Seasonal Variability in a Distributed Lag Model

The Review of Economic Studies, Ltd.

Seasonal Variability in a Distributed Lag ModelAuthor(s): P. K. Trivedi and B. M. S. LeeSource: The Review of Economic Studies, Vol. 48, No. 3 (Jul., 1981), pp. 497-505Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297161 .

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Review of Economic Studies (1981) XLVIII, 497-505 0034-6527/80/00420497$02.00

? 1981 The Society for Economic Analysis Limited

Seasonal Variability in a

Distributed Lag Model P. K. TRIVEDI

Australian National University

and

B. M. S. LEE Bureau of Agricultural Economics, Canberra

1. INTRODUCTION

To explain the kind of problem that this paper deals with it seems easiest to begin with an illustration. Consider a lending institution that approves applications for new home loans which are subsequently either cancelled or drawn. Suppose that typically a loan is not fully drawn immediately on approval; rather the sequence of payments is distributed over time and takes the form of periodic progress payments to builders. Further suppose that the time unit we are dealing with is a month. Then the total loan drawings in any one month would be a function of current and past approvals net of cancellations, i.e. a standard distributed lag model. Now, however, consider the additional complication that the dependence between the current loan drawings and any previous month's approvals is not fixed but varies in a systematic way with the month (or season) we are dealing with. In other words we wish to allow the distributed lag weights themselves to vary seasonally. Such seasonal variation in the distributed lag weights may be induced by the seasonal pattern of work and completion of building including the incidence of public holidays, or by random fluctuations in weather patterns which in turn affect building activity, or by seasonal variation in the "technology" of payments. Other reasons could also be thought of. It is emphasised that we are considering not just a seasonal intercept shift but also changes in the lag weights themselves. This paper considers a variety of specifications of such a model and tackles the estimation problem from a Bayesian viewpoint. But first we shall develop a suitable notation and framework for analysing this problem. This will highlight the bi-dimensional nature of the estimation problem. We shall then corsider several alternative representations of prior information regarding the lag coefficients in either the lag or the seasonal dimension. In particular we shall argue that exchangeable priors and smoothness priors which have been previously developed in other contexts are useful notions for representing prior information. Moreover it will be argued that such Bayesian priors can be usefully combined and not simply used separately. It is the emphasis on and systematic exploitation of the bi-dimensional nature of the specification and estimation within a Bayesian framework that constitutes the main novel feature of the paper. The main body of the paper, Section 2, shows how the results of previous work of Lindley and Smith (1972) on Bayesian estimation of a linear model can be adapted and extended in such a way as to yield a family of shrinkage estimators for the seasonally varying distributed lag model. Note however thatLindley and Smith are essentially dealing with a "one-dimensional" problem and we wish to deal with a model in which there are two. It would be necessary therefore to adapt their ideas to our situation. The analysis of Section 2 is based on the assumption of known prior variances. The final section

497



498 REVIEW OF ECONOMIC STUDIES

of the paper presents a brief discussion of how to extend it to cover the case of unknown variances in the prior distributions.

2. A FAMILY OF SHRINKAGE ESTIMATORS

We now formally consider the problem of estimating a finite distributed lag model with n lags where the data collected relate to s seasons under the assumption that the lag coefficients vary seasonally. The model then is of the following type:

Yt = =EL = I,Dtit_i+ Ut, t =-n, ...,~1, ... T.(1

where D1, ... ,Ds are seasonal dummy variables of the zero-one type. Under the restriction of a common distributed lag response for all s seasons the model reduces to

Yt = Ei= 1P' XM-E + ut (2)

where

iil =A2= * ** is = 6

2.1. Notation

We assume that after allowing for "loss" of observations due to lagging, we have a common number r, r> n, observations on each season. This assumption is only for notational convenience. The order of the lag is the same for each season. The observations are so arranged that yg denotes the (-r x 1) vector of observations on y in season g, (g = 1, . . . s), Xg the (-r x n) matrix of observations in season g, ,Bg the (n x 1) vector of unknown parameters in season g and ug the (-r x 1) vector of disturbances in season g. We now define the following matrices and vectors:

Y=[yl *.. Ys], y=vec Y Or xs) (sr x 1)

K? X2 ..0] (sr x ns) K ~ . .~ [: (3=vec B, [s, x ns

*n X]s() [wnl l

Gnsj n x

U=[ul us], u=vecU. (1-xs)

The complete model with ns coefficients may be written in the form

y = X3 + u (3) or

yg = Xg,Bg + ug (g = 1, . .s). (4)

Throughout the paper we shall assume u - N(O, _21T), (T= rs).

2.2. Alternative prior distributions

The usual approach to distributed lag estimation involves specification of a priori restrictions on the lag coefficients. For example, Almon's method specifies exact linear restrictions on the coefficients and they can be deduced from the assumption that the lag weights lie exactly on a polynomial of a specified degree. Shiller (1973) approached the same problem from a Bayesian standpoint and his treatment is much more in line with the treatment in the present paper. He specified a Bayesian prior regarding smoothness of the lag distribution, a smoothness prior of degree q being represented by a normal density with zero mean of the differences of order (q + 1) of lag coefficients. As is well known,



TRIVEDI & LEE SEASONAL VARIABILITY 499

the Almon representation is then obtained as a limit of Shiller's representation as the prior precision approaches infinity. Shiller's suggestion represents a more acceptable representation of prior information and has been enthusiastically endorsed even by non-Bayesians who can reinterpret the smoothness prior as simply a stochastic representation of the exact linear constraints which arise in the polynomial distributed lag model. See, for example, Taylor (1974). Smoothness priors seem to have some applicability in the context of the problem of this paper. First, rather obviously, prior information on the lag weights for any one season (that is, columns of matrix B above) can be represented by a smoothness prior. But also prior information on rows of B can be represented by such a distribution. This is the hypothesis of smooth seasonality due to Gersovitz and MacKinnon (1978). These authors considered a case (not a distributed lag model) in which the slope coefficient in the time series regression of y on X varies seasonally.' The time unit of data is often sufficiently small that one might expect smooth rather than arbitrary variation in the coefficient. (If the data were quarterly rather than monthly the hypothesis of smooth seasonality would be less acceptable). Then estimation of the relationships for all seasons simultaneously is feasible under the smoothness assumption. In the context of the seasonally varying distributed lag model one may represent smoothness in the lag as well as seasonal dimensions, such an assumption being termed bi-dimensional smoothness.

Another useful representation of prior information is the exchangeable prior. Sup- pose that from the viewpoint of the data analyst the distributed lag weights are a priori indistinguishable. In the terminology of de Finetti, also used by Lindley and Smith, the lag weights are taken to be exchangeable random variables in the sense that the marginal or joint distribution of any subset of them is symmetric. If one's prior expectation is that the lag distribution is rather flat then this could be represented by an exchangeable prior over those lag weights. Consider now the possibility that the lag distribution for any one season is a priori undistinguishable from that for any other season. (This is rather similar to the idea that the lag distribution varies over seasons in a random fashion.) This can be represented by an exchangeable prior distribution over columns of matrix B and we shall call this exchangeability between seasons. Exchangeability between seasons applies to vectors of random variables whereas exchangeability between lag coefficients in a given season applies to scalar random variables. The empirical usefulness of this idea arises from the fact that it enables us to pool information regarding different seasons for estimation purposes. While the assumption of exchangeability seems useful it appears to be even more so if combined with that of smoothness, as we shall do in one of our applications. (Other combinations are also possible.) This needs to be emphasised because by and large the value of smoothness priors in distributed lag estimation is recognised but the idea of exchangeability is relatively unfamiliar (to econometri- cians).

In summary, we shall consider three types of priors: vague or diffuse priors, exchangeable priors and smoothness priors. Vague priors are used to represent ignorance about either the columns of B (lag dimension) or rows of B (seasonal dimension). If the prior opinion about the i-th column of B is the same as for the j-th column (i $ j), their prior distributions are said to be exchangeable. We call this the exchangeability between seasons [EBs] assumption. If the prior opinion of fi3 is the same as for Pik (j $ k), we have the exchangeability within season [EWs] assumption. The assumption of full exchangeability, between and within seasons, [EBs x EWs], may be called bi-dimensional exchangeability. Analogously, the smoothness assumption may be applied, a la Shiller, to columns of B, or a la Gersovitz and MacKinnon, to rows of B or to both. We denote these three cases, respectively, by the acronyms SWs (smoothness within season), SBs (smoothness between seasons) and [SWs x SBs] (bi-dimensional smoothness). The assumptions of exchangeability and smoothness may be combined. We could assume smoothness within season and exchangeability between seasons-this case we denote by




the acronym [SWs x EBs]. The reverse combination [EWs x SBs] is also possible but is subject to some restrictions. We are thus left with the following cases:

Vague priors

Exchangeable priors: EBs; EWs; EBs x EWs.

Smoothness priors: SBs; SWs; SBs x SWs.

Mixed priors: SWs x EBs; EWs x SBs.

We now consider estimation under each set of assumptions.

2.2.1. Vague priors

Combining the normal likelihood for the y's with the diffuse prior over ,B yields a posterior distribution of ,B known to be of the multivariate Student-t form with the posterior mean

, = (X'X)-'X'y = b (5)

which in view of the block-diagonality of (X'X) reduces to

/3 *v= (X'gXg) YXgyg = bg, g = 1,.. s (6)

which is simply the least squares estimator for each season's regression equation.

2.2.2. Exchangeable priors

The concept of exchangeable prior distributions has been studied in the context of linear models by Lindley and Smith. We find it convenient to reproduce here some of their results which are used repeatedly in this paper.

Theorem (Lindley-Smith (1972), Smith (1973)). If, given O1,

y _ N(A 1 A, CO) (L-S. 1)

given 02,

01 - N(A202, C2), (L-S.2)

given 63,

02- N(A303, C3), (L-S.3)

then the posterior distribution of 01 given {Ai, C1, i = 1, 2, 3} 03 and y, is N(0*, D1) where

Di1 i0 = (A CI'A 1)01 + (C2 +A2C3A')-1A2A303 (L-S.4)

D 1 = (AiCi'A 1) + (C2+A2C3A )f' (L-S.5)

01 = (AiCj1'A1)-f(AjC-1y). (L-S.6)

Corollary If C_' -+0 , then

[AlCI'A, + C-1 - Cj1A2(A Cj1A2) '1A'Cj1 ]0* = [A'CI'A1] O. (L-S.7)

Lemma (Smith (1973)). If 0* is given by (L-S.7), then

=* (A'C-'Al + Cj1 )Y(AiCi'Ai6Ai + Cj'A2&2) (L-S.8)

where

2=[A2Al (Cl +AiC2A') 1AiA2P'[AA2 (C1 +AC2A7'y].

We refer the reader to the original sources for proofs.




EBs case. In dealing with the EBs assumption the set-up is as follows:

yg - N (XgBg, aI g = 13 . .. s (7)

i3g ~N(13, X) (8)

where # is a (n x 1) vector and S is a (known) nonsingular variance matrix. (8) says that the /3g (g = 1, . . . s) may be regarded as drawings from a common N(f3, S) distribution. This is equivalent to seasonal variation in ,B being random around the common location vector ,3. Assuming diffuse second stage priors and employing the result of the Lemma above we obtain the posterior mean and variance

/3 g,EBs = ( 2Xg,Xg + 11) -(F 2XgXgb + 1/3)(

Var (/3*,EBs) = (a-2XgXg +X 1) 1 (10)

where P = Zg=l Wgbg (11)

Wg = (Z1=1 (XXg& 2 + -1)-lXXg -2)-1(XXgt -2 +

-1)-lxgtx -2 (12

The EBs case just considered is essentially the L-S "exchangeability between m groups" case. Also note that as I + 0, /3*,EBs + bg.

EWs case. The EWs case has the following set up:

yg -N(Xg/3g, -2ir), g = 1 . .. s, (7)

p3ig -N(y, a2*g), i.=1 ... n, g=1 .... s (13)

and a diffuse second stage prior. The posterior mean in this case is

/3 ,EWs =g{In + (2/7,g) (XgXg)-[I -n-lEn]}-bg (14)

which under the restriction y = 0 specialises to the ridge estimator for each season, (En is a (n x n) matrix whose every element is unity),

13*EWs = (XXg + (O2/Ofc,g)In) Xgyg. (14a) 2 2

EWs x EBs case. For the EWs x EBs impose the additional restriction a2*,g = a* all g, and this leads to the posterior mean

/3 EWsxEBs = (X'X + (02/* )-YX'y (15)

This involves using the ridge-type estimator for all regression parameters simultaneously-a consequence of bi-dimensional exchangeability-whereas the estimator at (14) dealt with one seasonal regression at a time. The assumption of a common location parameter, 0 or y, for all lag coefficients is unrealistic and is relaxed in the next case considered.

2.2.3. Smoothness priors

SBs x SWs case. To achieve a concise treatment we begin by considering the concept of bi-dimensional smoothness which subsumes as special cases Shiller's smooth distributed lags and Gersovitz-McKinnon's smooth seasonality. Suppose3 we wish to impose p-degree smoothness priors on columns of B matrix and q-degree smoothness priors on rows of B, p < n and q < s. Let Jp and Jq denote, respectively, (n x p) and (s x q) Vandermonde matrices with (i - 1)"1 in the i-th row and j-th column (i = 1, . . . n, or i=1,. ..s andj=1,. ...p orj=1,.. .q). Let

B=JprJ + V (16)




where F is a (p x q) matrix of unknown (location) constants and V is a (n x s) matrix of random disturbances. Now define "differencing matrices" Rp and Rq of dimensions (n - p) x n and (s - q) x s chosen as in Shiller so that

RpJp = On-p,p

and RqJTq = Os-q,q

where q denotes a null matrix of dimensions indicated by the subscripts. Pre- and post-multiplying (16) by Rp and Rq we obtain

RpBRq = RpJpTJqR1 +R VRq (17)

=RpVRq and

(Rq ? Rp) vec B = (Rq ? Rp) vec V (18) or

SO =c

where S = (Rq ?&p) and co = S vec V. S has dimensions [(s - q)(n - p) 0 ns] whence it is clear that the specification (16), or equivalently (18), places (s - q)(n - p) restrictions on ,B. The restrictions (18) are in the same form as Shiller's and they may be conveniently thought as covering regular and seasonal smoothness requirement whereas Shiller's case only covers the former. From (16) we see that

vecB = [Jq?f Jp]vecF+vec V (19)

i.e.

/3 =Hy+ v, E(v) = 0, E(vv') = Q where

H = Jqf?Jp, y = vec F.

Also under normality of v we have

,B- ~N(Hy, Ql). (20)

The properties of matrix Ql depend partly on the nature of the seasonal smoothness assumption. Gersovitz-MacKinnon, see especially p. 266, consider a case in which the periodicity of the seasons implies linear-dependence between the seasonally varying coefficients. This in turn implies that IQlI = 0. More generally, one may simply wish to constrain fl in some particular way. Note also that given a singular fQ matrix the results of L-S and of Smith that were cited earlier would need to be modified. The approach involving the dropping of one of the seasons will in certain cases produce results which are not invariant to the-choice of the equation which is eliminated. To achieve invariance a generalized inverse of fl must be used. In what follows fl is assumed to be non-singular but in the event that fl is singular, its inverse should be interpreted as a Moore-Penrose generalized inverse which is unique. Given this qualification, we can now use the set-up consisting of (7) and (20) and employing the Smith Lemma we obtain the posterior mean

i3SBsxSWs = (J2 XX + Q1)'1(o 2XiXb +QH-1I3A) (21)

where

13A = [H'X'(o21T +XflX')'XH]P'[H'X'(o2IT +XQX'Y'y] (22)

which specialises to the OLS estimator b as Q-1 +0 (diffuse second stage prior) and to a bi-dimensional version of Almon estimator, /3A, as Q-1 0.




SWs case. To specialise to the SWs (of order p) we replace (19) by

,B=[Is? f Jp]jy+v =Hi+v (20a)

where - is a (ps x 1) vector of unknown coefficients and F is a (p x s) matrix. The posterior mean in this case, obtained by the same reasoning as for (19) is

SWs = GY_ -2X'X + W'6IF_X2Xb,+X 'H3 )- ) (21a)

where

/3B= [H'X'(o-2IT +XfIX')-XH]- [H X'(o2 I, +XQX')'y] (22a)

(As before, an alternative parameterisation of stochastic restrictions embodied in (20a) may be obtained. This is simply

[Is &S Rp]iG = [Is (D Rp]v. Compare (17)).

If Ql has a block diagonal structure (2 la) will simply reduce to a Shiller-type estimator applied to each seasonal regression. This is particularly obvious if we use the alternative parameterisation of restrictions just mentioned.

SBs case. To specialise to the Gersovitz-MacKinnon type smooth seasonality case, i.e. SBs case with q-degree smoothness priors we replace (20) by

13=H*y*+v (2.20b)

where H* = Jq? n, I * = vec F*, F* is a (n x q) matrix. The posterior mean in this case is

*s= ( 2X'X + '-l)-1(O_-2X,Xb +Wl_'H*3) (21b)

where

3B = H*'Xl(o- IT+XXfXP) XH*] [H*P(PXP(oX IT+XfIX') y] (22b)

This concludes our discussion of smoothness priors.

2.2.4. Mixed priors

SWs x EBs case. The specification

8g -N (Jpa, Q*) 9(23)'

where Q* is a (n x n) non-singular matrix and a is a (p x 1) unknown vector allows simultaneously for p-degree SWs and q-degree EBs because a is the same for all seasons. This assumption reflects the belief that lag coefficients between seasons are "similar" and each seasonal set of coefficients is smooth. Once again using a second stage diffuse prior we obtain the posterior mean for this case

a * (-2X I

g + fl*-l )-1( _-2 X b *1J & )(4 g3,SWsXEBs = (O- X~Xg +~''(~X Xgbg + *1f p (24)

where

a = (Z=1 (Xg(2 Igg+X *Xg)Xg)l() X'(o2Ig +Xf*Xgyg). (25)

The expression at (24) is essentially the same as for Shiller's estimator for distributed lag coefficients for each season. However, under the EBs assumption combined with the assumption of diffuse prior on a we obtain the estimator for & by pooling data on all seasons which then appears in (24).




3. EXTENSIONS AND QUALIFICATIONS

To get so far we have had to make quite strong assumptions about the priors. Exchange- ability, in particular, is a strong assumption. However, from a practical viewpoint it is especially important to relax the assumption of known variances. We have no new solutions here but simply wish to point to the relevance of some of the existing solutions to similar problems. In terms of the degree of difficulty posed by unknown variances, EWs, EBs x EWs are easier than the remaining cases-at least under the assumptions we entertain here. We shall therefore consider these cases first.

Staying in the natural conjugate framework assume that o2 and o_2 at (15) have independent x2 prior distributions, that is

p ((_2 s _2

) XC( 2 -( v1+2) (

2 )

-I ( ̂ 2+2) e p(_V1A 2 _ 2A 2) 26 p(o- o~3c(o2A(vl2)(4)(2? exp ( 12(26)

where vl, v2, A1 and A2 are prior parameters. Given this specification, the approach adopted by Fearn (1978) seems appropriate for studying the marginal posterior distribution of /3. First one transforms from o- , o-* to o- and k where k =2/o2. Let p (/ c_2, k data) be the joint posterior density. Integrating out o2 we obtain p (/3, k data), and integrating out both ,B and cr2 we get p(k|data). Now the marginal posterior distribution of ,B is obtained from

p (,B 1k, data) ocp (,/, k |data)/p (k data). (27)

The marginal posterior density of ,B can be found by numerically integrating 00

J p(,B|k, data) * p(k|data) dk (28)

for a range of values of /3. Since we are dealing with one dimensional integral computation is manageable.

For the case such as the EBs case we could follow L-S in contenting ourselves with an approximate solution in which case the approach is almost the same as theirs. This

-2 involves specifying, in addition to the priors on ,Bg and /3, a chi-square prior on -_ and a Wishart prior on 5-1. The resulting joint posterior distribution of (,/g(g = 1, . . . s), /3, cr2,

:51) can be differentiated to obtain joint modal equations for all the parameters. One possible approximation to the posterior mean would be the solution to these equations. Alternatively, and better still, as convincingly argued by O'Hagan (1976), ,Bg (g = 1, . . . s) and /3 could be integrated out from the joint posterior density yielding the marginal posterior distribution of (0_2, E1-). This can be used to obtain modal values of the unknown variances. These in turn can be substituted in equations like (9)-(12) to generate approximations to posterior mean and variance of /3g. See O'Hagan (1976) for further details. In practice, however, this may be expensive and/or inaccurate if the number of variance parameters is large. On the other hand, for the idea of smooth seasonality to be credible we should have a reasonable number of seasons.

Finally, we note that though the problem has been shown to be tractable, the model considered is somewhat restrictive in that it is a pure distributed lag model. In practice it is almost always the case that other variables enter distributed lag models.4

First version received March 1980, final version accepted February 1981 (Eds.).

The paper is a substantially revised version of a portion of a longer paper with the same title which was presented at the 1979 Warwick Summer Workshop. We are grateful to the participants at that Workshop, especially Christopher Sims, Timo Terasvirta, S. Yeo and two anonymous referees for this journal, for comments which have helped to improve this version. Of course we retain all responsibility for remaining errors.




NOTES 1. As they did not explicitly concern themselves with a distributed lag model, these authors did not need

to consider the possibility of simultaneous smoothing in the seasonal and the lag dimensions. 2. See Lindley and Smith (1972). 3. It is now well understood that the polynomial distributed lag model is the limiting form of the

smoothness model in thd non-seasonal case. If the seasonal part, i.e. the rows of B, converge similarly to the polynomial model, then for any given lag the seasonal variation in the lag coefficient is exactly described by a polynomial.

4. Another difficulty raised by an anonymous referee is that the lag length, n, may be different for different seasons. Potentially this appears to reduce some of the advantages of the exchangeability-between-seasons assumption. The question needs to be studied in depth.

REFERENCES ALMON, S. (1965), "The Distributed Lag between Capital Appropriations and Expenditure", Econometrica,

33, 178-196. FEARN, T. (1978), "Towards a Bayesian Package", COMPSTAT, 473-479. GERSOVITZ, M. and MACKINNON, J. G. (1978), "Seasonality in Regression: An Application of Smoothness

Priors", Journal of American Statistical Association, 73, 264-273. LINDLEY, DENNIS V. and SMITH, A. F. M. (1972), "Bayes Estimates for the Linear Model", Journal of

Royal Statistical Society, Series B, 1-41. O'HAGAN, A. (1976), "On posterior joint and marginal modes", Biometrika, 63, (2) 329-333. SHILLER, R. J. (1973), "A Distributed Lag Estimator Derived from Smoothness Priors", Econometrica, 41,

775-789. SMITH, A. F. M. (1973), "A General Bayesian Linear Model", Journal of Royal Statistical Society, Series B,

67-75. TAYLOR, W. E. (1974), "Smoothness Priors and Stochastic Prior Restrictions in Distributed Lag Estimation",

International Economic Review, 15, 803-804.



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Seasonal Variability in a Distributed Lag Model