Quantile Regression for Dynamic Panel Data

Transcription

1 Quantile Regressin fr Dynamic Panel Data Antni F. Galva University f Illinis at Urbana-Champaign June 03, 2008 Abstract This paper studies estimatin and inference in a quantile regressin dynamic panel mdel with fixed effects. T reduce the dynamic bias we develp an instrumental variables apprach that emplys lagged regressrs as instruments. We shw that the prpsed estimatrs are cnsistent and asympttically nrmal. We suggest Wald and Klmgrv-Smirnv type tests fr general linear restrictins. Mnte Carl studies are cnducted t evaluate the finite sample prperties f the estimatrs and tests. The instrumental variables apprach turns ut t be especially advantageus in terms f the bias, rt mean square errr, and pwer f the test statistics when innvatins are nn-gaussian and heavy-tailed. We illustrate the new apprach by testing fr the presence f time nn-separability in utility using husehld cnsumptin panel data. The results shw evidence f asymmetric persistence in cnsumptin dynamics. Key Wrds: Quantile regressin, dynamic panel, fixed effects, instrumental variables JEL Classificatin: C14, C23 Department f Ecnmics, University f Illinis at Urbana-Champaign, 484 Whlers Hall, 1206 Suth 6th Street, Champaign, IL 61820, USA. galva@uiuc.edu. The authr wuld like t express his appreciatin t Rger Kenker, participants in the seminars at City University Lndn, University f Illinis at Urbana-Champaign, and 17th Midwest Ecnmetrics Grup meeting fr helpful cmments and discussins. All the remaining errrs are my wn.

2 1 Intrductin This paper studies estimatin and inference in a quantile regressin dynamic panel data mdels with individual specific intercepts in additin t strictly exgenus explanatry variables. Cnsistency f the estimatrs in standard dynamic panel data depends critically n the assumptins abut the initial cnditins f the dynamic prcess. Andersn and Hsia (1981, 1982) and Arellan and Bnd (1991) have shwn that instrumental variables methds are able t prduce cnsistent estimatrs that are independent f the initial cnditins. T reduce the dynamic bias in the quantile regressin dynamic panel with fixed effects we develp an instrumental variables apprach that emplys lagged (r lagged differences f the) regressrs as instruments. Thus, the mdel cmbines the instrumental variables cncept fr dynamic panel data, and the quantile regressin instrumental variables framewrk prpsed by Chernzhukv and Hansen (2005, 2006, 2008). We shw that under sme mild regularity cnditins prvided N a /T 0, fr sme a > 0, the quantile regressin dynamic panel instrumental variables (QRIV) estimatrs are cnsistent and asympttically nrmal. Mnte Carl experiments shw that, even in shrt panels, the prpsed estimatrs can substantially reduce the dynamic bias. In additin, we prpse a Wald and Klmgrv-Smirnv tests fr general linear hypthesis, and derive the respective limiting distributins. Semiparametric panel data mdels have attracted cnsiderable interest in bth thery and applicatins since the Gaussian assumptins underlying classical least squares methds are smetimes implausible. Manski (1987), Hnre (1992), Kyriazidu (1997), Hnre and Kyriazidu (2000), Hnre and Lewbel (2002), and Kenker (2004) all discuss estimatin and inference in the semiparametric cntext. Kenker (2004) prpses a general apprach t estimate static quantile regressin mdels fr lngitudinal data with fixed effects, thus allwing ne t cntrl fr unbserved individual hetergeneity. Abrevaya and Dahl (2007) and Geraci and Bttai (2007) prpse different appraches fr quantile regressin static panel data. The latter uses a likelihd apprach under asymmetric Laplace distributins and the frmer cnsiders the crrelated randm effects mdel f Chamberlain (1982). Quantile regressin methds allw ne t explre a range f cnditinal quantiles expsing a variety f frms f cnditinal hetergeneity under weaker distributinal assumptins, and als prvide a framewrk fr rbust estimatin and inference. Hwever, it is ften desirable t use panel data t estimate behaviral relatinships that are dynamic in character, mdels cntaining lagged dependent variables, and this pses sme new prblems. As shwn in Kenker and 1

3 Xia (2006a), quantile regressin mdels with dynamic regressrs ffers a systematic strategy fr examining hw cvariates influence the lcatin, scale, and shape f the entire respnse distributin, ptentially expsing a variety f frms f hetergeneity in respnse dynamics. There is an extensive literature n estimatin f fixed effects dynamic panel data mdels fr cnditinal means, and varius instrumental variables (IV), generalized methd f mments (GMM), and least squares dummy variables estimatrs have been prpsed t attenuate the dynamic parameters bias; see, fr instance, Andersn and Hisa (1981, 1982), Arellan and Bnd (1991), Arellan and Bver (1995), Ahn and Schmidt (1995), Kiviet (1995), Ziliak (1997), Blundell and Bnd (1998), Hahn (1999), Bun and Carree (2005). Recently, a number f additinal appraches t reduce the bias in dynamic and nnlinear panels, that use the asympttic apprximatins derived as bth the number f individuals, N, and the number f bservatins per individual, T, g t infinity jintly, have been prpsed; see, fr example, Arellan and Hahn (2005) fr a survey, and Hahn and Kuersteiner (2002), Alvarez and Arellan (2003), Hahn and Kuersteiner (2004), Hahn and Newey (2004), Carr (2007), Bester and Hansen (2007), Fernandez-Val and Vella (2007) fr specific appraches. The apprach prpsed in this paper t reduce the bias in the quantile regressin dynamic panel data with fixed effects explits elements f the panel data instrumental variables literature. Mst instrumental variables estimatrs are at least partly based n the intuitin that first differencing yields a mdel free f fixed effects. Despite its appeal as a prcedure that avids estimatin f the individual specific effects parameters, differencing transfrmatin t eliminate the fixed effects is nt available in the quantile regressin framewrk, and we are required t deal directly with the full prblem. We develp a strategy based n Chernzhukv and Hansen (2006, 2008) instrumental quantile regressin methds t estimate the parameters f interest, and cnsider an asympttic thery where N can grwth at a cntrlled rate relative t T. Mnte Carl simulatin shws that the fixed effects panel quantile regressin is biased in the presence f lagged dependent variables, while the prpsed instrumental variables methd is able t reduce the bias even in shrt panels. In additin, the Mnte Carl experiments suggest that the quantile regressin IV apprach fr dynamic panel data perfrms better than rdinary least squares IV in terms f bias and rt mean squared errr fr nn-gaussian heavy-tailed distributins. Tests based n fixed effects QRIV turn ut t be especially advantageus when innvatins are heavy-tailed. 2

4 We illustrate the prpsed methds by testing fr the presence f time-nnseparability in utility using husehld cnsumptin data frm the Panel Study n Incme Dynamics (PSID) dataset. Previus wrks, as Dynan (2000), find n evidence f habit frmatin using PSID and rdinary least squares based estimatin at annual frequency. Hwever, the results using quantile regressin dynamic panel shw evidence f asymmetric persistence in cnsumptin dynamics. In additin, it is pssible t reject the null hypthesis f n effect f past cnsumptin grwth n subsequent cnsumptin grwth fr lw quantiles. Thus, fr lw quantiles f the cnditinal quantile functin f cnsumptin grwth, the results suggest that increasing in current cnsumptin grwth leads t decreasing in subsequent cnsumptin grwth, and fr these crrespnding quantiles, there is evidence f habit persistence. Mrever, the results shw imprtant evidence f hetergeneity in the determinants f cnsumptin such as difference in family size, age f the husehld, age f the husehld squared, and race. If an ecnmic dynamic panel data displays asymmetric dynamics systematically, then apprpriated mdels are needed t incrprate such behavir. The rest f the paper is rganized as fllws. Sectin 2 presents the quantile regressin dynamic panel data instrumental variables with fixed effects estimatin. Inference is described in Sectin 3. Sectin 4 describes the Mnte Carl experiment. In Sectin 5 we illustrate the new apprach using PSID dataset. Finally, Sectin 6 cncludes the paper. 2 The Mdel and Assumptins Cnsider the classical dynamic mdel fr panel data with individual fixed effects 1 y it = η i + αy it 1 + x itβ + u it i = 1,..., N; t = 1,..., T. (1) where y it is the respnse variable, η i dentes the individual fixed effects, y it 1 is the lag f the respnse variable, and x it is a p-vectr f the exgenus cvariates. It is pssible t write mdel (1) in a mre cncise matrix frm as, y = Zη + αy 1 + Xβ + u, (2) s that Z = I n ι T, and ι T is a T 1 vectr f nes. Nte that Z represents an incidence matrix that identifies the N distinct individuals in the sample. 1 T simplify the presentatin we fcus n the first-rder autregressive prcesses, since the main insights generalize in a simple way t higher-rder cases. 3

5 It is well knwn that in the dynamic mean regressin panel data mdel the fixed-effects (FE) estimatr can be severely biased when the number f individuals N is large relative t the time series dimensin T. Hahn and Kuersteiner (2002) and Alvarez and Arellan (2003) shw that even if the FE estimatr is cnsistent as T, its asympttic distributin may cntain an asympttic bias term when N, depending n the relative rates f increasing f T and N, in particular unless N 0 the bias term cannt be neglected. The T incnsistency f least squares based estimatrs with fixed T and large N has led t a fcus n the instrumental variables estimatin in the recent literature. Fr linear mdels with an additive individual effect, the prblems with the FE estimatr can be amelirated by using an apprpriate transfrmatin, such as differencing, t eliminate the unbserved effects. Then, instrumental variables (IV) can usually be fund fr implementatin f the estimatr, as in Arellan and Bnd (1991). Fr nnlinear in parameters mdels the inherent reliance n first differencing is prblematic. Except fr a small number f cases where cnditining n sme sufficient statistic eliminates fixed effects, there des nt seem t exist any general strategy fr ptentially nnlinear panel mdels. In the nnlinear case, Hahn and Newey (2004), in static mdels, and Hahn and Kuersteiner (2004), in dynamic mdels, prpsed crrected estimatrs and shw that the parameters f interest are cnsistently estimable when N and T increase t infinity at the same rate. 2.1 Quantile Regressin fr Dynamic Panel Data In this sectin we cnsider estimatin f the dynamic panel data quantile regressin that include individual specific fixed effects. Estimatin f this mdel is cmplicated by the nn-bserved initial cnditins. In least squares dynamic panel data applicatins the usual cmputatinal strategy t deal with the fixed effects has been t take the first difference f the mdel, and then cmpute the parameters f interest frm the transfrmed mdel applying the usual tw stage least squares instrumental variables apprach. In additin, using such transfrmatins, it is pssible t estimate cnsistent standard errrs fr the parameters f interest. In quantile regressin framewrk this transfrmatin t eliminate the individual effects is nt available and we are required t deal with the full prblem. 2 In spite f this 2 As Kenker (2004) bserves, in typical applicatins the design matrix [Z : y 1 : X] f the full prblem is very sparse, i.e. has mstly zer elements. Standard sparse matrix strage schemes nly require space fr the nn-zer elements and their indexing lcatins, and this cnsiderably reduces the cmputatinal effrt and memry requirements in large prblems. 4

6 difficulty we present an instrumental variables strategy t reduce the bias f the estimates and t make inference. Cnsider the analgus mdel fr the τ th cnditinal quantile functins f the respnse f the tth bservatin n the ith individual y it Q yit (τ z it, y it 1, x it ) = z it η(τ) + α(τ)y it 1 + x itβ(τ) (3) where y it is the utcme f interest, y it 1 is the lag f the variable f interest, x it are the exgenus variables, z it identifies the fixed effects, and η = (η 1,..., η N ) is the N 1 vectr f individual specific effects r intercepts. The effects f the cvariates (z it, y it 1, x it ) are allwed t depend upn the quantile, τ, f interest. 3 The dynamic panel quantile regressin mdel described in equatin (3) can be develped using the fllwing representatin, y it = z it η(u it ) + α(u it )y it 1 + x itβ(u it ), u it z it, y it 1, x it is Unifrm(0,1). (4) Kenker (2004) presents a penalized quantile regressin methd t estimate the parameters in mdel (3) when α(τ) = 0. Applying this principle t equatin (3) ne wuld slve (ˆη, ˆα, ˆβ) = min η,α,β N i=1 T ρ τ (y it z it η αy it 1 x itβ) where ρ τ (u) := u(τ I(u < 0)) as in Kenker and Bassett (1978). Hwever, the fixed effects quantile regressin estimatr, as in the rdinary least squares case, is ptentially biased in the presence f lagged variables. In standard estimatin f dynamic panel mdels the initial values f the dynamic prcess raise a prblem. Cnsistency f the estimatrs depends critically n the assumptins abut the initial cnditins, and mistaken chices f the initial cnditin will yield estimatrs that are nt asympttically equivalent t the crrect ne, and hence may nt be cnsistent. 4 Andersn and Hsia (1981, 1982) and Arellan and Bnd (1991) shw that cnsistent estimatrs fr dynamic panel data mdels can be btained by using instrumental variable methds that prduces estimatrs independent f the initial cnditins. 3 It is imprtant t nte that we allw the individual effects, η(τ), t depend n the specific quantile, τ, since we d nt estimate the mdel fr several quantiles at nce. In additin, cntrary t Kenker (2004), we d nt cnsider a penalized mdel. Thus, we wrk n the usual fixed effects quantile regressin estimatr. 4 See Hsia (2003) and Heckman (1981) fr mre details. 5

7 The prblem f bias fr the dynamic panel quantile regressin can be amelirated thrugh the use f instrumental variables, w, that affect the determinatin f lagged y but are independent f u. Assuming that the instrumental variable w is available the mdel described in equatin (4) can be written in the fllwing frm y it = z it η(u it ) + α(u it )y it 1 + x itβ(u it ), u it z it, w it, x it is Unifrm(0,1). (5) Fllwing Chernzhukv and Hansen (2006, 2008), frm (5) and the availability f IV, we cnsider estimatrs defined as: fr fixed α (ˆη(α), ˆβ(α), N T ˆγ(α)) = min ρ τ (y it z it η αy it 1 x itβ w itγ) η,β,γ and estimate α by slving fr, i=1 ˆα = min ˆγ(α) A α where x A = x Ax. Our final parameter estimates f interest are thus ˆθ(τ) (ˆα(τ), ˆβ(τ)) (ˆα(τ), ˆβ(ˆα(τ), τ)). The intuitin underlying the estimatr is that, since w is a valid instrument, it is independent f u and it shuld have a zer cefficient. Thus, fr given α, the quantile regressin f (y it αy it 1 ) n the variables (z it, w it, x it ) shuld generate cefficient zer fr the variable w it. Hence, by minimizing the cefficient f the variable w it ne can recver the estimatr f α. Therefre, the bias generated by inclusin f y it 1 in equatin (3) is reduced thrugh the presence f instrumental variables, w it, that affect the determinatin f y it but are independent f u it. Values f y lagged (r differences) tw perids r mre and/r lags f the exgenus variable x affect the determinatin f lagged y but are independent f u, s they can be used as instruments t estimate α and β by the quantile regressin dynamic panel instrumental variables (QRIV) methd. Given the estimates, the τ-th cnditinal quantile functin f y it, can be estimated by, ˆQ yit (τ z it, w it, x it ) = z itˆη(τ) + ˆα(τ)y it 1 + x ˆβ(τ). it In additin, given a family f estimated cnditinal quantile functins, the cnditinal density f y it qutients, at varius values f the cnditining cvariate can be estimated by the difference ˆf yit (τ η i, w it, x it ) = τ k τ k 1 ˆQ yit (τ k z it, w it, x it ) ˆQ yit (τ k 1 z it, w it, x it ), 6

8 fr sme apprpriately chsen sequence f τ s. It is imprtant t nte that, prvided that the right hand side f (5) is mntne increasing in u it, it fllws that the τth cnditinal quantile functin f y it can be written as (3). Implicit in the frmulatin f mdel (3) is the requirement that Q yit (τ F it 1 ) is mntne increasing in τ fr all F it 1, where F it is the σ-field generated by {y is, x is, s t}. Mntnicity f the cnditinal quantile functins impses sme discipline n the frms taken by the cefficients. This discipline essentially requires that the functin Q yit (τ F it 1 ) is mntne in τ in sme relevant regin f F it -space. In sme circumstances, this necessitates restricting the dmain f the dependent variables; in ther cases, when the crdinates f the dependent variables are themselves functinally dependent, mntnicity may hld glbally. In this paper, we assume that, fr sme relevant regin f F it, mntnicity f Q yit (τ F it 1 ) in τ hlds; see Kenker and Xia (2006a), and Kenker and Xia (2006b) fr mre details. The implementatin f the quantile regressin instrumental variables prcedure is straightfrward. Define the bjective functin N Q (τ, η i, α, β, γ) := i=1 T ρ τ (y it η i αy it 1 x itβ w itγ) (6) where y it 1 is, in general, a dim(α)-vectr f endgenus variables, η i are the fixed effects, x it is a dim(β)-vectr f exgenus explanatry variables, w it is a dim(γ)-vectr f instrumental variables such that dim(γ) dim(α). The quantile regressin instrumental variable estimatr fr dynamic panel can be implemented as fllws: 1) Fr a given quantile f interest τ, define a grid f values {α j, j = 1,..., J; α < 1}, and run the rdinary τ-quantile regressin f (y it y it 1 α j ) n (z it, w it, x it ) t btain cefficients ˆη(α j, τ), ˆβ(αj, τ) and ˆγ(α j, τ), that is, fr a given value f the autregressin structural parameter, say α, ne estimates the rdinary panel quantile regressin t btain (ˆη i (α j, τ), ˆβ(α j, τ), ˆγ(α j, τ)) := min η i,β,γ Q (τ, η i, α, β, γ). (7) 2) T find an estimate fr α(τ), chse ˆα(τ) as the value amng {α j, j = 1,..., J} that makes ˆγ(α j, τ) clsest t zer. Frmally, let ˆα(τ) = min[ˆγ(α, τ) ]Â(τ)[ˆγ(α, τ)] (8) α A where A is a psitive definite matrix. 5 The estimate ˆβ(τ) is then given by ˆβ(ˆα(τ), τ), which 5 As discussed in Chernzhukv and Hansen (2006), the exact frm f A is irrelevant when the mdel is exactly identified, but it is desirable t set A equal t the asympttic variance-cvariance matrix f ˆγ(α(τ), τ) therwise. 7

9 leads t the estimates ˆθ(τ) = ( ˆα(τ), ˆβ(τ) ) ( = ˆα(τ), ˆβ(ˆα(τ), ) τ). (9) The estimatr finds parameter values fr α and β thrugh the inverse step (8) such that the value f cefficient γ(α, τ) n w in the rdinary quantile regressin step (7) is driven as clse t zer as pssible. We shall shw that this estimatr is cnsistent and asympttically nrmal under sme regularity cnditins. The quantile regressin instrumental variables estimatr methd may be viewed as an apprpriate quantile regressin analg f the tw stage least squares (2SLS). The 2SLS estimates can be btained by using the same tw steps prcedure as described abve fr the QRIV. In the Appendix 1 we shw the details f the derivatin and illustrate the imprtance f matrix A in the exactly identified and veridentified mdels. Nw we discuss the asympttic prperties f the dynamic panel data quantile regressin instrumental variables estimatr. The existence f the parameter η i, whse dimensin N is tending t infinity, raises sme new issues fr the asympttic analysis f the prpsed estimatr. We derive cnsistency and asympttic nrmality f the estimatrs assuming that N a /T 0. We impse the fllwing regularity cnditins: A1 The y it are independent acrss individuals with cnditinal distributin functins F it, and differentiable cnditinal densities, 0 < f it <, with bunded derivatives f it fr i=1,...,n and,...,t ; A2 Let Z = I N ι T, and ι T a T -vectr f nes, y 1 = (y it 1 ) be a dim(α) matrix, X = (x it ) be a dim(β) matrix, and W = (w it ) be a dim(γ) matrix. Fr Π(η, α, β, τ) := E[(τ 1(Zη + y 1 α + Xβ)) ˇX(τ)] Π(η, α, β, γ, τ) := E[(τ 1(Zη + y 1 α + Xβ + W γ)) ˇX(τ)] ˇX(τ) := [Z, X, W ], Jacbian matrices Π(η, α, β, τ) and Π(η, α, β, γ, τ) are cntinuus and have full (η,α,β) (η,β,γ) rank, unifrmly ver E A B G T and the image f E A B under the map (η, α, β) Π(η, α, β, τ) is simply cnnected; A3 Dente Φ(τ) = diag(f it (ξ it (τ))), where ξ it (τ) = η i (τ) + α(τ)y it 1 + x itβ(τ) + w itγ(τ), M Z = I P Z and P Z = Z(Z Φ(τ)Z) 1 Z Φ(τ). Let X = [X, W ]. Then, the fllwing matrix 8

10 is invertible: J ϑ = ( X M Z Φ(τ)M Z X); Nw define [ J β, J γ] as a partitin f J 1 ϑ, J α = ( X M Z Φ(τ)M Z y 1 ) and H = J γa[α(τ)] J γ. Then, J αhj α is als invertible; A4 Fr all τ, (α(τ), β(τ)) int A B, and A B is cmpact and cnvex; A5 max it y it = O( ); max it x it = O( ); max it w it = O( ); A6 N a T 0 fr sme a > 0. Cnditin A1 is a standard assumptin in quantile regressin literature and impses a restrictin n the density functin f y it. Cnditin A2 is imprtant fr identificatin f the parameters. The identificatin is shwn thrugh the use f a versin f Hadamard s therem, as discussed in Chernzhukv and Hansen (2006). It requires that the instrument W impacts the cnditinal distributin f Y at many relevant pints. In additin, the cnditin that the image f the parameter space be simply cnnected requires that the image can be cntinuusly shrunk t a pint, this cnditin can be interpreted as ruling ut hles in the image f the set. Assumptin A3 states invertibility cnditins fr matrices in rder t guarantee asympttic nrmality. A4 impses cmpactness n the parameter space f α(τ). Such assumptin is needed since the bjective functin is nt cnvex in α. Assumptin A5 impses bund n the variables. Finally, cnditin A6 is the same assumptin as in Kenker (2004) and allws T t grw very slw relative t N. The recent literature analyzing bias in dynamic panel data mdels develps asympttic thery where N and T are large. In a linear case, Alvarez and Arellan (2003) establish cnsistency f within grup (WG), generalized methd f mments (GMM), and limited infrmatin maximum likelihd estimatrs fr a first rder autregressive mdel with individual effects when bth N and T tend t infinity and lim(n/t ) c <. Hahn and Kuersteiner (2002) and Hahn and Kuersteiner (2004) use the same relative rate fr N and T dynamic linear and nnlinear panels respectively. We can nw establish cnsistency and asympttic nrmality. Prfs appear in the Appendix 1. The fllwing therem states identificatin and cnsistency f ˆθ(τ). Therem 1 Given assumptins A1-A6, (η(τ), α(τ), β(τ)) uniquely slves the equatins E[ψ(Y Zη y 1 α Xβ) ˇX(τ)] = 0 ver E A B, and θ(τ) = (α(τ), β(τ)) is cnsistently estimable. Under cnditins A1-A6 we shw the asympttic prperties f the fixed effects QRIV 9

11 as N grws at a cntrlled rate relative t T. Therem 1 prvides a lwer bund fr the relative rate f increasing f T which is sufficient fr cnsistency. The intuitin behind this cnditin is that T must g t infinity fast enugh t guarantee cnsistent estimates fr the fixed effects, and then t the ther parameters. Under assumptin A6 the cnvergence is fast enugh t eliminate the incnsistency prblem fund fr very small T and large N asympttic apprximatins. An alternative view f this argument is that it applies t situatins in which T tends t infinity, and N is fixed. The limiting distributin f parameters f interest using the quantile regressin instrumental variables estimatr fr the dynamic panel mdel with fixed effects is given by Therem 2. Therem 2 (Asympttically Nrmality). Under cnditins A1-A6, fr a given τ (0, 1), ˆθ cnverges t a Gaussian distributin as (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 Z ], K = (J αhj α ) 1 J α H, H = J γa[α(τ)] J γ, L = J β M, M = I J α K, J α = ( X M Z ΦM Z y 1 ), [ J β, J γ ] is a partitin f J 1 ϑ = ( X M Z ΦM Z X) 1, Φ = diag(f it (ξ it (τ))), and X = [X, W ]. Remark 1. Fr a finite cllectin f quantile indexes we have (ˆθ(τj ) θ(τ j )) j J d N(0, Ω(τ k, τ l )) k,l J where Ω(τ k, τ l ) = (K(τ k ), L(τ k ) ) S(τ k, τ l )(K(τ l ), L(τ l ) ). Remark 2. When dim(γ) = dim(α), the chice f A(α) des nt affect asympttic variance, and the jint asympttic variance f α(τ) and β(τ) will generally have the simple frm Ω(τ) = (K, L ) S(K, L ), fr S, K and L as defined abve. As in Chernzhukv and Hansen (2008), when dim(γ) > dim(α), the chice f the weighting matrix A(α) generally matters, and it is imprtant fr efficiency. A natural chice fr A(α) is given by the inverse f the cvariance matrix f ˆγ(α(τ), τ). Nticing that A(α) is equal t ( J γ S J γ ) 1 at α(τ), it fllws that the asympttic variance f (ˆα(τ) α(τ)) is given by Ω α = (J α J γ( J γ S J γ) 1 Jγ Jα ) 1. The cmpnents f the asympttic variance matrix that need t be estimated include 10

12 J ϑ, J α and S. The matrix S can be estimated by its sample cunterpart Ŝ(τ, τ ) = (min(τ, τ ) ττ ) 1 N i=1 T V it V it. (10) Fllwing Pwell (1986), J ϑ and J α can be estimated as stated in Therem 2 abve, J ϑ = ( X M Z ΦM Z X) and Jα = ( X M Z ΦM Z y 1 ), where the cmpnent f the estimatrs, M Z Φ(τ)M Z, is given by the fllwing frm M Z ˆΦ(τ)MZ = 1 2 h n N T I( û(τ) h n )M Z M Z (11) i=1 where û(τ) Y Z ˆη(τ) ˆα(τ)y 1 X ˆβ(τ) and h n is an apprpriately chsen bandwidth, with h n 0 and h 2 n. By using the same prcedure we can estimate the element Z ˆΦ(τ)Z in P Z. The cnsistency f these asympttic cvariance matrix estimatrs are standard and will nt be discussed further in this paper. 3 Inference In this sectin, we turn ur attentin t inference in the quantile regressin dynamic panel instrumental variable (QRIV) mdel, and suggest a Wald type test fr general linear hyptheses, and a Klmgrv-Smirnv test fr linear hypthesis ver a range f quantiles τ T. In the independent and identically distributed setup the cnditinal quantile functins f the respnse variable, given the cvariates, are all parallel, implying that cvariates effects shift the lcatin f the respnse distributin but d nt change the scale r shape. Hwever, slpes estimates ften vary acrss quantiles implying that it is imprtant t test fr equality f slpes acrss quantiles. Wald tests designed fr this purpse were suggested by Kenker and Bassett (1982a), Kenker and Bassett (1982b), and Kenker and Machad (1999). It is pssible t frmulate a wide variety f tests using variants f the prpsed Wald test, frm simple tests n a single quantile regressin cefficient t jint tests invlving many cvariates and distinct quantiles at the same time. General hyptheses n the vectr θ(τ) can be accmmdated by Wald type f tests. The Wald prcess and assciated limiting thery prvide a natural fundatin fr the hypthesis Rθ(τ) = r, when r is knwn. We first cnsider a Wald type test where we test the cefficients 11

13 fr selected quantiles f intest. Later we intrduce a test fr linear hypthesis ver a range f quantiles τ T, instead f fcusing nly n a selected quantile. The fllwing are examples f hyptheses that may be cnsidered in the frmer framewrk. Fr simplicity f presentatin we use the mdel stated in equatin (3) with a single variable in the x it matrix. Example 1 (N dynamic effect). If there is n dynamic effect in the mdel, then under H 0 : α(τ) = 0. Thus, θ(τ) = (α(τ), β(τ)), R = [1, 0] and r = 0. Example 2 (Lcatin-scale shifts). The hyptheses f lcatin-scale shifts fr α(τ) and β(τ) can be accmmdated in the mdel. Fr the first case, H 0 : α(τ) = α, fr α < 1, s θ(τ) = (α(τ), β(τ)), R = [1, 0] and r = α. Fr the latter case, H 0 : β(τ) = β, s that R = [0, 1] and r = β. Prtny (1984) and Gutenbrunner and Jureckva (1992) shw that the quantile regressin prcess is tight and thus the limiting variate viewed as a functin f τ is a Brwnian Bridge ver τ T. Therefre, under the linear hypthesis H 0 : Rθ(τ) = r, cnditins A1-A6, and letting Σ = (K, L ) EV V (K, L ), we have V = [RΣ(τ)R ] 1/2 (Rˆθ(τ) r) B q (τ), (12) where B q (τ) represents a q-dimensinal standard Brwnian Bridge. Fr any fixed τ, B q (τ) is N(0, τ(1 τ)i q ). The nrmalized Euclidean nrm f B q (τ) Q q (τ) = B q (τ) / τ(1 τ) is generally referred t as a Keifer prcess f rder q. Thus, fr given τ, the regressin Wald prcess can be cnstructed as W = (Rˆθ(τ) r) [RˆΩ(τ)R ] 1 (Rˆθ(τ) r) (13) where ˆΩ is a cnsistent estimatrs f Ω, and Ω is given by Ω(τ) = (K (τ), L (τ)) S(τ, τ)(k (τ), L (τ)). If we are interested in testing Rθ(τ) = r at a particular quantile τ = τ 0, a Chi-square test can be cnducted based n the statistic W (τ 0 ). Under H 0, the statistic W is asympttically χ 2 q with q-degrees f freedm, where q is the rank f the matrix R. The limiting distributins f the test is summarized in the fllwing therem: 12

14 Therem 3 (Wald Test Inference). Under H 0 : Rθ(τ) = r, and cnditins A1-A6, fr fixed τ, W (τ) a χ 2 q. Prf. The prf f Therem 3 is very simple and it fllws frm bserving that fr any fixed τ, by Therem 2 (ˆθ(τ) θ(τ)) N(0, Ω(τ)) under the null hypthesis, (Rˆθ(τ) r) N(0, RΩ(τ)R ) since ˆΩ(τ) is a cnsistent estimatr f Ω(τ), by Slutsky therem W = (Rˆθ(τ) r) [RˆΩ(τ)R ] 1 (Rˆθ(τ) r) a χ 2 q. In rder t implement the test it is necessary t estimate Ω(τ) cnsistently. It is pssible t btain such estimatr as suggested in Therem 2 in the previus sectin, and the main cmpnents f ˆΩ(τ) can be btained as in equatins (10) and (11). Mre general hyptheses are als easily accmmdated by the Wald apprach. Let υ = (θ(τ 1 ),..., θ(τ m ) ) and define the null hypthesis as H 0 : Rυ = r. The test statistic is the same Wald test as equatin (13). Hwever, Ω is nw the matrix with klth bck Ω(τ k, τ l ) = (K (τ k ), L (τ k )) S(τ k, τ l )(K (τ l ), L (τ l )), and S(τ k, τ l ) = (τ k τ l τ k τ l )E[V V ], Φ(τ j ) = diag(f it (ξ it (τ j ))), and the ther variables are as defined abve. The statistic W is still asympttically χ 2 q under H 0 where q is the rank f the matrix R. This frmulatin accmmdates a wide variety f testing situatins, frm a simple test n single quantiles regressin cefficients t jint tests invlving several cvariates and several distinct quantiles. Thus, fr instance, we might test fr the equality f several slpe cefficients acrss several quantiles. Example 3 (Same dynamic effect fr tw distinct quantiles). If there is same dynamic effect fr tw given distinct quantiles in the mdel, then under H 0 : α(τ 1 ) = α(τ 2 ). Thus, υ = (θ(τ 1 ),..., θ(τ m ) ) = (α(τ 1 ), β(τ 1 ), α(τ 2 ), β(τ 2 ), ), R = [1, 0, 1, 0] and r = 0. 13

15 Anther imprtant class f tests in the quantile regressin literature invlves the Klmgrv- Smirnv (KS) type tests, where the interest is t examine the prperty f the estimatr ver a range f quantiles τ T, instead f fcusing nly n a selected quantile. Thus, if ne has interest in testing Rθ(τ) = r ver τ T, ne may cnsider the KS type sup-wald test. Fllwing Kenker and Xia (2006a), we may cnstruct a KS type test n the dynamic panel data regressin quantile prcess in the fllwing way KSW = sup W (τ). (14) τ T The next example shws a pssible applicatin f KS test in dynamic panel quantile regressin. Example 4 (Asymmetric dynamic effect). It is particularly interesting t analyze data displaying asymmetric dynamics. Thus, ne may cnsider testing the hypthesis that α(τ) = cnstant ver τ. In rder t perfrm such test fr α(τ) ne can use the Klmgrv-Smirnv test given by (14). The implementatin is dne by cmputing the statistic f test W (τ), given by (13), fr each τ T, and then calculating the maximum ver τ. The limiting distributin f the Klmgrv-Smirnv test is given in the fllwing therem: Therem 4 (Klmgrv-Smirnv Test). Under H 0 and cnditins A1-A6, KSW = sup τ T W (τ) sup Q 2 q(τ). τ T The prf f Therem 4 fllws directly frm the cntinuus mapping therem and equatin (12). Critical values fr sup Q 2 q(τ) have been tabled by DeLng (1981) and, mre extensively, by Andrews (1993) using simulatin methds. 4 Mnte Carl Simulatin 4.1 Mnte Carl Design In this sectin, we describe the design f sme simulatin experiments that have been cnducted t assess the finite sample perfrmance f the quantile regressin estimatr and inference prcedure discussed in the previus sectin. 6 Tw simple versins f the basic 6 The experiment shwn in this sectin is heavily based in Hsia, Pesaran, and Tahmisciglu (2002). 14

16 mdel (1) are cnsidered in the simulatin experiment. In the first, reprted in Tables 1 and 2, the scalar cvariate, x it, exerts a pure lcatin shift effect. In the secnd, reprted in Tables 3 and 4, x it exerts bth lcatin and scale effects. In the frmer case the respnse y it is generated by the mdel, y it = η i + αy it 1 + βx it + u it while in the latter case, y it = η i + αy it 1 + βx it + (γx it )u it. We emply tw different schemes t generate the disturbances u it. Under Scheme 1, we generate u it as N(0, σu), 2 and we als used a heavier distributin scheme t generate u it. Under Scheme 2 we generate u it as t-distributin with 3 degrees f freedm. The regressr x it is generated accrding t x it = µ i + ζ it (15) where ζ it fllws the ARMA(1, 1) prcess (1 φl)ζ it = ɛ it + θɛ it 1 (16) and ɛ it fllws the same distributin as u it, that is, nrmal distributin and t 3 fr Schemes 1 and 2, respectively. In all cases we set ζ i, 50 = 0 and generate ζ it fr t = 49, 48,..., T accrding t ζ it = φζ it 1 + ɛ it + θɛ it 1. (17) This ensures that the results are nt unduly influenced by the initial values f the x it prcess. In generating y it we als set y i, 50 = 0 and discarded the first 50 bservatins, using the bservatins t = 0 thrugh T fr estimatin. 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 ), T η i = e 2i + T 1 x it, e 2i N(0, σe 2 2 ). The abve methd f generating µ i and α i ensures that the usual randm effects estimatrs are incnsistent because f the crrelatin that exists between the individual effects and the errr term r the explanatry variables. 15

17 In the simulatins, we experiment with T = 10, 20 and N = 50, 100. We set the number f replicatins t 2000, and cnsider the fllwing values fr the remaining parameters: (α, β) = (0.4, 0.6), (0.8, 0.2); φ = 0.6, θ = 0.2, γ = 0.5, σu 2 = σe 2 1 = σe 2 2 = 1. In the Mnte Carl study, we cmpare the estimatrs cefficients in terms f bias and rt mean squared errr. We als investigate the small sample prperties f the tests based n different estimatrs paying particular attentin t the size and pwer f these tests. 4.2 Mnte Carl Results We study fur different estimatrs in the Mnte Carl experiment, the within grup estimatr (WG), OLS instrumental variables estimatr (OLS-IV), the fixed effects panel quantile regressin (PQR) prpsed by Kenker (2004), and the quantile regressin dynamic panel instrumental variables (QRIV) prpsed in this paper. The quantile regressin based estimatrs are analyzed fr the median case. Fr the OLS-IV and QRIV we use tw different instruments, say y it 3 and x it 1 and the results are essentially the same in bth cases. We present results fr the x it 1 case. We als cnsider different sample sizes in the experiments, hwever, due t space limitatins we reprt results nly fr T = 10 and N = 50. The results are similar t the ther sample size schemes Bias and RMSE In the first part f the Mnte Carl we study the bias and rt mean squared errr (RMSE) f the estimatrs. Tables 1 and 2 present bias and RMSE results fr estimates f the autregressin cefficient, α, and the exgenus variable cefficient, β, fr the lcatin-shift mdel and the Nrmal and t 3 distributins f the innvatins, respectively. In general, the bias f the estimates increase as parameters α and β becme larger. Regarding the autregressin cefficient the bias in the WG and PQR estimatrs, as a percentage f the true values, are rughly cnstants. Mrever, the cefficient f the exgenus variable is slightly biased in the WG and PQR cases. Table 1 shws that when the disturbances are sampled frm a Gaussian distributin, as expected, the autregressin cefficient is biased dwnward fr the the WG case, but the OLS-IV is apprximately unbiased. In the same way, in the presence f lagged variables 16

18 WG OLS-IV PQR QRIV α = 0.8 Bias RMSE β = 0.2 Bias RMSE α = 0.4 Bias RMSE β = 0.6 Bias RMSE Table 1: Lcatin-Shift Mdel: Bias and RMSE f Estimatrs fr Nrmal Distributin (T = 10 and N = 50) the fixed effects quantile regressin estimatr prpsed by Kenker (2004), PQR, is biased dwnward. Hwever, the instrumental variables prpsed in this paper is able t reduce the bias. As shwn in Table 1, the QRIV estimatr is apprximately unbiased fr bth selectins f the parameters α and β. In summary, estimates are biased in bth the WG and the PQR cases, and the instrumental variables strategy is able t cnsiderably diminish the bias fr bth rdinary least squares and quantile regressin cases. Regarding the RMSE, in the Gaussian cnditin, the OLS based estimatrs perfrm better than the respective quantile regressin estimatrs. Thus, the QRIV is capable t reduce the bias but it has a larger RMSE when cmpared with the PQR. Table 2 presents the results fr the t 3 -distributin case. The autregressive estimates f WG and PQR are biased, and the WG has a larger bias when cmpared with the same estimatr in the Gaussian case. In additin, the OLS-IV presents a small bias fr the autregressin cefficient. The QRIV is apprximately an unbiased estimatr fr bth cefficients. Mrever, the RMSE is smaller fr quantile estimatrs when cmpared with the respective OLS based estimatrs in this nn-gaussian case. Tables 3 and 4 present bias and RMSE results f the estimatrs fr the lcatin-scaleshift mdel fr the Nrmal and t 3 distributins, respectively. As in the lcatin-shift case, the bias f the estimatrs increase as the parameters α and β becme larger. Again, it is pssible t nte that the quantile regressin instrumental variables estimatr presents a much smaller bias when cmpared with PQR, and a much imprved precisin when cmpared with OLS-IV, in the t 3 case. Table 3 shws that in the Gaussian cnditin the WG and PQR estimatrs are biased dwnward and the OLS-IV and QRIV are apprximately unbiased. As in the lcatin-shift 17

19 WG OLS-IV PQR QRIV α = 0.8 Bias RMSE β = 0.2 Bias RMSE α = 0.4 Bias RMSE β = 0.6 Bias RMSE Table 2: Lcatin-Shift Mdel: Bias and RMSE f Estimatrs fr t 3 Distributin (T = 10 and N = 50) WG OLS-IV PQR QRIV α = 0.8 Bias RMSE β = 0.2 Bias RMSE α = 0.4 Bias RMSE β = 0.6 Bias RMSE Table 3: Lcatin-Scale Shift Mdel: Bias and RMSE f Estimatrs fr Nrmal Distributin (T = 10 and N = 50) case, in the presence f dynamic variables, the fixed effects quantile regressin estimatr prpsed by Kenker (2004) is biased dwnward, and the instrumental variables prpsed in this paper is able t dramatically reduce the bias. The RMSE s present the same features as in the previus lcatin case. Table 4 presents the results fr the t 3 -distributin case, and the results are qualitatively similar t thse in Table Size and Pwer Nw we turn ur attentin t the size and pwer f the asympttic inference given in the previus sectin. First, we cncentrate n tests fr selected quantiles, latter we mve t tests ver a range f quantiles. Fr the frmer case, in rder t calculate the pwer curves we use the same setup as in the presented calculatin f bias and RMSE. We present the results fr QRIV as well as fr OLS-IV in rder t cmpare the finite sample perfrmance f the estimatrs. Thus, we cnsider the QRIV mdel in equatin (1) and test the hypthesis 18

20 WG OLS-IV PQR QRIV α = 0.8 Bias RMSE β = 0.2 Bias RMSE α = 0.4 Bias RMSE β = 0.6 Bias RMSE Table 4: Lcatin-Scale Shift Mdel: Bias and RMSE f Estimatrs fr t 3 (T = 10 and N = 50) Distributin that ˆα(τ) = α and als that ˆβ(τ) = β fr given τ. We present the results fr α = 0.4 and β = Fr mdels under the alternative, we cnsidered linear deviatins frm the null as α + d/ and β + d/. The cnstructin f the test uses estimatr f the density as given in equatin (11). The prcedure prpsed by Pwell (1986) entails a chice f bandwidth. We cnsider the default bandwidth suggested by Bfinger (1975) h n = [Φ 1 (τ + c n ) Φ 1 (τ c n )] min(ˆσ 1, ˆσ 2 ) where the bandwidth c n = O(( ) 1/3 ), ˆσ 1 = V ar(û), and ˆσ 2 = ( ˆQ(û,.75) ˆQ(û,.25))/1.34. We als use a Gaussian bandwidth, but present results fr the first chice f bandwidth nly. The results fr T = 10 and N = 50 in the experiments are presented. The results are presented in Figures 1 and 2. Figure 1 shws the finite sample size and pwer fr the estimated α and β cefficients cnsidering Nrmal distributins and QRIV and OLS-IV estimatrs. Part 1 f the figure cncerns α and Part 2 shws the results fr β. We can bserve that the size is very clse t the established five percent fr all estimatrs. When cmparing QRIV and OLS-IV estimatrs with respect t the Nrmal distributin ne can see that the OLS based estimatrs perfrm better than the quantile regressin estimatr in terms f pwer. [Figure 1 abut here] Figure 2 presents the results fr finite sample size and pwer fr the estimated α and β cefficients cnsidering t 3 distributin and QRIV and OLS-IV estimatrs. As in the previus 7 The results fr α = 0.8 and β = 0.2 are similar. 19

21 case, the size is very clse t the established five percent fr all estimatrs. When the nise in the mdel cmes frm a heavier distributin, t 3, the QRIV estimatrs have a strngly superir perfrmance vis-a-vis the OLS-IV estimatrs, shwing that there are large gains in pwer by using a rbust estimatr in the case f a nn-gaussian heavy tail distributin. In summary, the results fr the pwer curves shw that the OLS-IV presents mre pwer than QRIV in the Nrmal case, but in the t 3 case the inverse ccurs such that QRIV estimatrs have mre pwer than OLS-IV. The QRIV presents a higher pwer vis-a-vis the OLS-IV estimatr in the nn-gaussian case. In additin, as expected, cmparisn f the same estimatrs using different distributins shw that the quantile regressin estimatr perfrms better in a t 3 distributin cnditin, and the OLS-IV estimatr has mre pwer under Gaussian cnditin. The results fr the ther sample cases are qualitatively similar t thse f Figures 1 and 2, but als shw that, as the sample sizes increase, the tests d have imprved pwer prperties, crrbrating the asympttic thery. [Figure 2 abut here] We als cnduct a Mnte Carl experiment t examine the QRIV based inference prcedures, where we are particularly interested in mdels displaying asymmetric dynamics. Thus, we cnsider the QRIV mdel t test the hypthesis that α(τ) = cnstant ver τ. The data in these experiments were generated frm mdel (1) in the same manner as in Sectin 4.1, where u it are i.i.d. randm variables. We cnsider the Klmgrv-Smirnv test KSW given by (14) fr different sample sizes and innvatin distributins, and chse T = [0.1, 0.9]. Bth Nrmal innvatins and student-t innvatins are cnsidered. The number f repetitins is Representative results f the empirical size and pwer f the prpsed tests are reprted in Table 5. We reprt the empirical size f this test fr tw chices f α(τ): (1) α = 0.45; (2) α = Fr mdels under the alternative, we cnsider the fllwing tw chices: { 0.35, u it 0, α = ϕ 1 (u it ) 0.55, u it < 0, { 0.15, u it 0, α = ϕ 2 (u it ) 0.75, u it < 0. Table 5 reprts the empirical size and pwer fr the case with Gaussian innvatins and sample size T = 10 and N = 50, as well as the results fr the student-t innvatins (with 3 20

22 Mdel Nrmal t 3 Size α = α = Pwer ϕ ϕ Table 5: Size and Pwer fr Nrmal Distributin degrees f freedm) and same sample size. Results in Table 5 shw that the size f the test is clse t the 5% that was set and als cnfirm that, using the quantile regressin based apprach, pwer gain can be btained in the presence f heavy-tailed disturbances. (Such gains bviusly depend n chsing quantiles at which there is sufficient cnditinal density.) 5 Applicatin In this sectin we illustrate the new apprach by applying the prpsed estimatr and testing prcedures t test fr the presence f time nn-separability in utility using husehld cnsumptin panel data. A simple mdel f habit frmatin implies a cnditin relating the strength f habits t the evlutin f cnsumptin ver time. We build n previus wrk by testing the time separability f preferences with husehld panel data using quantile regressin dynamic panel instrumental variables (QRIV) framewrk. There is a large literature using husehld panel data n cnsumptin t examine behavir when preferences are assumed t be time-separable, fr instance Hall and Mishkin (1982), Shapir (1984), and Zeldes (1989). Mre recently, there has been grwing interest in the implicatins f preferences that are time-nnseparable and several papers have used aggregate cnsumptin data t lk fr empirical evidence f such preferences. A very imprtant class f time-nnseparable preferences is that exhibiting habit frmatin. Studies f time-nnseparable preferences based n aggregated cnsumptin data prduce mixed cnclusins abut the strength f habit frmatin. Dunn and Singletn (1986) and Eichenbaum, Hansen, and Singletn (1988) find very weak evidence f habit frmatin in US mnthly aggregated data. Fersn and Cnstantinides (1991) find strng evidence f habit frmatin in mnthly, quarterly, and annual US cnsumptin data. Using panel data Dynan (2000) finds n evidence f habit frmatin in US cnsumptin data. With habit frmatin, current utility depends nt nly n current expenditures, but als n a habit stck frmed by lagged expenditures. Fr a given level f current expenditure, 21

23 a larger habit stck lwers utility. Mre frmally, husehld i chses current cnsumptin expenditures, c it, t maximize [ T ] E β s u ( c it+s ; ψ it+s ) s=0 where c it is the cnsumptin services in perid t, β is a time discunt factr, and ψ i,t crrespnds t taste-shifters - variables that mve marginal utility - at time t. Cnsumptin services in perid t are psitively related t current expenditures and negatively related t lagged expenditures in the fllwing way c it = c it αc it 1. The parameter α measures the strength f habit frmatin, thus when α is larger, the cnsumer receives less lifetime utility frm a given amunt f expenditure. Frm the first rder cnditin, assuming that the utility functin is f the fllwing iselastic frm 8 c 1 ρ it u( c it, ψ it ) = ψ it 1 ρ, the interest rates are cnstant, and fllwing Dynan (2000) and apprximating ln(c it αc it 1 ) with ( ln c it α ln c it 1 ), ne can derive the fllwing equatin 9 ln(c it ) = γ 0 + α ln(c it 1 ) + γ 1 ln(ψ it ) + ɛ it. (18) Habit persistence enters the Euler equatin thrugh lagged cnsumptin grwth and habit persistence cefficient α, with its magnitude reflecting the fractin f past expenditures that make up the habit stck and indicating the imprtance f habit frmatin in behavir. In ther wrds, the equatin shws that habit frmatin creates a link between current and lagged expenditure grwth, which stems frm cnsumers gradual adjustment t permanent incme shcks. In cntrast t traditinal mdels in which cnsumptin adjusts immediately t permanent incme innvatin, habits cause cnsumers t prefer a number f small cnsumptin changes t ne large cnsumptin change. We estimate a quantile regressin mdel using data frm Panel Study n Incme Dynamics (PSID), which cntains annual infrmatin abut the incme, fd cnsumptin, 8 We prvide the details f the derivatin in Appendix 2 9 Althugh γ 0 is a functin f the real interest rates, the time discunt factr, and the frecast errr variance (see Dynan (2000) fr details), mst Euler equatin analyses with husehld data have assumed these terms cnstant acrss husehlds and time perids. 22

24 emplyment, and demgraphic characteristics f individual husehlds. The PSID has limited cnsumptin infrmatin, and we fllw a substantial bdy f the literature in using fd expenditures t explre cnsumptin behavir. In rder t test the cnsumptin habit frmatin hypthesis, estimatin and testing are based in the fllwing equatin where C it = ln c it, and X it is a set f cvariates. Q Cit (τ F it 1 ) = η i + αc it 1 + X it β (19) The base line sample cntains 2132 husehlds, each with 13 bservatins n fd expenditure grwth. Althugh the PSID began in 1968 and cntinues tday, the sample uses spending data nly frm the perid 1974 thrugh 1987 because f the interpretatin prblems in the early years and the suspensin f the fd questins in As cvariates, fllwing Dynan (2000), we include as taste shifters in the estimated equatins: difference in family sizes, age f the head f the husehld, and age f the head f the husehld squared. All specificatins als include time dummies t ensure that aggregate shcks d nt lead t incnsistent estimates. In a secnd rund f estimatins we als include a variable race in the mdel. The cnsumptin variables are measured in lgs. We als apply the OLS based tw stage least squares (2SLS) estimatin and testing fr cmparisn reasns. We use tw different sets f instruments, C it 2 and C it 3, in bth QRIV and 2SLS estimatin and the results are essentially the same, thus we reprt the estimatins fr the frmer ne. 10 We have interest in testing H 0 : α(τ) = 0, that is, there is n evidence f habit persistence. Equatin (19) captures the dynamics f cnsumptin, the estimatin results will nt nly serve as a test f this particular mdel, but will als prvide evidence regarding the general imprtance f habit frmatin and the determinants f grwth in cnsumptin, as well as shwing evidence f asymmetric persistence in these variables. The results fr pint estimates and cnfidence intervals fr QRIV and 2SLS are presented in Figures 3 and 4. The slid straight lines represent the 2SLS estimates and the dashed lines the 90-percent cnfidence band fr the estimates. The gray lines shw the results fr pint estimatins and the cnfidence intervals fr all cefficients in the QRIV case. Figure 3 presents the estimates fr results fr the autregressive cefficient, family size, age and age squared. In Figure 4 we als include race as an additinal regressr. 10 We als estimate the mdel using dummies fr incme grwth as instruments, where incme is measured as real labr incme f head and wife r real dispsable incme, the results remain the same if we use this additinal instrument. 23