a simple, flexible distributed lag technique: the polynomial inverse lag

Journal of Econometrics 31 (1986) 329-340. North-Holland

A S I M P L E , F L E X I B L E D I S T R I B U T E D LAG T E C H N I Q U E The Polynomial Inverse Lag

Douglas W. M I T C H E L L and Paul J. SPEAKER*

West Virginia Unioersity, Morgantown, WV 26506-6025, USA

Received January 1985, final version received February 1986

This paper presents a new distributed lag technique, the polynomial inverse lag, which has two useful characteristics. It has a flexible shape, allowing both humped and monotonically declining lag weight distributions; and it can be easily implemented with a small number of nested OLS regressions. The lag is similar in spirit to the Almon lag, but it is an infinite lag and thus does not require specification of a fixed lag length. Experiments with simulated data show a good ability to replicate the true lag structure even in the face of lag structure misspecification. The technique is also demonstrated using the well-known St. Louis equation.

1. Introduction

There current ly exists a variety of techniques for estimating distributed lags. See Judge et al. (1980, chs. 15, 16) for a survey. The Almon (1965) lag is the principal one of those which assume a finite lag length that must be chosen by the researcher. The difficulty of choosing the correct lag length, and the misspecification bias which results f rom an incorrect choice, lead many people to prefer the use of infinite distributed lag techniques. These allow the lag weights to decline to zero asymptotically at a pace chosen by the data itself.

However , in general an infinite distributed lag cannot be estimated in simple fashion using OLS. For instance, a geometric lag subjected to a Koyck t ransformat ion will generally give biased results when estimated with OLS because of correlat ion between the resulting serially correlated errors and the lagged endogenous variable. 1 The same problem applies to generalizations of the geometr ic lag: the Pascal lag and the rational lag. Other distributed lags, such as the g a m m a lag of Tsurumi (1971) and Schmidt (1974a), require either

* We are grateful to Jean Rosales, Adriaan Dierx, Christopher Cornwell, and Kern Kymn for helpful comments.

~This holds true unless the disturbances of the original equation have very specific serial correlation properties, based on the same parameter as appears in the structural model. This is very unlikely. An exception is the stock adjustment version of the geometric lag, which can be estimated with OLS; but this is a limited case which is not usually applicable. Note that the geometric lag imposes a relatively inflexible shape on the weight distribution.

0304-4076/86/$3.50© 1986, Elsevier Science Publishers B.V. (North-Holland)

330 D. W. Mitchell and P.J. Speaker, The polynomial inverse lag

non-linear estimation or an extensive search procedure in which each step of the search involves a separate OLS regression with separately constructed right-hand-side variables.

The purpose of this paper is to present a new distributed lag technique which is as simple to implement as the Almon lag but which is infinite, so that no lag length need be specified. As with the Almon lag in the unlikely case of a known lag length, the estimation involves a brief and simple search for the polynomial degree, using a series of nested OLS regressions.

Thus this technique, referred to as the polynomial inverse lag (PIL), is much less cumbersome to use than are other infinite distributed lag techniques, implying that it will be attractive to a wider variety of researchers. Further, the proposed lag has a flexible shape. Since the lag is a modification ol the Almon lag idea to make it infinite, it is similar in spirit (though not so much in implementation) to the geometric polynomial lag of Schmidt (1974b).

Section 2 presents the PIL technique. Section 3 subjects the PIL to experiments with simulated data, to find whether it can reasonably accurately estimate a true lag structure which is not itself PIL. Two competing techniques are also subjected to such experiments. As with any lag technique, such misspecification guarantees bias, so the problem involves the severity of the bias. A final comparison of the PIL with existing techniques is made in section 4, where the well-known St. Louis equation is estimated by PIL and compared to the common use of the Almon lag technique. Concluding comments are in section 5.

2. The technique

Consider the following regression equation:

oo

yt=b+ ~_.wiXt_i+et, (1) i = O

where Yt is the dependent variable, X t is the independent variable, e t is the disturbance, and the wi's are the distributed lag weights. Assume that the wi's can be described by

n a j

W'=, ~--'~'2 ( i + 1 ) j ' i = 0 . . . . . OC, (2)

where the aj 's are the parameters to be estimated. The weights are thus assumed to fall on an n th degree polynomial in 1/(i + 1), where i is the lag number. Compare this to the Almon lag, in which the weights are assumed to fall on a polynomial in i. Thus, the similarity of the PIL to the Almon lag is apparent.

D.W. Mitchell and P.J. Speaker, The polynomial inverse lag 331

Several observations can be made about (2). First, ( i+ 1) instead of i appears in the denominators so the expression will be defined for i = 0. Second, the smallest exponent in (2) is two rather than one to ensure that the sum of lag weights, from zero to infinity, is finite. Third, all parameters (except the polynomial degree, n) enter linearly, allowing the use of OLS. Fourth, the lag weight distribution in (2) can take on both of the general shapes widely considered desirable: humped or monotonically declining. A humped distribution generally requires at least a fourth-degree polynomial. And fifth, the limit of w i as i approaches infinity is z e ro - an essential property for infinite distributed lags.

The form of the equation to be estimated is obtained by substituting" (2) into (1) and rearranging,

where

Y~=b+ ~ a j Z j t + R t + e t, (3) j = 2

t -1 gt_i

Zj,=i=0E ( i + l ) j , j = 2 . . . . . n, (4)

j=2 (i + 1) j" (5)

Data are not available for the computation of the remainder term R,. Since this term is negligible for t greater than about eight, (3) should be estimated without the firsteight Yt, and R, can simply be dropped from (3) for the remaining data points. This treatment of the remainder term is the same as in Schmidt (1974a, b).

Thus regression (3) is estimated using OLS, with R, omitted and with the right-hand-side variables constructed as in (4). Again, the similarity to the Almon lag procedure should be obvious in (31 and (4). The differences are that: (a) no first-degree term appears in (3), for the reason given earlier; (b) the summations in (4) go to t - 1, instead of to a fixed endpoint; and (c) in each term in (4) the coefficient is 1/(i + 1)J instead of iL Note that the regression in (3) is amenable to the use of the Cochrane-Orcutt adjustment for serial correlation, if necessary.

The remaining issue is the choice of n, the degree of the polynomial. The appropriate degree can be chosen by regressing (3) a number of times in succession, starting from a high degree and then successively dropping the highest-degree terms. In this procedure, for the PIL as for the Almon lag, the successive regressions are nested. Given the results of these nested regressions, the choice of the appropriate degree follows from a suggestion by Schmidt

J.Econ D

332 D. 14/. Mitchell and P.J. Speaker, The polynomial inverse lag

(1974b, p. 681). Since we do not know the true model, selection of the degree of the polynomial is based on the ability of the model to fit the data. This amounts to choosing the degree according to the model that minimizes the estimated variance, the sum of squared residuals divided by the degrees of freedom.

3. Experiments with simulated data under misspecification

If the true lag structure is of the polynomial inverse type, then OLS estimation of (3) with sufficiently high polynomial degree will give unbiased results. As with any distributed lag technique, however, the estimation will be biased if the true lag structure does not conform to what is imposed on the data. In practice it is likely that true lag structures do not conform precisely to the structure which any technique might impose. And as Griliches (1967) pointed out, we can 'not expect the data to give a clear-cut answer about the exact form of the lag. The world is not that benevolent.'

Thus bias is likely to be present regardless of what distributed lag model is used. The relevant question then concerns the severity of the bias. This question can only be answered by using simulated data experiments, in which the true (constructed) data are known to have come from a lag structure other than the one assumed by the estimation technique.

In this section, we report results from several experiments on the PIL estimation of the lag structure. As a means of comparison, estimation was also conducted for two competing infinite lag techniques: the Gamma lag proposed by Tsurumi (1971) and revised by Schmidt (1974a); and the Geometric Polynomial lag (hereafter, GPL) of Schmidt (1974b).

Estimation of (1) by use of the Gamma lag technique involves a respecifica- tion of the w,. The weights are expressed as

w i = a v i = a ( i + l ) ~ / ( l - " ) ~ ', 0 < a < l , 0 < h < l . (6)

Substituting (6) into (1) we obtain

where

Y~-=-b+ aVt+ Rt + e ,,

V~ = Y'~ viXt_ i and R, = a viXt_ i. i = 0 i=t

(7)

Estimation of eq. (1) is replaced by the actual estimation of expression (7) without the remainder terms.

D. W. Mitchell and P.J. Speaker, The polynomial inverse lag 333

For the GPL, the w i are replaced by

P

w i = Y i~_, aji j, 0 < y < l . (8) j=0

Expression (1) can be rewritten as

P

Y , = b + E a j S j , + R , + e , , (9) j = 0

where

t - 1 t - 1

Sot = £ iX = _ = t-i and Sj, ~_,Y iijX, i, J 1 . . . . . p, i=0 i s 0

and

p oo

R, = a o ~ "YiXt_ i + E aj E yiiJxt-i • i=t j s l i=t

As with the Gamma lag technique, estimation involves the creation of the newly defined right-hand-side variables, followed by OLS estimation with the R t t e r m s dropped.

For the simulated data experiments, in eacti case the data on the independent variable (X) were the 100 values given in Judge et al. (1980, table 16.2). These data are highly serially correlated, as is usually true of real world data. Left-hand-side data (Y) were generated according to

Yt = b + ~ w i S t _ i + et, (10 ) i=0

where e, is a serially uncorrelated, normally distributed disturbance with mean zero and variance nine, adaped from Kmenta (1971, table D-6). The different experiments involve different sets of true weights, of either the GPL or the Gamma type.

For all sets of experiments, eq. (10) was estimated using the polynomial inverse lag technique, and also using the GPL technique in the case of true Gamma weights or the Gamma technique in the case of true GPL weights. In each case the first eight data points were dropped, because these are the ones that contribute the biggest terms to the truncated remainder R t in (3), (7), and (9). For the remaining data points, R, is miniscule.


For the first set of experiments, the left-hand-side variables were calculated alternatively from three different true GPL weight structures. Three experiments were conducted to test the ability of the PIL to emulate the shapes (monotonically decreasing and humped) commonly found in lag structures. As a means of comparison, Gamma lags were estimated over the same sets of Y data. In the first experiment of this set, the Y data were generated using a monotonically decreasing GPL in (10) with b = 13 and the w, from (8) with p = 2, y = 0.1, a 0 = 3, a~ -- 0, and a 2 = 15. The second and third experiments employ humped GPL lag structures with the peaks of the lags occurring at one and two lags, respectively. In each ease b = 13; the weights in the second experiment follow from (8) for p = 1, , /= 0.2, a 0 = 2, and a 1 --- 14; and the weights in the third experiment come from (8) for p = 2, 3' = 0.4, a 0 = 0.5, a 1 = 1, and a 2 = 5.

The estimation procedure for each of these three experiments was fairly simple to conduct for the PIL because the OLS regressions are nested. Only one set of right-hand-side variables (Zj,, j = 2 . . . . . n) had to be constructed for all three experiments. Once these Zj , were obtained, a series of nested OLS regressions was conducted starting with a high-degree polynomial and successively dropping the highest-degree term. The reported results (table 1) corre- spond to the specification with the lowest estimated variance.

The estimation by the Gamma lag procedure, however, was much more tedious to conduct. This occurs since a grid search must be conducted over (a, ?,) and the right-hand-side variable, Vt, must be reconstructed for each pair (a, ?~). Further, as the degree of precision on a and )t is increased by one decimal place, the number of points in the grid is increased by a multiplicative factor of 10 2. As a result, we limited the precision of a and )t to two decimal places. Initially, a grid search of 100 combinations of a and ~ was conducted (i.e., selection to one decimal point) and the corresponding 100 sets of V, were obtained. Upon estimation by OLS it was discovered that for any given value of a, the mapping of the estimated variance over all ~ generated a U-shaped curve. Likewise, for any given 2~, the mapping of the estimated variance over all a provided a U-shaped curve. This regular behavior allowed the estimation procedure to be reduced from the 10,000 potential regressions (for two decimal point accuracy on a and 2,) to consideration of the valley created by the combinations of these U-shaped curves.

The results of this first set of three experiments are reported in table 1. In all three experiments, the PIL and the Gamma lag technique performed well at mimicking the shape of the true lag structure. In each experiment both techniques distinguished the correct peaks of the lag. A further positive result would be reasonably accurate estimation of the sum of lag weights. As the last line of the table indicates, both procedures did very well in estimating the total lag weights. Finally, notice that some of the PIL estimated lag weights are very

D. W. Mitchell and P.J. Speaker, The polynomial inverse lag,

Table 1

Estimation of three true GPL series by PIL and Gamma lag techniques.

335

Weight True PIL a Gamma b True PIL a Gamma c True PIL a Gamma

w l

w

w

% W

w

W

w O

2

~ 3

4

5

½ 6

7

8

9

9

.,w, 0

3.000 3.075 2.927 1.800 1.717 2.051 0.630 0.599 0.518 0.138 0.181 0.086 0.024 0.052 0.011 0.004 0.011 0.001 0.001 - 0.003 0.000 0.000 - 0.007 0.000 0.000 - 0.007 0.000 0.000 - 0.006 0.000 0.000 - 0.005 0.000 0.000 -0.004 0.000 0.000 -0.004 0.000 0.000 - 0.003 0.000 0.000 - 0.002 0.000 0.000 - 0.002 0.000 0.000 - 0 . 0 0 1 0 . 0 0 0

0.000 - 0.001 0.000 0.000 - 0.001 0.000 0.000 - 0.001 0.000

5.597 5.589 5.595

2.000 2.111 2,137 3.200 2.982 2.899 1.200 1.201 1,333 0.352 0.412 0.393 0.093 0.147 0.091 0.023 0.052 0.003 0.006 0.016 0.001 0.001 0.001 0.000 0.000 - 0.000 0.000 0.000 - 0.001 0.000 0.000 - 0.001 0.000 0.000 - 0.001 0.000 0.000 - 0.001 0.000 0 . 0 0 0 - 0.000 0.000 0 . 0 0 0 - 0 . 0 0 0 0 . 0 0 0

0.000 - 0.000 0.000 0.00o -0.0oh 0.00o 0.000 - 0 .000 0 .000 0.000 - 0.000 0.000 0.000 - 0.000 0 .000

6.875 6.874 6.874

0.500 0.336 0.581 2.600 2.540 2.443 3.600 4.682 3.480 3.104 2.826 3.184 2.163 1.659 2.285 1.336 1.011 1.407 0.764 0.642 0.779 0.414 0.422 0.399 0.215 0.286 0.193 0.109 0.197 0.089 0.054 0.138 0.039 0.026 0.098 0.017 0.012 0.070 0.007 0.006 0.050 0.003 0.003 0.035 0.001 0.001 0.025 0.000 0.001 0.017 0.000 0.000 0.0!1 0.000 0.000 0.006 0.000 0.000 0.003 0.000

14.907 15.054 14.906

a PIL estimated using a sixth-degree polynomial. UGamma estimated using a = 0.78 and h = 0.06. CGamma estimated using a = 0.79 and ~ = 0.10. dGamma estimated using a = 0.79 and h = 0.31.

slightly negative; in contrast, the corresponding true weights are approxi- mately zero to three decimal places. While a purist would prefer the absence of any negative weights, this situation is hardly cause for alarm. The estimated negative weights are always very close to the true value of zero.

In the second set of experiments, a similar procedure was followed. How- ever, the true lag structure was chosen as a Gamma lag with the three general shapes illustrated in the first set of experiments. The PIL estimates of these structures are compared with the estimation by the GPL technique. In the first experiment in this set, Y data were generated from (10) using a monotonically decreasing Gamma lag with b = 13 and the w~ from (6) with a = 1, a = 0, and 2, = 0.5. Humped shaped lag structures were generated in the second and third experiments with the peaks occurring at the first and second lags, respectively. In each case b = 13; the weights in the second experiment were obtained from (6) for a = 1, a = 2/3~ and A = 0.3; and the weights in the third experiment have a = 0 . 1 , a = 5 / 6 , and ~, = 0 . 2 .


Table 2

Estimation of a true Gamma series by PIL and G P L la techniques.

Weigh t T r u e PIL a G P L b True PILc G P L d True PIL ~ G P L e

1.000 0.996 1.154 0.500 0.599 0.349 0.250 0.210 0.158 0.125 0.091 0.127 0.063 0.045 0.113 0.031 0.025 0.089 0.016 0.015 0.057 0.008 0.009 0.026 0.004 0.006 0.003 0.002 0.004 - 0.012 0.001 0.002 - 0.019 0.000 0.001 - 0.020 0.000 0.001 - 0 . 0 1 8 0.000 0.000 - 0.015 0.000 0.000 - 0.010 0.000 - 0.000 - 0.007 0.000 - 0.000 - 0.003 0.000 - 0.000 - 0.001 0.000 - 0.000 0.001 0.000 - 0.000 0.002

2.000 2.004 1.975

w b

w l

We w 3

w4 w5

W 7 w~ ~b

w10

Wl 1

w12

w13

w14

Wl 5

w16

W17

w1S w19

19

E wt t=O

1.000 1.027 1.212 1.200 1.087 0.963 0.810 0.974 0.694 0.432 0.405 0.462 0.203 0.171 0.284 0.087 0.076 0.158 0.036 0.034 0.076 0.014 0.016 0.025 0.005 0.007 - 0.003 0.1302 0.002 - 0 . 0 1 7

0.001 0.001 - 0.021 0.000 - 0.001 - 0.021 0.000 - 0 . 0 0 1 - 0 . 0 1 8 0.000 - 0.001 - 0 . 0 1 3 0.000 - 0.001 - 0.009 0.000 - 0.001 - 0.006 0.000 - 0.001 - 0;003 0.000 - 0.001 - 0.001 0.000 - 0.001 0.001 0.000 - 0.001 0.002

3.790 3.788 3.764

0.100 0.197 0.230 0.640 0.309 0.567 0.972 1.473 0.821 0.819 0.723 0.773 0.500 0.356 0.569 0.249 "0.180 0.348 0.108 0.098 0.174 0.042 0.055 0.061 0.015 0.033 0.001 0.005 0.019 - 0.025

0.002 0.012 0.030 0.001 0.007 - 0.026 0.000 0.004 - 0.018 0.000 0.002 - 0.011 0.000 0.001 - 0.005 0.000 0.000 0.001 0.000 - 0.000 0.001 0.000 - 0.000 0.003 0 . 0 0 0 - 0 . 0 0 0 0.003 0.000 0.001 0.003

3.452 3.462 3.438

" P I L estimated using a fourth-degree polynomial. b G P L estimated using "t = 0.65 and p = 4.

P I L estimated using a sixth-degree polynomial. a G P L estimated using 7 = 0.71 and p = 3. ~ G P L estimated using 7 = 0.58 and p = 4.

The GPL estimation in this second set of experiments is hampered by the same problem as the Gamma estimates in the first set of experiments, in that new right-hand-side variables must be constructed for each iteration of 7. )~s with the first set of experiments, the choice parameter "t was calculated to two decimal places which left the potential for 100 sets of right-hand-side variables to be constructed. The polynomial degree was calculated for a maximum of a fifth-degree polynomial. Thus, estimation could require up to 600 series of Sir Unfortunately, the estimated variance did not display the regular behavior in the (-/, p) grid that was observed with the Gamma lag in the first ~et of experiments. For any given p, it was found that a mapping of the estimated variance over the potential 7 resulted in multiple local minima. Thus, the grid search of the two parameters required the calculation of the Sit in the neighborhood of each of these local minima and led to the subsequent OLS estimation of as many as two hundred of the potential model specifications.

The results from these three experiments are reported in table 2. As with the first set of experiments, the PIL did a good job of estimating the general shape

D. W. Mitchell and P.J. Speaker, The polynomial inverse lag

Table 3

PIL estimation of a Geometric lag and a contemporaneous structure.

337

Geometric Contemporaneous-effect-only

Weight True PIL ~ True PIL b

v~ b 1.000 1.082 3.000 3.049 w~ 0.300 0.239 0.000 - 0.057 w z 0.090 0.071 0.000 0.007 w 3 0.027 0.027 0.000 0.010 w 4 0.008 0.012 0.000 0.006 w~ 0.002 0.005 0.000 0.003 w 6 0.001 0.002 0.000 0.002 w7 0.000 0.001 0.000 0.000 w~ 0.000 0.000 0.000 - 0.000 w o 0 . 0 0 0 - 0.000 0.000 - 0.001 wlo 0.000 - 0.001 0.000 - 0.001 WI1 0.000 -- 0.001 0.000 -- 0.001 wl 2 0.000 - 0.001 0.000 - 0.001 w~ 3 0.000 - 0.001 0.000 - 0.001 wt4 0.000 - 0.001 0.000 - 0.001 wl 5 0.000 - 0.001 0.000 - 0.001 wl6 0.000 -0.001 0.000 -0.001 w : 0.000 - 0.001 0.000 - 0.001 w 1 ~ 0.000 - 0.001 0.000 - 0,001 w ~ 0 . 0 0 0 - 0 . 0 0 1 0 . 0 0 0 - 0 . 0 0 1

19

W, 1.428 1.432 3.000 3.011 i = 0

a PIL estimated using a fourth-degree polynomial. b PIL estimated using a fifth-degree polynomial.

o f e a c h lag s t ruc ture . In all three cases the cor rec t shape and p e a k were

o b t a i n e d . T h e G P L also p e r f o r m e d r e a s o n a b l y well on these s a m e s t ruc tures ,

a l t h o u g h it m i s sed the peak va lue in the second expe r imen t . Fu r the r , for b o t h

t e c h n i q u e s in all cases the sum of the weights c losely a p p r o x i m a t e d the t rue

w e i g h t to ta l .

T h e th i rd set o f e x p e r i m e n t s a l lows a c o m p a r i s o n of the P I L t e c h n i q u e to

b o t h the G a m m a lag and the G P L t e c h n i q u e t h rough the c o n s i d e r a t i o n o f two

spec ia l cases o f these t echn iques : a g e o m e t r i c lag and a pu re ly c o n t e m p o r a -

n e o u s effect . T h e g e o m e t r i c lag serves as a special case o f the G a m m a lag

w h e r e a = 0 in (6) and for the G P L wi th a j = 0 ( j = 1 . . . . . p ) in (8). Likewise ,

the p u r e l y c o n t e m p o r a n e o u s effect fo l lows f r o m a G a m m a lag wi th ~ = a = 0

in (6) o r f r o m the G P L for y = 0 in (8).

S ince o n l y the P I L has theore t i ca l spec i f ica t ion bias in these cases, on ly the

resu l t s for P I L e s t i m a t i o n are given. These resul ts are in table 3. In b o t h cases

the P I L c o r r e c t l y m i m i c s the shape o f the lag s t ruc tu re and the sums of the

first t w e n t y lag weigh ts a re ve ry close to the t rue sum.

In s u m m a r y , the e x p e r i m e n t s o f this sec t ion have tes ted the p e r f o r m a n c e o f

the P I L u n d e r c o n d i t i o n s o f misspec i f i ca t ion bias. T h e P I L did an exce l len t


job of estimating the general shape of the lag distribution, the sum of lag weights, and the individual lag weights when the true lag structure was humped, monotonically declining, or contemporaneous-effect-only. These tests also indicate that the competing infinite lag techniques provided similar results, but only at substantial cost in terms of artificial data construction and parameter search.

4. Application to the St. Louis equation

While the previous section demonstrated that the PIL performed well in simulated data tests, it is also interesting to note how the PIL compares with traditional distributed lag techniques in the estimation of unknown lag structures. In this section we report the results of PIL estimation of a frequently considered model, the St. Louis equation.

As with any application of a distributed lag technique to real data, we will not be able to make a comparison of the estimated PIL weights to a true weight structure. However, we are able to compare the results with traditional distributed lag techniques. The St. Louis equation was chosen because of the critical attention it has received since its introduction. Although there have been many criticisms of the model, we merely concentrate on the criticism of the choice of lag length and accompanying constraints on the lag weights. The traditional representation of the St. Louis equation is presented as

4 4

GblP t = a + ~_, rni)f4t_ i + ~., giGt_i "4- et, (11) i=o i = 0

where

G N P = growth rate in nominal G N P , A;/ = growth rate in M1,

= growth rate in nominal government expenditures, m i = weights on M from a fourth-degree polynomial with constraints

m 1 -----m 5 = 0 ,

g i = weights on G from a fourth-degree polynomial with constraints g_ ~ = g~ = 0.

Traditionally the St. Louis equation has been regressed with a four-quarter Almon lag. For purpose of comparison with the PIL, we will briefly present results using alternately a four-quarter and an eight-quarter Almon lag. Further, in their original specification of the St. Louis equation, Anderson and Jordan (1968) constrained the lag distribution to zero at the endpoints, and we will adhere to this practice. [See Schmidt and Waud (1973) for an argument against the use of endpoint constraints.]

D. W. Mitchell and P.J. Speaker, The polynomial inverse lag

Table 4

Es t ima tes of the St. Louis equa t ion by A l m o n lag and PIL techniques.

339

Four -qua r t e r A l m o n a E igh t -quar te r A l m o n b PIL ~

L a g ( i ) m i gi m i gi m i gl

0 0.3887 0.1913 0.3346 0.0545 0.6553 0.1165 1 0.4021 0.0662 0.3882 0.0520 0.0467 0.1299 2 0.2363 - 0 . 0 7 1 3 0.2883 0.0243 0.4797 - 0 . 1 8 4 8 3 0.0457 - 0 , 0 2 6 9 0.1340 - 0 . 0 0 4 9 - 0.0000 - 0 . 0 2 2 8 4 - 0.0569 0.0052 - 0.0045 - 0.0210 - 0.0835 0.0071 5 - 0 . 0 8 5 3 - 0 . 0 1 8 1 - 0 . 0 6 5 1 0.0021 6 - 0.0955 0.0013 - 0.0331 - 0.0073 7 - 0.0504 0.0256 - 0.0068 - 0.0147 8 0.0057 0.0348 0.0116 - 0.1950 9 0.0237 - 0.0222

~The four -quar te r A l m o n lag is a fourth-degree po lynomia l for cons t r a in t s m _ 1 = m 5 = g - t = g5 = 0.

b T h e e igh t -qua r t e r A l m o n lag is a fourth-degree po lynomia l for cons t r a in t s m _ 1 = m 9 = g - 1 = g 9 = 0 .

CThe PIL is a s ixth-degree po lynomia l for bo th sets of weights.

bo th sets of weights wi th

bo th sets of weights wi th

Recently, Ahmed and Johannes (1984) addressed some of the problems with the St. Louis equation. We follow their arguments for the consideration of the appropriate monetary and fiscal variables and the time period for estimation by the PIL technique. Quarterly data were collected from the Survey of Current Business on the levels of nominal GNP and actual government expenditures for final goods and services from 1959.1 to 1979.II1. The monetary aggregate (new MI) was provided directly by the Federal Reserve (January 1981) for the same time period. Because of the truncated remainder term in the PIL representation, the actual period of estimation covered the period 1961.1 to 1979.Ill, which reflects the omission of the first eight data points.

Table 4 contains the results of the estimation by the Almon lag and PIL techniques. The shape of the lag structure found by the Almon lag technique, for both the four-quarter and eight-quarter lag lengths, closely resembles the shape found by other studies. That is, the weights are presented as a humped- shaped structure with the peak weight occurring at m 1 for the rate of change in the monetary variable with rather meager effects from the changes in the fiscal variable.

The PIL estimates, however, do not present the same picture of the structure of lag weights. The PIL suggests the greatest weight occurs with the contemporaneous rate of change in the monetary aggregate and shows a trough at m 1, where the Almon lag suggested a peak. Interestingly, the finding of a trough in the monetary weights at a lag of one is in keeping with the Almon lag results of Ahmed and Johannes (1984) in a case in which they include exports as an

J.Econ E

340 1). 14/. Mitchell and P.J. Speaker, The polynomial inverse lag

additional explanatory variable. Likewise, differences exist between the shape of the lag structures on the fiscal change estimated by each technique. While the Almon weights immediately declined, the PIL estimates of the gi suggest a peak at g r The subsequent lag weight values for both the fiscal and monetary rates of change display a series of peaks and troughs that begin to hover around zero after the second period lag. While the magnitudes of our PIL weight estimates seem to support the notion that 'only money matters ' some caution should be exercised in the interpretation of our St. Louis equation since we have ignored many of the model specification issues surrounding this body of literature.

5. Conclusion

This paper has presented a new distributed lag technique, the polynomial inverse lag, which has two useful characteristics. It has a flexible shape (with no fixed endpoint), allowing both humped and monotonic lag weight distributions; and it can be easily implemented with a small number of nested OLS regressions. Simulated data experiments showed a good ability to replicate the true lag structure even in the face of lag structure misspecification. The technique was also employed on the St. Louis equation, and suggested a reversal of the peaks and troughs in the typical weight structure found by the use of the constrained Almon lag technique.

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a simple, flexible distributed lag technique: the polynomial inverse lag

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