Analysis ApproachBringing together a collection of previously published work, this bookprovides a timely discussion of major considerations relating to the con-struction of econometric m
Trang 2This page intentionally left blank
Trang 3Analysis Approach
Bringing together a collection of previously published work, this bookprovides a timely discussion of major considerations relating to the con-struction of econometric models that work well to explain economicphenomena, predict future outcomes, and be useful for policy-making.Analytical relations between dynamic econometric structural modelsand empirical time series MVARMA, VAR, transfer function, and uni-variate ARIMA models are established with important application formodel-checking and model construction The theory and applications
of these procedures to a variety of econometric modeling and ing problems as well as Bayesian and non-Bayesian testing, shrinkageestimation, and forecasting procedures are also presented and applied.Finally, attention is focused on the effects of disaggregation on forecast-ing precision and the new Marshallian macroeconomic model ()that features demand, supply, and entry equations for major sectors ofeconomies is analyzed and described This volume will prove invaluable
forecast-to professionals, academics and students alike
is H G B Alexander Distinguished Service sor Emeritus of Economics and Statistics, Graduate School of Business,University of Chicago and Adjunct Professor, University of California
Profes-at Berkeley He has published books and many articles on the theory andapplication of econometrics and statistics to a wide range of problems
is Professor of Econometrics, Faculty of Economicsand Business Administration, Maastricht University He has publishedmany articles on the theory and application of econometrics and statis-tics to a wide range of problems
Trang 5The Structural Econometric Time Series Analysis Approach
Edited by
Arnold Zellner and Franz C Palm
Trang 6
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São PauloCambridge University Press
The Edinburgh Building, Cambridge , UK
First published in print format
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© Cambridge University Press, 2004
2004
Information on this title: www.cambridge.org/9780521814072
This publication is in copyright Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press
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Cambridge University Press has no responsibility for the persistence or accuracy of sfor external or third-party internet websites referred to in this publication, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate
Published in the United States of America by Cambridge University Press, New Yorkwww.cambridge.org
hardback
eBook (EBL)eBook (EBL)hardback
Trang 7List of contributors page ix
1 Time series analysis and simultaneous equation
3 Structural econometric modeling and time series
Trang 8vi Contents
4 Time series analysis, forecasting, and econometric
modeling: the structural econometric modeling, time
5 Large-sample estimation and testing procedures for
6 Time series and structural analysis of monetary models
7 Time series versus structural models: a case study of
Canadian manufacturing inventory behavior (1975) 288
Trang 911 A comparison of the stochastic processes of structural
and time series exchange rate models (1987) 405
12 Encompassing univariate models in multivariate time
13 Macroeconomic forecasting using pooled international
-,
, ,
14 Forecasting international growth rates using Bayesian
15 Turning points in economic time series, loss structures,
, ,
16 Forecasting turning points in international output
growth rates using Bayesian exponentially weighted
autoregression, time-varying parameter, and pooling
, ,
-
17 Bayesian and non-Bayesian methods for combining
models and forecasts with applications to forecasting
19 Forecasting turning points in countries’ output growth
rates: a response to Milton Friedman (1999) 612
-
Pooling in dynamic panel data models: an application to
Trang 10viii Contents
20 Using Bayesian techniques for data pooling in regional
21 Forecasting turning points in metropolitan employment
growth rates using Bayesian techniques (1990) 637
Trang 11, ., Department of Economics, University of necticut, Storrs, CT
Con-, ., Professor of Economics, Department of nomics, Boston College, Boston, MA
Eco-, , Chicago Partners, LLC, Chicago, IL
, ., Professor of Economics, Department of nomics, Princeton University, Princeton, NJ
Eco-, ., Professor Emeritus of Economics, Department ofEconomics, Johns Hopkins University, Baltimore, MD
, , Professor of Economics, Ohio State University, bus, OH
Colum--, , Professor of Economics, Departamento
de Analsis Economico: Economia Cuantitava, Universidad Autonoma
de Madrid
, ., Georgetown University, Law Center, ington, DC
Wash-, ., Dean, School of Business Administration
364, School of Business, New York University at Albany, NY
, , Department of Economics, Sookmyung Women’sUniversity, Seoul
, ., Ministry of Justice, Netherlands ForensicInstitute, Rijswijk
, , Professor of Economics, Sloan School of Management,MIT, Cambridge, MA
, ., Professor of Economics, Department of Economics,University of Toledo, Toledo, OH
ix
Trang 12, , SIMC, Banco de Espa˜na, Madrid
, ., Department of Economics, College of NevadaLas Vegas, Las Vegas, NV
, -, Department of Economics, Hankuk University ofForeign Studies, Seoul
, ., Professor of Econometrics, Faculty of Economics andBusiness Administration, Universiteit Maastricht
, ., Professor of Econometrics, Faculty of Economicsand Business Administration, Universiteit Maastricht
, ., Professor of Economics, Simon ate School of Business Administration, University of Rochester,Rochester, NY
Gradu-, ., Professor of Econometrics, Department of nomics, London School of Economics and Political Science, London
Eco-, ., Professor of Economics, Department ofEconomics, University of California, Berkeley, CA
, ., Professor of Economics, Department of nomics, Princeton University, Princeton, NJ
Eco-, , Department of Economics University of California,Irvine, CA
, ., Professor of Economics, Department of nomics, Indiana University, Bloomington, IN
Eco-, ., McIntire School of Commerce, University ofVirginia, Charlottesville, VA
, , H G B Alexander Distinguished Service sor Emeritus of Economics and Statistics, Graduate School of Busi-ness, University of Chicago, Chicago, IL
Trang 13Profes-The following chapters are reprinted with the kind permission of the
publishers listed below Chapter 1, from the Journal of Econometrics, and chapters 1, 5, 10, 12, 16, 17, 19, and 20, from the Journal of Econometrics,
by North-Holland Publishing Company; chapter 2, from the Journal of the American Statistical Association, and chapters 13, 18, and 20, from the Journal of Business and Economic Statistics, by the American Statistical
Association; chapters 3 and 9, from the Bureau of the Census, US ment of Commerce, by the US Government Printing Office; chapters 4
Depart-and 22, from the Journal of Forecasting, by John Wiley & Sons; chapter 6, from Sankhya: The Indian Journal of Statistics, by the Indian Statistical Institute; chapters 7 and 8, from the International Economic Review, by the
Wharton School of Finance and Commerce, University of Pennsylvaniaand the Osaka University Institute of Economic Research Association;
chapter 11, from the Review of Economics and Statistics, by MIT Press; chapter 21, from the Journal of Regional Science, by the Regional Science
Research Institute, in cooperation with the Wharton School of Financeand Commerce, University of Pennsylvania, Blackwell Publishing Ltd;chapter 15, by North-Holland Publishing Company; chapter 23, byKluwer Academic Publishers; and chapter 24 by Cambridge UniversityPress
xi
Trang 15In the early 1970s we were concerned about the relationships betweenmultivariate and univariate time series models, such as those brilliantlyanalyzed by Quenouille (1957) and Box and Jenkins (1970) and mul-tivariate dynamic structural econometric models that had been and arewidely employed in explanation, prediction and policy-making Fortu-nately, we discovered the relationships and reported them in our paper,Zellner and Palm (1974) that is included in part I of this volume (chapter1) See also the other general chapters in part I discussing general features
of our approach, the reactions of leading researchers, and many usefulreferences to the literature
Having discovered the algebraic relations connecting statistical timeseries and structural econometric models, we next considered how thisdiscovery might be used to produce improved models In this connec-tion, we thought it important not only to emphasize a philosophicalpreference for sophisticatedly simple models that is discussed in sev-eral chapters in part I and Zellner, Keuzenkamp, and McAleer (2001),but also operational techniques that would help researchers actually pro-duce improved models As illustrated in the chapters included in thisvolume, our approach involves (1) deducing algebraically the impliedmarginal processes and transfer functions for individual variables in amulti-equation model, e.g a vector autoregression (VAR) or a structuraleconometric model (SEM), and (2) comparing these derived equations’forms and properties with those derived from the data by use of empir-ical model identification and testing techniques See Palm and Zellner(1980), included in part I (chapter 5) for some early estimation and test-ing procedures that have been improved over the years If the information
in the data is compatible with the empirically determined, simple timeseries models and not with those implied by a VAR or SEM, then weconclude that the VAR or SEM needs reformulation and improvement.See, for example our (1975) paper in part II (chapter 6) analyzing mone-tary models of the US economy and other papers for applications of thisapproach to many other problems including Trivedi (1975) on modeling
xiii
Trang 16xiv Introduction
inventory behavior (chapter 7), Evans (1978) on the German flation (chapter 8), Plosser (1976) on seasonality (Chapter 9), Webb(1985) on behavior of speculative prices (chapter 10), Ahking and Miller(1987) on exchange rate models (chapter 11), and Maravall and Mathis(1994) on diagnosis of VAR models using French macroeconomic data(chapter 12) These studies demonstrate well the usefulness of ourSEMTSA approach in analyzing, comparing, and improving models.Since there is often no satisfactory model available, in part III we illus-trate how relatively simple forecasting equations have been developed,studied, and tested in point and turning point forecasting experimentsusing modern estimation and forecasting techniques Here the objective
hyperin-is to get forecasting equations that work well in point and turning pointforecasting and have reasonable dynamic properties Then the objective
is to produce reasonable economic models to rationalize the good cal performance of these empirical forecasting equations Thus we do not
empiri-in the present empiri-instance go from theory to the data but reverse the process
by going from what works well empirically to theory that explains thisunusual empirical finding As mentioned in several chapters in part III,the empirical forecasting equations for countries’ annual GDP growthrates have been rationalized by Hong (1989) in terms of a Hicksian [IS-
LM macroeconomic model, by Min (1992) in terms of a generalizedreal business cycle model that he formulated and by Zellner and Anton(1986) in terms of an aggregate demand and supply model Thus theempirical relations studied intensively in the chapters included in partIII have some theoretical as well as empirical support Note, too, thatmany methodological tools were developed and tested in the chapters
on empirical forecasting work in part III – namely, Bayesian shrinkageestimation and prediction, optimal turning point forecasting techniques,optimal Bayesian model-combining or pooling methods, etc Also, com-parisons of forecasting root mean-squared errors (RMSEs) and meanabsolute errors (MAEs) indicate that various simple forecasting equa-tions’ performance is competitive with that of certain large-scale macroe-conometric models for many economies See, for example some compar-
isons reported in Garcia-Ferrer et al (1987) and Hoogstrate, Palm, and
Pfann (2000) (chapter 13 and 18) for some improved results that utilizevarious pooling techniques in analysis and forecasting of panel data foreighteen countries
While the studies in part III provide useful, improved macroeconomicresults, it is the case that aggregation of output and other kinds of data,say over sectors of an economy, can involve a loss of valuable informa-tion, as has been discussed many times in the past Thus part IV presentschapters dealing with disaggregation, forecasting, and modeling A simple
Trang 17experiment, reported in Zellner and Tobias (2000) (chapter 22) showsempirically how disaggregation can result in improved forecasting pre-cision in connection with forecasting the annual medians of eighteencountries’ growth rates In chapters 20 and 21 by LeSage and Magura(1990) and LeSage (1990), it is shown how shrinkage point and turn-ing point forecasting procedures perform using regional data Then inZellner (2000) and in Zellner and Chen (2000) (chapters 23 and 24),Marshallian sector models of industrial sectors are formulated, building
on the earlier work of Veloce and Zellner (1985), and tested in ing experiments using annual data for eleven sectors of the US economy.The annual output forecasts of the sectors are added to get a forecast
forecast-of total GDP and its growth rate year by year Such forecasts are pared with forecasts derived from models implemented with aggregatedata In this instance, it was found that it pays to disaggregate Furtherwork to improve and expand the Marshallian sector model in line withthe SEMTSA approach is described in these chapters
com-In summary, pursuing the SEMTSA approach over the years has been
an exciting experience that has led to new empirical findings, improvedand novel methodological tools, and improved models We thank all thosewho have contributed to these positive developments and hope that futuredevelopments will be even better Also, thanks to the US National ScienceFoundation and the H G B Alexander Endowment Fund, University ofChicago, for financial support Ashwin Rattan at Cambridge UniversityPress, provided much help in arranging for the publication of our book,for which we are most grateful
Zellner, A., H Keuzenkamp, and M McAleer, 2001, Simplicity, Inference and
Econometric Modeling (Cambridge, Cambridge University Press)
Trang 19The SEMTSA approach
Trang 21equation econometric models (1974)
Arnold Zellner and Franz C Palm
In this chapter we take up the analysis of dynamic simultaneous equationmodels (SEMs) within the context of general linear multiple time seriesprocesses 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 ables, namely the “final equations” of Tinbergen (1940), and these are
vari-in the autoregressive-movvari-ing average (ARMA) form ARMA processeshave been studied intensively by Box and Jenkins (1970) Further, if ageneral multiple time series process is appropriately specialized, we obtain
a usual dynamic SEM in structural form By algebraic manipulations, theassociated reduced form and transfer function equation systems can bederived In what follows, these equation systems are presented and theirproperties and uses are indicated
It will be shown that assumptions about variables being exogenous,about lags in structural equations of SEMs, and about serial correlationproperties of structural disturbance terms have strong implications forthe properties of transfer functions and final equations that can be tested.Further, we show how large sample posterior odds and likelihood ratioscan be used to appraise alternative hypotheses In agreement with Pierceand Mason (1971), we believe that testing the implications of structuralassumptions for transfer functions and, we add, final equations is animportant element in the process of iterating in on a model that is rea-sonably in accord with the information in a sample of data To illustratethese general points and to provide applications of the above methods,
Research financed in part by NSF Grant GS-2347 and by income from the H.G.B der Endowment Fund, Graduate School of Business, University of Chicago Some of the ideas in this chapter were presented in econometrics lectures and at a session of the Econo- metric Society’s meeting in 1971 by one of the authors The second author received financial support from the Belgian National Science Foundation.
Alexan-Originally published in the Journal of Econometrics 2 (1974), 17–54.
3
Trang 224 Arnold Zellner and Franz C Palm
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 multipletime series model is specified, its final equations are obtained, and theirproperties set forth Then the implications of assumptions needed tospecialize the multiple time series model to become a dynamic SEMfor transfer functions and final equations are presented In section 3,the algebraic analysis is applied to a small dynamic SEM Quarterly USdata are employed in sections 4 and 5 to analyze the final and transferequations of the dynamic SEM Section 6 provides a discussion of theempirical results, their implications for the specification and estimation
of the structural equations of the model, and some concluding remarks
each p × p matrices, assumed of full rank, whose elements are finite nomials in the lag operator L, defined as L n z t = z t −n Typical elements
poly-of H(L) and F(L) are given by h i j =r i j
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
1In (2.1), z t is assumed to be mean-corrected, that is z tis a deviation from a population mean vector Below, we relax this assumption.
Trang 23moving 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= maxi,j r i j and q= maxi,j q i j
Since H(L) in (2.1) is assumed to have full rank, (2.1) can be solved
be invertible, the roots of| H(L) |= 0 have to lie outside the unit circle.
Then (2.5) expresses z tas an infinite MA process that can be equivalentlyexpressed as the following system of finite order ARMA equations:
i is the ith row of H∗(L)F(L).
The following points regarding the set of final equations in (2.7) are ofinterest:
(i) Each equation is in ARMA form, as pointed out by Quenouille(1957, p 20) Thus the ARMA processes for individual variablesare compatible with some, perhaps unknown, joint process for a set
of random variables and are thus not necessarily “naive,” “ad hoc”alternative models
(ii) The order and parameters of the autoregressive part of each tion,|H(L)| z i t , i = 1, 2, , p, will usually be the same.2
equa-(iii) Statistical methods can be employed to investigate the form andproperties 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
2 In some cases in which|H(L)| contains factors in common with those appearing in all
elements of the vectorsα
i , e.g when H is triangular, diagonal or block diagonal, some
cancelling will take place In such cases the statement in (ii) has to be qualified.
Trang 246 Arnold Zellner and Franz C Palm
errors, have been determined, they can be estimated and used forprediction
(iv) The equations of (2.7) are in the form of a restricted “seeminglyunrelated” autoregressive model with correlated moving averageerror 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 asbeing endogenous and the remaining variables as being exogenous, that
is, generated by an independent process To represent this situation, wepartition (2.1) as follows:
If the p1× 1 vector y t is endogenous and the p2× 1 vector x tis exogenous,
this implies the following restrictions on the submatrices of H and F:
With the assumptions in (2.9), the elements of e 1tdo not affect the
ele-ments of x t and the elements of e 2t affect the elements of y tonly 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 tgenerated by an ARMA process The usual structuralequations, from (2.8) subject to (2.9), are:4
where H 11l , H 12l and F 11lare matrices the elements of which are
coeffi-cients of L l Under the assumption that H11 0is of full rank, the reduced
3 See Nelson (1970) and Akaike (1973) for estimation results for systems similar to (2.7).
4 Hannan (1969, 1971) has analysed the identification problem for systems in the form of (2.10).
Trang 25form equations, which express the current values of endogenous variables
as functions of the lagged endogenous and current and lagged exogenousvariables, are:
The reduced form system in (2.13) is a system of p1stochastic difference
equations of maximal order r.
The “final form” of (2.13), Theil and Boot (1962), or “set of damental dynamic equations” associated with (2.13), Kmenta (1971),which expresses the current values of endogenous variables as functions
fun-of only the exogenous variables, is given by:
y t = −H−1
11(L)H12(L)x t + H−1
11(L)F11(L)e 1t (2.14)
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:
|H11(L) |y t = −H∗
11(L)H12(L)x t + H∗
11(L)F11(L)e 1t, (2.15)
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 theexogenous variables, with or without lags, has been called the “sepa-rated form,” Marschak (1950), “autoregressive final form,” Dhrymes(1970), “transfer function form,” Box and Jenkins (1970), or “funda-mental dynamic equations,” Pierce and Mason (1971).5 As in (2.7),
the p1 endogenous variables in y t have autoregressive parts with tical order and parameters, a point emphasized by Pierce and Mason(1971)
iden-Having presented several equation systems above, it is useful to sider their possible uses and some requirements that must be met forthese uses As noted above, the final equations in (2.7) can be used to
con-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
5If some of the variables in x tare non-stochastic, say time trends, they will appear the final equations of the system.
Trang 268 Arnold Zellner and Franz C Palm
parameters have been estimated However, these final equations cannot
be used for control and structural analysis On the other hand, the reducedform equations (2.13) and transfer equations (2.15) can be employed forboth 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 tinguish endogenous and exogenous variables Further, if data on some
dis-of the endogenous variables are unavailable, it may be impossible to usethe reduced form equations whereas it will be possible to use the transferequations relating to those endogenous variables for which data are avail-able When the structural equation system in (2.10) is available, it can beemployed for structural analysis and the associated “restricted” reducedform or transfer equations can be employed for prediction and control.Use of the structural system (2.10) implies not only that endogenous andexogenous variables have been distinguished, but also that prior informa-tion is available to identify structural parameters and that the dynamicproperties of the structural equations have been determined Also, struc-tural analysis of the complete system in (2.10) will usually require thatdata be available on all variables.6For the reader’s convenience, some ofthese considerations are summarized in table 1.1
Aside from the differing data requirements for use of the various tion systems considered in table 1.1, it should be appreciated that beforeeach of the equation systems can be employed, the form of its equationsmust be ascertained For example, in the case of the structural equationsystem (2.10), not only must endogenous and exogenous variables bedistinguished, but also lag distributions, serial correlation properties oferror terms, and identifying restrictions must be specified Since these areoften difficult requirements, it may be that some of the simpler equationsystems will often be used although their uses are more limited than those
equa-of structural equation systems Furthermore, even when the objective equa-of
an analysis is to obtain a structural equation system, the other equationsystems, particularly the final equations and transfer equations, will befound useful That is, structural assumptions regarding lag structures,etc have implications for the forms and properties of final and trans-fer equations that can be checked with data Such checks on structuralassumptions can reveal weaknesses in them and possibly suggest alterna-tive structural assumptions more in accord with the information in thedata In the following sections we illustrate these points in the analysis of
a small dynamic structural equation system
6 This requirement will not be as stringent for partial analyses and for fully recursive models.
Trang 27Table 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
and parameter estimates
2 Reduced form
equations (2.13)
classification of variables, forms of equations, and parameter estimates
3 Transfer equationsb
(2.15)
classification of variables, forms of equations, and parameter estimates
4 Final form equationsc
(2.14)
classification of variables, forms of equations, and parameter estimates
5 Structural equations
(2.10)
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,” mental dynamic,” and “rational distributed lag” equations.
“funda-d That is, information in the form of restrictions to identify structural parameters.
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
Trang 2810 Arnold Zellner and Franz C Palm
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
is a polynomial lag operator that makes r tdepend 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 tare:
y t = [1 − µ(L)](c t + x t)− v − w t (3.3b)With respect to the disturbance terms in (3.3), we assume:
t = (c t , y t , x t ), the general multiple time series model for z t,
in the matrix form (2.1), is:
Trang 29To specialize (3.6) to represent the dynamic version of Haavelmo’s
model in (3.3) with x texogenous, we must haveθ1= β, θ2= v,
Note that the process on the exogenous variable is h33(L)x t = f33(L)e 3t+
θ3 and the fact that x t is assumed exogenous requires that h31(L)≡
h32(L) ≡ 0 and that F(L) be block diagonal as shown in (3.8).
In what follows, we shall denote the degree of h i j (L) by r i j and the
(1− h12h21)h33y t = θ
2+ ( f21− f11h21)h33e 1t + ( f − f h )h e − f h e , (3.10)
Trang 3012 Arnold Zellner and Franz C Palm
Table 1.2 Degrees of lag polynomials in final equations
Degrees of MA polynomials for errorsb
Degrees of AR
(3.9): c t r33 and r33+ q11 and r33+ q12 and r12+ r23+ q33
2being new constants Note that the AR parts
of (3.9) and (3.10) have the same order and parameters The degrees of
the lag polynomials in (3.9) and (3.10) and in the process for x t,
are indicated in table 1.2
As mentioned above, the AR polynomials in the final equations for
c t and y t are identical and of maximal degree equal to r12+ r21+ r33,
as shown in table 1.2, where r12= degree of α(L) in the consumption equation, r21is the degree of µ(L) in the business saving equation, and
r33is the degree of h33, the AR polynomial in the process for x t Also, if
the disturbance terms u t and w t are serially uncorrelated and if all the q i j
in table 1.2 are zero, the following results hold:
(i) In the final equation for c t, the degree of the AR part is larger than orequal to the order of the MA process for the disturbance term; that is
r12+ r21+ r33 max(r12+ r23, r33+ r12), with equality holding if
r33= 0, since r21= r23, or if r21= r23= 0
(ii) In the final equation for y t, the degree of the AR polynomial is largerthan or equal to the order of the MA process for the disturbance
term; i.e., r12+ r21+ r33 r33+ r21with equality holding if r12= 0
Thus if the process for x tis purely AR and the structural disturbance
terms u t and w tare not serially correlated, (i) and (ii) provide usefulimplications for properties of the final equations that can be checkedwith data as explained below
Further, under the assumption that the structural disturbance
terms u t and w t are serially uncorrelated, all q i j other than q33 in
table 1.2 will be equal to zero If the process for x is analyzed to
Trang 31determine the degree of h33, r33, and of f33, q33, this information can
be used in conjunction with the following:
(iii) In the final equation for c t, the degree of the AR polynomial will be
smaller than or equal to the order of the MA disturbance if q33 r33 (Note r21= r23.) If q33< r33, the degree of the AR polynomial will
be greater than the order of the MA disturbance term
(iv) In the final equation for y t, the degree of the AR polynomial will
be greater than the order of the MA disturbance term given that
q33or if r12+ r33= q33 The latter will be greater if r12+ r33< q33
In what follows, post-Second World War quarterly data for the United
States, 1947–72, are employed to analyze the final equations for c t , y tand
x tand to check some of the implications mentioned above
From (3.8), the dynamic structural equations of the dynamizedHaavelmo model are:
0
0
Trang 3214 Arnold Zellner and Franz C Palm
(a) The AR parts of the two transfer equations are identical Since h12is
of degree r12and h21of degree r21, the order of the autoregression in
each equation is r12+ r21
(b) In (3.14) the degree of the operator h12h23 hitting x t is r12+ r23=
r12+ r21, the same as that for the autoregressive part of the equation,
1 − h12h21
(c) In (3.15), the degree of the lag operator,−h23, applied to x t is r23=
r21, which is less than or equal to the degree of 1 − h12h21, the ARpolynomial
(d) The lag operator acting on x t in the equation for c t , h12h33, is a
mul-tiple of that acting on x t in the equation for y t and thus the formerhas degree larger than or equal to that of the latter
(e) If the structural disturbance terms are serially uncorrelated, i.e f i j
has degree zero in L for i, j = 1, 2, the orders of the MA error terms
in (3.14) and (3.15) are r12 0 and r21 0, respectively Thus forboth equations, the order of the MA error process is less than orequal to the order of the AR part of the equation
By use of appropriate statistical techniques and data, the transfer tions in (3.14)–(3.15) can be analyzed to determine the degrees of lagpolynomials and to estimate parameter values With these results in hand,
equa-it is possible to check the points (a)–(e) relating to the transfer equationsassociated with Haavelmo’s dynamic model The results of such calcula-tions are reported below
In this subsection, we report the results of applying [Box–Jenkins] (BJ)identification and estimation procedures to the final equations of thedynamized Haavelmo model Box and Jenkins (1970, p 175) provide thefollowing relations between the autocorrelation and partial autocorrela-tion functions associated with stationary stochastic processes for a singlerandom variable:8
(1) For a purely AR process of order p, the autocorrelation function tails
off and the partial autocorrelation function9has a cut-off after lag p (2) For a purely MA process of order q, the autocorrelation function has a cut-off after lag q and the partial autocorrelation function tails
off
8 See Box and Jenkins (1970, pp 64–5) for a definition of this function.
9 Autocorrelation functions have been formerly used in econometrics, see, e.g., Wold (1953).
Trang 331970 1965
1960 1955
Figure 1.1 Plots of data for y t , c t , and x t, 1949–1970
(3) For a mixed ARMA process, with the order of the AR being p and that of the MA being q, the autocorrelation function is a mixture
of exponential and damped sine waves after the first q − p lags and
the partial autocorrelation function is dominated by a mixture of
exponentials and damped sine waves after the first p − q lags.
Box and Jenkins suggest differencing a series until it is stationary andthen computing estimates of the autocorrelation and partial autocorrela-tion functions Using (1)–(3), it may be possible to determine or identifythe nature of the process for the differenced series as well as values of
p and q Once the process or model and p and q have been determined,
the model’s parameters can be estimated, usually by use of a non-linearestimation procedure
Plots of the data for the variables of Haavelmo’s model, c t , y t , and x t, areshown in figure 1.1.10From this figure, it is seen that the variables appar-ently have trends and thus are non-stationary First or second differencing
10 The variables have been defined above The data are seasonally adjusted quarterly,
price-deflated, per capita aggregates, expressed in dollars at an annual rate, for the US economy,
1947I–1972II, obtained from official sources cited in the appendix.
Trang 3416 Arnold Zellner and Franz C Palm
($)
80
55 1950
In figure 1.3, we present the estimated autocorrelation function for the
series c t − c t−1, the first difference of consumption.11Also indicated infigure 1.3, is a±2 ˆσ confidence band for the autocorrelations where ˆσ is
a large sample standard error for the sample autocorrelations.12It is seenthat all estimated autocorrelations lie within the band except for that oflag 2 This suggests that the underlying process is not purely AR If theautocorrelation estimate for lag 2 is regarded as a cut-off, the results
11 The computer program employed was developed by C R Nelson and S Beveridge, Graduate School of Business, University of Chicago.
12σˆ 2is an estimate of the following approximate variance of r k , the kth sample serial
corre-lation, given in Bartlett (1946) Withρ v = 0 for v > q, var (r k) ˙ = (1 + 2q
ρ = 0, v > 0 For k > 12, they are calculated assuming ρ = 0, v > 12.
Trang 35Estimated autocorrelation function
Estimated partial autocorrelation function
Figure 1.3 Estimated autocorrelation function and estimated partial
autocorrelation function for c t − c t−1
Estimated autocorrelation function
Estimated partial autocorrelation function
Figure 1.4 Estimated autocorrelation function and estimated partial
autocorrelation function for y t − y t−1
suggest that a second order MA process may be generating the first
differences of c t The estimated partial autocorrelation function, alsoshown in figure 1.3, does not appear to contradict this possibility Estima-tion of a second order MA model for the first differences of consumption,led to the following results using the BJ non-linear algorithm:
c t − c t−1= e t + 0.0211e t−1+ 0.278e t−2 + 10.73 s2= 530,
(4.1)
where s2 is the residual sum of squares (RSS) divided by the number
of degrees of freedom and the figures in parentheses are large samplestandard errors
For income, y t, a plot of the first differences is given in figure 1.2 Fromthe plot of the estimated autocorrelations for the first differences in figure
Trang 3618 Arnold Zellner and Franz C Palm
Estimated autocorrelation function
Estimated partial autocorrelation function
Figure 1.5 Vertical areas of figures 1.3–1.5 for x t − x t−1: on the right:ˆ
φ k ˆˆk , on the left: r l
1.4, it appears that none of the autocorrelations is significantly differentfrom zero, a finding that leads to the presumption that the underlyingmodel is not AR Estimates of the partial autocorrelations for lag 4 andlag 10 lie close to the limits of the ±2 ˆσ band – see figure 1.4 Other
partial autocorrelations appear not to differ significantly from zero If allautocorrelations and partial autocorrelations are deemed not significantlydifferent from zero, then the conclusion would be that the first differences
of income are generated by a random walk model which was estimatedwith the following results:
y t − y t−1= e t + 10.03 s2= 842.
For the first differences of investment, x t − x t−1 – see the plot in
figure 1.2 – the estimated autocorrelation and partial autocorrelationfunctions are given in figure 1.5 The autocorrelations alternate in signand show some significant values for lags less than or equal to 5 whichsuggests an AR model The partial autocorrelation function has a cut-off
at lag 4, supporting the presumption that the model is AR and ing a fourth order AR scheme Also, the partial autocorrelation functionfor the second differences has a cut-off at lag 3 while the autocorrelationsalternate in sign for lags less than 11, findings which support those derivedfrom analysis of first differences In view of these findings, a fourth order
indicat-AR model has been fitted with the data:
(1+ 0.263L − 0.0456L2+ 0.0148L3+ 0.376L4)(x t − x t−1)
Trang 37In contrast to the processes for the first differences of c t and y tin (4.1)
and (4.2), that for the first differences of investment, x t, in (4.3) has an
AR part Thus the requirement of the structural form that all endogenousvariables have identical AR parts of order equal to or greater than that
for x t– see (3.9)–(3.10) above – is not satisfied given the results in (4.1)–
(4.3) Using the notation of table 1.2 with h i j of degree r i j regarded as
an element of H(L) /(1 − L), the degree of the AR polynomial in (4.31)
is r33= 4 while that of the error process is q33= 0 In the case where nocancelling occurs in (4.1)–(4.2), it is clear that the conditions (3.9) and
(3.10) of table 1.2 can not be met Even if h23in (3.8) satisfies h23≡ 0
so that c t and y t are generated independently of x t, the conditions onthe final equations are not met by the results for the final equations in(4.1)–(4.3).13 Thus while (4.1)–(4.3) appear to be consistent with theinformation in the data, they are not compatible with the dynamizedHaavelmo model specified in section 3, (3.2a)–(3.2c)
At this point, the following are considerations that deserve attention:(1) Although the fits of the models in (4.1)–(4.3) are fairly good, it may
be that schemes somewhat more complicated than (4.1)–(4.3) areequally well or better supported by the information in the data andare compatible with the implications of the Haavelmo model Thispossibility is explored below
(2) To compare and test alternative final equations for each variable, itwould be desirable to have inference methods that are less “judgmen-tal” and more systematically formal than are the BJ methods In thenext subsection, we indicate how likelihood ratios and posterior oddsratios can be used for discriminating among alternative final equationmodels
(3) It must be recognized that there are some limitations on the class of
AR models that can be transformed to a stationary process throughdifferencing That is, only those AR models whose roots lie on theboundary or inside the unit circle can be transformed to stationarymodels by differencing Other transformations, say logarithmic, have
to be used for models with roots outside the unit circle
(4) Differencing series may amplify the effects of measurement errorspresent in the original data and seriously affect estimates of theautocorrelation and partial autocorrelation functions Of course, thisproblem arises not only in the BJ approach but also in any analysis ofARMA processes, particularly those of high order
13If h23≡ 0, then (4.1)–(4.2) imply r12+ r21= 1; r12+ q21, q11, q12, r12+ q22 2
(with at least one equality); and r21+ q11, q21, r21+ q12, q22 0 These
condi-tions imply q11= q12= q21= q22= r21= 0, r21= 1, and r21 = 2 which cannot hold simultaneously.
Trang 3820 Arnold Zellner and Franz C Palm
posterior odds
The purpose of this section is to provide additional procedures for tifying or determining the forms of final equations These proceduresinvolve use of likelihood ratios and Bayesian posterior odds After show-ing how to obtain likelihood ratios and posterior odds, some of the resultsare applied in the analysis of Haavelmo’s model
iden-Consider the following ARMA model for a single random variable z t,
whereφ(L) and θ(L) are polynomials in L of degree p and q, respectively.
Assume that theε t’s are normally and independently distributed, eachwith zero mean and common variance,σ2 Let u t ≡ θ(L)ε t Then giventhe “starting values” forε t and z t , ε0and z0, the vector u= (u1, u2 ,
u T ) has a T-dimensional multivariate normal distribution with zero mean
vector and covariance matrixΣ, that is:
p(u |φ, θ, σ2, z0, ε0)= (2Π) −T/2exp|Σ|− 1
−1
2uΣ−1u
, (4.5)
where φ= (φ1,φ2, ,φ p ) and θ= (θ1,θ2, ,θ q) The matrixΣ is a
T × T positive definite symmetric matrix with elements given by:
Trang 39Since the Jacobian of the transformation from theε t s to the z ts is equal
to one, the joint pdf for the z ts, the likelihood function, is:
compu-If we have an alternative ARMA model,
φ a (L)z t = θ a (L) ε at, t = 1, 2, , T, (4.10)whereφ a (L) is of degree p a,θ a (L) of degree q a and the error processε at
If model (4.10) is nested in model (4.4), i.e p a p and/or q a q,
with at least one strict inequality, and under the assumption that (4.10)
is the true model, 2lnλ is approximately distributed as χ2
r with r being
Trang 4022 Arnold Zellner and Franz C Palm
the number of restrictions imposed on (4.4) to obtain (4.10); that is,
r = p + q − (p a + q a) – see Silvey (1970, pp 112–13) In choosing asignificance level for this test, it is very important, as usual, to considererrors of the first and second kind Rejecting the nested model when it
is “true” appears to us to be a less serious error than failing to reject itwhen the broader model is “true” That is, using the restricted modelwhen the restrictions are not “true” may lead to serious errors Use ofthe broader model, when the restricted model is “true,” involves carryingalong some extra parameters which may not be as serious a problem
as giving these parameters incorrect values This argues against usingextremely low significance levels, e.g.α = 0.01 or α = 0.001 Also, these
considerations rationalize somewhat the usual practice of some degree ofover-fitting when the model form is somewhat uncertain More systematicanalysis and study of this problem would be desirable
In order to compare (4.4) and (4.10) in a Bayesian context, we have
to specify a prior distribution on the parameter space In the problem
of comparing nested models, this prior distribution has a mixed form
with weights whose ratio is the prior odds on alternative models – see,e.g., Jeffreys (1961, p 250), Zellner (1971, pp 297ff.), and Palm (1972).Formally, the posterior odds ratio relating to (4.4) and (4.10) is given by:
where K 1ais the posterior odds ratio,Π/Π a is the prior odds ratio, and
(4.14) can be made operational, it is necessary to formulate the prior pdfsand to evaluate the integrals, either exactly or approximately.14
We now compute likelihood ratios to compare alternative formulations
of the final equations of Haavelmo’s model The information in table 1.2and empirical results in the literature on quarterly consumption relationssuggest higher order AR and MA schemes than those fitted in section4.1 For example, a fourth order AR model for the second differences of
14 Note however, as pointed out by Lindley (1961), the likelihood functions in the ator and denominator of (4.14) can be expanded about ML estimates If just the first
numer-terms of these expansions are retained, namely l ( ˆ φ, ˆ θ, ˆ σ |z) and l( ˆ φ a, ˆθ a, ˆσ a |z), and if
the prior pdfs are proper, (4.14) is approximated by:
K 1a = [Π/Π˙ a [l ( ˆ φ, ˆ θ, ˆ σ |z)/l( ˆ φ a, ˆθ a, ˆσ a |z)],
i.e a prior odds ratio,Π/Π a, times the usual likelihood ratio As Lindley points out, additional terms in the expansions can be retained and the resulting expression will involve some prior moments of parameters Thus on assigning a value toΠ/Π a, the prior odds ratio, the usual likelihood ratio is transformed into an approximate posterior odds ratio for whatever non-dogmatic, proper prior pdfs employed.