Economic Modelling and Forecasting via Adaptive Estimation

Copyright © IFAC Identification and System Parameter Estimation 1982 . Washington D.e. . USA 1982

ECONOMIC MODELLING AND FORECASTING VIA ADAPTIVE ESTIMATION D. G. Lainiotis* and L. S. Segal** ·State University of New York at Buffalo, Amherst, N. Y., USA; Present address: University of Patras, College of Engineering, Patras, Greece ··State University of New York at Buffalo, Amherst, N. Y. , USA; Present address: MITI Lincoln Laboratory, p.a. Box 73, Lexington, MA 02173, USA

Abstract. This paper will present new results on adaptive economic forecasting. Specifically, a new stochastic macro-economic model, based on Pindyck's model of the post-Korean war United States economy will be formulated. This model will then be extended into an adaptive framework. In addition, a new adaptive filter for time-varying unknown parameters will be developed. This filter will be derived from the partitioned adaptive filter of Lainiotis. The filter will then be applied to the adaptive economic model for the purpose of economic forecasting. Keywords. Economic Forecasting, Adaptive Estimation, Partitioned Filters, Economic Modelling, Economics, Adaptive Systems, Modelling, Filtering. model. This allows him to use the powerful state-variable techniques of control, estimation and parameter identification in analyzing the model. For further discussion on the differences between a control and economics theory point of view regarding modelling see Mehra [5].

INTRODUCTION The forecasting of economic time series is an extremely difficult problem. Most of the important econometric models are too large and have too high a computational burden to be implemented on a medium size computer with a moderate amount of processor time. For example, the Wharton [1] and St. Louis [2] models are large, complex models which produce excellent forecasts. However, these models have over a hundred state variables and it is useful to determine what kind of economic forecasting results may be achieved using a smaller econometric model.

This paper will concern itself with the application of modern control theory or (state variable theory) to economic forecasting. Chow [6] is an excellent general reference on state variable theory applied to economics. In addition, Segal [7] contains a complete bibliography of the subject. Specifically, this paper deals with the application of Lainiotis' [8] adaptive filter to economic forecasting. First, the optimal linear filter as derived by Kalman [9] will be presented and applied to a modification of Pindyck's economic model. An adaptive form of Pindyck's model will be developed and Lainiotis' adaptive filter will be applied to it. Finally, a generalization of the adaptive flter developed by Segal and Lainiotis [10,11] will be presented. This adaptive filter was shown to perform very well when applied to the adaptive forecasting of economic time series.

Pindyck [3] has developed a small, quarterly, linear econometric model of the postKorean war United States economy. The model consists of nine behavioral equations together with a tax relation and an income identity. It has twenty-eight state variables, three control variables and one exogenous variable. Pindyck was interested in studying optimal control theory with his model. As such, the model, although performing well for a model of its size, does not forecast as well as the larger models mentioned earlier. The forecasts obtained from these models could be improved via the use of linear and nonlinear estimation theory. Most of the previous work by economists involves the use of both linear and non linear regression techiques, for example see Johnston [4]. A control engineer when asked to model a dynamic system, would be more inclined to use a state-variable model than a regression

LINEAR ESTIMATION ALGORITHMS The linear estimation problem - both filtering and smoothing - has attracted considerable attention in the past quarter century, as can be seen in some recent surveys 1501

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of the subject [12,13]. Specifically, there have been numerous approaches to linear filtering resulting essentially in the fundamental Kalman filter [9]. In a different approach to nonlinear estimation in general and linear estimation in particular, Lainiotis [14,15], derived the partitioning estimation algorithms. These algorithms reduce the optimal linear estimator into a partitioned or decomposed form.

PARTITIONED ADAPTIVE FILTER The design of optimal estimators requires the availability of three elements: a mathematical model, a cost function, and the optimization constraints imposed on the problem by physical, economic, or realization considerations. Of the above elements, the model constitutes a cardinal part since it is the vital link with the physical problem in which the resulting estimator must be designed. The applicability of the estimator depends integrally on the realism with which the associated mathematical model represents the underlying physical situation. Unfortunately, in most physical situations, complete knowledge of the model is not available and one is confronted with the task of designing an estimator in the face of incomplete model knowledge. If, moreover, the estimator design is to be done in real-time, it constitutes an adaptive estimation problem. Adaptive estimation problems may be classified according to: the nature of the basic mathematical model used, e.g., linear, nonlinear, etc.; the nature of the model uncertainty, e.g., parametric or functional uncertainty, and time-invariant or timevarying uncertainties. Namely, the model uncertainty can be summarized by an unknown finite parameter vector-parametric uncertainty - or the function form of the model may be unknown in the case of structural model uncertainty. Moreover, the parametric of functional uncertainty may be time-varying or time-invariant. Lainiotis [8] has derived the optimal adaptive estimator for time-invariant parametric uncertainty. This algorithm is based on Lainiotis' partitioning approach to estimation. The optimal adaptive filter consists of a parallel bank of elemental estimators, which for linear models are optimal linear filters. Each of these optimal linear filters are matched to a particular value of the unknown parameter set theta. These linear filters are therefore "model conditioned" linear filters where each filter is conditioned on a particular value of theta. The optimal filter. requires an infinite number of optimal linear filters. In practice, the parameter 9 will be discretized,

and only a finite number of linear filters will be used. The optimal estimate is given as a weighted-sum of model-conditional discrete linear filters which are completely decoupled from each other. This constitutes a parallel realization structure, which is ideally suited to parallel processing which can greatly increase the computational speed of the algorithm.

TIME-VARYING PARAMETER UNCERTAINTY The partitioned adaptive filter of Lainiotis is not optimal if the parameter vector is a random process instead of a time invariant random variable. This paper will only consider parametric uncertainty where the parameter vector, 9, can vary within a finite number of values(r). For a general parameter process, the optimal adaptive solution requires that the number of optimal linear filters grows exponentially with time. The optimal adaptive filter requires a matched linear filter for every possible sequence of parameter values. The number of elemental estimators grows exponentially. Due to computational complexity, the optimal adaptive filter is difficult to implement. It would therefore be desirable to obtain a suboptimal filter realization whose performance is close to the optimal adaptive implementation but does not have its complexity. The partitioned adaptive filter of Lainiotis has been found to perform well for both time-invariant and time-varying model parameters [8]. In addition, the adaptive filter designed for the above problem only requies r elemental linear filters. It has been found to have several desirable computational properties such as robustness to modelling errors, speed of convergence, and also to rapidly yield goo~ parameter estimates. They have also been shown to track parameter changes fairly quickly. However, since the partitioned adaptive filter is suboptimal, it would be desirable to find an adaptive filter implementation whose performance is closer to optimal but without the computaional requirements of the optimal filter, or requiring more than r linear filters. In this section a new adaptive filter implementation will be introduced. This filter will be based on a modification of the partitioned adaptive filter of Lainiotis [8]. At specified time intervals the model conditional linear filters will be initialized with the adaptive estimate and its covariance matrix. In addition, the calcution of the a posteriori probability includes information about the markov nature

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of the parameter random process. Since for discrete time models this initialization can take place at each sample, the filter will be called the "Per-Sample Initialized Partitioned Adaptive Estimator" or PSIPAE. This algorithm has been developed by Segal and Lainiotis [10-11].

Lainiotis' original partitioned adaptive filter. This is true even though the initialization procedure "couples" the outputs of the bank of linear filters. However, the major c o mputati o nal burden, which occurs in the linear filters may still be done in parallel.

The PSIPAE adaptive filter is formed by initializing each of the model conditional linear filters with the adaptive estimate and its covariance. This makes intuitive sense, since it means each of the model conditional filters are using the "best" estimate of the process and its covariance instead of the model conditional estimates.

This new adaptive filter implementation is suboptimal. This can easily be seen since for time-invariant theta it does not reduce to Lainiotis' partitioned adaptive filter which is optimal. In terms of numerical performance, for time-invariant theta, the partitioned adaptive filter performs better than the PSIPAE although the PSIPAE results were c los e t o optimal. For time-varying theta, the PSIPAE perfo rmed much better than the partitioned adaptive filter.

The partitioned adaptive filter of Lainiotis is optimal for time-invariant unknown parameters. However, this optimality results in a strong bias in favor of the old model which delays the identification of a model change. The bias is evident in the model conditional state estimates, their covariances and the a posteriori model probabilities. The PSIPAE removes the effects of the bias in favor of the old model. By initializing the linear filter with the adaptive state estimate, the model conditional state estimate for the previous model will no longer be the closest to the actual state. This will prevent a false model from generating state estimates with a smaller mean-square-error. An additional sensitivity to model changes occurs by initalizing the covariance equation. As the error increases for a model with high probability, which indicates a model change, the adaptive covariance will increase. This will result in an increase in the filter gains which increases the sensitivity of the linear filters to the data.

By taking into account the markov nature of the parameter process in the a posteriori probability the bias in favor of the previous model as shown by the model probability will be replaced by a "predictive" model probability based on its known markov nature. The three initializations can be used in any combination. Extensive numerical simulation has shown that the state esti.mate initialization produces the greatest improvement in mean square estimate, while the filter does not appear to be highly sensitive to the covariance initialization. It should be noted that the use of the covariance initialization prevents the offline calculation of the filter gains. This results in an increase in real time computation due to the data dependence of the covariance. The PSIPAE adaptive filter maintains the parallel structure of

SEGAL-LAINI OTIS' VERSI ON OF PINDYCK'S ECONOMIC MODEL Pindyck's [3] macroeco nomic model of the United States generates excellent economic forecasting results f o r a mode l of its size. However, the mo del suffers from the inability to track some of the economic variables. The use of linear filters improves the dynamic response of the model, but this does not help when the problem of interest is the multi-period forecasting of e c onomic time-series. Since Pindyck's mo del does perform reasonably well, it woul d b e desirable to derive another mo del using the same economic state variables that Pindyck used. A different set of parame ter values can be estimated by performing the regr e ssion in a different manner, Segal [ 10 ] . Specifically, the matrices i n Pindyck's regression model are s parse matrices. This is due to Pindyck a s suming a specific form f o r ea c h of the nine behavi o ral equati o ns. This gives Pindyck o nly forty-eight c o efficients to estimate. However, due to the matrix inversion which is necessary to compute the state variable f o rm o f the model, the sparseness f o und in the regression equations does n o t appear in the stat variable model. It see ms r e aso nable that a "better" economi c mo d el might be generated by estimating the c oefficients of the state variable model directly. This model will be called the SL mo de l •

ADAPTIVE ECONOMI C MODELS Due to the nature of ec o n omic models, some of the paramters in the model may be sto chastic or unknown. These parameters may be either time-varying in a random manner o r time-invariant. Two adaptive e con omi c models will n ow be de rived b ased on the assumption that conditi o ned o n the unknown paramter, the true model is linear and

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time-invariant. This will allow the Partitioned Adaptive estimator of Lainiotis (8) or the Per-Sample Initialized Partitioned Adaptive filter of Segal and Lainiotis [ 10-11) to be used. The first adaptive model will be called the PAPE (Partitioned Adaptive Parameter Estimate) and is derived by reprocessing the parameter estimates from the partitioned adaptive filter of Lainiotis (8). The second model will be callen the TVIC (TimeVarying-Initial-Condition) and is derived from the time-varying initial condition partioned identification algorithm of Lainiotis [10,16). Both of these models will use as the basis the linear models the Segal-Lainiotis economic model. The unknown parameters are the coefficients of disposable income. For the purpose of these experiments only five linear models were used in the adaptive filters. This was done in order to reduce the computation cost. For improved forcasting ability the number of linear models could be increased. The parameters of the PAPE model are estimated by reprocessing the data with the "best" estimate of the unknown parameter. This will be done until the parameter estimates converge to some value. This is done in the following manner. A series of simulations will be run using five models. One model will be the current best estimate of the model parameters, (6 1n , 6 2n ). Four other models will be chosen with the following parameters: (6 1n + 6 1d , 6 2n + 6 2d ), (6 1n + 6 1d , 6 2n - 6 2d ), (6 1n - 6 1d , 6 2n + 6 2d ), (81 n - 6 1d , 6 2n - 6 2d ), where 6 1d and 6 2d are to be determined at the beginning of each simulation. When these two parameters converge two more parameters are estimated in the same manner. This procedure is repeated until all the parameters have been estimated. At this point all the parameters are again estimated using the above procedure. The entire parameter set will be estimated using this procedure until all the parameters converge. This procedure can be generalized for any number of unknown parameters of sample models. For example, if N is the number of unknown parameters, than 2N + 1 is the number of models needed for the parameter estimation algorithm. If this procedure generates too many models the N parameters can be divided into M sub-, sets, with Lj parameters in subset j(j = 1, M) • . Parameters estimation will be run using 2Lj + 1 models in each subset and holding their subsets at their current nominal values. Each subset will be varied sequen-

tially and the sequence will be repeated until the parameter set converges. The TVIC model parameters are estimated using Lainiotis' Time-Varying-Initial-Condition Partitioned identification algorithm [10,16) to estimate the nine unknown parameters in the SL model. This algorithm treats the effect of the unknown parameters as a timevarying-initial condition. It then uses the partitioned linear filter of Lainiotis to estimate the unknown initial condition due to the parameters to be estimated.

ECONOMIC FORECASTING In this paper, a series of digital simulations will be discussed which compared the forecasting ability of the Pindyck, and adaptive econometric models. The forecasts from these models will be compared with the historical economic time series. Two classes of data will be used. The first class will be training data or data used to create the model. Training data is from the time period 1953-3 to 1965-4. The second classs of data is test data. Test data will be considered prediction of data from outside the training interval. For this example, the test data is from the time period 1966-1 to 1970-4. The simulations being presented will consist of 50 quarters of data, the first 30 quarters being training data and the last 20 being test data, the simulations time period was therefore 1958-3 to 1970-4. The results for the individual experiments will now be presented. First, Pindyck's and the SL economic models will be compared. Looking at each state's error independently, it can be seen that for eight out of the ten states, the SL models has a smaller mean--square-error. Next the performance of adaptive models will be studied. All the adaptive filter-model combinations performed significantly better than the models using a linear filter. Figures 1 - 4 show the predicted and historical values of consumption. The PSIPAE filter performed better in terms of mean square error with the PAPE model while the partitioned adaptive filter gave a better performance when using the TVIC adaptive mode. This is possibly due to a more accurate model estimation in the PAPE model which allowed the filter to identify model changes quickly even though the TVIC model had a smaller mean mean square error.

CONCLUSION This paper presented some new results on adaptive economic forecasting. In addition, a new adaptive filter was developed based on Lainiotis' partitioned adaptive filter. These results showed that the use of both

Economic Modelling and Forecasting v ia Adaptive Es timatio n adaptive estimation and partitioned identification techniques could be used to reduce the mean-square-error in economic forecasting.

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in partial fulfillment of the requirements for the degree of Doctor of Philosophy. While at Buffalo, the research was supported by a Henry M. Woodburn Graduate School Fellowship.

ACKNOWLEDGMENT The research for this paper was conducted at the State University of New York at Buffalo

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Consumption Non Residential Investment Residental Investment Inventory Investment Short Term Interest Long Term Interest Price Deflator Unemployment Wage Rate Disposable Income

269.39 12.03 1.494 32.77 1.123 0.114 24.06 7.054E-5 1.21 337.8

40.25 6.282 2.722 73.35 0.937 0.337 2.041 1.924E-5 0.060502 56.85

28.08 3.365 2.062 18.86 0.7421 0.1318 1.561 2.138E-5 0.0579 18.129

41.827 1.9298 2.2312 15.4885 0.41516 0.1103 1.5061 1.662E-5 0.034677 47.736

15.433 2.4986 0.73541 9.7429 0.5538 0.96635 0.19743 4.065E-5 0.031798 29.4178

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BIBLIOGRAPHY (1) Klein, L. R. and M. Evans (1968). The Wharton Econometric Forecasting Model, Univ. of Penn, Philadelphia. [2] Anderson, L.C., K.M. Carlson (1975). A Monetarist Model for Economic Stabilization, Review Federal Reserve Bank of ~ Louis, pp. 7-25. [3] Pindyck, R.S. (1973). Optimal Planning for Economic Stabilization, North Holland Press, Amsterdam . (4) Johnston, J. (1972). Econometric Methods, McGraw-Hill, N.Y. [5] Mehra, R.K. (1974). Identification in Control and Econometrics: Similarities and Differences, Annals of Economic and Social Measurements, Vol. 3, pp. 21. (6) Chow, G. C., (1975). Analysis and Control of Dynamic Economic Systems, Wiley, N.Y. (7) Segal, L.S., P.D. Scott, D.G. Lainiotis (to be published) Applications of Modern Control Theory to Economics: Annotated Bibliography, ~ Cybernetics(USA). (8) Lainiotis, D.G. (1976). Partitioning: ~ Unifying Framework for Adaptive Systems ~ Estimation, Proc. IEEE (USA), 64 pp. 1126.

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[9 ] Kalman, R.E. (1960). A New Approach to Linear Filtering and Prediction Theory, ASME Trans. ~ Basic Engineering (USA), Vol. 82 pp. 35. [10] Segal, L.S. (1979). partitioning Algorithms with Applications !£ Economics, Ph.D. Dissertation, State University of New York at Buffalo. (11) Segal, L.S. and D.G. Lainiotis (1979). Partitionined Adaptive Estimation of Time-Varying Random Parameters with Applications to Economic Forecasting, Proc. Joint Automatic Control Conference, Denver (USA). (12) Meditch, J.S. (1973). A Survey of Smoothing for Linear and Nonlinear Dynamic Systems Automatica, (USA) Vol. 9 pp. 151. [13) Lainiotis, D.G. (1974) Estimation; A Brief Survey, Inf. Science, Vol. 7, pp. 191. (14) Lainiotis, D.G. (1974). Partitioned Estimation Algorithms, I; Nonlinear Estimation, J. Inf. Science, Vol. 7 pp. 203. (15) Lainiotis, D.G. (1974). Partitioned Estimation Algorithms 11: Linear Estimation, ~ Inf. Science, Vol. 7 pp. 317. (16) Lainiotis, D.G. (1973). Identification of Linear and Non-linear Systems, Proc. IEEE Decision and Control Conference (USA)

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