A Modification of the Almon Distributed LagAuthor(s): Peter SchmidtSource: Journal of the American Statistical Association, Vol. 69, No. 347 (Sep., 1974), pp. 679-681Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286001 .

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A Modification of the Almon Distributed Lag PETER SCHMIDT*

This article modifies the Almon polynomial distributed lag to allow for a lag of infinite length. Estimation of the model is considered, and an empirical example is given to illustrate the usefulness of the model

1. INTRODUCTION

Virtually all common distributed lag specifications assume a lag of infinite length. The one notable exception is the Almon (polynomial) distributed lag,' which assumes that the independent variable has an influence on the dependent variable for a finite number of time periods.

The problems which this causes are fairly well known. In general, estimation requires the specification of the lag length, and this is often difficult. Specification error and resulting inconsistent estimates can occur either because the lag is specified to be shorter than it actually is or because it is specified to be longer than it actually is.2 This is a serious problem because one's a priori information on the length of the lag is often not very precise.

Of course, one may treat the lag length as an unknown parameter and "search" over it, picking the value that, say, minimizes the standard error of estimate. However, in this case the (estimated) variances of the resulting estimates are conditional on the final choice; this results in an understatement of their true variances, and hence in an overstatement of the level of significance of the results.3

A further problem is that, in the Almon specification, one cannot test the existence of a lag. The Almon lag, unlike the other common distributed lag specifications, does not contain a strictly contemporaneous influence as a special case. Hence, if there really were no lag present, the data could not reveal this fact if the Almon specifica- tion were used.4

In addition to these statistical problems, this author finds it conceptually difficult, in most applications, to justify a lag of finite length. The alternative assumption, implicit in other lag specifications, that the influence of one variable on another simply approaches zero as the time span increases, seems more reasonable. This allows the influence to be effectively zero, without forcing it to exactly zero, past some point in time. And in this way the data are allowed to indicate how quickly the influence dies

* Peter Schmidt is associate professor, Department of Economics, University of North Carolina, Chapel Hill, N.C. 27514.

1 Due to Almon [1]. 2 On this point, see Schmidt and Waud [9, p. 13]. 3 For an elaboration, see Schmidt [7]. 4 Schmidt and Waud [9, pp. 12-13].

out, rather than having the choice be made on (often tenuous) a priori grounds.

In this article, therefore, the Almon lag specification will be modified to assume a lag of infinite length.

2. THE MODIFIED ALMON LAG

The Almon lag model may be written as

Yt = a + =o wiXti + e, t = 1,2, ,T; (2.1) where the "weights" wi are constrained to lie on a poly- nomial of degree p < n:

L -r0 2x,ii. I i=O, 1, *, p. (2.2) The modification that this article suggests is, first, to assume a lag of infinite length:

Yt = a + Et=O WiXt-i + ?t. (2.3)

Now, so that wi -+ 0 as i -* o, we will specify the weights as follows:

Wi t= i Ejo2ijij 0 < <1 i< = 0,1*. (2.4)

The parameter y is essentially the same as the parameter in the geometric lag model. Indeed, this specification contains the geometric lag as the special case correspond- ing to 21 = 12 = * * * = lp = 0. It also contains the case of no lag as the special case corresponding to y = 0, if we take o -= 1 even for y = O,B

For large values of i, the term yi will dominate the expression (2.4), so that the influence of X on y will essentially decrease geometrically to zero. However, for small values of i the term yi need not dominate; one or more "humps" are possible in the lag distribution.

It may be noted that the interpretation as a modified geometric lag can also be given to several other dis- tributed lag models. For example, the weights for both the Pascal lag [10] and the Gamma lag [11, 8] are of the form: wi = f (i)yi, 0 < y <K 1, where f (i) is in each case a particular function. In our present case, of course, f (i) is taken to be a polynomial in i. This is a rather general

6 In the term corresponding to i = j = 0, ii = 1. 6 This modification of the Almon lag is similar in spirit to a lag structure proposed

by Hall and Sutch [4]. However, the Hall and Sutch lag structure was a combi- nation of the polynomial and geometric lags in a different sense; namely, the first q weights were assumed to lie on a polynomial, whereas later weights decline geo- metrically.

? Journal of the American Statistical Association September 1974, Volume 69, Number 347

Theory and Methods Section

679

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680 Journal of the American Statistical Association, September 1974

specification since any analytic function f (i) can be ap- proximated by such a polynomial.

3. ESTIMATION OF THE MODEL

We now turn to the estimation of the model specified by (2.3) and (2.4). First of all, we note that we can rewrite (2.3) as

Yt a + '5 jSt* + st(3.1) where

Sot = Ei=O tiXt-i, Sit = Z'0o YTiiXt-i, j=1,2, .Ip. (3.2)

Now, it is inconvenient that the Sjt contain unobserv- able values of X. However, as in the geometric and Pascal lag, we can truncate these infinite sums and decompose the remainder into "initial conditions" terms. To do so, define

Sot = Ei- yixt-i; Sjt = FZ2 tiijxt.i,

j-1,... p, (3.3) and note that

E P

0%s*= j_o yjSjt + R, (3.4) where

R = Eo ti yiXt-i + Ep=1ij E=t tit i. (3.5)

Now it can be shown, through simple (but tedious) algebra, that

R-= J'o?,(titrt), (3.6)

where the qj are constant-i.e., they are independent of t. This then yields the equation

yt= a + ?Lo Xjj + JT=o (t'yt) + et. (3.7)

As in the geometric and Pascal lags, it is clear that we cannot obtain consistent estimates of the qj. However, we can simply drop the initial conditions terms without affecting the asymptotic properties of the resulting estimates.7 It will make the ensuing discussion simpler if we assume that this is done.

Given a value of y, then, the Sjt can be constructed and (3.7) can be estimated by ordinary least squares. It is easily verified that maximum likelihood estimation amounts to "searching" over y, picking the value that minimizes the sum of squared residuals in the least squares estimation of (3.7). These estimates of a and of the 2j are consistent and asymptotically efficient. Estimates of their asymptotic variances can be gotten from the inverse of information matrix, which turn out to be of the form:

=- (/Ar2)A'A (3.8)

where A is a matrix of dimension T X (p + 3) whose

7 For a rigorous discussion in the case of the geometric lag, see Dhrymes C2 or 3].

tth row is given by A, = (1, Sot, Sit, , Sp,, Zt). Here we have

zt P 0 ljst (3.9)

where

So, = dSot/dy = Ei=o iyiT'Xt-i;

St ~~~~~t-1 it = dSjt/dy = Eio Yilij+lXt ,j = 1, *** p. (3.10)

The rows and columns of ? correspond to (, 0 1, ** *, 2p, ), in that order. The maximum likelihood estimator of a72 is asymptotically independent of the other estimators.

4. AN EXAMPLE

This section compares the results given by the tech- nique proposed earlier to those given by other common distributed lag models. The data used is that of Almon [1]. It consists of 60 observations on capital expenditures (the dependent variable) and net new appropriations (the independent variable). The modified Almon tech- nique is applied with p = 1, 2, and 3, both with initial conditions terms dropped and with them included. (The model is precisely as in the preceding, except that there

1. Sums of Squared Errors and Estimated Variances

Model SSE S2 No. parameters d.f.

A. 60 observations

Geometrica .09378 .001617 2 58

Pascal, r = 2a .09519 .001641 2 58 Pascal, r = 3a .09703 .001673 2 58 Pascal, r = 4a .09840 .001697 2 58 Pascal, r = 5a .10002 .001724 2 58 Modified Almon, p = ia .09348 .001640 3 57 Modified Almon, p = 2a .08986 .001605 4 56 Modified Almon, p = 3a .08978 .001632 5 55 Geometricb .01994 .000350 3 57 Pascal, r = 2b .01073 .000192 4 56 Pascal, r = 3b .00859 .000156 5 55 Pascal, r=4b .00797 .000148 6 54 Pascal, r = 5b .00755 .000142 7 53

Modified Almon, p = 1b .00801 .000146 5 55 Modified Almon, p = 2b .00782 .000148 7 53 Modified Almon, p = 3b .00518 .000101 9 51

Rational, m = 0, n = 2b .00913 .000166 5 55 Rational, m = 1, n = 2b .00880 .000163 6 54 Rational, m = 2, n = 2b .00846 .000160 7 53

B. 52 observations

Almon, n = 8, p = 4c .00742 .000158 5 47

Modified Almon, p = 2a .03011 .000627 5 47 Modified Almon, p = 1b .00498 .000106 5 47 Modified Almon, p = 2b .00411 .000091 7 45 Modified Almon, p = 3b .00258 .000060 9 43

aInitial conditions terms not included. bInitisl conditions terms included. No endpoint constrsints.

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Almon Lag 681

is no constant term.) The other lag structures vwith which comparison is made are the geometric lag (both with and without initial conditions), Pascal lag with r = 2, 3, 4, 5 (both with and without initial conditions), and rational lag with n (degree of denominator polynomial) = 2 and m (degree of numerator polynomial) = 0, 1, 2 (with initial conditions terms).8

Since it is not clear which model, if any, is a "true" model for these data, we will look only at how well the various models "fit." Since we would expect smaller error sums of squares for models with more parameters, we will look at the "estimated variance" S2, defined as the sum of squared errors divided by degrees of freedom. (Degrees of freedom here means sample size minus num- ber of parameters.)

These results are given in Table 1A. As is clear, the modified Almon lag technique does very well, in the present sense of giving a good "fit" without using too many parameters. It is also clear that it "pays," in the same sense, to include the initial conditions terms.

A final comparison of interest is to the Almon technique itself. To make this comparison, the Almon technique was applied with values of p from two to four and values of n (length of lag) from four to ten. The minimum value of S2 was attained for p = 4, n = 8, with no endpoint constraints, and is given in Table 1B.

This value is perhaps not directly comparable to those in Table 1A since only 52 observations are used here to compute SSE. Therefore the modified Almon techniuque was rerun over these 52 observations, and these new results are given in Table 1B as well. Note the superior

8 The author is indebted to G.S. Maddala for the data, and for the geometric, Pascal, and rational lag results, some of which are in [5].

performance of this technique, even with p only equal to one, as long as the initial conditions terms are included.

[Received Septe?mber 1973. Revised January 1974.]

REFERENCES [1] Almon, S., "The Distributed Lag Between Capital Appropria-

tions and Net Expenditures," Economnetrica, 33 (January 1965), 178-96.

[2] Dhrymes, P.J., "On the Strong Consistency of Estimators for Certain Distributed Lag Models with Autocorrelated Errors," International Economic Review, 12 (June 1971), 329-42.

[3] - , Distributed Lags: Problems of Estimation and Formula- tion, San Francisco: Holden-Day, Inc., 1971.

[4] Hall, R.E. and Sutch, R.C., "A Flexible Infinite Distributed Lag," Presented at 1967 Econometric Society Meetings; Abstract in Econometrica, 36 (1968 supplementary issue), 91-2.

[5] Maddala, G.S. and Rao, A.S., "Maximum Likelihood Esti- mation of Solow's and Jorgenson's Distributed Lag Models," The Review of Economics and Statistics, 52 (February 1971), 80-88.

[6] Pesaran, M.H., "The Small Sample Problem of Truncation Remainders in the Estimation of D)istributed Lag Models, with Autocorrelated Errors," International Economic Review 14 (February 1973), 120-31.

[7] Schmidt, P., "On the Difference Between the Conditional and Unconditional Asymptotic Distributions of Estimates in Distributed Lag Models with Integer-Valued Parameters," Econometrica, 21 (January 1973), 165-9.

[8] , "An Argument for the Usefulness of the Gamma Distributed Lag," International Economic Review, 15 (February 1974), forthcoming.

[9] , and Waud, R.N., "The Almon Lag Technique and the Monetary versus Fiscal Policy Debate," Journal of the Ameri- can Statistical Association, 68 (March 1973), 11-19.

[10] Solow, R.M., "On a Family of Lag Distributions," Econo- metrica, 28 (April 1960), 393-406.

[11] Tsurumi, H., "A Note on Gamma Distributed Lags," Inter- national Economic Review, 12 (June 1971), 317-24.

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