The purpose of this paper is to focus on the rigidity model and illustrate some relationships between the typical use of the model and a version of it involving first differences in the dependent variable. These relationships will be extended to least squares estimation of rigidity models. A useful correspondence between least squares estimates of the typical model and the first difference model will be demonstrated.Research Methods/ Statistical Methods,

Houck, James P.

Subotnik, Abraham

English

Research Papers in Economics

Staff Paper Series
      Staff Paper P70-19                                                                         November   1970
LEAST SQUARES ESTIMATION OF DISTRIBUTED LAG MODELS:
RELATIONSHIPS BETWEEN ACTUAL AND FIRST
DIFFERENCE EQUATIONS
By
Abraham Subotnik and James P. Houck
Department of Agricultural Economics
University of Minnesota
Institute of Agriculture
St. Paul, Minnesota 55108StaffPaperP70-19 November1970
LEASTSQ(JARES RSTJIV?ATI(’J14 OF DXST’RJWTED MG MODELS;
RELATIONSHIPS BETwEENACITJALANDFIRST
DIFFERENCE EQUATIONS
Abraham%botnik and ,James P. HouckNovember1970
LEASTSQUARESESTIMATION OF DISTRIBUTED LAG MODELS; SOME
RELATIONSHIPS BETWEENACTUALAND FIRSTDIFFERENCE EQUATIONS
AbrahamSubotnik
JamesP. Houck
University of Minnesota
Distributed lagmodelsof econom5.c behaviorhavewon an important
placein appliedeconometric research. The two modelsmost oftenused
in agricultural economics researchwerebroughtto the profession’s
attention by Nerlove~~. Following the terminology in Goldberger’s
text,thesetwo modelscan be calledthe ‘expectations model”and the
“rigidity model.”~,P,275]. The formerinvolveshypotheses about
the way producers or consumersformtheirexpectations of prices,incomes!
or othervariableson whichtheybase economicdecisions. The latter
model involves hypotheses aboutthe way producersor consumers adjust
overtime toward“desired” or long-runoutputor consumption.
The purposeof thispaperis to focuson the rigiditymodeland
illustrate somerelationships betweenthe typicaluse of the modeland
a versionof it involving firstdifferences in the dependent variable.
Theserelationships willbe extendedto leastsquaresestimation of
rigiditymodels. A usefulcorrespondence betweenleastsquaresesti-
matesof the typicalmodeland the firstdifference modelwillbe
demonstrated. U
~ Afterthe completion of thispaper,the authorsdiscovered a
similardemonstration of theseproperties in a bulletinby George
I-add ~, p. 836~. The discussion in thispaperis more ex-
tensivethan the terseproofprovidedby Laddand may be a useful
adjunctto thatwork.2
Sincethe relationships
usefulness in the development
simpleinventory equationbe
(1) s
to be discussed here can haveparticular
of Inventory or stockequations, let a
where
+(
= desiredlevelof stockat the end of time periodt
‘t
‘t = an exogenous variable(or set of
affectingthe stocklevel
‘t = an unobserved, mean-zero$ random
variance.
variables) at time t
disturbance with finite
The parameters ~0 and o(, are unknown,and generally y: cannotbe
observed. The rigiditymodel specifies the processof adjustment of
stocklevelsbetweentimeperiodsas
(2)
= Yt.~ + W(Y; - YJ
‘t
where ~ i,s the familiar“adjustment coefficient” hypothesized to lie
betweenO and +1. The usualprocedure
is to insertequation(1) into (2)and
{..\
to obtainan estimable equation
clearalgebraically.This gives
\d)
= If’#. + ‘2(<, xt+(l-~)ytol+ yet
‘t
with sufficient observations on y and X1$ the parametersc#~ ,0(, and ~
can be estimated with leastsquares.
Now let
(4)
A Yt = Yt - yt-lIn this formulation,
The chanqein stocks
variablein economic
(2)algebraically by
(5)
y~ = %(.
3
AYt is the changein stockduringperiodt.
may be specified as an important dependent
modelsGee for example,2~. Manipulate equation
bringingyt ~ to the rightside. Thenusing (4)
+ lfdI ~t - //yt.l + Yet
This equationexpresses the changein stocksin periodt as a linear
functionof Xt and yt-l. Equation(!3) also can be estimated by least
squares. Whetherequation(5) or equation(3) is selectedfor esti-
mationnormallywoulddependupon the purposeof the investigation
and whetheror not the stockequationis part of a largersystem.
Imagine,for example,thatthe inventory equationis part of a
largermodelwhich includes a market-clearing identityto insurethat
consumption in a givenperiodis equalto production plus inventory
change. In this case,an equation,like(5),with
specified mightbe lesscumbersome to dealwith in
stocklevelequationlike (3).
stockchanqe
the modelthan a
Next it willbe shownthat,givena seriesof observations on y
and X, leastsquaresestimates of~a, d~ and $ are identical Ino
matterwhetherequation(3) or equation(5) is specified. Employing
standard matrixnotationlet the regression modelindicated by equation
(3)be writtenas
where
Y = levelof stocks;a vectorof dimension Txl4
‘1 = a row vectorof 1’s and the independent variable; a
matrixof dimension Tx2
‘2 = laggedvalueof y; a vectorof dimension Txl
[1 p, .::
u = disturbance term;a vectorof dimension Txl witha
typicalelementbeing tiet
The leastsquaresestimatesof the ~’sare well-known as
‘;
?
,X2
Y H.
1 -1 ‘;xl‘1X2
= 8 t
‘2X1 ‘2X2
Now definethe following variable
(8)
z =Y-X2
Noticethat z = by
Then the regression modelof equation(5)becomes
(9)
z =
[ 1[]
xx IT, +U
12
lr~
where
I’q=[-?fq
and X1, X2, and u are the sameas in equation(6)5
The leastsquaresestimatesof (9) are
X’x X’x
11 12
t
‘;xl‘2X2
Thenusingequation(8)
(10
-1
[
x’
1
*
‘2
-1
x’
1
z
[’
X’x XIX
11 12
Y- , !
‘2X1 ‘2X2 [
-1’”X1X2
!
‘2X2
to Noticethatthe firstexpression in equation(10)reducesdirectly
the leastsquaresestimatesof 8, and ~q l The secondexpression
can be viewwdas an inversematrixmultiplied by part of its original
matrix. As such,it
this case,the final
Therefore
(11)
reducesto a part of an identity matrix-- in
columnvectorof an identity matrix.
Both ~ 8 and ~ are identical estimatesof the parameter vector
ryf. X4,]’ Similarly&a is an estimateof - r ) and F?
is an estimateof (1 - Y), Sincefromequation(11) t~= ~a -19
both provideidentical estimatesof ~ .
g
~ This particular propertyof the leastsquaresmethodis not con-
finedto the economicillustration whichhas been employedhere.
It holdsany timethe dependentvariableis alteredby the sub-
traction(oraddition)of one or more of the specifiedindependent
variables.6
Thisdirectcorrespondence betweenthe estimatedcoefficients
Of thesetwo models extends to the estimated gtandard errors of the
regression coefficients.The usualexpression of the standarderror
of the regression coefficient$g 69 i$
“ ‘[-O(’’’)-!’
where{ is the calculated residualvectorbasedon # ~ X is the matrix
bl, ~~, and d.f. indicates degreesof freedom. Similarity for the
A
estimatedcoefficients, ~ , the expression is
“ ‘Eo(’’x(l’
whereall is the sameexceptthat% is the calculated residualvector
basedon # . If it can be shownthat~’~ is equalto fi’~t hen
S$=s$,
Firstlookat~’~. Observing that~ =y - X6 and usinga
familiarrelationship fromleastsquaremethodology~, p. 159~.
(12)
Similarly
(13)
A */l
Uu= Y’Y -y’x~
*9*
Uu= 2’2 - z’ x$
Recallthat % = # - K whereK is a column
for the lastelementwhichis in this case 1,
vectorof zero’sexcept
see equation(11). Then
(14)
[
5%=Z’, z’-x (p
1
- K)
= z
[‘-’? +’2]7
Substituting for z from’ equation(8)
& *N
Uu = (y. x*) ’[y .x*-x@ +X23
= (y ’Y-Y’ x~) -x; (y-x$ )
~ *A = Uu-x; (y .x))
(15)
=U’u. x;y+x; xp
BecauseX2 is a column
the finalexpression in
Xi y. Thus
and the standard errors
are identical.This is
-1 ‘
vectorpartitionof X and since ~ = (XIX) X y
equation(1!3), namelyX; X ~ , collapses to
of the estimatedcoefficients in eachmodel
appropriate sincethe errorin estimating
fromequation(7)and the errorin estimating~~ fromequation(10)
both stemfromthe errorattachedto estimating~ (recallthat
In this illustration eitherthe actuallevelof stocksor the
changein stockscan be used as the dependent variablein a least
squaresestimation process. Parameter and standard errorestimates
basedon one specification are identical with thosebasedon the
otherspecification and canbe used interchangeably.References
1. Goldberger, A. S. Econometric Theory,(J.Wileyand Sons,Inc.:
New York),1964.
2. Houck,J. P. and J. S. Mann,An Analysisof Domesticand Foreicm
Demandfor U,S. Soybeansand SoybeanProducts, Agr. Exp’tStation
Tech.BulletinNo. 256, University of Minnesota, 1968.
3. Ladd,G. W.$ Distributed Laq Inventory Analysis, Agricultural and
HomeEconomics Experiment StationResearchBulletin515, Iowa
StateUniversity, April 1963.
4. Nerlove,M. “Distributed Lags and Estimation of Long-RunSupply
and DemandElasticities,” Journalof FarmEconomics$ Vol.XL,
No. 2, hkly 1958,pp. 301-310.

LEAST SQUARES ESTIMATION OF DISTRIBUTED LAG MODELS: RELATIONSHIPS BETWEEN ACTUAL AND FIRST DIFFERENCE EQUATIONS

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