The Structural Econometric Time Series Analysis Approach

by Arnold Zellner, Franz C. Palm

Overview

This book assembles key texts in the theory and applications of the Structural Econometric Time Series Analysis (SEMTSA) approach. The theory and applications of these procedures to a variety of econometric modeling and forecasting problems as well as Bayesian and non-Bayesian testing, shrinkage estimation and forecasting procedures are presented and applied. Finally, attention is focused on the effects of disaggregation on forecasting precision.

Product Details

ISBN-13:	9780521187435
Publisher:	Cambridge University Press
Publication date:	02/17/2011
Edition description:	New Edition
Pages:	736
Product dimensions:	5.98(w) x 9.02(h) x 1.46(d)

About the Author

Arnold Zellner is H. G. B. Alexander Distinguished Service Professor Emeritus of Economics and Statistics, Graduate School of Business, University of Chicago and Adjunct Professor, University of California at Berkeley. He has published books and many articles on the theory and application of econometrics and statistics to a wide range of problems.

Franz C. Palm is Professor of Econometrics, Faculty of Economics and Business Administration, Maastricht University. He has published many articles on the theory and application of econometrics and statistics to a wide range of problems.

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Cambridge University Press
0521814073 - The Structural Econometric Time Series Analysis Approach - by Arnold Zellner and Franz C. Palm
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Part I
The SEMTSA approach

1 Time series analysis and simultaneous equation econometric models (1974)

Arnold Zellner and Franz C. Palm

1 Introduction

In this chapter we take up the analysis of dynamic simultaneous equation models (SEMs) within the context of general linear multiple time series processes such as studied by Quenouille (1957). As noted by Quenouille, if a set of variables is generated by a multiple time series process, it is often possible to solve for the processes generating individual variables, namely the "final equations" of Tinbergen (1940), and these are in the autoregressive-moving average (ARMA) form. ARMA processes have been studied intensively by Box and Jenkins (1970). Further, if a general multiple time series process is appropriately specialized, we obtain a usual dynamic SEM in structural form. By algebraic manipulations, the associated reduced form and transfer function equation systems can be derived. In what follows, these equation systems are presented and their properties and uses are indicated.

It will be shown that assumptions about variables being exogenous, about lags in structural equations of SEMs, and about serial correlation properties of structural disturbance terms have strong implications for the properties of transfer functions and final equations that can be tested. Further, we show how large sample posterior odds and likelihood ratios can be used to appraise alternative hypotheses. In agreement with Pierce and Mason (1971), we believe that testing the implications of structural assumptions for transfer functions and, we add, final equations is an important element in the process of iterating in on a model that is reasonably in accord with the information in a sample of data. To illustrate these general points and to provide applications of the above methods, a dynamic version of a SEM due to Haavelmo (1947) is analyzed using US post-Second World War quarterly data.

The plan of the chapter is as follows. In section 2, a general multiple time series model is specified, its final equations are obtained, and their properties set forth. Then the implications of assumptions needed to specialize the multiple time series model to become a dynamic SEM for transfer functions and final equations are presented. In section 3, the algebraic analysis is applied to a small dynamic SEM. Quarterly US data are employed in sections 4 and 5 to analyze the final and transfer equations of the dynamic SEM. Section 6 provides a discussion of the empirical results, their implications for the specification and estimation of the structural equations of the model, and some concluding remarks.

2 General formulation and analysis of a system of dynamic equations

As indicated by Quenouille (1957), a linear multiple time series process can be represented as follows:¹

where z′_t = (z_1t,z_2t, . . . ,z_pt is a vector of random variables, e′_t=(e_1t,e_2t, . . . , e_pt is a vector of random errors, and H(L) and F(L) are each p × p matrices, assumed of full rank, whose elements are finite polynomials in the lag operator L, defined as Lⁿz_t = z_t-n. Typical elements of H(L) and F(L) are given by and . Further, we assume that the error process has a zero mean, an identity covariance matrix and no serial correlation, that is:

Eet = 0,

(2.2)

for all t and t′,

Eete′t′ = δtt′ I,

(2.3)

where I is a unit matrix and δ_tt′ is the Kronecker delta. The assumption in (2.3) does not involve a loss of generality since correlation of errors can be introduced through the matrix F(L).

The model in (2.1) is a multivariate autoregressive-moving average (ARMA) process. If H(L) = H₀, a matrix of degree zero in L, (2.1) is a moving average (MA) process; if F(L) = F₀, a matrix of degree zero in L, it is an autoregressive (AR) process. In general, (2.1) can be expressed as:

where H_l and F_l are matrices with all elements not depending on L, r = max_i,jr_ij and q = max_i,jq_ij.

Since H(L) in (2.1) is assumed to have full rank, (2.1) can be solved for z_t as follows:

zt = H-1(L)F(L)et,

(2.5a)

or

zt = [H*(L)/|H(L)|]F(L)et,

(2.5b)

where H^*(L) is the adjoint matrix associated with H(L) and ∣H(L)∣ is the determinant which is a scalar, finite polynomial in L. If the process is to be invertible, the roots of ∣H(L)∣ = 0 have to lie outside the unit circle. Then (2.5) expresses z_t as an infinite MA process that can be equivalently expressed as the following system of finite order ARMA equations:

|H(L)|zt = H*(L)F(L)et,

(2.6)

The ith equation of (2.6) is given by:

|H(L)|zit = ɑ′iet, i = 1,2, . . . , p,

(2.7)

where ɑ′_i is the ith row of H(L)F(L).

The following points regarding the set of final equations in (2.7) are of interest:

1. Each equation is in ARMA form, as pointed out by Quenouille (1957, p. 20). Thus the ARMA processes for individual variables are compatible with some, perhaps unknown, joint process for a set of random variables and are thus not necessarily "naive," "ad hoc" alternative models.

2. The order and parameters of the autoregressive part of each equation, ∣H(L)∣ z_it, i = 1, 2, . . . , p, will usually be the same.²

3. Statistical methods can be employed to investigate the form and properties of the ARMA equations in (2.7). Given that their forms, that is the degree of ∣H(L)∣ and the order of the moving average errors, have been determined, they can be estimated and used for prediction.

4. The equations of (2.7) are in the form of a restricted "seemingly unrelated" autoregressive model with correlated moving average error processes.³

The general multiple time series model in (2.1) can be specialized to a usual dynamic simultaneous equation model (SEM) if some prior information about H and F is available. That is, prior information may indicate that it is appropriate to regard some of the variables in z_t as being endogenous and the remaining variables as being exogenous, that is, generated by an independent process. To represent this situation, we partition (2.1) as follows:

If the p₁ × 1 vector y_t is endogenous and the p₂ × 1 vector x_t is exogenous, this implies the following restrictions on the submatrices of H and F:

With the assumptions in (2.9), the elements of e_1t do not affect the elements of x_t and the elements of e_2t affect the elements of y_t only through the elements of x_t. Under the hypotheses in (2.9), (2.8) is in the form of a dynamic SEM with endogenous variable vector y_t and exogenous variable vector x_t generated by an ARMA process. The usual structural equations, from (2.8) subject to (2.9), are:⁴

while the process generating the exogenous variables is:

with p₁ + p₂ = p.

Analogous to (2.4), the system (2.10) can be expressed as:

where H_11l, H_12l and F_11l are matrices the elements of which are coefficients of Ll. Under the assumption that H₁₁₀ is of full rank, the reduced form equations, which express the current values of endogenous variables as functions of the lagged endogenous and current and lagged exogenous variables, are:

The reduced form system in (2.13) is a system of p₁ stochastic difference equations of maximal order r.

The "final form" of (2.13), Theil and Boot (1962), or "set of fundamental dynamic equations" associated with (2.13), Kmenta (1971), which expresses the current values of endogenous variables as functions of only the exogenous variables, is given by:

If the process is invertible, i.e. if the roots of ∣H₁₁(L)∣ = 0 lie outside the unit circle, (2.14) is an infinite MA process in x_t and e_1t. Note that (2.14) is a set of "rational distributed lag" equations, Jorgenson (1966), or a system of "transfer function" equations, Box and Jenkins (1970). Also, the system in (2.14) can be brought into the following form:

where H^*₁₁(L) is the adjoint matrix associated with H₁₁(L) and ∣H₁₁(L)∣ is the determinant of H₁₁(L). The equation system in (2.15), where each endogenous variable depends only on its own lagged values and on the exogenous variables, with or without lags, has been called the "separated form," Marschak (1950), "autoregressive final form," Dhrymes (1970), "transfer function form," Box and Jenkins (1970), or "fundamental dynamic equations," Pierce and Mason (1971).⁵ As in (2.7), the p₁ endogenous variables in y_t have autoregressive parts with identical order and parameters, a point emphasized by Pierce and Mason (1971).

Having presented several equation systems above, it is useful to consider their possible uses and some requirements that must be met for these uses. As noted above, the final equations in (2.7) can be used to predict the future values of some or all variables in z_t, given that the forms of the ARMA processes for these variables have been determined and that parameters have been estimated. However, these final equations cannot be used for control and structural analysis. On the other hand, the reduced form equations (2.13) and transfer equations (2.15) can be employed for both prediction and control but not generally for structural analysis except when structural equations are in reduced form (H₁₁₀ ☰ I in (2.12)) or in final form [H₁₁ ☰ I in (2.10)]. Note that use of reduced form and transfer function equations implies that we have enough prior information to distinguish endogenous and exogenous variables. Further, if data on some of the endogenous variables are unavailable, it may be impossible to use the reduced form equations whereas it will be possible to use the transfer equations relating to those endogenous variables for which data are available. When the structural equation system in (2.10) is available, it can be employed for structural analysis and the associated "restricted" reduced form or transfer equations can be employed for prediction and control. Use of the structural system (2.10) implies not only that endogenous and exogenous variables have been distinguished, but also that prior information is available to identify structural parameters and that the dynamic properties of the structural equations have been determined. Also, structural analysis of the complete system in (2.10) will usually require that data be available on all variables.⁶ For the reader's convenience, some of these considerations are summarized in table 1.1.

Aside from the differing data requirements for use of the various equation systems considered in table 1.1, it should be appreciated that before each of the equation systems can be employed, the form of its equations must be ascertained. For example, in the case of the structural equation system (2.10), not only must endogenous and exogenous variables be distinguished, but also lag distributions, serial correlation properties of error terms, and identifying restrictions must be specified. Since these are often difficult requirements, it may be that some of the simpler equation systems will often be used although their uses are more limited than those of structural equation systems. Furthermore, even when the objective of an analysis is to obtain a structural equation system, the other equation systems, particularly the final equations and transfer equations, will be found useful. That is, structural assumptions regarding lag structures, etc. have implications for the forms and properties of final and transfer equations that can be checked with data. Such checks on structural assumptions can reveal weaknesses in them and possibly suggest alternative structural assumptions more in accord with the information in the data. In the following sections we illustrate these points in the analysis of a small dynamic structural equation system.

Table 1.1. Uses and requirements for various equation systems

Uses of equation systems
				Structural	Requirements for use of
	Equation system	Prediction	Control	analysis	equation systems
1.	Final equationsa (2.7)	yes	no	no	Forms of ARMA processes and parameter estimates
2.	Reduced form equations (2.13)	yes	yes	no	Endogenous-exogenous classification of variables, forms of equations, and parameter estimates
3.	Transfer equationsb (2.15)	yes	yes	no	Endogenous-exogenous classification of variables, forms of equations, and parameter estimates
4.	Final form equationsc (2.14)	yes	yes	no	Endogenous-exogenous classification of variables, forms of equations, and parameter estimates
5.	Structural equations (2.10)	yes	yes	yes	Endogenous-exogenous variable classification, identifying information,d forms of equations, and parameter estimates
Notes:
a This is Tinbergen's (1940) term.
b These equations are also referred to as "separated form" or "autoregressive final form" equations.
c As noted in the text, these equations are also referred to as "transfer function," "fundamental dynamic," and "rational distributed lag" equations.
d That is, information in the form of restrictions to identify structural parameters.

3 Algebraic analysis of a dynamic version of Haavelmo's model

Haavelmo (1947) formulated and analyzed the following static model with annual data for the United States, 1929-41:

ct = αyt + β + ut,	(3.1a)
rt = μ (ct + xt) + v + wt	(3.1b)
yt = ct + xt - rt	(3.1c)

where c_t, y_t and r_t are endogenous variables, x_t is exogenous, u_t and w_t are disturbance terms, and α, β, μ and v are scalar parameters. The definitions of the variables, all on a price-deflated, per capita basis, are:

c_t = personal consumption expenditures,
y_t = personal disposable income,
r_t = gross business saving, and
x_t = gross investment.⁷

Equation (3.1a) is a consumption relation, (3.1b) a gross business saving equation, and (3.1c) an accounting identity.

In Chetty's (1966, 1968) analyzes of the system (3.1) employing Haavelmo's annual data, he found the disturbance terms highly autocorrelated, perhaps indicating that the static nature of the model is not appropriate. In view of this possibility, (3.1) is made dynamic in the following way:

ct = α(L)yt + β + ut,	(3.2a)
rt = μ(L) (ct + xt) + v + wt	(3.2b)
yt = ct + xt - rt	(3.2c)

In (3.2a), α(L) is a polynomial lag operator that serves to make c_t a function of current and lagged values of income. Similarly, μ(L) in (3.2b) is a polynomial lag operator that makes r_t depend on current and lagged values of c_t + x_t, a variable that Haavelmo refers to as "gross disposable income." On substituting for r_t in (3.2b) from (3.2c), the equations for c_t and y_t are:

ct = α(L)yt + β + ut,	(3.3a)
yt = [1 - μ(L)](ct + xt) - v - wt.	(3.3b)

With respect to the disturbance terms in (3.3), we assume:

where the f_ij(L) are polynomials in L, e_1t and e_2t have zero means, unit variances, and are contemporaneously and serially uncorrelated.

Letting z^′_t = (c_t, y_t, x_t), the general multiple time series model for z_t, in the matrix form (2.1), is:

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© Cambridge University Press

Table of Contents

Introduction; Part I. The SEMTSA Approach: 1. Time series analysis and simultaneous equation econometric models A. Zellner and F. C. Palm; 2. Statistical analysis of econometric models A. Zellner; 3. Structural econometric modeling and time series analysis: an integrated approach F. C. Palm; 4. Time series analysis, forecasting and econometric modeling: the structural econometric modeling, times series analysis (SEMTSA) approach A. Zellner; 5. Large sample estimation and testing procedures for dynamic equation systems F. Palm and A. Zellner; Part II. Selected Applications: 6. Time series and structural analysis of monetary models of the US economy A. Zellner and F. Palm; 7. Time series versus structural models: a case study of Canadian manufacturing inventory behavior P. K. Trivedi; 8. Time series analysis of the German hyperinflation P. Evans; 9. A time series analysis of seasonality in econometric models C. I. Plosser; 10. The behavior of speculative prices and the consistency of economic models R. I. Webb; 11. A comparison of the stochastic processes of structural and time series exchange rate models F. W. Ahking and S. M. Miller; 12. Encompassing univariate models in multivariate times series: a case study A. Maravall and A. Mathis; Part III. Macroeconomic Forecasting and Modeling: 13. Macroeconomic forecasting using pooled international data A. Garcia-Ferrer, R. A. Highfield, F. Palm and A. Zellner; 14. Forecasting international growth rates using Bayesian shrinkage and other procedures A. Zellner and C. Hong; 15. Turning points in economic time series, loss structures and Bayesian forecasting A. Zellner, C. Hong and G. M. Gulati; 16. Forecasting turning points in international output growth rates using Bayesian exponentially weighted autoregression, time-varying parameter and pooling techniques A. Zellner, C. Hong and C. Min; 17. Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates C. Min and A. Zellner; 18. Pooling in dynamic panel data models: an application to forecasting GDP growth rates A. J. Hoogstrate, F. C. Palm and G. A. Pfann; 19. Forecasting turning points in countries' output growth rates: a response to Milton Friedman A. Zellner and C. Min; 20. Using Bayesian techniques for data pooling in regional payroll forecasting J. P. LeSage and M. Magura; 21. Forecasting turning points in metropolitan employment growth rates using Bayesian techniques J. P. LeSage; 22. A note on aggregation, disaggregation and forecasting performance A. Zellner and J. Tobias; 23. The Marshallian macroeconomic model A. Zellner; 24. Bayesian modeling of economies and data requirements A. Zellner and B. Chen.