Methodological Issues in Global Modelling: Structural vs. Data-Analytic Approaches

C LO B A L MO D ELI l\ ( ;

© I F.\ ( : l hll;ttllic \ ICJdell illg- alld COlltro l o t :\a li Olla l Eco ll om ie s , \\'i\ shill gtoll DC . l 'S, \ I q:-t~

( : o p ~ri,L{ h t

METHODOLOGICAL ISSUES IN GLOBAL MODELLING: STRUCTURAL VS. DATA-ANALYTIC APPROACHES S. Schleicher University

of Cra z.

Em ll IJlllics DI'partlllfllt. Grill. Austria

Abstract. After rrore than a decade of global econometric modelling and after rrore than four decades of national econometric modelling the methodological discussion about the parametrization of nonexperimental databases in economics is still controversial. The paper presents an overview about alternative parametrization procedures and suggests an extension of research and reporting strategies in the direction of data-analytic approaches complementary to the traditional structural modelling teclmiques. In addition reporting of sensitivity analyses of the effects of variations in the sample size and alternative structural prior restrictions is proposed. An example of a data-analyti c modelling approach for a national economic database is included.

Keyw:>rds. Fa:>nomics; data reduction and analysis; hierarchical systems; modelling; sensivity analysis.

by a coordination system describing the interaction of national economies via prices, t rade, and capital flONS

S'rnUC'IURE OF GIDBAL ~Cr.rnEIS

(lb)

Fa:>nometric studies of the global interdependence of national economies have been inspired and shaped by the pioneering work of Project LINK (Ball, 1973, waelbroeck, 1976, Sawyer, 1979) which started alrrost 15 years Oommon feature of these global econometric models is a hierarchical system structure linking the subsystem models of the national economies

n

i

i

fj (Yj'Yj'xj,v j ) j = l, ••• ,n .

g(y i ,x i ,xc ,w)

o,

where v. and w denote unobservable stochastic disturb~nces, y and x are observable system variables specified as endogenous rsp. exogenous with superscripts i, n, and c describing their interacting, noninteracting, and coordination system character (Schleicher, 1983).

ago.

(la)

identification;

Meanwhile this typical econometric approach to systems modelling which heavily relies on hypotheses about the internal system structure instead of follONing rather the system science approach of modelling the transfer function of a system has faced a number of critical challenges which will be discussed in th is paper.

0/

205

S. Sch l e i che r

206

ALTERNATIVE SYSTEM PARAMEl'RI ZAT IONS

Alternative methods to parametrize the system which is assumed to generate an observable database D are classified for our purfX)ses into three groups. (1) Parametrization without structural prior information specifies the likelihood function of the parameters P for the database D (2)

1

(P;D)

without imfX)sing any restrictions among the variables of D. As a typical example of this parametrization we may consider a multivariate ARMA-model (3)

A(L).D

= B(L).u

,

the parameter set P consisting of the lag PJlynomials A(L) and B(L). (2) Parametrization with structural prior information dichotomizes the database into endogenous (output) variables Y and exogenous (input) variables X yielding the likelihood tunctioo (4)

1 (P;Y/X) .

A multivariate ARW\X-rrOOel serves as an example: (5)

A(L).Y

= B(L).X

+ C(L).u

If in addition it is assumed that the parameter set P depends only on a smaller number of structural parameters S, (6)

P = P(S) ,

we arrive at what may be cal1e:1 the structural modeling approach of econometrics with likelihood function (7)

l(S;Y/X).

(3) Parametrization with structural and parameter prior information in addition includes nonsamp]e information N about the system parameters represented in the prior parameter density (8)

p(S;N)

yielding because of Bayes theorem the following posterior density for the parameters: (9)

p(S;Y/X,N) = c.p(S;N).l (S;Y/X) ,

c being a normalizing constant.

EXTENSIONS OF RESEAIOi AND REPORTING SI'RATEl3IES

Almost 45 years after the contribution of J. Tinbergen (1939) which initiated national econometric rrOOelling and almost 15 years after the first global econometric model initiated by Project LINK a new discussion 00 the methodology of econometrics seems to emerge which can be traced back to Keynes' a:mnent 00 Tinbergen's work (1940). Still controversial is the parametrization of a nonexperimental database which characterizes data about economic p,enomena. M:>st econometric models both with a national or global scope usa the second parametrization approach, namely parametrizatioo with structural prior information reflecting the authors' beliefs about the interactions between economic variables based 00 an "economic theory". This model building tradition reflects both the need of aciHtional "prior" information because of the paucity of available observations and the arguments against "measurement without theory" as presented by Koopmans (1949) in his debate with Vining. Translated into the practice of global rrodels with several thousams of equations and model variables based 00 time series of usua l.ly not more than thirty observations this means that the structural restrictions put 00 the parameter set almost a:::mpletely determine the system properties of the model whereas the statistical process of parameter estimation is of comparativly minor iffifX)rtance.

Sirns (1981) claillE that this sty1.e Ot "identification" is uncredible since the restrictions imposed on the structure of a model are neither essential nor innocuous and instead proposes an unrestricted autoregressive dynamic system (3), namely parametrization without structural prior information. The procedure proposed, however, suffers at least from two drawbacks. Limitations in degrees of freedom prevent applicatioo exept on very small databases am system resfX)nses with respect to externdl shocks are not unique without prior information about the ordering of the equatioo. It is our oplnloo that given the paucity of nonexperimenta1 databases econometric rrOOelling is inevitably dependent 00 add itional informatioo not contained in the database. In contrast to ITOSt of current econometric practice, however, the interaction between sample and nonsample information should be made more transparent along the following lines:

Me thodo lo gi ca l Issues in Gl oba l Mo de lli ng

First, by methcrls of exploratory data analysis (Mosteller and Tukey, 1977) diagnostics about the database should be reported indicating general cha~acteristics of the data like similiarities and dissimilarit.ies of the variables and predictability from the database alone. Second, sensitivity of the parameter estimates should be reported with respect to changes in the size of the sarrple pericrl by applying recursive stability analysis (Dufour, 1982). These procedures are in adHtion very poNerful against overfitting and offer evidence about the predictive performance of a specified mcrlel equation. 'Third, a general sensitivity analysis of the effects of alternative structural prior restrictions on the datebase should be reported as proposed by Learner (1983). Instead of pretending to offer a "true" rrodel of the real world ecorometricians should feel eoouraged "to develop a oorresPJl1dence between regions in the assumption space and reg ions in the inference sp3.ce". l\o::ord ingly the main job of a researcher v.ould be "to report economically and informatively the mapping from assumptions to inferences".

AN EXAMPLE

we

demonstrate a data-analytic rrodeling approach for a national ecoromic database D containing quarterly data from 1956 to 1981 for sixteen variables from the Austrian national income accounts. All variables except for the interest rate are transformErl into relative fourth differences.

In the first step we reduce the dimension of the database matrix D with K oolumns by calculating princip3.1 oomPJ!1ents p. which by definition are orthogonal and of l~ngth one: (lOa)

Pi'Pi

1

i

(lOb)

p. 'po

0

i,j

1

J

i

1, ••• ,K

i = 1, ••• ,K ,

though the dimension of the matrix DD' is ccnsiderably larger (Theil, 1971).

Table 1 shows for the principal components the oorresPJl1ding eigenvalues the sum of which is equal to tr (D'D). Therefore the size of an eigenvalue is an indicator for its relative importance in the explanation of the variance of the database matrix D. Table 1 makes evident that the principal component belonging to the largest eigenvalue alone explains 70 percent and the first four principal oomponents belonging to the four largest eigenvalues expJain 94 percent of the variance of the database. This result confirms the proposition that very few ind ices, in our case expressed as princ ipal components, are sufficient to explain a multidimensional macroeconomic database. what extent the individual principal con]Xlnents approximate the individual variables of the database can be seen from Table 2. 'Thus com]Xlnent one is sufficient to explain 92 percent of the variance of real GDP. If by lcoking at the columns of this table each principal com]Xlnent is analyzErl for its relative im]Xlrtance for the variables of the database an ecoromic interpretation of the individual principal oom]Xlnents is suggestErl: com]Xlnent one summarizes real activity of the economy, oomPJ!1ents tv.o and four domestic and foreign prices, com]Xlnent three monetary ]Xllicy, oomponents six and ten fiscal ]Xllicy, and components five and seven the specific shape of the series for inventory changes. Thus it becomes evident which variables in the database show similarities and which contain atypical movements. 'Tb

Each time series of the database can be represented by a regression on principal canPJl1ents. Table 3 indicates that very few princip3.1 com]Xlnents suffice to obtain a gcod approximation of the time series of the database.

1, ••. ,K

=j

These princip3.1 components are obtained ty solving the eigenwert problem (11)

i = 1, .•• ,K ,

where ei denotes the i-th eigenvalue and ai the corresponding (not normalizErl) eigenvector. The i-th princip3.1 oomponent results fron (12)

(13)

207

i

= 1, ••. ,K

•

Alternatively this princip3.1 component can be obtained directly from the eigenwert problem

In the second step we attempt to parametrize the individual principal components by time series methods. AR(4)-rrodels are specified for each principal com]Xlnent and the results are presented in Table 4. The coefficient of determination (R2) indicates to what extent this parametrization succeErls. A rather weak fit of the time series rrodel for a principal component evidently signals that those . variables which are strongly effected by thlS oomponent can not be well predicted by the information oontained in the database. With respect to the considerErl database these variables therefore should be rather considerErl as exogenous.

208

S. Schleicher

Eiqenvalue

Principal comnonent

PercentaQe of

explainrd variance

Tab!e 1 Eigenvalues of principal components

PI

68383.60 1

70.09

r2

11336.854

11.62

r] P4

84%.874

8. Il

3561.871

3.65

r5

2211.595

2.27

r6

1532.18 1

1. 57

P7 P8

556 . 733

0.57

43 2. 478

0 . 44

P9

402.502

0.41

rlO

263.277

0.27

P 11

203.741

0.21

P12

166.793

0.17

n13

13.409

0.0 1

P14

2. 438

0.00

P15

0.564

0.00

P16

0. 187

0.00

Princ i ra 1 component

C3tabase variable

I

2

3

4

5

7

6

8

9

la

c 0.984

Sr:J Ss :::.mestic product Pr; lIa t~ consumpt i ') n

0 . 924

0.008

0.012

0 .002

0 . 007

0.014

0 . 014

0.001

0.001

0.007

0.810

0.000

0.002

0.010

0 . 007

0.055

0.094

0.017

0.000

0.003

O. ,98

GC',er~ent

0 . 633

0.09 4

0.067

0.001

0.000

0.009

0.054

0.055

0.000

0.037

0.950 0 .99S

consumntion

0.644

0.229

0.004

0.027

o. ~94

0.001

0.001

0 . 000

0.000

O.COO

0 . 03:)

0.246

0.038

0.017

0.286

0.0 14

0.169

n.033

0.023

0.017

0.873

E .'(~orts

0 . 675

0.080

0.15 2

0.080

0.004

0.009

0.000

0.000

0.000

0 . 000

0 .999

Gross

~j.(ed

r~'1ent:ry

capital formation

chanQes

I-:!)or!s

0.804

0.126

0 . 027

0.002

0.040

0.000

0.000

0.000

0 .000

0.000

0.999

·. . :ney "1 (current "rices)

0.710

0.001

0.264

0.003

0.000

0.020

0.000

0.000

0.000

0.000

0.998

'loney "' 1 (constant "rices) Di sDos,~le income

0.3 18

0.070

0.439

0 .095

0.000

0 .022

0.000

0 .000

0.003

0.000

0.997

0 . 762

0.000

0.016

0.035

0.002

0.141

0 .003

0.002

0.02 1 0.015

0.997

Tax ra:e

0.028

0.041

0.247

0.089

0.009

0.283

0.000

0.029

0 .018

0.173

0.917

0.796

0.142

0.017

0 . 008

0.00 1 0.014

0.000

0.011

0.000

0.005

0.994

0 . 734

0 . 152

0.019

0.05 1 0.00\

0.000

0.001

0.008

0.021

0.000

0.987

I:noort :;ri ce index

0.233

0.362

0.047

0.250

0.00 1

0.051

0.002

0.005

0.047

0.002

0 . 999

199regate dema nd or"ic e ; ndex

0.71 1 0.170

0.002

0.098

0.00 1 0.003

0.00 1 0.002

0 . 004

0.001

0.993

[n~eres:

0.787

0.023

0 .005

0.002

0.r08

0.009

0.005

0.994

:~a

qe

G~ P

r~ ~e pr~::e

; ndex

rate

0. 130

O. OOB

0.017

Table 2 Percentage of variance by first ten principal components

Table 3 Approximati on of database variables by principal components

R2

Database variab l e

Pr; ne i Dd 1 comnone nt

(;ross domestic produc t

1

Pr; va te consumpt ion

I

7

r.904

Government consum!1t i on

I

2

0.727

(;ross fixed capital formation

1

2

Inventory Chi'! nl'Jes

5

2

Exrorts Importc;

1

3

I

2

n.930

Honey HI (curre nt nricec;)

1

3

0 . 974

'1oney '11 (co nsta nt orices)

3

I

0.807

Disrosab1e income

I

6

T.:u r" te

6

3

0.924

0.873 7

0.7a2 0 . 827

10

n. 9~3 r. 7~3

WaQe ra te

I

2

n.938

GOP price index

I

2

0.885

Imnort nrice index

2

4

Aqoreqa te dema nd price ; ndex

I

2

0.882

Tnterec;t rate

I

2

0.917

1

0.845

Me thod ol og i c al Issue s 1n Glob a l Mode lling

209

Pr i nc i pa 1 COlllpon~n

t

Pi. t- I

Pi I t - 'l.

u i • t-)

Pi, t-4

(onsldnt

R?

PI . t

0.531

0. 173

0.177

- 0.278

- 'J. 034

0 .446

P2. t

0.561

0. 11 9

0.381

-0.429

- 0.012

0.561

P3. t

0.742

0. 154

0 .109

-0.407

- 0.006

0.700

P4. t

0.532

0.273

0.316

- 0.5 15

1l.002

1l.609

P5. t

0.539

- 0 . 0 12

0 . 177

-0.297

0 .001

0.349

P6. t

0.604

- 0.054

0 .1 68

-0.4 31

0 . 007

0.488

P7 • t

0 .1 86

0. 140

0.253

- 0.380

-0.009

0 . 221

0.367

0 .073

0.345

-0 . 2j7

-0.007

0.284

P9 . t

0.313

0. 122

0.195

- 0.280

0.0 10

0 . 212

PlO. t

0.586

0.2 15

-0.007

-0.152

- 1l.OO I

0.488

PS . t

Tabl e 4 AR(4) - models of f irs t t e n principal compon e nts

Re 1a t i ve Endogenei ty

Database va ri ab I e

0.910

W<,lqC tJ le

Gr oss domestic product

0 . 906

Interest rate

0 . 885

Money MI (c ur r en t prices )

0.874

oi sposab 1~

income

0.865

Agq r-ega t e demand price i ndex

0.8 55

GDP pr ice index

0.854

Impo rt s

0 .854

Priv ate consump ti on

0.788

Gross fixeJ cap i ta l f orma tion

0 .733

Money MI (co nsta nt 9r i ces)

0.725

Gove rnment co n sumpt i on

0.682

Import pr ice i ndex

0.649

Ex port s

O. t.33

Tax ra te

0.490

J nventor y chanqes

0.384

Ta b l e 5 Meas ures for r e l a t i ve e n dogenity for sel ected Austrian economic da t a

Summing up the data-analytic macroeconometric rrodel based on priocip3l
Table 5 summarizes in increasing order the goodness of fit for the database variables as approximated by the data analytic model.

(A) ~he princip3l oomponents Pi, i 1, ••• ,K, obtaIned frcrn the database D I:Jf (11) am

Defining exogeneity of a variable as predictability frcrn a given database, this table can be oonsidered as a stochastic measure of end:>geneity respectively exogeneity of the time series analyzed.

(12) ,

(B) the AR(4)-models for the princip3l oomponents

(14a)

= 1, ••• ,K

i

,

(C) the representatioo of the database variables d j by (a few) princip3l oomponents Pj

(14b)

j

= 1, .•• ,K

•

If we substitute the forecasts for the princip3l oomponents produced by the autoregressive rrodel (14a) into the rrodel for the database (14b) then we obtain predictions for the various variables of the database which again can be oonpared with their actual values.

The results may be surprIsIng in the oontext of textbook macroecoocmics, they are not Unplausible, however, in the light of the peculiarities of Austrian ecoocmic PJlicy. MOst endogenous in the sense of predictability is the wage rate, followed by GDP and interest rate together with nominal noney stock. At the end of the scale the most exogenous variables can be found, namely public consllllPtioo, imPJrt prices, eXPJrts, tax rates, and as a curiosity inventory changes because of the ioc1usioo of statistical discref~ncies in this variable.

2 10

S. Schl e i cher

OJNCWSION

After more than a decade of global econometric modelling activities both successes and failures should encourage a number of extensions in research strategies: (1) Data-analytic approaches which provide insight into the information structure of the data should be considered as
Ball, R.J., ed . (lq73\. '!he International Linkage of National Eronomic M:lde Is . J\msterdam et al . Dufour, J.-M. (19821. Recursive Stability Analysis of Linear Regression Relationships. ~~,:!!:~~. of ~metr~~s, ]1., 31-76. Herx1ry, O. F. (1980). Eronometrics - Alchemy or Science? EkX>nomica, 1.7, 387-406. Hickmann. B.G •• Schleicher , S. {1978) . 'n'le Interdependence of National Eoonomies and the Synchronization of Economic Fluctuations: E.Vidence from the LINK Project. Weltwirtschaftliches Archiv . 114,

(3) '!he fact that inferences drawn from structural econometric models rely heavily on the structural prior restrictions should be made more transparent by reporting the sensitivity of the model properties with respect to alternative model structures.

Keynes. J.M. (1940) . Professor Tinbergen's Method . EkX>nomic Journal, 49, 558-568.

(4) In general it should be observed that all diagnostic tests made or inferences drawn from a model are useless without making explicit alternative hypotheses or the sensitivity of the inferences with respect to sample size and prior structural restrictions.

Learner, E. E. ;'1983). Let's Take the Con out of Eronometrics. 'Ihe American Eronomic Review. 73. 31-43 .

642-708.

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Kcopnans. T.e. (1947). Measurement without Theory. Rev iew of Eronomics and Statistics, 29. 161-179.

M::steller, F., 'l\.Jkey, J .W. (1977). Data AnalY§i~ _.and ~!,~s_l:?~on. Read ing. Sargent , T.J., Sims, C.A. '1977). Business CYCle M:xleling without Pretending 'lb Have Too Much A Pr ior i Economic Theorv. In: C.A. Sims (ed.) , New Methods in Business Cycle Research. Minneapolis . Sawyer, J . A., ed. (1979). fobdeling the International Transmission Mechanismn. J\msterdam et al. Schleidler, S. (1983\. Forecasting Theory for Hierardlical Systems with Applications to M.1lty-nometrics - Essays in Honor oTLaWrence R. Klein. Boston et al. Sims, C.A. (1980). Macroecon::mics and Reality . Eronometrica. 1.8, 1-48 Tinbergen, J. (1939\. A Method and its Apolication to Investment Activity . Geneva. waelbroeck, J . L., ed. (1976). 'Ihe r-tldels of Project LINK. J\msterdam et al. Zel1ner, A. (1979) . Statistical Analysis of Eronometric M:rlels. Journal of the American Statistical Association, 74, -628-641.