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Introduction To Econometrics 3rd Global Edition James Stock, Mark Watson - Solutions

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) are distributed independently of Xi.a. Let b nOLS 1 denote the OLS estimator of b1 given in Equation
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 = x) = e1 2(a + bx2).
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 - (sXY>s2 X)(x - mX).[Hint: Use the definition of the conditional probability density fY 0X=
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: Use Equation (17.43) with W = u n - u.4
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 , where 0Xi 0 is the absolute value of Xi, u0 7 0, and u1 Ú 0.a. Is the OLS estimator of b1 BLUE?b.
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, 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 0Xi, Xj)
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 that su>su ¡ p 1.
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 the central limit theorem, the law of large numbers, and Slutsky’s theorem to show that the final
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 in Equation (3.24). (Hint: Use the strategy outlined in Appendix 3.3 and the Cauchy–Schwarz
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 model Yi = b1Xi + ui. This is called the restricted least squares estimator(b nRLS 1 ) of b1
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 10, and then using the average the two OLS of estimators. Is this estimator more efficient than WLS?
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 … x … 10 and var(ui 0 Xi = x) = 16 for 10 6 x … 20. Draw a hypothetical scatterplot of the
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 is the asymptotic distribution of AnBn? Use this asymptotic distribution to compute an approximate
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 interval be valid asymptotically in the sense that it contained the true value of b1 in 95% of all
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 = 400 × [ln(GDPt) −ln(GDPt−1)], the growth rate of GDP.a. Using data on GDPGRt from 1960:1 to
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 quarterly data on several macroeconomic series for the United States; the data are described in the file
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 vector error correction model[Equations (16.22) and (16.23)] for X and Y.
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 - a12. (Hint: Use the law of iterated expectations E(u2t) = E[E(u2t ut - 1)].)b. Extend the result
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 for Y can be written as Ytt - 2 = d1Yt - 2 + d2Xt - 2 and derive values for d1 and d2 in terms of
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: Analyze the numerator of b n using analysis like that in Equation (16.21). Analyze the denominator
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). What are the values of u,d- 1, d0, and d1?b. i. Suppose that Xt is I(0) and ut is I(0). Are Y and X
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 - 1 + h•t + g + bpYt - p + h•t for h > p.
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, E1u2t 2 = E3E1u2t• ut - 124.)b. Suppose that the distribution of ut conditional on lagged values of
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, plus a term premium that is I(0). Specifically, let Rkt denote a k-period interest rate, let R1t
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. Suppose that Xt is related to Yt by Xt = g i = 0 diYt + it, where d < 1.Show that Xt = [mY>(1 -
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 equal to 1. Explain how you would investigate this implication by (a) plotting the data and (b) using a
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 for h = 20 seem reasonable to you?
16.1 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
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 Empirical Exercise 14.1 and the Consumer Price Index (CPI). These series are alternative measures of
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 Exercise 15.1. The data can be found on the textbook website, www.pearsonglobaleditions.com/Stock_
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. Derive the first five cumulative multipliers.d. Derive the long-run cumulative dynamic multiplier.
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 1.a. Show that the OLS estimator is b n0 = T -1gTt= 1Yt.b. Show that the (infeasible) GLS estimator
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 estimator b nGLS 1 ¡ p b1 -f1 1 + f21.[Hint: Use the omitted variable formula (6.1) applied to the
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 Xt is independent of u j for all t and j. Is Xt exogenous(past and present)? Is Xt strictly
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 follows the stationary AR(1)model Xt = g1Xt - 1 + et with et i.i.d. with mean 0 and variance s2e
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, including changes in the oil price. In experiment B, oil prices are set randomly, and the central
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 distributed lag regression relating the change in Rt (Rt) to Ot estimated over 1960:Q1–2013:Q4 isa.
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 those discussed in this chapter. Let GDPt denote the value of quarterly gross domestic product in the
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 change if FDD is exogenous but not strictly exogenous?
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 estimate a new ADL model with additional lags or simply use HAC standard errors for the ADL(1,1)
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 assumptions will this regression estimate the causal effects of money on nominal GDP? Are these
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 data set described in the boxes;the data are in the file Stock_Returns_1931_2002 and are described in
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 States; the data are described in the file USMacro_Description. The variable PCEP is the price index for
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 with 15% Trimming”)and Appendix Table 4 (“Critical Values of the Fm, Distribution”).a. The
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 E1Yt2 = b0.b. Show that the variance of Yt is var(Yt) = s2e(1 + b21+ b22+g+ b2 q).c. Show that rj = 0
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, housing starts are low in January and high in June. Let mJan denote the average value of housing starts
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 14.1.)b. Compute the first two autocovariances of Yt. (Hint: Read Appendix 14.2.)c. Compute the first two
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 computer, and then those artificial data are used to calculate the statistics being studied. This
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 = s2 w + (mW - c)2.b. Consider the problem of forecasting Yt, using data on Yt - 1, Yt - 2, c .Let ft
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 points at an annual rate).a. The Granger-causality F-statistic on the four lags of Rt is 4.16. Do
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 the homoskedasticity-only formula and the regressor t is a linear time trend.a. Use the ADF statistic
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 United States. All regressions are estimated over the sample period 1986:M1 to 2013:M12 (that is,
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 intercept? How would you test for a change in the intercept if you did not know the date of the change?
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 financial analyst claims to have a new model that makes better predictions than the random walk
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. Which of these series appears to be non-stationary? Which of them appears to resemble a random walk?
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 applicant to arrange an interview? Marianne Bertrand and Sendhil Mullainathan carried out a randomized
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) denote the treatment effect for individual i.a. Can you consistently estimate E [Yi(1)] and E[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 cardiac catheterization by the distance between a patient’s home and a hospital that offers the
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 that the model can be written as Yi = b0 + b1Xi + ui, where ui = (b0i - b0) + (b1i - b1)Xi + vi.b.
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 + b2Gi + b3Dt + uit, where Gi = 1 if the individual is in the treatment group and Gi = 0 if the
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 characteristics Wi that do not change over time, such as gender. Let the treatment be binary, so that Xit = 1 for
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 = 2) is for the posttreatment period. Suppose that the treatment is binary; that is, suppose that
13.5 Consider a study to evaluate the effect on college student grades of dorm room Internet connections. In a large dorm, half the rooms are randomly wired for high-speed Internet connections (the treatment group), and final course grades are collected for all residents. Which of the following
13.4 Read the box “What Is the Effect on Employment of the Minimum Wage?” in Section 13.4. Suppose, for concreteness, that Card and Krueger collected their data in 1991 (before the change in the New Jersey minimum wage) and in 1993 (after the change in the New Jersey minimum wage).Consider
13.3 Suppose in a randomized controlled experiment of the effect of a SAT preparatory course on SAT scores, the following results are reported:a. Estimate the average treatment effect on test scores.b. Is there evidence of nonrandom assignment? Explain. Treatment Group Control Group Average SAT
13.2 For the following calculations, use the results in column (3) of Table 13.2.Consider two classrooms, A and B, with identical values of the regressors in column (3) of Table 13.2, except that:a. Classroom A is a “small class,” and classroom B is a “regular class.”Construct a 90%
13.1 How would you calculate the small class treatment effect from the results in Table 13.1. Can you distinguish this treatment effect from the aide treatment effect? How would you change the program to correctly estimate both effects?
13.5 Consider the quasi-experiment described in Section 13.4 involving the draft lottery, military service, and civilian earnings. Explain why there might be heterogeneous effects of military service on civilian earnings; that is, explain why b1i in Equation (13.9) depends on i. Explain why there
13.4 What are experimental effects? How can they create biased treatment effects and what can a researcher do to reduce the bias?
13.3 Researchers studying the STAR data report anecdotal evidence that school principals were pressured by some parents to place their children in the small classes. Suppose that some principals succumbed to this pressure and transferred some children into the small classes. How would such
13.2 A clinical trial is carried out for a new cholesterol-lowering drug. The drug is given to 500 patients, and a placebo is given to another 500 patients, using random assignment of the patients. How would you estimate the treatment effect of the drug? Suppose that you had data on the weight,
13.1 A researcher studying the effects of a new fertilizer on crop yields plans to carry out an experiment in which different amounts of the fertilizer are applied to 100 different 1-acre parcels of land. There will be four treatment levels. Treatment level 1 is no fertilizer, treatment level 2 is
E12.3 (This requires Appendix 12.5) On the textbook website, www.pearsonglobaleditions.com/Stock_Watson, you will find the data set WeakInstrument which contains 200 observations on (Yi, Xi, Zi) for the instrumental regression Yi = b0 + b1Xi + ui.a. Construct b nTSLS 1 , its standard error, and the
E12.2 Does viewing a violent movie lead to violent behavior? If so, the incidence of violent crimes, such as assaults, should rise following the release of a violent movie that attracts many viewers. Alternatively, movie viewing may substitute for other activities (such as alcohol consumption)that
E12.1 How does fertility affect labor supply? That is, how much does a woman’s labor supply fall when she has an additional child? In this exercise you will estimate this effect using data for married women from the 1980 U.S. Census.4 The data are available on the textbook website,
12.10 Consider the instrumental variable regression model Yi = b0 + b1Xi +b2Wi + ui, where Zi is an instrument. Suppose that data on Wi are not available and the model is estimated omitting Wi from the regression.a. Suppose that Zi and Wi are uncorrelated. Is the IV estimator consistent?b. Suppose
12.9 A researcher is interested in the effect of military service on human capital.He collects data from a random sample of 4000 workers aged 40 and runs the OLS regression Yi = b0 + b1Xi + ui, where Yi is a worker’s annual earnings and Xi is a binary variable that is equal to 1 if the person
12.8 Consider a product market with a supply function Qsi= b0 + b1Pi + usi, a demand function Qdi= g0 + udi, and a market equilibrium condition Qsi= Qdi, where usi and udi are mutually independent i.i.d. random variables, both with a mean of zero.a. Show that Pi and usi are correlated.b. Show that
12.7 In an instrumental variable regression model with one regressor, Xi, and two instruments, Z1i and Z2i, the value of the J-statistic is J = 2.2.a. Does this suggest that E(ui 0Z1i, Z 2i) 0? Explain.b. What steps would you take if the J-statistic was 22.2?
12.6 In an instrumental variable regression model with one regressor, Xi, and one instrument, Zi, the regression of Xi onto Zi has R2 = 0.1 and n = 50.Is Zi a strong instrument? [Hint: See Equation (7.14).] Would your answer change if R2 = 0.1 and n = 150?
12.5 Consider the instrumental variable regression model Yi = b0 + b1Xi + b2Wi + ui, where Xi is correlated with ui and Zi is an instrument. Suppose that the first three assumptions in Key Concept 12.4 are satisfied. Which IV assumption is not satisfied when:a. Zi is independent of (Yi, Xi, Wi)?b.
12.4 Consider TSLS estimation with a single included endogenous variable and a single instrument. Then the predicted value from the first-stage regression is X ni = pn 0 + pn 1Zi. Use the definition of the sample variance and covariance to show that sXn Y = pn 1sZY and s2 Xn = pn1 2 s2 Z. Use this
12.3 A classmate is interested in estimating the variance of the error term in Equation (12.1).a. Suppose that she uses the estimator from the second-stage regression of TSLS: sn 2a= 1 n - 2gni= 1(Yi - b nTSLS 0 - b nTSLS 1 X ni)2, where X ni is the fitted value from the first-stage regression. Is
12.2 Consider the regression model with a single regressor: Yi = b0 + b1Xi + ui.Suppose that the least squares assumptions in Key Concept 4.3 are satisfied.a. Show that Xi is a valid instrument. That is, show that Key Concept 12.3 is satisfied with Zi = Xi.b. Show that the IV regression assumptions
12.1 This question refers to the panel data regressions summarized in Table 12.1.a. Suppose that the Federal Government is considering a new tax on cigarettes that is estimated to increase the retail price by $0.25 per pack. If the current price per pack is $6.75, use the regression in column (1)
12.4 In their study of the effectiveness of cardiac catheterization, McClellan, McNeil, and Newhouse (1994) used as an instrument the difference in distance to cardiac catheterization and regular hospitals. How could you determine whether this instrument is relevant? How could you determine whether

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