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Econometrics 1st Edition Bruce Hansen - Solutions
Take the linear spline from the previous question. Find the inequality restrictions on the coefficients ¯j so thatmK (x) is concave.
Take the linear spline with three knots mK (x) Æ ¯0 ů1x ů2 (x ¡¿1)1{x ¸ ¿1}ů3 (x ¡¿2)1{x ¸ ¿2}ů4 (x ¡¿3)1{x ¸ ¿3} .Find the inequality restrictions on the coefficients ¯j so thatmK (x) is non-decreasing.
Take the estimated model Y Æ ¡1Å2X Å5(X ¡1)1{X ¸ 1}¡3(X ¡2)1{X ¸ 2}Åe.What is the estimated marginal effect of X on Y for X Æ 3?
We will consider a nonlinear AR(1) model for gdp growth rates(a) Create GDP growth rates Yt . Extract the level of real U.S. GDP (gdpc1) from FRED-QD and make the above transformation to growth rates.(b) Use Nadaraya-Watson to estimatem(x). Plot with 95% confidence intervals.(c) Repeat using the
The RR2010 dataset is from Reinhart and Rogoff (2010). It contains observations on annual U.S. GDP growth rates, inflation rates, and the debt/gdp ratio for the long time span 1791-2009.The paper made the strong claim that gdp growth slows as debt/gdp increases and in particular that this
Take the Invest1993 dataset and the subsample of observations with Q · 5. (In the dataset Q is the variable vala.)(a) Use Nadaraya-Watson to estimate the regression of I on Q. (In the dataset I is the variable inva.)Plot with 95% confidence intervals.(b) Repeat using the Local Linear estimator.(c)
Take the cps09mar dataset and the subsample of individuals with education=20 (professional degree or doctorate), with experience between 0 and 40 years.(a) Use Nadaraya-Watson to estimate the regression of log(wage) on experience, separately for men and women. Plot with 95% confidence intervals.
Take the DDK2011 dataset and the subsample of boys who experienced tracking. As in Section 19.21 use the Local Linear estimator to estimate the regression of testscores on percentile but now with the subsample of boys. Plot with 95% confidence intervals. Comment on the similarities and differences
Prove Theorem 19.11: Show that when d ¸ 1 the AIMSE optimal bandwidth takes the form h0 Æ cn¡1/(4Åd) and AIMSE is O¡n¡4/(4Åd)¢.
Supposem(x) Æ ® is a constant function. Find the AIMSE-optimal bandwith (19.6) forNW estimation? Explain.
Suppose the true regression function is linearm(x) Æ ®Å¯x and we estimate the function using the Nadaraya-Watson estimator. Calculate the bias function B(x). Suppose ¯ È 0. For which regions is B(x) È 0 and for which regions is B(x) Ç 0? Now suppose that ¯ Ç 0 and re-answer the
Describe in words how the bias of the local linear estimator changes over regions of convexity and concavity inm(x). Does this make intuitive sense?
Show that (19.6) minimizes the AIMSE (19.5).
For kernel regression suppose you rescale Y , for example replace Y with 100Y . How should the bandwidth h change? To answer this, first address how the functions m(x) and ¾2(x) change under rescaling, and then calculate how B and ¾2 change. Deduce how the optimal h0 changes due to rescaling Y .
Use the datafile BMN2016 on the textbook webpage. The authors report results for liquor sales. The data file contains the same information for beer and wine sales. For either beer or wine sales, estimate diff-in-diff models similar to (18.7) and (18.8) and interpret your results. Some relevant
Use the datafile DS2004 on the textbook webpage. The authors argued that an exogenous police presence would deter automobile theft. The evidence presented in the chapter showed that car theft was reduced for city blocks which received police protection. Does this deterrence effect extend beyond the
Use the datafile CK1994 on the textbook webpage. Classical economics teaches that increasing the minimum wage will increase product prices. You can therefore use the Card-Krueger diffin-diff methodology to estimate the effect of the 1992 New Jersey minimum wage increase on product prices. The data
An economist is interested in the impact ofWisconsin’s 2011“Act 10” legislation on wages.(For background, Act 10 reduced the power of labor unions.) She computes the following statistics5 for average wage rates inWisconsin and the neighboring state ofMinnesota for the decades before and after
For the specification tests of Section 18.4 explain why the regression test for homogeneous treatment effects includes only N2 ¡1 interaction dummy variables rather than all N2 interaction dummies. Also explain why the regression test for equal control effects includes only N1 ¡1 interaction
Take the basic difference-in-difference model Yi t Æ µDi t Åui űt Å"i t .Instead of assuming that Di t and "i t are independent, assume we have an instrumental variable Zi t which is independent of "i t but is correlated with Di t . Describe how to estimate µ.Hint: Review Section 17.28.
In regression (18.1) with T Æ 2 and N Æ 2 suppose the time variable is omitted. Thus the estimating equation is Yi t Æ ¯0 ů1Statei ŵDi t Å"i t .where Di t ÆStateiTimet is the treatment indicator.(a) Find an algebraic expression for the least squares estimator bµ.(b) Show that bµ is a
In the text it was claimed that in a balanced sample individual-level fixed effects are orthogonal to any variable demeaned at the state level.(a) Show this claim.(b) Does this claim hold in unbalanced samples?(c) Explain why this claim implies that the regressionsyield identical estimates of µ.
Use the datafile Invest1993 on the textbook webpage. You will be estimating the model Di t Æ ®Di ,t¡1 ů1Ii ,t¡1 ů2Qi ,t¡1 ů3CFi ,t¡1 Åui Å"i t .The variables are debta, inva, vala, and cfa in the datafile. See the description file for definitions.(a) Estimate the above regression
Use the datafile Invest1993 on the textbook webpage. You will be estimating the panel AR(1) Di t Æ ®Di ,t¡1 Åui Å"i t for D Ædebt/assets (this is debta in the datafile). See the description file for definitions.(a) Estimate the model using Arellano-Bond twostep GMMwith clustered standard
This exercise uses the same dataset as the previous question. Blundell and Bond (1998)estimated a dynamic panel regression of log employment N on log real wages W and log capital K. The following specification1 used the Arellano-Bond one-step estimator, treatingWi ,t¡1 and Ki ,t¡1 as
In this exercise you will replicate and extend the empirical work reported in Arellano and Bond (1991) and Blundell and Bond (1998). Arellano-Bond gathered a dataset of 1031 observations from an unbalanced panel of 140 U.K. companies for 1976-1984 and is in the datafile AB1991 on the textbook
In Section 17.33 verify that in the just-identified case the 2SLS estimator b¯2sls simplifies as claimed: b¯1 and b¯2 are the fixed effects estimator. b°1 and b°2 equal the 2SLS estimator froma regression of bu on Z1 and Z2 using X 1 as an instrument for Z2.
Take the fixed effects model Yi t Æ Xi t¯1 Å X2 i t¯2 Åui Å"i t . A researcher estimates the model by first obtaining the within transformed ˙ Yi t and ˙X i t and then regressing ˙ Yi t on ˙X i t and ˙X 2 i t . Is the correct estimation method? If not, describe the correct fixed effects
Show (17.57).Exercise 17.11(a) For b¾2i defined in (17.59) show E£b¾2i j X i¤Æ ¾2i.(b) For eV fe defined in (17.58) show E£eV fe j X¤ÆV fe.Exercise 17.12(a) Show (17.61).(b) Show (17.62).(c) For eV fe defined in (17.60) show E£eV fe j X¤ÆV fe.
Develop a version of Theorem 17.2 for the differenced estimator b¯¢. Can you weaken Assumption 17.2.3? State an appropriate version which is sufficient for asymptotic normality.
Verify that b¾2"defined in (17.37) is unbiased for ¾2" under (17.18), (17.25) and (17.26).
In Section 17.14 it is described how to estimate the individual-effect variance ¾2 u using the between residuals. Develop an alternative estimator of ¾2 u only using the fixed effects error variance b¾2"and the levels error varianceb¾2eÆ n¡1PN iÆ1 Pt2Si be2 i t where bei t Æ Yi t ¡X0 i t
Show that when T Æ 2 the differenced estimator equals the fixed effects estimator.
Show (17.28).
Show (17.24).
Show that var£˙X i t¤· var[Xi t ].
Is E["i t j Xi t ] Æ 0 sufficient for b¯fe to be unbiased for ¯? Explain why or why not.
For each of the following monthly pairs from FRED-MD test the hypothesis of no cointegration using the Johansen trace test. For each, you need to consider the VAR order p and the trend specification.(a) 3-month treasury interest rate (tb3ms) and 10-year treasury interest rate (gs10). Note: In the
For each of the series in the previous exercise implement the KPSS test of stationarity.For each, you need consider the lag truncation M and the trend specification.
For each of the following monthly series from FRED-MD implement theDickey-Fuller unit root test. For each, you need to consider the AR order p and the trend specification.(a) log real personal income: log(rpi)(b) industrial production index: indpro(c) housing starts: houst(d) help-wanted index:
An economist wants to build an autoregressive model for the number of daily tweets by a prominant politician. For a model with an intercept they obtain ADF Æ ¡2.0. They assert “The number of tweets is a unit root process.” Is there an error in their reasoning?
An economist takes Yt , detrends to obtain the detrended series Zt , applies a ADF test to Zt and finds ADF Æ ¡2.5. They assert: “Stata provides the 5% critical value ¡1.9 with p-value less than 1%. Thus we reject the null hypothesis of a unit root.” Is there an error in their reasoning?
An economist estimates the model Yt Æ ®Yt¡1 Å et and finds b®Æ 0.9 with s (b®) Æ 0.04.They assert: “The 95% confidence interval for ® is [0.82,0.98] which does not contain 1. So ® Æ 1 is not consistent with the data.” Is there an error in their reasoning?
An economist estimates the model Yt Æ ®Yt¡1Ået and finds b®Æ 0.9 with s (b®) Æ 0.05. They assert: “The t-statistic for testing ® Æ 1 is 2, so ® Æ 1 is rejected.” Is there an error in their reasoning?
Take the VECM(1) model ¢Yt Æ ®¯0Yt¡1Ået . Showthat Zt Æ ¯0Yt follows an AR(1) process.
Take the AR(1) model Yt Æ ®Yt¡1 Ået with i.i.d. et and the least squares estimator b®. In Chaper 14 we learned that the asymptotic distribution when j®j Ç 1 is pn (b®¡®) ¡!d N¡0,1¡®2¢. How do you reconcile this with Theorem 16.9, especially for ® close to one?
LetUt ÆUt¡1 Ået , Yt ÆUt Åvt and Xt Æ 2Ut Åwt , where (et , vt ,wt ) is an i.i.d. sequence.Find the cointegrating vector for (Yt ,Xt ).
Let Yt Æ et be i.i.d. and Xt Æ ¢Yt .(a) Show that Yt is stationary and I (0).(b) Show that Xt is stationary but not I (0).
Suppose Yt Æ Xt Åut where Xt Æ Xt¡1 Ået with (et ,ut ) » I (0).(a) Is Yt I (0) or I (1)?.(b) Find the asymptotic functional distribution of n¡1/2Ybnr c.
Find the Beveridge-Nelson decomposition of ¢Yt Æ et Å£1et¡1 Å£2et¡2.
Take St Æ St¡1 Ået with S0 Æ 0 and et i.i.d. (0,¾2).(a) Calculate E[St ] and var[St ].(b) Set Yt Æ (St ¡E[St ])/p var[St ]. By construction E[Yt ] Æ 0 and var[Yt ] Æ 1. Is Yt stationary?(c) Find the asymptotic distribution of Ybnr c for r 2 [±,1].
Shapiro and Watson (1988) estimated a structural VAR imposing long-run constraints.Replicate a simplified version of their model. Take the quarterly series hoanbs (hours worked, nonfarm business sector), gdpc1 (real GDP), and gdpctpi (GDP deflator) from FRED-QD. Transform the first two to growth
Take the quarterly series gdpc1 (real GDP), m1realx (real M1 money stock), and cpiaucsl(CPI) fromFRED-QD. Create nominalM1 (multiply m1realx times cpiaucsl), and transformreal GDP and nominal M1 to growth rates. The hypothesis of monetary neutrality is that the nominal money supply has no effect on
Take the Kilian2009 dataset which has the variables oil (oil production), output (global economic activity), and price (price of crude oil). Consider a structural VAR based on short-run restrictions.Use a structure of the form Aet Æ "t . Impose the restrictions that oil production does not respond
Take the quarterly series gpdic1 (Real Gross Private Domestic Investment), gdpctpi (GDP price deflator), gdpc1 (real GDP), and fedfunds (Fed funds interest rate) from FRED-QD. Transform the first three into logs, e.g. gdpÆ 100log(gdpc1). Consider a structural VAR based on short-run
Take themonthly series permit (building permits), houst (housing starts), and realln (real estate loans) from FRED-MD. The listed ordering is motivated by transaction timing. A developer is required to obtain a building permit before they start building a house (the latter is known as a “housing
Take the Kilian2009 dataset which has the variables oil (oil production), output (global economic activity), and price (price of crude oil). Estimate an orthogonalized VAR(4) using the same ordering as in Kilian (2009) as described in Section 15.24. (As described in that section, multiply “oil”
Take the quarterly series gdpc1 (real GDP), gdpctpi (GDP price deflator), and fedfunds(Fed funds interest rate) from FRED-QD. Transform the first two into growth rates as in Section 15.13.Estimate the same three-variable VAR(6) using the same ordering. The identification strategy discussed in
You read an empirical paper which estimates a VAR in a listed set of variables and displays estimated orthogonalized impulse response functions. No comment is made in the paper about the ordering or the identification of the system, and you have no reason to believe that the order used
Cholesky factorization(a) Derive the Cholesky decomposition of the covariance matrix·¾21½¾1¾2½¾1¾2 ¾21 ¸.(b) Write the answer for the correlation matrix (the special case ¾21Æ 1 and ¾22Æ 1).(c) Find an upper triangular decomposition for the correlation matrix. That is, an
Let bet be the least squares residuals from an estimated VAR,b§be the residual covariance matrix, and bB Æ chol(b§). Show that bB can be calculated by recursive least squares using the residuals.
Suppose that you have 20 years of monthly observations onm Æ 8 variables. Your advisor recommends p Æ 12 lags to account for annual patterns. Howmany coefficients per equation will you be estimating? How many observations do you have? In this context does it make sense to you to estimate a
Continuting the previous exercise, suppose that both Y2t does not Granger-cause Y1t , and Y1t does not Granger-cause Y2t . What are the implications for the VAR coefficient matrices A1 and A2?
Let Yt Æ (Y1t ,Y2t )0 be 2£1 and consider a VAR(2) model. Suppose Y2t does not Grangercause Y1t . What are the implications for the VAR coefficient matrices A1 and A2?
Derive a VAR(1) representation of a VAR(p) process analogously to equation (14.33) for autoregressions.Use this to derive an explicit formula for the h-step impulse response IRF(h) analogously to (14.42).
In the VAR(2) model Yt Æ A1Yt¡1 Å A2Yt¡2 Ået find explicit expressions for the moving average matrix £h from (15.3) for h Æ 1, ...4.
In the VAR(1) model Yt Æ A1Yt¡1 Ået find an explicit expression for the h-step moving average matrix £h from (15.3).
Suppose Yi t , i Æ 1, ...,m, are independent AR(p) processes. Derive the form of their joint VAR representation.
Suppose Yt Æ AYt¡1 Åut and ut Æ But¡1 Ået . Show that Yt is a VAR(2) and derive the coefficient matrices and equation error.
Take the VAR(2) model Yt Æ A1Yt¡1ÅA2Yt¡2Ået with A1 Æ·0.3 0.2 0.2 0.3¸and A2 Æ·0.4 ¡0.1¡0.1 0.4¸.Assume et is i.i.d. Is Yt strictly stationary? Use mathematical software if needed.
Take the VAR(1) model Yt Æ AYt¡1 Ået . Assume et is i.i.d. For each specified matrix A below, check if Yt is strictly stationary. Use mathematical software to compute eigenvalues if needed.(a) A Æ·0.7 0.2 0.2 0.7¸(b) A Æ·0.8 0.4 0.4 0.8¸(c) A Æ·0.8 0.4¡0.4 0.8¸
Take the quarterly series gdpc1 (real GDP) and houst (housing starts) from FRED-QD.“Housing starts” are the number of new houses on which construction is started.(a) Transformthe real GDP series into its one quarter growth rate.(b) Estimate a distributed lag regression of GDP growth on housing
Take the quarterly series unrate (unemployment rate) and claimsx (initial claims) from FRED-QD. “Initial claims” are the number of individuals who file for unemployment insurance.(a) Estimate a distributed lag regression of the unemployment rate on initial claims. Use lags 1 through 4. Which
Take themonthly series unrate (unemployment rate) from FRED-MD.(a) Estimate AR(1) through AR(8) models, using the sample starting in 1960m1 so that all models use the same observations.(b) Compute the AIC for each AR model and report.(c) Which ARmodel has the lowest AIC?(d) Report the coefficient
Take the quarterly series oilpricex (real price of crude oil) from FRED-QD.(a) Transformthe series by taking first differences.(b) Estimate an AR(4)model. Report using heteroskedastic-consistent standard errors.(c) Test the hypothesis that the real oil prices is a random walk by testing that the
Take the quarterly series pnfix (nonresidential real private fixed investment) fromFRED-QD.(a) Transformthe series into quarterly growth rates.(b) Estimate an AR(4)model. Report using heteroskedastic-consistent standard errors.(c) Repeat using the Newey-West standard errors, using M Æ 5.(d)
Take the nonlinear process Yt Æ Y ®t¡1u1¡®t where ut is i.i.d. with strictly positive support.(a) Find the condition under which Yt is strictly stationary and ergodic.(b) Find an explicit expression for Yt as a function of (ut ,ut¡1, ...).
Assume that Yt is a Gaussian AR(1) as in the previous exercise. Calculate the momentsA colleague suggests estimating the parameters (®0,®1,¾2) of the Gaussian AR(1) model by GMM applied to the corresponding sample moments. He points out that there are three moments and three parameters, so it
A Gaussian AR model is an autoregression with i.i.d. N(0,¾2) errors. Consider the Gaussian AR(1) model Yt Æ ®0 Å®1Yt¡1 Ået et »N¡0,¾2¢with j®1j Ç 1. Show that the marginal distribution of Yt is also normal:Hint: Use the MA representation of Yt . Y N (1-a'1-a
Suppose that Yt Æ Xt Ået Xt Æ ®Xt¡1 Åut where the errors et and ut are mutually independent i.i.d. processes. Show that Yt is an ARMA(1,1)process.
Suppose that Yt Æ et Åut ŵut¡1 where ut and et are mutually independent i.i.d. (0,1)processes.(a) Show that Yt is aMA(1) process Yt Æ ´t Åôt¡1 for a white noise error ´t .Hint: Calculate the autocorrelation function of Yt .(b) Find an expression for à in terms of µ.(c) Suppose µ Æ
Take themodel®(L)Yt Æ ut¯(L)ut Æ et where ®(L) and ¯(L) are p and q order lag polynomials. Show that these equations imply that°(L)Yt Æ et for some lag polynomial °(L). What is the order of °(L)?
Show that the models®(L)Yt Æ ®0 Ået and®(L)Yt Æ ¹Åut®(L)ut Æ et are identical. Find an expression for ¹ in terms of ®0 and ®(L).
Take the AR(2) model Yt Æ ®1Yt¡1 Å®2Yt¡1 Ået .(a) Find expressions for the impulse responses b1, b2, b3 and b4.(b) Let (b®1,b®2) be the least squares estimator. Find an estimator of b2.(c) Let bV be the estimated covariance matrix for the coefficients. Use the delta method to find a
Take the AR(1) model with no intercept Yt Æ ®1Yt¡1 Ået .(a) Find the impulse response function bj Æ @@et YtÅj .(b) Let b®1 be the least squares estimator of ®1. Find an estimator of bj .(c) Let s (b®1) be a standard error for b®1. Use the delta method to find a 95% asymptotic confidence
Suppose Yt Æ Yt¡1 Ået with et i.i.d. (0,1) and Y0 Æ 0. Find var[Yt ]. Is Yt stationary?
Verify the formula ½(k) ƳP1 jÆ0 µjÅkµj´/³Pq jÆ0 µ2j´for aMA(1) process.
Verify the formula ½(1) Æ µ/¡1ŵ2¢for aMA(1) process.
A stochastic volatility model is Yt Æ ¾t et log¾2tÆ !ůlog¾2t¡1 Åut where et and ut are independent i.i.d. N(0,1) shocks.(a) Write down an information set for which Yt is aMDS.(b) Show that if¯¯¯¯¯Ç 1 then Yt is strictly stationary and ergodic.
A stochastic volatility model isExpress v2 in terms of themoments of et . Y = tet logo w+Blogo-1+ut
Continuing the previous exercise, show that if E£e4 t¤Ç1thenExpress v2 in terms of themoments of et . -1/2 n2 (e -)N (0, v). t=1 d
Let ¾2tÆ E£e2 t jFt¡1¤. Show that ut Æ e2 t ¡¾2t is aMDS.
Show that if (et ,Ft ) is a MDS and Xt is Ft-measurable then ut Æ Xt¡1et is aMDS.
For a scalar time series Yt define the sample autocovariance and autocorrelationAssume the series is strictly stationary, ergodic, strictly stationary, and E £Y 2 t ¤Ç1.Show that b°(k) ¡!p °(k) and b½(k) ¡!p °(k) as n!1. (Use the Ergodic Theorem.) (k) = n (-Y) (Y-k-Y) t=k+1 (k) +1 Yt +(-Y)
Continuation of Exercise 12.24, which involved estimation of a wage equation by 2SLS.(a) Re-estimate the model in part (a) by efficient GMM. Do the results changemeaningfully?(b) Re-estimate the model in part (d) by efficient GMM. Do the results changemeaningfully?(c) Report the J statistic for
Continuation of Exercise 12.22, based on the empiricalwork reported in Acemoglu, Johnson, and Robinson (2001).(a) Re-estimate the model estimated in part (j) by efficient GMM. Use the 2SLS estimates as the firststep for the weight matrix and then calculate the GMMestimator using this weight matrix
Take the model Y Æ X0¯Åe with E[e j X] Æ 0 and E£e2 j X¤Æ ¾2. An econometrician more enterprising than the one in previous question notices that this implies the nk moment conditions E[Xi ei ] Æ 0, i Æ 1, ...,n.We can write the moments using matrix notation as E hX 0 ¡Y ¡X ¯¢i
Take themodel Y Æ X0¯Åe with E[Xe] Æ 0 where X contains an intercept so E[e] Æ 0. An enterprising econometrician notices that this implies the n moment conditions E[ei ] Æ 0, i Æ 1, ...,n.Given an n £n weight matrixW, this implies a GMMcriterion J (¯) Æ¡Y ¡X ¯¢0W¡Y ¡X ¯¢.(a) Under
Take themodel Y Æ µÅe with E[Xe] Æ 0, Y 2 R, X 2 Rk and (Yi ,Xi ) a random sample.(a) Find the efficient GMMestimator of µ.(b) Is this model over-identified or just-identified?(c) Find the GMMtest statistic for over-identification.
Take the model Y Æ X0¯Åe with E[Xe] Æ 0 and ¯ ÆQµ, where ¯ is k £1, Q is k £m with m Ç k, Q is known, and µ ism£1. The observations (Yi ,Xi ) are i.i.d. across i Æ 1, ...,n.Under these assumptions what is the efficient estimator of µ?
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