Statistical Analysis and Forecasting of Economic Structural Change

The role of statistics in the detection and assimilation of structural change in econometric models is analyzed. Detection of structural change has been made much easier and more sophisticated by recent developments in graphical analysis and recursive estimation and testing techniques, particularly for use on microcomputers. A typology of models incorporating structural change is presented, and methods for discriminating between these models are considered. It is also argued that statistical tests for the hypothesis of structural constancy play an important role in the evaluation of econometric models. In addition, it is noted that major changes in the sample correlations between variables, rather than being a nuisance for econometric model builders, is in fact an important stimulus to model evaluation and improvement.

Tests for changes in the coefficients of linear regression models, particularly the analysis of covariance and the Chow tests, are well known to econometricians and they are widely used. This paper demonstrates that analogous tests can also be constructed in static simultaneous equation models when equations are estimated by common k-class estimators, e.g., OLS, 2SLS, and LIML. The tests are based on the residuals obtained when the estimated endogenous part of a simultaneous equation is regressed on all the exogenous variables in the system. The tests have many of the characteristics of the regression based tests although the nature of the residuals used makes it more difficult to analyse their power properties.

We present results of Monte Carlo simulations for model misspecification. Several variations of Stone’s (1974) Q ² statistic are computed using the jackknife procedure. These statistics measure the relative error in predicting the omitted observations using simple extrapolations. We compare the ability of these “predictive” statistics to detect model misspecification with the usual test statistics. The misspecifications we examined include omitted variables, nonlinearities, and structural change (shifts in coefficients during the sample period). Although none of the predictive statistics perform uniformly better over the entire range of misspecifications, in some cases, they detect departures from the “true model” with substantially greater frequency.

The chapter considers the rejection probability of the Chow test when there is unaccounted for autocorrelation among the disturbances in a linear regression model. The Chow test proved to be extremely nonrobust to autocorrelation. Its true size can even be one for the special case of an AR(1) disturbance process.

Procedures based on various types of counting statistics are considered for testing a sequence of independent random variables against trend alternatives. Power comparisons with standard parametric tests, are also performed, partly by analytical means and partly by Monte Carlo estimation. Some of the tests turn out to be strong competitors to the CUSUM procedure. Finally, the use of the nonparametric tests for the detection of parameter nonconstancy in regression models is discussed.

A variety of nonparametric and robust tests for the “change point” model relating to the location as well as regression problems is available in the literature. The main objective of the current study is to provide a broad coverage of the main stream of these develop­ments, encompassing both rank (R-) and maximum likelihood type (M-) procedures in a recursive as well as nonrecursive setup. Along with some interpretations of (asymptotic) optimality properties of nonparametric and robust tests for the change-point problems, suitable adaptive procedures are suggested, which achieve this optimality in a meaningful sense.

A Bayesian predictive density for the mean squared errors of post-sample forecasts is derived within the linear regression framework. The kernel of the predictive density is an F distribution. In the process of deriving the predictive density, we use a degenerate hyperbolic function to express the distribution of quadratic forms in normal variables. The Bayesian predictive density is then used to detect a join point by the highest posterior density interval criterion. Numerical examples are given to compare the Bayesian predictive density procedure with the maximum likelihood and Bayesian posterior density procedures for detecting the join point. When the join point is at either the beginning or ending edge of the sample period, the Bayesian predictive density procedure detects the join point whereas the maximum likelihood and Bayesian posterior density procedures cannot.

One way to represent a structural change when modeling an economic system is to allow for parameter changes. Besides verifying the existence of such parameter changes, the ultimate purpose of the structural analysis will be to further characterize them. This chapter includes a partial analysis of such a characterization process. Approximate expected values and variances of CUSUM-SQ and MOSUM-SQ statistics are given for various cases of parameter changes, instantaneous as well as gradual. A numerical simulation study of the statistics is also given. The theoretical study and the simulations together demonstrate that identification of time for structural changes is more intricate in the cases of gradual changes, and when the changes occur early during the observed time period.

This chapter considers the limiting behavior of the well-known CUSUM SQ test for suitably defined sequences of local alternatives describing heteroscedasticity. It is shown that under certain circumstances the asymptotic behavior of the CUSUM-SQ test can be computed even for alternatives describing changes of the conditional variance of the error term (e.g., ARCH-processes).

A sequence of independent random variables X ₁,...,X _N is said to have a change point n, if X ₁,...,X _N have a common distribution F and X _n+1,..., X _N have a common distribution G, G ≠ F. Consider the problem of testing the null hypothesis of no change, against the alternative of a one-sided change at an unknown change point n, when both F and G are normal with equal variance σ². Most of the test statistics for this problem can be interpreted as generalizations of two-sample statistics (n known). In this chapter we derive the Bahadur efficiencies for two classes of statistics that are generalizations of the two-sample likelihood ratio statistics. The asymptotic results are compared with some small-sample power estimates based on Monte Carlo experiments.

Recently developed techniques for graphic analysis of residuals have made them more legible. Things to look for in a plot are: (a) the largest residuals; (b) progressive change in the variability of the residuals; (c) a curved regression of residuals on fitted values or the number of cases; and (d) the subsets of successive residuals with significantly different configuration. In this chapter we concentrate on (d) which indicates possible structural changes. Using Monte Carlo experiments, we examine smoothing scatterplots of the transformed least squares residuals. To interpret the smoothed scatterplots of standardized residuals, we propose plotting a kind of confidence envelope around the smoothed curve, based on Atkinson (1981).

Heuristic and model-based approaches to adaptive estimation in regression models are reviewed in this chapter. We describe a model-based approach that introduces time-varying coefficients explicitly and assumes that the coefficients follow certain autoregressive integrated moving average time series processes. We show how these time-varying coefficient models can be written in state space form, we illustrate how the Kalman filter approach can be used to update the coefficient estimates and forecasts, and we discuss why the resulting estimates are more responsive to structural change than the standard least squares estimates. The parameters in the underlying stochastic processes that generate the time-varying coefficients are needed to update the coefficient estimates. It is shown how these parameters can be estimated from historic observations. These parameters determine how adaptive the resulting coefficient estimates are to changes in the coefficients.

A new adaptive method of analyzing a linear regression with time-varying coefficients is presented. The coefficients are adapted by means of exponentially weighted moving averages (EWMA). The coefficients’ trajectories imply possible improvements of the model specification. Aspects of suitable preparation of the time series, such as the elimination of time trends and parameter estimation, are also considered. The method is illustrated on the basis of both artificially generated and real economic data.

This chapter contains a survey of various econometric model formulations in which it is assumed that coefficients vary across time. Depending on the accepted parameter variation structure one may classify such models into two main groups: models with variable but nonstochastic parameters and models with randomly varying coefficients. The latter group consists of two types — models where coefficients are generated from stationary and models in which coefficients are generated from nonstationary stochastic processes. All three groups are surveyed. Several representative models from each group are shown with special emphasis on estimation, testing the specification and possible fields of implementation. Justification for the various model formulations is given. A detailed list of references ends the survey.

A sequence of observations y_t, t = 1, 2,..., N, is generated by the time-varying multiple regression model where, for t = 1, 2,..., N, u _t is an unobservable random variable with zero mean and unit variance, x _t is an observable p-vector-valued variable, and σ _t and β _t are, respectively, unobservable scalar and p-vector-valued parameters. No model (stochastic or nonstochastic) is assumed for the σ _t or β _t ; instead they are assumed to be smoothly varying over t, in a certain sense. A class of estimators of the β _t , σ _t is proposed, for each value of t; the estimators optimize a criterion prompted by Gaussian maximum likelihood considerations, and may be viewed as analogous to certain nonparametric function fitting estimators, employing a kernel function and band-width parameter, both selected by the practitioner. Consistency and asymptotic normality are established in case of independent u _t , and a consistent estimator of the asymptotic covariance matrix of the β _t estimators is given. Such results are also possible for serially correlated u _t . We discuss questions of implementation, in particular the choice of kernel function and band-width. Generalization of the class of estimators to include certain robust estimators is possible, as is generalization of the methods to more general models involving time-varying parameters.

Attention is drawn to the fact that a number of results from econometric analysis of regression models with unobservable variables can be readdressed using traditional regression analysis techniques. This observation is of importance in the choice of comparatively simple methods for handling corresponding problems, particulary in cases when unobservable explain structural changes in the final regression models.

This investigation introduces changing-parameter ARMA processes as a way to model a time series. Many time series exhibit a changing trend or a changing autocorrelation structure; that is to say, they have certain nonstationary characteristics that cannot be modeled by the usual ARMA representation. The analysis of a changing parameter process is accomplished by a Bayesian approach, where the posterior distributions of the parameters are derived, and the analysis is illustrated with a moving average model that has a changing autocorrelation function.

The central theme in this Chapter is unconventional analysis of time series data, the conventional one being that based on linear models (e.g., autoregressive/moving average models) and second-order moments (e.g., spectral analysis). After the natural emergence of thresholds, attention is focused on the stability of the global system in connection with that of each constituent subsystem delineated by the thresholds. Exotic results are obtained by relying on simple linear algebraic analysis of the system, which may be considered an application of symbolic dynamics. Some unexpected results are described in nonlinear forecasting, which expose a myth generated by linear mentality. Finally, comments are made about nonlinear modeling of irregularly sampled data.

The problem of optimal forecast combination is considered in situations of structural change. We develop a rather general approach, which combines the time-varying-parameter models of Diebold and Pauly (1987a) with allowance for prediction-error serial correlation as in Diebold (1988). The methodology is based on the regression-based paradigm of Granger and Ramanathan (1984), so that many earlier results emerge as special (and often restrictive) cases. Both deterministic and stochastic parameter variations are considered, with and without allowance for serial correlation. The results are illustrated in a series of examples.

A method by Gill, Golub, Murray, and Saunders for updating matrix factorizations is used to improve the computations necessary for detecting a change point of the regression line in linear models. The method shows considerably higher speed and better numerical stability than using standard routines for linear regression. It is based on updating the residual sum of squares and the least squares estimators for the regression parameters if one regression equation gets added to or deleted from the model.

An index of poverty reflects the extent to which individuals in a society or community fall below a minimal acceptable standard of living. It is generally framed in terms of a set poverty line, the income distribution of the poor, and other social welfare functions relevant to the poverty structure; the Gini coefficient plays a vital role in this context. The income distribution and other measures based on this distribution rarely remain stationary over time, so that in studying the poverty structure over a period of time, one essentially encounters a time-dependent model that may be analyzed in a parametric or nonparametric manner. In this context, the change point problem is very relevant, and the related methodology is considered in a systematic manner.

A mathematical model is introduced to explain the dynamics of the Hicksian IS-LM paradigm, in which the difference between the attitudes of Keynesians and monetarists is representable by a difference in the parameters of the model. Statistical identification procedures are introduced for both this model and for the nonstationary model of a time-varying Hicksian IS-LM structure. Application of the models to simulation data is also discussed, and numerical results are given.

Technical change greatly contributes to the explanation of economic structural changes. Numerous studies attempt to quantify and model this essential aspect of economic growth. This study surveys the models, their estimation techniques, and the problems and pitfalls in the applications. Particular emphasis is laid on aggregated production functions, factor productivity, and input-output approaches.

Macroeconomic time series often exhibit various nonstationary influences, such as outliers, breaks or jumps in levels. These imply very sensitive estimation not only for univariate and multivariate autoregressive time series models, but also for a Wiener-Granger causality analysis. This chapter investigates the impact of nonstationary behaviour by estimating local stationary AR processes. Two types of local stationarity analysis are proposed: a so-called consecutive bisectrix method, where time series are repeatedly halved as long as reasonable estimation is possible; and a certain span or estimation window moving along the time axis. For both methods, the Geweke (1982) causality measures are derived by comparing univariate and multivariate AR models for the same time spans. An example involving Austrian interest rates for the 1970s demonstrates the two approaches. It is shown that causalities changed considerably in the 1970 decade and that at least three different causality periods can be detected.

The aggregate investment schedule may be used to study the impact of various policy measures, such as changes in corporate tax rates, depreciation allowances, and investment tax credits. Its parameters should be invariant with respect to the policy changes themselves, a point forcefully stressed by Lucas (1976). On the impact of investment tax credits, Lucas makes two predictions: first, if the model is implemented under an assumption of static expectations (versus rational expectations) and estimated from a period during which policy rules changed appreciably, it will exhibit parameter instability; second, the impact of tax credits is likely to be heavily underestimated. This chapter presents empirical evidence on both these effects by studying a version of the Hall-Jorgenson model estimated from US data (1956–1972). For this purpose, we use recursive stability analysis, an exploratory methodology that makes very weak assumptions on the form of the instability to be detected and provides indications on the direction of prediction errors. The main finding is a discontinuity associated with the first imposition of the tax credit (1964–1966); further, this shift led to underprediction of investment. The results thus support Lucas’s hypothesis.