Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00049278/00001
## Material Information- Title:
- Updating and forecasting with a varying parameter recursive model
- Series Title:
- Technical report Florida Agricultural Market Research Center
- Creator:
- Shonkwiler, J. S ( John Scott )
Florida Agricultural Market Research Center - Place of Publication:
- Gainesville Fla
- Publisher:
- University of Florida, IFAS, Food and Resource Economics Dept.
- Publication Date:
- [1979]
- Language:
- English
- Physical Description:
- iv, 26 p. : ; 28 cm.
## Subjects- Subjects / Keywords:
- Agricultural prices -- Forecasting -- Mathematical models ( lcsh )
Agricultural prices -- Mathematical models ( lcsh ) Beef cattle -- Prices -- Forecasting -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (p. 25-26).
- General Note:
- "November 1979"--Cover.
- Funding:
- Technical report (Florida Agricultural Market Research Center) ;
- Statement of Responsibility:
- J. Scott Shonkwiler.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 027808633 ( ALEPH )
26812849 ( OCLC ) AJG5556 ( NOTIS )
## UFDC Membership |

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HISTORIC NOTE The publications in this collection do not reflect current scientific knowledge or recommendations. These texts represent the historic publishing record of the Institute for Food and Agricultural Sciences and should be used only to trace the historic work of the Institute and its staff. Current IFAS research may be found on the Electronic Data Information Source (EDIS) site maintained by the Florida Cooperative Extension Service. Copyright 2005, Board of Trustees, University of Florida NOVEMBER 1979 TECHNICAL REPORT 79 1 Updating and with a Varying Recursive I ting ler 2 L*.^ UPDATING AND FORECASTING WITH A VARYING PARAMETER RECURSIVE MODEL J. Scott Shonkwiler Assistant Professor University of Florida IFAS, Food and Resource Economics Department Gainesville, Florida 32611 The Florida Agricultural Market Research Center is a service of The Food and Resource Economics Department of the Institute of Food and Agricultural Sciences The purpose of this Center is to provide timely, applied research on current and emerging marketing problems affecting Florida's agricultural and marine industries. The Center seeks to provide research and information to production, marketing, and processing firms, groups and organizations concerned with improving and expanding markets for Florida agricultural and marine products. The Center is staffed by a basic group of economists trained in agriculture and marketing. In addition, cooperating personnel from other IFAS units provide a wide range of expertise which can be applied as determined by the requirements of individual pro- jects. TABLE OF CONTENTS Page LIST OF TABLES .................................................... iv LIST OF APPENDIX TABLES........................................... iv INTRODUCTION ...................................................... 1 The Varying Parameter Model .................................. 2 The Varying Parameter Recursive Model ........................ 7 The Recursive Structural Model............................... 13 The Varying Parameter Recursive Model and Forecasting Performance............................................. 15 SUMMARY AND CONCLUSIONS .......................................... 20 APPENDIX .......................................................... 22 FOOTNOTES ......................................................... 24 REFERENCES ........................................................ 25 iii LIST OF TABLES Table 1 Estimated structural parameters for constant and varying parameter specifications..................................... 2 Derived and unrestricted reduced forms for Choice steer prices........................................................ 3 Choice steer price predictive interval tests................. LIST OF APPENDIX TABLES 1 Variable definitions.......................................... Page UPDATING AND FORECASTING WITH A VARYING PARAMETER RECURSIVE MODEL J. Scott Shonkwiler INTRODUCTION Accurately forecasting prices of agricultural commodities during the present decade has been made difficult in light of severe shocks to the U.S. agricultural economy. The cattle sector has apparently undergone substantial disruption which was manifested, in part, by the sharp reduction of breeding cow inventories. In conjunction, quarterly average prices of Choice steers have displayed considerable variability during the 1970's. Specifically the run-up in Choice steer prices in the 1978-79 period has been unmatched by any other livestock price movements in recent history. The present study develops a four equation recursive model capable of forecasting Choice steer prices two quarters ahead. The model admits a limited varying parameter structure in an effort to capture possible structural change. The varying parameter technique adopted permits re- formulation of the model in terms of the Kalman filter time and measure- ment updating algroithms. Thus, updating the recursive model with recent data is handled systematically and forecast accuracy may be improved by I weighting recent observations differently than the weighting that occurs J. SCOTT SHONKWILER is assistant professor of food and resource economics, University of Florida. when simply re-estimating an augmented observation set. The Kalman filter updating technique is developed for both the structural econometric model and its restricted reduced form. To assess the relative performance of this approach, a comparison of the forecasting accuracy of the two varying parameter models and their constant parameter counterpart is presented. The following section outlines the varying parameter model, its implications and correspondence to a particular type of Kalman filter model and extends the varying parameter structure to it. Then, the subsequent sections present the specification and estimated parameters of the recursive model both under constant parameter and varying parameter regimes. Finally, the forecasting accuracy of the different models will be presented and discussed. The Varying Parameter Model The rationale for incorporating parameter variation stems from the lack of controlled effects and numerous unobservable forces inherent to modeling economic systems. Economists are typically constrained to analyzing secondary data with its attendant errors of reporting and collection with no assurance that the assumption of constant parameters holds unambigously. The reasons for this uncertainty are twofold. First, the actual coefficients may be generated by an underlying non-stationary process. Or secondly, the true parameters may be stable within the appropriate or ideal model context but factors such as omitted variables, errors in variables, aggregation bias and improper functional form may preclude the formulation of the appropriate model. A varying parameter -3- specification may reduce the effects of these factors (Cooley and Prescott, 1973). Additionally, Cooley and Prescott (1976) have re- marked that constant parameter formulations are inconsistent with theoretical specifications in the sense that the dynamics of economic behavior do not suggest constant-parameter behavioral equations. The type of varying parameter structure adopted allows one or more coefficients to follow a first order Markov process.1 Specifi- cally, the tth observation for the model may be written yt = xtat + Ct t = 1, 2, ..., T (1) t t- + Pt (2) where xt is a lxk vector of observations on the independent variables, and at is a kxl vector of coefficients at time t. The stochastic assmuptions for the model are E(et) = E(pt) = E(Esst) = 0 (3a) E(2S) 2 (3b) s : st% E(st) stQ (3c) where Q is a kxk covariance matrix assumed known and 6st denotes the Kronecker delta. The specification of Q is not as difficult as might seem. In the constant parameter case Q is identically equal to a null matrix. Otherwise, the variances and covariances may be specified in a manner similar to that used in mixed estimation (Cooper). That is, qii represents the variance of the varying parameter process of Bi -4- and it.2 V/qi would represent an approximate 95 percent confidence interval for the successive coefficient it+l" Clearly estimation of the varying parameters must be referenced to some point during the sample period. Because interest is focused on forecasting, we desire the value of the parameters given the most recent observation. The values taken by the parameters given all observations through the most recent will be denoted T*. To estimate BT, Sant (1977) has formulated a generalized least squares model of the form YT = X TT + ET ATUT (4) where X1 1X X1 X1 112 AT = 0 x2. x2 x2 and UT= 3 0 0 .0 xT-1 "lT-1 0 0 0 0 0 Alternatively (4) may be written compactly YT = XT8T + VT (5) Here YT and XT represent the customary TXl regressand vector and txk design matrix. The complex structure incorporated into the disturbance vector VT results from the successive solution of (2) in terms of BT" The matrix AT has dimension Tx(T-1)k and UT is a (T-l)kxl vector of parameter disturbances. The stochastic specification of VT is E(VT) = 0 (6a) E(VTVT) = a T + AT(I T-1Q)AT = T, (6b) where a 2 denotes the variance of the individual structural disturbances E contained in Et. If IT is known estimation of the parameter vector at time T is given by T = (XT XT) XT2T YT (7) 2 Of course, a is rarely known and cannot be estimated given BT since T is conditioned by a 2. Cooper has suggested an iterative procedure to calculate a2 which can be efficiently implemented. The method starts with an initial estimate of 2 and then derives T and BT. Then a new estimate of a2 is given by ^2 A ), l 1 = (YT-XT T)T1 (8) and the procedure is repeated until convergence is achieved. The forecasting and updating problem for the varying parameter model may be solved by using the appropriate Kalman filter recursions. The relationship between the Kalman filter model and generalized least squares models has been developed by Duncan and Horn and Sant (1977). The one step ahead prediction is given by the parameter time update aT+1/T = T (9) The evolution of the parameters may be referenced to previous parameter values and the new observations by the measurement update BT+l = 6T + KT+1 (YT+ xT+l T) (10) where K represents the filter and is determined according to i2 - KT+l :T+ /TXT+lI T+l T+1/TxT+I + 2 (11) The matrix E denotes the parameter covariance matrix. It is given sequentially by T+1I/,T T + Q (12) with measurement update ET+1 = T+l/T KT+l XT+l ET+l/T (13) For interpretation of the expressions(9)-(13) the interested reader is directed to Chow, Duncan and Horn, or Anderson and Moore. Implementation of the updating recursions in (9)-(13) is straightforward given the model developed in expressions (l)-(7). As stressed by Athans, the Kalman filter algorithm should not be confused with the underlying econometric model -- rather it should be viewed only as a means for refining a model. The value of the Kalman filter model stems from the systematic manner in which new data may be incorporated. Expression (10) shows that as the forecast error increases, greater weight is given to the new observation in the determination of the updated parameters. However, -7- the filter adjusts this weighting according to equation (11) which contains an expression for the (inverse of the) forecast variance. That is, if the system indicates "large" forecast errors due to lack of resolution, then the forecast errors are weighted less because K is "smaller". However, if the forecast variance is (in some sense) small while the forecast error is large, then the updated parameter vector will depend much less on its previous value, i.e., the last observation will be weighted heavily. The Varying Parameter Recursive Model The extension of single-equation varying parameter techniques to simultaneous equation systems has been presented in several studies (Mariano and Schleicher, Narasimham et al., Mahajan and Mahajan), how- ever, none of these studies showed the effect of a varying parameter structure on the evolution of the restricted reduced form. Further, it is not immediately clear whether the structure should be updated with predictions being formed from the restricted reduced form, or whether knowledge of the structure should be used to specify a re- stricted reduced form that can be updated directly. In view of the fact that combinations of restricted and unrestricted reduced forms (Maasoumi, Sant,1978) may provide desirable properties, this latter approach will be developed as well. The recursive simultaneous system represents a useful vehicle for development of the multiequation varying parameter model because the structural equations may be consistently estimated via immediate application of ordinary least squares. Thus, questions associated with the appropriate means to generate instrumental variables within a varying parameter system are avoided. This section proceeds by first briefly developing the general simultaneous equation model and the recursive system in particular. This system is extended to admit vary- ing parameters and use of the updating alogorithms. Let the Txm matrix of m jointly dependent variables and the Txz matrix of predetermined variables be written Yr + XB = n (14) where r and B are coefficient matrices of dimension mxm and zxm re- spectively and n is a Txm matrix of structural disturbances. The restricted reduced form of (14) is, of course, Y = Xn + E, (15) where n = -Br and E = nr . The traditional stochastic assumptions concerning (14) and (15) include E(n) = E(z) = 0 (16a) E(n'n) = D (16b) E(g'g) = r" -1 -1 (16c) The matrix D represents the structural covariances between the disturbances of the m equations in the system. In order to satisfy the conditions of a truly recursive model, r is upper triangular and , is diagonal (Goldberger). The recursive system may be conveniently written (17) Vec(Y) = ZA + Vec(n) where Y :X m. m and A = B. (r2 m(r : B:2) : B'.M ) m The notation Y. and X. indicates the endogenous and predetermined variables designated as regressors in the ith equation. Estimation of the structural parameters is given by A = (Z'Z)-1Z'Vec(Y) with the variance of the estimated parameters in the ith equation determined as Var(Ai) = ii(Z 'Zi) (18) (19) = yi If we let T represent the block diagonal matrix whose ith block is T. then the variance of the restricted reduced form parameter may be written as (Schmidt) Var(6) = DW'W'D' = () Y2 X2 (20) -10- where -1 D = (r- )' I and W represents the block diagonal matrix such that the ith block of W is Wi = plim (X'X)-Ix'(Yi Xi). The importance of obtaining the restricted reduced form parameters and their estimated variances will become clear when the varying para- meter recursive model is set forth. The other measures developed are traditionally estimated and reported in most studies. The method adopted by Schmidt for obtaining E is particularly attractive due to its ease of implementation. This method,however,is based on the original Goldberger, et al. approach which is a Taylor series approximation based on large sample theory (Dhrymes). To incorporate a first order Markov process as a parameter variation scheme, the recursive structural model may be written as2 Vec(YT) = ZTAT + Vec(nT) ATVec(UT) (21) which introduces the structure of expression (4) to the system in (17). The matrix AT is a block diagonal matrix whose ith block would th be AiT and would correspond to UiT, the partition of UT for the i equation. Again, the full model may be expressed Vec(YT) = ZTAT + VT (22) -11- with the stochastic specification that E(VT) = 0 (23a) E(VTVT') = 4IT + AT(IT-1Q )AT = T (23b) In (23b) Q is a block diagonal matrix whose i block consists of the varying parameter covariance matrix of the ith equation. Unless Qi = 0, note that Dii in equation (23b) will differ from the same expression in (16b). Since nT is the sum of a diagonal and a block diagonal matrix, this result preserves the recursiveness of the system. Estimation of the parameters at time T now follows from appli- cation of GLS to the system. Specifically AT = (ZT1T ZT) ZT T Vec(YT) (24) with the structural parameter covariance matrix for the ith equation denoted by Var(AiT) = (ZiT iTZiT)1 T (25) Forecasts are generated from the reduced form implied by the varying structure. Thus, the reduced form parameters not only carry information concerning structural exclusion restrictions but also translate the varying parameter process from the structure to the reduced form. In general, the structural coefficients can be updated using the sequential algorithms presented in expression (9)-(13) and forecasts would be made using the updated restricted reduced form parameter matrix 11T+/T = T+ T+/T (26) -12- Alternatively the time varying structural model can be viewed as a convenient vehicle by which some appropriate structural specifi- cations are employed to produce restrictions on both the parameter space and the parameter evolution process. The reduced form model implied by (22) is YT = XTT + T (27) S* -th * such that 11T+l/T = 11T and *iT represents the i column of ET where E('T T) = 1rT T( + RT (28) The matrix.,RT represents a remainder term which is not required by the Kalman filter algorithm. Once (27) and (28) are given at time T additional information about the reduced form parameters is required in order to update expression (27) directly. Specifically the variance- covariance of the parameter nT and its evolutionary covariance must be determined. This first measure is derived by noting that Var(nT) = DW TTW D = :T (29) Lastly the time update for the reduced form parameters' covariance follows directly as ET+l/T = T + DWQ W (30) Expressions (27)-(30) permit updating the restricted reduced form directly. As additional sample observations are included it is expected that the restricted reduced form derived from the updated structure will diverge (26) from the directly updated reduced form. The rate of this divergence will be conditioned by numerous factors -13- with a substantial influence possibly attributed to the compatability of the restricted reduced form with the process generating the additional data. Remember that the structural parameters are not derived from minimizing the reduced form errors. The directly updated reduced form model, however, will be conditioned more by reduced form errors since the strength of the structural restrictions will deteriorate as additional measurement updates are made. The Recursive Structural Model Simultaneous equation models may offer advantages to least squares estimation of an unrestricted reduced form if the structural model provides useful restrictions that lead to more precise parameter estimates over the forecast interval. A problem typically hampering the use of many simultaneous equation models as forecasting tools is the requirement that many so-called predetermined variables must be forecast over the prediction period as well (Johnson). While methods are availbalbe for evaluating forecast variances for such models (Feldstein) they appear to weaken the structural approach particularly when the predetermined variables must be forecast for example using time series methods (Granger and Newbold). In view of this, the recursive model specified uses predetermined variables which are either deterministic or lagged two or more quarters. The model relates quarterly Choice steer prices to current levels of fed and non-fed cattle slaughter and pork production. A trend variable is included to account for all other factors, particularly growth in consumer income given that fed beef is generally assumed to be a -14- superior good. Additional equations are required in order to predict levels of the livestock output variables two quarters ahead. Non-fed slaughter is largely composed of cull cow slaughter and some grass-fed steers and heifers. Slaughter levels for this category typically increase when cow-calf operations are being reduced and when the price-cost outlook does not favor feeding out young animals. Thus, lagged prices for the fed product and feed costs (representing a major input) are expected to influence relative levels of non-fed slaughter. As fed steer prices fall and/or costs increase, non-fed cattle slaughter should increase. Movements in the price and cost variables mentioned above are expected to have an opposite effect on fed cattle slaughter. There- fore, non-fed slaughter levels are hypothesized to be inversely related to slaughter of the fed category. In addition, variables which represent the number of feeder animals put on feed quarterly and seasonal patterns in fed cattle marketing are included. Although not all animals put on feed are automatically slaughtered within a fixed amount of time, lagged levels of this variable provide valuable information as to current fed slaughter levels. For pork production, farrow to finish operations require about six months so lagged values of farrowings are good indicators of current production. Data are available on sow farrowings by quarter for the major pork producing states. Lagged prices provide a measure of the expected profitability foreseen by producers and, thus, should be positively correlated with current production. -15- Incorporation of these notions and an awareness of the seasonality inherent in the production side of the model led to the specification adopted. The four equation system is structurally recursive and the parameter matrix is triangular. The matrix of covariances between structural equations was assumed to be diagonal -- an assumption not particularly contradicted by the data given that the largest correlation between structural disturbance vectors was only .277 with 28 observations. Thus, the model is assumed to meet the theoretical conditions for a recursive system so that ordinary least squares may be applied to the estimation of the structure (Goldberger). The model initially is estimated over the period 1971-I through 1977-IV. A longer period could have been chosen but it is likely that this would not benefit forecasting accuracy in view of the probable structural changes likely over even this short period. The structural specification and estimated constant parameters are presented in Table 1 and corresponding variable definitions may be found in the Appendix. Table 2 presents the derived reduced form and its estimated standard errors as well as similar estimates for the unrestricted reduced form. The Varying Parameter Recursive Model and Forecast Performance The parameter evolution structures adopted are presented in Table 1. The choice of coefficients was conditioned by the rationale that producer response to lagged prices may be subject to substantial vari- ability over time.3 Thus, the non-fed cattle slaughter and pork pro- duction equations admit a plus or minus 8 unit and 1.4 unit change in -16- TLble I.--Estimated structural parameters for constant and varying parameter specifirationsa Constant para eter structural I odels Varyin pcarameter models variable endent arales Oependent ariables Variable NFCs FCS PORK SRP :-FCS ;O-RK SP"R - XFCS -1 -.150 -.00453 .1 -.00319 (2.52) (5.20) (3.09) FCS -1 -.00844 -.00825 (5.7S) (5. 1) PORK -1 -.00675 .1 ..00564 (4.14) (3.48) SPR .1 -1 INTERCEPT .1314 261Z -130 124 2106 -110 113 S195 -145 (1.02) (1.01) 03 364 319 (1 .86) (2.02) Q4 657 623 (3.37) (3.99) 1FCSt.2 .335 .186 (3.74) (1.35) ZSPRt- i -60.6 -69.1 (3.00) (2.75) 3FC ~. 26.0 25.5 (7.00) (S.73) Placet.2 .324 (2.30) Qz*Place -.122 St-2 (2.71) ,*Pl:ca -.192 3 p2 ( .7) Q4*Plac -.022 4 t-2 (.76) Place. .398 ct-3 (2.80) Fart-2 .691 .687 (8.4) (9.13) Ql*Farrt-z -.153 -.151 (2. 79) (3.00) Farrt.3 .611 599 (4.50) (4.68) iPn1 5.71 5.65 .45 S(.3) (1.45)49 Time .388 .463 .01 (3.59) (4.45) R2 .936 .763 .828 .799 c 126632 83706 29967 6.03 COS6 23056 2.19 Ow 1.54 2.04 1.11 1.63 aFigures in parentheses represent approximate t-values. -17- Table 2.-Derived and unrestricted reduced forms for choice steer prices Method Restricted reduced forms Ordinary least squares constant Panmeters Varying Parameters Unrestricted Variable Parameter (Standard Error) Parameter (Standard Error) Parameter (Standard Error) Intercept Q2 Q3 Q4 NFCSt-2 ZSPRt-i ZSFCt-i Placet-2 Q2*Placet-2 Q3*Placet-2 Q4*P acet-2 P1 acet-3 Farrt-2 Q *Farrt-2, Farrt.3 ZHPRt. Time 88.07 (11.54) .2837 ( .319) -.625 ( .450) -1.221 ( .707) -.000365 (.000331) .1355 (.0862) .0500 (.0275) -.00268 (.00111) .00100 (.000365) .00158 (.000600) .000179 (.00021) -.00328 (.00116) -.00387 (.00119) .000849 (.000374) -.00338 (.0121) -.0319 (.0238) .463 (.104) 7.43 28 53.59 45.50 35.67 48.06 -.00420 .0670 .0490 .00237 -.00206 -.000724 -.00295 -.000602 -.00846 .0153 -.0103 -.299 .495 (56.35) (32.53) (36.24) (40.83) (.0017) (.479) (.167) (.00402) (.00405) (.00603) (.00659) (.00514) (.00626) (.0124) (.0081) (.228) (.931) 16.47 12 (10.71) ( .562) (.625) (.759) (.00037) (.0746) (.0231) (.00112) (.000266) (.000603) (.00021) (.00117) (.00114) (.000404) (.00123) (.0245) (.0980) 98.4 .6367 -1.191 -2.147 -.00109 .198 .0849 -.00274 .00103 .00162 0001 83 -.00336 -.00466 .00103 -.00412 -.0385 .388 12.83 28 -18- their respective price coefficients from period to period with 95 percent confidence in the limiting interval. These stochastic speci- fications are exhibited in the a2 column of Table 1. Additionally, due to uncertainty on both the sign and the magnitude of the time coefficient in the Choice steer price equation it was specified to change by as much as plus or minus .2 with 95 percent confidence. Interestingly, the structural results in Table 1 show little discrepancy between the constant parameter and varying parameter schemes. This may indicate that the variances attached to the varying parameters may be too small to effectively change the constant parameter values by very much. The restricted reduced forms for both models also reflect this similarity in coefficient magnitudes. The contribution of the varying parameter specification adopted can be evaluated by the predictive performance of this method versus the constant parameter model. The predictive interval tests are pre- sented in Table 3. It should be noted that the estimation of each model was based on data which occurred before the sharp run-up in Choice steer prices. Through the 1977-IV estimation period the highest price observed for was the $48.64 during 1975-111. Table 3 separates the forecasts into two categories -- the one period ahead forecast based on parameter estimates made through the previous quarter, and the two period ahead forecast defined similarly. The "Naive" column corresponds to the structural model which is naively updated by adding additional unweighted observations successively. The "VARY-S" and "VARY-RF" columns correspond to using the updating recursions for the structural model and the reduced form model, respectively. Of course, Table 3.--Choice steer price predictive interval tests Estimation period Models estimated thru previous quarter Models estimated two quarters previous Forecast Actual Models Models Period Value Naive Vary-S Vary-RF Naive Vary-S Vary-RF 1978 I 45.77 39.72 42.10 42.10 1978 II 55.06 42.45 46.76 44.95 41.13 42.69 42.69 1978 III 53.75 46.18 53.76 52.24 43.29 47.42 45.50 1978 IV 54.76 46.20 51.60 52.85 44.93 53.36 51.84 1979 I 65.42 50.07 54.60 54.36 48.02 51.44 53.06 1979 II 72.51 57.75 63.56 64.32 54.21 57.35 56.93 1979 III 62.86 68.37 68.47 0f 11.40 6.95 7.20 14.41 11.14 11.18 a 5.82 6.08 3.98 9.85 0.30 MAE 10. 5.82 6.08 13.98 9.85 10.30 aMAE designates the mean absolute error of the forecasts. -20- the varying parameter structural model uses its restricted reduced form for prediction, but this reduced form is obtained by updating the structure -- not by updating the reduced form directly as in the second case. The results in Table 3 indicate that the varying parameter approach achieves a respectable improvement in forecast accuracy over the constant parameter model. However, the substantial increase in the forecast errors from the one period ahead predictions to the two-period ahead case suggests that the variances on the varying parameters are not allow- ing sufficiently rapid adjustment of the parameters over the forecast interval. ,This is evident by considering the results in Table 3. Recall that for both forecast intervals the reduced form design matrix either consists of variables lagged two periods or deterministic components. Thus, the only difference between the one and two period ahead fore- casts is the amount of information available with which to estimate the coefficients. Apparently the varying parameters are not adjusting rapidly enough to the new information conveyed by the measurement up- dates with the result that two periods ahead forecast accuracy suffers dramatically. SUMMARY AND CONCLUSIONS The varying parameter, generalized least squares, and Kalman filter models may all be related algebraically. By imposing a varying parameter structure on a behavioral relation, the corresponding filtering equation may be derived so as to permit efficient updating. A properly specified -21- varying parameter model should give increased forecast performance (Athans). Anderson and Moore (p. 52) state that the traditional constant parameter formulation with Q = 0 may not be wise since "there is the possibility that the smallest of modeling errors can lead, in time, to overwhelming errors" which were not predictable from the estimated error covariance. Within a multi-equation context the varying parameter structure - may be introduced. Given this framework updating may take place with regard to the structural parameters or the (initially) re- stricted reduced form parameters. The results for predicting quarterly Choice steer prices indicate that both methods gave improved fore- casting accuracy over the naively updated constant parameter model, but with no clear advantage for either updating scheme evidenced. APPENDIX -23- Appendix Table 1.--Variable definitions. Dependent variables FCS Fed cattle marketed, 23 states (1,000 head) NFCS Non-fed cattle slaughter, equal to difference of total commercial cattle slaughter and FCS (1,000 head) PORK Total commercial pork production (millions of pounds) SPR Choice steer price, Omaha (dollars per cwt) Predetermined variables Q1-Q4 Quarterly dummy variables ESPRt-i = .2SPRt-2 + .5SPRt-3 + .3SPRt-4 .SFCt. = .2SFCt-2 + .5SFCt.3 + .3SFCt4, where SFC is a steer feed cost index Place Cattle placed on feed quarterly, 23 states (1,000 head) Farr Sows farrowing, 14 states (1,000 head) ZHPRti = .3HPRt-3 + .7HPRt-4, where HPR is the 7 market price of barrows and gilts (dollars per cwt) Time Linear trend, 1971-1 has value 1 -24- FOOTNOTES 1. Numerous other varying, switching, or random parameter structures may be hypothesized, however, the present treatment is attractive because of its generality. 2. The stacking scheme presented simply permits the treatment of all equations jointly. 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