Division of Research Graduate:School of Business Administration The University of Michigan December 1973 Adaptive Forecasting Models for Interrelated Time Series Working Paper No. 87 Archer McWhorter W. Allen Spivey William J. Wrobleski The University of Michigan Q The University of Michigan FOR DISCUSSION PURPOSES ONLY None of this material is to be reproduced without the express of the Div'ision of Research. quoted or permission p~~~~~....

INTRODUCTION The forecasting of interest rates has been a complex problem for many years, and satisfactory methods for accurately forecasting interest rates seem to be few and of limited generality. There are many reasons for this scarcity; theories of interest rates often involve expectations of the~ future and the statistical testing and validation of the theories has proved to be extraordinarily difficult. Moreover, there are several additional problems inherent in forecasting: (1) the problem of explaining movements of interest rates for sample time periods for which relevant data are available, (2) the problem of making and evaluating a forecast for future time periods, and (3) the problem of how a manager can use a forecast in his decision making. This paper is concerned with forecasting short-term interest rates (3-onth treasury bills and 4- to 6-month commercial paper) by means of econometric models. Some of these models are standard econometric models in that they consist of fixed coefficient regression equations estimated by ordinary least squares (OLS). Others are adaptive econometric models having regression equations in which coefficients are allowed to vary. The latter are models of the Kalman type and are discussed in Section 2 below. These models are relatively new in econometric research and have not, to our knowledge, been applied previously to the forecasting of interest rates...... Our main purpose here is to contrast the results of standard and adaptive econometric models on the basis of their performance with regard to forecast errors. - Specifically, each type of model will be rear de1 wilb

-2 - used to explain the movements of interest rates in the sample time period (the second quarter of 1962 through the fourth quarter of 1972) and each will be used to forecast short-term interest rates for the first and second quarters of 1973. It is not our intention to present forecasting models which, in their present stage of development, can be used directly in financial decision making; our view is that much researchmust be done before adequate models for decision making can:be developed. DISCUSSION OF THE MODELS There are several well-known standard econometric models of the U.S. economy which include treasury bill and commercial paper rates as current endogenous variables in their financial subsectors. We have chosen to use the financial subsector of the Office of Business Economics (OBE) quarterly model as it appeared in [4]. This subsector contains nine equations and among these are two equations in which 3-month treasury bill rates are related to several aggregate monetary and fiscal variables and 4- to 6-month commercial paper rates are related in turn to treasury bill rates.1 These two equations, considered in isolation from the rest of the financial subsector as well as from 1This version of the OBE model contains 56 equations and is an outgrowth of an earlier 36-equation model described in [8]. The Office of Business Economics has been renamed the Bureau of Economic Analysis, and the most recent version of the model, reported on in [5], is called the BEA model. We refer to the version discussed in this paper as the OBE model.:

-3 -the remaniing subsectors of the full OBE model, will be taken as the standard econometric model for the purposes of the experimental studies in this paper.; The behavioral equations of this model are as follows, where t denotes the t-th quarter:: y1(t) - 0+ of x(t) + a2x2(t) + a3x3(t) + el(t) (1) ~1 3. 3 + ' Y2(t) = o + lYl(t) + %yl(t-1) e2(t) (2) and where Y.(t) denotes 3-month treasury bill market yields,: Y2(t) denotes 4- to 6-month commercial paper rates, xl(t) denotes the Federal Reserve discount rate, x2(t) denotes the amount of free reserves divided by the sum of demand deposits and currency, both for the previous quarter, and x3(t) denotes the fiscal balance, federal net surplus (NIA basis) divided by GNP in 1958 dollars for the previous: -. quarter. In [8], equation (2) also contains a dummy variable which assumes the value of 1 or 0, respectively, according to whether or not the quarter was one for which certificates of deposit were issued. The period for which our experiments were conducted (including the experiments for' forecasting these interest rates into the future) was a period in which certificates of deposits were issued during each quarter, so the effect of the dummy variable is absorbed into the intercept term 60 in (2) and does not appear explicitly in this equation.

-4 - system matrix The behavioral equations (1) and (2) can be regarded as a of stochastic equations and can be expressed in the standard form for such (constant coefficient) econometrit c -systems as 1. ol 'il (t) b1 2 1:.Y(t + 0.La J 2. 20 b 11 2 13 ~ 0 0 0 b24 2. 1 x(t) x3(t) y1 ct-i) e(t)]; L e2( or as Ay(t) + Bz(t) =e(t), (t.. n). (4) The conventional reduced form of (3) or (4) cana be written f ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~... Y (t)] 0. ~ -. 2o.11.21 212 $22. 0 13..23 24 2.J.. s(t).. 2:.:(5) ~2 (t or y(t) = IIz(t) + E(t), (t = 1,..., n) (6)

i -5 — The adaptive econometric model used in our experiments is a version of the Kalman-Bucy model [7]. We write this model as follows, i1 ^ ''l::. 1., **:~~~~..' 1 ' '10(t).:yyD 8ti(t) Yi(ft)) 0 0 0 1 t x1(t) x (t) x3() y(.,..... - -, i:,,,: --.24 (t) 0 0 0 0 I Y 1 xl ) x24(t)x*' ' ' ' ' ** ' * '*.,. '* *' '. * ' "*" '*.. * *' '*'*',~' i *~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~.. ' *.. * ' ** ' * * '* * '-.12 )i.'*\* ' /'^ i::, ' 1^; 1;/ 1'-.' 1;.: ' ' - \ ^^^^^^^^^~ ) e,(t e (t) 2 ^-. where s' " (t) 24;(t Alternatively, li 12 19: 10 t-l ', t '.. t91 t92 t99 i 24 ti (7) and (8) can be written in matrix y(t) X(t)S(t) + e(t) (t) = T(t(t-l) + u(t). ti ult(t) u2(t) -. form as (8) (9) (10)

-6 -The model represented by (9) and (10) requires that the matrix T(t) be specified, as well as an initial value 8(0) and the variancescovariances of the error vectors:(t) and u(t). It should be noted that the vector S(t) in (9) varies with t and is not a constant parameter. Thus, the model (9) and (10) is not a constant coefficient model as is the standard econometric model (3), (4) or (,, (6). NATURE OF FORECASTING EXPERIMENTS In forecasting with a standard econometric model for T periods into the future, one uses an estimate ir of the reduced form coefficient matrix I in (6) as well as an estimate z(t + T) of the values of the independent variables T periods into the future and then develops a forecast y(t + T) of the dependent variables by y(t + T) = t z(t + T). (11) In contrast to this procedure, the Kalman-Bucy varying parameter regression model recognizes the changing coefficient vector i(t) in (9) and adaptively updates this vector through (10) to obtain B(t + t). Then a forecast X(t + T) of the matrix of independent variables is developed and the forecast of the dependent variables is then given by y(t + T) = X(t + T) 3(t + T). (12) As noted earlier, two kinds of forecast experiments are made in this paper: forecasts for the sample period (second quarter of 1962 through fourth quarter 1972) and forecasts for the quarters 1973.1 and 1973.2. In the former case, X(t + T) is known and is, of course, taken to be the matrix of the actual values of the independent variables for

7 the given quarter. In the latter case, the elements of this matrix must be forecast if they are not available. The elements of X(t + T) could be forecast, for example, using exponential smoothing or some other time series forecasting method, or one could obtain them by solving the full OBE model. The latter would require reestimation of the OBE model through the fourth quarter of 1972. This reestimated model then would be used to forecast z(t + T) and this in turn would require the specification of the values of several other current endogenous variables which in the OBE model are assumed to be predetermined in a manner external to the model itself. Therefore these variables would have to be estimated in some way, perhaps by time series methods or by means of a so-called hidden model. Since in this paper our goal is to contrast the forecasting performance of fixed and varying parameter regression models, we used the actual values of the independent variables for our forecasting experiments for 1973.1 and 1973.2 rather than forecasting them by either of the above methods. We plan to examine the problem of forecasting independent variables and its effect upon the forecasting performance of varying parameter regression models more completely in a subsequent paper. ESTIMATION IN THE KALMAN-BUCY MODEL In order to use the Kalman-Bucy model (9), (10), we indicated earlier that one must specify the matrix T(t), the: initial value of the coefficient vector 0(0), and the variances and covariances of the error vectors E(t) and u(t). Our computer experiments were conducted in two s

modes, a single-equation and a simultaneous-equation mode, In the former, we took the first equation of (7) for 3-month treasury bill rates (Y1) and placed it in the context of the Kalman-Bucy model (9), (10). A similar procedure was followed for the second equation of (7) in terms of 4- to 6-month commercial paper rates (y2). This produced a single-equation Kalman-Bucy model for each of these interest rates. Because of the various specifications that are necessary, we characterize the former model by the parameter set (8i(0), Tl(t), Var(el), Cov(ul)) and the latter by (2(0), T2(t), Var(e), Cov(u2)). In the simultaneous-equation model (7), (8) or (9), (10) the two interest rates are considered jointly and this model is characterized by the parameter set (8(0), T(t), Cov(s), Cov(u)). Estimation in either case began with an examination of the residuals around the OLS fit for each of the interest rate time series using the two individual equations implicit in (5). The sample 2 2 variances of these residuals, sl and s respectively, were calculated 1 2 1 2 12 and Var(E1) was set equal to S and Var(2) equal to2. Then 2-s was allocated equally to each of the four diagonal elements of the matrix Cov(u) —that is, each of these elements was set equal to 12 12 -sl —and -s2 was allocated in similar fashion to the five diagonal elements of the matrix Cov(u2). All off-diagonal elements of the matrices Cov(u1) and Cov(u2) were set equal to zero (i.e., the e re set -.. * -.2

-9 -coefficient disturbances were assumed to be mutually uncorrelated) giving 12 0- o 0 0 1.2 o 1s2 o 0 Cov(u1) = o 0 -: 2 T5 0 0 0 0 00.i. 0 12 0 0 0 1 2 Cov(u 2) 0 0 T O 0 i 2 o o ~ i-~.o 12 0 0 O 0: 0 1T In the simultaneous-equation models Cov(e) and Cov(u) were specified as above except that we used sample covariance sl2 (= 21) of the two OLS residuals as the off-diagonal elements of the matrix Coy(s) instead of setting these values equal to zero —thus there is contemporaneous correlation between the equation errors sl(t) and s2(t). In other words, we used 2. -i

-10- fl;t1 j12 ri2 2 -: - GCov (&) = I:: 1 2 diagonal elements are equal to;sl, whose remaining five diagonal 12 elements are equal to l-s2, and whose off-diagonal elements are equal to zero. The vectors e(t) and u(t) were assumed to be uncorrelated in the Kalman-Bucy models, and the problem of possible autocorrelation in E(t) and u(t) has been ignored. Since little is known about the sensitivity of the KalmanBucy model in economic applications to the specification of the vector (0)3, we estimated 8(0) in two different ways to investigate this matter., ' ' ' 1) OLS regressions using the individual equations implicit in (5) were calculated using the first 15 quarters.of data (1962.2 through 1965.4) for each of the two interest rate series. The estimated values of 6 (0) and 62(0) were then taen to be the components of the vector (0O) in (9). 2) OLS regressions using the individual equations implicit in (5) for 43 quarters (1962.2 through 1972.4) were calculated and the resulting estimates 61(0) and 62(0) were taken to be the components of the vector 6(0).

-11-... The remaining parameter in the Kalman.-Bucy model, the matrix T(t), offers some interesting challenges in the application of this model to economic time series. In an engineering context in which the Kalman-Bucy model first found applications, this matrix is typically given by known physical properties of some relevant real world system, but in econometrics one must estimate this matrix and a tractable and generally applicable method of estimation is not yet available. We assumed T(t) to be a constant matrix chosen to be either the identity matrix or a matrix estimated by OLS. In the latter case, two steps were necessary. First, the time-varying coefficients were estimated using the Kalman-Bucy model with T(t) specified as the identity matrix. Then these coefficient estimates were used as the actual values for 8(t) in (10), and the rows of T were estimated using stepwise regression (significance levels of.10 were used for the addition and deletion of variables). RESULTS OF THE FORI A summary of the result presented in Tables 1, 2, and 3 I term interest rate time series at gression coefficients for the vai treasury bill rates are also pre, behavior of a set of time-varying with corresponding OLS estimates Figures 1 and 2, the asterisks rE ICASTING EXPERIMENTS s of our forecasting experiments are elow. CalComp graphs of the two short-.d the four series of- estimated reying parameter model for 3-month ented, the latter to illustrate the coefficients and to compare them (dashed line in each graph). In ipresent the actual values of the.~~~~~~~~~~.:

-12 -respective interest rates for the first two quarters of 1973. In Figures 3, 4, 5, and 6, the asterisks represent the values of the coefficients that were used in generating the forecasts for the model KB(1, OLS(2),I). Single Equat.on Models The results presented in Table 1 for single-equation models reveal that for both time series the varying parameter single-equation models of the Kalman-Bucy type fit the sample period data much better than the corresponding fixed coefficient regression models estimated by OLS. Moreover, these models appear to provide similar fits for the sample period data regardless of the choice made for the initial value S(0) or the coefficient updating matrix T featured in the Kalman-Bucy models (however, the choices of 8(0) and T do have considerable effect on the time paths of the coefficients themselves). In terms of the forecasting experiments related to predicting the actual levels of the two interest rates for the first and second quarters of 1973, however, only the models which used an initial value for 8(0) based on OLS(1) rather than OLS(2) provided reasonable forecast errors substantially smaller than those realized by using ordinary least squares. It is interesting to note that although the single-equation Kalman-Bucy models fit the commercial paper series better in the sample period, the 1973.1 and 1973.2 forecasts of treasury bill rates are better (in terms of average squared error for the two quarters) than the corresponding forecasts of commercial paper rates.

-13 -Simultaneous Equation Models Table 2 contrasts the forecasting performance of simultaneousequation varying parameter models of the Kalman-Bucy type for the two interest rate time series. The results are generally similar to those observed for the various single-equation models. Examination of Table 2 reveals that one simultaneous varying parameter model, denoted by KB3(2, OLS(1),T), is clearly superior to any of the others in fitting the two interest rate time series over the sample period. Furthermore, this model performs well with respect to forecasting both interest rates for the first and second quarters of 1973, with the average squared forecast error for this period being smaller for the treasury bill rates than for the commercial paper rates. Summary...... Table 3 presents a comparison of how well 3-month treasury bill rates and 4- to 6-month commercial paper ratesare forecast jointly (rather than individually) by the single-equation and simultaneousequation models. For the purpose of providing a joint analysis of the movements of these two time series, the simultaneous-equation model KB(2,OLS(1),T) again fits the sample period data better than any other model. Furthermore, this model performs quite well when treasury bill rates and commercial paper rates are forecast jointly for the first and second quarters of 1973, although the simultaneous-equation model KB(2,0LS(l), I) provides slightly better joint forecasts of the two interest rates during this forecast period.

:14 -Table 1 SINGLE EQUATION MODELS THREE-MONTH TREASURY BILL RATES Time Period __1962.2-1972.4 j 1973.1 1973.2 1973.1-19730.2 Forecasting Method* SSESu Sq. Errors) Forecast ForecastASE(Averae Sq. Error).Sum. Sq Er ) Forecast AS: KB (1,OLS(),I) 0.0933 5.878 6.454 0.0269 KB(1,OLS(2),I) 0.0911 6.654 7.308 0.7036 KB(1,OLS(I),T) 0.0792 5.833 6.350 0.0408 KB(1,OLS(2),T) 0.0816 6.568 7.142 0.5220.* Squares) a Actual Rates 5 5l70.0 '6603 ActualBlOS, t 0.0933 - - 58700 ' 6.456j.00 6. 63

I -15 -Table 1 (Continued) 4- TO 6-MONTH COMMERCIAL PAPER RATES.~~~~~~~~~ F Time Period 1962.2-1972.4 1 973.1 1973.2 1973.1-1973.2 Forecasting Method* SSE(Sum Sq. Errors) Forecast Forecast ASE(Average Sq. Error) KB(1,OLS(1),) 0.0444 6.310 i6.949 0.1345 KB(1,OLS(2),I) 0.0483 7 7.2828 0.5088 K(OL (1,T 0 03.1.4.15 KB (I,OLS (1),T) I 0.04.32 6.311: 6.947 0.1356 KB(1,OLS.(2),T) 0.0485 i 7.227 l 7.833 0.5125 I IOLS(Ordinary Least 1 7Vi \ a ll7 i nf R7QO A -I 11M 7 911; R-f!W~ nn-'' I. fV 'aaaQ. Squares) I Actual Rates - 6.283 7467 *In Tables 1, 2, and 3, OLS(1) refers to estimating B(0) by procedure 1 described on page 10; OLS(2) refers to estimating (0O) by procedure 2 described on page 10. KB(1,OLS(1), signifies single-equation Kalman-Bucy estimation with T(t) = I; KB(2,OLS(1),T) signifies simultaneous-Equation Kalman-Bucy estimation with T estimated by OLS (see page 11). s~~~~ ~~~ ~ ~ Al -uc,s I.),

I -16 -Table 2 SIMULTANEOUS EQUATION MODELS THREE-MONTH TREASURY BILL RATES.. Time Period 1962.2-1972.4 1973.1 1 1973.2 1973.1-1973.2 Forecasting Method* SSE(Sum Sq. Errors) Forecast Forecast ASE(Aerage Sq. Error) KB(2,OLS(1),I) 0.0804 5.936 6.493 0.0339 i KB(2,OLS(2),I) 0.0965 6.724 7.366 0.8154 i KB(2,OLS(1),T) 0.0036 5.949 6.513 0.0351 ____.___ 0. ___( 0x ' KB(2,OLS(2),T) 0.0374 6.290 6.705 0.1793 OLS(Ordinary Least 6.2893 6.470 76.104 0.4219 Squares) _ Actual — Rate s_ 5.-7 -- — _-0! Actual Rates I -- 5.700 6.603 L:! i 6 6 03

-17- - Table 2 (Continued) 4- TO 6-MONTH COMMERCIAL PAPER RATES Time Period 1962.2-1972.4 1731 1973.1-1973.2 Forecasting Method* SSE(Sum Sq. Errors) Forecast Forecast ASE(Average Sq. Error) KB(2,OLS(1),) 0.0629 6.310 7.027 0.0972 KB (2, OLS (2), I) 0.0830 7.185 | 7.903 0.5018 KB(2,OLS(1),T) 0.0031 6.299 6.977 0.1202 KB(2.0LS(2) T) 0 0592 7.127 7.767 0.4012 OLS(Ordinary Least OLS(Ordinary Least 6.1308 7.236 8.007 0.5999 Squares) Actual Rates -- 6.283 7.467 *In Tables 1, 2, and 3, OLS(1) refers to estimating S(0) by procedure 1 described on page OLS(2) refers to estimating (O0) by procedure 2 described on page 10. KB(lOLS'(l),I) sign fies single-equation Kalman-Bucy estimation with T(t) I; KB(2, OLS(1),T) signifies simultaneous-equation Kalman-Bucy estimation with T estimated by OLS (see page 1B. tinwt. esiae y-.S (e a 10; i

-18 -Table 3 SHORT TERM INTEREST RATES JOINTLY FORECAST A. SINGLE EQUATION MODELS Time Period 1962.2-1972.4 1973. Forecasting Method* Combined SSE Combj KB(1,OLS(1),I) 0.1377 0. KB(, OLS (2)) 0.1394 1 KB(1,OLS(1),T) 0.1224 0. B(1, OLS(2),T) 0.1301 1. OLS(Ordinary Least Squares) 12.420 1. 1-1973.2 -ned ASE,1614 2124 1764 ~10345 0218

, 19 -B. SIMULTANEOUS EQUATION MODELS Time Period 1962.2-1972.4 1973.1-1973.2 Forecasting Method* Combined SSE Combined ASE KB (2,OLS(1),I) 0.1433 0.1311** KB(2,OLS(2), I) 0.1795 1,3172 KB(2,OLS(1),T) 0.0067** 0.1553 KB(2. OLS(2).T) 0.0966 0. 5805 I *In Tables 1, 2, and 3, OLS(1) refers to estimating (0) by procedure 1 described on page 10; OLS(2) refers to estimating 8(0) by procedure 2 described on page 10. KB(1,OLS(1),I) signifies single-equation Kalman-Bucy estimation with T(t) 1; KB(2,OLS(1),T) signifies simultaneous-equation Kalman-Bucy estimation with T estimated by OLS (see page 11). **Minimum values of SSE and ASE for forecast errors for the sample period and the forecas period when three-month treasury bill rates and 4- to 6-month commercial paper rates are sidered jointly. Table 3-Continued -C it con

4 t 0 c) M X i >^ u -20- THREE-MONTH TREASURY BILLS —MARKET YIELD QUARTERLY DATA-1962.1 THROUGH 1973,2 * *.. 0D. I I I I MA/ *:.o1v V 1962.00 1964.00 1966.00 1968.00 197000 1972.0 1974.00 YEAR Figure 1 j. I

0 ' t C C -21 -4- to 6-MONTH CCMMERCIAL PAPER RATES QUARTERLY DATA —1962.1 THROUGH 1973.2 0 o * * 0..o. 'C. ' CD * *. I i -... I V I.:....* - ft. 0..SBB I.... -lara a-r.... f. N. * X I. X.I 1962.00 1964.00 1966.00 1968.00 1970.00 YEAR Figure 2. -170 ----0 1974.00- 1972.00 1974, 00

a; ' O* 2 -KALMAN-BUCY ESTIMATES OF B (0) IN TREASURY BILL EQUATION (SINGLE EQUATION APPROACH WITH T=I) 1.. 0 OC 0C PA 0 0 N r-,. H' E-4 t: ----------------- eCp- — Y~9asgllb V ** - I 0 M" F II.1 ---- I ---- I ---- p ---1 - -- - t ---- - * 1962.00 1964.00 1966. 00 1968.00 1970.0 1900 1974.00 YEAR Figure 3 7

T: * -23 -KALMAN-BUCY ESTIMATES OF B(1) IN TREASURY BILL EQUATION (SINGLE EQUATION APPROACH WITH T=I) q "o. 14' C' CV^ [. rOs: p- _ _..=k.h __ J__ III-b;. -rra i~~r~PP _ -= _ _r-,~ '1"_ -— ~~~~~~~~~~ w-N-n -i X. '. "...... e-. 1962.00 1964.00 1966,00 1968.00 1970.00 1972.00 1974.00 YEAR Figure 4 rn.. g9 ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1920 ig.a- 16,0 95.0.g01: 17.0.94 ~- - a:P.~ ~ ~ ~ ~ ~ Fgr.

I* 0 -24 -KALMAN-BUCY ESTIMATES OF B(2) IN TREASURY BILL EQUATION (SINGLE EQUATION APPROACH WITH T=I) r — O%>4! 7. C C,4 ia~ P-I C CP ar a ) IN D e e D \ 4 q1 ~ - ** <- ------- --. ', 1962.00 1964.00 1966.00 1968.00 1970.00 1972.00 1974.00 YEAR Figure 5

( a Er I 1 A; p -25 -KALMAN-BUCY ESTIMATES OF B(3) IN TREASURY BILL EQUATION (SINGLE EQUATION APPROACH WITH T=I) C.o. c.:4.:.. *. ' m. Ln r4rH Cr) Q Fw 0 r-I E-l Ln O 0, o 0< **.. &I. - - iaBrzBli. a 1974.00 j ~ ________ - ~ ~ -,,r,,~,,,~P%-~bbl~ "CI-~ ~.F~d-j — lsa~~~l~~IRO-A- - - 1 ~ --- —---— ~ --- —----. -- --!. i-. 962.00 1964.00 1966.00 1968.00 1970.00 1972.00 YEAR.1.I I I Ii.. Figure 6

I New York: REFERENCES 1. Astrom, K.J. Introduction to Stochastic Control Theory. Academic Press, 1970. 2. ___ _ and Eykoff, P. "System Identification —A Survey." Automatica 7 (1971): 123-162. 3. Duncan, D.B., and Horn, S.D. "Linear Dynamic Recursive Estimation from the Viewpoint of Regression Analysis." Journal of the American Statistical Association 67 (December 1972): 815-821. 4. Green, G.R., Liebenberg, M., and Hirsch, A. "Short and Long Term Simulations with the OBE Econometric Model" In Econometric Models of Cyclical Behavior, edited by B.G. Hickman, Vol. I. New York: Columbia University Press, 1972. 5. Hirsch, A. "The BEA Model Quarterly Model as a Forecasting Instrument." Survey of Current Business. August 1973, 24-52. 6. _, Liebenberg, M., and Green, G.R. "The BEA Quarterly Model.'" Bureau of Economic Analysis Staff Paper No. 22, July -197 3.. 7. Kalman, R.E., and Bucy, R.S. "New Results in Linear Filtering and Prediction Theory," Journal of Basic Engineering 83 (961) 95-108. 80 Liebenberg, M., Hirsch, A., and Popkin, J. "A Quarterly Econometric Model of the United States: A Program Report." Survey of Current Business, May 1966. a