ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2017 Instructor: Victor Aguirregabiria

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1 ECOOMETRICS II ECO 40S Unversty of Toronto Department of Economcs Wnter 07 Instructor: Vctor Agurregabra SOLUTIO TO FIAL EXAM Tuesday, Aprl 8, 07 From :00pm-5:00pm 3 hours ISTRUCTIOS: - Ths s a closed-book exam - o study ads, ncludng calculators, are allowed - Please, answer all the questons TOTAL MARKS = 00 PROBLEM 50 ponts Let y t be the logarthm of output produced by frm at perod t The researcher has a panel dataset {y t : =,, ; t =,,, T } where the number of frms s large and the number of perods T s small The researcher s nterested n the relatonshp between frm s growth, y t y t y,t, and frm sze, y,t She proposes the followng model: y t = δ y,t + α + γ t + u t where Eα = Eu t = 0; Eα u t = 0; varα = σ α; and u t s not serally correlated ote that ths model s equvalent to y t = δ y,t + α + γ t + u t wth β + δ Queston 5 ponts] Accordng to Gbrat s hypothess often referred as Gbrat s Law, there s not a causal relatonshp between frm sze and frm growth, e, δ = 0 or β = However, emprcal applcatons presentng a Wthn-Group estmator of the regresson of y t on y,t typcally provde negatve and a sgnfcant estmate of the parameter δ Explan why ths negatve relatonshp can be spurous and not causal For the case wth T = 3, show that the asymptotc bas of the Wthn-Groups estmator s negatve Hnt: when T=3 the WG estmator s the same as OLS n frst dfferences ASWER: The model s a dynamc panel data model In ths class of model, the Fxed-Effects FE or Wthn-Groups WG estmator s nconsstent as goes to nfnty and T s fxed More specfcally, ths s an AR panel data model and the WG estmator of β or of δ s asymptotcally downward based: p lm βw G < β ckell Econometrca, 98 obtaned the closed form expresson of the asymptotc based of the WG estmator for ths AR panel data model and showed that ths based s negatve For general T, ths expresson s convoluted Here we consder the smpler case wth T = 3

2 We have the model n levels at t = and t = 3: y = β y + α + γ + u The WG transformaton s: y + y 3 y 3 = β y y 3 = β y + α + γ 3 + u 3 y + y ] + γ 3 ] γ + γ 3 + u 3 ] u + u 3 Multplyng RHS and LHS by, we have that for T = 3 ths transformaton s equvalent to the transformaton n frst dfferences: y 3 y = β y y ] + γ 3 γ ] + u 3 u ] Therefore, for T = 3, the WG estmator s equvalent to OLS n the regresson of y 3 y ] on y y ] ote that γ 3 γ ] s smply a constant term Therefore, β W G = = y 3 y ] y y ] = y y ] p Ey 3 y ] y y ] Ey y ] and ths s equal to β + Eu 3 u ] y y ] Ey y ] Therefore, the sgn of the asymptotc bas s equal to the sgn of Eu 3 u ] y y ] In the AR model, we have that, y t = u t + β u,t + β u,t + β 3 u,t 3 + Ths mples that, Eu 3 u ] y y ] = Eu 3 y Eu 3 y Eu y + Eu y = 0 0 σ u + 0 = σ u < 0 Queston 0 ponts] Descrbe n detal the Arellano-Bond moment condtons and the correspondng GMM estmator of β n ths model Present the closed-form expresson for ths estmator n ths model ASWER: To descrbe the AB estmator, t s convenent to see the model as a system of T equatons n frst dfferences: Equaton at t = 3 : y 3 = β y + γ 3 + u 3 Equaton at t = 4 : y 4 = β y 3 + γ 4 + u 4 Equaton at t = T : y T = β y T + γ T + u T

3 For each equaton we have dfferent vald nstruments or moment condtons: For t = 3 : E u 3 ] = 0; Ey u 3 ] = 0 For t = 4 : E u 4 ] = 0; Ey u 4 ] = 0; Ey u 4 ] = 0 For t = T : E u T ] = 0; Ey u T ] = 0; ; Ey,T u T ] = 0 The total number of moment condtons s q = T + q moment condtons n vector form as: E Y β Y ] T T We can represent these where Y and Y are T vectors, and s the q T matrx of nstruments Let m β be the q vector of sample moment condtons, e, the sample counterpart of E ] Y β Y m β = The Arellano-Bond GMM estmator s: = ˆβ AB = arg mn {β} Y β Y m β ˆΩ m β where ˆΩ s a consstent estmator of the optmal weghtng matrx, Ω = E Z U U Z, wth U = u 3, u 4,, u T Dervng the frst order condtons and solvng for ˆβ AB we get: Y ˆΩ Y = = ˆβ AB = = Y ˆΩ = Y Queston 3 0 ponts] Obtan the expresson of the asymptotc varance of the Arellano-Bond estmator ote: For smplcty, consder that T = 3] Show that ths varance s nfnte f Gbrat s hypothess holds ASWER: When T = 3 we have only one equaton n frst dfferences, y 3 = β y + γ 3 + u 3, and only two moment condtons: E u 3 ] = 0 and Ey u 3 ] = 0 In ths case, AB estmator s equvalent to the Anderson-Hsao estmator where the endogenous regressor y s nstrumented wth y The expresson of ths estmator s: y y 3 = ˆβ AB = y y = 3

4 Its asymptotc dstrbuton s: ˆβAB β = d as / = y u 3 = y y 0, V ar y u 3 E y y Therefore, V ar ˆβAB = V ar y u 3 E y y ow, we show that E y y = σ u + β and V ar y u 3 = σ u V ar y Frst, note that V ary u 3 = Ey u 3, and by the law of teratve expectatons, Ey u 3 = Ey E u 3 Gven that E u 3 = σ u, and E y σ = α β + σ u β, we have that V ary u 3 = σ u σ α β + σ u ] ext, we have that: β E y y = E u + βu 0 + ] u + β u + ββ u 0 + ] = β σ u β = σ u + β Then, AV arˆβ AH = = V ary u 3 E y y = + β + β σ u + β β σ ] α β + σ u β σ u ] σ α σ u + β Under Gbrat s hypothess, we have that β =, and t s clear that ths varance s equal to nfnte Queston 4 0 ponts] Descrbe n detal the Blundell-Bond moment condtons and the correspondng System-GMM estmator of β n ths model ASWER: To descrbe BB moment condtons, t s convenent to see the model as a system of T equatons n levels: Equaton at t = 3 : y 3 = β y + γ 3 + α + u 3 Equaton at t = 4 : y 4 = β y 3 + γ 4 + α + u 4 Equaton at t = T : y T = β y T + γ T + α + u T 4

5 For each equaton we have dfferent vald nstruments or moment condtons: For t = 3 : Eα + u 3 ] = 0; E y α + u 3 ] = 0 For t = 4 : Eα + u 4 ] = 0; E y α + u ] = 0; E y 3 α + u 4 ] = 0 For t = T : Eα + u T ] = 0; E y α + u T ] = 0; ; E y T α + u T ] = 0 The total number of moment condtons s q = T + q moment condtons n vector form as: E Y β Y ] T T We can represent these where Y and Y are T vectors, and s the q T matrx of nstruments The system-gmm estmator combnes AB and BB moment condtons We can represent these moment condtons as: ] Z AB E 0 Y β Y 0 Y β Y Z AB = E ] ] Y β Y ] = 0 Y β Y Let m β be the q vector of sample moment condtons The sample counterpart s: The System-GMM estmator s: ] Z m β = AB Y β Y ] Y β Y = ] ˆβ Sys = arg mn {β} m β ˆΩ m β where ˆΩ s a consstent estmator of the optmal weghtng matrx Dervng the frst order condtons and solvng for ˆβ Sys we get: = ˆβ Sys = = Y Y Y Y ˆΩ ˆΩ ] Z AB Y = = Y Y Y Queston 5 0 ponts] Obtan the expresson of the asymptotc varance of Blundell- Bond estmator ote: For smplcty, consder that T = 3 and use only Blundell-Bond moment condtons] Show that ths varance s fnte for any value of β + δ wthn 0, ], ncludng β = 5

6 ASWER: When T = 3 we have only one equaton n levels wth vald BB nstruments: y 3 = β y + γ 3 + α + u 3, and two moment condtons: Eα + u 3 ] = 0 and E y α + u 3 ] = 0 In ths case, the BB estmator s equvalent to the followng IV estmator: ˆβ BB = y y 3 = y y = Its asymptotc dstrbuton s: ˆβBB β = / = y α + u 3 = y y d 0, V ar y α + u 3 ] E y y Therefore, V ar ˆβBB = V ar y α + u 3 ] E y y Gven that: V ar y α + u 3 ] = E y E α + u 3 = σ u E y y = σ u/ + β; we have that: V ar ˆβBB = σ u σ u + σ α / + β σ 4 u/ + β = + β + σ α σ u σ u + σ α / + β; and In contrast to the AB estmator, the varance of ths estmator s fnte when β = Queston 6 5 ponts] Gven the System-GMM estmator, consder testng the null hypothess β = Comment potental problems wth the standard t-test for ths null hypothess ASWER: The dervaton of Blundell and Bond moment condtons requre the assumpton that the model s statonary such that the condton β < should hold Therefore, when β s exactly equal to, the AB moment condtons do not have any dentfcaton power and the BB moment condtons do not hold Ths mples that when n the DGP wth true β = the parameter β s not dentfed, at least from the AB and BB moment condtons The standard regularty condtons for the t-test and for Wald test, LM test, and LR test do not hold Andrews Econometrca, 00 shows that under these condtons the t-test statstc does not have the well-known standard normal dstrbuton and t should be corrected to construct a vald test 6

7 PROBLEM 50 ponts Consder the random coeff cents multnomal choce model, Y n = arg max β n X j + ε nj ] j {0,,,J} where n s the ndex for ndvduals/observatons, and j s the ndex for choce alternatves X j s a contnuous characterstc of choce alternatve j, eg, the prce of product j β n s a random coeff cent wth the followng propertes: β n = β + σ v n, where v n s d standard normal Varables ε nj are d over n, j wth a Type I extreme value dstrbuton and ndependent of v n Defne the condtonal choce probabltes CCPs P j X PrY n = j X wth X = X j : j = 0,,, J The researcher s nterested n the estmaton of ths condtonal choce probabltes, and especally n the estmaton of the partal effects P jx The dataset conssts of a cross-secton of ndvduals wth nformaton about ther choces: {y n : n =,,, } The value of the vector of product characterstcs, X, s the same for every ndvdual/observaton n the sample, e, all the ndvduals have the same set of feasble choce alternatves The sample sze s large, e, asymptotcs n Queston 5 ponts] Descrbe a nonparametrc estmator of the CCPs P j X = PrY n = j X Explan why ths nonparametrc approach cannot dentfy the partal effects P jx ASWER: ote that the vector of product characterstcs X does not have any sample varaton Therefore, we can obtan a nonparametrc estmator of P j X only for the value of X that we observe Ths nonparametrc estmator can be smply the frequency estmator of PrY n = j: P j X = n= {Y n = j} It s clear that ths nonparametrc estmator does not provde any nformaton about how P j X vares when we change a component of X More specfcally, t does not provde any nformaton about P jx Queston 0 ponts] Consder the Random utlty model wth parameters θ = β, σ Descrbe the log-lkelhood functon of ths model and data Obtan the frst order condtons lkelhood equatons that defne the MLE of θ Show that, for any Random Utlty Model, these lkelhood equatons have the followng form: J P j θ ] j P j θ P jθ = 0 where j s the number of observatons n the sample wth Y n = j 7

8 ASWER: The log-lkelhood functon s: lθ = n= The frst order condtons of optmalty are: lθ = n= J J {Y n = j} ln P j θ {Y n = j} P jθ Interchangng the sums, and takng nto account that j = We also have that J J j P j θ P jθ =, and therefore J J j J P j θ P j θ P j θ = 0 P j θ P j θ J P j θ ] j P j θ P jθ P j θ = 0 n= {Y n = j}, we have: = 0, such that: = 0 = 0 Queston 3 0 ponts] Suppose that the model s restrcted to a standard logt model, e, no random coeff cents, σ = 0 Obtan the expressons for P j β and P jβ β n ths model Show that the frst order condton that defnes the MLE of β s: J ] j X j P jβ = 0 Hnt: P jβ = P jβ δ j β δ j β + P j β δ k k j δ k β, wth δ j β X j ] Propose an algorthm to compute the MLE of β Comment on the computatonal propertes of ths algorthm ASWER: When σ = 0 we have the standard logt model wth: P j β = Λ j β = exp {β X j } J k=0 exp {β X k} = exp {δ j } J k=0 exp {δ k} wth δ j β X j For the logt model, we have that Λ j δ j Λ k Therefore, P j β = Λ jβ δ j β δ j β + k j = Λ j Λ j ; and for k j, Λ j δ k = Λ j Λ j β δ k δ k β = Λ j Λ j X j k j Λ jλ k X k = Λ j Xj Xβ ] 8

9 wth Xβ J k=0 Λ kβ X k Ths mples that P jβ β P j β = P jβ Xj Xβ ] = X j Xβ P j β The frst order condtons becomes: ote that J Xβ j J ] P jβ wrte the frst order condtons as: Xj Xβ ] ] j P jβ = 0 = Xβ ] J j P jβ = Xβ ] = 0 And we can J ] j X j P jβ = 0 We can use a ewton s method to compute the MLE of β The expresson of the ewton s method s: ] ] β K+ = β l β K K l β K β β where lβ β = ] J X j j P jβ, and l β K β = J X j P j β X j Xβ ] The loglkelhood functon of the logt model s globally concave n β, such that the ewton s method always converges to the MLE Queston 4 0 ponts] In the context of ths standard logt model, descrbe the property of Independence of Irrelevant Alternatves Explan why ths s not an attractve property when the researcher s nterested n the estmaton of the Average Partal Effects P jx ASWER: The logt model mposes the restrcton that the rato between the probabltes of two alternatves, say j and, depends OLY on the utltes / characterstcs of these alternatves: P j X P k X = exp{βx j X k ]} Therefore, f we change the choce set by addng or/and removng alternatves, the ratos between probabltes should not change Ths property can generate unrealstc predctons, eg, blue bus / red bus example The IIA property restrcts also the partal effects P jx For any alternatve j k: P j X = β P j X P k X Suppose that X k s the prce of product k Ths expresson mples that a change n the prce of product k affects the "market shares" CCPs of two products, j and j, proportonally to the ther 9

10 market shares If P j X = P j X, then the effect s the same, regardless of whether product k s very smlar to product j but very dfferent to product j n terms of product characterstcs X Queston 5 5 ponts] Consder the Random Coeff cents Logt model e, σ > 0 The log-lkelhood functon of ths model s not globally concave n β and σ Ths complcates the computaton of the MLE Instead, consder the followng smulaton-based estmator For every observaton n n the sample, let v n be a random draw from the standard normal dstrbuton Then, usng v ns as f they were observable explanatory varables, estmate β and σ n the standard logt model Y n = arg max j {0,,,J} β X j + σ X j v n + ε nj ] a Wrte the expresson of the pseudo log-lkelhood functon that treats smulated v ns as f they were observable b Show that the frst order condtons that defne the estmator of β and σ are: n= n= J X j {Y n = j} Λ j v n, θ] = 0 J X j v n {Y n = j} Λ j v n, θ] = 0 wth Λ j v n, θ exp{βx j + σx j v n }/ J k=0 exp{βx k + σx k v n } c Show that ths estmator s consstent as goes to nfnty ASWER: a The log-lkelhood functon s: lθ; v = n= J {Y n = j} ln Λ j v n, θ where Λ j v n, θ exp{βx j + σx j v n }/ J k=0 exp{βx k + σx k v n } b The frst order condtons of optmalty are: lθ; v β lθ; v σ = 0 = 0 Defne δ jn = βx j + σx j v n Then, Λ j v n, θ β n= n= J J {Y n = j} Λ jv n, θ β {Y n = j} Λ jv n, θ σ = Λ jv n, θ δ jn δ jn β + k j Λ j v n, θ Λ j v n, θ Λ j v n, θ δ kn δ kn β = 0 = 0 = Λ jn Λ jn X j k j Λ jnλ kn X k = Λ jn Xj X n v n, θ ] 0

11 wth Xv n, θ J k=0 Λ kv n, θ X k And, Λ j v n, θ σ = Λ jv n, θ δ jn δ jn σ + k j Λ j v n, θ δ kn δ kn σ = Λ jn Λ jn X j v n k j Λ jnλ kn X k v n = Λ jn v n Xk X n v n, θ ] ote that J Xv n, θ {Y n = j} Λ j v n, θ] = Xv n, θ J {Y n = j} Λ j v n, θ] = Xv n, θ ] = 0 Then, we can wrte the frst order condtons as: n= n= J X j {Y n = j} Λ j v n, θ] = 0 J X j v n {Y n = j} Λ j v n, θ] = 0 c Consstency of the estmator as goes to nfnty ote that as goes to nfnty, J J X j {Y n = j} Λ j v n, θ] p X j E {Y n = j} Λ j v n, θ n= J J X j v n {Y n = j} Λ j v n, θ] p X j E v n {Y n = j} Λ j v n, θ] n= ow, we show that, at the true parameters θ 0, the expectatons E {Y n = j} Λ j v n, θ 0 and E v n {Yn = j} Λ j v n, θ 0 ] are equal to zero E {Y n = j} Λ j v n, θ 0 = E {Y n = j} E vn Λj v n, θ 0 And usng teratve expectatons, = E {Y n = j} P j θ 0 = 0 E v n {Yn = j} Λ j v n, θ 0 ] = E vn vn E εn v n {Yn = j} Λ j v n, θ 0 ] = E vn vn Λj v n, θ 0 Λ j v n, θ 0 ] = 0 Ths result mples that ths pseudo maxmum lkelhood estmator can be nterpreted as a Method of Moments estmator where the moments are vald and dentfy the parameters β, σ The model satsfes standard regularty condtons of the Method of Moments estmator Therefore, ths estmator of θ 0 s root- consstent and asymptotcally normal estmator The man computatonal advantage of the estmator s that the pseudo log-lkelhood functon s globally concave n β, σ such that ewton s method always converges to the consstent pseudo-mle