A SIMPLE VARIABLE LENGTH DISTRIBUTED LAG MODEL

A SIMPLE VARIABLE LENGTH DISTRIBUTED

LAG MODEL

D. J. S. RUTLEDGE*

Macquarie University

I. THE PROBLEM

It has long been widely recognized that the transmission of changes from one economic variable to another may not be instantaneous. This has given rise to the common practice of constructing and estimating distributed lag models of the form

where ut is an appropriately specified stochastic error term. In such a formulation Pi re- presents the impact on Y of a unit change in X which occurred i time periods earlier.

In estimating equation (1) the [Pi] are usually assumed to be constants. It has, however, been acknowledged that in some cases this assumption may not be appropriate, and several authors (see, for example, [3], [5] , [6] ) have proposed procedures which permit the lag weights to vary in some fashion, typically as a linear function of some other variables. It is suggested below that even this modification of the traditional fmed weight method may be insufficient in some circumstances.

Consider, for example, a financial market in which holders of a financial asset are assumed to respond to fluctuations in the rate of interest. Because of costs of adjustment or habit formation, it is assumed that adjustment is not instantaneous but rather is spread over n periods subsequent t o the change in the interest rate. This assumption gives rise to the traditional futed weights model defined by (1).

Suppose there is a one unit increase in X (in this case the interest rate) in period T. By proposing a model of the form (1) we are assuming that this change in the interest rate will influence Y in periods T+1, T+2, . . . , T+n, regardless of any subsequent change in the interest rate. But suppose that the interest rate changes again in period T+k (0 < k < n). Is it then reasonable to assume that in periods T+k+l, T+k+2, . . . , T+n, Y is still res- ponding to the interest rate change in period T (as well as the change in period T+k)? In cases such as this where the distributed lag is not a reflection of some physical process, is it not more reasonable to assume that the change in X in period T+k now completely dominates any previous lag effects? If so, a reformulation of the estimating equation is required.

*While retaining full responsibility for any errors, the author would like to thank V. E. Argy for sug-

185

gesting the problem considered in this paper.

186 AUSTRALIAN ECONOMIC PAPERS JUNE

11. A FORMULATION AND SOLUTION

The above discussion suggests that in modelling some financial markets a formulation along the following lines may be appropriate

k

i = O Yt = a t c f l j X t - i t U t ; k = 0 , 1 , 2 , . . . , n

k

1 =o i.e. Y, = u t (.2 pi) xt t ut

for those t such that

X, = x , - ~ = x , - ~ = . . . = x , - ~ and X t - k f xt.-k-.l (0 < k < n). In this specification n denotes the maximum lag length. Written in this way the estima-

tion problem has an obvious and simple solution. Define a set of dummy variables, [Dj, j =O, 1,2, . . . , n] such that

1 when X t = X t - l = . . .=Xt-jandXt-jfX,-j-l

0 otherwise

Then (2) can be rewritten as

Djt =

n Yt = a t 2 y j Zit t ut

i= 0 (3)

where

Equation (3) can be estimated using ordinary least squares or some other method depend- ing on the assumed covariance structure of the [ut] . Estimates of the p’s can be easily obtained from the estimated 7’s.

111. AN APPLICATION

In order to assess the possible practical value of the procedure described above, we consider the demand for certain financial assets in Australia for the period 1958-1969. The basic equations to be estimated are similar to those estimated by Norton, Cohen and Sweeny [2] and a discussion of the underlying economic theory may be found there.

We introduce the following notation: A

GNP gross national product at market prices DUB domestic official borrowing

rfd

value of holdings of a particular asset

interest rate on fHed deposits with twelve-month term.

mechanism to a theory of portfolio behaviour and are essentially of the form’ The equations estimated in [2] are based on the application of the partial adjustment

‘Norton, Cohen and Sweeney use &it as the dependent variable rather than A t ,

187 1978 DISTRIBUTED LAG MODEL

A t = a 0 + a 1 GNPt+a2 DOB, + a 3 rfd, + A A,-1 + u,

As a bench-mark against which estimates using the variable length lag procedure can be compared, equation (4) is estimated on the assumption that u, is a first order autoregressive process using a maximum likelihood search procedure of the kind described by Zellner and Geisel [7] . Three financial assets are considered-currency, current deposits and fixed deposits. The data used are quarterly observations for the period 1958 (4) to 1960 (3) and are obtained from [2]. The results are reported in Table I and appear, at

TABLE I Demand for Financial Assets: Partial Adjustment Model

Dependent Currency Current Fixed Variable Deposits Deposits

(At) Independent

Variable

0.9969 0.7732 0.8434 (0.03683) (0.07986) (0.03072) *t-1

GNPt

DOB,

0.00825 1 0.1018 0.07 589 (0.002494) (0.02788) (0.01556)

0.02974 0.15073 0.09759 (0.0 1030) (0.06028) (0.02956)

rfdt -7.300 -69.98 39.59 (1.9 18) (12.52) (7.322) -

RZ 0.9972 0.9930 0.9989 s. e. e. 8.015 32.333 20.402

Numbers in brackets are estimates of asymptotic standard errors.

first glance at any rate, to support the partial adjustment formulation. All coefficients have signs which accord with Q priori notions and all are significant at commonly used levels of significance. The overall goodness of fit of the estimated relationships is very hgh. However, the implied average lag for currency demand is 321.6 quarters which is clearly unacceptable. For current and fured deposits the average lags are 3.4 quarters and 5.0 quarters, respectively. Even these may be considered implausibly high. They imply that, after eight quarters, 9.9 per cent of the adjustment in demand for current deposits and 21.6 per cent of the adjustment in the demand for fured deposits has yet to occur.’

Demand functions .are now estimated under the ‘variable length lag’ assumptions of Section 11. This involves estimation of equations of the form

n

r=O A , = 6, + 6, GNP, + 6, GNPt-l + 6 , DOB, + .z yi Dit rfd, + ut ( 5 )

’ Although peripheral to the main thrust of this paper, it is interesting to note that recent research on the demand for money persists in using the partial adjustment formulation despite the extremely slow rate of adjustment that is implied by empirical analysis. See, in particular, [ 11.

188 AUSTRALIAN ECONOMIC PAPERS JUNE

where [Dit] is a set of dummy variables of the kind described in Section III.' It is again assumed that ut is a first order autoregressive process and the estimation procedure is as for equation (4). The results are reported in Table I1 while the implied lag distribution of rfd (the 0's of equation (2)) are shown in Table 111. Although the goodness of fit (as measured by E2) in equation (5) is below that of equation (4), it is still very satisfactory and it would probably be difficult to choose between the alternative specifications on this ground alone.4 Yet the implications of the two specifications are strikingly different. As noted before the geometric lag implies relatively slow adjustment while the variable length lag implies virtually instantaneous adjustment. It is not the purpose of this paper to choose amongst alternative models of the demand for financial assets. However, the above analysis illustrates clearly the sensitivity of distributed lag estimation to the lag specification and provides further warning of the care required in selecting a lag speci- ficat ion.

TABLE I1 Demand for Financial Assets: Variable Length Lag


(A*) Independent

Variable

GNP, 0.07202 0.1356 0.2693 (0.01021) (0.04269) (0.1228)

D l t rfdt

D2t rfdt

- 0.1994 0.2232 (0.04387) (0.1296)

0.00987 3 -0.02848 0.1621 (0.01945) (0.05915) (0.1 526)

-0.3264 -14.27 30.22 (2.167) (6.822) (12.92)

-1.495 -16.87 28.74 (2.608) (8.051) (14.64)

-1.908 -17.49 30.17 (2.671) (8.292) (14.39)

-0.9713 -15.35 33.54 (2.619) (8.400) (16.70)

-0.1478 -6.736 48.50 (2.835) (8.872) (19.06) 0.3036 -25.31 58.09

(3.779) (12.81) (29.84)

R2 s. e. e.

0.9082 0.9473 0.9764

15.405 47.302 95.899


'In the estimation to be discussed below we have concentrated on imposing a distributed lag on interest rates. The results would possiily be improved by considering more complicated kg structures

Pesaran [.4] has described the general difficulties associated with using i2 as the basis of model selection.

income as well as on the interest rate.

1978 DISTRIBUTED LAG MODEL 189

TABLE I11 Variable Length Lag: Implied Lag Distribution on rfd


(At) Estimated Coefficient

P O - 0.3 2 64 -14.27 30.22 (2.167) (6.822) (12.92)

-1.169 -2.606 -1.477 (1.509) (4.394) (12.78)

P 1

P2 0.4 126 -0.6168 1.421 (1.409) (4.45 7) (1 3.22) 0.9363 2.135 3.378

( 1.5 22) 4.199 (14.21) 0.8234 8.619 14.95

(1.762) (5.781) (1 7.33) 0.4515 -18.58 9.588

(3.2 82) (10.94) (29.94) P 5


IV. CONCLUSION

In this paper a variant of the traditional futed weight distributed lag model is proposed which may be useful in the modelling of certain financial markets. The method has been illustrated by estimating the demand for certain financial assets and is found to yield plausible results although ones which are markedly different from those found using alternative specifications. In the final analysis the real value of the proposed formulation can only be assessed in the light of further experience.

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REFERENCES G. S. Laumas, and Y. P. Mehra, “The Stability of the Demand for Money Function: The Evidence from Quarterly Data”, The Review of Economics and Statistics, vol. LVIII, 1976. W. E. Norton, A. M. Cohen, and K. M. Sweeny, A Model of the Monetary Sector, Occasional Paper No. 30, Restrve Bank of Australia, Sydney, 1970. J. E. Pesando, “Seasonal Variability in Distributed Lag Models”, Journal of the American Statis- ticalAssociation, vol. 67,1972. M. H. Pesaran, “On the General Problem of Model Selection”, Review of Economic Studies, VOL 56, 1974. P. A. Tinsley, “A Application of Variable Weight Distributed Lags”, Journal of the American Statistical Association, vol. 62, 1967. R. A. Williams, “Variable Weight Distributed Lags: Housing Completions in Australia”, Australian Journal of Statistics, vol. 15, 1973. A. Zellner, and M. S. Geisel, ‘‘Analysis of Distributed Lag Models with Applications to Consump tion Function Estimation”, Econometrica, vol. 38, 1970.

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A SIMPLE VARIABLE LENGTH DISTRIBUTED LAG MODEL