Conditional Sum of Squares Estimation of Multiple Frequency Long Memory Models

Beaumont, Paul and Smallwood, Aaron (2019): Conditional Sum of Squares Estimation of Multiple Frequency Long Memory Models.

Abstract

We review the multiple frequency Gegenbauer autoregressive moving average model, which is able to reproduce a wide range of autocorrelation functions. Extending the result of Chung (1996a), we propose the asymptotic distributions for a conditional sum of squares estimator of the model parameters. The parameters that determine the cycle lengths are asymptotically independent, converging at rate T for finite cycles. This result does not hold generally, most notably for the differencing parameters associated with the cycle lengths. Remaining parameters are typically not independent and converge at the standard rate of T1/2. We present simulation results to explore small sample properties of the estimator, which strongly support most distributional results while also highlighting areas that merit additional exploration. We demonstrate the applicability of the theory and estimator with an application to IBM trading volume.

• Item Type: MPRA Paper
• Language: English
• Keywords: k-factor Gegenbauer processes, Asymptotic distributions, ARFIMA, Conditional sum of squares
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C40 - General; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics; G - Financial Economics > G1 - General Financial Markets; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates

References

Adenstedt, R. K. (1974). On large-sample estimation for the mean of a stationary random sequence. The Annals of Statistics, 2(6):1095–1107.

Andrews, D. W. and Sun, Y. (2004). Adaptive local polynomial Whittle estimation of long-range dependence. Econometrica, 72(2):569–614.

Asai, M., Peiris, S., McAleer, M., and Allen, D. (2018). Cointegrated dynamics for a generalized long memory process. Technical Report EI2018-32, Econometric Institute Research Papers.

Beaumont, P. M. and Smallwood, A. D. (2019). Inference for likelihood-based estimators of generalized long-memory processes. Working Paper, University of Texas-Arlington.

Caporale, G. M. and Gil-Alana, L. (2014). Long-run and cyclical dynamics in the US stock market. Journal of Forecasting, 33(2):147–161.

Caporale, G. M. and Gil-Alana, L. A. (2011). Multi-factor Gegenbauer processes and European inflation rates. Journal of Economic Integration, pages 386–409.

Chan, N. H. and Wei, C. Z. (1988). Limiting distributions of least squares estimates of unstable autoregressive processes. The Annals of Statistics, 16(1):367–401.

Cheung, Y.-W. and Diebold, F. X. (1994). On maximum likelihood estimation of the differencing parameter of fractionally-integrated noise with unknown mean. Journal of Econometrics, 62(2):301 – 316.

Chung, C.-F. (1996a). Estimating a generalized long memory process. Journal of Econometrics, 73(1):237 – 259.

Chung, C.-F. (1996b). A generalized fractionally integrated autoregressive moving average pro- cess. Journal of Time Series Analysis, 17:111–140.

Chung, C.-F. and Baillie, R. T. (1993). Small sample bias in conditional sum-of-squares estimators of fractionally integrated ARMA models. Empirical Economics, 18(4):791–806.

Diongue, A. K. and Ndongo, M. (2016). The k-factor GARMA process with infinite variance innovations. Communications in Statistics-Simulation and Computation, 45(2):420–437.

Dissanayake, G. S., Peiris, M. S., and Proietti, T. (2018). Fractionally differenced Gegenbauer processes with long memory: A review. Statistical Science, 33(3):413–426.

Gil-Alanaa, L. A. (2007). Testing the existence of multiple cycles in financial and economic time series. Annals of Economics & Statistics, 8(1):1–20.

Giraitis, L., Hidalgo, J., and Robinson, P. M. (2001). Gaussian estimation of parametric spectral density with unknown pole. The Annals of Statistics, 29(4):987–1023.

Gradshteyn, I. S. and Ryzhik, I. M. (1980). Tables of integrals, series, and products. Academic Press, New York, NY, 4th edition.

Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1(1):15–24.

Gray, H. L., Zhang, N., and Woodward, W. A. (1989). On generalized fractional processes. Journal of Time Series Analysis, 10:233–257.

Hidalgo, J. and Soulier, P. (2004). Estimation of the location and exponent of the spectral singularity of a long memory process. Journal of Time Series Analysis, 25(1):55–81.

Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68:165–76.

Leschinski, C. and Sibbertsen, P. (2019). Model order selection in periodic long memory models. Econometrics and Statistics, 9:78 – 94.

Lu, Z. and Guegan, D. (2011). Estimation of time-varying long memory parameter using wavelet method. Communications in Statistics—Simulation and Computation, 40(4):596–613.

McElroy, T. S. and Holan, S. H. (2016). Computation of the autocovariances for time series with multiple long-range persistencies. Computational Statistics & Data Analysis, 101:44–56.

Peiris, M. and Asai, M. (2016). Generalized fractional processes with long memory and time dependent volatility revisited. Econometrics, 4(4):37.

Ramachandran, R. and Beaumont, P. (2001). Robust estimation of GARMA model parameters with an application to cointegration among interest rates of industrialized countries. Computational Economics, 17(2/3):179–201.

Smallwood, A. D. and Norrbin, S. C. (2006). Generalized long memory processes, failure of cointegration tests and exchange rate dynamics. Journal of Applied Econometrics, 21(4):409–417.

Woodward, W. A., Cheng, Q. C., and Gray, H. L. (1998). A k-factor GARMA long-memory model. Journal of Time Series Analysis, 19(485-504).

Yajima, Y. (1991). Asymptotic properties of the LSE in a regression model with long-memory stationary errors. The Annals of Statistics, 19(1):158–177.