Updating and forecasting with a varying parameter recursive model

Material Information

Updating and forecasting with a varying parameter recursive model
Series Title:
Technical report Florida Agricultural Market Research Center
Shonkwiler, J. S ( John Scott )
Florida Agricultural Market Research Center
Place of Publication:
Gainesville Fla
University of Florida, IFAS, Food and Resource Economics Dept.
Publication Date:
Physical Description:
iv, 26 p. : ; 28 cm.


Subjects / Keywords:
Agricultural prices -- Forecasting -- Mathematical models ( lcsh )
Agricultural prices -- Mathematical models ( lcsh )
Beef cattle -- Prices -- Forecasting -- Mathematical models ( lcsh )
bibliography ( marcgt )
non-fiction ( marcgt )


Includes bibliographical references (p. 25-26).
General Note:
"November 1979"--Cover.
Technical report (Florida Agricultural Market Research Center) ;
Statement of Responsibility:
J. Scott Shonkwiler.

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University of Florida
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University of Florida
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Resource Identifier:
027808633 ( ALEPH )
26812849 ( OCLC )
AJG5556 ( NOTIS )


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79 1

Updating and
with a Varying
Recursive I






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-



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




1 Estimated structural parameters for constant and varying
parameter specifications.....................................

2 Derived and unrestricted reduced forms for Choice steer

3 Choice steer price predictive interval tests.................


1 Variable definitions..........................................



J. Scott Shonkwiler


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


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


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


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)


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
contained in Et.
If IT is known estimation of the parameter vector at time T is

given by

T = (XT XT) XT2T YT (7)

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,


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)


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


Vec(Y) = ZA + Vec(n)


Y :X
m. m

and A =




: B:2)

: B'.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)



= 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




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
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)


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)


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


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


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.


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

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

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
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.


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)




Q4*P acet-2
P1 acet-3


Q *Farrt-2,


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)



















( .562)











0001 83





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.


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



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


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 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



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. Given a recursive system, single equation
methods would yield identical results.

3. Alternatively,it could be assumed that the way in which expectations
are formulated varies over time. Or, it could be argued that the
model is mispecified in terms of incorporating expectations appro-


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John B. Moore.

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