Problem Set 2 Solution

Transcription

1 Problem Set 2 Solution Aril 22nd, 29 by Yang. More Generalized Slutsky heorem [Simle and Abstract] su L n γ, β n Lγ, β = su L n γ, β n Lγ, β n + Lγ, β n Lγ, β su L n γ, β n Lγ, β n + su Lγ, β n Lγ, β su su β Bβ,ε = o + o = o L n γ, β Lγ, β + su Lγ, β n Lγ, β where the third line can be obtained with robability aroaching, and the fourth line follows from the assumtions i, ii and iii. [Formal and Detailed] ake any ξ >. Note that by lim Pr n su L n γ, β n Lγ, β > ξ lim n Pr su L n γ, β n Lγ, β n + su Lγ, β n Lγ, β > ξ lim Pr su n γ, n L β n Lγ, β n > ξ + lim 2 Pr su n Lγ, β n Lγ, β > ξ 2 }{{}}{{} =:A =:A 2 Partitioning the event in A yields A = lim n Pr su L n γ, β n Lγ, β n > ξ 2 and β n β > ε + lim n Pr su L n γ, β n Lγ, β n > ξ 2 and β n β ε lim n Pr β n β > ε + lim n Pr su su n γ, β Lγ, β > β Bβ,ε L ξ 2 where the last equality follows from the assumtions i and ii. Also by the assumtion iii, there exists δ > such that for any β Bβ, δ su Lγ, β Lγ, β ξ 2 =

2 his imlies, by contraositive, A 2 lim n Pr β n β > δ = where the last equality follows from the assumtion i. herefore, we have lim Pr n for any ξ >, which comletes the roof. su L n γ, β n Lγ, β > ξ = 2. Eicker-White Robust Covariance Estimation a By β LS β theorem, d Z N, Ω, we have β LS Û t = Y t X t β LS his imlies in turn, by continuous maing theorem, EÛ 2 t X t X t β. Y t X tβ = U t EU 2 t X t X t his imlies, by continuous maing Note that the LHS term is a random variable, while the RHS term is a scalar. Alying LLN, under the following assumtions. 2 Ût 2 EÛ t 2 EU 2 t < 3 E U 2 t X ti X tj < for all i, j =,, K where K = dimx t. E X 2 ti X tjx tk < for all i, j, k =,, K. Combining these two, Ût 2 EUt 2 Also without any further assumtion, by alying LLN, E o see this, note that β LS β = O /2 as we will rove in roblem set 3. So β LS β = o. 2 We may not aly iid version of LLN directly to Û t 2. his is because {Ût : t } are not indeendent. Instead, we aly uniform LLN to Y t X tθ 2 for θ Bβ, ε. Since β LS is consistent for β, we can get the desired result under the described assumtions. See White Econometrica, 98 for details. 3 Note this is not imlied by EUt 2 X t <. 2

3 herefore, by multilication rule, Ω := Ût 2 under the additional assumtion E EUt 2 EX t X t = Ω EX t X t is nonsingular b Suose the assumtion of conditional homoskedasticity holds so that EU 2 t X t = σ 2. hen the variance of β LS simlifies as follows. Ω = EX t X t Eσ 2 X t X tex t X t = σ 2 EX t X t So if we believe this assumtion, we may use the following formula to estimate a covariance matrix. Ω CH = σ 2 where CH denotes conditional homoskedasticity, and σ 2 can be estimated by σ 2 = Û 2 = Y t X t β LS 2 If we are more dubious of the assumtion, we can use the Eicker-White robust covariance estimator. Ω EW = Ût 2 Let us talk a little on how to rogram this. Denote by X, the matrix which stacks X t by row. Define Y, U and Û in the same way. hen, X tx t = X X, and Û 2 = Û Û. herfore, to estimate Ω CH, we can use OmegaCH = Uhat * Uhat / * invx * X /, or simly dro both s. However, Û t 2 cannot be simlified using matrix notation. 4 he most straightforward way to calculate it is using a loo. But there is a simler way to do this in MALAB, using element-wise multilication. Define W t := ÛtX t, and denote by W, the matrix which stacks W t by row. hen, W W = W t W t = Ût 2 Note that W is the matrix obtained by multilying Ût to each element in t th row of X. his is equivalent to element-wise multilication of ÛL and X, where L is a k row vector of s. So by doing W = Uhat * ones,k.* X, we get such a matrix. herefore Ω EW can be calculated by OmegaEW = invx * X / * W * W / * invx * X /. 4 Note that Û ÛX X = Û 2 t X tx t is totally different from Û 2 t X t X t. 3

4 See the MALAB code osted for details. It would be good to run the code with different choices of R,, k, and most imortantly, σ 2. he following shows the result with two sets of arameters with R =, =. On the left, k = and σ 2 = 9, and on the right, k = 3 and σ 2 = rue beta estimates of beta Mean Standard Dev rue Omega Mean Bias estimates of Omega: correct formula estimates of Omega2: White formula Standard Dev estimates of Omega: correct formula estimates of Omega2: White formula R M S Error estimates of Omega: correct formula estimates of Omega2: White formula Norm Stats Omega Omega2 Omega Omega Mean Standard Dev Minimum Median Maximum You may try another set of assumtions on β, EX tx t, the distribution of X t, and that of U t. But when you change the distributions of X t and U t, make sure that the correct variances of X t and U t are used in the code. 4

5 o summarize the result, the standard deviation of each element of Eicker-White estimator is 3-4% bigger than that of the estimator using the correct formula. So is the root mean squared error. But looking at the mean bias, Eicker-White estimator is less biased by mean in many cases, esecially on diagonal elements. hese roerties hold for many choices of R,, k, and σ 2 that have ever been tried. In the table, Norm stats are also reorted. he norm is the distance between true Ω and estimates Ω, calcualted by taking square root of the summed squares of difference in each element. he distributions of 4 sets of norms are summarized as seen. his imlies that the Eicker-White estimator is bigger in general, but not that bad. Moreover it is robust to heteroskedasticity of errors, so it would be better to use this formula if we are not sure of conditional homoskedasticity. 3. Asymtotic heory a Additionally we need to assume. [CF i] θ intθ 2. [EE2 ii] θ Q n θ n = o n /2 3. gθ C 2 Θ for some neighborhood Θ Θ of θ with robability one 4. θ gθ has full column rank. Note that we already assumed 5. n π n π d N, V 6. A n A where A is nonsingular. 7. here exists a unique value θ Θ such that π = gθ. 8. Θ is comact. 9. [EE] θ n Θ and Q n θ n inf Q n θ + o g is continuous. π n π Since the last two are imlied by 3 and 5 reectively, we need assumtions -9. Now let us check how the above assumtions imly CF and EE2. EE2 is imlied by 2-3 and 5-9. CF i is assumed in. CF ii is imlied by 3. CF iii is imlied by 3, 5 and 6. o see what Ω is, consider n θ Q nθ = θ g θ A na n n πn gθ = θ g θ A na n n πn π 5

6 Define Γ := θ gθ. hen by 5 and 6, n θ Q d nθ N, Γ A AV A AΓ Let us check that CF iv is imlied by 3-7. he candidate for Bθ can be obatined by 3, 5 and 6. 2 [Bθ] mj = lim Q n θ = θ m θ j g θa A gθ θ m θ j 2 θ m θ j g θa Aπ gθ By 3, this is continuous in θ. Moreover, 2 θ θ Q n θ is also continuous in θ, and by choice, Θ is comact. So uniform convergence follows from ointwise convergence. su 2 θ θ Q nθ Bθ Finally, Bθ is nonsingular by 4 and 6. When evaluated at θ, the second summand of [Bθ ] mj is, so B := Bθ = θ g θ A A θ gθ = Γ A AΓ So the asymtotic normality follows. d n θn θ N, Γ A AΓ Γ A AV A AΓ Γ A AΓ b We need to assume the following.. [EE] θ n Θ and Q n θ n inf Q n θ + o 2. [ID i] Θ is comact. 3. Gθ, τ is continuous in θ at τ = τ. 4. here exists a unique value θ Θ such that Gθ, τ =. 5. A n A where A is nonsingular. 6. τ n τ 7. su su τ Bτ,ε G n θ, τ Gθ, τ for some ε > 8. Gθ, τ is continuous in τ at τ uniformly over θ Θ. Let us first determine what Qθ is. By 5-7, we have ointwise convergence such that for any θ Θ, Q n θ AGθ, τ 2 /2 =: Qθ For this Qθ, ID holds by 2-5. In articular, 4 and 5 imly ID iii. o see this, note first that Qθ = and that this is the minimum since Qθ for any θ. Suose that θ satisfies Qθ =. hen, by 6

7 the roerty of norm oerator, AGθ, τ =. By 5, Gθ, τ =, and thus by 4, θ = θ, which roves ID iii. Now let us turn to UWCON. 2-3 and 5-8 imly UWCON. First, alying Question, 6-8 imly Using 2, 3, 5, 6 and this, su G n θ, τ n Gθ, τ su Q n θ Qθ su 2 Q n θ G θ, τ A na n Gθ, τ /2 + su G θ, τ A na n Gθ, τ /2 Qθ 2 su G θ, τ A na n su G n θ, τ n Gθ, τ w aroaching + 2 su G θ, τ A na n A A su Gθ, τ = O O o + O o O = o 4. Monte Carlo Simulation: Goodness of Aroximation to the Assymtotic Distribution i As we derived before, under conditional homoskedasticity, β 2SLS has the asymtotic variance of σ 2 Ex i z i Ez iz i Ez i x i. Since σ 2 =, Ez i z i = I K and Ez i u i =, Ex i z i = Ez iπz i + u i z i = π Ez i z i + Eu i z i = π and thus we can calculate the asymtotic variance as follows. It deends on K and η but not on ρ. σ 2 Ex i z iez i z i Ez i x i = π I K π = Kη 2 ii Looking at the first figure, we can see many huge outliers of estimates in the case where K = and η =.5. his is consistent with the fact that the 2SLS estimator has no finite first moment when it is exactly identified. Even when η =, we can observe some outliers although they are not very big. he 2SLS estimator tends to be biased when ρ =.95 and the instruments are weak. K and η affect the variance of estimates in general. iii In the second figure, the histogram of estimates was normalized so that the area of histogram bars is, and the df of the distribution aroximated by the asymtotic distribution is lotted together. 6 o focus on the shae of the distribution, the outliers are droed from this figure. When η =, the asymtotic distribution aroximates the simulated distribution quite well, while it does not if η =.5. When K = and ρ =, there does not seem to be a bias, but there are outliers and estimates which are not outliers are centered more than the aroximated distribution. In the thrid figure, the numbers drawn from the aroximated distribution are lotted, along with the histogram of estimates without rescaling. It looks similar to the second figure. 6 Since the asymtotic distribution is obtained by scaling β 2SLS u by n, we always need to scale the asymtotic variance down by n to get the aroximated variance. 7

8 FIGURE. Histogram of Estimates 6 K=,eta=.5,rho=. 6 K=,eta=.5,rho= K=,eta=.,rho=. 2 K=,eta=.,rho= K=,eta=.5,rho=. 3 K=,eta=.5,rho= K=,eta=.,rho=. 2 K=,eta=.,rho=

9 FIGURE2. Histogram and Aroximated Distribution.3 K=,eta=.5,rho=. K=,eta=.5,rho= K=,eta=.,rho=. K=,eta=.,rho= K=,eta=.5,rho= K=,eta=.5,rho= K=,eta=.,rho=. K=,eta=.,rho= Estimates Aroximate 9

10 FIGURE3. Histogram and Aroximated Distribution simulated K=,eta=.5,rho=. K=,eta=.5,rho= K=,eta=.,rho=. K=,eta=.,rho= K=,eta=.5,rho=. K=,eta=.5,rho= K=,eta=.,rho=. K=,eta=.,rho= Estimates Aroximate