Quantile Regression with Integrated Time Series

Transcription

1 Quantile Regression with Integrated Time Series hijie Xiao Department of Economics Boston College November 6, 25. Abstract This paper studies quantile regression with integrated time series. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modi ed quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald test are constructed based on the fully modi ed quantile regression estimators. A robust test for cointegration based on quantile regression is proposed. JEL: C22, G. Key Words: Bahadur Representation, Cointegration, Endogeneity, Quantile Regression, Unit Root. Introduction Since Granger (98) and Engle and Granger (987), cointegration has become a common econometric tool for empirical analysis in numerous areas (see, inter alia, Phillips and Ouliaris 988; Johansen 995; Sims, Stock, and Watson 99; and Hsiao 997, among others), especially in macroeconomic and nancial applications. Wellknown nancial applications of cointegration include Campbell and Shiller (987) in the study of bubbles in asset prices, Cochrane (994) and Lettau and Ludvigson (2) on the predictability of stock prices, Hall, Anderson and Granger (992) on Preliminary and Incomplete. Address correspondence: Department of Economics, Boston College, Chestnut Hill, MA Tel: xiaoz@bc.edu. The author wish to thank Roger Koenker, Steve Portnoy, Xumin He, and Peter Phillips for helpful comments.

2 term structure of interest rates, Pindyck and Rothemberg (992), Lucas (997) and Alexander (999) on portfolio allocation. Also see, inter alia, Evans (99), Campbell, Lo, and MacKinley (997), Cerchi and Havenner (988), Chowdhury (99), Hendry (996), on other applications in nance. In applications of portfolio management, cointegration measures long run comovements in prices, leading to hedging methodologies that may be more e ective than traditional correlation analysis-based approach in the long term. Recent research (e.g. Alexander (999)) indicate that mis-pricing and over-hedging can occur if cointegration is ignored. When portfolios are allocated using risk criteria such as the conditional value at risk (CVaR), the optimization problem leads to a quantile cointegrating regression. Quantile regression method has recently attracted an increasing amount of research attention in nance. Taylor (999) applies quantile regression approach to estimating the distribution of multiperiod returns. Engle and Manganelli (999) proposes estimating value at risk (VaR) using quantile regression. Quantile regression is now an important tool in modern risk management operations. For example, the popular risk measure, value at risk (VaR), is simply a concept of quantile and can be naturally estimated using quantile regression method. In recent years, motivated by regulatory reasons in the nancial sector, an in uential axiomatic foundation raised by Artzner, Delbaen, Eber, and Heath (999) is the concept of coherence". A widely used coherent risk measure is the conditional value at risk (CVaR) (Rockafellar and Uryasev (2)) (or, in di erent names, Expected Shortfall, Acerbi and Tasche (22); tail conditional expectation, Artzner, Delbaen, Eber, and Heath (999)). Bassett, Koenker and Kordas (24) recently show that, when the portfolio risk is measured by CVaR, the managers operation can be formulated as a quantile regression of cointegrated time series. Although there has been a large amount of recent attempts in applying quantile regression to nancial time series models, there is little investigation on the statistical validity and properties of these methods and models. This paper studies the statistical properties of quantile regression estimation and inference of cointegrated time series. Limiting distribution of the regression quantiles is derived. In the presence of endogenous regressors, a fully-modi ed quantile regression estimator is proposed to remove the second order bias and nuisance parameters. We develop statistical inference based on the quantile regression estimators. Asymptotic properties of the proposed fully-modi ed quantile regression estimator and testing procedure based on this estimator are studied. In matters of notation, we use ) to signify weak convergence of the associated VaR is not a coherent risk measure. 2

3 probability measures, [nr] to signify the integer part of nr, := to signify de nitional equality, and I(k) to denote integration of order k. Continuous stochastic process such as the Brownian motion B(r) on [; ] are usually written simply as B and integrals R are understood to be taken over the interval [; ], unless otherwise speci ed. 2 Quantile Regression on Cointegration Model In this section, we consider quantile regression of the following cointegration model: y t = + x t + u t = z t + u t ; () where x t is a k-dimension vector of integrated regressors, z t = (; x t), and u t is mean zero stationary. The quantile regression estimator of the cointegrating vector can be obtained by solving the problem where (u) = u( b () = arg min 2R p nx (y t z t > ); (2) t= I(u < )) as in Koenker and Bassett (978). In the special case = :5, the above quantile regression delivers the least absolute deviation (LAD) estimation of the cointegration model (). 2. Limiting Distribution of the Quantile Regression Estimator To derive the limiting distribution of the quantile regression estimator of the cointegrating vector. we follow the approach of Knight (99) (also see Herce (996), Hasan and Koenker (997), Koenker and Xiao (23) for related results). Let f() and F () be the p.d.f. and c.d.f. of u t, denoting (u) = () = ((); ), and u t = y t () z t = u t F (); we have Q ut () =, where Q ut () is the -th quantile of u t, and E (u t ) = : I(u < ); () = + F (), To facilitate the asymptotic analysis, we make the following assumptions. Assumption A: Let v t = x t, fu t ; v t g is a zero-mean, stationary sequence of (k+)- dimensional random vectors. The partial sums of the vector process ( (u t ); v t ) follow a multivariate invariance principle [nr] X n =2 t= (u t ) v t ) B (r) B v (r) = BM(; ) 3

4 where is the covariance matrix of the Brownian motion (B (r); B v (r) ). Assumption B: The distribution function of u t, F (u), has a continuous density f(u) with f(u) > on fu : < F (u) < g: Assumption C: The conditional distribution function F t (u) = Pr[u t < uju t j ; j ; v t k ; k ] has derivative f t (); a:s:, and f t (s n ) is uniformly integrable for any sequence s n! F (), and E[f t (F ())] < for some > : Conformable to ( (u t ); v t ), we partition into! 2 v = v The asymptotic distribution of the quantile regression estimator is closely related to the asymptotic behavior of n P n t= x t (u t ). Under Assumption A, it is easy to verify (e.g. Phillips and Durlauf (986, Lemma 3.); and Hansen 992)) that n nx x t (u t ) ) t= vv B v db + v ; where v is the one-sided long-run covariance between v t and (u t ). Due to the nonstationarity of x t ; the two components in b () = (b(); b () ) have di erent rates of convergence. In particular, the estimate of cointegrating vector b () converges at rate n, while the intercept b() converges at rate p n. Thus, we introduce the standardization matrix D n = diag( p n; ni k ), where I k is a k k identity matrix. The limiting distribution of the quantile regression estimator for the cointegration model is summarized in the following Theorem: Theorem. Under Assumptions A, B, and C, D n ( b () ()) ) f(f B v B > v B v db + v ; ()) where B v (r) = (; B v (r) ), and v = (; v ). In particular, n( b () ) ) where B v (r) = B v (r) f(f B ()) v B > v B v db + v rb v () is a k-dimensional demeaned Brownian motion. (3) The above limiting result is very similar to that of the conventional cointegrating regression estimators: (i) The quantile regression estimator of the cointegrating vector is consistent at the usual O(n) rate. (ii) Like OLS, the quantile regression estimator su ers from second order bias ( v ) coming from the correlation between the regressor 4

5 x and the residual u. (iii) In addition, the Brownian motions B v (r) and B (r) are in general correlated (as long as v 6= in ). (iv) Similar to the usual limit theory for the LAD estimator in both stationary and nonstationary time series regression, the limiting distribution (3) depends on the sparsity function =f(f ()). In the special case when v = and v = (x t and u s are independent), the limiting distribution (3) is a mixed normal. 2.2 A Fully-Modi ed Quantile Regression Estimator We are interested in developing estimation and inference procedures based on the quantile regression in cointegration models. As will become clear in later analysis, the asymptotic behavior of quantile regression-based inference procedures depends on the limiting distribution of b (). However, as shown by Theorem, the limiting processes B v (r) and B (r) are correlated Brownian motions whenever contemporaneous correlation between v t and (u t ) exists. Despite super-consistency, b () is second-order biased and the miscentering e ect in the limit distribution is re ected in v. Consequently, the distribution of the test based on the quantile regression residual will be dependent on nuisance parameters. To restore the asymptotic nuisance parameter free property of inference procedure, we need to modify the original quantile regression estimator so that we obtain a mixed normal limiting distribution. In this paper, we propose two approaches to achieve this goal: () Nonparametric fully-modi cation on the original quantile regression estimator and (2) Parametrically augmented quantile regression using leads and lags. We propose a nonparametric fully-modi ed quantile regression estimator to deal with the endogeneity problem in this section. In Section 3, we introduce a parametrically augmented quantile regression using leads and lags and extend the conventional cointegration model to the case with time-varying coe cients. We develop a fully-modi ed quantile cointegrating regression estimator in the spirit of Phillips and Hansen (99). We rst decompose the limiting distribution (3) into the following two components: f(f B ()) v B > v B v db :v, and f(f B ()) v B > v B v db v > vv v + v where B :v(r) = B (r) vvv B v (r) is Brownian motion with variance! 2 :v =! 2 vvv v. Notice that B :v(r) is independent of B v (r) and the rst term in the above decomposition, h R B vb > v i R B vdb :v, is a mixed Gaussian variate. 5

6 The basic idea of fully modi cation on () b (or b ()) is to construct a nonparametric correction to remove the second term in the above decomposition. To facilitate the nonparametric correction, we consider the following kernel estimates of vv, v, v ; vv : b v = b v = MX k( h M )C v (h); b MX vv = k( h M )C vv(h); h= MX h= k( h M )C v (h); b MX vv = h= M h= M k( h M )C vv(h); where k() is the lag window de ned on [ ; ] with k() =, and M is the bandwidth parameter satisfying the property that M! and M=n! (say M = O(n =3 ) for many commonly used kernels, as in Andrews, 99) as the sample size n! : The quantities C v (h) and C vv (h) are sample covariances de ned by C v (h) = n P v t (bu t+h; ), C vv (h) = n P v t v t+h, where P signi es summation over t; t + h n. Candidate kernel functions can be found in standard texts (e.g., Hannan, 97; Brillinger, 98; and Priestley, 98). Let \ f(f ()) be a nonparametric sparsity estimator of f(f ()) (see, e.g., Siddiqui (96), Bo nger (975), Sheather and Maritz (983), and Welsh (987)), we de ne the following nonparametric fully modi ed qunatile regression estimators: b b() () + = b() + where b() + = b () " # " X X x t x f(f\ t x t vt b vv b v + n b + v ()) t t # (4) and b + v = b v b vv b vv b v : Like the fully modi ed OLS estimators, the fully modi ed quantile regression estimator of the cointegrating vector has a mixed normal distribution in limit. Theorem 2. Under Assumptions A, B, and C, D n b() + () ) f(f B v B > v B v db :v ())! 2! :v MN ; f(f ()) 2 B v B > v : 6

7 In particular n( () b + ) ) f(f B ()) v B > v B v db :v! 2! :v MN ; f(f ()) 2 B v B > v : 2.3 Regression Wald Test The fully modi ed quantile regression estimator and resulting asymptotic mixture normal distribution facilitates statistical inference based on quantile cointegrating regression. In this section, we consider the classical inference problem of linear restrictions on the cointegrating vector : H : R = r; where R denotes an q k-dimensional matrix and r is an q-dimensional vector. Under the null hypothesis H : R = r and the assumptions of our previous theorem, we have f(f ())! :v " # =2 R B v B > v R > n(r() b r) ) N(; I q ); (5) where N(; I q ) represents a q-dimensional standard Normal. Therefore, let M X = nx (x t x)(x t x) ; t= a regression Wald statistic can be constructed as f(f W n () = \ ()) (R() b h r) > RM X R>i (R() b b! :v r); where \ f(f ()) and b! :v are consistent estimators of f(f ()) and! :v. limiting distribution of the Wald statistic is summarized in the following Theorem. The Theorem 3. Under the assumptions of Theorem 2 and the linear restriction H, W n () ) 2 q; where 2 q is a centered Chi-square random variable with q-degrees of freedom. 7

8 3 Cointegrating Regression with Leads and Lags An alternative way to deal with the endogeneity problem is to introduce leads and lags as Saikkonen (99). In particular, we make the following assumptions: Assumption A : Let v t = x t, fu t ; v t g is a zero-mean, stationary sequence of (k + )-dimensional random vectors and for some K; u t has the following representation u t = KX j= K where " t is a stationary process such that v t j j + " t ; (6) E(v t j " t ) =, for any j: The partial sums of the vector process ( (" t ); v t ) follow a multivariate invariance principle [nr] X n =2 t= (" t ) v t ) B(r) = B (r) B v (r) = BM(; ) Assumption B : The distribution function of " t, F " ("), has a continuous density f " (") with f " (") > on f" : < F " (") < g: Assumption C : The conditional distribution function F t (u) = Pr[u t < uju t j ; j ; v t k ; k ] has derivative f t (); a:s:, and f t (s n ) is uniformly integrable for any sequence s n! F (), and E[f t (F ())] < for some > : Assumption D: Let t = ( t ; ; kt ), the cointegrating coe cients it are monotone functions of the innovation process " t. The idea of using leads and lags to deal with endogeneity in traditional cointegration model was proposed by Saikkonen (99). It can be veri ed that, under Assumption A, f "" () = f uu () f uv ()f vv () f vu () where f "" (), f uu (), f vv () are spectral densities of ", u, v, and f uv () is the cross spectral of u and v, implying that the long run variance of " is! 2 "" =! 2 uu uv vv vu. Notice that the Brownian motion B (r) is now independent with B v (r). We partition the covariance matrix (of the Brownian motion B(r)) into! 2 = vv : 8

9 as: Under Assumption A, the original cointegrating regression (??) can be re-written y t = + tx t + KX j= K x t j j + " t : If we denote the -th quantile of " t as Q " (), let F t = fx t ; x t conditional on F t, the -th quantile of y t is given by Q yt (jf t ) = + () x t + KX j= K j ; 8jg, then, x t j j + F " (); (7) where F " () is the CDF of " t. Let t be the vector of regressors consisting z t = (, x t ) and (x t j, j = K; ; K), = (; ; K ; ; K ), and () = ((); () ; K; ; K) where () = + F " (), then, we can re-write the above regression as y t = t + " t and Let " t = " t F " (); then Q yt (jf t ) = () t : (8) Q "t () =! We now consider the following modi ed quantile cointegrating regression: b() = arg min nx (y t t ); (9) t= Denote G n = diag(d n ; p n; ; p n) = diag( p n; n; ; n; p n; ; p n). Conformable with (); we partition () b as follows: h i b() = b(); () b ; b K () ; ; K b () : Given b (), the -th conditional quantile function of y t ; conditional on x t, can be estimated by, ^Q yt (jf t ) = z > t b (); and the conditional density of y t can be estimated by the di erence quotients, ^f yt (jf t ) = ( i i )=( ^Q yt ( i jf t ) ^Qyt ( i jf t )); for some appropriately chosen sequence of s. 9

10 The limiting distribution of this estimator is given in the following Theorem: Theorem 4. Under Assumptions A, B, C, and D, the RQ process b () has the following Bahadur representation: G n ( b () ()) = f(f ()) nx t= Gn X t (" t ) + o p() where In particular G n ( b () ()) ) R Bv B = v ; " R f " (F" B vb > v ()) # " R B vdb # : n( b () ()) ) f " (F" B v B > v B v db ; ()) and where B v (r) and B v (r) are the same as those de ned in Theorem, = E(V t V t ) and V t = (x t K ; ; x t+k ), and is a multivariate normal with dimension conformable with ( K () ; ; K () ). 3. Inference on Quantile Cointegration Models Notice that the limiting distribution of b () is mixture normal, statistical inference procedures can be constructed based on the above augmented quantile regression. If we consider again the inference problem in Section 3; H : R = r, let () b be estimated from the augmented quantile regression, and f \ " (F" ()) and b! are consistent estimators of f " (F" ()) and!, we may construct the following regression Wald statistic: W n () = \ f " (F " ()) b! (R b () r) > h RM X R>i (R b () r) where M X is de ned as in Section 3, then, we obtain a similar result as Theorem 3. Theorem 5. Under the assumptions of Theorem 4 and the linear restriction H, W n () ) 2 q; where 2 q is a centered Chi-square random variable with q-degrees of freedom.

11 Another interesting inference problem in the quantile cointegration model is the hypothesis test on constancy of the cointegrating vector. In particular, we are interested in the hypothesis H 2 : () =, over 2 T, where is a vector of unknown constants. A natural preliminary candidate for testing constancy of the cointegrating vector is a standardized version of b() Under the null, n b() ) f " (F" B v B > v B v db ; ()) by the result of Theorem 4. In practice, the vector of constants is unknown and appropriate estimator of is needed. In many econometrics applications, a n-consistent preliminary estimator of is available. Denote b as a preliminary estimator of ; we look at the process bv n () = n( () b ) b Under H 2 ; bv n () = n b() n b () ) f " (F" B v B > v B v db ()) p lim n b () which depend on the preliminary estimation b. In the case that b is the OLS estimator of in (??), p lim n b = B v B > v B v db" where B" () is the limit of partial sum of " t. Thus, under H 2 ; sup V b n () ) sup f " (F" B v B > v B v d B f " (F" ())B " ()) ) in ad- The necessity of estimating introduces a drift component (p lim n b dition to the limit of n b(). We may generate critical values for the statistic sup Vn b () using simulation or resampling methods. Using the usual notation to signify the bootstrap samples and P for the probability conditional on the original sample, we may consider the following resampling procedure:

12 () First, obtain estimates () b and b by quantile regression and OLS regression respectively from KX y t = + x t + x t j j + " t : j= K Construct V b n () = n( () b ), b and obtain residuals but, bu t = y t b b x t, t = ; ::::; n; [Here, bu t may be constructed using any n-consistent estimator and p n-consistent estimator of.] (2) De ne bw t = (v t ; bu t ); v t = x t, apply a sieve (autoregression) estimation on bw t qx bwt = bb j bw t j + be t, t = q + ; ::::; n; and get tted residuals j= be t = bw t qx j= bb j bw t j, t = q + ; ::::; n; (3) Draw i.i.d. variables fe t g n t=q+ from the centered residuals be t and generate wt from e t using the tted autoregression: n q P n j=q+ be j w t = qx j= bb j w t j + e t, t = q + ; ::::; n; with w j = bw j for j = ; :::; q: (4) De ne w t =(v t ; u t ) in conformable with bw t = (v t ; bu t ); and generate x t from: x t = x t + v t, with x = x. Generate y t = b + b x t + u t Thus, we obtain the bootstrapped samples (y t ; x t ). (5) We now construct bootstrap version of b (), b, and b V n () using the bootstrapped samples (y t ; x t ). We rst calculate b () and b from quantile and OLS regression on then, we construct y t = + x t + KX j= K x t j j + " t ; bv n () = n( b () b ): 2

13 In the above procedure, to make the subsequent bootstrap test valid, we generate y t under the null hypothesis of constant. The limiting null distribution of the test statistics can then be approximated by repeating steps 2-5 many times. Let C t (; ) be the ()-th quantiles, i.e., h i P V b n () Ct (; ) = ; then the unit root hypothesis will be rejected at the ( ) level if b V n () C t (; ). Alternatively, instead of using resampling methods, we may directly simulate the Brownian motions f " (F " ()) B v B > v B v d B f " (F " ())B " : In particular, we may replace the regressions in step 5 by, say, directly approximating R B vb > v and R B vdb using X n 2 (yt y ) 2 and X (yt y ) n (" t ) t where y = n P yt ; and " t = " t ef " (); where F e " () is the quantile function of " t. Thus, the limiting null distribution of t n () can be approximated based on the following quantities p ( ) " X t (y t y ) 2 # =2 " X t t (y t y ) (u t ) Since we simply calculate sample moment and avoid solving the linear programming in each repetition in this alternative procedure, computationally this is faster. 4 Robust Inference on Cointegration # : To be written... References Alexander, C.O., 999, Optimal hedging using cointegration, Philosophical Transactions of the Royal Society A 357, pp Artzner, P., F. Delbaen, J. Eber, and D. Heath, 999, Coherent Measures of Risk, Mathematical Finance, 9,

14 Bassett, G., R. Koenker, and G. Kordas, 24, Choquet Risk and Portfolio Optimization, Journal of Financial Econometrics. Campbell, J. Y., and R. J. Shiller, 987, Cointegration and Tests of Present Value Models, Journal of Political Economy, 95(5), Campbell, J. Y., and R. J. Shiller, 988, The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors, Review of Financial Studies,, Campbell, J. Y., A. Lo, and MacKinley, 997, The Econometrics of Financial Market, Princeton University Press. Cerchi, M. and A. Havenner, 988, Cointegration and Stock Prices, Journal of Economic Dynamics and Control, 2, p Chowdhury, A.R., 99, Futures market e ciency: evidence from cointegration tests, The Journal of Future Markets :5, pp Cochrane, J.Y., 994, Permanent and Transitory components of GNP and Stock Prices, Quarterly Journal of Economics, 9, Engle, R.F. and C.W.J. Granger (987), Cointegration and Error Correction: Representation, Estimation, and Testing, Econometrica 55, Engle, R. F., and S. Manganelli, 999, CAViaR: Conditional autoregressive value at risk by regression quantiles, working paper, University of California, San Diego. Evans, G., 99, Pitfalls in Testing for Explosive Bubbles in Asset Prices, American Economic Review, 8, 4, Granger, C.W.J. (98), Some Properties of Time Series Data and Their Use in Econometric Model Speci cation, Journal of Econometrics, 2-3 Hall, A.D., H.M. Anderson and C.W.J. Granger, 992, A cointegration analysis of treasury bill yields, The Review of Economics and Statistics, 74, page Hannan, E. J., 97, Multiple Time Series, Wiley, New York. Hendry, D., 996, Dynamic Econometrics, Oxford University Press. Hsiao, C., 997, Cointegration and Dynamic Simultaneous Equations Models, Econometrica, 65, pp

15 Johansen, S. (995), Likelihood-based inference in cointegrated vector autoregressive models, Oxford: Oxford University Press. Hasan, M.N. and R. Koenker, 997, Robust rank tests of the unit root hypothesis, Econometrica 65, No., Koenker, R. and G. Bassett, 978, Regression Quantiles, Econometrica, V46, Koenker, R. and. Xiao, 22, Inference on the Quantile Regression Processes, Econometrica, 7, Lettau, M. and S. Ludvigson, 2, Consumption, Aggregate Wealth, and Expected Stock Returns, Journal of Finance, 56, Lucas, A., 997, Strategic and Tactical Asset Allocation and the E ect of Long- Run Equilibrium Relations", Research Memorandum , Vrije Universiteit Amsterdam. Park, J., and S. Hahn, 999, Cointegrating Regressions with Time Varying Coe - cients, Econometric Theory, 5, Phillips, P. C. B. and B. E. Hansen (99). Statistical inference in instrumental variables regression with I() processes, Review of Economic Studies 57, Phillips, P. C. B. and S. Ouliaris (99), Asymptotic properties of residual-based tests for cointegration, Econometrica, 58, Phillips, P.C.B. and V. Solo, 992, Asymptotics for linear processes, Annals of Statistics, 2, 97. Pindyck, R.S. and J.J. Rothemberg, 992, The comovement of stock prices, Quarterly Journal of Economics, pp Ploberger, W., and W. Kramer, 992, The CUSUM test with OLS residuals, Econometrica, 6, Robinson, P. M. (99), Automatic frequency domain inference on semiparametric and nonparametric models, Econometrica 59, Rockafellar, R., and S. Uryasev, 2, Optimization of Conditional VaR, Journal of Risk, 2, 2-4. Shin, Y., 994, A residual based test of the null of cointegration against the alternative of no cointegration, Econometric Theory,,

16 Sims, C., J. Stock and M. Watson, 99, Inference in linear time series models with some unit roots, Econometrica, 58, Stock, J. H. and M. W. Watson (988), Testing for common trends, Journal of American Statistical Association, 83, Taylor, J. W., A Quantile Regression Approach to Estimating the Distribution of Multiperiod Returns, Journal of Derivatives, Vol. 7, 64-75, 999. Xiao,., 2, Testing the null hypothesis of stationarity against an autoregressive unit root alternative, Journal of Time Series Analysis, 22, Xiao,. and P.C.B. Phillips, 22, A CUSUM test for Cointegration Using Regression Residuals, Journal of Econometrics, 8,