Concepts and tools for nonlinear time series modelling

Amendola, Alessandra and Francq, Christian (2009): Concepts and tools for nonlinear time series modelling. Forthcoming in: Handbook of Computational Econometrics (July 2009)

Abstract

Tools and approaches are provided for nonlinear time series modelling in econometrics. A wide range of topics is covered, including probabilistic properties, statistical inference and computational methods. The focus is on the applications but the ideas of the mathematical arguments are also provided. Techniques and concepts are illustrated by various examples, Monte Carlo experiments and a real application.

• Item Type: MPRA Paper
• Language: English
• Keywords: Consistency and asymptotic normality; MCMC algorithms; Mixing; Nonlinear modelling; Stationarity; Time-series forecasting.
• 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

References

H. Akaike, Information theory and an extension of the maximum likelihood principle, {in Second International Symposium in Information Theory}, eds. B.N. Petrov and F. Csaki, Akademia Kiado, Budapest (2003) 267--281.

M.S. Al-Qassem, J.A. Lane, Forecasting exponential autoregressive models of order 1, Journal of Time Series Analysis, {10} (1989) 95--113.

A. Amendola, M. Niglio, Predictor distribution and forecast accuracy of threshold models, {Statistical Methods \& Applications}, 13 (2003) 3--14.

A. Amendola, M. Niglio, C. Vitale, The moments of SETARMA models, {Journal of Statistics \& Probability Letters}, 76 (2006) 625--633.

A. Amendola, M. Niglio, C. Vitale, The multi-step ahead predictors of SETARMA models, mimeo (2006).

T.G. Andersen, T. Bollerslev, F.X. Diebold, P. Labys, Modeling and Forecasting Realized Volatility, {Econometrica}, 71 (2005) 579--625.

T.G. Andersen, T. Bollerslev, Volatility and Correlation Forecasting, {Handbook of Economic Forecasting}, ed. C.W.J. Granger and A. Timmermann, Elsevier, Amsterdam, (2006).

D.W.K. Andrews, Heteroskedasticity and autocorrelation consistent covariance matrix estimation, { Econometrica}, 59 (1991) 817--858.

D.W.K. Andrews, W. Ploberger, Optimal tests when a nuisance parameter is present only under the alternative, { Econometrica}, 62 (1994) 1383--1414.

J. Bai, Testing Parametric Conditional Distributions of Dynamic Models, {Review of Economics and Statistics}, 85 (2003) 531--549.

R.T. Baillie, T. Bollerslev, Prediction in dynamic models, {Journal of Econometrics}, 52 (1992) 91--113.

Y. Bao, T.H. Lee, B. Saltoglu, Comparing Density Forecast Models, {Journal of Forecasting}, 26 (2007) 203--225.

I.V. Basawa, R. Lund, Large sample properties of parameter estimates for periodic ARMA models, Journal of Time Series Analysis, 22 (2001) 651--663.

J.M. Bates, C.W.J. Granger, The combination of forecasts, {Operational Research Quarterly}, 30 (1969) 451--468.

L.E. Baum, An inequality and associated maximization technique in statistical estimation for probabilistic functions of Markov processes, { Inequalities}, 3 (1972) 1--8.

L.E. Baum, T. Petrie, Statistical inference for probabilistic functions of finite state Markov chains, { Annals of Mathematical statistics}, 30 (1966) 1554--1563.

J. Berkowitz, Testing Density Forecasts with Applications to Risk Management, {Journal of Business and Economic Statistics}, 19 (2001) 465--474.

A. Bibi, A. Gautier, Propriétés dans L$^2$ et estimation des processus purement bilinéaires et strictement superdiagonaux à coefficients périodiques, La revue Canadienne de Statistique, (2006) to appear.

P. Billingsley, The Lindeberg-Levy theorem for martingales, { Proceedings of the American Mathematical Society}, 12 (1961) 788--792.

P. Billingsley, {Probability and Measure,} Wiley, New-York, (1995).

T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, { Journal of Econometrics}, 31 (1986) 307--327.

P. Bougerol, N. Picard, Strict stationarity of generalized autoregressive processes, { Annals of Probability}, 20 (1992a) 1714--1729.

P. Bougerol, N. Picard, Stationarity of GARCH processes and of some nonnegative time series, { Journal of Econometrics}, 52 (1992b) 115--127.

G.E.P. Box, G.M. Jenkins, Time series analysis, forecasting and control, San Francisco, Holden-Day (1976).

G.E.P. Box, D.A. Pierce, Distribution of Residual Autocorrelations in Autoregressive-Integreted Moving Average Time Series Models, { Journal of the American Statististical Association}, { 65} (1970) 1509--1526.

A. Brandt, The stochastic equation $Y_{n+1}=A_nY_n+B_n$ with stationary coefficients, { Advances in Applied Probability,} 18 (1986) 221--254.

F.J. Breidt, R.A. Davis, Time-reversibility, identifiability, and independence of innovations for stationary time series, Journal of Time Series Analysis, 13 (1992) 377--390.

F.J. Breidt, R.A. Davis, A. Trindade, Least Absolute Deviation Estimation for All-Pass Time Series Models, Annals of Statistics, 29 (2001) 919--946.

P.J. Brockwell, R.A. Davis, Time Series: Theory and Methods, Springer-Verlag, New-York (1991).

S.P. Brooks, G.O. Roberts, Assessing Convergence of Markov Chain Monte Carlo Algorithms, Statistics and Computing, 8 (1998) 319--335.

B.Y. Brown, R.S. Mariano, Predictors dynamic nonlinear models: large sample behaviour, {Econometric Theory}, 5 (1989) 430--452.

C. Chatfield, Calculating interval forecasts, {Journal of Business and Economic Statistics}, 11 (1993) 121--144.

R. Chen, R.S. Tsay, Functional-Coefficient Autoregressive Models, Journal of the American Statistical Association, 88 (1993) 298--308.

Q. Cheng, On time-reservibility of linear processes, Biometrika, 86 (1999) 483--486.

P. Christoffersen, Evaluating Interval Forecasts, {International Economic Review}, 39 (1998) 841--62.

P. Christoffersen, F.X. Diebold, How Relevant is Volatility Forecasting for Financial Risk Management?, {Review of Economics and Statistics}, 82 (2000) 12--22.

M.P. Clements, J. Smith, The performance of alternative forecasting methods for SETAR models, { International Journal of Forecasting}, 13 (1997) 463--475.

M.P. Clements, P.H. Franses, L. Smith, D. Van Dijk, On SETAR non-linearity and forecasting, { Journal of Forecasting}, 22 (2003) 359--375.

M.P. Clements, P.H. Franses, N.R. Swanson, Forecasting economic and financial time-series with non-linear models, International Journal of Forecasting, 20 (2004) 169--183.

D.B.H. Cline, Evaluating the Lyapounov exponent and existence of moments for threshold AR-ARCH models, Preprint (2006).

D.B.H. Cline, H.H. Pu, Stability and the Lyapounov exponent of threshold AR-ARCH models, The Annals of Applied Probability, 14 (2004) 1920--1949.

V. Corradi, N.R. Swanson, Predictive Density Evaluation,{ Handbook of Economic Forecasting}, Vol. 1, ed. Elliott G., Granger C. and Timmermann A., Amsterdam: North-Holland, (2006).

J. Davidson, Stochastic Limit Theory, Oxford University Press, Oxford (1994).

J. Davidson, Stochastic Limit Theory, Oxford University Press, Oxford (1994).

R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative, Biometrika, (1977) 64 247--254.

R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative, { Biometrika}, 74 (1987) 33-43.

Y.A. Davydov, Convergence of Distributions Generated by Stationary Stochastic Processes, {Theory of Probability and Its Applications}, 13 (1968) 691--696.

P. Devijver, Baum's forward-backward algorithm revisited, Pattern Recognition Letters, 3 (1985) 369--373.

F.X. Diebold, T.G. Gunther, A.S. Tay, Evaluating Density forecast with applications to Financial Risk Management, International Economic Review, 39 (1998) 863--883.

Z. Ding, C. Granger, R.F. Engle, A long memory property of stock market returns and a new model, { Journal of Empirical Finance}, 1 (1993) 83--106.

R.F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation, { Econometrica}, 50 (1982) 987--1007.

R.F. Engle, Wald, likelihood ratio, and Lagrange multiplier tests in econometrics, {Handbook of Econometrics, Vol. 2, ed. Z. Griliches and M.D. Intriligator}, Amsterdam: North-Holland, (1984) 775--826.

R.F. Engle, D.M. Lilien, R.P. Robins, Estimating time varying riak premia in the term structure: the ARCH-M mode, {Econometrica}, 55 (1987) 391--407.

R.F. Engle, V.K. Ng, Measuring and Testing the Impact of News on Volatility, The Journal of Finance, 48 (1993) 1749--1778.

J. Fan, Q. Yao, Nonlinear Time Series : Nonparametric and Parametric Methods. Springer-Verlag, New York (2003).

P.D. Feigin, R.L. Tweedie, Random Coefficient Autoregressive Processes: a Markov Chain Analysis of Stationarity and Finiteness of Moments, { Journal of Time Series Analysis}, 6 (1985) 1--14.

D.F. Findley, The overfitting principles supporting AIC, Statistical Research Division Report RR 93/04, Bureau of the Census (1993).

C. Francq, M. Roussignol, J.M. Zakoïan, Conditional heteroskedasticity driven by hidden Markov chains. { Journal of Time Series Analysis}, 22 (2001) 197--220.

C. Francq, R. Roy, J.M. Zakoïan, Diagnostic Checking in ARMA Models with Uncorrelated Errors, Journal of the American Statistical Association, 100 (2005) 532--544.

C. Francq, J.M. Zakoïan, Stationarity of Multivariate Markov-switching ARMA Models, { Journal of Econometrics}, 102 (2001) 339--364.

C. Francq, J.M. Zakoïan, Recent results for linear time series models with non independent innovations, in { Statistical Modeling and Analysis for Complex Data Problems}, Duchesne, P. et Rémillard, B., Éditeurs, Kluwer, (2004) 241--266.

P.H. Franses, D. van Dijk, Non-linear time series Models in empirical finance, Cambrige University Press, Cambrige (2002).

S. Geman, D. Geman, Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 45 (1984) 721--741.

L.R. Glosten, R. Jagannathan, D. Runkle, On the relation between the expected value and the volatility of the nominal excess return on stocks, { Journal of Finance}, 48 (1993) 1779--1801.

L.G. Godfrey, { Misspecification tests in econometrics: the Lagrange multiplier principle and other approaches,} Cambridge University Press, Cambridge (1988).

J.B. de Gooijer, P.T. de Bruin, On forecasting SETAR processes, Statistics \& Probability Letters, 37 (1998) 7--14. \bibitem{Granger78} C.W.J. Granger, A.P. Andersen, { An introduction to bilinear time series models.} Vandenhoek and Ruprecht, Gottingen (1978).

C.W.J. Granger, R. Ramanathan, Improved methods of combining forecasts, {Journal of Forecasting}, 3 (1984) 197--204.

C.W.J. Granger, T. Teräsvirta, {Modelling nonlinear relationships}, Oxford University Press, Oxford (1993).

V. Haggan, T. Ozaki, Modelling Nonlinear Random Vibrations Using an Amplitude-Dependent Autoregressive Time Series Model, Biometrika, 68 (1981) 189--196.

S.G. Halla, J. Mitchell, Combining density forecasts, {International Journal of Forecasting}, 23 (2007) 1--13.

J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, {Econometrica}, 57 (1989) 357--384.

J.D. Hamilton, Time Series Analysis, Princeton University Press ed, (1994).

B.E. Hansen, Inference when a nuisance parameter is not identified under the null hypothesis, { Econometrica}, 64 (1996) 413-430.

W. Härdle, Applied Nonparametric Regression, Cambridge University Press: New York (1990).

W.K. Hastings, Monte Carlo Sampling Methods using Markov Chains and their Applications, Biometrika, 57 (1970) 97--109.

N. Herrndorf, A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables, { The Annals of Probability}, 12 (1984) 141--153.

Y. Hong, Evaluation of out-of-sample Probability Density Forecasts with Applications to S\&P 500 Stock Prices, Working Paper, (2001) Cornell University.

Y. Hong, Y.J. Lee, Consistent Testing for Serial Correlation of Unknown Form Under General Conditional Heteroskedasticity, { Preprint, Cornell University}, (2003).

R.J. Hyndman, Highest-density forecast regions for non-linear and non-normal time series models, {Journal of Forecasting}, 14 (1995) 431--441.

R.J. Hyndman, Computing and graphing highest density regions, {American Statisticians}, 50 (1996) 120--126.

H.K. Krolzig, Markov Switching Vector Autoregressions. Modelling Statistical Inference and Application to Business Cycle Analysis, Springer Verlag ed (1997).

W.K. Li, Diagnostic Checks in Time Series. Chapman \& Hall/CRC, Boca Raton, Florida (2004).

W.K. Li, K. Lam, Modelling asymmetry in stock returns by threshold autoregressive conditional heteroscedastic model, {The Statistician}, 44 (1995) 333--341.

C.W. Li, W.K. Li, On a double-threshold heteroscedastic time series model, {Journal Applied Econometric}, 11 (1996) 253--274.

J.L. Lin, C.W.J. Granger, Forecasting from non-linear models in practice, {Journal of Forecasting}, 13 (1994) 1--19.

J. Liu, W.K. Li, C.W. Li, On a threshold autoregression with conditional heteroscedastic variances, {Journal of Statistical Planning and inference}, 62 (1997) 279--300.

G.M. Ljung, G.E.P. Box, On the Measure of Lack of Fit in Time Series Models, { Biometrika}, {65} (1978) 297--303.

I.N. Lobato, Testing for Zero Autocorrelation in the Presence of Statistical Dependence, { Econometric Theory}, 18 (2002) 730--743.

R. Luukkonen, P. Saikkonen, T. Teräsverta, Testing linearity against smooth transition autoregression, { Biometrika}, 75 (1988) 491--499.

A.I. McLeod, W.K. Li, Diagnostic checking ARMA time series models using squaredresidual autocorrelations, Journal of Time Series Analysis, 4 (1983) 269--273.

J.R. Magnus, H. Neudecker, { Matrix Differential Calculus with Application in Statistics and Econometrics,} New-York, Wiley (1988).

M. Marcellino, Forecasting EMU macroeconomic variables, International Journal Of Forecasting, 20 (2004) 359--372.

N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equations of State Calculations by Fast Computing Machines, Journal of Chemical Physics, 21 (1953) 1087--1092.

S.P. Meyn, R.L. Tweedie, { Markov Chains and Stochastic stability,} 3rd Edition, Sprin\-ger, London (1996).

R. Moenaddin, Approximating multi-step non-linear least squares prediction of threshold autoregressive models, Paper presented at the IMS Philadelphia meeting, (1991).

D.B. Nelson, Conditional Heteroskedasticity in Asset Returns : a New Approach, { Econometrica}, 59 (1991) 347--370.

D.F. Nicholls, B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction. Lecture Notes in Statist. Springer-Verlag, New York (1982).

A. Patton, Volatility Forecast Comparison using Imperfect Volatility Proxies, Research Paper Series 175, Quantitative Finance Research Centre, University of Technology, Sydney (2006).

B.M. Pötscher, I.R. Prucha, Dynamic Nonlinear Econometric Models, Springer, Berlin (1997).

S.M. Potter, Nonlinear time series modelling: an introduction. Journal of Economic Survey, 13 (1999) 505--528.

M.B. Priestley, { Nonlinear and non-stationary time series analysis,} Academic press, London, 1988.

L. Rabiner, B. Juang, An introduction to hidden Markov models, ASSP Magazine, IEEE, 3 (1986) 4--16.

C.P. Robert, G. Casella, Monte Carlo Statistical Methods. 2nd Edition. New York: Springer (2004).

J.P. Romano, L.A. Thombs, Inference for autocorrelations under weak assumptions, Journal of the American Statistical Association, 91 (1996) 590--600.

E. Rio, Covariance inequalities for strongly mixing processes, Ann. Inst. Henri Poincaré, 29 (1993) 587--597.

E. Sentana, Quadratic ARCH Models, { Review of Economic Studies}, 62 (1995) 639--661.

J.H. Stock, M.W. Watson, A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series, in R.F. Engle and h. White (eds), Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger, Oxford University Press, Oxford, (1999) 1--44.

T. Subba Rao, M.M. Gabr, { An introduction to bispectral analysis and bilinear time series models.} Lecture Notes in Statistics, Springer, New-York (1984).

S.J. Taylor, { Modelling Financial Time Series}, New-York: Wiley (1986).

T. Teräsvirta, Smooth Transition Regression Modeling, in { Applied Time Series Econometrics} Eds. H. Lütkepohl and M. Krätzig, Cambridge University Press, Cambridge (2004) 222--242.

T. Teräsvirta, Forecasting economic variables with nonlinear models, {Handbook of Economic Forecasting}, ed. G. Elliott, C.W.J. Granger and A. Timmermann, Elsevier, Amsterdam, (2006).

T. Teräsvirta, D. Tjøstheim, C.W.J. Granger, Aspects of Modelling Nonlinear Time Series, { Handbook of Econometrics, Vol. 4, ed. R.F. Engle and D.L. McFadden}, Elsevier, Amsterdam, (1994) 2919--2960.

A. Timmermann, Forecast Combination, { Handbook of Economic Forecasting}, { Vol.1}, ed. Elliott G., Granger C. and Timmermann A., Amsterdam: North-Holland, (2006) 135--194.

D. Tj\o stheim, Estimation in nonlinear time series models, { Stochastic Processes and their Applications}, 21 (1986) 251--273.

D. Tj\o stheim, Non-linear time series and Markov chains, { Advances in Applied Probability}, 22 (1990) 587--611.

D. Tj\o stheim, B.H. Auestad, Nonparametric identification of nonlinear time series: projections, Journal of the American Statistical Association, 89 (1994) 1398--1409.

H. Tong, A note on using threshold autoregressive models for multi-step-ahead prediction of cyclical data, Journal of the Time Series Analysis, 3 (1982) 137--140.

H. Tong, Threshold models in nonlinear time series analysis, Springer-Verlag, New York (1983).

H. Tong, Non-Linear Time Series. A dynamical System Approach, Oxford University Press, Oxford (1990).

H. Tong, K.S. Lim, Threshold autoregression, limit cycles and cyclical data (with discussion), { Journal of the Royal Statististical Society, Series B}, 42 (1980) 245--292.

H. Tong, R. Moennaddin, On multi-step non-linear least squares prediction, {The Statistician}, {37} (1988) 101--110.

R.S. Tsay, Analysis of Financial Time Series, Wiley 2nd Edition (2005).

K.F. Wallis, Asymmetric density forecasts of inflation and the Bank of England's fan chart, National Institute Economic Review, 167 (1999) 106--112.

K.F. Wallis, Combining Density and Interval Forecasts: A modest proposal, {Oxford Bulletin of Economics and Statistics}, 67 (2005) 983--994.

H. Wold, {A study in the analysis of stationary time series,} Almqvist and Wiksell, Stocholm (1938).

Q. Yao, H. Tong, Quantifying the influence of initial values on nonlinear prediction, Journal of the Royal Statistical Society, Series B, 56 (1994) 701--725.

J.M. Zakoïan, Threshold Heteroskedastic Models, { Journal of Economic Dynamics and Control}, 18 (1994) 931--955.