Econometrics II - EXAM Outline Solutions All questions have 25pts Answer each question in separate sheets

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1 Econometrics II - EXAM Outline Solutions All questions hae 5pts Answer each question in separate sheets. Consider the two linear simultaneous equations G with two exogeneous ariables K, y γ + y γ + x δ + x δ u y γ + y γ + x δ + x δ u where, u u, u, E uu Σ σ σ σ σ, Γ γ γ γ γ. a Using the standard normalization, write the general form of the order and rank conditions for single equation identification and for system identification. hen, stating the restrictions on the system parameters, check the identification of the aboe system in the following cases. Imposing the normalization γ γ, these are: Single equation: rankr B G, with R β 0. Order condition: rankr G. System: rankr I G B G G, with restrictions Rβ 0, β ecb. Order condition: rankr G G. b SUR model. γ γ 0. System always just identified, taking for granted that E xx is full rank and E u i x 0, i,, x x, x. For equation : γ γ γ R B γ δ δ γ γ 0 δ δ using the normalization and the exclusion restrictions, which is of rank G. c A different exogeneous ariable is omitted from each structural equation. For example, δ δ 0. Each equation just identified if these ariables are in the other equations, i.e. if δ 0 for eq. and δ 0 for eq.. d he ariable x does not appear in the system. δ δ 0. Rank conditions fail: no single equation identified. e Neither x nor x appear in the first equation. δ δ 0. First equation oeridentified if δ δ 0 just identified if δ 0 or and δ 0. Second equation not identified. f Γ is constrained to be symmetric and the coefficient in x is the same in both equations.

2 γ γ, δ δ. his corresponds to R, so, with β γ, γ, δ, δ, γ, γ, δ, δ, we obtain that γ R I G B γ γ γ γ γ δ δ δ δ δ δ δ δ which is of rank, if any two ectors are linearly independent, ie. if δ 0 and γ, so the system would be just identified. g Γ is constrained to be lower triangular with diagonal elements equal to and Σ is diagonal. In this case explain how you would estimate the structural form parameters from the estimation of the reduced form. Are these estimates efficient in general? And if you further assume the restrictions on a? γ 0 : this is a triangular system: just identified since y is exogeneous in the second equation. We can estimate the reduced form y Π x +, where Π Γ, by OLS and the coariance matrix E Λ Γ ΣΓ using OLS residuals. In this case we hae that γ Γ γ 0, so Γ 0, γ γ γ γ and Σ σ σ σ σ σ 0 0 σ and therefore from Π Γ we obtain 4 equations and from Λ Γ ΣΓ we obtain another 3 equations because of symmetry, and we hae 7 unknowns element in Γ, 4 in. and in Σ. In particular note that Γ ΣΓ γ 0 σ γ σ 0 σ σ 0 0 σ 0 γ we obtain three equations for three unknowns σ, γ, σ. h Which is your recommended estimation method in each case? 0 γ γ σ γ σ γ σ σ When the system is identified, and if we can not assume some form of conditional homoskedasticity, i.e. E u u x E u u Σ, then we should rely on system efficient system GMM chi-square estimates using Ŵn Ê Xu u X with X I x and the corresponding restrictions. Otherwise we could also use 3SLS. In case e we can only use efficient single equation GMM for the first one.,

3 In the just identified cases we hae that system GMM is equal to System IV and 3SLS to SLS. In b all estimates are equal to SOLS, and therefore to equation by equation OLS as in g if we impose diagonal Σ.. Consider a nonlinear SUR system, E y g z E y g z g m g z g, θ 0g, g,..., G. a Propose a single equation consistent estimate θ n,g for the parameters θ 0g of each equation gien a random sample of n obserations. State the additional regularity conditions you may need. NLS equation by equation, θn,g arg min Q n θ g arg min θ g Θ g θ g Θ g E n y g m g z g, θ g. Gien identification of θ 0g with conditions such as θ 0g arg min θg Θ g E and consistency of θ n,g so Q n θ g conerges uniformly in Θ g to E y g m g z g, θ g y g m g z g, θ g to study the asymptotic distribution of θ n,g we need that m g is smooth with second continuous deriaties and that u g θ m g z g, θ 0g satisfies a CL, i.e., n / where D g V u g θ m g z g, θ 0g, θ Q n θ 0g n / E n u g θ m g z g, θ 0g d N 0, D g, p lim u g y g E y g z, and that θ θ Q n ˆθng E g, ˆθng p θ 0g, where E g E θ m g z g, θ 0g θ m g z g, θ 0g is not singular, so we obtain that n / θn,g θ 0g d N 0, E g D g E g. In case of homoskedasticity we hae that E g D g E g σ ge g, where σ g E ug. b Suppose that V y z Ω 0 > 0, y y,..., y G. How can Ω 0 be estimated consistently? Using the single equation residuals in ˆΩ n E n ũn ũ n, ũgn y g m g z g, θ ng. c Let ˆθ n be the nonlinear SUR estimate that soles the problem min E n y m z, θ ˆΩ n y m z, θ θ where ˆΩ n is a consistent estimate of Ω 0 and m z, θ is the G ector of conditional expectations m g x g, θ 0g, g,..., G. Find the asymptotic distribution of ˆθ n. Using the same arguments as in single equation NLS, we find that the first order conditions are θ Q nˆθ n E n θ m z, ˆθ n ˆΩ n y m z, ˆθ n 0 { where θ m z, θ means } θ m z, θ n θ Q nθ 0 ne n p θ m z, θ 0 ˆΩ n u ne n θ m z, θ 0 Ω 0 u d N 0, 4Ē, 3

4 where Ē E θ m z, θ 0 Ω 0 θm z, θ 0 and u y m z, θ 0. Similarly, we hae that the second order deriatie of the objectie function satisfies that θ θ Q nθ 0 E n θ m z, θ 0 ˆΩ n θ m z, θ 0 E n θ { θ m z, θ 0 } ˆΩ n y m z, θ 0 p Ē + 0 θ {} indicates the terms inoling second deriaties of m so n / ˆθn θ 0 d N 0, Ē. d Describe a numerical method to obtain the estimate ˆθ n and how you would estimate AVar ˆθn? We could use Newton-Raphson, but preferably Gauss Newton, dropping the second term in the second deriatie of the objectie function which has zero expectation, so the s-iteration is ˆθns ˆθ ns Ē s θ Q nˆθ ns, ˆΩ ns where for example Ē s E n θ m z, ˆθ ns ˆΩ ns θm z, ˆθ ns where in the first iteration ˆθ ns and ˆΩ ns are based on the single equation estimates θn,g, g,..., G. he last alue of Ē s can be used to approximate the AVar. e If Ω 0 is diagonal and if the assumptions stated preiously hold, which estimation method would be preferred? Noting that, if there are no common parameters among the different equations SUR, θ m z, ˆθ n is block-diagonal and Ω 0 diag { σ,..., σg}, we can obtain that Ē diag {E,..., E G } so single equation NLS is asymptotically as efficient as system estimation but not numerically. 3. Consider the econometric model which relates y t to the expected alue of x t, x t, where the expectation is based on all obsered information at time t, y t α 0 + α x t + u t. A natural assumption on {u t } is that E u t I t 0, where I t denotes all information on {y t } and {x t } obsered at time t. o complete the model, we assume that x t is formed according to x t x t λ x t x t, where λ 0,. his equation implies that the change in expectations reacts to whether last period s realized alue was aboe or below its expectation. he assumption 0 < λ < implies that the change in expectations is a fraction of last period s error. 4

5 a Show that the two equations imply that y t δ 0 + δ y t + δ x t + t and gie appropriate definitions of δ δ 0, δ, δ and of t in terms of {u t }. What are the dynamic properties of the errors t? We hae that x t x t λ x t x t implies because λ <, so x t x t + λ x t x t λx t + λ x t λ λ L x t, where y t α 0 + α x t + u t α λ α 0 + λ L x t + u t α 0 { λ} + λ y t + α λx t + λ L u t α 0 λ }{{} + λ }{{} δ 0 δ y t + α λ }{{} δ x t + λ L u t }{{} t t λ L u t u t λ u t u t δ u t is a MA, inertible because δ λ <. b How would you estimate the ector δ consistently? Study the asymptotic properties of your estimate. OLS estimates are inconsistent because of the correlation between t and y t, which is a function of u t as t. A possibly solution is to use IV, with instrument x t for y t adding possibly more lags in x t x t,..., x t q to improe efficiency using SLS. In this case we hae that ˆδ z t w t z t y t t where z t, x t, x t, w t, y t, x t. We hae that, using that E z t t E E z t t I t E z t E t I t 0, since z t I t depends only on past obserations of x t, / ˆδ δ d N t t z t w t / z t t t 0, E z t w t SE w t z t, assuming weak stationarity and E z t w t full rank, where S j Γ j, Γ j E t t+j z t z t+j. 5

6 Note that if we further assume some sort of dynamic conditional homoskedasticity we hae that E t t+j z t z t+j E E t t+j z t z t+j I t E zt z t+je t t+j I t E z t z t+j E t t+j E z t z t E t E zt z t σ, j 0 E z t z t+ E t t+ E z t z t+ γ, j 0, j >, so S { E z t z t+ + E zt+ z t } γ + E z t z t σ, and estimates of S could use this structure. Otherwise we hae to use general estimates with a bandwidth m. hen, we can think on alternatie, more efficient solutions, such as GLS, transforming the noise into an uncorrelated sequence and simultaneously orthogonal with the regressors. In this case the solution is not so neat as with ARp disturbances, because Ω is not band diagonal and we require additionally at all lags strong exogeneity of the x ts. Note also that Ω only depends on δ or λ up to scale. c Gien a consistent estimate of δ, how would you estimate α and λ? Just use the definition of δ in terms of α 0, α, λ, and find the solution. hen use the the delta method to obtain the asymptotic distribution. d If you estimate by OLS, how would you check the correct specification of the model, i.e. how can you test the assumptions required for consistency of the OLS estimates. Apart from usual checkings using alternatie estimates, as Hausman tests since we can not check directly the orthogonality condition E y t t 0 with OLS estimation, and gien the aboe structure, with a lagged dependent ariable on the rhs of the model, a key condition for OLS consistency is that the errors should be uncorrelated. his can be checked using OLS residuals, either by means of LM or Portmanteau statistics using an exogeneity assumption, or correcting for possible endogeneity, which is not ruled out by the conditions gien. Using IV estimation we can also directly check that δ 0 or λ using part c. 4. Consider the linear regression model y t z tβ + t where E t I t 0, E t I t : σ t σ + δ t, where I t denotes the information set of current and past obserations of z t and past obserations of t. Assume that t and z t are stationary. a Are the t independent? Uncorrelated? And/or form a martingale difference? Not independent at least second order conditional moments are not equal to marginal ones and is a MD sequence because E t t,... E E t I t t,... 0, since { t,...} I t. b Find the asymptotic properties of the OLS estimate of β and propose an estimate of its asymptotic coariance matrix if consistent. We hae that / β n β d N 0, E z t z t E t z t z t E zt z t 6

7 assuming that because z t t / z t z t p E z t z t > 0 t t z t t d N 0, E t z t z t is zero mean and serially uncorrelated. c Using the additional assumption that the t I t are normally distributed, study the first order conditions for ML estimation of all the parameters of the model, θ δ, σ, β, and gie some approximation for the asymptotic distribution of ˆθ. In this case we hae that the t are conditionally independent, so the likelihood is L δ, σ, β l t δ, σ, β t c t log σ t θ y t z tβ t σ t θ with σ t θ σ + δ t β σ + δ y t z t β, t β y t z tβ and pretending we hae initial conditions on the ariables required, the score is gien by s δ, β, σ L θ θ t t t β σ t θ + t β yt z t β σ 4 t θ δ t βz t y t z σ t θ δ t β z t β t σ 4 t θ t β σ t θ + σ t θ t β tβz t σ t θ δ σ t θ σ t θ t+ β σ t+ θ σ t θ y t z t β σ 4 t θ σ t+ θ, t β σ t θ + yt z t βzt σ t θ the last step ignoring end effects since β σ t θ β δ y t z t β δ yt z t β z t δ t β z t which shows that s θ 0 s t θ 0 t t tz t σ t t σ t δ σ t σ t t σ t σ t+ t+ σ t+ t σ t, so the estimation equation is Es t θ 0 Es t θ 0 I t 0 and under the aboe conditions on t the elements of the sum are a zero mean conditionally independent sequence so a MD een if Gaussianity is dropped, so / s θ 0 is asymptotically normal under standard conditions. hen, using the usual aylor-expansion argument, / ˆθ θ0 d N 0, I, 7

8 where L I : E θ θ θ 0 : E H θ 0 AsyV ar / s θ 0 is the information matrix, and ignoring end effects, we hae for example, H δ, β, σ L θ θ θ t θ t t β σ t θ t β tβz t σ t θ δ σ t θ σ t θ t+ β σ t+ θ σ t θ t β t β σ t θ σ t θ t β σ t+ θ t β σ t θ σ t θ. t β t β σ t θ σ t θ Note that the conditional expectation of the, term is zero, so we hae that I t β E t β σ t θ σ E t β t θ σ t θ t because E t β /σ t θ I t. d Propose an LM test for the lack of ARCH effects in t and describe its asymptotic properties. he restricted estimates when δ 0 are gien by s 0, β, σ 0, and are equialent to the OLS coefficient and the residual ariance, β z t z t z t y t σ so the restricted score is equal to t t t ˆ t, ˆ t t β y t z t β, t s 0, β, σ Note that under the null σ t σ and s θ 0 t t { } t β t β σ σ t σ t tz t σ σ t 0 0 σ σ which is asymptotically normal with Asymptotic Var-Co matrix I 0, which can be estimated by H 0, β, σ which can be showed to be diagonal under the null. In this case we can use Ĥ, 0, β, σ t β t σ. 8

9 herefore the LM statistic is LM s 0, β, σ s 0, β, σ p d H 0 χ, t Ĥ Ĥ, 0, β, σ s 0, β, σ 0, β, σ { } { } t β σ t β σ t t β which checks the first order auto-correlation between the OLS-squared-residuals t Using the further approximation, we hae that LM p t t t β σ ˆρ, t β p E Z 4 E Z, Z N 0, { } { } t β σ t β σ which can also be interpreted as usual as a R statistic. e Dropping the Gaussianity assumption, consider the moment conditions where x t β E y t β z t zt 0 E t σ t xt β 0, y t β z t, to define GMM estimates. t t β. β Compare GMM estimation and GMM LM test for the same hypothesis with ML estimation. he basic difference is that the first moment condition which identifies β does not use the information on the conditional ariance to set up a GLS-type estimate for β as ML estimation does as is equialent to OLS estimation. Note also that we are in a just identified case, so weighting is not releant. herefore, under the null of no ARCH effects both estimates are the same, OLS for β no GLS correction and the LM test is the same. f Set up an iteratie procedure to approximate the alue of the GMM estimates from an initial point. We can use the linearized GMM estimate, ˆθ iter GMM θ n { ξ θ, θ ξ θ, θ } ξ θ, θ n ξ θ, t t β where θ is an initial estimate, where ξ θ yt β z t zt t σ t xt β t t t β z t t β σ t θ t β σ t θ t β, 9

10 and, θ β, σ, δ, ξθ,n θ θ ξ θ t z t z t t β z t δ t β z t t β z t + δ t β z t t β t β σ t θ t β z t 0 t β 0 t β t 4 β, from where you can show that, under appropriate conditions, Ξ 0 p lim ξ θ,n θ 0 is block diagonal, so asymptotic properties are ery easy to establish. 0