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Time Series: Economic

Brockwell P J, Davis R A 1991 Time Series: Theory and Panel Data; Simultaneous Equation Estimation: OŠer- Methods, 2nd edn. Springer, New York Šiew). Time-series models typically forecast the vari- Crame! r H 1942 On harmonic analysis of certain function spaces. Š W able(s) of by implicitly extrapolating past Arki fur Matematik, Astronomi och Fysik. 28B(12): 1–7 policies into the future, while structural models, Dagum E B 1980 The X-11 ARIMA Seasonal Adjustment Method. Research Paper, , Canada because they rely on economic theory, can evaluate Geweke J F 1978 The revision of seasonally adjusted time series. hypothetical policy changes. In this light, perhaps it Proceedings of the and Section— is not surprising that time-series models typically American Statistical Association, pp. 320–25 produce forecasts as good as, or better than, far more Granger C W J, Newbold P 1977 Forecasting Economic Time complicated structural models. Still, it was an intel- Series. Academic Press, New York lectual watershed when several studies in the 1970s Hamilton J 1994 Time Series Analysis. Princeton University (reviewed in Granger and Newbold 1986) showed that Press, Princeton, NJ simple univariate time-series models could outforecast Harvey A 1993 Time Series Models, 2nd edn. MIT Press, the large structural models of the day, a result which Cambridge, UK Kolmorgorov A N 1940 Kurven in Hilbertschen Raum die continues to be true (see McNees 1990). This good gegenu$ ber eine einparametrigen Gruppe von Bewegungen forecasting performance, plus the relatively low cost of invariant sind. C. R. (Doklady) de L’Academie des Sciences de developing and maintaining time-series forecasting l’IRSS, New Series 26: 6–9 models, makes time-series modeling an attractive way Nerlove M, Grether D M, Carvalho J L 1979 Analysis of to produce baseline economic forecasts. Economic Time Series. Academic Press, New York At a general level, time-series forecasting models Priestly M B 1981 Spectral Analysis and Time Series. Academic can be written, Press, New York Wallis K F 1974 Seasonal adjustment and relations between y l g(X , θ)jε (1) variables. Journal of the American Statistical Association 69: t+h t t+h 18–31 where yt denotes the variable or variables to be M. W. Watson forecast, t denotes the date at which the forecast is made, h is the forecast horizon, Xt denotes the variables used at date t to make the forecast, θ is a ε vector of parameters of the function g, and t+h denotes the forecast error. The variables in Xt usually in- clude current and lagged values of yt. It is useful to Time Series: define the forecast error in (1) such that it has ε Q l conditional mean zero, that is, E( t+h Xt) 0. Thus, Time-series forecasts are used in a wide range of given the predictor variables Xt, under mean-squared economic activities, including setting monetary and error loss the optimal forecast of y is its conditional θ t+h fiscal policies, state and local budgeting, financial mean, g(Xt, ). Of course, this forecast is infeasible , and financial engineering. Key elements because in practice neither g nor θ are known. The task of economic forecasting include selecting the fore- of the time-series forecaster therefore is to select the θ casting model(s) appropriate for the problem at hand, predictors Xt, to approximate g, and to estimate in assessing and communicating the asso- such a way that the resulting forecasts are reliable and ciated with a forecast, and guarding against model have mean-squared forecast errors as close as possible instability. to that of the optimal infeasible forecast. Time-series models are usefully separated into univariate and multivariate models. In univariate 1. Time Series Models for Economic Forecasting models, Xt consists solely of current and past values of yt. In multivariate models, this is augmented by data Broadly speaking, statistical approaches to economic on other time series observed at date t. The next forecasting fall into two categories: time-series subsections provide a brief survey of some leading methods and structural economic models. Time-series time-series models used in economic forecasting. For methods use economic theory mainly as a guide to simplicity attention is restricted to one-step ahead variable selection, and rely on past patterns in the data forecasts (h l 1). Here, the focus is on forecasting in a to predict the future. In contrast, structural economic stationary environment; the issue of nonstationarity in models take as a starting point formal economic theory the form of structural breaks or time varying para- and attempt to translate this theory into empirical meters is returned to below. relations, with parameter values either suggested by theory or estimated using historical data. In practice, time-series models tend to be small with at most a 1.1 UniŠariate Models handful of variables, while structural models tend to be large, simultaneous equation systems which some- Univariate models can be either linear, so that g is times incorporate hundreds of variables (see Economic linear in Xt, or nonlinear. All linear time-series models 15721 Time Series: Economic Forecasting can be interpreted as devices for modeling the co- data that switches between the ‘regimes’ α(L) and o q l β variance structure of the data. If yt , t 1, … ,T has a (L). Variations of Eqn. (2) interact dt with only some Gaussian and if this distribution is sta- autoregressive coefficients, the intercept, and\or the tionary (does not depend on time), then the optimal error variance. Various functions are available for dt, forecast is a linear combination of past values of the either a sharp indicator function (the threshold auto- data with constant weights. Different linear time-series regressive model) or a smooth function (smooth l models provide different parametric approximations transition autoregression). For example, setting dt j γ jγh ζ −" to this optimal linear combination. (1 exp [ ! " t]) , yields the logistic smooth tran- The leading linear models are autoregressive sition autoregression (LSTAR) model, where ζ de- ζ t models, autoregressive–integrated moving-average notes current or past data, say, t might equal yt−k for (ARIMA) models, and unobserved components some k. See Granger and Tera$ svirta (1993) for models. additional details. For an application of threshold An autoregressive model of order p (AR(p)) is autoregressions (and other models) to forecasting US written, , see Montgomery et al. (1998). The Markov switching models are conceptually lµjα j ( jα jε lµjα jε yt+" "yt pyt−p+" t+" (L)yt t+", similar, except that the regime switch depends on an unobserved series. In the context of Eqn. (2), dt is where (µ, α ,…,α ) are unknown parameters, L is the modeled as unobserved and following a two-state " pα lag operator, and (L) is a lag polynomial. If yt is Markov process; see Hamilton (1994). observed at all dates, t l 1, … , T, these parameters An alternative approach is to treat g itself as are readily estimated by ordinary least squares. unknown, which leads to nonparametric methods. ARIMA models extend autoregressive models to These include nearest neighbor, kernel, and artificial include a moving-average term in the error, which has neural network models for the conditional expecta- the effect of inducing long lags in the forecast when it tion. In practice these nonparametric approaches to is written as a function of current and past values of economic forecasting have met with success which is yt. These are discussed in Time Series: ARIMA mixed at best. There are several possible reasons for Methods. this, including that there are insufficiently many In an unobserved components model, yt is repre- observations in typical sets to support sented as the sum of two or more different stochastic the use of these data-intensive methods. components. These components are then modeled, and the combined model provides a parametric ap- proximation to the autocovariances; see Harvey (1989) 1.2 MultiŠariate Models for a detailed modern treatment. The unobserved components framework can provide a useful frame- In multivariate time-series models, Xt includes mul- work for extracting cyclical components of economic tiple time-series that can usefully contribute to fore- time-series and for seasonal adjustment (see Time casting yt+". The choice of these series is typically Series: Seasonal Adjustment). guided by both empirical experience and by economic o q If yt is not Gaussian, in general the optimal theory, for example, the theory of the term structure of forecast will not be linear, which suggests the use of interest rates suggests that the spread between long forecasts based on nonlinear time-series models. Para- and short term interest rates might be a useful metric nonlinear time-series models posit a functional predictor of future inflation. form g, and take this as known up to the finite- The multivariate extension of the univariate auto- dimensional parameter vector θ. Two leading ex- regression is the (VAR), in amples of parametric nonlinear time-series models, which a vector of time-series variables, Yt+", is repre- currently popular for economic forecasting, are thres- sented as a linear function of Yt,…,Yt−p+", perhaps hold autoregressive models and Markov switching with deterministic terms (an intercept or trends). An models. Both are similar in the sense that they posit interesting possibility arises in VARs that is not two (or more) regimes. In the threshold autoregres- present in univariate autoregressions, specifically, it sion, switches between the regimes occur based on past might be that the time-series are cointegrated (that is, values of the observed data; in Markov switching the individual series are nonstationary in the sense that models, the switches occur based on an unobserved or they are integrated, but linear combinations of the latent variable. series are integrated or order zero); see Time Series: The threshold autoregressive family of models can Co-integration and Watson (1994). be written Multivariate time-series models involve a large number of unknown parameters, a problem which is l α j k β jε yt+" dt (L)yt (1 dt) (L)yt t+" (2) greatly exacerbated when nonlinearities are intro- duced. Conceptually, the extension of univariate where the mean is suppressed, α(L) and β(L) are lag nonlinear models to the multivariate setting is straight- polynomials, and dt is a nonlinear function of past forward. In practice, however, because of the relatively 15722 Time Series: Economic Forecasting small number of time-series observations available to 2.2 OŠerfitting economic forecasters, it is unclear how best to im- Overfitting poses a particular threat in economic plement nonlinear multivariate models and there are forecasting given the impossibility of creating new currently no definite conclusions in this area. data sets through experiments and given the relatively short number of observations on many economic time-series. Overfitting can be addressed in part by 1.3 Models for Forecasting Higher Moments relying on automatic methods for pruning a model. The foregoing categorization of forecasting models Leading such methods are model selection using has focused on prediction of conditional means. In information criteria such as the Akaike or Bayes Model Testing and Selection, some economic applications, especially in financial information criteria (see Theory of , interest is also in forecasting conditional ). Alternatives include Bayes or Empirical higher moments. The most developed tools in this area Bayes estimation. Combining forecasts by taking are models of conditional variances. Two families of averages or weighted averages can also be interpreted such models are autoregressive conditional hetero- as a method for addressing the overfitting that arises in skedasticity (ARCH) models (Engle 1982; see model selection. Bollerslev et al. 1994) and stochastic volatility models. Both approaches provide parametric models for the volatility of the series, which can then be forecasted 2.3 Difficulties Modeling Trends given parameters estimated from historical data. An important characteristic of economic data is that many economic time-series contain considerable per- sistence, sometimes resembling a trend. Starting with 2. When Do Time-series Forecasts Fail? the seminal work of Nelson and Plosser (1982), there has been considerable debate about whether these Regrettably, only by luck will economic forecasts be trends are best modeled as deterministic or stochastic ‘right’: there are inevitable sources of uncertainty that (that is, arising from unit autoregressive roots), see make perfectly accurate predictions of the future Stock (1994). If a forecasting model incorrectly in- impossible. corporates deterministic trends, but the series is highly persistent, spuriously large weight can be placed on an estimated trend component and out-of-sample fore- 2.1 Shock, Model, and Estimation Uncertainty casts can suffer. Such considerations lead to the use of methods that explicitly address the possibilities of Sources of uncertainty that comprise the error term stochastic trends, an example being preliminarily ε t+h in (1) (‘shock uncertainty’) consist of unknowable testing for a unit root in a univariate time-series and future events, random measurement error in the series constructing a univariate model in levels or in differ- being forecast, and unforeseeable data revisions. ences depending on the outcome of the pretest. Additional forecast uncertainty arises from model approximation error (that is, error in the choice of g) and from error in the estimation of the model 2.4 Model Instability parameters θ. Sensible use of economic forecasts requires that this Clements and Hendry (1999) argue that many major forecast uncertainty be accurately communicated. failures of economic forecasts arise because of struc- Prediction intervals can be computed either directly tural breaks to which conventional forecasting models based on historically estimated parameters and resi- fail to adapt. In some cases, model instability can arise duals, or by simulation for more complicated non- because of slow changes in parameter values, perhaps linear models. Ideally these prediction intervals should arising from evolution in underlying technologies in incorporate parameter estimation uncertainty. There the . In other, more dramatic cases, model are several ways to do this, but a natural one is to instability appears as a sudden break, such as a change construct forecast error distributions using a simulated in or the collapse of an exchange rate out of sample prediction methodology. In this ap- regime. proach, the model is estimated using data through Whether structural change is gradual or abrupt, it is some date, say th, then a forecast is made for the next critical to monitor forecast performance and to test for period (thjh, for h-step ahead forecasts). The model is model instability. Various statistical methods are then reestimated using data through date thj1, and available for detecting structural change; these typi- the forecast is made for thjhj1. This is repeated until cally involve testing for evidence of instability either in a series of simulated out of sample forecast errors is estimated coefficients or in simulated forecast errors. constructed, which in turn can be used to construct In practice, these methods need to be augmented by prediction intervals around the point forecast of expert judgment that draws on information not interest. incorporated in the model, such as knowledge of

15723 Time Series: Economic Forecasting particularly severe weather (which could lead to a See also: , History of; Explanation: structural break being spuriously detected) or of a Conceptions in the Social Sciences; Quantification in known change in a policy regime (so a break might be the History of the Social Sciences; Time Series: imposed even if it is not detected statistically). Unfor- Advanced Methods; Time Series: General tunately, expert judgment can be wrong, and knowing when to intervene in an economic forecast is a rather murky art. Bibliography Bollerslev T, Engle R F, Nelson D B 1994 ARCH Models. In: Engle R, McFadden D (eds.) Handbook of Econometrics, 3. Which Methods Work Best? Elsevier, Amsterdam, Vol. IV, pp. 2959–3038 Clements M P, Hendry D F 1999 Forecasting Non-Stationary It is perhaps overly ambitious to hope for a simple Economic Time Series. MIT Press, Cambridge, MA prescription that provides a good forecasting method Engle R F 1982 Autoregressive conditional heteroskedasticity for all economic time-series. Nonetheless, a number of with estimates of the variance of UK inflation. Econometrica studies have attempted to provide some general 50: 987–1008 guidance on this question. Two general lessons Granger C W J, Newbold P 1986 Forecasting Economic Time emerge: among univariate models, simple linear Series, 2nd edn. Academic Press, Orlando, FL Granger C W J, Tera$ svirta T 1993 Modelling Non-linear Eco- models often forecast as well or better than more nomic Relationships. Oxford University Press, Oxford, UK complicated nonlinear models; and the gains from Hamilton J D 1994 Time Series Analysis. Princeton University moving from univariate to multivariate models are Press, Princeton, NJ often (but not always) small. Harvey A C 1990 Forecasting, Structural Time Series Models and This is illustrated by forecasts of the rate of the Kalman Filter. Cambridge University Press, Cambridge, inflation over six months in the , as UK measured by the percentage rate of inflation of the McNees S K 1990 The role of judgment in macroeconomic Consumer Price (this example is taken from forecasting accuracy. International Journal of Forecasting 6: Stock and Watson’s (1999) comparison of linear and 287–99 Montgomery A L, Zarnowitz V, Tsay R S, Tiao G C 1998 nonlinear forecasting models for 215 US macroecon- Forecasting the US unemployment rate. Journal of the omic time-series). Simulated out of sample six-month- American Statistical Association 93: 478–93 ahead forecasts over the period March 1971 to June Nelson C R, Plosser C I 1982 Trends and random walks in 1996 (the full sample begins January 1959) produce a macroeconomic time series—some evidence and implications. root mean-square forecast error (RMSFE) of 2.44 Journal of 10: 139–62 percentage points (at an annual rate) for a univariate Stock J H 1994 Unit roots, structural breaks, and trends. In: AR(4) fit recursively to the annualized rate of inflation. Engle R F, McFadden D (eds.) Handbook of Econometrics. If the autoregressive lag length is selected recursively Elsevier: Amsterdam, Vol. IV, pp. 2740–843 using the Bayes information criterion and a recursive Stock J H, Watson M W 1999 A comparison of linear and nonlinear univariate models for forecasting macroeconomic unit root protest is used to determine whether the AR time series. In: Engle R F, White H (eds.) Cointegration, is fit to the rate of inflation or the change in the rate, Causality, and Forecasting: a festschrift in honour of CliŠe the RMSFE drops to 2.05. Forecasts based on artificial W. J. Granger. Cambridge University Press, Cambridge, UK, neural networks and LSTAR models have RMSFEs pp. 1–44 exceeding 2.15 and in some cases exceeding 2.5, Watson M W 1994 Vector Autoregressions and cointegration. depending on the specification. A VAR which adds the In: Engle R, McFadden D (eds.) Handbook of Econometrics. unemployment rate and short-term (recur- Elsevier, Amsterdam, Vol. IV, pp. 2844–915 sive Bayes information criterion lag selection) pro- duces a RMSFE of 2.31. For these models, then, J. H. Stock introducing nonlinearity or adding these other pre- dictors does not improve upon a simple autoregression with a unit root pretest and data-dependent lag length selection. Time Series: General The future of economic forecasting promises the development of new, computer-intensive methods and the real-time availability of very large data sets with A time series is a stretch of values on the same scale complex patterns of historical dependence. These indexed by a time-like parameter. The basic data and developments could prove invaluable for advancing parameters are functions. economic forecasting, and the lessons from the past Time series take on a dazzling variety of shapes and four decades of economic forecasting might not extend forms, indeed there are as many time series as there are to these new datasets. Yet those lessons do underscore functions of real numbers. Some common examples of the importance of having a simple and reliable time series forms are provided in Fig. 1. One notes benchmark forecast against which alternative fore- periods, trends, wandering and integer-values. The casts can be assessed. time series such as those in the figure may be

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Copyright # 2001 Elsevier Science Ltd. All rights reserved. International Encyclopedia of the Social & Behavioral Sciences ISBN: 0-08-043076-7