The quantilogram: With an application to evaluating directional predictability

Abstract We propose a new diagnostic tool for time series called the quantilogram. The tool can be used formally and we provide the inference tools to do this under general conditions, and it can also be used as a simple graphical device. We apply our method to measure directional predictability and to test the hypothesis that a given time series has no directional predictability. The test is based on comparing the correlogram of quantile hits to a pointwise confidence interval or on comparing the cumulated squared autocorrelations with the corresponding critical value. We provide the distribution theory needed to conduct inference, propose some model free upper bound critical values, and apply our methods to S&P500 stock index return data. The empirical results suggest some directional predictability in returns. The evidence is strongest in mid range quantiles like 5–10% and for daily data. The evidence for predictability at the median is of comparable strength to the evidence around the mean, and is strongest at the daily frequency.

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

[2]  J. Powell,et al.  Nonparametric and Semiparametric Methods in Econometrics and Statistics , 1993 .

[3]  T. Nijman,et al.  Temporal Aggregation of GARCH Processes. , 1993 .

[4]  R. Koenker,et al.  Asymptotic Theory of Least Absolute Error Regression , 1978 .

[5]  W. Andrew,et al.  LO, and A. , 1988 .

[6]  Jana Jurečková,et al.  Asymptotic Relations of $M$-Estimates and $R$-Estimates in Linear Regression Model , 1977 .

[7]  A. Lo,et al.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1996, Macroeconomic Dynamics.

[8]  Ralf Runde,et al.  The asymptotic null distribution of the Box-Pierce Q-statistic for random variables with infinite variance An application to German stock returns , 1997 .

[9]  D. Pollard.  Convergence of stochastic processes , 1984 .

[10]  W. Steiger,et al.  Least Absolute Deviations: Theory, Applications and Algorithms , 1984 .

[11]  Jushan Bai,et al.  Weak Convergence of the Sequential Empirical Processes of Residuals in ARMA Models , 1994 .

[12]  T. Mikosch,et al.  Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process , 2000 .

[13]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[14]  Anthony S. Tay,et al.  Evaluating Density Forecasts with Applications to Financial Risk Management , 1998 .

[15]  H. Koul.  A weak convergence result useful in robust autoregression , 1991 .

[16]  Enno Mammen,et al.  Estimating Semiparametric Arch (∞) Models by Kernel Smoothing Methods , 2003 .

[17]  Neil Shephard,et al.  Dynamics of Trade-by-Trade Price Movements: Decomposition and Models , 1999 .

[18]  M. Hallin,et al.  Rank-based autoregressive order identification , 1999 .

[19]  R. Engle,et al.  CAViaR , 1999 .

[20]  D. Pollard.  Asymptotics for Least Absolute Deviation Regression Estimators , 1991, Econometric Theory.

[21]  D. Andrews.  Generic Uniform Convergence , 1992, Econometric Theory.

[22]  S. Lahiri.  Resampling Methods for Dependent Data , 2003 .

[23]  R. Koenker,et al.  Pessimistic Portfolio Allocation and Choquet Expected Utility , 2004 .

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

[25]  Jonathan B. Hill.  Gaussian Tests of "Extremal White Noise" for Dependent, Heterogeneous, Heavy Tailed Time Series with an Application ∗ , 2005 .

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

[27]  Yongmiao Hong,et al.  Are the directions of stock price changes predictable? A generalized cross-spectral approach , 2004 .

[28]  Jean-Marie Dufour,et al.  Generalized Portmanteau Statistics and Tests of Randomness , 1985 .

[29]  A. Cowles,et al.  Some A Posteriori Probabilities in Stock Market Action , 1937 .

[30]  C. Gutenbrunner,et al.  Regression Rank Scores and Regression Quantiles , 1992 .

[31]  H. Koul.  Weighted Empirical Processes in Dynamic Nonlinear Models , 2002 .

[32]  Richard A. Davis,et al.  The sample autocorrelations of heavy-tailed processes with applications to ARCH , 1998 .

[33]  I. Mizera,et al.  Generalized run tests for heteroscedastic time series , 1998 .

[34]  Galen R. Shorack,et al.  The weighted empirical process of row independent random variables with arbitrary distribution functions , 1979 .

[35]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .