404 A. Harvey serial correlation in volatility is by means of the GARCH class in which it is assumed that the conditional variance of the observations is an exact function of the squares of past observations and previous variances. An alternative approach is to model volatil- ity as an unobserved component in the variance. This leads to the class of stochastic volatility (SV) models. The topic is covered Chapter 15 by Andersen et al. in this Hand- book so the treatment here will be brief. Earlier reviews of the literature are to be found in Taylor (1994) and Ghysels, Harvey and Renault (1996), while the edited volume by Shephard (2005) contains many of the important papers. The stochastic volatility model has two attractions. The first is that it is the natural dis- crete time analogue (though it is only an approximation) of the continuous time model used in work on option pricing; see Hull and White (1987) and the review by Hang (1998). The second is that its statistical properties are relatively easy to determine and extensions, such as the introduction of seasonal components, are easily handled. The disadvantage with respect to the conditional variance models of the GARCH class is that whereas GARCH can be estimated by maximum likelihood, the full treatment of an SV model requires the use of computer intensive methods such as MCMC and im- portance sampling. However, these methods are now quite rapid and it would be wrong to rule out SV models on the grounds that they make unreasonably heavy computational demands. 10.1. Basic specification and properties The basic discrete time SV model for a demeaned series of returns, y t , may be written as (180)y t = σ t ε t = σ e 0.5h t ε t ,ε t ∼ IID(0, 1), t = 1, ,T where σ is a scale parameter and h t is a stationary first-order autoregressive process, that is, (181)h t+1 = φh t + η t ,η t ∼ IID 0,σ 2 η where η t is a disturbance term which may or may not be correlated with ε t .Ifε t and η t are allowed to be correlated with each other, the model can pick up the kind of asymmetric behaviour which is often found in stock prices. The following properties of the SV model hold even if ε t and η t are contempora- neously correlated. Firstly y t is a martingale difference. Secondly, stationarity of h t implies stationarity of y t .Thirdly,ifη t is normally distributed, terms involving expo- nents of h t may be evaluated using properties of the lognormal distribution. Thus, the variance of y t can be found and its kurtosis shown to be κ ε exp(σ 2 h )>κ ε where κ ε is the kurtosis of ε t . Similarly, the autocorrelations of powers of the absolute value of y t , and its logarithm, can be derived; see Ghysels, Harvey and Renault (1996). Ch. 7: Forecasting with Unobserved Components Time Series Models 405 10.2. Estimation Squaring the observations in (180) and taking logarithms gives (182)log y 2 t = ω + h t + ξ t where ξ t = log ε 2 t − E log ε 2 t , and ω = log σ 2 + E log ε 2 t , so that ξ t has zero mean by construction. If ε t has a t ν -distribution, it can be shown that the moments of ξ t exist even if the distribution of ε t is Cauchy, that is ν = 1. In fact in this case ξ t is symmet- ric with excess kurtosis two, compared with excess kurtosis four and a highly skewed distribution when ε t is Gaussian. The transformed observations, the log y 2 t s, can be used to construct a linear state space model. The measurement equation is (182) while (181) is the transition equa- tion. The quasi maximum likelihood (QML) estimators of the parameters φ, σ 2 η and the variance of ξ t , σ 2 ξ , are obtained by treating ξ t and η t as though they were normal in the linear SSF and maximizing the prediction error decomposition form of the like- lihood obtained via the Kalman filter; see Harvey, Ruiz and Shephard (1994). Harvey and Shephard (1996) show how the linear state space form can be modified so as to deal with an asymmetric model. The QML method is relatively easy to apply and, even though it is not efficient, it provides a reasonable alternative if the sample size is not too small; see Yu (2005). Simulation based methods of estimation, such as Markov chain Monte Carlo and efficient method of moments, are discussed at some length in Chapter 15 by Andersen et al. in this Handbook. Important references include Jacquier, Polson and Rossi (1994, p. 416), Kim, Shephard and Chib (1998), Watanabe (1999) and Durbin and Koopman (2000). 10.3. Comparison with GARCH The GARCH(1, 1) model has been applied extensively to financial time series. The variance in y t = σ t ε t is assumed to depend on the variance and squared observation in the previous time period. Thus, (183)σ 2 t = γ + αy 2 t−1 + βσ 2 t−1 ,t= 1, ,T. The GARCH(1, 1) model displays similar properties to the SV model, particularly if φ is close to one (in which case α + β is also close to one). Jacquier, Polson and Rossi (1994, p. 373) present a graph of the correlogram of the squared weekly returns of a portfolio on the New York Stock Exchange together with the ACFs implied by fitting SV and GARCH(1, 1) models. The main difference in the ACFs seems to show up most at lag one with the ACF implied by the SV model being closer to the sample values. The Gaussian SV model displays excess kurtosis even if φ is zero since y t is a mixture of distributions. The σ 2 η parameter governs the degree of mixing independently of the degree of smoothness of the variance evolution. This is not the case with a GARCH 406 A. Harvey model where the degree of kurtosis is tied to the roots of the variance equation, α and β in the case of GARCH(1, 1). Hence, it is very often necessary to use a non-Gaussian distribution for ε t to capture the high kurtosis typically found in a financial time series. Kim, Shephard and Chib (1998) present strong evidence against the use of the Gaussian GARCH, but find GARCH-t and Gaussian SV to be similar. In the exchange rate data they conclude on p. 384 that the two models “ fitthedatamoreorlessequally well”. Further evidence on kurtosis is in Carnero, Pena and Ruiz (2004). Fleming and Kirby (2003) compare the forecasting performance of GARCH and SV models. They conclude that “. GARCH models produce less precise forecasts ”, but go on to observe that “ in the simulations, it is not clear that the performance differences are large enough to be economically meaningful”. On the other hand, Sec- tion 5.5 of Chapter 1 by Geweke and Whiteman in this Handbook describes a decision theoretic application, concerned with foreign currency hedging, in which there are clear advantages to using the SV model. 10.4. Multivariate models The multivariate model corresponding to (180) assumes that each series is generated by a model of the form (184)y it = σ i ε it e 0.5h it ,t= 1, ,T with the covariance (correlation) matrix of the vector ε t = (ε 1t , ,ε Nt ) being de- noted by ε . The vector of volatilities, h t , follows a VAR(1) process, that is, h t+1 = h t + η t , η t ∼ IID(0, η ). This specification allows the movements in volatility to be correlated across different series via η . Interactions can be picked up by the off-diagonal elements of .Asimple nonstationary model is obtained by assuming that the volatilities follow a multivariate random walk, that is = I.If η is singular, of rank K<N, there are only K components in volatility, that is each h it in (184) is a linear combination of K<N common trends. Harvey, Ruiz and Shephard (1994) apply the nonstationary model to four exchange rates and find just two common factors driving volatility. Other ways of incorporating factor structures into multivariate models are described in Andersen et al. Chapter 15 in this Handbook. 11. Conclusions The principal structural time series models can be regarded as regression models in which the explanatory variables are functions of time and the parameters are time- varying. As such they provide a model based method of forecasting with an implicit weighting scheme that takes account of the properties of the time series and its salient features. The simplest procedures coincide with ad hoc methods that typically do well Ch. 7: Forecasting with Unobserved Components Time Series Models 407 in forecasting competitions. For example the exponentially weighted moving average is rationalized by a random walk plus noise, though once non-Gaussian models are brought into the picture, exponentially weighting can also be shown to be appropriate for distributions such as the Poisson and binomial. Because of the interpretation in terms of components of interest, model selection of structural time series models does not rely on correlograms and related statistical devices. This is important, since it means that the models chosen are typically more robust to changes in structure as well as being less susceptible to the distortions caused by sampling error. Furthermore, plausible models can be selected in situations where the observations are subject to data irregularities. Once a model has been chosen, problems like missing observations are easily handled within the state space framework. Indeed, even irregularly spaced observations are easily dealt with as the principal structural time series models can be set up in continuous time and the implied discrete time state space form derived. The structural time series model framework can be adapted to produce forecasts – and ‘nowcasts’ – for a target series taking account of the information in an auxiliary series – possibly at a different sampling interval. Again the freedom from the model selection procedures needed for autoregressive-integrated-moving average models and the flexibility afforded by the state space form is of crucial importance. As well as drawing attention to some of the attractions of structural time series mod- els, the chapter has also set out some basic results for the state space form and derived some formulae linking models that can be put in this form with autoregressive inte- grated moving average and autoregressive representations. In a multivariate context, the vector error correction representation of a common trends structural time series model is obtained. Finally, it is pointed out how recent advances in computer intensive methods have opened up the way to dealing with non-Gaussian and nonlinear models. Such models may be motivated in a variety of ways: for example by the need to fit heavy tailed distributions in order to handle outliers and structural breaks in a robust fashion or by a complex nonlinear functional form suggested by economic theory. Acknowledgements I would like to thank Fabio Busetti, David Dickey, Siem Koopman, Ralph Snyder, Allan Timmermann, Thomas Trimbur and participants at the San Diego conference in April 2004 for helpful comments on earlier drafts. The material in the chapter also provided the basis for a plenary session at the 24th International Symposium of Forecasting in Sydney in July 2004 and the L. Solari lecture at the University of Geneva in November 2004. The results in Section 7.3 on the VECM representation of the common trends model were presented at the EMM conference in Alghero, Sardinia in September, 2004 and I’m grateful to Soren Johansen and other participants for their comments. 408 A. Harvey References Anderson, B.D.O., Moore, J.B. (1979). Optimal Filtering. Prentice-Hall, Englewood Cliffs, NJ. Andrews, R.C. (1994). “Forecasting performance of structural time series models”. Journal of Business and Economic Statistics 12, 237–252. Assimakopoulos, V., Nikolopoulos, K. (2000). “The theta model: A decomposition approach to forecasting”. International Journal of Forecasting 16, 521–530. Bazen, S., Marimoutou, V. (2002). “Looking for a needle in a haystack? A re-examination of the time series relationship between teenage employment and minimum wages in the United States”. Oxford Bulletin of Economics and Statistics 64, 699–725. Bergstrom, A.R. (1984). “Continuous time stochastic models and issues of aggregation over time”. In: Griliches, Z., Intriligator, M. (Eds.), Handbook of Econometrics, vol. 2. North-Holland, Amsterdam, pp. 1145–1212. Box, G.E.P., Jenkins, G.M. (1976). Time Series Analysis: Forecasting and Control, revised ed. Holden-Day, San Francisco. Box, G.E.P., Pierce, D.A., Newbold, P. (1987). “Estimating trend and growth rates in seasonal time series”. Journal of the American Statistical Association 82, 276–282. Brown, R.G. (1963). Smoothing, Forecasting and Prediction. Prentice-Hall, Englewood Cliffs, NJ. Bruce, A.G., Jurke, S.R. (1996). “Non-Gaussian seasonal adjustment: X-12-ARIMA versus robust structural models”. Journal of Forecasting 15, 305–328. Burridge, P., Wallis, K.F. (1988). “Prediction theory for autoregressive-moving average processes”. Econo- metric Reviews 7, 65–69. Busetti, F., Harvey, A.C. (2003). “Seasonality tests”. Journal of Business and Economic Statistics 21, 420– 436. Canova, F., Hansen, B.E. (1995). “Are seasonal patterns constant over time? A test for seasonal stability”. Journal of Business and Economic Statistics 13, 237–252. Carnero, M.A., Pena, D., Ruiz, E. (2004). “Persistence and kurtosis in GARCH and stochastic volatility models”. Journal of Financial Econometrics 2, 319–342. Carter, C.K., Kohn, R. (1994). “On Gibbs sampling for state space models”. Biometrika 81, 541–553. Carvalho, V.M., Harvey, A.C. (2005). “Growth, cycles and convergence in US regional time series”. Interna- tional Journal of Forecasting 21, 667–686. Chambers, M.J., McGarry, J. (2002). “Modeling cyclical behaviour with differential–difference equations in an unobserved components framework”. Econometric Theory 18, 387–419. Chatfield, C., Koehler, A.B., Ord, J.K., Snyder, R.D. (2001). “A new look at models for exponential smooth- ing”. The Statistician 50, 147–159. Chow, G.C. (1984). “Random and changing coefficient models”. In: Griliches, Z., Intriligator, M. (Eds.), Handbook of Econometrics, vol. 2. North-Holland, Amsterdam, pp. 1213–1245. Clements, M.P., Hendry, D.F. (1998). Forecasting Economic Time Series. Cambridge University Press, Cam- bridge. Clements, M.P., Hendry, D.F. (2003). “Economic forecasting: Some lessons from recent research”. Economic Modelling 20, 301–329. Dagum, E.B., Quenneville, B., Sutradhar, B. (1992). “Trading-day multiple regression models with random parameters”. International Statistical Review 60, 57–73. Davidson, J., Hendry, D.F., Srba, F., Yeo, S. (1978). “Econometric modelling of the aggregate time-series relationship between consumers’ expenditure and income in the United Kingdom”. Economic Journal 88, 661–692. de Jong, P., Shephard, N. (1995). “The simulation smoother for time series models”. Biometrika 82, 339–350. Durbin, J., Quenneville, B. (1997). “Benchmarking by state space models”. International Statistical Re- view 65, 23–48. Durbin, J., Koopman, S.J. (2000). “Time series analysis of non-Gaussian observations based on state-space models from both classical and Bayesian perspectives (with discussion)”. Journal of Royal Statistical Society, Series B 62, 3–56. Ch. 7: Forecasting with Unobserved Components Time Series Models 409 Durbin, J., Koopman, S.J. (2001). Time Series Analysis by State Space Methods. Oxford University Press, Oxford. Durbin, J., Koopman, S.J. (2002). “A simple and efficient simulation smoother for state space time series models”. Biometrika 89, 603–616. Engle, R.F. (1978). “Estimating structural models of seasonality”. In: Zellner, A. (Ed.), Seasonal Analysis of Economic Time Series. Bureau of the Census, Washington, DC, pp. 281–308. Engle, R., Kozicki, S. (1993). “Testing for common features”. Journal of Business and Economic Statistics 11, 369–380. Fleming, J., Kirby, C. (2003). “A closer look at the relation between GARCH and stochastic autoregressive volatility”. Journal of Financial Econometrics 1, 365–419. Franses, P.H., Papp, R. (2004). Periodic Time Series Models. Oxford University Press, Oxford. Frühwirth-Schnatter, S. (1994). “Data augmentation and dynamic linear models”. Journal of Time Series Analysis 15, 183–202. Frühwirth-Schnatter, S. (2004). “Efficient Bayesian parameter estimation”. In: Harvey, A.C., et al. (Eds.), State Space and Unobserved Component Models. Cambridge University Press, Cambridge, pp. 123–151. Fuller, W.A. (1996). Introduction to Statistical Time Series, second ed. Wiley, New York. Ghysels, E., Harvey, A.C., Renault, E. (1996). “Stochastic volatility”. In: Maddala, G.S., Rao, C.R. (Eds.), Handbook of Statistics, vol. 14. Elsevier, Amsterdam, pp. 119–192. Grunwald, G.K., Hamza, K., Hyndman, R.J. (1997). “Some properties and generalizations of non-negative Bayesian time series models”. Journal of the Royal Statistical Society, Series B 59, 615–626. Hamilton, J.D. (1989). “A new approach to the economic analysis of nonstationary time series and the business cycle”. Econometrica 57, 357–384. Hang, J.J. (1998). “Stochastic volatility and option pricing”. In: Knight, J., Satchell, S. (Eds.), Forecasting Volatility. Butterworth-Heinemann, Oxford, pp. 47–96. Hannan, E.J., Terrell, R.D., Tuckwell, N. (1970). “The seasonal adjustment of economic time series”. Inter- national Economic Review 11, 24–52. Harrison, P.J., Stevens, C.F. (1976). “Bayesian forecasting”. Journal of the Royal Statistical Society, Series B 38, 205–247. Harvey, A.C. (1984). “A unified view of statistical forecasting procedures (with discussion)”. Journal of Fore- casting 3, 245–283. Harvey, A.C. (1989). Forecasting, Structural Time Series Models and Kalman Filter. Cambridge University Press, Cambridge. Harvey, A.C. (2001). “Testing in unobserved components models”. Journal of Forecasting 20, 1–19. Harvey, A.C., Chung, C H. (2000). “Estimating the underlying change in unemployment in the UK (with discussion)”. Journal of the Royal Statistical Society, Series A 163, 303–339. Harvey, A.C., de Rossi, G. (2005). “Signal extraction”. In: Patterson, K., Mills, T.C. (Eds.), Palgrave Hand- book of Econometrics, vol. 1. Palgrave MacMillan, Basingstoke, pp. 970–1000. Harvey, A.C., Fernandes, C. (1989). “Time series models for count data or qualitative observations”. Journal of Business and Economic Statistics 7, 409–422. Harvey, A.C., Jaeger, A. (1993). “Detrending, stylized facts and the business cycle”. Journal of Applied Econometrics 8, 231–247. Harvey, A.C., Koopman, S.J. (1992). “Diagnostic checking of unobserved components time series models”. Journal of Business and Economic Statistics 10, 377–389. Harvey, A.C., Koopman, S.J. (1993). “Forecasting hourly electricity demand using time-varying splines”. Journal of American Statistical Association 88, 1228–1236. Harvey, A.C., Koopman, S.J. (2000). “Signal extraction and the formulation of unobserved components mod- els”. Econometrics Journal 3, 84–107. Harvey, A.C., Koopman, S.J., Riani, M. (1997). “The modeling and seasonal adjustment of weekly observa- tions”. Journal of Business and Economic Statistics 15, 354–368. Harvey, A.C., Ruiz, E., Shephard, N. (1994). “Multivariate stochastic variance models”. Review of Economic Studies 61, 247–264. 410 A. Harvey Harvey, A.C., Scott, A. (1994). “Seasonality in dynamic regression models”. Economic Journal 104, 1324– 1345. Harvey, A.C., Shephard, N. (1996). “Estimation of an asymmetric stochastic volatility model for asset re- turns”. Journal of Business and Economic Statistics 14, 429–434. Harvey, A.C., Snyder, R.D. (1990). “Structural time series models in inventory control”. International Journal of Forecasting 6, 187–198. Harvey, A.C., Todd, P.H.J. (1983). “Forecasting economic time series with structural and Box–Jenkins models (with discussion)”. Journal of Business and Economic Statistics 1, 299–315. Harvey, A.C., Trimbur, T. (2003). “General model-based filters for extracting cycles and trends in economic time series”. Review of Economics and Statistics 85, 244–255. Harvey, A.C., Trimbur, T., van Dijk, H. (2006). “Trends and cycles in economic time series: A Bayesian approach”. Journal of Econometrics. In press. Hillmer, S.C. (1982). “Forecasting time series with trading day variation”. Journal of Forecasting 1, 385–395. Hillmer, S.C., Tiao, G.C. (1982). “An ARIMA-model-based approach to seasonal adjustment”. Journal of the American Statistical Association 77, 63–70. Hipel, R.W., McLeod, A.I. (1994). Time Series Modelling of Water Resources and Environmental Systems. Developments in Water Science, vol. 45. Elsevier, Amsterdam. Holt, C.C. (1957). “Forecasting seasonals and trends by exponentially weighted moving averages”. ONR Research Memorandum 52, Carnegie Institute of Technology, Pittsburgh, PA. Hull, J., White, A. (1987). “The pricing of options onassets with stochastic volatilities”. Journal of Finance 42, 281–300. Hyndman, R.J., Billah, B. (2003). “Unmasking the Theta method”. International Journal of Forecasting 19, 287–290. Ionescu, V., Oara, C., Weiss, M. (1997). “General matrix pencil techniques for the solution of algebraic Riccati equations: A unified approach”. IEEE Transactions in Automatic Control 42, 1085–1097. Jacquier, E., Polson, N.G., Rossi, P.E. (1994). “Bayesian analysis of stochastic volatility models (with discus- sion)”. Journal of Business and Economic Statistics 12, 371–417. Johansen, S. (1995). Likelihood-Based Inference in Co-Integrated Vector Autoregressive Models. Oxford University Press, Oxford. Johnston, F.R., Harrison, P.J. (1986). “The variance of lead time demand”. Journal of the Operational Research Society 37, 303–308. Jones, R.H. (1993). Longitudinal Data with Serial Correlation: A State Space Approach. Chapman and Hall, London. Kalman, R.E. (1960). “A new approach to linear filtering and prediction problems”. Journal of Basic Engi- neering, Transactions ASME. Series D 82, 35–45. Kim, C.J., Nelson, C. (1999). State-Space Models with Regime-Switching. MIT Press, Cambridge, MA. Kim, S., Shephard, N.S., Chib, S. (1998). “Stochastic volatility: Likelihood inference and comparison with ARCH models”. Review of Economic Studies 65, 361–393. Kitagawa, G. (1987). “Non-Gaussian state space modeling of nonstationary time series (with discussion)”. Journal of the American Statistical Association 82, 1032–1063. Kitagawa, G., Gersch, W. (1996). Smoothness Priors Analysis of Time Series. Springer-Verlag, Berlin. Koop, G., van Dijk, H.K. (2000). “Testing for integration using evolving trend and seasonals models: A Bayesian approach”. Journal of Econometrics 97, 261–291. Koopman, S.J., Harvey, A.C. (2003). “Computing observation weights for signal extraction and filtering”. Journal of Economic Dynamics and Control 27, 1317–1333. Koopman, S.J., Harvey, A.C., Doornik, J.A., Shephard, N. (2000). “STAMP 6.0 Structural Time Series Analyser, Modeller and Predictor”. Timberlake Consultants Ltd., London. Kozicki, S. (1999). “Multivariate detrending under common trend restrictions: Implications for business cycle research”. Journal of Economic Dynamics and Control 23, 997–1028. Krane, S., Wascher, W. (1999). “The cyclical sensitivity of seasonality in U.S. employment”. Journal of Monetary Economics 44, 523–553. Ch. 7: Forecasting with Unobserved Components Time Series Models 411 Kuttner, K.N. (1994). “Estimating potential output as a latent variable”. Journal of Business and Economic Statistics 12, 361–368. Lenten, L.J.A., Moosa, I.A. (1999). “Modelling the trend and seasonality in the consumption of alcoholic beverages in the United Kingdom”. Applied Economics 31, 795–804. Luginbuhl, R., de Vos, A. (1999). “Bayesian analysis of an unobserved components time series model of GDP with Markov-switching and time-varying growths”. Journal of Business and Economic Statistics 17, 456–465. MacDonald, I.L., Zucchini, W. (1997). Hidden Markov Chains and other Models for Discrete-Valued Time Series. Chapman and Hall, London. Makridakis, S., Hibon, M. (2000). “The M3-competitions: Results, conclusions and implications”. Interna- tional Journal of Forecasting 16, 451–476. Maravall, A. (1985). “On structural time series models and the characterization of components”. Journal of Business and Economic Statistics 3, 350–355. McCullagh, P., Nelder, J.A. (1983). Generalised Linear Models. Chapman and Hall, London. Meinhold, R.J., Singpurwalla, N.D. (1989). “Robustification of Kalman filter models”. Journal of the Ameri- can Statistical Association 84, 479–486. Moosa, I.A., Kennedy, P. (1998). “Modelling seasonality in the Australian consumption function”. Australian Economic Papers 37, 88–102. Morley, J.C., Nelson, C.R., Zivot, E. (2003). “Why are Beveridge–Nelson and unobserved components de- compositions of GDP so different?”. Review of Economic and Statistics 85, 235–244. Muth, J.F. (1960). “Optimal properties of exponentially weighted forecasts”. Journal of the American Statis- tical Association 55, 299–305. Nerlove, M., Wage, S. (1964). “On the optimality of adaptive forecasting”. Management Science 10, 207–229. Nerlove, M., Grether, D.M., Carvalho, J.L. (1979). Analysis of Economic Time Series. Academic Press, New Yo rk. Nicholls, D.F., Pagan, A.R. (1985). “Varying coefficient regression”. In: Hannan, E.J., Krishnaiah, P.R., Rao, M.M. (Eds.), Handbook of Statistics, vol. 5. North-Holland, Amsterdam, pp. 413–450. Ord, J.K., Koehler, A.B., Snyder, R.D. (1997). “Estimation and prediction for a class of dynamic nonlinear statistical model”. Journal of the American Statistical Association 92, 1621–1629. Osborn, D.R., Smith, J.R. (1989). “The performance of periodic autoregressive models in forecasting U.K. consumption”. Journal of Business and Economic Statistics 7, 117–127. Patterson, K.D. (1995). “An integrated model of the date measurement and data generation processes with an application to consumers’ expenditure”. Economic Journal 105, 54–76. Pfeffermann, D. (1991). “Estimation and seasonal adjustment of population means using data from repeated surveys”. Journal of Business and Economic Statistics 9, 163–175. Planas, C., Rossi, A. (2004). “Can inflation data improve the real-time reliability of output gap estimates?”. Journal of Applied Econometrics 19, 121–133. Proietti, T. (1998). “Seasonal heteroscedasticity and trends”. Journal of Forecasting 17, 1–17. Proietti, T. (2000). “Comparing seasonal components for structural time series models”. International Journal of Forecasting 16, 247–260. Quenneville, B., Singh, A.C. (2000). “Bayesian prediction MSE for state space models with estimated para- meters”. Journal of Time Series Analysis 21, 219–236. Rosenberg, B. (1973). “Random coefficient models: The analysis of a cross-section of time series by stochas- tically convergent parameter regression”. Annals of Economic and Social Measurement 2, 399–428. Schweppe, F. (1965). “Evaluation of likelihood functions for Gaussian signals”. IEEE Transactions on Infor- mation Theory 11, 61–70. Shephard, N. (2005). Stochastic Volatility. Oxford University Press, Oxford. Shephard, N., Pitt, M.K. (1997). “Likelihood analysis of non-Gaussian measurement time series”. Bio- metrika 84, 653–667. Smith, R.L., Miller, J.E. (1986). “A non-Gaussian state space model and application to prediction of records”. Journal of the Royal Statistical Society, Series B 48, 79–88. 412 A. Harvey Snyder, R.D. (1984). “Inventory control with the Gamma probability distribution”. European Journal of Op- erational Research 17, 373–381. Stoffer, D., Wall, K. (2004). “Resampling in state space models”. In: Harvey, A.C., Koopman, S.J., Shephard, N. (Eds.), State Space and Unobserved Component Models. Cambridge University Press, Cambridge, pp. 171–202. Taylor, S.J. (1994). “Modelling stochastic volatility”. Mathematical Finance 4, 183–204. Trimbur, T. (2006). “Properties of higher order stochastic cycles”. Journal of Time Series Analysis 27, 1–17. Visser, H., Molenaar, J. (1995). “Trend estimation and regression analysis in climatological time series: An application of structural time series models and the Kalman filter”. Journal of Climate 8, 969–979. Watanabe, T. (1999). “A non-linear filtering approach to stochastic volatility models with an application to daily stock returns”. Journal of Applied Econometrics 14, 101–121. Wells, C. (1996). The Kalman Filter in Finance. Kluwer Academic Publishers, Dordrecht. West, M., Harrison, P.J. (1989). Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York. Winters, P.R. (1960). “Forecasting sales by exponentially weighted moving averages”. Management Sci- ence 6, 324–342. Young, P. (1984). Recursive Estimation and Time-Series Analysis. Springer-Verlag, Berlin. Yu, J. (2005). “On leverage in a stochastic volatility model”. Journal of Econometrics 127, 165–178. Chapter 8 FORECASTING ECONOMIC VARIABLES WITH NONLINEAR MODELS TIMO TERÄSVIRTA Stockholm School of Economics Contents Abstract 414 Keywords 415 1. Introduction 416 2. Nonlinear models 416 2.1. General 416 2.2. Nonlinear dynamic regression model 417 2.3. Smooth transition regression model 418 2.4. Switching regression and threshold autoregressive model 420 2.5. Markov-switching model 421 2.6. Artificial neural network model 422 2.7. Time-varying regression model 423 2.8. Nonlinear moving average models 424 3. Building nonlinear models 425 3.1. Testing linearity 426 3.2. Building STR models 428 3.3. Building switching regression models 429 3.4. Building Markov-switching regression models 431 4. Forecasting with nonlinear models 431 4.1. Analytical point forecasts 431 4.2. Numerical techniques in forecasting 433 4.3. Forecasting using recursion formulas 436 4.4. Accounting for estimation uncertainty 437 4.5. Interval and density forecasts 438 4.6. Combining forecasts 438 4.7. Different models for different forecast horizons? 439 5. Forecast accuracy 440 5.1. Comparing point forecasts 440 6. Lessons from a simulation study 444 Handbook of Economic Forecasting, Volume 1 Edited by Graham Elliott, Clive W.J. Granger and Allan Timmermann © 2006 Elsevier B.V. All rights reserved DOI: 10.1016/S1574-0706(05)01008-6 . different forecast horizons? 439 5. Forecast accuracy 440 5.1. Comparing point forecasts 440 6. Lessons from a simulation study 444 Handbook of Economic Forecasting, Volume 1 Edited by Graham Elliott,. of Economic Dynamics and Control 23, 997–1028. Krane, S., Wascher, W. (1999). “The cyclical sensitivity of seasonality in U.S. employment”. Journal of Monetary Economics 44, 523–553. Ch. 7: Forecasting. Journal of Forecasting 6, 187–198. Harvey, A.C., Todd, P.H.J. (1983). Forecasting economic time series with structural and Box–Jenkins models (with discussion)”. Journal of Business and Economic