Introduction To Econometrics 4th Global Edition James H. Stock, Mark W. Watson - Solutions

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19.17 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 MW = I - W(WW)-1W.a. Show that the OLS estimators of B and G can be written as cB nGn d = c XX
19.16 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 term/intercept is omitted for convenience.a. Suppose that the least squares assumptions in Key Concept 4.3 are
19.15 (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 and strengthen assumption 3, so that Xit and uit have eight nonzero finite moments. Let M = IT -T
19.14 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 = 3Ik1 0k1 * k24.a. Show that B n (XX)B n= (RB n)3R(XX)-1R4-1(RB n).b. Consider the regression described in Equation
19.13 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 B be the value of b that solves the constrained minimization problem.a. Show that the Lagrangian for the minimization problem is L(b, g) = (Y - Xb)
19.12a. Show that BEff.GMM is the efficient GMM estimator—that is, that BEff.GMM in Equation (19.66) is the solution to Equation (19.65).b. Show that 2n (B nEff.GMM - BEff.GMM) ¡p 0.c. Show that JGMM ¡d x2 m- k.
19.11 Suppose that C is an n * n symmetric idempotent matrix with rank r, and let V N(0n, In).a. Show that C = A A, where A is n * r with AA = Ir. (Hint: C is positive semidefinite and can be written as Q Q, as explained in Appendix 19.1.)b. Show that AV N10r, Ir2.c. Show that VCV x2r..
19.10 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. Show that trace(C) = rank(C).c. Let d be an n * 1 vector. Show that dCd Ú 0.
19.9 This exercise shows that the OLS estimator of a subset of the regression coefficients is consistent under the conditional mean independence assumption stated in Key Concept 6.6. Consider the multiple regression model in matrix form Y = XB + WG + U, where X and W are, respectively, n * k1 and n
19.8 Consider the regression model Yi = b0 + b1Xi + ui, where u1 = u 1 and ui = 0.5ui - 1 + u i for i = 2, 3,c, n. Suppose that u i are i.i.d. with mean 0 and variance 1 and are distributed independently of Xj for all i and j.a. Derive an expression for E(U U) = .b. Explain how to estimate the
19.7 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 0.Suppose that Xi is distributed independently of (Wi, ui) but Wi and ui might be correlated, and let b n1 and b n2 be the OLS estimators for this
19.6 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 19.17, the OLS estimator B n can be expressed Bn= (XMWX )-1(XMWY ).Now let b nBV 1 be the “binary variable” fixed
19.5 Let PX and MX be as defined in Equations (19.24) and (19.25).a. Prove that PXMX = 0n * n and that PX and MX are idempotent.b. Derive Equations (19.27) and (19.28).c. Show that rank(PX) = k + 1 and rank(MX) = n - k - 1. [Hint: First solve Exercise 19.10, and then use the fact that trace(AB) =
19.4 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 (19.2) and (19.3).b. Show that assumptions 1 through 4 in Key Concept 19.1 are satisfied.c. Use the
19.3 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 0m. Show that 0 6 var(Q) 6 .
19.2 Suppose that a sample of n = 20 households has the sample means and sample covariances below for a dependent variable and two regressors:Exercises 743 Sample Means Sample Covariances Y X1 X2 Y 6.39 0.26 0.22 0.32 X1 7.24 0.80 0.28 X2 4.00 2.40a. Calculate the OLS estimates of b0, b1, and b2.
19.1 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(19.5). Define Y, X, U, and B.b. Explain how to test the null hypothesis that the relationship between test scores
19.5 Construct an example of a regression model that satisfies the assumption E(ui Xi) = 0 but for which E(U X ) 0n.
19.4 When is the GLS estimator more efficient than the OLS estimator within the class of linear conditionally unbiased estimators?
19.3 Suppose that assumptions 1 through 5 in Key Concept 19.1 are true but that assumption 6 is not. Does the result in Equation (19.31) hold? Explain.
19.2 You are analyzing a linear regression model with 500 observations and one regressor. Explain how you would construct a confidence interval for b1 ifa. Assumptions 1 through 4 in Key Concept 19.1 are true but you think assumption 5 or 6 might not be true.b. Assumptions 1 through 5 are true but
19.1 A researcher studying the relationship between earnings and workers’ sex specifies the regression model Yi = b0 + X1ib1 + X2ib2 + ui, where X1i is a binary variable that equals 1 if the ith person is a female and X2i is a binary variable that equals 1 if the ith person is a male. Write the
18.15 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, thata. W>n ¡ p 1.b. Z 2W>n ¡ d N(0, 1). Use the result to explain why the t distribution is the same as the standard normal distribution.c. V/m W>n ¡ d x2 m > m. Use the result to
18.14 Suppose that Yi, i = 1, 2,c, n, are i.i.d. with E(Yi) = m, var(Yi) = s2, and finite fourth moments. Show the following:a. E1Y2i 2 = m2 + s2.b. Y ¡ p m.c.1 na ni = 1 Y2i¡ p m2 + s2.d.1 na ni = 1(Yi - Y)2 =1 na ni = 1 Y2i- Y2.e.1 na ni = 1(Yi - Y)2 ¡ p s2.f. s2 =1 n - 1a ni = 1(Yi - Y)2 ¡ p
18.13 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 Xi) = 0 and (b0i, b1i) are distributed independently of Xi and that the observational units are randomly drawn from the
18.12a. Suppose that u N(0, s2 u). Show that E(eu) = e1 2s2u.b. 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 X = x) = e1 2(a + bx2).
18.11 Suppose that X and Y are distributed bivariate normal with the density given in Equation (18.38).a. Show that the density of Y given X = x can be written as fYX= x(y) =1 sYX22p expc -1 2a y - mYX sYX b2 dwhere sYX = 2s2 Y(1 - r2 XY) and mYX = mY + (sXYs2 X)(x - mX).[Hint: Use the
18.10 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, if the mean squared error of un tends to 0), then u n ¡ p u. [Hint: Use Equation (18.43) with W = u n - u.]
18.9 Prove Equation (18.16) under assumptions 1 and 2 of Key Concept 18.1 plus the assumption that Xi and ui have eight moments.
18.8 Consider the regression model in Key Concept 18.1, and suppose that assumptions 1, 2, 3, and 5 hold. Suppose that assumption 4 is replaced by the assumption that var(ui Xi) = u0 + u1 Xi , where Xi is the absolute value of Xi, u0 7 0, and u1 Ú 0.a. Is the OLS estimator of b1 BLUE?b.
18.7 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, Xi) f(uj, Xj) for i j, where f(ui, Xi) is the joint p.d.f. of ui and Xi.b. Show that E(uiuj Xi,
18.6 Show that if b n1 is conditionally unbiased, then it is unbiased; that is, show that if E(b n1 X1,c, Xn) = b1, then E(b n1) = b1.
18.5 Suppose that W is a random variable with E(W4) 6 . Show that E(W2) 6 .
18.4 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 s2 u > s2 u ¡ p 1 implies that su > su ¡ p 1.
18.3. This exercise fills in the details of the derivation of the asymptotic distribution of b n1 given in Appendix 4.3.a. Use Equation (18.19) to derive the expression 2n(b n1 - b1) =A 1n an i = 1 vi 1n an i = 1(Xi - X)2-(X - mX)A 1n an i = 1 ui 1n an i = 1(Xi - X)2, where vi = (Xi - mX)ui.b. Use
18.2 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, that sXY ¡ p sXY, where sXY is defined in Equation (3.24).(Hint: Use the strategy outlined in Appendix 3.3 and the Cauchy–Schwarz
18.1 Consider the regression model without an intercept term, Yi = b1Xi + ui (so the true value of the intercept, b0, is 0).a. Derive the least squares estimator of b1 for the restricted regression model Yi = b1Xi + ui. This is called the restricted least squares estimator (b nRLS 1 ) of b1 because
18.4 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 10, and then using the average the two OLS of estimators. Is this estimator more efficient than WLS?
18.3 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 Xi = x) = 1 for 0 … x … 10 and var(ui Xi = x) = 16 for 10 6 x … 20. Draw a hypothetical scatterplot of the
18.2 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 is the asymptotic distribution of AnBn? Use this asymptotic distribution to compute an approximate
18.1 Suppose that assumption 4 in Key Concept 18.1 is true but you construct a 95%confidence interval for b1 using the heteroskedastic-robust standard error in a large sample. Would this confidence interval be valid asymptotically in the sense that it contained the true value of b1 in 95% of all
E17.2 On the text website, http://www.pearsonglobaleditions.com, you will find the data file USMacro_Quarterly, which contains quarterly data on real GDP, measured in 2009 dollars. Compute GDPGRt = 400 * [ln(GDPt) -ln(GDPt-1)], the GDP growth rate.a. Using data on GDPGRt from 1960:Q1 to 2017:Q4,
E17.1 This exercise is an extension of Empirical Exercise 14.1. On the text website, http://www.pearsonglobaleditions.com, you will find the data file USMacro_Quarterly, which contains quarterly data on several macroeconomic series for the United States; the data are described in the file
17.10 Consider the cointegrated model Yt = uXt + v1t and Xt = Xt - 1 + v2t, where v1t and v2t are mean 0 serially uncorrelated random variables with E(v1t v2j) = 0 for all t and j. Derive the vector error correction model [Equations (17.22) and(17.23)] for X and Y.
17.9a. Suppose that E(ut ut - 1, ut - 2,c) = 0, that var(ut ut - 1, ut - 2,c) follows the ARCH(1) model s2t= a0 + a1u2t- 1, and that the process for ut is stationary. Show that var(ut) = a0>(1 - a1). (Hint: Use the law of iterated expectations, E(u2t) = E3E(u2t ut - 1)4.)b. Extend the result
17.8 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 for Y can be written as Ytt - 2 = d1Yt - 2 + d2Xt - 2, and derive values for d1 and d2 in terms of
17.7 Suppose that Yt = ut, where ut is i.i.d. N(0, 1), and consider the regression Yt = bXt + error, where Xt = Yt + 1 and error is the regression error. Show that b n ¡d 12(x21- 1). [Hint: Analyze the numerator of b n using analysis like that in Equation (17.21). Analyze the denominator using
17.6 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 (17.25).What are the values of u, d-1, d0, and d1?i. Suppose that Xt is I(0) and ut is I(0). Are Y and X
17.5 Verify Equation (17.20). [Hint: Use gTt= 1Y2t= gTt= 1(Yt - 1 + Yt)2 to show that gTt= 1Y2t= gTt= 1Y2t- 1 + 2gTt= 1Yt - 1 Yt + gTt= 1Y2t, and solve for gTt= 1 Yt - 1 Yt.]
17.4 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 + ht = E(Yt + h Yt, Yt - 1, c). Show that Yt + ht = b0 + b1Yt - 1 + ht + g+ bpYt - p + ht for h 7 p.
17.3 Suppose that E(ut ut - 1, ut - 2, . . .) = 0 and 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, E(u2t) = E3Eu2t ut - 1)4.)b. Suppose that the distribution of
17.2 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 shortterm interest rates into the future plus a term premium that is I(0). Specifically, let Rkt denote a k-period interest rate, let R1t
17.1 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 + bh1(Yt - mY), where mY = b0>(1 - b1).b. Suppose that Xt is related to Yt by Xt = g i = 0 diYt + it, where d 6 1.Show that Xt = 3mY>(1 - d)4
17.5 What is a unit root? How does a researcher test for the presence of a unit root in the data?
17.4 What is volatility clustering? Explain two models that are used to describe data processes with volatility clustering.
17.3 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 equal to 1. Explain how you would investigate this implication by (a) plotting the data and (b) using a
17.2 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 for h = 20 seem reasonable to you?
17.1 A macroeconomist wants to construct forecasts for the following macroeconomic variables: GDP, consumption, investment, government purchases, exports, imports, short-term interest rates, long-term interest rates, and the rate of price inflation. He has quarterly time series for each of these
E16.3 In the data file USMacro_Quarterly, you will find the data on U.S. real GDP(GDPC1) that was analyzed in Chapter 15. In this exercise, you will construct a 95% confidence interval for the mean growth rate of real GDP in the United States; in addition, you will construct a 95% forecast interval
E16.2 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), which you used in Empirical Exercise 15.1, and the Consumer Price Index (CPI). These series are alternative measures of
E16.1 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 Exercise 16.1. The data can be found on the text website, http://www.pearsonglobaleditions.com, in
16.12 Suppose Yt = b0 + ut, where ut follows a stationary stationary AR(1)ut = f1ut - 1 + u t with u t i.i.d. with mean 0 and variance s2 u and |f1| 6 1.a. Show that b0 = μY = E(Yt).b. Let Y1:T = 1T gTt= 1Yt denote the sample mean of Yt using observations from t = 1 through t = T. Show that the
16.11 Suppose that a(L) = (1 - fL), with f1 6 1, and b(L) = 1 + fL +f2L2 +f3L3 c.a. Show that the product b(L)a(L) = 1, so that b(L) = a(L)-1.b. Why is the restriction f1 6 1 important?
16.10 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. Derive the first five cumulative multipliers.d. Derive the long-run cumulative dynamic multiplier.
16.9 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 f1 6 1.a. Show that the OLS estimator is b n0 = T-1gTt= 1Yt.b. Show that the (infeasible) GLS estimator is b
16.8 Consider the model in Exercise 16.7 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 estimator b nGLS 1 ¡p b1 - f1 1 + f21 .[Hint: Apply the omitted variable formula in Equation (6.1)
16.7 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 f1 6 1.a. Suppose that Xt is independent of u j for all t and j. Is Xt exogenous (past and present)? Is Xt strictly
16.6 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 f1 6 1; the regressor Xt follows the stationary AR(1) model Xt = g1Xt - 1 + et with et i.i.d. with mean 0 and variance s2e
16.5 Derive Equation (16.7) from Equation (16.4), 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.)
16.4 Suppose that oil prices are strictly exogenous. Discuss how you could improve on the estimates of the dynamic multipliers in Exercise 16.1.
16.3 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, including changes in the oil price. In experiment B, oil prices are set randomly, and the central
16.2 Macroeconomists have also noticed that interest rates change following oil price jumps. Let Rt denote the interest rate on three-month Treasury bills (in percentage points at an annual rate). The distributed lag regression relating the change in Rt (Rt) to Ot estimated over 1960:Q1–2017:Q4
16.1 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 those discussed in this chapter. Let GDPt denote the value of quarterly real GDP in the United States,
16.4 Suppose that you added FDDt + 1 as an additional regressor in Equation (16.2).If FDD is strictly exogenous, would you expect the coefficient on FDDt + 1 to be 0 or nonzero? Would your answer change if FDD is exogenous but not strictly exogenous?
16.3 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.
16.2 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 estimate a new ADL model with additional lags or simply use HAC standard errors for the ADL(1, 1)
16.1 In the 1970s, a common practice was to estimate a distributed lag model relating changes in nominal GDP (Y) to current and past changes in the money supply (X). Under what assumptions will this regression estimate the causal effects of money on nominal GDP? Are these assumptions likely to be
E15.2 Read the box “Can You Beat the Market?” Next go to the course website, where you will find an extended version of the data set described in the box;the data are in the file Stock_Returns_1931_2002 and are described in the file Stock_Returns_1931_2002_Description.a. Repeat the calculations
E15.1 On the text website, http://www.pearsonglobaleditions.com, you will find the data file USMacro_Quarterly, which contains quarterly data on several macroeconomic series for the United States; the data are described in the file USMacro_Description. The variable PCEP is the price index for
15.13 Suppose Yt follows a random walk, Yt = Yt−1 + ut, for t = 1,c, T, where Y0 = 0 and ut is i.i.d. with mean 0 and variance s2 u.a. Compute the mean and variance of Yt.b. Compute the covariance between Yt and Yt−k.c. Use the results in (a) and (b) to show that Yt is nonstationary.
15.12 Consider the stationary AR(1) model Yt = b0 + b1Yt−1 + ut, where ut is i.i.d.with mean 0 and variance s2 u . The model is estimated using data from time periods t = 1 through t = T, yielding the OLS estimators b n0 and b n1. You are interested in forecasting the value of Y at time T +
15.11 Suppose 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.
15.10 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 15.5(“Critical Values of the QLR Statistic with 15% Trimming”) and Appendix Table 4 (“Critical Values for the Fm, Distribution”).a. The
15.9 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 E1Yt2 = b0.b. Show that the variance of Yt is var1Yt2 = s2e 11 + b21+ b22+ g+ b2 q2.c. Show that rj =
15.8 Suppose 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, housing starts are low in January and high in June. Let mJan denote the average value of housing starts in
15.7 Suppose 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 15.1.)b. Compute the first two autocovariances of Yt. (Hint: Read Appendix 15.2.)c. Compute the first two
15.6 In this exercise, you will conduct a Monte Carlo experiment to study the phenomenon of spurious regression discussed in Section 15.7. In a Monte Carlo study, artificial data are generated using a computer, and then those artificial data are used to calculate the statistics being studied. This
15.5 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 E31W - c2 24 = s2 w + 1mW - c2 2.b. Consider the problem of forecasting Yt, using data on Yt - 1, Yt - 2, c.Let
15.4 The forecaster in Exercise 15.2 augments her AR(4) model for IP growth to include four lagged values of Rt, where Rt is the interest rate on three-month U.S. Treasury bills (measured in percentage points at an annual rate).a. The F-statistic on the four lags of Rt is 3.91. Do interest rates
15.3. Using the same data as in Exercise 15.2, a researcher tests for a stochastic trend in ln(IPt), using the following regression:ln1IPt2 = 0.026 + 0.000097t - 0.0070 ln1IPt - 12 + 0.068ln1IPt - 12 10.0132 10.0000672 10.00372 10.0502+ 0.169ln1IPt - 22 + 0.219ln1IPt - 32 + 0.173ln1IPt - 42,
15.2 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 United States. All regressions are estimated over the sample period 1986:M1–2017:M12 (that is,
15.1 Consider the AR(1) model Yt = b0 + b1Yt - 1 + ut. Suppose the process is stationary.a. Show that E1Yt2 = E1Yt - 12. (Hint: Read Key Concept 15.3.)b. Show that E1Yt2 = b0 > 11 - b12.
15.4 Suppose you suspected that the intercept in Equation (15.15) changed in 1992:Q1. How would you modify the equation to incorporate this change?How would you test for a change in the intercept? How would you test for a change in the intercept if you did not know the date of the change?
15.3 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.
15.2 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 financial analyst claims to have a new model that makes better predictions than the random walk
15.1 Look at the four plots in Figure 15.2—U.S. unemployment rate, U.S. dollar/British pound exchange rate, logarithm of Japan index of industrial production, and the percentage change in daily values. Which of these series appears to be nonstationary? Which of them appears to resemble a random
E14.1 On the text website, http://www.pearsonglobaleditions.com, you will find a data set CASchools_EE14_InSample that contains a subset of n = 500 schools from the data set used in this chapter. Included are data on test scores and 20 of the primitive predictor variables; see CASchools_EE141_
14.12 Consider the fixed-effects panel data model Yjt = aj + ujt for j = 1,c, k and t = 1,c, T. Assume that ujt is i.i.d. across entities j and over time t with E1ujt2 = 0 and var1ujt2 = s2 u.a. The OLS estimator of aj is the value of aj that makes the sum of squared residuals gk j = 1gTt= 11Yjt -
14.11 Let X1 and X2 be two positively correlated random variables, both with variance 1.a. (Requires calculus) The first principal component, PC1, is the linear combination of X1 and X2 that maximizes var1w1X1 + w2X22, where w21+ w22= 1. Show that PC1 = 1X1 + X22 >22 . (Hint: First derive an
14.10 You have a sample of size n = 1 with data y1 = 2 and x1 = 1. You are interested in the value of b in the regression Y = Xb + u. (Note there is no intercept.)a. Plot the sum of squared residuals (y1 - bxl)2 as function of b.b. Show that the least squares estimate of b is b nOLS = 2.c. Using

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Introduction To Econometrics 4th Global Edition James H. Stock, Mark W. Watson - Solutions

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