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Economics Letters 21 (1986) 21-25 
North-Holland 
21 
ON USING DURBIN’S h-TEST TO VALIDATE THE PARTIAL-ADJUSTMENT MODEL 
H.E. DORAN 
University of New England, Armdale, N.S. W. 2351, Australia 
Received 20 November 1985 
Final version received 18 December 1985 
A Lagrange multiplier test for diagnostically checking the partial-adjustment model is presented. This test is more powerful 
than the commonly used Durbin h-test because a non-linear restriction on the parameters is taken into account. 
1. Introduction 
The partial-adjustment (PA) model, originally proposed by Nerlove (1958) has had long and 
continued usage in empirical research. Its attraction lies in its simplicity - it involves few parameters 
and can be estimated by ordinary least squares (OLS). It still has an important role in situations 
where inadequate data make it necessary to invest in a simple model. However, if simple models are 
to be used to represent complex reality, it is necessary to have valid diagnostic procedures available 
to detect when the models are inappropriate. 
The PA model may be formulated as follows: 
y:=a+px,-,, Y,- Y-1 = (1 - Y)(y: - y,-,> + u,> (1)7(2) 
where Y,* is the desired or optimal level of the variable Y, (often production or investment), u, is i.i.d. 
N(0, a*) and 0 5 y < 1. When Y,* is eliminated from (1) and (2), the estimating equation 
r,=a(l-y)+P(l-Y)X,-,+vrc-1+u, (3) 
is obtained. 
A common method of diagnostically validating the PA model is to regress Y on X,_ , and Y_ , 
and examine the residuals for autocorrelation using Durbin’s (1970) h-statistic. There is theoretical 
justification for this procedure. 
Suppose the PA model is nested in the more general partial-adjustment-adaptive-expectations 
(PAAE) model, in which X,_ 1 is replaced by X:, given by 
x;c - x:_, = (1 -6)(X,-, - X7-i). ’ (4) 
‘Eq. (4) imphes X,! = (l- s)c~$‘X,_,_~ 
0165-1765/86/$3.50 0 1986, Elsevier Science Publishers B.V. (North-Holland) 
22 H. E. Dorm / Validating the partial-adjustment model 
The estimating model then becomes 
y=b,+b,X,-, + b,Y,_, +&I’_, + u,, where 
b, = a(1 - y)(l - S), b, = P(1 - Y)(I - 61, b,=y+6, 6,= -ys, 
(5) 
and 
u,=u,--u,_,, O$S<l. (6) 
The PA model may be validated by testing the hypothesis S = 0 in (5) and (6). These equations 
represent a dynamic model with a first-order moving average disturbance and Breusch (1978) has 
shown that the h-test is asymptotically equivalent to a Lagrange multiplier (LM) test of the 
hypothesis 13 = 0. 
However, an important feature of the PAAE model is that there is a non-linear restriction on the 
parameters. 
From the definition of 6, and b, above, 
b, = -S(b, -S). (7) 
Use of Durbin’s h-test ignores this restriction and therefore results in a loss of power. 
The purpose of this note is to derive an LM test which does incorporate the restriction (7) and to 
demonstrate its relationship to the h-test. 
2. A Lagrange multiplier test 
When x,_, is replaced by X;C = (1 - S)~~OgS’X,_,--l in (3), the PAAE model becomes 
v,=&+P,( .F,SX_,_‘) +P*r,-, +u,> where 
PO = 41- Y>, P, = P(l - Y)(l - a)? P*=u, 
and the symbols have been chosen so that under the null 6 = 0, p, = b, (i = 0, 1, 2). The restriction 
(7) is implicit in this formulation. 
If we define 8 = (&, &, &)‘, then the likelihood function L(6, 8, a*) is given by 
L= -+ ln(2s)-t Ino’---$u’U, 
where N is the sample size and u = (v,, v2, . . . , uN)‘. The Lagrange multiplier statistic LM has the 
form 
where [see, for example, 
+* = 0, and ‘*’ indicates 
asymptotically distributed 
the form 
H. E. Dorm / Validating the partial-adjumnent model 23 
Breusch and Pagan (1980)] 3,, = -plim(N-‘a*L/&#+Q~) with +, = S, 
evaluation under the null hypothesis. When the null is true, LM is 
as x:. Doran (1985) has shown that when testing the PA model, LM takes 
plim( N- 'RSS ), (10) 
where RSS is the residual sum of squares which is obtained when aC/&V is regressed on afi/M. 
From (8) 
ac - 
--#= -p,x_*, ai 
^ 
1 a6 P, a; 
-zz as -,O’,=,im,, a= -[jt x-1, y-113 
j being a vector of ones. Thus, in finite samples 
LM = ( NC?‘)-‘&( iYX_,)2/( N-‘& RSS*), 
where RSS* = D*‘O*,D* being the residual vector obtained when X_, is regressed on j, X_ , and 
Y -1’ 
That is, on writing NG* = 26, 
LM = N( D’Xp,)*,‘[ (;‘;)( fi*‘O*)] . 
Also, because D* can be written in the form D* = X_, - ( j, X_ t, Y_ , )? for some vector P, 
fi’fi* = $X_, - C’( j, X_,, Y_,)F = 2X_,, 
by the OLS properties of D, which ensure that C’j = D’X-, = D’Y-, = 0. 
Finally then, 
LM= Nr’, 
where r2 is the ‘R* ’ which is obtained when i3 is regressed on O*. 
The LM test of the PA model thus consists of the following simple procedure: 
(11) 
(i) regress Y on X_, and Y-, to obtain the residuals D, 
(ii) regress X_, on X_, and Y_, to obtain the residuals G*, 
(iii) regress G on 6* to obtain r2, and 
(iv) compare Nr* with Xf. 
It should be noted that the additional information that S 2 0 can be utilised by constructing a 
one-sided test in which N’/* r is compared with the standardised normal random variable, with the 
null being rejected when N112 r exceeds the positive critical value. 
The above procedure can be justified intuitively by recognising that the three regressions are 
equivalent to estimating the coefficient & in the multiple regression model 
yI=Po+P,X,-1 +P2Y-l+&xI-2+~,. 
24 H. E. Doran / Valrdating the part&adjusrmenr model 
A ‘large’ r2 is directly related to a ‘large’ &. Now from (S), the PAAE model may be written as 
~=~,+~,X,_,+~,~_,+~,SX,_,+ (higherordersina)+u,, 
and the coefficient of XI+, is seen to be proportional to S. 
3. Connection between the LM and h tests 
In order to recognise the role that the non-linear restriction (7) plays in the LM statistic, it will be 
convenient to consider the PAAE model in the form (5) and (6). The presence of the restriction 
implies that au/a8 is non-zero. 
Expressing the moving-average process (6) in the form 
u=M(6)u, 2 
where M,, = 1, M, ,_] = -6 and all other elements of M are zero, we have 
(12) 
au awl au -= -I___ 
as as u+M a6 _ _M-l$++M-‘~. 
Thus. 
where V, ,p1 = 1 and otherwise y, = 0. 
Substituting into (lo), 
(13) 
where RSS is obtained by regressing Vii + i3G/at3 on Z = (j, X_ ,, Y_,). This expression shows how 
ait, enters both the numerator and denominator of LM. 
Suppose now that the restriction is ignored, and consequently, au/a8 is taken to be zero. Then the 
LM statistic, call it LM*, is given by 
LM* = (;l’Vii)*/[ Ns’plim( N-'RSS*)] , 
where RSS is obtained by regressing Vii on Z. That is, 
RSS* = ?I”[ I - Z( Z’Z)-‘Z’] Vii. 
From the definition of V, 
N N-l 
B’Vfi = c ii_,& = Ne2;, plimN_‘( ii’V’V;i) = plimN_’ c n: = u2, 
t=2 I=1 
plimN_‘( Z’Vii) = [0, 0, u’]. 
*It is clear M(g) = I, and hence ic = 8. 
Thus, 
plimN-‘RSS* = a’plim 1 [ 
For finite samples, we use 
H. E. Dorm / Validating the partial-adjustment model 25 
(15) 
When the square-root of LM* is taken, enabling one-sided tests to be performed, we have precisely 
Durbin’s h-statistic. 
Preliminary Monte Carlo studies reported in Doran (1985) indicate that, at least in cases in which 
X, has strong autocorrelation, the loss of power incurred from using the h-statistic instead of the LM 
statistic (11) is very considerable. 
References 
Breusch, T.S., 1978, Testing for autocorrelation in dynamic linear models, Australian Economic Papers 17, 334-355. 
Breusch, T.S. and A.R. Pagan, 1980, The Lagrange multiplier test and its application to model specification in econometrics, 
Review of Economic Studies 47, 239-253. 
Doran, H.E., 1985, Diagnostic tests for the partial adjustment and adaptive expectations models, University of New England 
working papers in econometrics and applied statistics,no. 17 (University of New England, Armidale). 
Durbin, J., 1970, Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent 
variables, Econometrica 38, 410-421. 
Nerlove, M., 1958, The dynamics of supply: Estimation of farmers’ response to price (The Johns Hopkins University, 
Baltimore, MD).