Threshold quantile autoregressive models

This article studies estimation and asymptotic properties of Threshold Quantile Autoregressive processes. In particular, we show the consistency of the threshold and slope parameter estimators for each quantile and regime, and derive the asymptotic normality of the slope parameter estimators. A Monte Carlo experiment shows that the standard ordinary least squares estimation method is not able to detect important nonlinearities produced in the quantile process. The empirical study concentrates on modelling the dynamics of the conditional distribution of unemployment growth after the second world war. The results show evidence of important heterogeneity associated with unemployment and strong asymmetric persistence of unemployment growth in the higher quantiles.

[1]  H. Tong,et al.  Threshold Autoregression, Limit Cycles and Cyclical Data , 1980 .

[2]  B. Hansen.  Sample Splitting and Threshold Estimation , 2000 .

[3]  Jianqing Fan,et al.  Quantile autoregression. Commentary , 2006 .

[4]  Roger Koenker,et al.  Quantile Autoregression , 2006 .

[5]  Zhijie Xiao,et al.  Testing for parameter stability in quantile regression models , 2008 .

[6]  S. Geer.  Empirical Processes in M-Estimation , 2000 .

[7]  Ruey S. Tsay,et al.  Testing and Modeling Threshold Autoregressive Processes , 1989 .

[8]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

[9]  Cathy W. S. Chen,et al.  On a threshold heteroscedastic model , 2006 .

[10]  J. Coakley,et al.  Evaluating the Persistence and Structuralist Theories of Unemployment from a Nonlinear Perspective , 2000 .

[11]  Herold Dehling,et al.  Empirical Process Techniques for Dependent Data , 2002 .

[12]  V. Chernozhukov,et al.  QUANTILE AND PROBABILITY CURVES WITHOUT CROSSING , 2007, 0704.3649.

[13]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[14]  Mark P. Taylor,et al.  The Role of Asymmetries and Regime Shifts in the Term Structure of Interest Rates , 2005 .

[15]  D. Andrews.  Tests for Parameter Instability and Structural Change with Unknown Change Point , 1993 .

[16]  Lucio Sarno,et al.  Nonlinearity in Deviations from Uncovered Interest Parity: An Explanation of the Forward Bias Puzzle , 2006, SSRN Electronic Journal.

[17]  Holger Dette,et al.  Non‐crossing non‐parametric estimates of quantile curves , 2008 .

[18]  L. Glosten,et al.  On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks , 1993 .

[19]  Politis,et al.  [Springer Series in Statistics] Subsampling || Subsampling for Stationary Time Series , 1999 .

[20]  Gary Koop,et al.  Do recessions permanently change output , 1993 .

[21]  D. Andrews.  Tests for Parameter Instability and Structural Change with Unknown Change Point , 1993 .

[22]  Zhongjun Qu.  Testing for Structural Change in Regression Quantiles , 2007 .

[23]  Roger Koenker,et al.  An Empirical Quantile Function for Linear Models with | operatornameiid Errors , 1982 .

[24]  E. Mammen.  Nonparametric regression under qualitative smoothness assumptions , 1991 .

[25]  K. Chan,et al.  Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model , 1993 .

[26]  Qi Li,et al.  On Hotelling's Approach to Hypothesis Testing When a Nuisance Parameter Is Present Only under the Alternative , 2007 .

[27]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[28]  Hung Man Tong,et al.  Threshold models in non-linear time series analysis. Lecture notes in statistics, No.21 , 1983 .

[29]  D. Ruppert,et al.  Trimmed Least Squares Estimation in the Linear Model , 1980 .

[30]  Roger Koenker,et al.  Inequality constrained quantile regression , 2005 .

[31]  Yuzhi Cai.  Forecasting for quantile self-exciting threshold autoregressive time series models , 2010 .

[32]  S. R. Jammalamadaka,et al.  Empirical Processes in M-Estimation , 2001 .

[33]  C. Nelson,et al.  Trends and random walks in macroeconmic time series: Some evidence and implications , 1982 .

[34]  Jerry Coakley,et al.  Numerical Issues in Threshold Autoregressive Modeling of Time Series , 2003 .

[35]  H. Tong,et al.  ON ESTIMATING THRESHOLDS IN AUTOREGRESSIVE MODELS , 1986 .

[36]  Xuming He.  Quantile Curves without Crossing , 1997 .

[37]  Kengo Kato,et al.  Asymptotics for argmin processes: Convexity arguments , 2009, J. Multivar. Anal..

[38]  Christian Hansen,et al.  Instrumental quantile regression inference for structural and treatment effect models , 2006 .

[39]  Roger Koenker,et al.  An empirical quantile function for linear models with iid errors , 1981 .

[40]  P. Massart,et al.  Invariance principles for absolutely regular empirical processes , 1995 .

[41]  D. Andrews,et al.  Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative , 1992 .

[42]  R. Koenker,et al.  Regression Quantiles , 2007 .

[43]  B. Hansen,et al.  Inference in TAR Models , 1997 .

[44]  J. Zakoian.  Threshold heteroskedastic models , 1994 .

[45]  Kung-Sik Chan,et al.  Limiting properties of the least squares estimator of a continuous threshold autoregressive model , 1998 .

[46]  J. Stander,et al.  Quantile self‐exciting threshold autoregressive time series models , 2007 .

[47]  Bruce E. Hansen,et al.  Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis , 1996 .

[48]  M. Kosorok.  Introduction to Empirical Processes and Semiparametric Inference , 2008 .

[49]  R. Davies.  Hypothesis testing when a nuisance parameter is present only under the alternative , 1977 .

[50]  Mehmet Caner.  A NOTE ON LEAST ABSOLUTE DEVIATION ESTIMATION OF A THRESHOLD MODEL , 2002, Econometric Theory.

[51]  Roger Koenker,et al.  Conditional Quantile Estimation and Inference for Arch Models , 1996, Econometric Theory.

[52]  J. Coakley,et al.  Testing for Sign and Amplitude Asymmetries Using Threshold Autoregressions , 2006 .

[53]  Paul Waltman,et al.  A Threshold Model , 1974 .

[54]  M. A. Arcones,et al.  Central limit theorems for empirical andU-processes of stationary mixing sequences , 1994 .

[55]  R. Davies.  Hypothesis testing when a nuisance parameter is present only under the alternative , 1977 .

[56]  T. Teräsvirta.  Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models , 1994 .

[57]  Cathy W. S. Chen,et al.  Semi-parametric quantile estimation for double threshold autoregressive models with heteroskedasticity , 2013, Comput. Stat..

[58]  Jose Olmo,et al.  A Nonlinear Threshold Model for the Dependence of Extremes of Stationary Sequences , 2012 .