Bayesian analysis of periodic asymmetric power GARCH models

Abdelhakim Aknouche; Nacer Demmouche; Stefanos Dimitrakopoulos; Nassim Touche

doi:10.1515/snde-2018-0112

Abstract

In this paper, we set up a generalized periodic asymmetric power GARCH (PAP-GARCH) model whose coefficients, power, and innovation distribution are periodic over time. We first study its properties, such as periodic ergodicity, finiteness of moments and tail behavior of the marginal distributions. Then, we develop an MCMC algorithm, based on the Griddy-Gibbs sampler, under various distributions of the innovation term (Gaussian, Student-t, mixed Gaussian-Student-t). To assess our estimation method we conduct volatility and Value-at-Risk forecasting. Our model is compared against other competing models via the Deviance Information Criterion (DIC). The proposed methodology is applied to simulated and real data.

Keywords: Bayesian forecasting; Deviance Information Criterion; Griddy-Gibbs; periodic asymmetric power GARCH model; probability properties; Value at Risk

MSC 2010: AMS 2000 Primary 62M10; Secondary 60F99

Appendix A

Proofs

Proof of Theorem 1

(i) Sufficiency: Equation (3) may be cast in the following system of S recurrence equations

(20)YnS+v=AnS+vY(n−1)S+v+BnS+v,n∈Z,v∈{0,…,S−1},

where AnS+v=∏i=0S−1AnS+v−i and BnS+v=∑j=0S−1∏i=0j−1AnS+v−iBnS+v−j, so {(AnS+v,BnS+v),n∈Z} is iid for all v∈{0,…,S−1}. The top Lyapunov exponent γv(S) associated with (20) is given for all v∈{0,…,S−1} by (Bougerol and Picard, 1992)

(21)γv(S)=inf{1nElog⁡‖AnS+vA(n−1)S+v…AS+v‖,n≥1}=inf{1nElog⁡‖AnS+vAnS+v−1…Av+1‖, n≥1},=limn→∞1nlog⁡‖AnS+vAnS+v−1…Av+1‖ a.s.

Since Elog⁡|ηv|δv<∞ for all 0≤v≤S−1, it follows that Elog+⁡‖Av‖<∞ and Elog+⁡‖Bv‖<∞. Therefore, by Theorem 2.5 of Bougerol and Picard (1992), equation (20) admits a unique nonanticipative strictly stationary and ergodic solution {YnS+v,n∈Z} provided that γv(S)<0. The solution is given for all v∈{0,…,S−1} by

(22)YnS+v=∑j=0∞∏i=0j−1A(n−i)S+vB(n−j)S+v,n∈Z,v∈{0,…,S−1},

where the series in equality (22), which is exactly (6), converges absolutely a.s. This shows that {Yt,t∈Z} is the unique causal strictly periodically stationary and periodically ergodic solution of (3). Note finally that by a sandwitching argument, it is easy to see that for all v∈{0,…,S−1}

γv(S)=limn→∞1nlog⁡‖AnS+vAnS+v−1…Av+1‖=limn→∞1nlog⁡‖AnSAnS−1…A1‖:=γ(S).

Necessity: Assume that model (3) admits a nonanticipative strictly periodically stationary solution {Yt,t∈Z}. From the non-negativity of the coefficients of A_t in (3) it follows that for all k > 1,

Yv≥∑j=0k∏i=0j−1Av−iBv−j,a.s.,

implying that the series ∑j=0∞∏i=0j−1Av−iBv−j converges a.s. Therefore,

∏i=0j−1Av−iBv−j→0,a.s. as j→∞,

from which we have to show

(23)∏i=0j−1Av−i→0,a.s. as j→∞.

This holds whenever

(24)limj→∞∏i=0j−1Av−iem=0,a.s. for all 1≤m≤r,

where r=p+2q−2 and (em)1≤m≤r is the canonical basis of ℝ^r. Since A_t has the same “sparsity” as the matrix A_t in Pan, Wang, and Tong (2008, p. 373), then (24) follows from their results using similar arguments (see also Aknouche and Bibi (2009) for the particular P-GARCH case).

(ii) Since {At,t∈Z} is nonnegative then

(25)γS(A)≥γS(β):=log⁡ρ(∏v=0S−1βS−v).

If (3) has a strictly periodically stationary solution, then γS(A)<0. In view of (25), it follows that γS(β)<0 establishing (7). ■

Proof of Theorem 2

(i) The proof is similar to that of Lemma 2.3 of Berkes, Horvàth, and Kokoskza (2003). First, we have to show that if γS(A)<0 then there is δ > 0 and n₀ such that

(26)E(‖An0SAn0S−1…A1‖δ)<1.

Since γS(A)=infn∈N∗{1nE(log⁡‖AnSAnS−1…A1‖)} is strictly negative, there is a positive integer n₀ such that

E(log⁡‖An0SAn0S−1…A1‖)<0.

On the other hand, working with a multiplicative norm and by the ipd_S property of the sequence {At,t∈Z} we have

E(‖An0SAn0S−1…A1‖)=‖E(An0SAn0S−1…A1)‖=‖E(ASAS−1…A1)n0‖≤‖E(ASAS−1…A1)‖n0<∞.

Let f(x)=E(‖An0SAn0S−1…A1‖x). Since f′(0)=E(log⁡‖An0SAn0S−1…A1‖)<0, f(x) decrease in a neighborhood of 0 and since f(0)=1, it follows that there exists 0 < δ < 1 such that (26) holds. Now from (6) we have for some v∈{1,…,S}

‖Yv‖≤∑k=1∞‖∏j=0k−1Av−j‖‖Bv−k‖+‖Bv‖.

Since 0 < κ < 1, then

‖Yv‖κ≤∑k=1∞‖∏j=0k−1Av−j‖κ‖Bv−k‖κ+‖Bv‖κ,

which, by the independence of A_v−j and B_v−k for j < k, implies that

E‖Yv‖κ≤∑k=1∞E(‖∏j=0k−1Av−j‖κ)E(‖Bv−k‖κ)+E(‖Bv‖κ)≤B(κ)∑k=1∞E(‖∏j=0k−1Av−j‖κ)+E(‖Bv‖κ),

where B(κ)=max0≤v≤S−1E(‖Bv−k‖κ). In view of (26) there exist a_v > 0 and 0 < b_v < 1 such that

E(‖∏j=0k−1Av−j‖κ)≤avbvk≤abk,

where abk=max0≤v≤S−1{avbvk}. This proves that E‖Yv‖κ<∞, showing (8).

(ii) Define {Y~t,t∈Z} by

(27){Y~t=AtY~t−1+Btt≥1Y~t=0t≤0,

and let Y(v)(0≤v≤S−1) be a random variable having the same distribution as the term YnS+v of the unique periodically stationary solution given by (22). It is clear that Y~nS+v→LY(v) as n → ∞. Let m = 2. From the weak convergence theory (Billingsley 1968), to show that E(vec(Y(v)Y(v)′)) is finite for all v, it is sufficient to show that liminfn→∞E(vec(Y~nS+v′Y~nS+v))<∞ for all v. Set VnS+v=E(vec(Y~nS+v′Y~nS+v)). From (27) we get the following first-order S-periodic difference equation

(28)VnS+v=E(Av⊗2)VnS+v−1+[E(Av⊗Bv)+E(Bv⊗Av)]E(Y~nS+v)+vec(E(BvBv′)),

where E(At⊗2), E(At⊗Bt) and vec(E(BtBt′)) are finite S-periodic matrices in t. Since, the last two terms of the right-hand side of (27) are bounded, it follows that limn→∞VnS+v exists for every 1 ≤ v ≤ S whenever (9) holds, which completes the proof for m = 2. For general m, the proof is similar. ■

Proof of Theorem 3

The proof is very similar to that of Corollary 3.5 of Basrak, Davis, and Mikosch (2002). ■

A

MCMC diagnostic tools

In order to assess the convergence of the proposed algorithm, we use some MCMC diagnostic tools, such as the autocorrelation of posterior draws, the Relative Numerical Inefficiency (RNI, Geweke 1989) and the Numerical Standard Error (NSE, Geweke 1989). The autocorrelations of parameter draws indicate how the posterior draws mix. The RNI measures the degree of the inefficiency due to the serial correlation of the MCMC draws. It is given by

RNI=1+2∑h=1BK(hB)ρ^h,

where B = 500 is the bandwidth, K(.) is the Parzen kernel (e.g. Kim, Shephard, and Chib 1998) and ρ^h is the sample autocorrelation for the lag h of the parameter draws.

The NSE is the square-root of the estimated asymptotic variance of the MCMC estimator. It is given by

NSE=1L(γ^0+2∑h=1BK(hB)γ^h),

where γ^h is the sample autocovariance at lag h of the parameter draws.

Acknowledgement

The authors are deeply grateful to the Editor-in-Chief and two referees for their relevant comments and suggestions that were very helpful in improving the first draft.

References

Aknouche, A. 2017. “Periodic Autoregressive Stochastic Volatility.” Statistical Inference for Stochastic Processes 20: 139–177.10.1007/s11203-016-9139-zSearch in Google Scholar

Aknouche, A., and E. Al-Eid. 2012. “Asymptotic Inference of Unstable Periodic ARCH Processes.” Statistical Inference for Stochastic Processes 15: 61–79.10.1007/s11203-011-9063-1Search in Google Scholar

Aknouche, A., and A. Bibi. 2009. “Quasi-Maximum Likelihood Estimation of Periodic GARCH and Periodic ARMA-GARCH Processes.” Journal of Time Series Analysis 30: 19–46.10.1111/j.1467-9892.2008.00598.xSearch in Google Scholar

Aknouche, A., and N. Touche. 2015. “Weighted Least Squares-Based Inference for Stable and Unstable Threshold Power ARCH Processes.” Statistics & Probability Letters 97: 108–115.10.1016/j.spl.2014.11.011Search in Google Scholar

Aknouche, A., E. Al-Eid, and N. Demouche. 2018. “Generalized Quasi-Maximum Likelihood Inference for Periodic Conditionally Heteroskedastic Models.” Statistical Inference for Stochastic Processes 21: 485–511.10.1007/s11203-017-9160-xSearch in Google Scholar

Ambach, D., and C. Croonenbroeck. 2015. “Obtaining Superior Wind Power Predictions from a Periodic and Heteroskedastic Wind Power Prediction Tool.” In Stochastic Models, Statistics and Their Applications, Springer Proceedings in Mathematics & Statistics Vol. 122, 225–232. Cham: Springer.10.1007/978-3-319-13881-7_25Search in Google Scholar

Ambach, D., and W. Schmid. 2015. “Periodic and Long Range Dependent Models for High Frequency Wind Speed Data.” Energy 82: 277–293.10.1016/j.energy.2015.01.038Search in Google Scholar

Ardia, D. 2008. “Bayesian Estimation of a Markov-Switching Threshold Asymmetric GARCH Model with Student-t Innovations.” The Econometrics Journal 12: 105–126.10.1111/j.1368-423X.2008.00253.xSearch in Google Scholar

Basrak, B., R. A. Davis, and T. Mikosch. 2002. “Regular Variation of GARCH Processes.” Stochastic Processes and Their Applications 99: 95–115.10.1016/S0304-4149(01)00156-9Search in Google Scholar

Bauwens, L., and M. Lubrano. 1998. “Bayesian Inference on GARCH Models Using Gibbs Sampler.” Journal of Econometrics 1: 23–46.10.1111/1368-423X.11003Search in Google Scholar

Bauwens, L., A. Dufays, and J. V. K. Rombouts. 2014. “Marginal Likelihood for Markov-Switching and Change-Point GARCH Models.” Journal of Econometrics 178: 508–522.10.1016/j.jeconom.2013.08.017Search in Google Scholar

Berkes, I., L. Horvàth, and P. Kokoskza. 2003. “GARCH Processes: Structure and Estimation.” Bernoulli 9: 201–227.10.3150/bj/1068128975Search in Google Scholar

Billingsley, P. 1968. Probability and Measure. New York: Wiley.Search in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–327.10.1016/0304-4076(86)90063-1Search in Google Scholar

Bollerslev, T., and E. Ghysels. 1996. “Periodic Autoregressive Conditional Heteroskedasticity.” Journal of Business & Economic Statistics 14: 139–152.10.2307/1392425Search in Google Scholar

Bollerslev, T., J. Cai, and F. M. Song. 2000. “Intraday Periodicity, Long Memory Volatility, and Macroeconomic Announcement Effects in the US Treasury Bond Market.” Journal of Empirical Finance 7: 37–55.10.1016/S0927-5398(00)00002-5Search in Google Scholar

Bougerol, P., and N. Picard. 1992. “Stationarity of GARCH Processes and some Nonnegative Time Series.” Journal of Econometrics 52: 115–127.10.1016/0304-4076(92)90067-2Search in Google Scholar

Chan, J. C. C., and A. L. Grant. 2016. “On the Observed-Data Deviance Information Criterion for Volatility Modeling.” Journal of Financial Econometrics 14: 772–802.10.1093/jjfinec/nbw002Search in Google Scholar

Chen, C. W. S., and M. K. P. So. 2006. “On a Threshold Heteroscedastic Model.” International Journal of Forecasting 22: 73–89.10.1016/j.ijforecast.2005.08.001Search in Google Scholar

Ding, Z., C. W. J. Granger, and R. F. Engle. 1993. “A Long Memory Property of Stock Market Returns and a New Model.” Journal of Empirical Finance 1: 83–106.10.1016/0927-5398(93)90006-DSearch in Google Scholar

Engle, R. F. 1982. “Autoregressive Conditional Heteroskedasticity with Estimates of Variance of U.K. Inflation.” Econometrica 50: 987–1008.10.2307/1912773Search in Google Scholar

Francq, C., and J. M. Zakoïan. 2008. “Deriving the Autocovariances of Powers of Markov-Switching GARCH Models, with Applications to Statistical Inference.” Computational Statistics & Data Analysis 52: 3027–3046.10.1016/j.csda.2007.08.003Search in Google Scholar

Francq, C., and J. M. Zakoïan. 2013. “Optimal Predictions of Powers of Conditionally Heteroskedastic Processes.” Journal of Royal Statistical Society B75: 345–367.10.1111/j.1467-9868.2012.01045.xSearch in Google Scholar

Francq, C., and J. M. Zakoïan. 2019. GARCH Models: Structure, Statistical Inference and Financial Applications. 2nd ed. Hoboken, NJ: John Wiley.10.1002/9781119313472Search in Google Scholar

Franses, P. H., and R. Paap. 2000. “Modeling Day-of-the-Week Seasonality in the S&P 500 Index.” Applied Financial Economics 10: 483–488.10.1080/096031000416352Search in Google Scholar

Geweke, J. 1989. “Bayesian Inference in Econometric Models Using Monte Carlo Integration.” Econometrica 57: 1317–1339.10.2307/1913710Search in Google Scholar

Granger, C. W. J. 2005. “The Past and Future of Empirical Finance: Some Personal Comments.” Journal of Econometrics 129: 35–40.10.1016/j.jeconom.2004.09.002Search in Google Scholar

Haas, M. 2009. “Persistence in Volatility, Conditional Kurtosis, and the Taylor Property in Absolute Value GARCH Processes.” Statistics & Probability Letters 79: 1674–1683.10.1016/j.spl.2009.04.017Search in Google Scholar

Haas, M., S. Mittnik, and M. S. Paolella. 2004. “A New Approach to Markov Switching GARCH Models.” Journal of Financial Econometrics 4: 493–530.10.1093/jjfinec/nbh020Search in Google Scholar

Hamadeh, T., and J. M. Zakoïan. 2011. “Asymptotic Properties of LS and QML Estimators for a Class of Nonlinear GARCH Processes.” Journal of Statistical Planning and Inference 141: 488–507.10.1016/j.jspi.2010.06.026Search in Google Scholar

Hoogerheide, L., and H. K. van Dijk. 2010. “Bayesian Forecasting of Value at Risk and Expected Shortfall Using Adaptive Importance Sampling.” International Journal of Forecasting 26: 231–247.10.1016/j.ijforecast.2010.01.007Search in Google Scholar

Hwang, S. Y., and I. V. Basawa. 2004. “Stationarity and Moment Structure for Box-Cox Transformed Threshold GARCH(1, 1) Processes.” Statistics and Probability Letters 68: 209–220.10.1016/j.spl.2003.08.016Search in Google Scholar

Kim, S., N. Shephard, and S. Chib. 1998. “Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models.” The Review of Economic Studies 65: 361–393.10.1111/1467-937X.00050Search in Google Scholar

Osborn, D. R., C. S. Savva, and L. Gill. 2008. “Periodic Dynamic Conditional Correlations between Stock Markets in Europe and the US.” Journal of Financial Econometrics 6: 307–325.10.1093/jjfinec/nbn005Search in Google Scholar

Pan, J., H. Wang, and H. Tong. 2008. “Estimation and Tests for Power-Transformed and Threshold GARCH Models.” Journal of Econometrics 142: 352–378.10.1016/j.jeconom.2007.06.004Search in Google Scholar PubMed PubMed Central

Regnard, N., and J. M. Zakoïan. 2011. “A Conditionally Heteroskedastic Model with Time-Varying Coefficients for Daily Gas Spot Prices.” Energy Economics 33: 1240–1251.10.1016/j.eneco.2011.02.004Search in Google Scholar

Ritter, C., and M. A. Tanner. 1992. “Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler.” Journal of the American Statistical Association 87: 861–870.10.1080/01621459.1992.10475289Search in Google Scholar

Rossi, E., and D. Fantazani. 2015. “Long Memory and Periodicity in Intraday Volatility.” Journal of Financial Econometrics 13: 922–961.10.1093/jjfinec/nbu006Search in Google Scholar

Smith, M. S. 2010 . “Bayesian Inference for a Periodic Stochastic Volatility Model of Intraday Electricity Prices.” In: Kneib T., Tutz G. (eds) Statistical Modelling and Regression Structures. Heidelberg, 353–376. Berlin: Physica-Verlag HD.10.1007/978-3-7908-2413-1_19Search in Google Scholar

Spiegelhalter, D. J., N. G. Best, B. P. Carlin, and A. van der Linde. 2002. “Bayesian Measures of Model Complexity and Fit.” Journal of Royal Statistical Society B64: 583–639.10.1111/1467-9868.00353Search in Google Scholar

Tsay, R. S. 2010. Analysis of Financial Time Series: Financial Econometrics, 3rd ed. New York: Wiley.10.1002/9780470644560Search in Google Scholar

Tsiakas, I. 2006. “Periodic Stochastic Volatility and Fat Tails.” Journal of Financial Econometrics 4: 90–135.10.1093/jjfinec/nbi023Search in Google Scholar

Xia, Q., H. Wong, J. Liu, and R. Liang. 2017. “Bayesian Analysis of Power-Transformed and Threshold GARCH Models: A Griddy-Gibbs Sampler Approach.” Computational Economics 50: 353–372.10.1007/s10614-016-9588-xSearch in Google Scholar

Ziel, F., R. Steinert, and S. Husmann. 2015. “Efficient Modeling and Forecasting of Electricity Spot Prices.” Energy Economics 47: 98–111.10.1016/j.eneco.2014.10.012Search in Google Scholar

Ziel, F., C. Croonenbroeck, and D. Ambach. 2016. “Forecasting wind Power Modeling Periodic and Non-Linear Effects Under Conditional Heteroscedasticity.” Applied Energy 177: 285–297.10.1016/j.apenergy.2016.05.111Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2018-0112).

Published Online: 2019-10-19

You are currently not able to access this content.

Supplementary Material