show that the likelihood inequality in Theorem 14
> Using the gasoline market data in Appendix Table F2.2, use the partially linear regression method in Section 7.4 to fit an equation of the form
> Limited Information Maximum Likelihood Estimation. Consider a bivariate distribution for x and y that is a function of two parameters, and b. The joint density is f(x, y | , ). We consider maximum likelihood estimation of the two parameters. The full i
> The following data were generated by the Weibull distribution of Exercise 4: a. Obtain the maximum likelihood estimates of a and , and estimate the asymptotic covariance matrix for the estimates. b. Carry out a Wald test of the hypoth
> Suppose that x has the Weibull distribution a. Obtain the log-likelihood function for a random sample of n observations. b. Obtain the likelihood equations for maximum likelihood estimation of and . Note that the first provides an expli
> Mixture distribution. suppose that the joint distribution of the two random variables x and y is a. Find the maximum likelihood estimators of  and θ and their asymptotic joint distribution. b. Find the maximum likelihood
> In random sampling from the exponential distribution // find the maximum likelihood estimator of θ and obtain the asymptotic distribution of this estimator.
> Assume that the distribution of x is In random sampling from this distribution, prove that the sample maximum is a consistent estimator of θ. Note: You can prove that the maximum is the maximum likelihood estimator of θ. But the
> Consider GMM estimation of a regression model as shown at the beginning of Example 13.8. 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 th
> Consider the probit model analyzed in Chapter 17. The model states that for given vector of independent variables, Consider a GMM estimator based on the result that This suggests that we might base estimation on the orthogonality conditions Construct a G
> In the classical regression model with heteroscedasticity, which is more efficient, ordinary least squares or GMM? Obtain the two estimators and their respective asymptotic covariance matrices, then prove your assertion.
> For the Wald distribution discussed in Example 13.3, we have the following results / a. Derive the maximum likelihood estimators of  and  and an estimator of the asymptotic variances of the MLEs. (Hint: Expand the qua
> In Example 7.1, the cES function is suggested as a model for production, / (7-36) Example 6.19 suggested an indirect method of estimating the parameters of this model. The function is linearized around = 0, which produces an intrinsically linear approx
> The data listed in Table 3.5 are extracted from Koop and Tobias’s (2004) study of the relationship between wages and education, ability, and family characteristics. (see Appendix Table F3.2.) Their data set is a panel of 2,178 individua
> Using the bond from Practice Exercise 14-5A, journalize the first interest payment and the amortization of the related bond premium. Round to the nearest dollar. Data from Practice Exercise 14-5A: On the first day of the fiscal year, a company issues an
> On the first day of the fiscal year, a company issues a $5,300,000, 8%, five-year bond that pays semiannual interest of $212,000 ($5,300,000 × 8% × ½), receiving cash of $5,520,390. Journalize the bond issuance.
> Desmond Co. is considering the following alternative financing plans: Income tax is estimated at 40% of income. Determine the earnings per share of common stock, assuming that income before bond interest and income tax is $400,000.
> On January 31, Outback Coast Resorts Inc. reacquired 18,700 shares of its common stock at $45 per share. On April 20, Outback Coast Resorts sold 10,600 of the reacquired shares at $52 per share. On October 4, Outback Coast Resorts sold the remaining shar
> Red Market Corporation has 370,000 shares of $27 par common stock outstanding. On June 8, Red Market Corporation declared a 5% stock dividend to be issued August 12 to stockholders of record on July 13. The market price of the stock was $51 per share on
> Top-Value Corporation has 900,000 shares of $26 par common stock outstanding. On September 2, Top-Value Corporation declared a 3% stock dividend to be issued November 30 to stockholders of record on October 3. The market price of the stock was $44 per sh
> The declaration, record, and payment dates in connection with a cash dividend of $195,000 on a corporation’s common stock are February 1, March 18, and May 1. Journalize the entries required on each date.
> The declaration, record, and payment dates in connection with a cash dividend of $428,000 on a corporation’s common stock are February 28, April 1, and May 15. Journalize the entries required on each date.
> On January 22, Micah Corporation issued for cash 125,000 shares of no-par common stock at $6. On February 14, Micah Corporation issued at par value 32,000 shares of preferred 2% stock, $80 par for cash. On August 30, Micah Corporation issued for cash 7,0
> On May 23, Washburn Realty Inc. issued for cash 45,000 shares of no-par common stock (with a stated value of $4) at $16. On July 6, Washburn Realty Inc. issued at par value 12,000 shares of preferred 1% stock, $75 par for cash. On September 15, Washburn
> In teams, select a public company that interests you. Obtain the company’s most recent annual report on Form 10-K. The Form 10-K is a company’s annually required filing with the Securities and Exchange Commission (SEC). It includes the company’s financia
> Castillo Nutrition Company has 14,000 shares of cumulative preferred 1% stock, $130 par, and 70,000 shares of $5 par common stock. The following amounts were distributed as dividends: Year 1 ……………….. $35,000 Year 2 …………………… 6,300 Year 3 ………………… 80,500 De
> Financial statement data for the years ended December 31 for Brown Cow Inc. follow: a. Determine the earnings per share for 20Y6 and 20Y5. b. Does the change in the earnings per share from 20Y5 to 20Y6 indicate a favorable or unfavorable trend?
> Financial statement data for the years ended December 31 for Cottontop Corporation follow: a. Determine the earnings per share for 20Y3 and 20Y2. b. Does the change in the earnings per share from 20Y2 to 20Y3 indicate a favorable or unfavorable trend?
> Haggen Cruises Inc. reported the following results for the year ended October 31, 20Y9: Retained earnings, November 1, 20Y8 …………. $11,775,000 Net income ………………………………………………….. 2,232,000 Cash dividends declared ………………………………….. 166,000 Stock dividends decla
> Using the following accounts and balances, prepare the Stockholders’ Equity section of the balance sheet using Method 1 of Exhibit 8. Five hundred thousand shares of common stock are authorized, and 35,000 shares have been reacquired. C
> Using the following accounts and balances, prepare the Stockholders’ Equity section of the balance sheet using Method 1 of Exhibit 8. Two hundred thousand shares of common stock are authorized, and 5,000 shares have been reacquired. Com
> On May 27, Idress Clothing Inc. reacquired 64,000 shares of its common stock at $12 per share. On August 3, Idress Clothing sold 41,000 of the reacquired shares at $17 per share. On November 14, Idress Clothing sold the remaining shares at $9 per share.
> Swilley Furniture Company has 50,000 shares of cumulative preferred 2% stock, $75 par, and 100,000 shares of $10 par common stock. The following amounts were distributed as dividends: Year 1 …………… $ 45,000 Year 2 ……………. 123,000 Year 3 …………… 130,000 Deter
> Prior to liquidating their partnership, Cameron and Solivita had capital accounts of $44,000 and $92,000, respectively. Prior to liquidation, the partnership had no cash assets other than what was realized from the sale of assets. These partnership asset
> Todd has a capital balance of $170,600 after adjusting assets to fair market value. Zanetti contributes $45,500 to receive a 40% interest in a new partnership with Todd. Determine the amount and recipient of the partner bonus.
> The following financial data (in thousands) were taken from recent financial statements of Office Depot, Inc.: 1. Determine the times interest earned ratio for Office Depot in Year 3, Year 2, and Year 1? Round to one decimal place. 2. Evaluate this rati
> Patel has a capital balance of $310,000 after adjusting assets to fair market value. Killingsworth contributes $490,000 to receive a 60% interest in a new partnership with Patel. Determine the amount and recipient of the partner bonus.
> Marquis Westbury invested $119,100 in the Trenton and Rainwater partnership for ownership equity of $119,100. Prior to the investment, equipment was revalued to a market value of $77,400 from a book value of $57,300. Daniel Trenton and Ann Marie Rainwate
> Greg Thomas purchased one-half of Ian Hamilton’s interest in the Freidman and Hamilton partnership for $49,500. Prior to the investment, land was revalued to a market value of $189,200 from a book value of $116,400. Adam Freidman and Ian Hamilton share n
> Bruce Delew and Nadia Comatof formed a partnership, dividing income as follows: 1. Annual salary allowance to Delew, $18,000, and Comatof, $51,000. 2. Interest of 6% on each partner’s capital balance on January 1. 3. Any remaining net income divided equa
> Adriana Gonzalez and Sylvester Van Horne formed a partnership, dividing income as follows: 1. Annual salary allowance to Gonzalez of $25,000. 2. Interest of 5% on each partner’s capital balance on January 1. 3. Any remaining net income divided to Gonzale
> Xi Lin contributed land, inventory, and $38,600 cash to a partnership. The land had a book value of $128,500 and a market value of $187,400. The inventory had a book value of $53,500 and a market value of $45,100. The partnership also assumed a $37,600 n
> Makeman Architects earned $4,400,000 during 20Y1 using 40 employees. During 20Y2, the firm reduced revenues to $3,894,000 and reduced the staff to 33 employees. a. Determine the revenue per employee for each year. b. Interpret the results.
> Schwartz and Beer, CPAs earned $9,338,000 during 20Y4 using 46 employees. During 20Y5, the firm grew revenues to $11,825,000 and expanded the staff to 55 employees. a. Determine the revenue per employee for each year. b. Interpret the results.
> Prior to liquidating their partnership, Jacobs and Sanchez had capital accounts of $320,000 and $424,000, respectively. The partnership assets were sold for $52,000. The partnership had no liabilities. Jacobs and Sanchez share income and losses equally.
> Prior to liquidating their partnership, Heller and Warren had capital accounts of $128,000 and $67,000, respectively. The partnership assets were sold for $49,000. The partnership had no liabilities. Heller and Warren share income and losses equally. a.
> Philip Morris International Inc. has numerous pages dedicated to describing contingent liabilities in the notes to recent financial statements. These pages include extensive descriptions of multiple contingent liabilities. Use the Internet to research Ph
> Prior to liquidating their partnership, Kim and Cheyenne had capital accounts of $304,000 and $190,000, respectively. Prior to liquidation, the partnership had no cash assets other than what was realized from the sale of assets. These partnership assets
> Melissa Shallowford contributed a patent, accounts receivable, and $15,000 cash to a partnership. The patent had a book value of $6,000. However, the technology covered by the patent appeared to have significant market potential. Thus, the patent was app
