Multivariate subset autoregressive modelling with zero constraints for detecting ‘overall causality’

Journal

of Econometrics

24 (1984) 311-330.

North-Holland

MULTIVARIATE SUBSET AUTOREGRESSIVE MODELLING WITH ZERO CONSTRAINTS FOR DETECTING ‘OVERALL CAUSALITY’

J.H.W. PENM and R.D. TERRELL* The Australian National University, Canberra, A.C.T. 2601, Australra Received

February

1983, final version received July 1983

The necessary and sufficient condition to test for ‘overall causality’, i.e.. the presence of Grangercausality and instantaneous causal relations, in a bivariate and trivariate autoregressive model with recursive form is discussed. It is argued that the conventional AR model (the reduced form AR) is a more straightforward and effective means of testing for ‘overall causality’. To detect instantaneous causality it is proposed to select the best subset system in a residual regression system in conjunction with model selection criteria. The Canadian money-income-bank rate system is re-examined in this way and by using a previously proposed algorithm we identify the optimum multivariate subset AR with constraints to detect whether there is ‘overall causality’ in that system.

1. Introduction

In a recent paper Ltitkepohl(l982) took a multivariate autoregressive system with a special a priori structure for the current matrix of coefficients to analyze Granger causality in a trivariate system linking the Canadian money, income and bank rate series. With X, a zero-mean, wide-sense stationary time series of dimension m, he considered the AR system of the form

whereA,,n=0,1,2 ,..., p, are m X m parameter matrices. The a priori condition is that A,, is upper triangular with unit diagonal and E{ {,c;_ 7} = 52 if T = 0 and = 0 if T > 0, where D is a diagonal matrix. In this paper we refer to the model (1) as a recursive AR form. Lutkepohl then studied a recursive AR form with m = 3 to find Granger-causal relations among those three economic variables, but neglecting to establish the instantaneous causal relations. The algorithm he followed to test Granger-causality patterns can be very mislead*The authors would like to thank one anonymous improvements in this paper. 0304-4076/84/$3.0001984,

referee for helpful

Elsevier Science Publishers

suggestions

B.V. (North-Holland)

which led to

312

J. H. W. Penm and R. D. Terrell, Multiounute A R modeling

ing, because economic time series may not be observed so frequently that one can ignore the instantaneous relations. Hatanaka (1982) has supported this viewpoint. It also seems to us that the conventional vector AR form provides an effective and straightforward test of Granger-causality including the instanto ‘Grangertaneous relations, i.e., ‘overall causality’, which is equivalent causality at all’ as defined by Hatanaka (1982) (or = in the symbols presented in section 4). The conventional AR form (often referred to as the canonical or AR reduced form) is

X,+

f: B,,X,_.=q. II=1

(2)

where B,, n = 1,2,. . ,p, are m x m parameter matrices, and E{ e,.$ 7 } - G if T = 0 and = 0 if 7 > 0, where G is a general matrix. The reason that the reduced form AR is more attractive for testing Granger-causality and for testing whether one vector of variables is an ‘overall’ cause of another set of variables [see Hatanaka (1982)] is given in section 2. Section 3 presents a proposed algorithm to select the subset AR with zero constraints and so to detect whether there is ‘overall causality’, i.e., Granger-causality and/or instantaneous causal relations. To establish whether there is instantaneous causality it is proposed to select the best subset in the residual regression system by using OLS estimation in conjunction with the model selection criteria. Section 4 provides the results of re-examining the Canadian money, income and bank rate series [investigated by Liitkepohl (1982)] using the proposed procedures, and compares the results with those found by analyzing only the money and income series. Section 5 contains brief concluding remarks. 2. Granger-causality 2.1. Bivariate

patterns in AR forms

block AR forms

If we partition X, into [Xi, 1X2,]‘, where Xi, consists nents of X, and X,, of the remaining m - n components, form of (2) becomes

and G’=

G,, [ 61

1

G,, 6, .

of the first n compothen the reduced AR

J. H. W. Penm and R. D. Terrell, Multivariate

AR modeling

313

Granger (1963, 1969) Sims (1972, 1980), Caines (1981) Geweke (1978, 1981) Hsiao (1979a, b, 1980, 1981) and Tjsstheim (1981) have shown that X,, does not cause Xi, in the Granger sense if and only if B,,, n = 0, n = 1,2,. . . ,p. Also X,, does not Granger-cause Xi, instantaneously if and only if G,, = G,, = 0 (or ‘overall’ if B12,n = 0, n = 1,2,. . . ,p, and G,, = G,, = 0). We now return, to the recursive AR form of (1) with the representation

and

4, o

i-2=

0

D

[

22

1 .

The equivalent reduced AR form from (4) can be obtained by multiplying (4) with

1 1 42,o

[0

I

-l

’

which becomes

It should be noted that now X2, does not cause Xi, in the Granger sense if and only if A12,n- A12,0A22,n= 0, n = 1,. . .,p, in (5), i.e., X,, does not cause Xi, in the Granger sense does not imply that A12,n= 0, n = O,l,. . .,p. The condition on the current matrix A,,,, = 0 and A,,, n = 0, n = 1,2,. . . ,p, in (4) is only sufficient but not necessary for the absence of causality from X2, to Xi, in the Granger sense. 2.2. Trivariate block system If we partition X, into [Xi, X2, X3,]‘, the reduced form of (2) in trivariate block system is

J.H. W. Penm and R. D. Terrell, Multivariate AR modeling

314

and

G,, GI, 1 Gl3

G= [

G,, G,,

G23

G31

G33

G32

.

By multiplying (7) with the inverse of the coefficient matrix of X,, (7) becomes

II I I Ill Xl,

x2,

+f:

n=l

x3,

X 1rmmn

x

&F,

*

*

*

*

*

*

*

43,n

-

A12,OA23.n

+

M*.0A23,0

-

&,OL433,n

l*11

=

X 3t-”

*

[2*,

)

Lc* 31

with the residual variance - covariance matrix *

*

M,,OA23,0

-43,0w33

*

*

-A 23.0 il 33

*

*

*

(8)

where we only explicitly display the significant information which is necessary for the following explanation. We use t,‘;, i = 1,2,3, to represent the new residual vector after multiplication, and denote by * each remaining element of the matrices. Consider a particular case. Hsiao (1981) has shown X3, does not Grangercause XI, if and only if B,,, n = 0, n = 1,2,. . . ,p, in the reduced AR form of (6). Notice that the necessary and sufficient condition for X,, does not Granger-

315

J. H. W. Penm and R. D. Terrell, Multivariate AR modeling

cause Xi, in the recursive AR form becomes A 13,n -

A12,0A23,n +(A12,0A23,0

-A13,0)A33,.

= '5

n=l

,.*-,

P.

The condition A13,n = 0, n = O,l,..., p, is now neither a sufficient nor a necessary condition for the absence of causality from X3, to X1, in the Granger sense. In the instantaneous causality case, specifically X,, is not instantaneously caused directly by X,, if and only if G,, = 0, or alternatively G,, = 0, in (6). Accordingly this means (A,,,,A,,,, - A13,0) = 0 in (8). Note that A13,0 = 0 is neither a sufficient nor a necessary condition for Xi, not to be instantaneously caused directly by X,,. From the above observations it is clear that testing for Granger-causality and ‘overall causality’ from the recursive AR form is complicated and is increasingly difficult for m 2 3. Alternatively it is apparent that the reduced AR form (often referred to as the canonical or AR reduced form) is a more straightforward and effective means of testing for ‘overall causality’. 3. Methodology Three model selection criteria are employed by Penm and Terre11 (1982a) to identify the multivariate subset AR and these have again been used in selecting both the order and the most appropriate subset AR with constraints. They are AIC = log($;,( + [2/N]&

(9)

HC=log@J+[2loglogN/N]S,

(10)

SC = log@

(11)

+ [log N/N]&

where N is the sample size, S is the number of functionally independent parameters estimated, and (?p is the estimated residual variance-covariance matrix in the testing AR model. The interested reader is referred to Akaike (1973), Hannan (1981), Hannan and Quinn (1979), and Schwarz (1978). The procedures to select the optimum AR model are then summarized in the following steps: (I) To assign a maximum lag q. A maximum lag q is assigned so that one is confident that the order of the true model is less than this maximum lag q, i.e., 4’P. (II) To select the optimum subset AR for each criterion. The subset AR models include the full order AR models, and the AR models with intermediate lags constrained to zero matrices. The subset AR form with the

316

J. H. W. Penm and R. D. Terrell, Multivariate AR modeling

deleted lags i,, i,, . . . , i, has the representation

n=l

where Z, represents an integer set with elements i,, i,, . . . ,i,, 1 I i, I i, I i, I q and B,(I,) = 0 as n E I,, or 1, represents an empty set where no deleted lag exists. We then use a leaps and bounds algorithm based on Furnival and Wilson’s (1974) search principle to search for the ‘best’ AR model of size k, where k is the number of lags with non-zero coefficient matrices, k = 1,2,. . . , q, without evaluating all possible subset AR models. We then select the optimum subset AR for each criterion. Now Gp in (9)-(11) should be &,,(I,), which represents the estimated residual variance-covariance matrix of the testing subset AR model. The interested reader is referred to Penm and Terre11 (1982b), and to section A.1 of the appendix for a discussion of the multivariate subset AR model. (III) To select the optimum subset AR with zero constraints for each criterion. After selecting the optimum subset AR of (12) for each criterion, we now consider the zero constraints in the remaining matrices B,(I,). n P I,, of (12). For example, we have a bivariate subset AR with lags 1.3 of the form

I[

b,,(l) x,(t) x,(t) + ha(l) [

bdl) b,,(l)

I[

b12(3) M3)

Xd- 1) X,(t- 1)1 (13)

then some entries b,,(7), the (i, j)th entry of B, may be constrained to zero, there are eight entries, so the possible models are 2’ = 256 different models. Therefore by again using a leaps and bounds algorithm the ‘best’ multivariate subset AR with constraints (i.e., allowing zero elements in the matrices included in the optimum subset AR) is found for each size k, k = 1,2,. . . , S. Here, S is the number of functionally independent parameters arising from the optimum subset AR obtained in (II). We then search over these S parameters and calculate each of the proposed criteria for the sequence of ‘best’ k-subsets, to obtain the optimum multivariate subset AR with constraints for each criterion. Also the Gp estimate in (9)-(11) is the residual variance-covariance matrix for the subset AR with constraints in each case. The interested reader is again referred to Penm and Terre11 (1982~) and a discussion of the fitting of the multivariate subset AR with zero constraints is provided in section A.2 of the appendix.

J. H. W. Penm and R. D. Terrell. Multioariate A R modeling

317

In discussing instantaneous causality, (IV) Detecting instantaneous causality. we have to consider possible zero constraints on G (see section 1). Consider a bivariate system for (12) with

G,=

[“E’Rf*].

We have to investigate the following cases: (1) Xl(t) causes X2(t) instantaneously if and only if c Z 0, i.e., lGzl = g,,g,, C2.

(2) Xl(t)

does not cause X2(t) IG,l= g,,g,,.

instantaneously

if and only if c = 0. i.e.,

Notice that Xl(t) causing X2(t) instantaneously is equivalent to X2(t) causing X,(t) instantaneously. Also the number of functionally independent parameters, S, for each criterion is reduced by one for case (2) compared to case (1). Therefore we compare each of the proposed criteria for cases (1) and (2). We detect instantaneous causal relations from the case which minimizes the criterion. We can detect instantaneous causality in a higher-dimensional AR (say m 2 3) in the same manner. In a trivariate system of (12) with

G,=

g11

e

f

e

g22

h

I f

h

833

1 9

we shall have eight different cases to investigate to establish the pattern of instantaneous causality: (a) Xl(t), X2(t) and X3(t) are mutually instantaneously directly caused if and only if e f 0, f # 0 and h f 0 in G,, i.e., IGjj = gllg,,gj3 - h2gll -f 2g2, e2g,, + 2hef.

(b) Xl(t) and X2(t) are instantaneously indirectly caused via X3(t) if and only if e = 0, i.e., ICI = gllg22g33 - h2gll -f 2g22. Also, the number of functionally independent parameters, S, is now reduced by 1 compared to case (a).

(c) Xl(t) and X3(t) are instantaneously indirectly caused via X2(t) if and only

if f = 0, i.e., lG31=gllg2,g33 - h’g,, - e2g3,. S is again reduced by 1 compared to case (a).

(d) X,(t) and X,(t) are instantaneously indirectly caused via X,(t) if and only

if h = 0, i.e., ICI = gllg2,gs3 -f 2g22 - e2g3,. S is now reduced by 1 compared to case (a).

J. H. W. Penm and R. D. Terrell, Multivariate AR modeling

318

(e) Xi(t)

does not cause instantaneously X2(t) and X3(t) if and only if e =f= 0, i.e., lGsl = gllgz2g33 - h2g,,. S is now reduced by 2 compared to case (a).

(f) X2(t) does not cause instantaneously X,(t) and X3(t) if and only if e = h = 0, i.e., IG31= g,,g,,g,, - f 2g22. S is again reduced by 2 when compared with case (a). (g) X3(t)

does

not

cause

instantaneously

Xi(t) and X2(t) if and only if Also, S is now reduced by 2 comf = h = 0, i.e., lG31 = gllg22g33 - e2gj3. pared to case (a).

(h) Xi(t), X1(t) and X3(t) are independent if and only if f= h = g = 0, i.e., IGjI = g,,g22g33. Now S is reduced by 3 when compared with case (a). Again we evaluate the proposed criterion for the above eight cases and select that case which minimizes the criterion and so detect the pattern of instantaneous causal relations in a trivariate system.

4. Empirical results The quarterly, seasonally adjusted Canadian bank rate (BR), money (M2) and income (GNP) for the period 1955.1 to 1977.IV are re-examined. In the following analysis the variables actually used are Xi = A log GNP, X2 = A log M2, and X, = log BR, where A is the difference operator. All results of the examination reported below are performed on a Univac 1100/80. First we use Forsythe’s (1957) method for generating orthogonal polynomials to assess the raw data for suitable trends. The results of these regressions are reported in table 1. The standard errors of estimate of coefficients show that detrending using a 1st order polynomial is required before fitting the multivariate subset AR with zero constraints. After detrending, we then use the approach proposed in section 3 to select the optimum subset with constraints. First we assign 4 = 12 to search for the best subset AR,’ then select the optimum subset AR with zero constraints for each criterion. The results are provided in table 2. Zellner (1962) indicated that the GLS estimates of coefficient matrices with zero constraints are more efficient than the LS estimates as the regressors in each equation are no longer necessarily the same, so we use the estimate of the variance-covariance matrix (?,,(I,) of the optimum subset AR to approximate the true variance-covariance matrix for each subset AR with constraints.

‘One possible approach is to use the classical sequential method proposed in Penm and Terre11 (1982a), i.e.. choose y XZ-p, and use each criterion to select the best full order model among all full models with the order O,l, 2, , q. Set the value of p equal to the order of the chosen full order model for each criterion. Here we provide the results by assigning 4 = 12, and then go directly to step (II) proposed in section 3 to select the optimum subset AR for each criterion.

319

J. H. W. Penm and R. D. Terrell, Multivariate AR modeling Table 1 Orthogonal

polynomial

regression

on testing data for examining

Intercept

Orthogonal

stationary.a -_____ polynomial P,

Alog GNP

2.26 E-02 (1.36 E-03)

4.12 E-03 (1.17 E-03)

A log MZ

2.35 E-02 (1.26 E-03)

X.61 E-03 (1.08 E-03)

log BR

1.586 (2.40 E-02)

0.303 (2.068 E-02)

‘The cients.

values in parentheses

are the standard

errors

of estimates

of coeffi-

Hence we must find k such that $( Z,) = kk’, and then after premultiplication of the vector X(t) by Z? -l, least squares methods will produce the ‘feasible’ GLS estimates. Therefore by again using a leaps and bounds algorithm, the ‘best’ multivariate subset AR with constraints is found for each size k, k = 1,2,. . . , S. Here S is the number of functionally independent parameters arising from the optimum subset AR obtained in step (II) of section 3. We then search over these S parameters and calculate each of the proposed criterion for

Table 2 Summary

Order (k) 1 2 3 4 5 6 7 8 9 10 11 12

of results

for selection of the best AR for Canadian money-income-bank data by assigning the maximum lag q = 12.”

The best subset of size k based on the generalized 1 1 1 1 1 1 1 1 1 1 1 1

3 3 3 3 3 3 3 3 3 2 2

4 4 4 4 4 4 4 4 3 3

Criterion AIC HC SC aCPU time: 25.664 sec. The number models: 317/4096.

11 10 9 5 5 5 5 4 4

11 10 9 6 6 6 5 5

11 10 9 8 7 6 6

11 10 9 8 7 7

11 10 9 8 8

residual

11 10 9 9

11 10 10

11 11

rate

sum of squares

12

The optimum model selected 1 3 4 1 3 1 of the models

tested / the number

of candidate

320

J. H. W. Penm und R. D. Terrell, Multiuuriute AR modelirlg Table 3 The optimum

subset AR with zero constraints X,=[AlogGNP,.3logM.?,.logRR,]‘.

Criterion The optimum

AIC subset AR

1

Estimate

3

4

LS 0 0

The type of coefficient matrices selected

r

GLS

0 0.3x2 (0.0962) 0

b, I

-4.11 (1.308) -0.1~4

I

0 0

0.741 (0 0717)

0

r

0

(0 (KKl7) 03.05 (1.3616)

0 0.013

I 1

0

0

0 0.399 (0.0961) 0

0 234 (O.OY67)

0

0

I

0 0.162 (0.0866) 0.756 (0 077X) 0

0

0 0

0.734 (0.0416)

0

0 i

0.135 (0.0794)

0

L

0 1.36 ().1%X -6.21

of

0 (().1(X)3) 0.345 0

0.Y51 04.35 lxx

(0 0060) 0.017 0~~~ ~j II

0.305 (0 OY5U)

0

0

lY7Y7 f2’?

CPU time

13 7 005 SW

35Y.151 \cc

Patterns of causality

A log (i.4 P 8 \ log RR Q _I log \I’

I

“The values In parenthchcs the optimum AR model

the sequence of with constraints constraints are neous causality section 3. The tables 3-5 and

0.23x (0.0694) 0.226 (0.1056) 0

4 35 ‘lY.2 (>71 I

lj30/2~'

I

I

0.217 (0.1160)

0

1

&

No of subset5 checked Candidate subsets

0 0.01 I II ((1.0055)

0.01 Y (0 0054)

0

(O.OYX3)

b

The estimate GbvLS timis 10 ’

selected for AIC.”

are the standard

errors <>fcstimatc\

_I lop

Y

c;.\1’ \

lop RR - _\ log ,%I-’ of the nonhero

cc,cHicicnts

from

‘best’ k-subsets, to obtain the optimum multivariate subset AR for each criterion. The resulting optimum subset AR with also summarized in tables 3 through 5. Finally the instantais investigated by using the procedures proposed in step (IV) of detected instantaneous causal relationships are reported in we use the following notation proposed by Harvey (1981) to

‘There is no proper solution on the confidence intervals for the estimated coefficients of the optimal model. This has led the authors to provide only the standard error of estimate of the coefficients from the optimal model because the provision of t-statistics would be misleading. In fact the use of criteria as a basis for the choice of model can be seen as a way of avoiding the very difficult question of what is the appropriate distribution for the coefficient estimates in a causal investigation.

321

J. H. W. Penm and R. D. Terrell, Multivariate AR modeling

Table 4 The optimum subset AR with zero constraints selected for HC. X,=[AlogGNP,,AlogM2,,logBR,]‘. HC 1 3

Criterion The optimum subset AR Estimate The type of

0

coefficient matrices selected

0

B3

0

- 0.323 (0.1019)

GbyLS times 10 --4 The estimate of No. of subsets checked [ Candidate subsets CPU time

0

0

0

0

- 0.741 (0.0716)

- 0.184 (0.0984) 0

0

0.019 (0.0054) 0

1

0

- 3.05 (1.3616)

0.210 - 6.21 1.36

1.13 - 4.72 0.210

0

0.013 (0.0052)

-4.11 (1.309)

0

Patterns of causality

GLS

LS

0

0

-0.272 (0.0967) 0 0

0

0

-0.347 (0.0972)

0.162 (0.0726) -0.164 (0.0782)

0 0 - 0.247 (0.0968)

9.228 (0.0693) 0 0.141 (0.0792)

- 4.72 229.2 - 6.21 I

210/21x

501/2”

2.210 set

4.931 set

A log GNP

A log GNP r/ \ log BR - A log M2

0 \ log BR - A log M-’

present the causal relation between x and Y: Description

(1) instantaneous causality only

(2) x Granger-causes y only and not instantaneously (31 x Granger-causes Y only and instantaneously (4) feedback, not instantaneously (5) feedback and instantaneous causality, i.e., ‘overall causality’ (6) no causal relation

Notation

(x -Yl (x -+Y) (x2-Y)

(x “Y) (x _Yl (x- * -Y)

The first difference of the logarithm of a set of variables may be thought of as variables expressed in percentage changes. In looking for the causal relations among such variables, for LS estimates, both AIC and HC select a model which indicates instantaneous causality and feedback relations exist between the pair of A log GNP and log BR and the pair of A log M2 and log BR, and

J. H. W. Penm and R. D. Terrell, Multrvariate A R modeling

322

Table 5 The optimum

subset AR with zero constraints X,=[AlogGNP,,AlogM2,,logBR,]‘. SC

Criterion The optimum

1

subset AR LS

Estimate

1.66 0.117

No. of subsets checked Candidate

CPU time

(0.1019) 0

-4.72 (2.6516)

The estimate of GbyLS times 10 4 subsets

1

- 4.63

GLS

0 - 0.323

0 0

The type of coefficient matrices selected

1

-

0 0

0

0.0133 (0.0054) - 0.75 (0.0732)

0.117 1.13

~ 4.63 ~ 4.86

4.86

242.3

Ii

0

- 0.371

-0.269 (0.1043)

(0.0954) 0

25/512

0.202 set

0.1X0 set

log BRZ

A log M-’

0

0.176 (0.0743) - 0.711 (0.0726)

1

I

28/512

A log GNP Patterns of causality

selected for SC

A log GNP log BR 2 A log M?

only instantaneous causality exists between A log GNP and A log M2. For GLS estimates, AIC and HC favour a model which shows feedback between A log M2 and log BR, one way causation from log BR to A log GNP, and instantaneous causality among those three variables. However, SC supports a model which has one way causation from log BR to A log M2, instantaneous causality between log BR and A log M2, and one way causation from A log GNP to log BR in both LS and GLS estimates. Following Ltitkepohl (1982) we also examine the money-income relationship in a bivariate system. Now the variables are Xi = A log GNP and X, = Alog M2. Again we assign q = 12 and go through the procedures in section 3 to select the optimum AR model for each criterion, then detect causal relations from the optimum model. The results are provided in tables 6, 7 and 8. For GLS estimates, both AIC and HC select a model which shows feedback and instantaneous causal relation between A log GNP and A log It42. Using LS estimates, AIC also supports the above-mentioned model, whereas HC favours a model which indicates one way causation from A log M2 to A log GNP, and instantaneous causality between these two variables. Yet in the trivariate analysis, there is no direct Granger-causation between A log GNP and A logM2 although there is a link via instantaneous causality for AIC and HC. Based on the terminology introduced by Hsiao (1982) we do however find indirect Granger-causation and instantaneous causality holds between A log GNP and

J. H. W. Penm and R. D. Terrell, Multivariate A R modeling

323

A log M2 via log BR in LS estimates, and indirect causation from A log M2 to A log GNP via log BR for the GLS estimates. It is not our purpose at this stage to link our results to particular economic theories but to illustrate how this analysis can provide insights into indirect Granger causation. We conclude that ‘overall causality’ exists (directly or indirectly) between A log GNP and A log 442 in bivariate and trivariate analysis for AIC and HC. For the SC criterion, there is no causal relation between A log GNP and A log M2 in the bivariate analysis. However, in the trivariate analysis, it appears that there is causation from A log GNP to A log M2 via log BR. We are cautious about the different causal relations detected by SC because of its apparent propensity to underspecify the order of the autoregressive model [see Penm and Terre11 (1982a)J. We would wish to emphasize that more detailed information is available through an approach which establishes instantaneous causal links and through the use of a higher-dimensional system.

Table 6 The optimum

subset AR with zero constraints X,=[AlogGNP,,AlogM2,]‘.

Criterion The optimum

selected for AK.

AIC subset AR

1

Estimate

3

4

LS

The type of coefficient matrices selected

0 ii

”

B3

GLS

I

- 0.245 (0.1012) 0

0

. 0

0

-0.139 (0.0874)

- 0.413 (0.0902)

- 0.237 (0.1022)

- 0.396 (0.0876)

- 0.232 (0.1074)

- 0.282 (0.1003)

-0.190 (0.0879)

0

I

0.185

0

0.156 (0.1006)

0

0

GbyLS The estimate of times 10 4 No. of subsets checked Candidate subsets

0.222 1.45

1

1.03 0.222

I

I

0.363 (0.0896) I

1

140/4096

120/4096

CPU time

1.02 set

0.80 set

Patterns of causality

A log GNP - A log M2

A log GNP 0 A log MZ

324

J. H. W. Penm und R. D. Terrell, Multioariate

A R modeling

Table 7 The optimum

subset AR with zero constraints selected for HC X, = [A log GNP,, A log M2,] ‘.

Criterion The optimum

HC subset AR

1

Estimate

LS

The type of coefficient matrices

I

0

0

0

it

- 0.435 (0.0896)

3

I[il

GLS 0

- 0.237 (0.1021)

~ 0.396 (0.0876)

selected i - 0.245

[

GbyLS The estimate of times 10 4 No. of subsets checked Candidate subsets

I

(;;;;I

1 A log

II

(i,;l

0.238 1.45

CPU Patterns of causality

- 0.232

1.06 0.238

- 0.262

,

.i_i::

1

140/4096

105,‘4096

1.02 set

0.78 set

t

A log

Q

a significant

Conclusion

In this paper, we have shown that the canonical reduced AR form is a straightforward and effective AR form to test Granger-causality and instantaneous causality among time series variables. Our effective and fast procedures to fit multivariate subset AR processes with zero constraints is simple to use and leads to a neat and efficient analysis of vector time series.

325

J. H. W. Penm and R.D. Terrell, Multir~ariateA R modelmg

Table 8 The optimum subset AR with zero constraints selected for SC. X,=[AlogGNP,,AlogMI,]‘. SC

Criterion The optimum

1

subset AR

GLS

LS

Estimate The type of coefficient matrices selected GbyLS The estimate of

0.014 1.66

1.20 0.014

times 10 _ 4

[

No. of subsets checked Candidate subsets

CPU time

1

Patterns of causality

1

E/16

lo/16

0.06 set

0.05 set

AlogGNP-

* -AlogM?

AlogGNP-

* -AlogMZ

Appendix A. 1. Fitting of the multivariate subset AR model In (2) we consider the multivariate AR(q) model of the form

X,+

5 B,,Xr_,,=q. n=l

The sample lag covariance matrices,

obey the Yule-Walker

equations given by

(A.1)

.%$R4= -& where o,=

{ B1, B* ,..., S,},

r4= {r(-I),jY(-2),...,r(-q)},

J. H. W. Penm and R. D. Terrell, Multivariate AR modeling

326

and Q-1) R,=

W) ,

I.

T&2)

r(qL)

r(-q+l)

... ...

r(0)

1

6)

...

We now form a block Toeplitz matrix C4 of the form

rc4- 1)

r(-q+i)

r(-q+2)

...

thus we have the modified Choleski decomposition form

rio)

1.

for the matrix Cq+r of the

where Lq+l is a unit block lower triangular matrix, and Dq+1 is a block diagonal matrix with typical diagonal block matrix d,,I = 1,2,. . . , q + 1. The matrix C4+1 can also be written as

I

cY PA [ rqpqr(o) ’

Cq+1=

(A4

where Pqis the block permutation matrix, 0

0

0

0

:

:

0

p4=p;=

_L

0 IIn

InI 0

:

:

I,’

0

0

0

0

0

:

,

and of course Py' = I,,. Using a result for partitioned matrices in (A.2), we have IC,+,I =

~qir(o)- rqpqcq-lpqr;i

= Ic,llr(o)-r&q = IqlQ

J. H. W. Penm and R. D. Terrell, Multiwriare AR modeling

321

Therefore IGI is estimated by

In considering subset autoregressions, (12) has the form

x, + i l?,(z,)x*_, = El. n=l

The modified Yule-Walker

equation (Al) now has the form (A.3)

&J(ZS)R,(ZS) = -TJZ,L

where &JZ,) and r,(Z,) are formed by placing zero block matrices 0, in the (i r,. . . ,i,)th column of blocks of &.4 and r4. R,(Zs) is formed by placing Z, in the ((ir, 0 (i2, i2), . . . , (i,, i,))th diagonal block of R, and zero block matrices everywhere else in the (il, iI,..., i,)th row of blocks and also in the i,)th column of blocks of Rp. Thus the estimate of the lG41for the (i,,i,,..., AR with missing lags i,, i,, . . . , i, is estimated by

li;,UJ=

lcq+IvswlcqvJ~

(A.4)

wherej,=q+l-i,,Z=1,2 ,..., s,JScontainsj,,j2 ,..., j,,andmatricesC,+,(J,) and CJJ,) are formed by placing Z, in the ((ji, j,),( _i,, jz),...,(jS, j,))th diagonal blocks and zero matrices elsewhere in the (jr, j,, . . . ,j,)th row and column of blocks of C4+i and C4. A.2. Fitting of the multivariate subset AR model with zero constraints In considering a multivariate subset AR model with zero constraints, we allow for zero elements in the parameter matrices Z?,(Z,) of (12). We transpose and vectorize eq. (A.3) to be {

L@R,(l,)}vec{ a;&)} = -vet{ r,‘(I,)}

(A.5)

where the vet operation is obtained by stacking the columns of a matrix. Rewriting eq. (AS) in the form za=y, where Z= {ZPR,(Z,)},

,=vec{9?;(ZS)},

y= -vec{r;(ZS)}.

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J. H. W. Penm und R. D. Terrell, Multrounate

;‘he least constraints

square (LS) coefficient then obey the following

A R modelrng

estimates of multivariate relationship:

AR

with

zero

where C, is an integer set which contains ct, c?, . . . ,cr, and the (c,. c7,. . . , c,)th elements of (Y are constrained to the value 0. a(C,) and y( C,.) are formed by placing0 in (c1,c2,..., c,)th row elements of (Yand y, and Z(C) is formed by placing 1 in the ((c,, cI),(c2, c,), . . . , (cr, c,))th diagonal elements of Z and 0 everywhere else in the (cr. c2,. . . ,c,)th rows and columns of Z. G,( I,V,C,) is the estimate of Gq = E{ a(t)&‘(t)} in the multivariate AR model with zero constraints. We note that consideration of the contemporaneous correlation in e(t) cannot be ignored. Zellner (1962) indicated that the generalized least square (GLS) estimates of &(I,) of the multivariate subset AR with zero constraints are more efficient than the LS estimates, as the regressors in each equation are no longer necessarily the same. Since ($ is positive definite, there exists an m x m non-singular matrix i? such that G = I?I? ‘. Therefore by premultiplying X, by K -’ and then following the proposed method in this section, we can obtain the GLS estimates of B,( Z,Y).

A.3. A.3.1.

Tree pruning algorithm Tree structure and relabelling

The root of the inverse tree represents the model which includes all the lags. The ath generation, LT= 1,2,. . . ,q - 1, is defined by interior nodes, of which there are C’: nodes in the ath generation. These nodes represent models which comprise the (q - a) remaining lags, i.e., models of size (q - a). Consider the first generation and define IG(k,)l as the determinant of the varianceecovariance matrix arising from the deletion of the k,th lag, k, = 1,2,. . . , q. The lags are then relabelled so that the integers in increasing order correspond to diminishing determinants, i.e.,

Thus the members of the first generation are ordered in the lag inverse tree on the basis of the above inequality. To move from one generation to the next we make use of the rule that the Ith offspring in generation (Yhas I- 1 offspring in generation ((Y + l), i.e., the next generation down the tree. In setting up the second and later generations the ordering of the nodes from left to right is controlled by the natural ordering, e.g., in the four-lag case we would have in the second generation the

J. H. W. Penm und R. D. Terrell, Multiouriate AR modeling

329

1234

Root

\

/m

23

A

23

3rd generation

Fig

1. Four-lag

inverse

24

34

1

tree; (a) integers at each node are the remaining represents the vector ‘white noise’ model.

two-lag remaining subsets, i.e., 12,13,14,23,24,34. model in terms of the remaining lags, see fig. 1.

lags,

and

(b) null

So a node describes a

A.3.2. Pruning principle The pruning is performed using the fundamental relation in multiple linear regression theory,

where C is any set of remaining lags of X, and D is a subset of C. A. 3.3. Tree traversal

There are many algorithms available to traverse the tree to obtain the best subset of each size. In the tree search procedures used in obtaining the best subset, subject to constraints and the instantaneous causation, we use Furnival and Wilson’s (1974) leaps and bounds method with slight modification.

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J. H. W. Penm and R. D. Terrell, Multwarrate A R modehnp

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