On univariate time series methods and simultaneous equation econometric models

Journal of Econometrics

5 (1977) 379-388. 0 North-Holland

Publishing Company

ON UNIVARIATE TIME SERIES METHODS AND SIMULTANEOUS EQUATION ECONOMETRIC MODELS

Franz PALM* University of Louvain, Core, Heverlee, Belgium Received December 1975, final version received July 1976 Systematic testing of the implications of the structural assumptions for the properties of the final equations and transfer functions associated with a dynamic econometric model, as proposed by Zellner and Palm (19741975), proved to be useful in model building. This paper contains several remarks on the use of univariate time series methods to empirically check out the implications of a linear dynamic simultaneous equation model.

1. Introduction

In their recent paper on ‘Time Series Analysis and Simultaneous Equation Econometric Models’ Zellner and Palm (1974), hereafter referred to as ZP, propose a procedure for testing the dynamics of a linear multiple time series process and apply it to a dynamic version of Haavelmo’s model II (1947). They specify a general multiple time series process for a vector of random variables. The final equations (FEs) associated with the general model are derived and are in autoregressive-moving average (ARMA) form. Box-Jenkins techniques and large sample likelihood ratio tests were used to determine the lag structures of the final equations and to discriminate between alternative schemes. The transfer functions (TFs) associated with the model are derived and analyzed along the same lines. The approach permits testing of the implications of the structural assumptions for the properties of transfer functions and final equations. It has recently been applied in the analysis of variants of a dynamic monetary model for the U.S. economy [ZP (1975)]. In this note, we comment on several aspects related to the FE and TF form associated with a multiple ARMA process. In section 2.1, we discuss the representation of a FE as a univariate normal ARMA process. The relationship between the distributed lags of the structural form and the properties of a FE’s *Aspirant du Fonds National de la Recherche Scientifique, Belgique. The author is very grateful to J.H. Dreze, J.F. Richard and A. Zellner for many helpful discussions. He also wishes to thank L. Phlips, P. Schiinfeld and K. Wallis for their useful remarks. This paper is a revised version of chapter III of my Ph.D. Dissertation. The work was started during my visit at the H.G.B. Alexander Research Foundation, G.S.B., University of Chicago.

380

F. Palm, Univariate time series models

disturbance and autoregressive (AR) part are discussed in sections 2.2 and 2.3. Section 3 is devoted to the properties of the TF form. In section 4, we comment on the exogeneity assumption in the framework of a linear multiple ARMA process. Finally, we close with some concluding remarks.

2. The system of final equations 2.1. An ARMA representation for a single final equation As in ZP (1974), we consider

the following

multiple

time series process:

f&9 zt = I;(L) e, , PXP

where z, is a H(L) and F(L) elements being Assume that

PXl

PXP

(1)

PXl

vector of random variables,r e, is a vector of random errors, are each matrix lag operators, assumed of full rank, with typical finite degree polynomials in L, namely IZij(L) andfij(L). e, is normally distributed with

Ee, = 0

Ee,e’,, = S,,J, ,

and

(2)

for all t and t’, where 6,,, is the Kronecker delta. H(L) is assumed to be invertible, a scalar polynomial in L of degree so that the roots of its determinant IH(L m, lie outside the unit circle. The system of FEs associated with (1) is given by

O(L)z, = A(L)e, = i

A,, ef-h

,

h=O

where O(L) = IH(L)[/c is a of z,, A(L) = H*(L)F(L)/c adjoint matrix of H(L) and in IH(L The disturbance sum of independent MAs,

8(L)z,, = f

(3)

scalar polynomial in L operating on each element is a matrix lag operator of order q, H*(L) is the the normalizing constant c is the coefficient of Lo of any equation of system (3) can be written as a

&Ai ef-h

h=O

(4) IFor reason of notational

simplicity, we assume Ezzt = 0.

F. Palm, Univariate time series models

381

where aLi

= ith row of the matrix Ah ,

xhij = i, jth element

of matrix

A,

,

4i

=

maxh {h = 0, 1,.

4ij

=

maxh {h = 0, 1, . . . qi(ahi, # O}.

. . qlC&i

#

0’},

In order to simplify the reading of this note, lemma the conditions under which the disturbance sented as a moving average (MA) in one variable.

we restate in the following of a FE in (4) can be repre-

Lemma. 2 The sum z, of two independent wide selzse stationary moving averages of order q1 and q2, respectively, is a wide sense stationary process and there exists some number q, 0 5 q 5 max(q,, q2) such that for the covariance function of z, denoted o(h), h = 0, + 1, . . ., we have o(q) # 0, a(h) = 0, for all Ihl > q. If the spectral density of z,, ES= -4 o(h)eViAh, is strictly positive for all 1 E [ - 71, z], then zt can be represented as an MA of order q, i.e., there exists a white noise process v, and a collection of reals PO, . . . pq, such that

Zt =

i

h=O

fih%-h.

The result is easily extended to sums with a finite number of terms. Yamamoto (1975) has shown that the spectral density of B(L)zi, is zero at I, if and only if there exists a common factor (1 - zoL) for the ith row of A(L) such that z. = eilo (i.e., lzol = 1). H e a 1so shows that situations of a common root exactly on the unit circle are excluded if the system is stable and invertible, i.e., the characteristic roots of H(L) and F(L) are less than unity in modulus, so that the requirement for the lemma given above is met. Therefore, the ith FE in (4) can usually be written as a univariate ARMA process,

e(L)zit

=

vit+

hq$l

fii#it-

h 3

i = 1,. . .,p,

where the Vif_h’S, h = 0, . . ., q:, are white noise and qT 2 qi = maxj(qij}. From the lemma we cannot derive a specific distribution for the Vi,‘S. But conditionally on vif-h, h = 1, . . ., q?, and given a normal distribution for zit, vit is also normally distributed. Furthermore, if the roots of the r.h.s. of (6) are outside the unit circle, vit can be expressed as a convergent series of Yke Granger (1972) or Rozanov (1967, pp. 39-43). A generalization of the lemma to m-dependent processes has been proved by P. Schijnfeld in an unpublished note.

382

F. Palm, Univariate time series models

normally distributed random variables zit, t = 1, 2, . . ., and thus is normally distributed. Applying the lemma in (5) to each equation of system (3) of the FEs, we have

B(L)z, = u, =

c

h=O

B,u,_, ,

(7)

where B, is a diagonal matrix, v,-h is a p x 1 white noise vector3 such that the diagonal elements of E(u,u:_,) are zero for h # 0. However the auto- and serial correlation structure of the u,‘s is not further determined in contrast to that of the error term in (3). In general it is not possible to reconstruct the joint density of U, from the marginal densities of the elements uir. To summarize the discussion, we can represent any variable out of a multivariate normal ARMA process as a univariate normal ARMA process. Thus the analysis of the likelihood function associated with a single FE is legitimate. As pointed out by ZP (1974), the single FE analysis, testing and estimation ignore part of the information: the cross equations covariances of the disturbances are not taken into account and the restrictions on the AR part are not incorporated. But it is possibly less subject to misspecification. Certainly it is computationally easier to implement than joint estimation methods. 2.2. The order of the MA of a FE’s disturbance As a result of the lemma in section 2.1, the of the FEs is smaller than or equal to the highest it, a point we shall now discuss. On multiplying taking expectations we see that the auto- and Zi and zj satisfy the difference equation,

yij(k)-Biyrj(k-l)-

. +. -8mlij(k-m)

degree of the MA polynomial degree of the terms composing throughout in (4) by zjt_k and cross covariance functions for

= 5 h=O

Q;liYez,(k-h),

(8)

whereYij(k) = E(ZitZjt-J, and L,(k) = (y,,,,(k), - . -, yepz,(k)),_i= 1, . . ., P, is a 1 xp vector whose elements y ehzj(k) = E(ehtzj,_k) are the cross covariances between eht and z~~_~. For i = j, expression (8) gives the auto-covariance

31n order to express the vector of-,, as a function of the basic random variables e, for h = 0, . _., q, a stronger condition has to hold for the parameters of the r.h.s. of (3), nr = XqhZo &z,_~. For example the following proportionality condition Ah = &A,,, B. = I and B,,h = 1,. . ., q, a non-zero diagonal matrix, will be sufficient and leads to an error covariance structure .Q = ELQV’~ = AoAf 0, Ev,v’,-B = 0, 0 # 0, where v,-,, is now expressed as a linear transformation of et-,,, i.e., v~._~= A,,e,_,,, and the knowledge of the set of marginal densities for zt is equivalent to knowing the joint density.

383

F. Palm, Univariate time series models

function for zi. As zjt is generated straight-forward that y,,,(k)

= E(e&je,)

by an ARMA

= aoj,

= 0,

for

k = 0,

for

k > 0,

so that for k > qi, the auto- and cross covariances difference equation, yij(k) = 8,yij(k-l)+

model as given in (4), it is

yij(k)

(9) follow the mth order

. . . +B,yij(k-m).

(IO)

If for example, the vectors abj and c& are orthogonal, then for k 2 qi the covariance function yij(k) is generated by the mth order difference equation (lo), since, by substituting (9) into (8), the right-hand term in (8) vanishes for k = qi. In particular, if abi and aJti are orthogonal, zit is generated by an mth order AR and at most (qi- 1)th order MA process, although the ith row of A(L), a;(L), is of order qi. Situations in which a simpler univariate model arises than might be expected are called ‘coincidental situations’ by Granger (1972). They arise from cancelling of common factors, a point already discussed in ZP (1974), and/or the presence of orthogonality in the ith FE. A simple example may illustrate this point. Consider a special case of system (l),

[I::] =[i Y][:::I+[: :][E::::]~

(11)

with [e,,e,,]’ N IN(0, I,). We have assumed that H(L) = I, SO that Ai E Fi and system (11) is a first-order multiple MA process in FE form. The autocovariances are E zfr = 2,

Ez~,z,~-~

= 0,

for

k # 0,

Ezg,

EzZt~Zr_k = 1,

for

k = 1,

= 0,

for

k # 0, 1.

and = 2,

Due to orthogonality process. The variable covariances are

of ah1 and ail, zlt can be represented as a white noise zpt is generated by a first-order MA model. The cross

E~ttz~t-k = 1, =

0,

for

k = 0, 1,

elsewhere.

384

F. Palm, Univariate time series models

For k = - 1, the cross covariance vanishes although the joint process is a firstorder MA. This is due to the orthogonality of c$,i = (I 0) and aiz = (0 1). In conclusion, the analysis of the p auto- and cross correlation functions for the variable Zit gives us p lower bounds for qi, the order of the ith row of A(L). We retain the highest one as a lower bound for qi and can rule out as not in accord with the information in the sample the structural model, for which the FE form infringes upon the empirically determined lower bound for qi, i = 1, . . .) p. Tests on auto- and cross correlations among variables generated by a joint process are helpful in specifying the system of FEs. Using the highest of the p empirically determined lower bounds for the MA part will be sufficient to reproduce the cross equations error covariance structure of a set of FEs whereas the lower bound obtained from the autocorrelation function analysis is sufficient to determine the error correlation structure for single FEs. The analysis of the cross correlation functions of a system of FEs is thereby a natural extension of the Box-Jenkins identification procedure for univariate models. 2.3. The AR part of a FE The FEs associated with (1) usually have identical AR part IH(L However if, as mentioned in section 2 of ZP (1974), the matrix H(L) is for example triangular, diagonal, block triangular or block diagonal, cancelling will take place for some or all equations out of system (3). The AR parts for the p FEs in (3) may then have different orders but may still have common roots. For example, when H(L) is a diagonal matrix, the system of structural equations is already in FE form and the AR parts may be different for each equation. Usually one might expect that the order of lH(L) 1, which is a sum of products of elements of I-I(L), dominates the order of the matrix H(L). If however H(L) has special features, such as being the product of a unimodular matrix times a general matrix lag operator, for example,

fa)

= [

1

yL+PL2

1

l+aL+/?L2

1’

with IH( = 1+ (a- y)L, then the order of IH( may be lower than the highest order of H(L). Notice if a = y, H(L) is unimodular. We conclude that in general the order of the AR part of any FE out of (3) gives a lower bound to the order of IH(L Cancelling in some or all FEs may explain why the AR parts are not identical, in which case the number of distinct roots in the AR parts for the p elements of z, is a lower bound to the order of IH( and can be used to rule out all structural form matrices H(L) with a determinant of smaller order as incompatible with the analysis of the FEs.

385

F. Palm, Univariate time series models

3. The transfer function form To specify a set of transfer functions (TFs), we partition the vector z, in (1) as follows, z: = (y;, xi) where yI is a p1 x 1 vector of endogenous variables and x, is a pz x 1 vector of exogenous variables with p1+p2 = p. With zt SOpartitioned, the system in (1) becomes

elt F,,(L) xt [F,,(L) H,I(O ff,,m I[e2t 1- (12) I[1 H,,(L)

[

HI,(L)

Yt

=

The assumption that X, is exogenous the system in (12):4

H21(L)= 0, F,,(L) With the restrictions

in (13) imposed

~I(L)Y,+~~~(L)x~ ff226%

= 0,

=

=

&I(L)

F,,(L)

gives rise to the following

F,,(L)

= 0.

on

restrictions

(13)

on (12), we have

Fl,(Lkl,,

F22(Oe2,y

(144 Wb)

where the current and lagged x,‘s are stochastically independent from the disturbances in (14a). The TFs associated with (14a) are obtained by multiplying both sides of (14a) by IIT,( the adjoint matrix associated with H,,(L), to obtain

where IHI ,(L) 1is the determinant of HI,(L). Notice 5 that the scalar polynomial IHI ,(L) 1operates on each element of y,. The subset of equations in (14a) is a dynamic system of simultaneous equations with matrix-MA disturbances and can be analyzed without specifying an ARMA process as in (14b) for the exogenous variables. Although the specification in (14b) is very useful in prewhitening the exogenous variables and in deriving the FEs for the endogenous variables, in some cases the analysis has to be carried out conditionally on given exogenous variables as already pointed out in ZP (1974). For example for a single historical event, a dummy variable representation may be more appropriate than an ARMA model. However to the extent that the exogenous variables are closed-loop control variables, their stochastic nature is hardly questionable. But even in the 4The exogeneity of x, requires the absence of restrictions marginal distribution of X, and the conditional distribution (195O)l. 5The set of TFs has to be normalized.

between the parameters of the of y, given X, [e.g. Koopmans

386

F. Palm, Univariate time series models

situation where it is reasonable to assume that the exogenous variables are generated by an ARMA model in the form of (14b), the analysis of the TFs (15) associated with (14a) will be very fruitful in order to determine the dynamics of the structural form. The points discussed in the present note in relationship with a system of FEs also apply to the set of TFs:

(9

The disturbance of each equation in (15) is a sum of MAs and thus, by the lemma of section 2.1, it can be represented as an MA in one variable.

(ii)

The conclusions of section 2.2 also hold for the TFs. Due to cancelling and/or orthogonality, the order of the disturbance MA of a given TF say, the ith one, might be smaller than the order of the corresponding row of

H,*,(-Wl,(O (iii) The discussion on identification of the order of jH(L) 1in section 2.3 carries over to the problem of determining the order of jH,,(L)I in (15). (iv) Further, the i, jth element of HT,(L)H, 2(L) is a sum of products of polynomials in L and thus some or all of its coefficients may vanish due to special features of the ith row of H:,(L) and the jth column of H,,(L). Therefore, the empirically determined order for the TF relating the ith endogenous variable and thejth exogenous variable ought to be considered as a lower bound to the order of the polynomial products composing the i, jth element of HT,(L)H, ,(L).

4. Exogeneity

in a multiple ARMA process

The TF form associated

with (1) has been derived

by imposing

restrictions

on the matrices H(L) and F(L). Denoting by H, and H(L), respectively, the matrices of the zeroth-order terms and of the homogeneous part of H(L), and by Ho the inverse of Ho, the system in (14) can be solved to give

Under assumption (2), the conditional density ables of the r.h.s. of (16), is N(m, C), where

of (yl, xj), given the lagged vari-

m = -H”H(L)z,+HoF(L)e,, and Z = H°FoF;Hot,

(17)

F. Palm, U&variate time series models

381

with F,, and F(L) being, respectively, the matrices of the zeroth-order terms and of the homogeneous part of F(L). The matrix C is usually not block diagonal since H’l” $ 0. E ven with the restrictions imposed on the structural parameters in (14), the endogenous and exogenous variables are determined by a joint normal process. The conditional mean of yt given x, and lagged endogenous and exogenous variables and lagged errors is identical with the reduced form of (14a). After appropriate partitioning of m and C, the conditional mean of x, given yr is the regression function m2+

z21

I:;,1

(Y,-ml>*

(18)

It is a function of the current values of yr and of lagged values of yt, X, and et.6 Thus the conditional mean of any element in zt, given some or all other elements of z,, is in TF form. Its parameters are functions of the structural parameters in (14) and may be consistently estimated by non-linear least squares. The exogeneity7 of X, means that if we are interested in the parameters of the distribution of yt for given xt, knowledge of the marginal distribution of X, is of no additional help. However, when searching for a model specification that is in accord with the information in the data, the analysis of the regression function in (18) promises to give further insight into the dynamic structure of the model.

5. Conclusions First, the remarks in this note should be taken into consideration in the analysis of a dynamic SEM and the implication of its structural form for the properties of the FEs and TFs. Second, as the discussion shows, many forms of parametrization are compatible with the multiple ARMA process in (I), even under the restrictions given in (13). In choosing a parametrization and a set of regressors, we are not selecting a true model but a specification that meets the objective of the analysis in the best possible way. Third, from their empirical analysis, ZP (1974, 1975) conclude that smallorder ARMA schemes fit their series well. On the other hand, for a reasonable size econometric model, one may expect that IH( and H*(L)F(L) are of high order. For example, in a model of 20 equations, if the elements of H(L) are of % is worthwhile to point out that, with the restriction in (13) imposed, the cross correlogram between prewhitened endogenous and exogenous variables remains double-sided. Therefore the testing procedure for causality proposed by Granger and Newbold (1975) will usually show instantaneous feedback rather than causality between exogenous and endogenous variables. ‘For a detailed discussion of exogeneity in a linear model, see Florens-Mouchart-Richard (1976).

388

F. Palm, Univariate time series models

degree one, the determinant of H(L) may be of degree 20. Cancelling and orthogonality as discussed in sections 2.2, 2.3 and 3 reconcile the empirical findings of low-order FEs and TFs with the a priori expected high orders of IH(L WI,(0I, * * . . But after all, it may be that the lag structure of realistic models is simple. For example, a random walk for stock prices predicted by the efficient market hypothesis may not be the exception.

References Box, G.E.P. and G.M. Jenkins, 1970, Time series analysis forecasting and control (Holden-

Day, San Francisco). Florens, J.P., M. Mouchart and J. F.Richard, 1976, Maximum likelihood analysis of linear models, mimeo. (CORE, University of Louvain, Heverlee). Granger, C.W.J., 1972, Time series modelling and interpretation, Paper presented to the European Econometric Society Meeting. Granger, C.W.J. and P. Newbold, 1975, Identification of two-way causal systems, mimeo. (University of California, San Diego, CA). Ha&elmo, T:, 1947, Methods of measuring .the marginal propensity to consume, Journal of the American Statistical Society 42, 105-122; reprinted in: W. Hood and T.C. Koopmans, eds., 1953, Studies in econometric methods (Wiley, New York). Koopmans, T.C., 1950, When is an equation system complete for statistical purposes?, in: T.C. Koopmans, ed., Statistical inference in dynamic economic models (Wiley, New York). Palm, F., 1975, Time series analysis and simultaneous equation models with macroeconomic applications, UnpubPshed Ph.D. Dissertation (Unive&ty of Louvain, Louvain). Ouenouille. M.H.. 1968. The analvsis of multiole time series. 2nd ed. (C. Griffin, London). Rozanov, +.A.. 1467, Siationary iandom processes (HoldenlDay, SanFrancisco; CA). Yamamoto, T., 1975, On the autoregressive representation of dynamic simultaneous equations models, mimeo. (CORE, University of Louvain, Heverlee). Zellner, A. and F. Palm, 1974, Time series analysis and simultaneous equation econometric models, Journal of Econometrics 2, 17-54. Zellner, A. and F. Palm, 1975, Time series and structural analysis of monetary models of the U.S. economy, Sankhya (The Indian Journal of Statistics) 37, Series C, Pt. 2, 12-56.