APPLYING LINEAR TIME-VARYING CONSTRAINTS
TO ECONOMETRIC MODELS:
AN APPLICATION OF THE KALMAN FrLTER
Howard E. Doran and Alicia N. Rambaldi
No. 83 -November 1995
ISSN
0 157 0188
ISBN
1 86389 291 5
Applying Linear Time-Varying Constraints to Econometric Models: An
Application of the Kalman Filter.
Howard E. Doran* and Alicia N. Rambaldi**
Department of Econometrics
University of New England, Australia
Abstract
When linear equality constraints are invariant through time they can be incorporated
into estimation by restricted least squares. If, however, the constraints are timevarying, this standard methodology cannot be applied. In this paper we show how to
incorporate linear time-varying constraints into the estimation of econometric models.
The method involves the augmentation of the observation equation of a state-space
model prior to estimation by the Kalman filter. Numerical optimisation routines are
used for the estimation. A simple example drawn from demand analysis is used to
illustrate the method proposed in this paper and its application.
JEL Classification: C51, C32.
Keywords: state space models, time-varying constraints, Kalman filter, numerical
optimisation.
Corresponding author. Department of Econometrics, University of New England, Armidale,
NSW 2351, Australia. Tel.: 61-67-732319, Fax: 61-67-733607, Email:
[email protected]
We wish to thank William Griffiths and D. S. Prasada Rao for helpful comments. The
responsibility for any errors is, of course, ours.
2
1. Introduction
When building econometric models, economic theory often provides what
Goldberger (1964) calls ’extraneous information’ in the form of constraints that the
parameters of a model should satisfy. In the case of time-series dal~a, the constraints
are usually identical in each time period. Such restrictions, particularly when linear in
the parameters, can be handled in a completely standard way by restricted least
squares.
However, when the constraints are different in each time period, the standard
method does not apply. Such cases occur, for example, in production theory. Three
examples from this area are presented as illustrations.
First, the zero-degree homogeneity condition of factor demand functions in a
simple two factor case takes the form:
r~Of / OX2 ~ r2Of / OxI
where, rl and r2 are factor prices, xl and x_ are input quantities and f(.) a production
function.
For common production functions such as the generalised Cobb-Douglas,
quadratic and transcendental, this condition results in restrictions which are linear in
parameters and also involve variables, and are therefore time-varying. Beattie and
Taylor (1985, 119) comment that this constraint ’should not be ignored, although it
often is, in empirical studies’
Second, the assumption of Hicks-neutral technology (see Lau, 1978, 20) results in
constraints of the form:
~Xix ~x~
-’xl =
~t ~ --’~
0
3
which for most commonly used profit functions (which incorporates technology) also
gives rise to linear time-varying constraints.
Third, in a discussion of several useful functional forms for variable profit
functions, Diewert (1973, 307-309) derives linear time varying constraints which
should be imposed during estimation.
Demand systems which are linear in logs are popular in applied economics because
of the ease of interpretation in terms of elasticities. In two later sections of this paper,
we discuss such demand systems as they furnish another example of the occurrence of
time-varying constraints. We use this illustration to elaborate on the methodology put
forward in this paper.
When the model is the conventional linear model with f’~xed parameters, the
fundamental problem involved in imposing time-varying constraints is that there are
more constraints than parameters. When this information is not ignored, the usual
approach of applied economists is to reduce the number of constraints by imposing
them ’at the mean’, (thus converting time-varying constraints into time-invadant
constraints). Such an approach would, at the least, imply a loss of ’efficiency’, in the
sense that relevant information has been ignored. However, when the constraints are
non-linear in variables, it would also result in imposing constraints that are not
consistent with the underlying economic theory and hence biases are introduced into
the estimation procedure.
The approach adopted in this paper is to postulate a more flexible model -one in
which the parameters can vary over time. Under certain assumptions, such a model
can be set up as a state-space model which incorporates the linear time-varying
constraints. This particular specification of state space models does not seem to have
4
been exploited so far. The Kalman triter, a standard tool in time series analysis, can be
used to estimate such models.
The purpose of this paper is toshow how to incorporate linear time-varying
of econometric models, thus extending standard
constraints into the estimation
practice.
In Section 2 we discuss the methodology. Section 3 provides an illustration drawn
from demand analysis. Some empirical results are described in Section 4 and
concluding remarks are made in Section 5.
2. Applying Time-Varying Constraints
Consider a model of the form
Yt =Xt~+ut (t= 1,2 .....T)
(1)
where Yt ,ut are rr~l (m>l), Xt is rnxK, I~ is a Kxl vector of unknown parameters
and ut is iid N(0,H). Suppose that the parameters I~ are required to satisfy time-varying
constraints of the form
Rtl~ = rt
(t = 1,2 .....T)
(2)
where Rt is JxK of rank J_<K, rt is Jxl and both Rt and r, are known for all t.
Such constraints can arise when economic restrictions applied to a model of the
form (1) involve both the parameters [~ and variables.
5
An inherent inconsistency exists between (1) and (2), in that the vector 13, of size
K, is required to satisfy JT constraints, and invariably JT > K. In practice, time-varying
constraints are likely to be ignored. However, when attempts are made to incorporate
such information into the estimation, they are usually applied ’at the mean’. Such
constraints are of the form
(3)
where R is J×K of rank J. Thus, the T constraints (2) have been replaced by the single
constraint (3) and estimation of 13 is accomplished by standard restricted least squares.
However, replacing (2) by (3) may have two consequences. First, even if ~ = ~ is
consistent with Rt13 = rt, in the sense that the latter implies the former, there is a loss of
information in using R and f for Rt and rt, respectively. Second, and potentially more
serious, is that it may well be the case that RI3 = f is not implied by the time-varying
constraints (2). In such cases using the restriction (3) will result in estimates of 13 which
are biased.
The fundamental problem is that the model (1) is not sufficiently flexible to
accommodate restrictions which change in each time period. The method outlined
above accommodates the inflexible nature of the model by compromising the
constraints, and in the process may lose touch with underlying economic theory.
In this paper, we are suggesting the opposite approach, namely, the model is made
flexible enough to allow time-varying restrictions.
6
Specifically, we will assume that 13, rather than being f’~ed over time, can vary in
each time period, and wil~ be denoted by I~t. Further, we will assume that I~t evolves
according to some linear process, for example,
~t = ~ + A~r-1 + ~t
(4)
where g and A are constant, and ~ is iid N(0,Q). This equation together with
Yt = X,l]t + ut
(5)
constitute a state-space model for the state vector I~t, where (4) and (5) are known as
the "transition equation" and "observation equation", respectively.
Let us, for the moment, ignore the constraints (2) and consider only the above
state-space model. It is well-known that provided 5, A, Q and H are known, optimal
estimates of the I~t can be obtained using the Kalman filter (Kalman, 1960). As
numerous descriptions of the Kalman filter exist (see, for example Harvey, 1990, Judge
et al, 1985, Hamilton, 1994) we will not describe it here.
A property of the Kalman filter, which is of fundamental importance to this
problem, is that if the observation equation (5) is appropriately augmented, the
estimates ~ t can be made to satisfy linear constraints.
Specifically, if we defme
Y’ /r,j
(6)
a new observation equation of the form
Yt = Zt~t +
(7)
7
can be associated with the transition equation (4). Estimates ~ t obtained by applying
the Kalman triter using the augmented observation equation (7) will satisfy
Rt~t = rt
(8)
(see Doran, 1992).
Thus, the Kalman filter appropriately augmented, provides a convenient
methodology for obtaining an estimate of I~t which satisfies linear time-varying
constraints. The smoothed Kalman filter estimates (see Harvey, 1990, page 154) also
have this property.
In practice, many of the elements of 5, A, Q and H will not be known and wi~
have to be estimated prior to obtaining [~t- Let us suppose that a vector 0 contains the
unknown elements.
It can be shown (see, for example, Harvey 1990, 126) that, for a given 0, the loglikelihood function can be expressed in the form
T
tnL(y*10) =C-zt=
~1 ~ enlFtl
1
T
2
Z)’t Ft-la.)t
(9)
t=l
where v, = Yt- ~rlt-1, ~tlt-! is the one-step ahead prediction of Yt and Ft is the
covariance of’Or. The vector ’I)t and matrix Ft are obtained as intermediate outputs from
the Kalman filter.
Thus, applying the Kalman f’flter to the series
,T) and gn L(y*I0).
Yl, Y2 .....y~-, assuming a value for 0, will produce both ~ t (t=l,2 ....
Maximum-likelihood estimates of 0 can be obtained by using a numerical optimisation
routine, such as the Newton-Raphson procedure. A well known practical problem
associated with the standard optimisation routines is that they may not be effective in
locating the global maximum of the likelihood function (See, for example, Greene,
8
1993, 350-352). It therefore becomes important to reduce the dimension of 0 as much
as possible before optimisation is attempted in order to make the search more efficient.
The elements of 0 come from two distinct sources. First, there are those elements
which come from the covariance matrices Q and H. Second, there may be elements
coming from the level of 13t, for example elements of ~ and A in equation (4). Simple
reparameterisations are available which allow the ~ parameters to be estimated by
Generalised Least Squares (GLS), enabling them to be removed from the numerical
optimisation procedure (see Harvey, 1990, 130-133, for details).
In the next section we introduce a simple illustration of the foregoing discussion.
o
An Illustration
As a simple example of the problem and procedures discussed in Section 2, we
will consider a set of constant elasticity demand equations.
Consider the demand for N commodities, assuming a model of the form
N
gn qit = ~i0 + ~ij gn Pjt + ~i,N+l £n Yt + uit
i = 1,2 .....
N
(10)
j=l
where
qit = quantity demanded of commodity i at time t,
pit = price of commodity j at time t,
and
yt = income at time t.
Demand theory requires the system to satisfy three types of equality constraints,
known as the Engel condition, the Slutsky symmetry conditions and the homogeneity
condition. When appfied to the model (10), these constraints take the form
Engel:
~ Wit~i,N+l "- 1
i=l
(11)
131J + 131,N+1
Slutsky:
i = 1,2 ..... N; j = i+l .....N
(12)
Wit
N+I
Homogeneity:
i= 1,2 .....N
~13ij = 0,
(13)
j--1
N
where
Wit ~ Pitqit
j=l
We note that the Engel and Slutsky conditions involve wit, and therefore vary across
time periods. The homogeneity condition (13), on the other hand, is time-invariant and
can be used to eliminate the parameters I~,N+I1 (i = 1,2 ..... N) to obtain the demand
system in the form
N
gn qit = ~i0 + Z~ij gn Pit + Hit
(10)"
j=l
where
P~t = P jr/Yt ¯ The constraints (1 1) and (12) now become
Engel:
Slutsky: ~ij
NN
2 ~ Wit~ij =-1
i=1 j=l
(11)"
(W;1 - 1)-~3ik --~ji(wTt1 - 1)-~[3jk
k*j
(12)"
k#i
i=1,2 .....N; j=i+l .....N
If the constraints were to be applied ’at the mean’, then wit, wit in (11)’ and (12)"
would be replaced by their sample means "~i and W j, respectively. The effect of such
a substitution is different for the two types of constraints. The Engel constraint is linear
NN
NN
in wit and therefore ZZWit~ij =--1 (t = 1,2 ..... T) implies ~Wil3ij =-1. Thus,
i=l j=l
i=l j=l
1 Elimination of time-invariant constraints is necessary. Otherwise the matrix Ft, referred to in
equation (9), will be singular.
10
replacing the T constraints (11)’ by this single constraint only involves a loss of
information (unless wit = Wi for all i and t).
On the other hand, the Slutsky condition (12)’ is non-linear in the wit and wit. This
implies that replacing wit and wit by W~ and Wj, respectively, in (12)¯ will result in
constraints which are not consistent with the Slutsky condition. Thus, this practice will
result in not only a loss of information, but also the application of incorrect constraints.
We now define
[gn qlt,gn q2t ..... gn qNt,] ,
,gn Plt,gn P2t ..... gn PNt ,
jN =
a column of l’s of length N,
jN ® xt,
.....
and
Ut "- [Hlt,U2t ..... UNt].
Then the system of equations (10)’ can be written in the standard form
Yt = Xt~ + ut"
Similarly, because the constraints (11)’ and (12)’ are linear in the I~i~ they can be
expressed in the form
Rt~ = rt,
where Rt depends on the wit. In this illustration the vector r, is in fact time-invariant,
but this has no effect on the analysis.
11
Following Section 2, we now allow the elasticity vector 13 to vary over time, and
postulate a transition equation of the form
~t = ~ + P~t-1 + ~t,
(14)
where p is a scalar, such that I~ <1. This vector AR(1) transition equation
incorporates two simplifying assumptions, namely
(i) the elasticities 13t are stationary, and
(i.i) following shocks, all elasticities adjust back to equilibrium at the same
rate, determined by p.
The transition equation (14) and the augmented observation equation (7) from
Section 2 form the state-space model which is being postulated for this illustration.
The parameters 5, p, Q and H are unknown and will have to be estimated. If we
assume Q and H are diagonal the parameter vector 0, described in Section 2, will have
(2N+I)(N+I) elements.
A simple reparameterisation will enable us to reduce the dimension of 0 by
N(N+I) leaving dimension equal to (N+I)=. If we set ~ = g(1-p), where I.t is the mean
of ~3t, and
~t "-- ~t -- ~
(15)
then the system can be written in the form
I~t = Pl3t-x +{t
(16)
Yt = Zt[’t + Zt~t -1- ]~t
(17)
The advantage of expressing the system in this form is that the parameter vector g
can be estimated by GLS independently of the optimisation routine needed to estimate
the remaining elements of 0.
12
In the next section we give an empirical example, using Australian data.
4. Empirical Results
The demand system described in Section 3 was estimated for the consumption
of meat in Australia, using data from Alston and Chalfant (1991) in which the meat
categories were beef, lamb, pork and chicken. The data consisted of quarterly
observations over the period 1977:1 to 1988:4. Thus, in this application N--4 and
T=48.
Following Murray (1984), we assumed that the consumer’s utility function was
wealdy separable between meats and all other commodity groupings (see also
Silberberg, 1990). This implied that the income variable yt could be defined as the total
expenditure on meat.
The specific series were:
pit = nominal average quarterly retail price of meat group i in S/kilogram
qit = per capita consumption of meat group i in kilograms
Yt = total meat expenditures
As in Alston and Chalfant (1991), the data were deseasonalised prior to
estimation. The vector 0 containing the unknown parameters, had dimension (N+I)2 =
25, with the elements 0j identified as follows:
1 < j < 20,
diagonal elements of Q;
21 <j < 24,
diagonal elements of H;
j = 25,
autoregressive parameter 9A constrained Newton-Raphson procedure, guaranteeing 0j > 0 (1 <j < 24)
and -1 < 0j < 1 (j = 25), was used to obtain the maximum-likelihood estimate ~. The
vector g of mean elasticities was estimated by GLS, as described in Section 3. The
13
estimate of 19 was [~ = 0.1003 and the other elements of ~ are reported in Table1
below. Values of ~t are given in Table 2.
Table 1
ML Estimates of Diagonal Elements of Q and H
Intercept
Beef
7.84E-04
Lamb
1.25E-04
Pork
1.39E-04
Chicken
1.25E-04
Beef
1.32E-04
1.23E-04
1.61E-04
1.23E-04
Lamb
1.44E-06
1.28E-04
0
0
Pork
8.1E-07
0
9.41E-05
0
Chicken
4.9E-07
0
0
1.93E-04
9.22E-05
1.21E-04
1.32E-04
1.23E-04
Q
H
Table 2
Estimated Average Elasticities ~t, with t-values
Beef
-0.91172
(-31.32469)
Lamb
-1.79667
(-44.21734)
Pork
-1.18018
(-30.81011)
Chicken
-1.78166
(-47.12815)
- 1.02327
(-92.74668)
0.00777
(0.52086)
0.05571
(4.00574)
0.00264
(0.18862)
£n PL
-0.01426
(-4.44825)
-0.88531
(70.61460)
-0.03000
(-15.58942)
-0.03000
(-15.58942)
£n pv
-0.02601
(-7.18041)
-0.06432
(-23.57320)
-0.83774
(-73.51619)
-0.06432
(-23.57320)
£n pc
-0.01683
(-7.65200)
-0.02924
(-20.52649)
-0.02924
(-20.52649)
-0.85603
(-93.00836)
gny
1.08037
(111.1954)
0.97110
(72.68703)
0.84127
(65.11874)
0.94772
(75.75877)
Intercept
14
Figures 1 to 3 below show the movements through time of selected price
elasticities conditional on the maximum likelihood estimate of 0.
-0.96
-0.97
-- average elasticity I
+ elasticity at time t
-0.98
I
-0.99
-1
-1.01
-1.02
-1.03
-1.04
-1.05
Fig. 1 Beef- Own Price Elasticity
43.79
-0.8
-0.81
-0.82
4
7
10 13 16 19 22 25 28 31 34 37 40 43 46
I average elasticity
1
~ elasticity at time t t
-0.83
-0.84
-0.85
-0.86
-0.87
-0.88
-0.89
Fig. 2 Chicken - Own Price Elasticity
15
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Time
Fig. 3 Pork/Beef- Cross Price Elasticity
In order to assess goodness-of-fit of each of the equations (10)’, we used a
conventional R2 measure, defined as the square of the correlation between gn qi and
where,
and the ~s were the smoothed Kalman filter estimates.
The values of R2 for the beef, lamb, pork and chicken equations were 0.9985,
0.9855, 0.9937 and 0.9948, respectively.
Finally, as stated in Section 2, the constraints were satisfied in every time
period. To illustrate, we arbitrarily chose t = 40. The values of ~t and wit for this
period are shown below in Table 3.
16
Table 3
~, and wit, t = 40
Beef
Beef
- 1.02962
Lamb
0.00456
Pork
0.04868
Chicken
-0.00112
Lamb
-0.01457
-0.89660
-0.03000
-0.03000
Pork
-0.02618
-0.06432
-0.84812
-0.06432
Chicken
-0.01692
-0.02924
-0.02924
-0.87565
Income
1.08729
0.98560
0.85868
0.97109
wi,4o
0.48900
0.15773
0.20283
0.14282
Then, from (11), the Engel condition was:
4
Engel:
~ wi,4ol~i5 = 1
i=l
Using the estimates from Table 3 we have,
0.4890xl.08729 + 0.15773x0.98560 + 0.20283x0.85868
+ 0.14282x0.97109 = 1
Slutsky:
W j,40
I~15W=i,40-J____!_ + 15j~
Choosing i = 1, j = 2 for example
~12
k~15 = -0.014557/0.15773 + 1.08729 = 0.9949
W 2,40
1~21 ’1" 1~25 --
0.00456/0.48890 + 0.98560 = 0.9949
W 1,40
These two results verified that the time-varying constraints have been satisfied.
17
5. Concluding Remarks
In this paper we have shown how time-varying constraints can be incorporated
into the estimation of econometric models.
The theory underlying the method is quite straight-forward, involving the
augmentation of the observation equation of a state-space model prior to estimation by
the Kalman filter.
Examples have been cited showing when such constraints may occur, and an
illustration from demand analysis is used for an empirical application.
In practice, the major problem comes from the need to use numerical search
procedures as a first step since optimisation routines may have convergence problems
when the parameter space is large. An important part of the method is using an
appropriate reparametrisation to minimise the number of parameters over which the
search takes place.
On-going research is concentrating on two lines. First, extension of the
methodology to non-linear constraints and second, ways for further reducing the
number of parameters required in the search routines.
18
References
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structural change in demand (Unpublished manuscript, Department of Agricultural
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Doran, Howard E., 1992, Constraining Kalman filter and smoothing estimates to
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19
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20
WORKING PAPERS IN ECONOMETRICS AND APPLIED STATISTICS
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ad2A ~eai~aad Neqo~ong~ ~. W.E. Griffiths and
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Helmut L~tkepohl, No. 42 - March 1990.
Howard E. Doran, No. 43 - March 1990.
I!ainq ffAe ~o/nu~ ~i&teat.e galimaieSaA-9~. Howard E. Doran,
No. 44 - March 1990.
Rea~. Howard Doran, No. 45 - May, 1990.
Howard Doran and Jan Kmenta, No. 46 - May, 1990.
and ~nn~ ~. D.S. Prasada Rao and E.A. Selvanathan,
No. 47 - September, 1990.
gcoa~mi~YOmi~_~iAe ~n~o~ Nea~ g~. D.M. Dancer and
H.E. Doran, No. 48 - September, 1990.
D.S. Prasada Rao and E.A. Selvanathan, No. 49 - November, 1990.
~2i~i~ic~ g~. George E. Battese,
No. 50 - May 1991.
~~ gan~m~o~ ~~ ~ozun. Howard E. Doran,
No. 51 - May 1991.
FexJin@ Non-Nealed~ad21~. Howard E. Doran, No. 52 - May 1991.
~emp~ ~gn~. C.J. O’Donnell and A.D. Woodland,
No. 53 - October 1991.
~(~%a~/~nZn9 Se~. C. Hargreaves, J. Harrington and A.M.
Siriwardarna, No. 54 - October, 1991.
Colin Hargreaves, No. 55 - October 1991.
~9~ic~Ze ~add@ ~~in ~ndie. G.E. Battese and T.J. Coelli,
No. 56 - November 1991.
2.0. T.J. Coelli, No. 57- October 1991.
Barbara Cornelius and Colin Hargreaves, No. 58 - October 1991.
Barbara Cornelius and Colin Hargreaves, No. 59 - October 1991.
Duangkamon Chotikapanich, No. 60 - October 1991.
Colin Hargreaves and Melissa Hope, No. 61 - October 1991.
Colin Hargreaves, No. 62 - November 1991.
25
O&~Z ~ ~u/v~e. Duangkamon Chotikapanich, No. 63 - May 1992.
V~. G.E. Battese and G.A. Tessema, No. 64 - May 1992.
gn~b~ ~a~hzhi~. C.J. O’Donnell, No. 65- June 1992.
~~. Guang H. Wan and George E. Battese, No. 66 - June 1992.
Ma. Rebecca J. Valenzuela, No. 67 - April, 1993.
Mo2~ ~ gh~ ~.~. Alicia N. Rambaldi, R. Carter Hill and
Stephen Farber, No. 68 - August, 1993.
~a~ g~. G.E. Battese and T.J. Coelli, No. 69 October 1993.
Yea/ StY. Tim Coelli, No. 70 - November 1993.
~~n Wealen~aaZasIian ~. Tim J. Coelli, No. 71 December, 1993.
G.E. Battese and M. Bernabe, No. 72 - December, 1993.
Getachew Asgedom Tessema, No. 73 - April, 1994.
#~ ~o~. W.E. Griffiths and A.T.K. Wan, No. 74 May, 1994.
O:~ian~ M~2iAp/~e ~x~k~e. C.J. O’Donnell and D.H. Connor, No. 75 September, 1994.
~o~un2.~. T.J. Coelli and G.E. Battese, No. 76 - September, 1994.
Z~ ~~ ~ ~u% ~(I) ~ ~o~. William E. Griffiths,
No. 77 - September, 1994.
Hector O. Zapata and Alicia N. Rambaldi, No. 78 - December, 1994.
~/~. Ma. Rebecca J. Valenzuela, No. 79- August, 1995.
Ro~ ~~ Y~. William E. Griffiths and
Ma. Rebecca J. Valenzuela, No. 80 - November, 1995.
~o~ on ~o2_~~.
No. 81- November, 1995.
Alan T.K. Wan and William E. Griffiths,
~Ae~ ~o~e. Alicia N. Rambaldi, Tony Auld and Jonathan Baldry,
No. 82 - November, 1995.