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with a Varying
UPDATING AND FORECASTING WITH A VARYING PARAMETER
J. Scott Shonkwiler
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
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
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-
TABLE OF CONTENTS
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
SUMMARY AND CONCLUSIONS .......................................... 20
APPENDIX .......................................................... 22
FOOTNOTES ......................................................... 24
REFERENCES ........................................................ 25
LIST OF TABLES
1 Estimated structural parameters for constant and varying
2 Derived and unrestricted reduced forms for Choice steer
3 Choice steer price predictive interval tests.................
LIST OF APPENDIX TABLES
1 Variable definitions..........................................
UPDATING AND FORECASTING WITH A VARYING PARAMETER
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
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
KT+l :T+ /TXT+lI T+l T+1/TxT+I + 2 (11)
The matrix E denotes the parameter covariance matrix. It is given
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)
and A =
: 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
Var(Ai) = ii(Z 'Zi)
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' = ()
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
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
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
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
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
SPR .1 -1
INTERCEPT .1314 261Z -130 124 2106 -110 113
03 364 319
(1 .86) (2.02)
Q4 657 623
1FCSt.2 .335 .186
ZSPRt- i -60.6 -69.1
3FC ~. 26.0 25.5
3 p2 ( .7)
4 t-2 (.76)
Fart-2 .691 .687
Ql*Farrt-z -.153 -.151
(2. 79) (3.00)
Farrt.3 .611 599
iPn1 5.71 5.65 .45
Time .388 .463 .01
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)
.2837 ( .319)
-.625 ( .450)
-1.221 ( .707)
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
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
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
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
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.
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)
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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