Questions and Answers of Econometrics

18.17 Consider the regression model in matrix form Y = XB + WG + U, where X and W are matrices of regressors and B and G are vectors of unknown regression coefficients. Let X= MWX and Y= MWY, where
18.16 This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model Yi = Xib + ui, i = 1,c, n, where all variables are scalars and the constant
18.15 (Consistency of clustered standard errors.) Consider the panel data model Yit =bXit + ai + uit, where all variables are scalars. Assume that Assumptions #1, #2, and #4 in Key Concept 10.3 hold
18.14 Consider the regression model Y = XB + U. Partition X as [X1 X2] and B as [B1 B2], where X1 has k1 columns and X2 has k2 columns. Suppose that X2Y = 0k2 * 1. Let R = [Ik1 0k1 * k2].a. Show
18.13 Consider the problem of minimizing the sum of squared residuals, subject to the constraint that Rb = r, where R is q * (k + 1) with rank q. Let Bbe the value of b that solves the constrained
18.12a. Show that B Eff.GMM is the efficient GMM estimator—that is, that B Eff.GMM in Equation (18.66) is the solution to Equation (18.65).b. Show that 2n1B nEff.GMM - B Eff.GMM2 ¡p 0.c. Show
18.11 Suppose that C is an n * n symmetric idempotent matrix with rank r and let V N(0n, In).a. Show that C = AA, where A is n * r with AA = Ir. (Hint: C is positive semidefinite and can be
18.10 Let C be a symmetric idempotent matrix.a. Show that the eigenvalues of C are either 0 or 1. (Hint: Note that Cq = gq implies 0 = Cq - gq = CCq - gq = gCq - gq = g2q - gq and solve for G.)b.
18.9 This exercise shows that the OLS estimator of a subset of the regression coefficients is consistent under the conditional mean independence assumption stated in Appendix 7.2. Consider the
18.8 Consider the regression model Yi = b0 + b1Xi + ui, where u1 = u 1 and ui = 0.5ui-1 + u i for i = 2, 3, . . . , n. Suppose that u i are i.i.d. with mean 0 and variance 1 and are distributed
18.7 Consider the regression model Yi = b1Xi + b2Wi + ui, where for simplicity the intercept is omitted and all variables are assumed to have a mean of zero.Suppose that Xi is distributed
18.6 Consider the regression model in matrix form, Y = XB + WG + U, where X is an n * k1 matrix of regressors and W is an n * k2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS
18.5 Let PX and MX be as defined in Equations (18.24) and (18.25).a. Prove that PXMX = 0n * n and that PX and MX are idempotent.b. Derive Equations (18.27) and (18.28).c. Show that rank(PX) = k + 1
18.4 Consider the regression model Yi = b0 + b1Xi + ui from Chapter 4 and assume that the least squares assumptions in Key Concept 4.3 hold.a. Write the model in the matrix form given in Equations
18.3 Let W be an m * 1 vector with covariance matrix W, where W is finite and positive definite. Let c be a nonrandom m * 1 vector and let Q = cW.a. Show that var(Q) = cW c.b. Suppose that c
18.2 Suppose that a sample of n = 20 households has the sample means and sample covariances below for a dependent variable and two regressors:a. Calculate the OLS estimates of b0, b1, and b2.
18.1 Consider the population regression of test scores against income and the square of income in Equation (8.1).a. Write the regression in Equation (8.1) in the matrix form of Equation(18.5). Define
18.5 Construct an example of a regression model that satisfies the assumption E(ui Xi) = 0 but for which E(U X ) 0n.
18.4 When is the GLS estimator more efficient than the OLS estimator within the class of linear conditionally unbiased estimators?
18.3 Suppose that Assumptions #1 through #5 in Key Concept 18.1 are true but that Assumption #6 is not. Does the result in Equation (18.31) hold?Explain.
18.2 You are analyzing a linear regression model with 500 observations and one regressor. Explain how you would construct a confidence interval for b1 if:a. Assumptions #1 through #4 in Key Concept
18.1 A researcher studying the relationship between earnings and gender for a group of workers specifies the regression model Yi =b0 + X1ib1 + X2ib2 + ui, where X1i is a binary variable that equals 1
17.15 Z is distributed N (0,1), W is distributed x2 n, and V is distributed x2 m. Show, as n S and m is fixed, that: a. W/n1. b. kc. Z VW/n d N(0,1). Use the result to explain why the to
17.14 Suppose that Yi, i = 1, 2,c, n, are i.i.d. with E(Yi) = m, var(Yi) = s2, and finite fourth moment. Show the following: a. E(Y)+0 = b. Y c. n P + d. ( - = e. n (YY) n 1 - ( - -
17.13 Consider the heterogeneous regression model Yi = b0i + b1iXi + ui, where b0i and b1i are random variables that differ from one observation to the next.Suppose that E(ui 0Xi) = 0 and (b0i, b1i)
17.12a. Suppose that u N(0, s2 u). Show that E(eu) = e1 2s2 ub. Suppose that the conditional distribution of u given X = x is N(0, a + bx2), where a and b are positive constants. Show that E(eu 0X
17.11 Suppose that X and Y are distributed bivariate normal with density given in Equation (17.38).a. Show that the density of Y given X = x can be written aswhere sYX = 2s2 Y(1 - r2 XY) and mYX = mY
17.10 Let u n be an estimator of the parameter u, where u n might be biased. Show that if E3(u n - u)24 ¡0 as n¡ (that is, the mean squared error of u n tends to zero), then u n ¡ p u. [Hint:
17.9 Prove Equation (17.16) under Assumptions #1 and #2 of Key Concept 17.1 plus the assumption that Xi and ui have eight moments.
17.8 Consider the regression model in Key Concept 17.1 and suppose that Assumptions #1, #2, #3, and #5 hold. Suppose that Assumption #4 is replaced by the assumption that var(ui 0Xi) = u0 + u1 0Xi 0
17.7 Suppose that X and u are continuous random variables and (Xi, ui), i =1,c, n, are i.i.d.a. Show that the joint probability density function (p.d.f.) of (ui, uj, Xi, Xj)can be written as f(ui,
17.6 Show that if b n1 is conditionally unbiased, then it is unbiased; that is, show that if E(b n1 0X1,c, Xn) = b1, then E(b n1) = b1.
17.5 Suppose that W is a random variable with E(W4) 6 . Show that E(W2) 6 .
17.4 Show the following results:a. Show that 2n(b n1 - b1) ¡d N(0, a2), where a2 is a constant, implies that b n1 is consistent. (Hint: Use Slutsky’s theorem.)b. Show that su 2 >su 2¡ p 1 implies
17.3. This exercise fills in the details of the derivation of the asymptotic distribution of b n1 given in Appendix 4.3.a. Use Equation (17.19) to derive the expressionwhere vi = (Xi - mX)ui.b. Use
17.2 Suppose that (Xi,Yi) are i.i.d. with finite fourth moments. Prove that the sample covariance is a consistent estimator of the population covariance—that is, sXY ¡ p sXY, where sXY is defined
17.1 Consider the regression model without an intercept term, Yi = b1Xi + ui(so the true value of the intercept, b0, is zero).a. Derive the least squares estimator of b1 for the restricted regression
17.4 Instead of using WLS, the researcher in the previous problem decides to compute the OLS estimator using only the observations for which x … 10, then using only the observations for which x 7
17.3 Suppose that Y and X are related by the regression Y = 1.0 + 2.0X + u.A researcher has observations on Y and X, where 0 … X … 20, where the conditional variance is var(ui 0 Xi = x) = 1 for 0
17.2 Suppose that An is a sequence of random variables that converges in probability to 3. Suppose that Bn is a sequence of random variables that converges in distribution to a standard normal. What
17.1 Suppose that Assumption #4 in Key Concept 17.1 is true, but you construct a 95% confidence interval for b1 using the heteroskedasticrobust standard error in a large sample. Would this confidence
E16.2 On the text website, www.pearsonglobaleditions.com/Stock_Watson, you will find the data file USMacro_Quarterly, which contains quarterly data on real GDP, measured in $1996. Compute GDPGRt =
E16.1 This exercise is an extension of Empirical Exercise 14.1. On the text website, www.pearsonglobaleditions.com/Stock_Watson, you will find the data file USMacro_Quarterly, which contains
16.10 Consider the cointegrated model Yt = uXt + v1t and Xt = Xt - 1 + v2t, where v1t and v2t are mean zero serially uncorrelated random variables with E1v1tv2j2 = 0 for all t and j. Derive the
16.9a. Suppose that E(ut ut - 1, ut - 2,c) = 0, that var1ut ut - 1, ut - 2, c2 follows the ARCH(1) model s2t= a0 + a1u2t- 1, and that the process for ut is stationary. Show that var1ut2 = a0> 11
16.8 Consider the following two-variable VAR model with one lag and no intercept:Yt = b11Yt - 1 + g11Xt - 1 + u1t Xt = b21Yt - 1 + g21Xt - 1 + u2t.a. Show that the iterated two-period-ahead forecast
16.7 Suppose that ΔYt = ut, where ut is i.i.d. N(0, 1), and consider the regression Yt = bXt + error, where Xt = ΔYt + 1 and error is the regression error.Show that b n ¡d 12 1x21 - 12. [Hint:
16.6 A regression of Yt onto current, past, and future values of Xt yields Yt = 2.0 + 1.5Xt + 1 + 0.9Xt - 0.3Xt - 1 + ut.a. Rearrange the regression so that it has the form shown in Equation(16.25).
16.5 Verify Equation (16.20). [Hint: Use gT t = 1Y2t= gTt= 11Yt - 1 + ΔYt22 to show that gTt= 1Y2t= gT t = 1Y2t- 1 + 2gT t = 1Yt - 1ΔYt + gT t = 1ΔY2t and solve for gTt= 1Yt - 1ΔYt.4
16.4 Suppose that Yt follows the AR(p) model Yt = b0 + b1Yt - 1 + g+bpYt - p + ut, where E(ut •Yt - 1, Yt - 2,c) = 0 Let Yt + h•t = E(Yt + h •Yt, Yt - 1, c). Show that Yt + h•t = b0 + b1Yt -
16.3 Suppose that ut follows the ARCH process, s2t= 1.0 + 0.5 u2t- 1.a. Let E(u2t) = var(ut) be the unconditional variance of ut. Show that var(ut) = 2. (Hint: Use the law of iterated expectations,
16.2 One version of the expectations theory of the term structure of interest rates holds that a long-term rate equals the average of the expected values of short-term interest rates into the future,
16.1 Suppose that Yt follows a stationary AR(1) model, Yt = b0 + b1Yt - 1 + ut.a. Show that the h-period-ahead forecast of Yt is given by Yt + ht = mY + b h1(Yt - mY), where mY = b0> 11 - b12.b.
16.5 What is a unit root? How does a researcher test for the presence of a unit root n in the data?
16.4 What is meant by volatility clustering? Briefly describe two models that are used to describe data processes with volatility clustering.
16.3 A version of the permanent income theory of consumption implies that the logarithm of real GDP (Y) and the logarithm of real consumption (C) are cointegrated with a cointegrating coefficient
16.2 Suppose that Yt follows a stationary AR(1) model with b0 = 0 and b1 = 0.5.If Yt = 10, what is your forecast of Yt + 2 (that is, what is Yt + 2t)? What is Yt + ht for h = 20? Does this forecast
16.1 A macroeconomist wants to construct forecasts for the following macroeconomic variables: GDP, consumption, investment, government purchases, exports, imports, short-term interest rates,
E15.2 In the data file USMacro_Quarterly, you will find data on two aggregate price series for the United States: the price index for personal consumption expenditures (PCEP) that you used in
E15.1 In this exercise you will estimate the effect of oil prices on macroeconomic activity, using monthly data on the Index of Industrial Production (IP)and the monthly measure of Ot described in
15.11 Suppose that a(L) = (1 - fL), with 0 f1 0 6 1, and b(L) = 1 + fL +f2L2 + f3L3c.a. Show that the product b(L)a(L) = 1, so that b(L) = a(L) - 1.b. Why is the restriction 0 f1 0 6 1 important?
15.10 Consider the ADL model Yt = 5.3 + 0.2Yt - 1 + 1.5Xt - 0.1Xt - 1 + u t, where Xt is strictly exogenous.a. Derive the impact effect of X on Y.b. Derive the first five dynamic multipliers.c.
15.9 Consider the “constant-term-only” regression model Yt = b0 + ut, where ut follows the stationary AR(1) model ut = f1ut - 1 + u t with u t i.i.d. with mean 0 and variance s2 u and 0 f1 0 6
15.8 Consider the model in Exercise 15.7 with Xt = u t + 1.a. Is the OLS estimator of b1 consistent? Explain.b. Explain why the GLS estimator of b1 is not consistent.c. Show that the infeasible GLS
15.7 Consider the regression model Yt = b0 + b1Xt + ut, where ut follows the stationary AR(1) model ut = f1ut - 1 + u t with u t i.i.d. with mean 0 and variance s2 u and 0 f1 0 6 1.a. Suppose that
15.6 Consider the regression model Yt = b0 + b1Xt + ut, where ut follows the stationary AR(1) model ut = f1ut - 1 + u t with ut i.i.d. with mean 0 and variance s2 u and 0 f1 0 6 1; the regressorXt
15.5 Derive Equation (15.7) from Equation (15.4) and show that d0 = b0, d1 = b1, d2 = b1 + b2, d3 = b1 + b2 + b3 (etc.). (Hint: Note that Xt =Xt + Xt - 1 + g + Xt - p + 1 + Xt - p .)
15.4 Suppose that oil prices are strictly exogenous. Discuss how you could improve on the estimates of the dynamic multipliers in Exercise 15.1.
15.3 Consider two different randomized experiments. In experiment A, oil prices are set randomly, and the central bank reacts according to its usual policy rules in response to economic conditions,
15.2 Macroeconomists have also noticed that interest rates change following oil price jumps. Let Rt denote the interest rate on 3-month Treasury bills (in percentage points at an annual rate). The
15.1 Increases in oil prices have been blamed for several recessions in developed countries. To quantify the effect of oil prices on real economic activity, researchers have run regressions like
15.4 Suppose that you added FDDt + 1 as an additional regressor in Equation(15.2). If FDD is strictly exogenous, would you expect the coefficient on FDDt + 1 to be zero or nonzero? Would your answer
15.3 Suppose that a distributed lag regression is estimated, where the dependent variable is Yt instead of Yt. Explain how you would compute the dynamic multipliers of Xt on Yt.
15.2 Suppose that X is strictly exogenous. A researcher estimates an ADL(1,1)model, calculates the regression residual, and finds the residual to be highly serially correlated. Should the researcher
15.1 In the 1970s a common practice was to estimate a distributed lag model relating changes in nominal gross domestic product (Y) to current and past changes in the money supply (X). Under what
E14.2 Read the boxes “Can You Beat the Market? Part I” and “Can You Beat the Market? Part II” in this chapter. Next, go to the course website, where you will find an extended version of the
E14.1 On the text website, www.pearsonglobaleditions.com/Stock_Watson, you will find the data file USMacro_Quarterly, which contains quarterly data on several macroeconomic series for the United
14.11 Suppose that Yt follows the AR(1) model Yt = b0 + b1Yt - 1 + ut.a. Show that Yt follows an AR(2) model.b. Derive the AR(2) coefficients for Yt as a function of b0 and b1.
14.10 A researcher carries out a QLR test using 30% trimming, and there are q = 5 restrictions. Answer the following questions, using the values in Table 14.5 (“Critical Values of the QLR Statistic
14.9 The moving average model of order q has the form Yt = b0 + et + b1et - 1 + b2et - 2 + g+ bqet - q, where et is a serially uncorrelated random variable with mean 0 and variance s2e.a. Show that
14.8 Suppose that Yt is the monthly value of the number of new home construction projects started in the United States. Because of the weather, Yt has a pronounced seasonal pattern; for example,
14.7 Suppose that Yt follows the stationary AR(1) model Yt = 2.5 + 0.7Yt - 1 + ut, where ut is i.i.d. with E1ut2 = 0 and var1ut2 = 9.a. Compute the mean and variance of Yt. (Hint: See Exercise
14.6 In this exercise you will conduct a Monte Carlo experiment to study the phenomenon of spurious regression discussed in Section 14.6. In a Monte Carlo study, artificial data are generated using a
14.5 Prove the following results about conditional means, forecasts, and forecast errors:a. Let W be a random variable with mean mW and variance s2 w and let c be a constant. Show that E3(W - c)24 =
14.4 The forecaster in Exercise 14.2 augments her AR(4) model for IP growth to include four lagged values of Rt, where Rt is the interest rate on 3-month U.S. Treasury bills (measured in percentage
14.3. Using the same data as in Exercise 14.2, a researcher tests for a stochastic trend in ln(IPt), using the following regression:where the standard errors shown in parentheses are computed using
14.2 The index of industrial production (IPt) is a monthly time series that measures the quantity of industrial commodities produced in a given month.This problem uses data on this index for the
14.1 Consider the AR(1) model Yt = b0 + b1Yt - 1 + ut. Suppose that the process is stationary.a. Show that E(Yt) = E(Yt - 1). (Hint: Read Key Concept 14.5.)b. Show that E(Yt) = b0>(1 - b1).
14.4 Suppose that you suspected that the intercept in Equation (14.16) changed in 1992:Q1. How would you modify the equation to incorporate this change? How would you test for a change in the
14.3 A researcher estimates an AR(1) with an intercept and finds that the OLS estimate of b1 is 0.88, with a standard error of 0.03. Does a 95% confidence interval include b1 = 1? Explain.
14.2 Many financial economists believe that the random walk model is a good description of the logarithm of stock prices. It implies that the percentage changes in stock prices are unforecastable. A
14.1 Look at the four plots in Figure 14.2—the US unemployment rate, the dollar-pound exchange rate, the logarithm of the index of industrial production, and the percentage change in stock prices.
E13.1 A prospective employer receives two resumes: a resume from a white job applicant and a similar resume from an African American applicant. Is the employer more likely to call back the white
13.12 Consider the potential outcomes framework from Appendix 13.3. Suppose that Xi is a binary treatment that is independent of the potential outcomes Yi(1) and Yi(0). Let TEi = Yi (1) – Yi (0)
13.11 Results of a study by McClelan, McNeill, and Newhouse are reported in Chapter 12. They estimate the effect of cardiac catheterization on patient survival times. They instrument the use of
13.10 Consider the regression model with heterogeneous regression coefficients Yi = b0i + b1iXi + vi, where (vi, Xi, b0i, b1i) are i.i.d. random variables with b0 = E(b0i) and b1 = E(b1i).a. Show
13.9 Derive the final equality in Equation (13.10). (Hint: Use the definition of the covariance and that, because the actual treatment Xi is random, b1i and Xi are independently distributed.)
13.8 Suppose that you have the same data as in Exercise 13.7 (panel data with two periods, n observations), but ignore the W regressor. Consider the alternative regression model Yit = b0 + b1Xit +
13.7 Suppose that you have panel data from an experiment with T = 2 periods(so t = 1, 2). Consider the panel data regression model with fixed individual and time effects and individual
13.6 Suppose that there are panel data for T = 2 time periods for a randomized controlled experiment, where the first observation (t = 1) is taken before the experiment and the second observation (t

Showing 1 - 100 of 1029