Inference and Forecasting for Fractional Autoregressive Integrated Moving Average Models, with an application to US and UK inflation

We discuss computational aspects of likelihood-based specification, estimation,inference, and forecasting of possibly nonstationary series with long memory. We use the \\ARFIMA$(p,d,q)$ model with deterministic regressors and we compare sampling characteristics of approximate and exact first-order asymptotic methods. We extend the analysis using a higher-order asymptotic method, suggested by \\cite{CoxRe.87}. Efficient computation and simulation allow us to apply parametric bootstrap inference as well. We investigate the relevance of the differences between the methods for the time-series analysis of monthly core consumer price inflation in the US and quarterly overall consumer price inflation in the UK. We concentrate on (stationarity) tests for the order of integration and on inference for out-of-sample forecasts of the price level.

[1]  P. Robinson,et al.  LONG AND SHORT MEMORY CONDITIONAL HETEROSKEDASTICITY IN ESTIMATING THE MEMORY PARAMETER OF LEVELS , 1999, Econometric Theory.

[2]  Richard T. Baillie,et al.  Analysing inflation by the fractionally integrated ARFIMA–GARCH model , 1996 .

[3]  Uwe Hassler,et al.  Long Memory in Inflation Rates: International Evidence , 1995 .

[4]  D. Dickey,et al.  RECOGNIZING OVERDIFFERENCED TIME SERIES , 1994 .

[5]  U. Hassler.  Sample autocorrelations of nonsationary fractionally integrated series , 1997 .

[6]  R. Baillie,et al.  Prediction in dynamic models with time-dependent conditional variances , 1992 .

[7]  Vance L. Martin,et al.  Indirect Estimation of Arfima and Varfima Models , 1999 .

[8]  Nalini Ravishanker,et al.  Exact Likelihood Function Forms for an ARFIMA Process , 1996 .

[9]  C. Z. Wei,et al.  Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes , 1988 .

[10]  Inference for Unstable Long-Memory Processes with Applications to Fractional Unit Root Autoregressions , 1995 .

[11]  P. Robinson,et al.  Advances in Econometrics: Time series with strong dependence , 1994 .

[12]  D. Cox,et al.  A note on the calculation of adjusted profile likelihood , 1993 .

[13]  J. Beran.  Statistical methods for data with long-range dependence , 1992 .

[14]  T. Bollerslev,et al.  Equity trading volume and volatility: Latent information arrivals and common long-run dependencies , 1999 .

[15]  Gene H. Golub,et al.  Matrix computations , 1983 .

[16]  J. MacKinnon,et al.  Bootstrap Testing in Nonlinear Models , 1999 .

[17]  Wilfredo Palma,et al.  State space modeling of long-memory processes , 1998 .

[18]  Wai Keung Li,et al.  On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity , 1997 .

[19]  Uwe Hassler,et al.  On the power of unit root tests against fractional alternatives , 1994 .

[20]  Anil K. Bera,et al.  ARCH Models: Properties, Estimation and Testing , 1993 .

[21]  Marius Ooms,et al.  A Package for Estimating, Forecasting and Simulating Arfima Models: Arfima package 1.0 for Ox , 1999 .

[22]  J. Geweke,et al.  Exact predictive densities for linear models with arch disturbances , 1989 .

[23]  Stanley E. Zin,et al.  Fractional integration with drift: estimation in small samples , 1997 .

[24]  P. Schmidt,et al.  On the Power of the KPSS Test of Stationarity Against Fractionally-Integrated Alternatives , 1996 .

[25]  Miguel A. Delgado,et al.  New methods for the analysis of long memory time series: application to Spanish inflation , 1994 .

[26]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[27]  Steve Beveridge,et al.  Estimating fractionally integrated time series models , 1993 .

[28]  Michael Hauser.  Maximum Likelihood Estimators for Arma and ARFIMA Models: A Monte Carlo Study , 1999 .

[29]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[30]  Jan F. Kiviet,et al.  Degrees of Freedom Adjustment for Disturbance Variance Estimators in Dynamic Regression Models , 1998 .

[31]  D. Cox,et al.  Inference and Asymptotics , 1994 .

[32]  On the effect of seasonal adjustment on the log-periodogram regression , 1997 .

[33]  Jan Beran,et al.  Local polynomial estimation with a FARIMA-GARCH error process , 2001 .

[34]  A. Harvey,et al.  Diagnostic Checking of Unobserved-Components Time Series Models , 1992 .

[35]  Jonathan R. M. Hosking,et al.  Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series , 1996 .

[36]  Shiqing Ling.  On the Probabilistic Properties of a Double Threshold ARMA Conditional Heteroskedastic Model , 1999 .

[37]  A. Scott.  Why is consumption so seasonal , 1995 .

[38]  R. Dahlhaus.  Efficient Location and Regression Estimation for Long Range Dependent Regression Models , 1995 .

[39]  M. King,et al.  Modified Likelihood and Related Methods for Handling Nuisance Parameters in the Linear Regression Model , 1998 .

[40]  Jan Beran,et al.  Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average Models , 1995 .

[41]  Nalini Ravishanker,et al.  BAYESIAN ANALYSIS OF VECTOR ARFIMA PROCESSES , 1997 .

[42]  Mitsuhiro Odaki.  On the invertibility of fractionally differenced ARIMA processes , 1993 .

[43]  W. D. Ray.  Time series models, second edition : Andrew C. Harvey, 1993, (Harvester-Wheatsheaf, New York) xviii + 308 pp., [UK pound]14.99 (paperback), ISBN 0-7450-1200-0 , 1993 .

[44]  P. Jeganathan.  ON ASYMPTOTIC INFERENCE IN COINTEGRATED TIME SERIES WITH FRACTIONALLY INTEGRATED ERRORS , 1999, Econometric Theory.

[45]  Philip Hans Franses,et al.  Long memory and level shifts: Re-analyzing inflation rates , 1999 .

[46]  Paul Newbold,et al.  LAGRANGE MULTIPLIER TESTS FOR FRACTIONAL DIFFERENCE , 1994 .

[47]  Fallaw Sowell.  Maximum likelihood estimation of stationary univariate fractionally integrated time series models , 1992 .

[48]  A. I. McLeod,et al.  Fractional time series modelling , 1986 .

[49]  J. Stock.  Unit roots, structural breaks and trends , 1986 .

[50]  G. T. Wilson,et al.  On the Use of Marginal Likelihood in Time Series Model Estimation , 1989 .

[51]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[52]  ASYMPTOTIC INFERENCE FOR NONSTATIONARY FRACTIONALLY INTEGRATED AUTOREGRESSIVE MOVING-AVERAGE MODELS , 2001, Econometric Theory.

[53]  Richard T. Baillie,et al.  Small sample bias in conditional sum-of-squares estimators of fractionally integrated ARMA models , 1993 .

[54]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[55]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.

[56]  James G. MacKinnon,et al.  THE SIZE DISTORTION OF BOOTSTRAP TESTS , 1999, Econometric Theory.

[57]  P. Robinson,et al.  Testing of unit root and other nonstationary hypotheses in macroeconomic time series , 1996 .