Predictive, finite-sample model choice for time series under stationarity and non-stationarity. (English) Zbl 1432.62323

The paper is concerned with the bias-variance trade-off problem in the context of stationarity versus non-stationarity modeling for prediction. The authors find that a simple stationary model can often perform better in terms of finite sample prediction than a more complicated non-stationary one.
The authors illustrate that phenomenon first at a time varying AR(2) process, where they compare the prediction performance with that of a time varying AR(1) process and with stationary AR(1) and AR(2) processes, with the findings that for not too large sample sizes, the stationary models may very well give more accurate prediction in terms of the empirical mean squared errors. The authors then continue to develop a general methodology to find a model for prediction of finite sample data. For that, they divide the observation sets into three sets, namely the training set \(M_0\), a validation set \(M_1\) and a final validation set \(M_2\). Based on the observations in \(M_0\), they calculate the locally stationary model that predicts best (among certain other locally stationary models) into the set \(M_1\) in terms of minimised empirical mean square prediction errors for linear \(h\)-step predictors, and similarly the best stationary model with this property. Whether to choose the best locally stationary model or the best stationary model is then decided by the comparison of the performance of both models on the second evaluation set \(M_2\).
The authors then show rigorously under certain assumptions that with high probability, the chosen of the two models (best locally stationary / best stationary) will perform empirically better in forecasting the future of not yet observed values. Various simulations are presented and an R package is provided. The authors apply their method to three real data sets, namely to London housing prices, to temperatures and to the volatility around the time of the EU referendum in the UK in 2016. As a further theoretical result, they prove that the localised Yule-Walker estimator in locally stationary models is strongly, uniformly consistent.


62M20 Inference from stochastic processes and prediction
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics


R; forecastSNSTS
Full Text: DOI arXiv Euclid


[1] Akaike, H. (1969). Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics, 21(1):243-247. · Zbl 0202.17301 · doi:10.1007/BF02532251
[2] Akaike, H. (1970). A fundamental relation between predictor identification and power spectrum estimation. Annals of the Institute of Statistical Mathematics, 22(1):219-223. · Zbl 0259.62077 · doi:10.1007/BF02506338
[3] Baillie, R. T. (1979). Asymptotic prediction mean squared error for vector autoregressive models. Biometrika, 66(3):675-8. · Zbl 0416.62072 · doi:10.1093/biomet/66.3.675
[4] Bercu, B. (2001). On large deviations in the Gaussian autoregressive process: stable, unstable and explosive cases. Bernoulli, 7(2):299-316. · Zbl 0981.62072 · doi:10.2307/3318740
[5] Bercu, B., Gamboa, F., and Lavielle, M. (2000). Sharp large deviations for Gaussian quadratic forms with applications. ESAIM: Probability and Statistics, 4:1-24. · Zbl 0939.60013 · doi:10.1051/ps:2000101
[6] Bercu, B., Gamboa, F., and Rouault, A. (1997). Large deviations for quadratic forms of stationary Gaussian processes. Stochastic Processes and their Applications, 71(1):75-90. · Zbl 0941.60050 · doi:10.1016/S0304-4149(97)00071-9
[7] Bercu, B. and Touati, A. (2008). Exponential inequalities for self-normalized martingales with applications. The Annals of Applied Probability, 18(5):1848-1869. · Zbl 1152.60309 · doi:10.1214/07-AAP506
[8] Bhansali, R. J. (1996). Asymptotically efficient autoregressive model selection for multistep prediction. Annals of the Institute of Statistical Mathematics, 48:577-602. · Zbl 0926.62086 · doi:10.1007/BF00050857
[9] Birr, S., Volgushev, S., Kley, T., Dette, H., and Hallin, M. (2017). Quantile spectral analysis for locally stationary time series. Journal of the Royal Statistical Society: Series B, 79(5):1619-1643. · Zbl 06840449 · doi:10.1111/rssb.12231
[10] Brillinger, D. R. (1975). Time Series: Data Analysis and Theory. Holt, Rinehart and Winston, Inc., New, York. · Zbl 0321.62004
[11] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New, York. · Zbl 0709.62080
[12] Brownlees, C. T. and Gallo, G. M. (2008). On variable selection for volatility forecasting: The role of focused selection criteria. Journal of Financial Econometrics, 6:513-539.
[13] Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. Ann. Statist., 15(3):1050-1063. · Zbl 0638.62082 · doi:10.1214/aos/1176350492
[14] Chen, Y., Hä rdle, W., and Pigorsch, U. (2010). Localized realized volatility modeling. Journal of the American Statistical Association, 105(492):1376-1393. · Zbl 1388.62305 · doi:10.1198/jasa.2010.ap09039
[15] Claeskens, G., Croux, C., and Van Kerckhoven, J. (2007). Prediction-focused model selection for autoregressive models. Australian & New Zealand Journal of Statistics, 49(4):359-379. · Zbl 1521.62144 · doi:10.1111/j.1467-842X.2007.00487.x
[16] Claeskens, G. and Hjort, N. L. (2003). The focused information criterion [with discussion]. Journal of the American Statistical Association, 98:900-916. · Zbl 1045.62003 · doi:10.1198/016214503000000819
[17] Dahlhaus, R. (1996). On the Kullback-Leibler information divergence of locally stationary processes. Stochastic Processes and their Applications, 62(1):139-168. · Zbl 0849.60032 · doi:10.1016/0304-4149(95)00090-9
[18] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Annals of Statistics, 25(1):1-37. · Zbl 0871.62080 · doi:10.1214/aos/1034276620
[19] Dahlhaus, R. (2012). Locally stationary processes. In Rao, T. S., Rao, S. S., and Rao, C., editors, Time Series Analysis: Methods and Applications, volume 30 of Handbook of Statistics, pages 351-413., Elsevier. · Zbl 1242.62005
[20] Dahlhaus, R. and Giraitis, L. (1998). On the optimal segment length for parameter estimates for locally stationary time series. Journal of Time Series Analysis, 19(6):629-655. · Zbl 0921.62107 · doi:10.1111/1467-9892.00114
[21] Das, S. and Politis, D. N. (2018). Predictive inference for locally stationary time series with an application to climate data., arXiv:1712.02383. · Zbl 1464.62379
[22] Dette, H., Preuß, P., and Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing. Journal of the American Statistical Association, 106(495):1113-1124. · Zbl 1229.62119
[23] Dwivedi, J. and Subba Rao, S. (2010). A test for second order stationarity based on the discrete fourier transform. Journal of Time Series Analysis, 32:68-91. · Zbl 1290.62059 · doi:10.1111/j.1467-9892.2010.00685.x
[24] Dzhaparidze, K., Kormos, J., van der Meer, T., and van Zuijlen, M. (1994). Parameter estimation for nearly nonstationary AR(1) processes. Mathematical and Computer Modelling, 19(2):29-41. · Zbl 0852.62080 · doi:10.1016/0895-7177(94)90047-7
[25] Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli, 17(1):320-346. · Zbl 1284.62550 · doi:10.3150/10-BEJ270
[26] Giraitis, L., Kapetanios, G., and Price, S. (2013). Adaptive forecasting in the presence of recent and ongoing structural change. Journal of Econometrics, 177(2):153-170. · Zbl 1288.91163 · doi:10.1016/j.jeconom.2013.04.003
[27] Hallin, M. (1978). Mixed autoregressive-moving average multivariate processes with time-dependent coefficients. Journal of Multivariate Analysis, 8(4):567-572. · Zbl 0394.62067 · doi:10.1016/0047-259X(78)90034-9
[28] Hong-Zhi, A., Zhao-Guo, C., and Hannan, E. J. (1982). Autocorrelation, autoregression and autoregressive approximation. Annals of Statistics, 10(3):926-936. · Zbl 0512.62087 · doi:10.1214/aos/1176345882
[29] Jirak, M. (2012). Simultaneous confidence bands for Yule-Walker estimators and order selection. Annals of Statistics, 40(1):494-528. · Zbl 1246.62187 · doi:10.1214/11-AOS963
[30] Jirak, M. (2014). Simultaneous confidence bands for sequential autoregressive fitting. Journal of Multivariate Analysis, 124:130-149. · Zbl 1360.62459 · doi:10.1016/j.jmva.2013.10.018
[31] Kley, T., Fryzlewicz, P., and Preuß, P. (2019a). forecastSNSTS: Forecasting for Stationary and Non-Stationary Time Series. R package version, 1.3-0. · Zbl 1432.62323
[32] Kley, T., Preuß, P., and Fryzlewicz, P. (2019b). Predictive, finite-sample model choice for time series under stationarity and non-stationarity., arXiv:1611.04460v3. · Zbl 1432.62323 · doi:10.1214/19-EJS1606
[33] Lai, T. L. and Wei, C. Z. (1982). Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Annals of Statistics, 10(1):154-166. · Zbl 0488.62071 · doi:10.1214/aos/1176345697
[34] Landregistry (2019). UK house price index., http://landregistry.data.gov.uk/app/ukhpi. Accessed: 2019-01-11.
[35] McDonald, D. J., Shalizi, C. R., and Schervish, M. (2016). Nonparametric risk bounds for time-series forecasting., arXiv:1212.0463. · Zbl 1437.62337
[36] Mikosch, T. and Starica, C. (2004). Non-stationarities in financial time series, the long range dependence and the IGARCH effects. The Review of Economics and Statistics, 86:378-390.
[37] Moulines, E., Priouret, P., and Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. Annals of Statistics, 33(6):2610-2654. · Zbl 1084.62089 · doi:10.1214/009053605000000624
[38] Nason, G. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. Journal of the Royal Statistical Society: Series B, 75:879-904. · Zbl 1411.62259 · doi:10.1111/rssb.12015
[39] Palma, W., Olea, R., and Ferreira, G. (2013). Estimation and forecasting of locally stationary processes. Journal of Forecasting, 32:86-96. · Zbl 1397.62319 · doi:10.1002/for.1259
[40] Paparoditis, E. (2009). Testing temporal constancy of the spectral structure of a time series. Bernoulli, 15:1190-1221. · Zbl 1200.62049 · doi:10.3150/08-BEJ179
[41] Paparoditis, E. (2010). Validating stationarity assumptions in time series analysis by rolling local periodograms. Journal of the American Statistical Association, 105:839-851. · Zbl 1392.62275 · doi:10.1198/jasa.2010.tm08243
[42] Peña, D. and Sánchez, I. (2007). Measuring the advantages of multivariate vs. univariate forecasts. Journal of Time Series Analysis, 28(6):886-909. · Zbl 1150.62061 · doi:10.1111/j.1467-9892.2007.00538.x
[43] Politis, D. N. (2015). Model-Free Prediction and Regression: A Transformation-Based Approach to Inference., Springer. · Zbl 1397.62008
[44] Preuß, P. and Vetter, M. (2013). Discriminating between long-range dependence and non-stationarity. Electron. J. Statist., 7:2241-2297. · Zbl 1293.62201 · doi:10.1214/13-EJS836
[45] Preuß, P., Vetter, M., and Dette, H. (2013). A test for stationarity based on empirical processes. Bernoulli, 19(5B):2715-2749. · Zbl 1281.62183 · doi:10.3150/12-BEJ472
[46] Priestley, M. B. (1965). Evolutionary spectra and non-stationary processes. Journal of the Royal Statistical Society. Series B (Methodological), 27(2):204-237. · Zbl 0144.41001 · doi:10.1111/j.2517-6161.1965.tb01488.x
[47] Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic, Press. · Zbl 0537.62075
[48] R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
[49] Reinsel, G. (1980). Asymptotic properties of prediction errors for the multivariate autoregressive model using estimated parameters. Journal of the Royal Statistical Society: Series B, 42(3):328-333. · Zbl 0454.62083 · doi:10.1111/j.2517-6161.1980.tb01131.x
[50] Richter, S. and Dahlhaus, R. (2017). Cross validation for locally stationary processes., arXiv:1705.10046. · Zbl 1433.62267 · doi:10.1214/18-AOS1743
[51] Rohan, N. and Ramanathan, T. V. (2011). Order selection in arma models using the focused information criterion. Australian & New Zealand Journal of Statistics, 53(2):217-231. · Zbl 1274.62616 · doi:10.1111/j.1467-842X.2011.00626.x
[52] Rossi, B. (2013). Advances in forecasting under instability. In Elliott, G. and Timmermann, A., editors, Handbook of Economic Forecasting, volume 2B, pages 1203-1324., Elsevier.
[53] Roueff, F. and Sanchez-Perez, A. (2016). Locally stationary processes prediction by auto-regression., arXiv:1602.01942v1. · Zbl 1404.62094 · doi:10.30757/ALEA.v15-45
[54] Roueff, F. and Sanchez-Perez, A. (2018). Prediction of weakly locally stationary processes by auto-regression., arXiv:1602.01942v3. · Zbl 1404.62094 · doi:10.30757/ALEA.v15-45
[55] Saulis, L. and Statulevičus, V. A. (1991). Limit Theorems for Large Deviations. Kluwer, Dordrecht. · Zbl 0744.60028
[56] Subba Rao, T. (1970). The fitting of non-stationary time-series models with time-dependent parameters. Journal of the Royal Statistical Society. Series B (Methodological), 32(2):312-322. · Zbl 0225.62109 · doi:10.1111/j.2517-6161.1970.tb00844.x
[57] Vogt, M. (2012). Nonparametric regression for locally stationary time series. Annals of Statistics, 40(5):2601-2633. · Zbl 1373.62459 · doi:10.1214/12-AOS1043
[58] Vogt, M. and Dette, H. (2015). Detecting smooth changes in locally stationary processes. Annals of Statistics, 43(2):713-740. · Zbl 1312.62045 · doi:10.1214/14-AOS1297
[59] von Sachs, R. and Neumann, M. H. (2000). A wavelet-based test for stationarity. Journal of Time Series Analysis, 21:597-613. · Zbl 0972.62085 · doi:10.1111/1467-9892.00200
[60] Xia, Y. and Tong, H. (2011). Feature matching in time series modeling. Statistical Science, 26(1):21-46. · Zbl 1219.62142 · doi:10.1214/10-STS345
[61] Yu, M. and Si, S. (2009). Moderate deviation principle for autoregressive processes. Journal of Multivariate Analysis, 100(9):1952-1961. · Zbl 1173.60010 · doi:10.1016/j.jmva.2009.06.005
[62] Zhang, Y. and Koreisha, S. (2015). Adaptive order determination for constructing time series forecasting models. Communications in Statistics - Theory and Methods, 44(22):4826-4847. · Zbl 1343.62074 · doi:10.1080/03610926.2013.800881
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