Instrumental Variables Quantile Regression for Panel Data with Measurement Errors

Transcription

1 Instrumental Variables Quantile Regressin fr Panel Data with Measurement Errrs Antni Galva University f Illinis Gabriel Mntes-Rjas City University Lndn Tky, August 2009

2 Panel Data Panel data allws the pssibility f fllwing the same individuals ver time, which allws t cntrl fr individual specific effects Recently, semiparametric panel data mdels have attracted cnsiderable interest in bth thery and applicatin since the errr distributin is nt specified Hnre and Lewbell (ECMT 2002), Kenker (Jurnal f Multivariate Analysis 2004), Geraci and Bttai (Bistatistics 2007), Abrevaya and Dahl (JBES 2008)

3 Quantile Regressin Panel Data Kenker (JMA 2004) - Quantile Regressin fr Panel Data Cnditinal quantile functins f the respnse f the y it Q yit (τ η i, x it ) = η i + β(τ)x it i = 1,..., n, t = 1,..., T In sme applicatins, it is f interest t explre a brad class f cvariate effects (lcatin and scale shifts), while still accunting fr individual specific effects Such mdels enable t explre varius frms f hetergeneity assciated with the cvariates under less stringent distributinal assumptins

4 Measurement Errr Ecnmic quantities are frequently measured with errr, particularly if lngitudinal infrmatin is cllected thrugh ne-time retrspective surveys, which are ntriusly susceptible t recall errrs If the regressrs are indeed subject t measurement errrs, it is well knwn that the slpe cefficient f the least squares regressin estimatr is biased attenuatin bias The bias may be exacerbated in panel data mdels This tpic has als attracted cnsiderable attentin in the quantile regressin literature: Chesher (2001) and Schennach (ET 2008)

5 IV slutin fr ME in Panel Data There is an extensive literature that use IV strategies t slve the ME bias See fr instance Griliches and Hausman (JE 1986), Hsia (1992, 2003), Wansbeek (JE 2001), Birn (ER 2000) The IV apprach emplys lagged (r lagged differences f the) regressrs as instruments fr the mismeasured variable

6 Our Cntributin We shw that measurement errrs prduce a similar attenuatin bias as in OLS, by analytically deriving an apprximatin t the bias in QR Prpse an instrumental variables strategy t estimate fixed effects quantile regressin panel data with measurement errr and reduce the bias The estimatr uses lagged dependent bservatins (r lagged differences) as instruments The estimatr emplys Chernzhukv and Hansen (ECMT 2005, JE 2006, JE 2008) We shw cnsistency and asympttic nrmality f the estimatr, prvided N a /T 0, fr sme a > 0

7 Our Cntributin Cnduct Mnte Carl experiments t shw that, even in shrt panels, the prpsed estimatrs can substantially reduce the bias Prpse Wald and Klmgrv-Smirnv tests fr general linear hypthesis, and derive the respective limiting distributins Illustrate the new apprach t the Tbin s q thery f investment

8 Measurement Errrs in OLS Panel Data Cnsider the fllwing representatin f a panel data mdel with individual fixed effects and measurement errrs y it = d itη + x itβ + u it i = 1,..., N; t = 1,..., T. (1) Suppse that we d nt bserve x it, but rather x it, which is a nisy measure f x it subject t an additive measurement errr ɛ it, x it = x it + ɛ it. (2) Using equatin (2) we can express (1) in terms f the bserved y s and x s as y it = d itη +x itβ +u it ɛ itβ i = 1,..., N; t = 1,..., T. (3)

9 Measurement Errrs in OLS Panel Data (Cnt.) Define v = [d, x ], Λ ɛ = [0, ɛ ], v = [d, x ] = v + Λ ɛ and ϕ = [η, β ] Tw estimatrs: ϕ = argmin ϕ E[y v ϕ] 2, (4) and ϕ = argmin ϕ E[y v ϕ] 2. (5) Effect f ME (signal t nise-rati) ϕ = ϕ ( E [ v v + Λ ɛ Λ ɛ]) 1 E[Λɛ ɛ β ]

10 Measurement Errrs in QR Panel Data Cnsider nw the τ th cnditinal quantile functin f the respnse y, Q y (τ d, x ) = d η (τ) + x β (τ) (6) Define v = [d, x ], Λ ɛ = [0, ɛ ], v = [d, x ] = v + Λ ɛ and ϕ = [η, β ] Tw estimatrs: ϕ (τ) = argmin ϕ E[ρ τ (y v ϕ)] (7) where ρ τ (u) := u(τ I(u < 0)) as in Kenker and Bassett (ECMT 1978), and ϕ (τ) = argmin ϕ E[ρ τ (y v ϕ)] (8)

11 The fllwing Lemma shws that the measurement errr bias in QR can be apprximated t an expressin similar t that in OLS, using the Angrist, Chernzhukv, and Fernandez-Val (ECMT 2006) apprximatin. Lemma 1. Assume that: (i) the cnditinal density functin f y(y v, ɛ) exists and is bunded a.s.; (ii) E[y], E[Q y(τ v, ɛ) 2 ], and E [v, ɛ ] 2 are finite; (iii) ϕ (τ) and ϕ (τ) uniquely slve equatins (7) and (8) respectively; (iv) ɛ is independent f (d, x, u). Then, where ϕ (τ) = ϕ (τ) ( E [ ω τ (v, ɛ) (vv ) ]) 1 E[ω τ (v, ɛ) vɛ β (τ)] ω τ (v, ɛ) = 1 0 f u(τ) (u τ (v, ɛ; ϕ (τ)) v, ɛ) du/2 is a weighting functin, and τ (v, ɛ; ϕ (τ)) = v (ϕ (τ) ϕ (τ)) + ɛ β (τ) is the quantile regressin specificatin errr.

12 In particular, under sme bundedness cnditins: ϕ (τ) ϕ (τ) ( E [ f y (q) ( v v + Λ ɛ Λ ɛ)]) 1 E[fy (q)λ ɛ ɛ β (τ)] where f y (q) = f y (Q τ (y v ) v )

13 Digressin n IV fr Quantile Regressin Cnsider the fllwing quantile regressin mdel, Q Y (τ X) = Xβ(τ) where Y is the utcme variable cnditinal n the exgenus variables f interest X Then nte that Y = Xβ(U) U X U(0, 1) Kenker and Bassett (ECMT 1978) shw that ˆβ = argmin β ρτ (y x β) where ρ τ (u) = u(τ I(u < 0))

14 Digressin n IV fr Quantile Regressin (Cnt.) ˆβ(τ) based n the mment cnditin ψ τ (R) X, where R is the residual Y Xβ and ψ τ (u) = τ I(u < 0) Suppse that ψ τ (R) X des nt hld But we can find ψ τ (R) W We want t estimate β But nw Y = Xβ(U) U W U(0, 1)

15 Digressin n IV fr Quantile Regressin (Cnt.) W des nt belng t the mdel Thus, fr fixed β, in the quantile regressin f (Y Xβ) n W, W shuld have cefficient f zer Estimatr is define as: ˆβ = argmin β ˆγ(β) A where ˆγ(β) = argmin γ ρτ (y x β w γ) ˆβ(τ) that makes ˆγ(τ) 0 is the instrumental variables estimatr

16 IVQRFE Nw we cnsider a finite-sample analg f the abve prcedure fr panel data Define Q y (τ η, x, z) = d η(τ) + x β(τ) + z α(τ), Q NT := N T i=1 t=1 ) ρ τ (y it d itη(τ) x itβ(τ) z itα(τ) w itγ(τ)

17 IVQRFE (Cnt.) The IVQRFE is defined as fllws: Fr a given value f the structural parameter, say β, ne estimates the panel QR t btain (ˆη(β, τ), ˆβ(β, τ), ˆα(β, τ), ˆγ(β, τ)) := argmin η,α,γ Q NT (τ, η, β, α, γ) T find an estimate fr β(τ), we lk fr a value β that makes the cefficient n the instrumental variable γ(β, τ) as clse t 0 as pssible ˆβ(τ) = argmin β B ˆγ(β, τ) A

18 Asympttics fr IVQRFE Assumptins A1 The y it are independent with cnditinal distributin functins F it, and differentiable cnditinal densities, 0 < f it <, with bunded derivatives f it fr i = 1,..., N and t = 1,..., T ;

19 Asympttics fr IVQRFE Assumptins (Cnt.) A2 Let D = I N ι T, and ι T a T -vectr f nes, X = (x it ) be a NT dim(β) matrix, Z = (z it ) be a NT dim(α) matrix, and W = (w it ) be a NT dim(γ) matrix. Fr Π(η, β, α, τ) := E[(τ 1(D η + X β + Z α)) ˇX] Π(η, β, α, γ, τ) := E[(τ 1(D η + X β + Z α + W γ)) ˇX] ˇX := [D, Z, W ], Jacbian matrices (η,β,α) Π(η, β, α, τ) and (η,α,γ) Π(η, β, α, γ, τ) are cntinuus and have full rank, unifrmly ver E B A G T and the image f E B A under the map (η, β, α) Π(η, β, α, τ) is simply cnnected;

20 Asympttics fr IVQRFE Assumptins (Cnt.) A3 Dente Φ(τ) = diag(f it (ξ it (τ))), where ξ it (τ) = d it η(τ) + x it β(τ) + z it α(τ) + w itγ(τ), M D = I P D and P D = D(D Φ(τ)D) 1 D Φ(τ). Let X = [W, Z]. Then, the fllwing matrix is invertible: J αγ = E( X M D Φ(τ)M D X); Nw define [ J α, J γ] as a partitin f J 1 αγ, J β = E( X M D Φ(τ)M D X) and H = J γa[β(τ)] J γ. Then, J β HJ β is als invertible;

21 Asympttics fr IVQRFE Assumptins (Cnt.) A4 Fr all τ, (β(τ), α(τ)) int B A, and B A is cmpact and cnvex; A5 max it x it = O( NT ); max it z it = O( NT ); max it w it = O( NT ); A6 N a T 0, fr sme a > 0.

22 Asympttics fr IVQRFE Results Let θ(τ) = (β(τ), α(τ)) Therem 1 Given assumptins A1-A6, (η(τ), β(τ), α(τ)) uniquely slves the equatins E[ψ τ (Y D η X β Z α) ˇX] = 0 ver E B A, and ϕ(τ) = (η(τ), β(τ), α(τ)) is cnsistently estimable.

23 Asympttics fr IVQRFE Results (Cnt.) Therem 2 (Asympttic Nrmality) Under cnditins A1-A6, fr a given τ (0, 1), ˆθ(τ) cnverges t a Gaussian distributin as NT (ˆθ(τ) θ(τ)) d N(0, Ω(τ)), Ω(τ) = (K, L ) S(K, L ) where S = τ(1 τ)e[v V ], V = X M D, K = (J β HJ β) 1 J β H, H = J γa[β(τ)] J γ, L = J α M, M = I J β K, J β = E( X M D Φ(τ)M D X), [ J α, J γ] is a partitin f Jαγ 1 = (E( X M D ΦM D X)) 1, Φ(τ) = diag(f it (ξ it (τ))), and X = [Z, W ].

24 Inference fr IVQRFE Test the hypthesis Rθ(τ) = r, when r is knwn Under the linear hypthesis H 0 : Rθ(τ) = r, assumptins A1-A6, we have V n = NT [RΣR ] 1/2 (Rˆθ r) B q (τ), where B q (τ) represents a q-dimensinal standard Brwnian Bridge Thus, fr given τ, the regressin Wald prcess can be cnstructed as W n = NT (Rˆθ r) [RˆΩR ] 1 (Rˆθ r) where ˆΩ is a cnsistent estimatrs f Ω

25 Inference fr IVQRFE (Cnt.) Therem 3 (Wald Test Inference). Under H 0 : Rθ(τ) = r and cnditins A1-A6, fr fixed τ, W n (τ) a χ 2 q. Therem 4 (Klmgrv-Smirnv test). Under H 0 and cnditins A1-A6, KSW n = sup τ T W n (τ) sup Q 2 q(τ). τ T Critical values fr sup Q 2 q(τ) have been tabled by DeLng (1981) and, mre extensively, by Andrews (1993) using simulatin methds.

26 Mnte Carl - Descriptin Evaluate the finite sample perfrmance f the quantile regressin instrumental variables estimatr Mdel: y it = η i + x itβ + z itα + u it, Tw schemes t generate the disturbances u it u it N(0, σ 2 u) u it t 3

27 Mnte Carl - Descriptin (Cnt.) In bth cases we have an additive measurement errr f the frm x it = x it + ɛ it, where x it fllws an ARMA(1,1) prcess (1 φl)x it = µ i + ε it + θε it 1 and ɛ it fllws the same distributin as u it, that is, nrmal distributin and t 3

28 Mnte Carl - Descriptin (Cnt.) The fixed effects, µ i and α i, are generated as T µ i = e 1i + T 1 ɛ it, e 1i N(0, σe 2 1 ), η i = e 2i + T 1 t=1 T t=1 x it, e 2i N(0, σ 2 e 2 ). In the simulatins, we experiment with T = 10 and N = replicatins Cnsider the fllwing values fr the remaining parameters: (β, α) = (1, 1); φ = 0.6, θ = 0.7, γ = 1, σ 2 u = σ 2 e 1 = σ 2 e 2 = 1.

29 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr nrmal distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

30 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr nrmal distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

31 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr nrmal distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

32 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr nrmal distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

33 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr nrmal distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

34 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr nrmal distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

35 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr t 3 distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

36 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr t 3 distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

37 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr t 3 distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

38 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr t 3 distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

39 Lcatin Shift Mdel: Bias and RMSE f estimatrs fr t 3 distributin (T = 10 and N = 100) FE IVFE QRFE IVQRFE β = 1 Bias RMSE α = 1 Bias RMSE

40 Empirical applicatin: Investment, Tbin s q and cash flw Tbin s q is the rati f the market valuatin f a firm and the replacement value f its assets Firms with a high value f q are cnsidered attractive as t the investment pprtunities, whereas a lw value f q indicates the ppsite Since the peratinalizatin f q is nt clear-cut and unambiguus, estimatin pses a measurement errr prblem Many empirical investment studies fund a very disappinting perfrmance f the q thery f investment, althugh this thery has a gd perfrmance when measurement errr is purged as in Ericksn and Whited (JPE 2000)

41 Empirical applicatin: Investment, Tbin s q and cash flw In parallel, investment thery is als interested in the effect f cash flw, as the thery predicts that financially cnstrained firms are mre likely t rely n internal funds t finance investment Alti (JF 2003), Almeida, Campell and Weisbach (JF 2004) Hwever, Ericksn and Whited (2000) argue that cash flw has n effect n investment nce measurement errr in Tbin s q is taken int accunt

42 Empirical applicatin: Investment, Tbin s q and cash flw QR is useful t describe differences in investment ratis acrss firms in the presence f unbserved hetergeneity Unbserved characteristics (fr the ecnmetrician) acrss firms may be due t several reasns: 1 The ability f managers (insiders) r the characteristics f investrs (utsiders) (Bertrand and Schar, QJE 2003; Walentin and Lrenzni, NBER 2007) 2 Firms capital structure (Chirink, JEDC 1993)

43 Main hypthesis In the upper cnditinal quantiles f investment, investment may be driven by insiders knwledge f business pprtunities r a particular capital structure that requires mre investment, and therefre, we expect that investment wuld be mre respnsive t changes in q and cash flw, as the firms wuld use all available resurces t finance its prjects.

44 The baseline mdel in the literature is I it /K it = η i + βq it + αcf it /K it + u it, where I dentes investment, K capital stck, CF cash flw, q well-measured Tbin s q, η is the firm-specific effects and u is the innvatin term. Our bjective is estimating the fllwing cnditinal quantile functin: Q I/K (τ η, CF/K, q) = η i (τ) + β(τ)q it + α(τ)cf it /K it.

45 Data We fllw Almeida, Campell and Weisbach (JF 2004) apprach by cnsidering a sample f manufacturing firms (SICs 2000 t 3999) ver the 1980 t 2005 perid with data available frm COMPUSTAT s P/S/T, Full Cverage Only firms with bservatins in every year are used, in rder t cnstruct a balanced panel f firms fr the 26 year perid (the chice f a balanced panel is made t reduce the cmputatinal burden) Mrever, fllwing thse authrs we eliminate firms fr which cash-hldings exceeded the value f ttal assets and thse displaying asset r sales grwth exceeding 100% Our final sample cnsists f 4550 firm-years and 175 firms

46 τ Variable FE QRFE q (0.010) (0.009) (0.009) (0.013) CF/K (0.006) (0.009) (0.009) (0.013) IVFE IVQRFE q (0.074) (0.045) (0.089) (0.104) CF/K (0.010) (0.008) (0.012) (0.025) Inst. q t 1 q t 1 q (0.051) (0.033) (0.057) (0.082) CF/K (0.008) (0.029) (0.046) (0.061) Inst. q t 1, q t 2 q t 1, q t 2

47 Figure: FE and QRFE QRFE Tbin's Q cefficient quantiles QRFE Cash Flw cefficient quantiles

48 τ Variable FE QRFE q (0.010) (0.009) (0.009) (0.013) CF/K (0.006) (0.009) (0.009) (0.013) IVFE IVQRFE q (0.074) (0.045) (0.089) (0.104) CF/K (0.010) (0.008) (0.012) (0.025) Inst. q t 1 q t 1 q (0.051) (0.033) (0.057) (0.082) CF/K (0.008) (0.029) (0.046) (0.061) Inst. q t 1, q t 2 q t 1, q t 2

49 Figure: IVFE and IVQRFE (Instrument q t 1 ) IVQRFE Tbin's Q cefficient quantiles IVQRFE Cash Flw cefficient quantiles

50 Figure: IVFE and IVQRFE (Instruments q t 1, q t 2 ) IVQRFE (DL1x DL2x) Tbin's Q cefficient quantiles IVQRFE (DL1x DL2x) Cash Flw cefficient quantiles

51 Cnclusins Prpse an instrumental variables quantile regressin estimatr fr panel data that slves measurement errr prblems Shw cnsistency and asympttic nrmality f the estimatrs Prpse a Wald and Klmgrv-Smirnv tests fr linear hyptheses Apply the estimatr and test t Tbin s q thery f investment