Instrumental Variables Quantile Regression for Panel Data with Measurement Errors

Transcription

1 Instrumental Variables Quantile Regressin fr Panel Data wh Measurement Errrs Antni F. Galva, Jr. Universy f Illinis Gabriel V. Mntes-Rjas Cy Universy Lndn January, 2009 Abstract This paper develps an instrumental variables estimatr fr quantile regressin in panel data wh fixed effects. Asympttic prperties f the instrumental variables estimatr are studied fr large N and T when N a /T 0, fr sme a > 0. Wald and Klmgrv-Smirnv type tests fr general linear restrictins are develped. Mnte Carl simulatins are cnducted t evaluate the fine sample prperties f the estimatr in terms f bias and rt mean squared errr. The estimatr is applied t the prblem f measurement errrs in variables. We derive an apprximatin t the bias in the quantile regressin fixed effects estimatr and shw s cnnectin t similar effects in standard least squares mdels. Finally, the methds are applied t a mdel f firm investment. The results shw interesting hetergeney in the Tbin s q and cash flw sensivies f investment. In bth cases, the sensivies are mntnically increasing alng the quantiles. Key Wrds: Quantile Regressin; Panel Data; Measurement Errrs; Instrumental Variables JEL Classificatin: C14, C23 The authrs wuld like t express their appreciatin t Rger Kenker, Her Almeida, Murill Campell, and participants in the seminars at Cass Business Schl, Universy f Alicante, and 18th Midwest Ecnmetrics Grup Meeting fr helpful cmments and discussins. All the remaining errrs are urs. Department f Ecnmics, Universy f Illinis at Urbana-Champaign, 419 David Kinley Hall, 1407 W. Gregry Dr., Urbana, IL 61801, USA. galva@illinis.edu Department f Ecnmics, Cy Universy Lndn, D308 Scial Sciences Bldg, Nrthamptn Square, Lndn EC1V 0HB, UK. Gabriel.Mntes-Rjas.1@cy.ac.uk

2 1 Intrductin Semiparametric panel data mdels have attracted interest in bth thery and applicatins since the Gaussian assumptins underlying classical least-squares (LS) methds are smetimes implausible. Kenker (2004) intrduced a general apprach t estimatin f quantile regressin (QR) mdels fr lngudinal data. Individual specific fixed effects (FE) are treated as pure lcatin shift parameters cmmn t all cndinal quantiles and may be subject t shrinkage tward a cmmn value as in the Gaussian randm effects paradigm. QR methds allw ne t explre a range f cndinal quantiles expsing a variety f frms f cndinal hetergeney under weaker distributinal assumptins and als prvide a framewrk fr rbust estimatin and inference. Cntrlling fr individual specific hetergeney via FE while explring hetergeneus cvariate effects whin the QR framewrk ffers a mre flexible apprach t the analysis f panel data than that affrded by the classical Gaussian fixed and randm effects estimatrs. Recent wrk by Lamarche (2006, 2008) and Geraci and Bttai (2007) have elabrated n these mdels using penalized QR estimatrs. Abrevaya and Dahl (2007) have intrduced an alternative apprach t estimating QR mdels fr panel data emplying the crrelated randm effects mdel f Chamberlain (1982). Ecnmic quanties are frequently measured wh errr, particularly if lngudinal infrmatin is cllected thrugh ne-time retrspective surveys, which are ntriusly susceptible t recall errrs. If the regressrs are indeed subject t measurement errrs (ME), is well knwn that the slpe cefficient f the least squares (LS) regressin estimatr is incnsistent because the measurement errr induces endgeney in the mdel. In the ne regressr case (r the multiple regressr case wh uncrrelated regressrs), under standard assumptins, the rdinary LS estimatr is biased tward zer, a prblem ften dented as attenuatin. The mst cmmn remedy t reduce this bias caused by the endgeney prblem is t use eher ecnmic thery r intuin t find addinal bservable variables that can serve as instrumental variables (IV). Mst f the lerature n the estimatin f fixed effects (FE) in panel data mdels wh measurement errrs is based n LS wh instrumental variables and the generalized methd f mments (GMM). See fr instance Hsia (2003, 1992), Wansbeek (2001), Birn (2000, 1992), Hsia and Taylr (1991), Wansbeek and Kning (1989) and Griliches and Hausman (1986). The gal f this paper is t cntribute t this lerature by develping an IV estimatr fr 1

3 QR wh FE (IVQRFE hereafter) t reduce the bias caused by the presence f endgeney in the mdel. In particular, the IV estimatr fr QR intrduced by Chernzhukv and Hansen (2006, 2008) will be adapted t the panel data setting f Kenker (2004). We derive cnsistency and asympttic distributin f this estimatr and suggest a Wald type test fr general linear hyptheses and a Klmgrv-Smirnv test fr linear hyptheses ver a range f quantiles τ T, and derive the respective liming distributins. Althugh the prpsed estimatr is general fr endgeney in QR wh FE, we fcus n the applicatin ME because this is an imprtant class f endgeney prblem in panel data frm the applied research standpint, and als, under certain cndins, the instruments are available whin the mdel. In particular, if the latent regressr is autcrrelated r nn-statinary, lags f the regressrs are valid instruments fr the mismeasured variable, and estimatrs f the cefficient f the latent regressr will be cnsistent. Based n Griliches and Hausman (1986), and Birn (2000) the IV apprach emplys lagged (r lagged differences f the) regressrs as instruments. Thus, the estimatr cmbines the usual cncept fr panel data wh ME and the IV QR framewrk. Recently, the tpic f ME in variables has als attracted cnsiderable attentin in the QR lerature. Chesher (2001) studies the impact f cvariate ME n quantile functins using small variance apprximatin, and Schennach (2008) discusses identificatin and estimatin issues fr general quantile functins based n Furier transfrms and previus results fr nnlinear mdels (see Schennach, 2004, 2008). As a by-prduct cntributin f the paper, we derive an apprximatin t the bias in the QR FE estimatr in the presence f a mismeasured variable. The effect f ME n the slpe cefficient estimatrs can be seen as an applicatin f Angrist, Chernzhukv, and Fernandez-Val (2006) mted variables frmula, which prvides a useful generalizatin f the prblem f ME bias discussed abve t the QR framewrk. This representatin prvides an explic frmulatin fr the bias in the slpe cefficients and cmplements the results in Chesher (2001). As first nted by Neyman and Sctt (1948), leaving the individual hetergeney unrestricted in a nnlinear r dynamic mdel generally results in incnsistent estimatrs f the cmmn parameters due t the incidental parameters prblem; that is, nise in the estimatin f the FE when the time dimensin is shrt results in incnsistent estimates f the cmmn parameters due t the nnlineary f the prblem. QR estimatin f panel data mdels wh FE suffers frm similar prblems t thse seen in the LS case when T is mdest. Reliance n the existing bias reductin strategies using LS differencing, eher temprally, 2

4 r via the usual deviatin frm individual means (whin) transfrmatin, is unsatisfactry in the QR setting. Linear transfrmatins that are cmpletely inncuus in the cntext f cndinal mean mdels are highly prblematic in the cndinal quantile mdels since they alter in a fundamental way what is being estimated. Expectatins enjy the cnvenient prperty that they cmmute wh linear transfrmatins; quantiles d nt. 1 there is n need t transfrm the QR mdel t cmpute the FE estimatr. Frtunately, Even when the number f FE is large, interir pint ptimizatin methds using mdern sparse linear algebra make direct estimatin f the QR mdel que efficient. Kenker (2004) vercmes this prblem by using a large N and T asympttics wh the restrictin that N a /T 0, fr sme a > 0. 2 Mnte Carl simulatins are used t assess the prperties f the estimatrs in terms f bias, rt mean squared errr, and t cmpare s perfrmance wh LS methds. The simulatins shw that the QRFE estimatr is severally biased in the presence f ME, while the prpsed IV estimatr is able t reduce the bias even in shrt panels. In addin, the Mnte Carl experiments suggest that IVQRFE perfrms better than LS-IV in terms f bias and rt mean squared errr fr nn-gaussian heavy-tailed distributins. Finally, we prvide an example t illustrate the new methds. We apply the IVQRFE estimatr t study the relatinship between investment and Tbin s q and cash flw, a tpic f cnsiderable attentin in crprate finance and applied ecnmetrics in general. Tbin s q is the rati f the market valuatin f a firm and the replacement value f s assets. Ecnmic thery predicts that firms wh a high value f q are attractive investment pprtunies, whereas a lw value f q indicates the ppse. As argued in Ericksn and Whed (2000), the pr perfrmance f the Tbin s q thery is prbably due t the measurement errr f q values. Related t this is the debate n whether cash flw has an effect n investment (Alti, 2003; Almeida, Campell, and Weisbach, 2004). We argue that QR methds capture imprtant hetergeney acrss firms in terms f their q and cash flw sensivies f investment. In the upper cndinal quantiles f investment, investment may be driven by 1 This intrinsic difficulty has been recgnized by Abrevaya and Dahl (2007), amng thers, and is clarified by Kenker and Hallck (2000, p.19): Quantiles f cnvlutins f randm variables are rather intractable bjects, and preliminary differencing strategies familiar frm Gaussian mdels have smetimes unanticipated effects. 2 Mre recently, a number f addinal appraches have been prpsed t reduce the bias in dynamic and nnlinear panels. These methds use asympttic apprximatins derived as bth the number f individuals, N, and the number time series, T, g t infiny jintly; see, fr example, Arellan and Hahn (2005) fr a survey, and Hahn and Kuersteiner (2002), Alvarez and Arellan (2003), Hahn and Newey (2004), Bester and Hansen (2007), fr specific appraches. 3

5 insiders knwledge f business pprtunies r a particular capal 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 addinal resurces t finance s prjects. Mrever, high investment ratis reduce the marginal prductivy f addinal investment, and therefre, wuld be difficult t get addinal resurces. Then, we expect that q and cash flw sensivy will be higher in the upper quantiles than in the lwer quantiles f investment. The results shw interesting hetergeney in the Tbin s q and cash flw sensivies f investment. In bth cases, the sensivies are mntnically increasing alng the quantiles. We shw that q values are subject t measurement errr, and we use past values f q in differences as instrumental variables t reduce the bias. The rest f the paper is rganized as fllws. Sectin 2 presents the instrumental variables quantile regressin panel data wh FE estimatr and derives s asympttic prperties and related inference. Sectin 4 has Mnte Carl experiments. In Sectin 5 illustrates the new apprach in the Tbin s q thery f investment. Finally, Sectin 6 cncludes the paper. 2 The Mdel and Assumptins 2.1 Measurement errr bias in quantile regressin Cnsider the fllwing representatin f a panel data mdel wh individual FE and ME y = d η + x β + u i = 1,..., N; t = 1,..., T. (1) where y is the respnse variable, d dentes a dummy variable that identifies the N distinct individuals in the sample, η dentes the N-vectr f individual FE, and x is a dim(x)-vectr f the mismeasured regressrs. Suppse that we d nt bserve x, but rather x, which is a nisy measure f x subject t an addive ME 3 ɛ, x = x + ɛ. (2) It is assumed that ɛ is independent and identically distributed (iid) wh V ar[ɛ ] = σɛ 2 <. Using equatin (2) we can express (1) in terms f the bserved y s and x s as y = d η + x β + u ɛ β i = 1,..., N; t = 1,..., T. (3) 3 We d nt cnsider Schennach (2008) case f a nnseparable errr structure, rather we restrict urselves t the tradinal framewrk f addivy. 4

6 It fllws that the bserved regressr x in (3) will be crrelated wh the cmpse errr, u ɛ β, inducing endgeney in the mdel. This prblem is f practical significance since the resulting bias may be large. In the fllwing paragraphs, we derive an apprximatin t the bias in the QR estimatr in the presence f ME using Angrist, Chernzhukv, and Fernandez-Val (2006) (dented ACFV hereafter) apprximatin t the mted variable bias in QR. This apprach shws that, as in LS, ME bias in QR can be derived analytically as an endgeney bias, and prvides a simpler representatin than that in Chesher (2001). Since this is the first attempt in the QR lerature t d s in this way, we d nt tackle simultaneusly the incidental parameter prblem, but rather we cnsider a standard QR structure. 4 The reader may skip these and g t sectin 2.2 fr the asympttic results f the paper. The standard result fr the LS estimatr wh ME can be seen as an mted variables prblem, where ɛ is the mted variable. In the rest f this sectin we m the indexes i and t t simplify the ntatin. Define v = [d, x ], Λ ɛ = [0, ɛ ], v = [d, x ] = v + Λ ɛ and ϕ = [η, β ]. In addin, let and ϕ = argmin E[y v ϕ] 2, (4) ϕ ϕ = argmin E[y v ϕ] 2, (5) ϕ where ϕ and ϕ are the parameters that slve the ppulatin minimizatin prblem. Under knwn cndins, f curse, these are the prbabily lim f the crrespnding estimatrs. Hwever, we fllw ACFV expsin a clse as pssible in this sub-sectin. In this case, ϕ = ϕ (E [vv ]) 1 E[vɛ β ] = ϕ (E [v v + Λ ɛ Λ ɛ]) 1 E[Λ ɛ ɛ β ]. (6) This shws the standard result that the bias depends n the nise t signal rati. Therefre, after sme algebra, the bias in the least squares FE estimatr f β can be als wrten as β = β [1 σɛ 2 ] V ar(x x i ), 4 Alternatively the results presented here can be seen as the special case where N is fixed and T ges t infiny. 5

7 where x i is the defined as 1 T T t=1 x. Cnsider nw the τ th cndinal quantile functin f the respnse y, Q y (τ d, x ) = d η (τ) + x β (τ). (7) In this frmulatin η, and β are allwed t depend upn the quantile, τ, f interest. 5 Using equatin (2) ne can rewre (7) as Q y (τ v ) = d η (τ) + (x ɛ) β (τ) = d η (τ) + x β (τ) ɛ β (τ) = Q y (τ v, ɛ). (8) As in the standard LS case wh measurement errr, the QR cunterpart can be seen as an mted variable prblem, where ɛ is the mted variable. We derive the apprximate bias using the ACFV mted variable bias frmula. Applying the standard estimatin prcedure in QR t equatin (7) wuld slve ϕ (τ) = argmin E[ρ τ (y v ϕ)], (9) ϕ where ρ τ (u) := u(τ I(u < 0)) as in Kenker and Bassett (1978), which is analgus t equatin (4). Hwever, the QR FE estimatr, as in the LS case, is biased in the presence f the mismeasured variables. In this case, in the prblem f slving (8) mting ɛ, the standard QR slves ϕ (τ) = argmin E[ρ τ (y v ϕ)]. (10) ϕ Here ϕ (τ) and ϕ (τ) are the parameters that slve the ppulatin minimizatin prblem, defined in an analg way t ACFV paper. The fllwing Lemma shws that the measurement errr bias in QR can be apprximated t an expressin similar t that in (6). Lemma 1 Assume that: (i) the cndinal densy 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 fine; (iii) ϕ (τ) and ϕ (τ) uniquely slves equatins (9) and (10) respectively; (iv) ɛ is independent f (d, x, u). Then, ϕ (τ) = ϕ (τ) (E [ω τ (v, ɛ) (vv )]) 1 E[ω τ (v, ɛ) vɛ β (τ)]. (11) where 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 QR specificatin errr. 5 This apprach is different frm that in Kenker (2004) where the individual effects are cnstant acrss quantiles. A straightfrward mdificatin f ur estimatr returns the cnstant individual effects case. We discuss briefly the details in Appendix 0. 6

8 Prf. The prf fllws ACFV results fr partial QR and mted variables bias (p ). Since the cndinal quantile functin is linear, Q y (τ v, ɛ) = v ϕ (τ) ɛ β (τ), where ϕ (τ) is defined as in (9). Then, the cndinal QR mdel in equatin (7) can be seen as ne wh [v, ɛ ] as cvariates. Mrever, the cndinal QR wh measurement errr Q y (τ v) = vϕ (τ), btaining the cefficient ϕ (τ) frm equatin (10), can be seen as a mdel wh mted variable ɛ. Recall that ω τ (v, ɛ) := 1 0 f u(τ) (u τ (v, ɛ; ϕ (τ)) v, ɛ) du/2 where f u(τ) (..) is the cndinal densy functin f u(τ) := y Q y (τ v, ɛ), and τ (v, ɛ; ϕ) := v ϕ Q y (τ v, ɛ) is the bias in the estimated quantile functin fr a given ϕ. Then, τ (v, ɛ; ϕ (τ)) = d (η (τ) η (τ)) + x (β (τ) β (τ)) + ɛ β (τ). Under the stated assumptins, by Therem 2 in ACFV, ϕ (τ) uniquely slves the equatin ϕ (τ) := argmin ϕ E[ω τ (v, ɛ) 2 τ(v, ɛ; ϕ)]. Slving fr ϕ (τ) we have ϕ (τ) = ϕ (τ) (E [ω τ (v, ɛ) (vv )]) 1 E[ω τ (v, ɛ) vɛ β (τ)]. Nte that the weighting functin ω(.) depends n bth v and ɛ and is a distinctinve feature f QR when cmpared wh LS case. Hwever, can be shwn that the leading term in the QR bias has the same frm as that in LS. In rder t shw this, assume that f y (y v, ɛ) has a first derivative in y that is bunded in abslute value by f (v, ɛ) and cnsider a Taylr expansin f the weights as in ACFV, p.546, ω τ (v, ɛ) = 1/2 f y (Q τ (y v, ɛ) v, ɛ) + ς(v, ɛ), where ς(v, ɛ) 1/4 τ (v, ɛ; ϕ (τ)) f (v, ɛ). Nte that by independence f y and ɛ, f y (Q τ (y v, ɛ) v, ɛ) = f y (Q τ (y v ) v ) (wh first derivative bunded by f (v )). Then, when eher τ (v, ɛ; ϕ (τ)) r f (v ) is small, ω τ (v, ɛ) 1/2 f y (Q τ (y v ) v ). 7

9 Then, the ACFV weighted LS apprximatin t QR implies that ϕ (τ) ϕ (τ) (E [f y (Q τ (y v ) v ) (v v + Λ ɛ Λ ɛ)]) 1 E[f y (Q τ (y v ) v )Λ ɛ ɛ β (τ)]. (12) It is imprtant t nte that a key factr in the cefficient bias apprximatin given by (12) is the cndinal densy functin f y (y v ). In addin, we can cmpare equatins (6) and (12), and as in the OLS case, the bias in the variable wh measurement errr parameter in the QR framewrk depends n the nise t signal rati, but in the QR case, weighted by the cndinal densy functin f y (y v ). 2.2 Estimatin We cnsider the panel data mdel wh measurement errrs expsed in eqs. (1) and (2), and we add a cvariate z whut measurement errr t the mdel. We repeat the equatins fr cnvenience wh y = d η + x β + z α + u i = 1,..., N; t = 1,..., T, (13) x = x + ɛ. (14) As in the LS case, the bias in QR can be amelirated thrugh the use f IV, w, that affect the determinatin f x but are independent f bth ɛ and u. If the latent regressr is autcrrelated, prvided that sme structure is impsed n the disturbances and measurement errrs, instruments are available whin the mdel. Examples f these are lagged values f the mismeasured variable (in levels r in differences) r ther cvariates. Fllwing Birn (2000) we use lags f the first differences f the mismeasured variable as IV. In this particular case, the bias can be slved whut relying n addinal exgenus variables that d nt belng t the mdel. If this is nt the case, the methd is still valid but relies n the availabily f valid exgenus instruments. 6 Other types f measurement errrs can be accmmdated by the use f apprpriate IV, althugh we cnsider this simple ME structure t cncentrate n the asympttic prperties f the IV estimatr. Cnsider nw the τ th cndinal quantile functin f the respnse y, Q y (τ d, x, z) = d η(τ) + x β(τ) + z α(τ). (15) 6 In a related wrk, Schennach (2008) discusses identificatin and estimatin fr general quantile functins in the presence f ME in the regressrs, establishing the availabily f IV that enables the identificatin and the cnsistent estimatin f nnparametric QR mdels. Our estimatr has the advantage ver Schennach (2008) estimatr that can be implemented using standard QR and IV techniques. 8

10 Fllwing Chernzhukv and Hansen (2006, 2008), frm the availabily f IV w, we cnsider estimatrs defined as: where (ˆη(β, τ), ˆα(β, τ), ˆγ(β, τ)) = argmin η,α,γ wh x A = x Ax and A is a psive define matrix. 7 ˆβ(τ) = argmin ˆγ(β, τ) A, (16) β N i=1 T ρ τ (y d η x β z α w γ), (17) t=1 As suggested by Chernzhukv and Hansen (2008), wh abuse f ntatin, in practice, a simple prcedure is t let the instruments w eher be w r the predicted value frm a least squares prjectin f x n w and z. Our final parameter estimatrs are thus ˆϕ(τ) := (ˆη( ˆβ(τ), τ), ˆβ(τ), ˆα( ˆβ(τ), τ)). (18) The intuin underlying the estimatr is that, since w is a valid instrument, is independent f ɛ and u, and therefre, shuld have a zer cefficient in (17). Thus, fr given β, the QR f (y x β) n the variables (d, z, w ) shuld generate a zer cefficient fr the variable w. Hence, by minimizing the cefficient f the variable w ne can recver a cnsistent estimatr f η, β and α. Values f lagged x and lags f the exgenus variable z affect the determinatin f x but are independent f ɛ, s they can be used as instruments. 8 The instrumental variables quantile regressin (IVQR) estimatr methd may be viewed as an apprpriate QR 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 IVQR. In Appendix 1 we shw the details f the derivatin. The instrumental variables quantile regressin wh fixed effects (IVQRFE) estimatr can be implemented as fllws: 1) Fr a given quantile f interest τ, define a grid f values {β j, j = 1,..., J}, and run the rdinary τ-qr f (y x β j ) n (d, z, w ) t btain cefficients ˆη(β j, τ), ˆα(β j, τ) and 7 As discussed in Chernzhukv and Hansen (2006), the exact frm f A is irrelevant when the mdel is exactly identified, but is desirable t set A equal t the asympttic variance-cvariance matrix f ˆγ(α(τ), τ) therwise. 8 It is knwn that if the distributin f the latent regressr vectr is nt time invariant, then cnsistent IV estimatrs f the cefficient f the latent regressr vectr exist (see e.g. Griliches and Hausman, 1986; Birn, 2000). The cnsistency f these estimatrs is rbust t ptential crrelatin between the individual hetergeney and the latent regressr. An interesting pint is that serial crrelatin r nn-statinary f the latent regressr is favrable frm the pint f view f identificatin and estimabily. 9

11 ˆγ(β j, τ), that is, by slving equatin (17) fr each value β j. 2) Find ˆβ(τ) as the value amng {β j, j = 1,..., J} that makes ˆγ(β j, τ) A the clsest t zer. This gives us the parameters in (18). We shall shw that this estimatr is cnsistent and asympttically nrmal under sme regulary cndins. Given the parameter estimates, fr an individual i, the τ-th cndinal quantile functin f y, can be estimated by, ˆQ y (τ d, x, z ) = d ˆη(τ) + x ˆβ(τ) + z ˆα(τ). In addin, given a family f estimated cndinal quantile functins fr an individual i, the cndinal densy f y at several values f the cndining cvariate can be estimated by the difference qutients, ˆf y (τ d, x, z ) = τ k τ k 1 ˆQ y (τ k d, x, z ) ˆQ y (τ k 1 d, x, z ), fr sme apprpriately chsen sequence f τ s. Nw we discuss the asympttic prperties f the IVQRFE estimatr. The parameter η, whse dimensin N is tending t infiny, raises sme new issues fr the asympttic analysis f the prpsed estimatr. Fllwing the recent lerature n panel data, we derive cnsistency and asympttic nrmaly assuming that bth N and T tend t infiny. We impse the fllwing regulary cndins: A1 The y are independent wh cndinal distributin functins F, and differentiable cndinal densies, 0 < f <, wh bunded derivatives f fr i=1,...,n and t=1,...,t ; A2 Let D = I N ι T, and ι T a T -vectr f nes, X = (x ) be a NT dim(β) matrix, Z = (z ) be a NT dim(α) matrix, and W = (w ) 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, W, Z ], Jacbian matrices Π(η, β, α, τ) and Π(η, β, α, γ, τ) are cntinuus and have full (η,β,α) (η,α,γ) rank, unifrmly ver E B A G T. The parameter space, E A B, is a cnnected set. Mrever the image f E A B under the map (η, α, β) Π(η, α, β, τ) is simply cnnected; 10

12 A3 Dente Φ(τ) = diag(f (ξ (τ))), where ξ (τ) = d η(τ) + x β(τ) + z α(τ) + w γ(τ), M D = I P D and P D = D(D Φ(τ)D) 1 D Φ(τ). Let X = [Z, W ]. 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 partin 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; A4 Fr all τ T = [c, 1 c] wh c (0, 1/2), (β(τ), α(τ)) int B A, and B A is cmpact and cnvex; A5 max x = O( NT ); max z = O( NT ); max w = O( NT ); A6 N a T 0, fr sme a > 0. Cndin A1 is a standard assumptin in the QR lerature and impses a restrictin n the densy functin f y. Cndin A2 is imprtant fr the parameter s identificatin. This 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 cndinal distributin f y at many relevant pints. In addin, the cndin that the image f the parameter space be simply cnnected requires that the image can be cntinuusly shrunk t a pint, and this cndin can be interpreted as ruling ut hles in the image f the set. 9 Assumptin A3 states invertibily cndins fr matrices in rder t guarantee asympttic nrmaly. A4 impses cmpactness n the parameter space f β(τ). Such assumptin is needed since the bjective functin is nt cnvex in β. Assumptin A5 impses a bund n the variables. Finally, cndin A6 is the same assumptin as in Kenker (2004) and allws T t grw very slwly relative t N. T further cmment n the nature f crrelatin between X and W required by A2, nte that by A1 we have that E[(τ 1(D η + X β + Z α)) ˇX]/ (η, β, α) = E[(D, Z, W ) Φ(τ)(D, X, Z)] Hence, the Jacbian in A2 takes a frm f densy-weighted cvariance matrix fr D, X and W, and A2 requires that this matrix has full rank. In addin, A2 impses that glbal identifiabily must hld; hence, the impact f W shuld be rich enugh t guarantee that the equatins are slved uniquely. 9 We assume that the image f E A B under the map (η, α, β) Π(η, α, β, τ) is cnnected fr ease f expsin. Hwever, is straightfrward t shw that the image f a cnnected set by an cntinuus functin is a cnnected set. 11

13 We can nw establish cnsistency and asympttic nrmaly. Prfs appear in Appendix 2. The fllwing therem states identificatin and cnsistency f ˆϕ(τ). Therem 1 Define ψ τ (u) := (τ I(u < 0)). 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. It is imprtant t ntice that even if the dimensin f the parameter space increases wh the number f crss-sectin, we nly need t impse that parameter space, E B A, is simple cnnected rather than cmpact. Therefre, by applying a Hadamards glbal univalence therem fr general metric spaces is pssible t shw that there is a ne-t-ne crrespndence between the parameter space and Π(E, B, A, τ), the image f E B A under Π(,,, τ). In addin, the identificatin fllws frm the glbal cnvexy f the quantile functin and the instrumental variables exclusin restrictin. Under cndins A1-A6 we shw that the asympttic prperties f the IVQRFE estimatr as N a /T 0 fr sme a > 0. Therem 1 prvides a lwer bund fr the rate at which T grws relative t N fr ensuring cnsistency. The intuin behind this cndin is that T must g t infiny fast enugh t guarantee cnsistent estimates fr the FE. The intuin behind the prf f cnsistency relies n the unifrm cnvergence f the bjective functin ver the parameter space. The basic technique used t shw this unifrm cnvergence is similar t Wei and He (2006) where we divide the grwing parameter space int small cubes. Then the ttal number f cubes grws at a plynmial rate s that the expnential bund btained at each cube hlds glbally and the unifrm cnvergence fllws. Using this technique we establish stchastic equicntinuy, and cnsistency fllws frm applicatin f an argmax therem as in van der Vaart and Wellner (1996). 10 In general, η is nly a nuisance parameter wh n particular interest. Define θ := (β, α) as the vectr f parameters f interest in this cntext. The liming distributin f these parameters IVQRFE estimatrs is given by Therem 2. Therem 2 (Asympttic Nrmaly) 10 It is imprtant t nte that we can use this technique f prf since the parameter space is grwing at a knwn rate, N. There is a large lerature shwing the asympttic prperties f QR estimatr fr infine dimensin parameter space as He and Sha (2000) and Prtny (1985) when the rate f the parameter increasing is nt knwn. Hwever, is pssible t achieve a better rate f N relative t T in the first case. 12

14 as Under cndins A1-A6, fr a given τ (0, 1), ˆθ(τ) cnverges t a Gaussian distributin 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 partin f J 1 αγ = (E( X M D ΦM D X)) 1, Φ(τ) = diag(f (ξ (τ))), and X = [W, Z]. The prf f asympttic nrmaly has sme elements f the nnlinear panel data lerature where we cncentrate ut the FE. Here we wre a asympttic representatin fr the FE, then we plug them int the representatin fr all the cefficients. Therefre, there will be a reminder term cming frm this tw step prcedure. It happens that the large N and T asympttics wh the restrictin that N a /T 0 is a sufficient cndin t ensure that the reminder term is negligible and the estimatr is asympttically nrmal centered at zer. Remark 1. Fr a fine cllectin f quantile indexes we have NT (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(β), the chice f A(β) des nt affect asympttic variance, and the jint asympttic variance f β(τ) and α(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(β), the chice f the weighting matrix A(β) generally matters, and is imprtant fr efficiency. A natural chice fr A(β) is given by the inverse f the cvariance matrix f ˆγ(β(τ), τ). Nticing that A(β) is equal t ( J γ S J γ ) 1 at β(τ), fllws that the asympttic variance f NT ( ˆβ(τ) β(τ)) is given by Ω β = (J β J γ( J γ S J γ) 1 Jγ Jβ ) 1. The cmpnents f the asympttic variance matrix that need t be estimated include J αγ, J β and S. The matrix S can be estimated by s sample cunterpart Ŝ(τ, τ ) = (min(τ, τ ) ττ ) 1 NT N i=1 T t=1 V V. (19) Fllwing Pwell (1986), J αγ and J β can be estimated as stated in Therem 2 abve, such as J αγ = E( X M D Φ(τ)M D X) and Jβ = E( X M D Φ(τ)M D X). The estimatr f J αγ is given 13

15 by the fllwing frm Ĵ αγ = 1 2NT h n N T I( û(τ) h n ) XM D M D X (20) i=1 t=1 where û(τ) := Y Dˆη(τ) X ˆβ(τ) Z ˆα(τ) and h n is an apprpriately chsen bandwidth, wh h n 0 and NT h 2 n. The estimatr f J β is analgus t Ĵαγ. By using the same prcedure we can estimate the element D ˆΦ(τ)D in P D. The cnsistency f these asympttic cvariance matrix estimatrs are standard and will nt be discussed further in this paper. 2.3 Inference In this sectin, we turn ur attentin t inference in the IVQRFE 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 cndinal 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 is imprtant t test fr equaly 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 QR 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 tests. The Wald prcess and assciated liming thery prvide a natural fundatin fr the hypthesis Rθ(τ) = r, when r is knwn. Fllwing the arguments f Prtny (1984) and Gutenbrunner and Jureckva (1992), the QR prcess is tight and thus the liming variate viewed as a functin f τ is a Brwnian Bridge ver τ T. 11 Therefre, under the linear hypthesis H 0 : Rθ(τ) = r, cndins A1-A6, and letting Σ = (K, L ) EV V (K, L ), we have V NT = NT [RΣ(τ)R ] 1/2 (Rˆθ(τ) r) B q (τ), (21) 11 In a related result, Wei and He (2006) establish tightness f the QR prcess in the lngudinal data cntext wh increasing parameter dimensin, see fr instance Lemma

16 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 Kiefer prcess f rder q. Thus, fr given τ, the regressin Wald prcess can be cnstructed as W NT (τ) = NT (Rˆθ(τ) r) [RˆΩ(τ)R ] 1 (Rˆθ(τ) r) (22) where ˆΩ is a cnsistent estimatr f Ω, and Ω is given by Ω(τ) = (K (τ), L (τ)) S(τ, τ)(k (τ), L (τ)). Under H 0, the statistic W NT is asympttically χ 2 q wh q-degrees f freedm, where q is the rank f the matrix R. The liming distributins f the test is summarized in the fllwing therem: Therem 3 (Wald Test Inference). Under H 0 : Rθ(τ) = r, fr fixed τ and cndins A1-A6, W NT (τ) a χ 2 q. Prf. The prf f Therem 3 is very simple and fllws frm bserving that fr any fixed τ, by Therem 2 NT (ˆθ(τ) θ(τ)) N(0, Ω(τ)) under the null hypthesis, NT (Rˆθ(τ) r) N(0, RΩ(τ)R ) since ˆΩ(τ) is a cnsistent estimatr f Ω(τ), by Slutsky therem W NT (τ) = NT (Rˆθ(τ) r) [RˆΩ(τ)R ] 1 (Rˆθ(τ) r) a χ 2 q. In rder t implement the test 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 (19) and (20). 15

17 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 (22). Hwever, Ω is nw the matrix wh 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 (ξ (τ j ))), and the ther variables are as defined abve. The statistic W NT is still asympttically χ 2 q under H 0 where q is the rank f the new matrix R. This frmulatin accmmdates a wide variety f testing suatins, 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 equaly f several slpe cefficients acrss several quantiles. Anther imprtant class f tests in the QR lerature 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 (2006), we may cnstruct a KS type test n the panel data regressin quantile prcess in the fllwing way: The liming distributin f the Klmgrv-Smirnv test is given in the fllwing therem: KSW NT = sup W NT (τ). (23) τ T Therem 4 (Klmgrv-Smirnv Test). Under H 0 and cndins A1-A6, KSW NT = sup τ T W NT (τ) sup Q 2 q(τ). τ T The prf f Therem 4 fllws directly frm the cntinuus mapping therem and equatin (23). Crical values fr sup Q 2 q(τ) have been tabled by DeLng (1981) and, mre extensively, by Andrews (1993) using simulatin methds. 16

18 3 Mnte Carl Simulatin 3.1 Mnte Carl design fr the perfrmance f IVQRFE In this sectin, we describe the design f sme simulatin experiments that have been cnducted t assess the fine sample perfrmance f the IVQRFE estimatr discussed in the previus sectins. 12 Tw simple versins f the basic mdel (1) are cnsidered in the simulatin experiment. In the first, reprted in Tables 1 and 2, the scalar cvariate, z, exerts a pure lcatin shift effect. In the secnd, reprted in Tables 3 and 4, z exerts bth lcatin and scale effects. In the frmer case the respnse y is generated by the mdel y = η i + x β + z α + u, while in the latter y = η i + x β + z α + (γz )u. In bth cases we have an addive measurement errr f the frm x = x + ɛ, where x fllws an ARMA(1,1) prcess (1 φl)x = µ i + ε + θε 1 (24) and ε fllws the same distributin as u, that is, nrmal distributin and t 3 fr Schemes 1 and 2 respectively. In all cases we set x i, 50 = 0 and generate x fr t = 49, 48,..., T accrding t x = µ i + φx 1 + ε + θε 1. (25) This ensures that the results are nt unduly influenced by the inial values f the x prcess. Given the structure f the measurement errr we use x 1 as instrumental variable. We emply tw different schemes t generate the disturbances (u, ε ). Under Scheme 1, we generate them as N(0, 1). Under Scheme 2 we generate them as t-distributin wh 3 degrees f freedm. The exgenus regressr z is generated accrding t the same distributin as the innvatins. In rder t estimate the mdel we discarded the first 50 time series 12 The experiment shwn in this sectin is based in Hsia, Pesaran, and Tahmisciglu (2002). 17

19 bservatins, using the bservatins t = 0 thrugh T fr estimatin. The FE, µ i and η i, are generated as T µ i = e 1i + T 1 ε, e 2i N(0, σe 2 2 ). t=1 T η i = e 2i + T 1 z, e 2i N(0, σe 2 2 ). t=1 The abve methd f generating µ i and η i ensures that the randm effects estimatr is incnsistent because f the crrelatin that exists amng the individual effects and the explanatry variables. In the simulatins, we experiment wh T = 10 and N = 100. We set the number f replicatins t 1000, and cnsider the fllwing values fr the remaining parameters: (β, α) = (1, 1); φ = 0.6, θ = 0.7, γ = 1, σu 2 = σe 2 1 = σe 2 2 = 1. In the Mnte Carl study, we cmpare the LS and QR based estimatrs in terms f bias and rt mean squared errr (RMSE). Mre specifically, we study fur estimatrs: the usual least squares whin grup estimatr (FE); the least squares instrumental variables estimatr (IVFE); the quantile regressin fixed effects estimatr prpsed by Kenker (2004) (QRFE); and finally, the instrumental variables quantile regressin fixed effects estimatr prpsed in this paper (IVQRFE). The QR estimatrs are studied nly fr the median case, althugh similar results are btained fr ther quantiles. 3.2 Mnte Carl Results Tables 1 and 2 present bias and RMSE simulatins fr estimates f the mismeasured cefficient, β, and the perfectly measured variable cefficient, α, fr the lcatin-shift mdel and the nrmal and t 3 distributins f the innvatins respectively. The results shw that the bias in the mismeasured variable cefficient f the FE and QRFE estimatrs are severe. Hwever the cefficients fr the IVFE and IVQRFE are apprximately unbiased. Mrever, since we d nt cnsider any crrelatin amng the regressrs, as expected, the cefficient f the perfectly measured variable is apprximately unbiased in all cases. Table 1 shws that when the disturbances are sampled frm a nrmal distributin, as expected, the mismeasured regressr cefficient is dwnward biased fr the FE case, but the 18

20 Table 1: 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 Table 2: 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 IVFE is apprximately unbiased. Similarly, in the presence f errrs in variables, the QRFE estimatr prpsed by Kenker (2004) is dwnward biased and IVQRFE is able t reduce the bias. In summary, estimates are biased in bth the FE and the QRFE cases, and the IV strategy is able t cnsiderably diminish the bias fr bth LS and QR cases. Regarding the RMSE, in the Gaussian cndin, the LS based estimatrs perfrm better than the respective QR estimatrs. Thus, under Gaussian cndins, the IVQRFE is capable t reduce the bias but has a larger RMSE when cmpared wh the IVFE. Table 2 presents simulatins fr the t 3 -distributin case. The β estimates f FE and QRFE are biased, and the FE has a larger bias when cmpared wh the same estimatr in the Gaussian case. The IVQRFE is apprximately an unbiased estimatr fr bth cefficients. Interestingly, in the nn-gaussian heavy-tailed cndin, the RMSE f IVQRFE is smaller than the RMSE f IVFE, evidencing that there are gains in efficiency by using a rbust estimatr. Tables 3 and 4 present bias and RMSE results fr the lcatin-scale-shift mdel fr the nrmal and t 3 distributins respectively. In bth cases the FE and QRFE estimatrs are dwnward biased and the IVFE and IVQRFE are apprximately unbiased. Nte that the IVQRFE estimatr presents a cnsiderable smaller bias than QRFE and much mre precisin when cmpared wh IVFE in the t 3 case. 19

21 Table 3: Lcatin-Scale 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 Table 4: Lcatin-Scale 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 Empirical applicatin: Investment, Tbin s q and cash flw As pinted ut in Wansbeek (2001), the mst frequent applicatin f measurement errr techniques is in the estimatin f investment mdels using Tbin s q. This variable is the rati f the market valuatin f a firm and the replacement value f s assets. Firms wh a high value f q are cnsidered attractive as t the investment pprtunies, whereas a lw value f q indicates the ppse. 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 Whed (2000). 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 (see fr instance Alti, 2003; Almeida, Campell and Weisbach, 2004). Hwever, Ericksn and Whed (2000) argue that cash flw has n effect n investment nce measurement errr in Tbin s q is taken int accunt. There are distinct advantages in expling panel data n individual firms (see fr instance Blundell, Bnd, Devereux and Schiantarelli, 1992). In the first place allws testing the thery, develped in the cntext f a representative firm, in the cntext f many firms, 20

22 s reducing ecnmetric prblems intrduced by aggregatin acrss firms. Aggregatin prblems may be imprtant here since the standard q mdel is specified in ratis and there are clear nnlinearies. Secndly, the estimates are btained by using bth the time-series and crss-sectinal variatin in the data. This cntributes t their precisin and als allws cnsistent estimatin in the presence f firm specific effects crrelated bth wh investment and q values. Althugh nt reprted, pled and FE estimates differ in ur estimatin, which determines that firm individual FE need t be cntrlled fr in ur mdel. We argue that QR is useful t describe differences in investment ratis acrss firms in the presence f unbserved hetergeney. Unbserved characteristics (fr the ecnmetrician) acrss firms may be due t several reasns. The abily f managers (insiders) r the characteristics f investrs (utsiders) can be named as the main unbservable cmpnent. Fr instance, Bertrand and Schar (2003) find that a significant extent f the hetergeney in investment, financial, and rganizatinal practices f firms can be explained by managers style. Walentin and Lrenzni (2007) develp a mdel f financially cnstrained firms financed partly by insiders, wh cntrl s assets, and partly by utside investrs. When their wealth is scarce, insiders earn a rate f return higher than the market rate f return, that is, they receive a quasi-rent n invested capal. Therefre, differences in insiders and utsiders characteristics acrss firms may cntribute t differences in investment behavir. Addinal hetergeney may be due t differences in firms capal structure. Chirink (1993) shws that when the q thery is expanded fr the pssibily that the value f the firm depends n tw r mre capal inputs wh differing adjustment cst technlgies, the ecnmetric equatin fllwing frm ptimizing behavir includes q as well as addinal variables (e.g. inventry, research and develpment). In the upper cndinal quantiles f investment, investment may be driven by insiders knwledge f business pprtunies r a particular capal 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 s prjects. Mrever, high investment ratis reduce the marginal prductivy f addinal investment, and therefre, wuld be difficult t get addinal resurces. In ther wrds, q and cash flw sensivies will be higher in the upper quantiles f investment than in the lwer quantiles. In cntrast wh the interpretatin f QR in crss-sectin, by cntrlling fr the firmspecific characteristics, the quantiles shuld be interpreted in terms f the pairs firm-year. In this case, lw (high) quantiles refer t firms in perids wh unusually lw (high) investment 21

23 ratis when cndining n ther time-variant bservable infrmatin. This determines that the same firm can be at different quantiles in different perids, which is in line wh the idea that firm s behavir will change depending n their investment needs. The baseline mdel in the lerature is I /K = η i + βq + αcf /K + u, (26) where I dentes investment, K capal 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 cndinal quantile functin: Q I/K (τ η, CF/K, q) = η i (τ) + β(τ)q + α(τ)cf /K. (27) As mentined earlier, if q is subject t measurement errr, estimates f β(τ) will be biased dwn (assuming β( ) > 0), but this prblem can be amelirated by using instrumental variables. We fllw Almeida, Campell, and Weisbach (2004) apprach by cnsidering a sample f manufacturing firms (SICs 2000 t 3999) ver the 1980 t 2005 perid wh data available frm COMPUSTAT s P/S/T, Full Cverage. Only firms wh bservatins in every year are used, in rder t cnstruct a balanced panel f firms fr the 26 year perid. 13 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. Summary statistics f the variables emplyed in ur ecnmetric specificatins appear in Table 5. Because we nly cnsider firms that reprt infrmatin in each f the 26 years, the sample cnsists mainly f relatively large firms. Althugh this reduces the sample hetergeney in terms f cred cnstraints (e.g. presumably firms wh market infrmatin fr cntinuus 26 years are cred uncnstrained), serves fr ur purpses f studying firm hetergeney alng different quantiles. Table 5: Summary statistics Variable Mean Std.Dev quantile Median 0.75 quantile I/K q CF/K The chice f a balanced panel is made t reduce the cmputatinal burden. 22

24 I/K q CF/K Table 6 reprt the regressin estimates. The first clumn reprts least squares whin grup (FE) estimatrs whut (first set f rws) and wh instruments (last set f rws, IVFE). The secnd clumn reprts least squares in first differences fr the same set f rws (FD and IVFD respectively). Fllwing Birn (2000) we use lags f the first differences f the mismeasured variable as instruments fr fixed effects, and lags in levels fr the first differences estimatr. The last five clumns reprt QR estimates wh fixed effects (QRFE). Our prpsed estimatr, IVQRFE, appears in the last rws where the same instruments used in FE are emplyed. IV estimates are reprted wh ne instrument ( q t 1 fr the mdel in levels, q t 2 fr FD) and wh tw instruments ( q t 1, q t 2 fr the mdel in levels, q t 2, q t 3 fr FD) t assess the sensivy f the estimates t the instrument selectin. 14 Table 6: Ecnmetric results τ = 0.10 τ = 0.25 τ = 0.50 τ = 0.75 τ = 0.90 Variable FE FD QRFE q (0.010) (0.018) (0.009) (0.009) (0.009) (0.013) (0.017) CF/K (0.006) (0.008) (0.007) (0.009) (0.009) (0.013) (0.014) IVFE IVFD IVQRFE q (0.074) (0.081) (0.033) (0.045) (0.089) (0.104) (0.219) CF/K (0.010) (0.008) (0.008) (0.008) (0.012) (0.025) (0.032) Inst. q t 1 q t 2 q t 1 q (0.051) (0.080) (0.022) (0.033) (0.057) (0.082) (0.167) CF/K (0.008) (0.008) (0.029) (0.029) (0.046) (0.061) (0.078) Inst. q t 1, q t 2 q t 2, q t 3 q t 1, q t 2 Ntes: Standard errrs in parenthesis. Sample f 4550 firm-years and 175 firms. FE: fixed effects least squares; FD: first differences least squares; QRFE: quantile regressin wh fixed effects; IVFE: instrumental 14 Althugh nt reprted, we als use least squares prjectins as a different set f instruments in the QR estimatins. We cnsider the prjectin f q n each respective instrument and the exgenus variables, i.e. the fixed effects and cash flw. In every case, we btain similar results. 23

25 variables fixed effects least squares; IVFD: instrumental variables fixed differences least squares; IVQRFE: instrumental variables quantile regressin fixed effects. Bth FE and FD estimatrs give similar results fr the effect f Tbin s q n investment, implying that firms wh higher q values wuld invest mre. In particular, the q-sensivy f investment is clse t Cash flw appear as psive and statistically significant. Hwever, the FD estimate f cash flw is half f that in FE. When instrumental variables are used, the effect f Tbin s q raises cnsiderably, which is in line wh the presence f measurement errr in this variable. By cntrasting the effect f the instrumental variables acrss different estimates, can be cncluded that measurement errr in Tbin s q reduces s effect by a half. Mrever, the pint estimates f cash flw d nt change, which can be interpreted as the fact that measurement errr in Tbin s q has n spillver measurement errr t cash flw. In turn, this cntradicts Ericksn and Whed (2000) statement that cash flw des nt matter as has a highly significant (mean) effect. FE, FD and QRFE pint estimates differ bth in terms f Tbin s q and cash flw effects. QRFE estimates are mntnically increasing in τ, ranging frm fr τ = 0.05 t 0.11 at τ = 0.95 (see figure 1). In terms f the Tbin s q pint estimates, QRFE is belw FE fr τ < 0.75 and belw FD almst everywhere. As a result, firms-years wh unusually high investment ratis will be very respnsive t changes in their market valuatin. In a similar tken, the effect f cash flw n investment is mntnically increasing in τ. This determines that firms wh cndinally high investment ratis have a higher prpensy t use addinal internal resurces t finance investment than thse wh lw investment ratis. The use f IVQRFE differ frm QRFE pint estimates in a similar way as IVFE and IVFD differ frm FE and FD respectively (see figures 2 and 3). Hwever, in this case, the mntnicy is weakened in the q-sensivy f investment. Fr lw quantiles, QRFE and IVQRFE prvide similar pint estimates. Hwever, bth estimatrs differ greatly as τ increases. This suggests an interesting asymmetry in terms f the measurement errr prblem, as may manifest self mstly fr firms-years in the upper quantiles f the investment distributin. Pint estimates fr IVQRFE are particularly high fr τ 0.90, reaching a q-sensivy f 0.4. Nevertheless, IVQRFE has bigger standard errrs than bth IVFE and QRFE. In fact fr τ > 0.25 the IVFE cnfidence interval is cntained in the IVQRFE interval, while QRFE interval is always cntained. In addin, QRFE and IVQRFE give similar estimates in terms f the cash flw sensivy f investment, which reassures the fact that 24