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Autoregressive modeling and causal ordering of economic variables
-Journal of Economic Dynamics and Control 4 (1982) 243-259. North-Holland
AUTOREGRESSIVE MODELING OF ECONOMIC
AND CAUSAL VARIABLES*
ORDERING
Cheng HSIAO Bell L.&oratories,
Murray
University of Toronto, Toronto,
Hill, NJ 07974, USA Ont. MSS IAI, Canada
Received December 1980, final version received December 1981 A multivariate generalization of the Wiener-&anger notion of causality is suggested. Propositions about population properties of various causal events are derived. These propositions may be used to interpret the statistical results as well as to check the empirical implications of various model variants and in ruling out a number of variants as being inconsistent with prior theories.
1. Introduction In a recent paper Sims (1980a) has criticized the haphazard way conventional econometric models were constructed. His main point is that the true structural relationship governing the probability distribution of economic variables is very complex, and yet in practice econometricians achieve identification of their models by imposing false or spurious a priori restrictions. If a model is specified according to a set of incorrect laws, statistical inference based on it will be meaningless. He then suggests that in the first stage of model construction we treat all variables as jointly dependent and fit a vector autoregressive (AR) model for these variables to avoid imposing false or spurious restrictions and, in the second stage, we test hypotheses concerning economic contents. This is an appealing approach because under fairly general conditions there exists an AR representation for a set of stationary variables. Thus, in the specification of a model economic theory is used only to the extent of selecting a proper set of variables for analysis. No other prior restrictions need to be used. *This paper is a revised and condensed version of the paper entitled ‘Time Series Modelling and Causal Ordering of Canadian Money, Income, and Interest Rate’, which was presented in the Fourth International Time Series Meetings in Valencia, Spain, June 1981. This work was completed while the author was visiting Bell Laboratories, Murray Hill, NJ. Research is supported in part by National Science Foundation Grant SES80-07576 at the Institute for Mathematical Studies in the Social Sciences, Stanford Universitv. and bv Social Sciences and Humanities Research Council of Canada Grant 410-80-0080 at the Instituie for Policy Analysis, University of Toronto. I am indebted to C.W. Keng and K.Y. Tsui for computational assistance. I also wish to thank the referees and T. Amemiya, T.W. Anderson, V. Bencivenga, J.D. Bossqns, J. Carr and G. Chow for helpful comments.
0165-1889/82/OOOO-OOOO/$O2.75 0 1982 North-Holland
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It is one thing to fit a vector AR model; it is another thing to interpret the empirical results. One of the central concepts in the discussion of economic laws and econometric models is the concept of ‘causality’. In a controlled experiment, this concept has an unambiguous meaning. By holding the effects of other factors constant and repeating the experiments, one can isolate the effects of a factor and attribute it as the causal factor. But in economics very rarely one can run a controlled experiment. We only observe certain events. Based on these observations, we try to make inferences. In such situations economists often hypothesize a model and assume that the observed data are generated by this model. In this context, Zellner (1978) has defined causality as ‘predictability according to well thought out economic laws’. Although this is a reasonable definition, there are dilliculties in applying it to model construction with time series techniques. To apply Zellner’s definition to establish a causal ordering, one must have a priori knowledge of a model. But an empirical model-building strategy such as that suggested by Sims (1980a) is recommended when economists disagree about the set of laws governing economic relationships. In this sense, a definition of causality not relying on the specification of an econometric model may be very useful. Granger (1969) has suggested a definition of causality that is not dependent on a specific model. It is based on the stochastic nature of the variables. Its central feature is the lead-lag relations between variables through time. Granger’s definition is at variance with certain philosophical definition of causality in certain important aspects [Feigl (1953), Zellner (1978)]. In certain cases it may even lead to inferences which contradict the causal relations implied by structural models. However, Sims (1977) has demonstrated that these incorrect inferences exist only under special and rather restrictive conditions. Furthermore, in a linear probability model if a relation follows the Granger one-way causality, it also satisfies the necessary condition for the classification of variables as endogenous and exogenous [e.g. Geweke (1978, 1980), Hsiao (1979b)]. Therefore, despite the disagreement about its appropriateness we shall adopt the Wiener-Granger notion of causality as a starting point of our investigation. We hope by generalizing Granger’s notion of causality we can make Sims’ approach more useful by, first, providing an interpretation of a vector autoregressive model, and second, using this generalized notion judicially to check the empirical implications of various variants and rule out a number of variants as being inconsistent with prior theories. In section 2 we generalize Granger’s notion of causality to make some provision for spurious and indirect causality which may arise in multivariate analysis. Section 3 characterizes mathematical properties of causal events and gives an empirical example. The proof of these propositions are put into the appendix. Conclusions are in section 4.
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2. Patterns of causality To provide a basic framework for interpreting a multivariate AR model, we shall generalize Granger’s notion of causality in this section. Our aim is to identify whether a variable causes another variable directly, indirectly, or spuriously. Of course, theoretically, the notions to be discussed below can be further generalized into (n- 1) different levels of causal ordering in an nvariable system in. a manner similar to that of McElroy (1978) in a different context. We shall not attempt this here in the interest of simplicity of exposition. Let {y,x,z} be full rank, zero mean, joint covariance stationary, purely linearly indeterministic processes. ’ The analysis will remain unchanged if we let x be an r-component, y be an s-component and z be a q-component stationary processes. However, for simplicity of exposition we shall assume that x, y, and z are univariate. It has been shown by Wold [e.g. see Rosanov (1961)] that a regular full rank stationary process {y, x, z} possesses a unique one-sided moving average (MA) representation of the form
where <, is a three-component zero mean orthogonal process (i.e., E&=0, E~,~~=6,,,52, 6,,,= 1 for t=s, 6,,,=0 for t#s). Under fairly general conditions [Masani (1966)], (1) also admits an autoregressive representation,
(z-YlL-Y2L*---) x, =&, [I Yl
(2)
Zl
where
1--1I/11(~) -11/l&) I--Y1L-.**= l-$,,(L) l-#33WI [ -~,A~) =(z+r,L+r,L*+-)-l, -$12v4)
-4521(-Q
--$*3(L)
-+32vJ
‘Note Hosoya because formula
that the theorems in (19771. We use the the weaker conditions used below depends on
section 3 may hold under weaker conditions than this [e.g. covariance stationarity condition for simplicity of exposition would be unfamiliar to many readers, and because the FPE the stationarity condition.
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and L is the lag operator, Ly, = y, _ 1. Typical elements of ei,{L) are given by CEl
*i&*
We note that if we multiply r by the lower triangular matrix P (such that PCW’ = I), (1) .can be transformed as
(3)
where
and QO is an invertible, lower triangular matrix with positive elements on the diagonal. Either (1) or (3) is an identified process [Hannan (1969)]. For ease of exposition, we shall take (2) and (3) as the AR and MA representation of {y, x, z} in this paper. Let X,= {x,: s< t}, and similarly define k, 2,. Let A, be the relevant information set accumulated since time t - 1. Specifically, A, in this case is a stochastic process consisting only of {y,, x,, z,}. We define A, - x, as the set of elements in A, without the element x,. The sets A#, A,-X,, are defined analogously to 8,. Denote by a2(y1 A) the mean square error of the minimum mean square linear prediction error of y, given information set A,. We follow Wiener-Granger in defining various patterns of causality in terms of predictability of a variable. We first strengthen Granger’s (1969) definition of causality as: I (Direct Causality). If a2(y 14 < a2(y 1A-Z) and 0~011 E a < a2(y 1Q, then we say z causes y directly relative to A, denoted by zay.
Definition
This definition says that z causes y directly only when present y can be better predicted, in the mean square prediction error sense, by using past values of z, no matter which information set is used. We emphasize this
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aspect because, as the following discussion will show, there are cases where past z may help in predicting present y when one information set is used, but may not help in predicting present y when another information set is used. Definition 2 (Direct Feedback). If z-y and y*z, feedback occurs between y and z, denoted by zoy.
then we say that direct
Definition ! (No Causality). z does not cause y when either (i) rr”(y 1if) =cr’(yIA-X-Z) or (ii) 0~(yI~)=~~(y1,4-2) and a2(xIA)=a2(xI~-~). Condition (i) implies that the best linear predictor of y makes use of past values of y only. Condition (ii) implies that past x and y are jointly sufficient for predicting present y and x.~ However, Definition 3 does not exhaust all the cases where one would normally consider as an indication of no causality. There are cases where we may find 02@ 1;T) < a’(y I A-Z) but y and z may be uncorrelated. For.example, consider the following model:
where (u,, vI, w,) are mutually orthogonal independent Gaussian process. In this system z does not cause y by virtue of their independence, hence - a2(yl Y,Z)=o’(yI n. Y et w h en past x is used to predict y, it is advantageous to also use past z. This can be seen by taking the inverse of (4),
(5) That is, a2(y I A) < a2(y I A -Z). An analogous case in regression analysis is where y depends on x*, but not on z. Suppose x* is not observable. Instead, a proxy for x*, x, which is a function of x* plus noise, is observed. If z is correlated with the noise in the proxy variable x, then we can use z to eliminate or reduce the noise in x so that a more accurate prediction for y may be achieved. However, z appears in the y equation not because z ‘In fact, a more natural definition of no causality would be to follow Gaines and Chan (1975) by specifying 0((L) and ‘P(L) as lower block triangular. However, we have been defining causality as a reduction in forecasting variance with respect to a given information set. To switch the definitions in terms of the forms of AR or MA operator would seem to be inconsistent. In anj case, these two ways of defining no causality are equivalent, as stated in Theorem 1, section 3.
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somehow causes y, but rather because it serves as a purifying We call this situation spurious causality.
variable for x.
Definition 4 (Type I Spurious Causality). If a2(y 1r z) = a2(y 1Q and a2(y 1A)
3. Characterization
of causal events
In this section we state some basic relationships among variables for various causal events. The proofs are in the appendix. We also give an
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applied example to illustrate the possible presence of various causal relations. Theorem I. z does not cause y if and only if the following conditions hold: (i)
The moving average operator @(L) is lower block triangular.
(ii)
The autoregressive operator Y(L) is lower block triangular.
equivalent
This theorem is a straightforward application of theorems proved by Caines and Chan (1975), Geweke (1978), Granger (1969), Pierce and Haugh (1977), and Sims (1972), etc. Corollary 1. If z does not cause y, then either z does not cause x, or neither z nor x causes y.
This corollary follows straightforwardly of Y(L) and Q(L). Corollary 2. not cause Y.~
If
a’(y 1A) = a2(y 1A -Z)
from the lower block triangularity
and a2(y 1Y, a = a2(y 1 n, then z does
Note that Corollary 2 is a sufficient condition for z not causing y. It is not a necessary condition. It is possible that z does not cause y, yet a2(y 1A) < a2(y I A -Z). This is referred to as Type I spurious causality. Theorem 2. ?)rpe I spurious. causality from z to y occurs if and only if the following equivalent conditions hold:
(i) In the MA representation (3) there exists a C(L)=c,+c,L+c2L2+...
such
that C&l(L)
432(L)l=WC411(L)
and +131=Ofor
dJ&)l,
(6)
all 1, 41~~,#0, +23,#0 for some 1.
(ii) In the AR representation
(2), +a21=0 for all 1, tk13t#0 for some 1, and there exists a non-zero C(L) such that cti1203
(m
$13W)l=W)CIcI22(L)
ti23m.
(7)
‘Skoog (1976) has derived conditions for results from mth and nth dimensional AR processes # n) to coincide.
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Corollary 3. A sujkient condition for the existence of 7jpe I spurious causality from z to y is that 4 31~=432~=h=0 for all 1 and h#Q &,#O for some 1.
The situations where z is merely a proxy variable for the important left-out variables and where z is a primary driving force may be distinguished according to the following theorem: Theorem 3. A necessary condition for the existence of Qpe causality ji-om z to y is:
II
spurious
(i) In the moving average representation (3), 4131 =0, &[=O
for all 1, +121#O for some 1, and there does not exist a C(L) such that (6) holds.
(ii) In the AR representation
(2), I,Q~~~=O,J/231=O for all 1, and tJIZ1#O,
tis2,#0 for some 1. Theorem 4.
z does not cause y directly, but causes y indirectly if and only if
the following equivalent conditions hold: (i) There exists a non-zero C(L) such that
(8) and &,#O
for some 1 in (3).
(ii) tiIS1=O for all 1, and tiIZ1#O, J1231#0 for some 1 in (2). Corollary 4. If ZJX, x*y, but z does not cause y directly, causes y indirectly or z causes y spuriously (in a Type I sense).
then either z
We can use these propositions to interpret a statistical time series model. As an illustrative example we consider the causal relations of Canadian money, income, and interest rate. Quarterly data of M2, nominal GNP, and bank rate (BR) from 1955.1 to 1977.IV are used. The basic model we have in mind relates rates of change of money and income to the level of interest rate [e.g. see Cagan (1966)]. We then take the first difference of each of these variables (i.e., the second differences of the logarithm of the money and income variables and the first difference of the logarithm of the bank rate variable) to remove any trend. There are many different criteria one may use in selecting an approximate AR representation based on a finite number of observations. Here, we use Akaike’s (1969a, b) final prediction error (FPE) criterion because it fits in nicely with the idea of evaluating the predictability in terms of mean square prediction error. The smallest FPE’s for one-, two-, and three-dimensional
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analysis with the maximum order of lag set equal to 13 are reported in table 1.4 Based on these results we choose the following representation as a finite-order approximation to Canadian income, money, and interest rate data:
(9) The superscript of *ii indicates the order of lag of that variable. Table 1 The optimal lag order and the FPEs of the controlled variable. Controlled variable BR GNP M2 BR BR GNP GNP M2 M2 BR GNP M2
(3) (6) (12) I:; (6) (6) (12) (12) (3) (6) (12)
First manipulated variable
M2 GNP M2 BR GNP %P BR BR
Second manipulated variable
(1) (5) (1)
(6) (2) 1:;
M2
(1)
FPE x 1o-4 131.4 1.701 1.333 133.5 131.353 1.716 1.633 1.332 1.284 132.7 1.662 1.299
“The number in parenthesis indicates the order of lags of each variable.
The specification (9) indicates that GNPoBR, and BR*M2. According to Theorem 1, we also conclude that eq. (9) indicates that A42 neither causes GNP nor BR, and according to Theorem 4, GNP causes M2 indirectly. We then check the validity of these assertions by comparing the FPE's of the bivariate and trivariate analysis. Table 1 indicates that for the bivariate analysis, GNPaM2, GNPoBR, and BR=sM2. Yet in the trivariate analysis, GNP no longer appears in the M2 equation. However, the direct causalities 41 have used a step-wise procedure suggested by Hsiao (1979a) to reduce the computational burden. For detail, see Hsiao (1982).
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from GNP to BR, and BR to M2 still hold. This is precisely our definition of indirect causality. On the other hand, the bivariate and trivariate analysis treating either income or the interest rate as the output variable show that using money as an input variable increases the FPE. According to Corollary 2, they indicate that money neither causes income nor the interest rate. As one can see these results do confirm our assertion. Although different procedures may lead to different specifications, at least in this particular case using other methods also tend to point to the same qualitative conclusion that the bank rate is the primary driving force for income and money, and to a lesser degree responds to income changes.5 These results are consistent with the Bank of Canada’s primary policy concern of trying to maintain ‘appropriate exchange rates’ and ‘appropriate credit conditions’. To the extent to which the Bank primarily aims to regulate the structure of interest rates and not the money supply [e.g. see Courchene (1977) and Rasminsky (1967)], and Canadian interest rates cannot be independent of world (essentially U.S.) rates, movements in the money stock and interest rates can be expected to respond more to movements in nominal income. The fact that nominal income does not respond to changes in M2 might be due to volatile variations in M2 rather than representing a contradiction of the monetarists’ position that nominal income is closely related to broad aggregates of financial assets. The term structure of deposit rates has been less stable in Canada than in the U.S. With the existence of a great number of close substitutes issued by near-banks as well as by other financial institutions, structural shifts in intermediation can occur, the effect of which might be to merely alter the charter banks’ share but not the total value of these interest-bearing liabilities. In this sense, M2 might not be a good proxy for the broad aggregates of financial assets in Canada. In fact, the Bank has also maintained that M2 is an unsatisfactory definition of money in Canada (Bank of Canada Annual Report 1974). 4. Conclusions
In modeling economic time series data there is usually difficulty in discriminating finely between several models with different forms which appear consistent with the information in the data. In fact, in finite sample it often happens that if we do not restrict the parametric specifications, the problem of collinearity, shortage of degrees of freedom, etc. would so confound the interpretation of the model that we do not know what to make of it [Ando (1977), Klein (1977)]. Engle (197Q Hendry (1974), Granger and Newbold (1977), Wallis (1977), Zellner and Palm (1974) and many others 5For the results of usingthe conventionalmethod, see Hsiao (1982).
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have suggested approaches to blend traditional econometric and time series analysis to cohstruct better econometric models. In this paper we used Granger’s (1969) notion of causality to derive various propositions about population properties of various causal events. These propositions can be used to provide an interpretation for multivariate time series models as well as a means for developing and checking the empirical implications of various variants of models and for ruling out a number of variants as being inconsistent. with the prior theorems. 6 We have applied these concepts to the time series analysis of Canadian money, income, and interest rate data. We believe that if we are able to use the degrees of freedom more efficiently a purely statistical analysis of economic time series data is capable of yielding useful information. As Granger (1980) has remarked, the concept of causality and procedures to fit time series model are topics in which individual tastes predominate. It would be improper to try to force research workers to accept definitions and procedures with which they feel uneasy. There is clearly a need for more discussion of the appropriate definitions of causality and for more exploration of various multiple time series modeling procedures. This paper represents a preliminary attempt to analyze multivariate time series data. The empirical result may or may not stand further scrutiny. However the topic in my view is of sufficient importance and interest to justify further work in this area rather than to be brushed aside as ad hoc or not serious. Appendix: Proof of various characterizations
of causal eve&’
Let the notation H,(r) stand for the completion with respect to the meansquare norm of the linear space of random variables spanned by y, for ss t. Let u, be the difference between yI and the projection of y, on H,,x,n(t- 1). Let & be the difference between x, and the projection of x, on H,,,,$1). Let u, be that part of 5, which is orthogonal to u,. Similarly, let w, be that part of the difference between z, and the projection of z1 on Hy,x,z(t- 1) which is orthogonal to u, and u,. By definition, u,, u,, and w, are contemporaneously uncorrelated and are uncorrelated with past values of each other. Also H,,,,,(r) is identical to HySx,Jt), and {y,,x,,z,} has a moving average representation of the form (3) [Rozanov (1967)]. We similarly define u: as the difference between y, and the projection of y, on H&t1) and u: as that part of the difference between x, and the projection of x, on HY,.Jt- 1) which is orthogonal to u:; hence H,,,,(t) is identical to H,,,(t). We also let u:* denote the difference between y, and the “For an illustration, see Hsiao (1982). ‘Hosoya (1977) has proved the Granger non-causality in non-stationary cases. The result does not seem easily generalizable to the proof here because in the non-stationary case the processis no longer a Cauchy sequence in Hilbert space, thus making it very dilkult to operate be&en different dimensions.
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projection of y, on HY,&- 1) and w,** denote that part of the difference between q and the projection of z, on H,,.(t - 1) which is orthogonal to u:*. Thus, H,,,(t) is identical to HP,+(t). Lemma I. Zf c$l21# 0 bnd $1 31# 0 for some 1 in (3), then the {y, x} process has an MA representation of the form
0(
Yf 4:1(L) Xl = dJt,w
4:2(L) u: ML) >( 0: )
(10)
with $rz,#O for some 1. Proof. We denote by D(t) the orthogonal subspa= H,, x.A%
complement
of H,,,(t)
in the
Then, by construction, u: ED@-- l)@u,,
(12)
where @ denotes the direct sum. ~,,&)~&,&) and o(t)=H,,,,(t)BH,,,.(t). = H,(t) . Therefore,’ H,*(t)Gzf,(t)8H,(t-
By construction we know that Hence o(t)~H,,.,,(t)OH,,,(t)
I).
(13)
By Weld’s decomposition theorem [Rozanov (1967)], {y,x} from moving averages of uncorrelated processes as follows:
Given
are obtained
=~:,@Ju:+4:,(Lb:,
(14)
= 4;,w:
(15)
that H,(t)lH,,,(f)
+ 4M4u:. and (13), we know
that
{y, x} may not be
sWe note that if Cp,3,=0 for all I and &,#O for some I, then we may switch the roles of 1) and H,(t-1) and construct an MA representation with 4T2,=0 for all I under certain conditions. H,(t-
C. Hsiao, Autoregressive modeling and causal ordering
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represented as
x, = 62,w: Thus,
+ 622(~h.
(17)
. H”(t)EH:(t).
W
However, &I#0 for some 1 means that the projection of y, on H,(t - 1) is non-zero. Consequently, the projection of y, on H:(t- 1) is non-zero as well. Hence $&#O for some 1. Proof of Corollary 2. Expressing (10). Since a2(y 14 = a2(y 1A -Z), H,,.&1) lies in H&-l), and and u,lH,(t); hence, 4:,(~5)=4,,(L) then 4r2,=0 and $r3,= 0 for all y. If 4:21 #O for some 1, by (3) and
Letting C(L) = @2(L)/+f2(L),
{yt,x,} in terms of the @, $, we have we know that the projection of y, on therefore u:=u~. Furthermore u,lHV,,(t) and @i(L)=&(L). If &=O for all I, 1, and it is trivial that z does not cause (10) we have [Ansley et al. (1977)]
we have
Theorem 1 says that a2(y 1Y,Z)=a’(y 1Q if and only if there exists a moving average representation for {yl,z,} of the form
J+= 4Gw) 0 u:* 0Zf ( 4:w av) )( w:*>* By Lemma +131=0 for all From Theorem does not cause
(22)
1, a necessary condition to construct (22) from (3) is that 1. However, (21) says that 4ra1=0 for all 1, so are +231=O. 1 we know that 4 rsl=O and $J~~~=Ofor all 1 if and only if z y.
Proof of Theorem 2. We note that a2(y 1zZ)=a’(y 1p) implies that the moving average representation for {y,,z,} has the form (22) with #i=O for all 1. If both 4121=0 and r$131=0 for all 1, the projection of y, on H,,x,z(t- 1) lies in H,(t - 1) alone. That is, a2(y 1A) = 02(y I A -X-Z), and neither z nor x
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causes y. On the other hand, if both +I21 #O and rjr 31# 0 for some 1, then by Lemma 1, cbt$ # 0 for some 1. Therefore we assume 4121 #O for some I and 41sI=0 for all 1.’ With 4111#0, +121#0 for some I, the only way for (22) to hold is that
Such a representation exists if and only if there exists a C(L) such that
If 4231also equal zero for all 1, then Q(L) is lower block triangular, and hence a2(y 1A) = c2(y 1A -2). This is a contradiction and, therefore, 4231 # 0 for some 1. Together with bIzl#O for some 1, this implies that $r3r#0 for some 1, i.e., c2(y 1A) < fl”(y ) A -Z). Condition (ii) can be derived by taking the inverse of the Q(L) matrix. Proof of Corollary 3.
Condition
(i) of Theorem 2 is automatically
satisfied
with C(L) 3 0. Proof of Theorem 3. We first prove condition 02(y 17, Z) < cr2(y1Y) if and only if
(i). Theorem
y, = 4::(L) 9:w u,** 0Zt (W(L) MIL)>(w:*) ’
1 says that
(26)
with @zl#O for some 1. However, condition (ii) of no causality from z to y implies 4rS1=0 and 4231= 0 for all 1 in (3). Therefore, {y,,z,} has the following representation: Y, =
41
lw4
+
4J12uh
(27)
- From a2(z 1ii) < c2(z 1A-X) and g2(z I X, Z) < a2(z ( Z) we know that 4b321# 0 for some 1. Given (27) and (28), as the proof of Corollary 2 shows that a necessary condition for the existence of (26) is +t2, #0 for some 1, and no ‘Equivalently, we (6) should 13 .
yf$on
can assume be replaced
d, Izr=O for all I and 4131#0, &#O by the equivalent condition ol [4,,(L)
for some I; then q&,(L)]=C(L)[4,,(L)
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existence of C(L) such that
Condition (i) implies condition (ii) follows straightforwardly from taking the inverse of Q(L). To show that condition (ii) implies condition (i), we note that h31=0, ~~31=0 for all 1 and $121#0 for some 1 jointly imply a’(yIa =a2(yp-Z)Io2(yp-X) and a2(xIA~=a2(xIA-Z). That is c&,=0, 4231 = 0 for all I and 4i2r #O for some 1. $a2, #O for some I implies that there does not exist a C(L) such that (29) holds. Proof of Theorem 4. We first prove condition (i). We note that a2(y 1A) = 02(y1A-Z) if a n d only if (21) holds. If C(L) = 0, then this implies that +121= 0, r$isl =O for all 1. That is, y can be viewed as generated by an innovation process that is uncorrelated with the innovations in the x and z processes. Therefore C(L) must be non-zero for a2(y 1r z) < a2(y 1q to hold. On the other hand if dz3,= 0 for all 1, by (21) +131=0 for all 1. Then by Theorem 1, (r2(x 1A) = 02(x 1A -Z). (Neither can 4 i 21--0 for all I, since this will imply 4221=0 for all 1.) When 4121#0, 4131#0 for some 1, by Lemma 1 the {j&z,} process will have 473: # 0 for some 1 in (26), that is a2(y I z z) < a20, I q. Condition (ii) can be proved by taking the inverse of @(L).
Proof of Coro&ry 4. Definition 1 says that if z does not cause y directly either 02(y I a=a2(yl A-Z) or a2(y I ~Z)=02(yl 7) or both. We first show that 02(y I a=a2(y I A-Z) and a’(y I EZ)=a’b I 7) cannot hold simultaneously under the assumption that z-x and x-y. We note that a2(yp)=a2(yp-z) implies (21) holds. Therefore, 4i2r $0 for some 1 unless C(L) is identically equal to zero --which is ruled out by the assumption that x=y. Thus by Lemma 1, a2(y ) Y,Z) = a2(y I ?) implies that 4i3[ =0 for all 1. By (21), 4231=0 for all 1. This contradicts the assumption that z*x. Therefore, either a2(yIA)=aZ(yIA-Z) and o’(y( Y,Z)<02(y) Y) or and a2(y 1KZ) = a2(y ( n. The former is indicated as a2(yIAT
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References Akaike, H., 1969a, Statistical predictor identification, Annals of the Institute of Statistical Mathematics 21,203-217. Akaike, H., 1969b, Fitting autoregressions for prediction, Annals of the Institute of Statistical Mathematics 21, 243-247. Ando, A., 1977, A comment, in: New methods in business cycle research: Proceedings from a conference (Federal Reserve Bank of Minneapolis, MN) 209-212. Ansley, C., W.A. Spivey and W.J. Wrobleski, 1977, On the structure of moving average processes, Journal of Econometrics 6, 121-134. Bank of Canada, 1974, Annual report (Ottawa). Cagan, P., 1965, Determinant and elIects of changes in the stock of money, 1875-1960 (Columbia University Press, New York). Gaines, P.E., 1976, Weak and strong feedback free processes, IEEE Transactions on Automatic Control AC-21, 1137-l 139. Caines, P.E. and C.W. Chan, 1975, Feedback between stationary stochastic process, IEEE Transactions on Automatic Control AC-20, 4988508. Courchene, T.J., 1977, Money, inflation, and the Bank of Canada: An analysis of Canadian monetary policy from 1970 to early 1975 (C.D. Howe Research Institute). Engle, R., 1978, Testing price equations for stability across spectral frequency bands, Econometrica 46,869-881. Feigl, H., 1953, Notes on causality, in: H. Feigl and M. Brodbeck, eds., Readings in the philosophy of science (Appleton-Century-Crofts, Inc., New York) 4088418. Geweke, J., 1978, Testing the exogeneity specification in the complete dynamic simultaneous equation model, Journal of Econometrics 7, 163-186. Geweke, J., 1980, Inference and causality in economic time series models, Handbook of Econometrics (North-Holland, Amsterdam) forthcoming. Granger, C.W.J., 1969, Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37.424-438. Granger, C.W.J., 1980, Testing for causality: A personal view, Journal of Economic Dynamics and Control, forthcoming. Granger, C.W.J. and P. Newbold, 1977, Forecasting economic time series (Academic Press, New York). Hannan, E.J., 1969, The identification of vector mixed autoregressive-moving average systems, Biometrika 56, 223-225. Hendry, D.F., 1974, Stochastic specification in an aggregated demand model of the U&ted Kingdom, Econometrica 42, 547-558. Hosoya, Y., 1977, On the Granger condition for non-causality, Econometrica 45, 1735-1736. Hsiao, C., 1979a, Autoregressive modelling of Canadian money and income data, Journal of the American Statistical Association 74, 553-560. Hsiao, C., 1979b, Causality tests in econometrics, Journal of Economic Dynamics and Control 1, 321-346. Hsiao, C., 1982, Time series modelling and causal ordering of Canadian money, income and interest rate, in: O.D. Anderson, ed., Tie series analysis: Theory and practice (NorthHolland, Amsterdam) forthcoming. Klein, L., 1977, Comments on Sargent and Sims’ business cycle modelling without pretending too much a priori economic theory, New methods in business cycle research: Proceedings from a conference (Federal Reserve Bank of Minneapolis, MA) 203-208. Masani, P., 1966, Recent trends in multivariate prediction theory, in: P.R. Krishnaiah, ed., Multivariate analysis I (Academic Press, New York) 351-382. McElroy, F.W., 1978, A simple method of causal ordering, International Economic Review 19, l-24. Pierce, D.A. and L.D. Haugh, 1977, Causality in temporal systems: Characterizations and survey, Journal of Econometrics 5, 265-293. Rasminsky, L., 1967, Central banking in the Canadian financial system, Opening address by the Governor of the Bank of Canada to the 20th International Banking Summer School (Queen’s University, Kingston).
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modeling
and causal
or&ring
259
Rozanov, Yu A., 1967, Stationary random process (Holden-Day, San Francisco, CA). Sims, C.A., 1972, Money, income and causality, American Economic Review 62, 540-555. Sims, C.A., 1977, Exogeneity and causal orderings in marcoeconomic models, in: New methods of business cycle research (Federal Reserve Bank of Minneapolis, MN) 23-44. Sims, CA., 1980a, Macro-economics and reality, Econometrica, l-48. Sims, C.A., 1980b, Comparison of interwar and postwar business cycles: Monetarism reconsidered, American Economic Review, Papers and Proceedings 70, 250-257. Skoog, G., 1976, Causality characterizations: Bivariate, trivariate and multivariate propositions, Staff report no. T4 (Federal Reserve Bank of Minneapolis, MN). Wahis, K., 1977, Multiple time series analysis and the final form of econometric models, Econometrica 45, 1481-1497. Zellner, A., 1978, Causality and econometrics, Paper presented at a joint University of Rochester - Carnegie-Mellon University Conference. Zellner, A. and F. Palm, 1974, Tie series analysis and simultaneous equation econometric models, Journal of Econometrics 2, 17-54.
AUTOREGRESSIVE MODELING OF ECONOMIC
AND CAUSAL VARIABLES*
ORDERING
Cheng HSIAO Bell L.&oratories,
Murray
University of Toronto, Toronto,
Hill, NJ 07974, USA Ont. MSS IAI, Canada
Received December 1980, final version received December 1981 A multivariate generalization of the Wiener-&anger notion of causality is suggested. Propositions about population properties of various causal events are derived. These propositions may be used to interpret the statistical results as well as to check the empirical implications of various model variants and in ruling out a number of variants as being inconsistent with prior theories.
1. Introduction In a recent paper Sims (1980a) has criticized the haphazard way conventional econometric models were constructed. His main point is that the true structural relationship governing the probability distribution of economic variables is very complex, and yet in practice econometricians achieve identification of their models by imposing false or spurious a priori restrictions. If a model is specified according to a set of incorrect laws, statistical inference based on it will be meaningless. He then suggests that in the first stage of model construction we treat all variables as jointly dependent and fit a vector autoregressive (AR) model for these variables to avoid imposing false or spurious restrictions and, in the second stage, we test hypotheses concerning economic contents. This is an appealing approach because under fairly general conditions there exists an AR representation for a set of stationary variables. Thus, in the specification of a model economic theory is used only to the extent of selecting a proper set of variables for analysis. No other prior restrictions need to be used. *This paper is a revised and condensed version of the paper entitled ‘Time Series Modelling and Causal Ordering of Canadian Money, Income, and Interest Rate’, which was presented in the Fourth International Time Series Meetings in Valencia, Spain, June 1981. This work was completed while the author was visiting Bell Laboratories, Murray Hill, NJ. Research is supported in part by National Science Foundation Grant SES80-07576 at the Institute for Mathematical Studies in the Social Sciences, Stanford Universitv. and bv Social Sciences and Humanities Research Council of Canada Grant 410-80-0080 at the Instituie for Policy Analysis, University of Toronto. I am indebted to C.W. Keng and K.Y. Tsui for computational assistance. I also wish to thank the referees and T. Amemiya, T.W. Anderson, V. Bencivenga, J.D. Bossqns, J. Carr and G. Chow for helpful comments.
0165-1889/82/OOOO-OOOO/$O2.75 0 1982 North-Holland
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It is one thing to fit a vector AR model; it is another thing to interpret the empirical results. One of the central concepts in the discussion of economic laws and econometric models is the concept of ‘causality’. In a controlled experiment, this concept has an unambiguous meaning. By holding the effects of other factors constant and repeating the experiments, one can isolate the effects of a factor and attribute it as the causal factor. But in economics very rarely one can run a controlled experiment. We only observe certain events. Based on these observations, we try to make inferences. In such situations economists often hypothesize a model and assume that the observed data are generated by this model. In this context, Zellner (1978) has defined causality as ‘predictability according to well thought out economic laws’. Although this is a reasonable definition, there are dilliculties in applying it to model construction with time series techniques. To apply Zellner’s definition to establish a causal ordering, one must have a priori knowledge of a model. But an empirical model-building strategy such as that suggested by Sims (1980a) is recommended when economists disagree about the set of laws governing economic relationships. In this sense, a definition of causality not relying on the specification of an econometric model may be very useful. Granger (1969) has suggested a definition of causality that is not dependent on a specific model. It is based on the stochastic nature of the variables. Its central feature is the lead-lag relations between variables through time. Granger’s definition is at variance with certain philosophical definition of causality in certain important aspects [Feigl (1953), Zellner (1978)]. In certain cases it may even lead to inferences which contradict the causal relations implied by structural models. However, Sims (1977) has demonstrated that these incorrect inferences exist only under special and rather restrictive conditions. Furthermore, in a linear probability model if a relation follows the Granger one-way causality, it also satisfies the necessary condition for the classification of variables as endogenous and exogenous [e.g. Geweke (1978, 1980), Hsiao (1979b)]. Therefore, despite the disagreement about its appropriateness we shall adopt the Wiener-Granger notion of causality as a starting point of our investigation. We hope by generalizing Granger’s notion of causality we can make Sims’ approach more useful by, first, providing an interpretation of a vector autoregressive model, and second, using this generalized notion judicially to check the empirical implications of various variants and rule out a number of variants as being inconsistent with prior theories. In section 2 we generalize Granger’s notion of causality to make some provision for spurious and indirect causality which may arise in multivariate analysis. Section 3 characterizes mathematical properties of causal events and gives an empirical example. The proof of these propositions are put into the appendix. Conclusions are in section 4.
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2. Patterns of causality To provide a basic framework for interpreting a multivariate AR model, we shall generalize Granger’s notion of causality in this section. Our aim is to identify whether a variable causes another variable directly, indirectly, or spuriously. Of course, theoretically, the notions to be discussed below can be further generalized into (n- 1) different levels of causal ordering in an nvariable system in. a manner similar to that of McElroy (1978) in a different context. We shall not attempt this here in the interest of simplicity of exposition. Let {y,x,z} be full rank, zero mean, joint covariance stationary, purely linearly indeterministic processes. ’ The analysis will remain unchanged if we let x be an r-component, y be an s-component and z be a q-component stationary processes. However, for simplicity of exposition we shall assume that x, y, and z are univariate. It has been shown by Wold [e.g. see Rosanov (1961)] that a regular full rank stationary process {y, x, z} possesses a unique one-sided moving average (MA) representation of the form
where <, is a three-component zero mean orthogonal process (i.e., E&=0, E~,~~=6,,,52, 6,,,= 1 for t=s, 6,,,=0 for t#s). Under fairly general conditions [Masani (1966)], (1) also admits an autoregressive representation,
(z-YlL-Y2L*---) x, =&, [I Yl
(2)
Zl
where
1--1I/11(~) -11/l&) I--Y1L-.**= l-$,,(L) l-#33WI [ -~,A~) =(z+r,L+r,L*+-)-l, -$12v4)
-4521(-Q
--$*3(L)
-+32vJ
‘Note Hosoya because formula
that the theorems in (19771. We use the the weaker conditions used below depends on
section 3 may hold under weaker conditions than this [e.g. covariance stationarity condition for simplicity of exposition would be unfamiliar to many readers, and because the FPE the stationarity condition.
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and L is the lag operator, Ly, = y, _ 1. Typical elements of ei,{L) are given by CEl
*i&*
We note that if we multiply r by the lower triangular matrix P (such that PCW’ = I), (1) .can be transformed as
(3)
where
and QO is an invertible, lower triangular matrix with positive elements on the diagonal. Either (1) or (3) is an identified process [Hannan (1969)]. For ease of exposition, we shall take (2) and (3) as the AR and MA representation of {y, x, z} in this paper. Let X,= {x,: s< t}, and similarly define k, 2,. Let A, be the relevant information set accumulated since time t - 1. Specifically, A, in this case is a stochastic process consisting only of {y,, x,, z,}. We define A, - x, as the set of elements in A, without the element x,. The sets A#, A,-X,, are defined analogously to 8,. Denote by a2(y1 A) the mean square error of the minimum mean square linear prediction error of y, given information set A,. We follow Wiener-Granger in defining various patterns of causality in terms of predictability of a variable. We first strengthen Granger’s (1969) definition of causality as: I (Direct Causality). If a2(y 14 < a2(y 1A-Z) and 0~011 E a < a2(y 1Q, then we say z causes y directly relative to A, denoted by zay.
Definition
This definition says that z causes y directly only when present y can be better predicted, in the mean square prediction error sense, by using past values of z, no matter which information set is used. We emphasize this
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247
aspect because, as the following discussion will show, there are cases where past z may help in predicting present y when one information set is used, but may not help in predicting present y when another information set is used. Definition 2 (Direct Feedback). If z-y and y*z, feedback occurs between y and z, denoted by zoy.
then we say that direct
Definition ! (No Causality). z does not cause y when either (i) rr”(y 1if) =cr’(yIA-X-Z) or (ii) 0~(yI~)=~~(y1,4-2) and a2(xIA)=a2(xI~-~). Condition (i) implies that the best linear predictor of y makes use of past values of y only. Condition (ii) implies that past x and y are jointly sufficient for predicting present y and x.~ However, Definition 3 does not exhaust all the cases where one would normally consider as an indication of no causality. There are cases where we may find 02@ 1;T) < a’(y I A-Z) but y and z may be uncorrelated. For.example, consider the following model:
where (u,, vI, w,) are mutually orthogonal independent Gaussian process. In this system z does not cause y by virtue of their independence, hence - a2(yl Y,Z)=o’(yI n. Y et w h en past x is used to predict y, it is advantageous to also use past z. This can be seen by taking the inverse of (4),
(5) That is, a2(y I A) < a2(y I A -Z). An analogous case in regression analysis is where y depends on x*, but not on z. Suppose x* is not observable. Instead, a proxy for x*, x, which is a function of x* plus noise, is observed. If z is correlated with the noise in the proxy variable x, then we can use z to eliminate or reduce the noise in x so that a more accurate prediction for y may be achieved. However, z appears in the y equation not because z ‘In fact, a more natural definition of no causality would be to follow Gaines and Chan (1975) by specifying 0((L) and ‘P(L) as lower block triangular. However, we have been defining causality as a reduction in forecasting variance with respect to a given information set. To switch the definitions in terms of the forms of AR or MA operator would seem to be inconsistent. In anj case, these two ways of defining no causality are equivalent, as stated in Theorem 1, section 3.
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ordering
somehow causes y, but rather because it serves as a purifying We call this situation spurious causality.
variable for x.
Definition 4 (Type I Spurious Causality). If a2(y 1r z) = a2(y 1Q and a2(y 1A)
3. Characterization
of causal events
In this section we state some basic relationships among variables for various causal events. The proofs are in the appendix. We also give an
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applied example to illustrate the possible presence of various causal relations. Theorem I. z does not cause y if and only if the following conditions hold: (i)
The moving average operator @(L) is lower block triangular.
(ii)
The autoregressive operator Y(L) is lower block triangular.
equivalent
This theorem is a straightforward application of theorems proved by Caines and Chan (1975), Geweke (1978), Granger (1969), Pierce and Haugh (1977), and Sims (1972), etc. Corollary 1. If z does not cause y, then either z does not cause x, or neither z nor x causes y.
This corollary follows straightforwardly of Y(L) and Q(L). Corollary 2. not cause Y.~
If
a’(y 1A) = a2(y 1A -Z)
from the lower block triangularity
and a2(y 1Y, a = a2(y 1 n, then z does
Note that Corollary 2 is a sufficient condition for z not causing y. It is not a necessary condition. It is possible that z does not cause y, yet a2(y 1A) < a2(y I A -Z). This is referred to as Type I spurious causality. Theorem 2. ?)rpe I spurious. causality from z to y occurs if and only if the following equivalent conditions hold:
(i) In the MA representation (3) there exists a C(L)=c,+c,L+c2L2+...
such
that C&l(L)
432(L)l=WC411(L)
and +131=Ofor
dJ&)l,
(6)
all 1, 41~~,#0, +23,#0 for some 1.
(ii) In the AR representation
(2), +a21=0 for all 1, tk13t#0 for some 1, and there exists a non-zero C(L) such that cti1203
(m
$13W)l=W)CIcI22(L)
ti23m.
(7)
‘Skoog (1976) has derived conditions for results from mth and nth dimensional AR processes # n) to coincide.
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Corollary 3. A sujkient condition for the existence of 7jpe I spurious causality from z to y is that 4 31~=432~=h=0 for all 1 and h#Q &,#O for some 1.
The situations where z is merely a proxy variable for the important left-out variables and where z is a primary driving force may be distinguished according to the following theorem: Theorem 3. A necessary condition for the existence of Qpe causality ji-om z to y is:
II
spurious
(i) In the moving average representation (3), 4131 =0, &[=O
for all 1, +121#O for some 1, and there does not exist a C(L) such that (6) holds.
(ii) In the AR representation
(2), I,Q~~~=O,J/231=O for all 1, and tJIZ1#O,
tis2,#0 for some 1. Theorem 4.
z does not cause y directly, but causes y indirectly if and only if
the following equivalent conditions hold: (i) There exists a non-zero C(L) such that
(8) and &,#O
for some 1 in (3).
(ii) tiIS1=O for all 1, and tiIZ1#O, J1231#0 for some 1 in (2). Corollary 4. If ZJX, x*y, but z does not cause y directly, causes y indirectly or z causes y spuriously (in a Type I sense).
then either z
We can use these propositions to interpret a statistical time series model. As an illustrative example we consider the causal relations of Canadian money, income, and interest rate. Quarterly data of M2, nominal GNP, and bank rate (BR) from 1955.1 to 1977.IV are used. The basic model we have in mind relates rates of change of money and income to the level of interest rate [e.g. see Cagan (1966)]. We then take the first difference of each of these variables (i.e., the second differences of the logarithm of the money and income variables and the first difference of the logarithm of the bank rate variable) to remove any trend. There are many different criteria one may use in selecting an approximate AR representation based on a finite number of observations. Here, we use Akaike’s (1969a, b) final prediction error (FPE) criterion because it fits in nicely with the idea of evaluating the predictability in terms of mean square prediction error. The smallest FPE’s for one-, two-, and three-dimensional
251
C. Hsiao, Autoregressive modeling and causal ordering
analysis with the maximum order of lag set equal to 13 are reported in table 1.4 Based on these results we choose the following representation as a finite-order approximation to Canadian income, money, and interest rate data:
(9) The superscript of *ii indicates the order of lag of that variable. Table 1 The optimal lag order and the FPEs of the controlled variable. Controlled variable BR GNP M2 BR BR GNP GNP M2 M2 BR GNP M2
(3) (6) (12) I:; (6) (6) (12) (12) (3) (6) (12)
First manipulated variable
M2 GNP M2 BR GNP %P BR BR
Second manipulated variable
(1) (5) (1)
(6) (2) 1:;
M2
(1)
FPE x 1o-4 131.4 1.701 1.333 133.5 131.353 1.716 1.633 1.332 1.284 132.7 1.662 1.299
“The number in parenthesis indicates the order of lags of each variable.
The specification (9) indicates that GNPoBR, and BR*M2. According to Theorem 1, we also conclude that eq. (9) indicates that A42 neither causes GNP nor BR, and according to Theorem 4, GNP causes M2 indirectly. We then check the validity of these assertions by comparing the FPE's of the bivariate and trivariate analysis. Table 1 indicates that for the bivariate analysis, GNPaM2, GNPoBR, and BR=sM2. Yet in the trivariate analysis, GNP no longer appears in the M2 equation. However, the direct causalities 41 have used a step-wise procedure suggested by Hsiao (1979a) to reduce the computational burden. For detail, see Hsiao (1982).
252
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Hsiao, Autoregressive
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and causal
ordering
from GNP to BR, and BR to M2 still hold. This is precisely our definition of indirect causality. On the other hand, the bivariate and trivariate analysis treating either income or the interest rate as the output variable show that using money as an input variable increases the FPE. According to Corollary 2, they indicate that money neither causes income nor the interest rate. As one can see these results do confirm our assertion. Although different procedures may lead to different specifications, at least in this particular case using other methods also tend to point to the same qualitative conclusion that the bank rate is the primary driving force for income and money, and to a lesser degree responds to income changes.5 These results are consistent with the Bank of Canada’s primary policy concern of trying to maintain ‘appropriate exchange rates’ and ‘appropriate credit conditions’. To the extent to which the Bank primarily aims to regulate the structure of interest rates and not the money supply [e.g. see Courchene (1977) and Rasminsky (1967)], and Canadian interest rates cannot be independent of world (essentially U.S.) rates, movements in the money stock and interest rates can be expected to respond more to movements in nominal income. The fact that nominal income does not respond to changes in M2 might be due to volatile variations in M2 rather than representing a contradiction of the monetarists’ position that nominal income is closely related to broad aggregates of financial assets. The term structure of deposit rates has been less stable in Canada than in the U.S. With the existence of a great number of close substitutes issued by near-banks as well as by other financial institutions, structural shifts in intermediation can occur, the effect of which might be to merely alter the charter banks’ share but not the total value of these interest-bearing liabilities. In this sense, M2 might not be a good proxy for the broad aggregates of financial assets in Canada. In fact, the Bank has also maintained that M2 is an unsatisfactory definition of money in Canada (Bank of Canada Annual Report 1974). 4. Conclusions
In modeling economic time series data there is usually difficulty in discriminating finely between several models with different forms which appear consistent with the information in the data. In fact, in finite sample it often happens that if we do not restrict the parametric specifications, the problem of collinearity, shortage of degrees of freedom, etc. would so confound the interpretation of the model that we do not know what to make of it [Ando (1977), Klein (1977)]. Engle (197Q Hendry (1974), Granger and Newbold (1977), Wallis (1977), Zellner and Palm (1974) and many others 5For the results of usingthe conventionalmethod, see Hsiao (1982).
C. Hsiao,
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253
have suggested approaches to blend traditional econometric and time series analysis to cohstruct better econometric models. In this paper we used Granger’s (1969) notion of causality to derive various propositions about population properties of various causal events. These propositions can be used to provide an interpretation for multivariate time series models as well as a means for developing and checking the empirical implications of various variants of models and for ruling out a number of variants as being inconsistent. with the prior theorems. 6 We have applied these concepts to the time series analysis of Canadian money, income, and interest rate data. We believe that if we are able to use the degrees of freedom more efficiently a purely statistical analysis of economic time series data is capable of yielding useful information. As Granger (1980) has remarked, the concept of causality and procedures to fit time series model are topics in which individual tastes predominate. It would be improper to try to force research workers to accept definitions and procedures with which they feel uneasy. There is clearly a need for more discussion of the appropriate definitions of causality and for more exploration of various multiple time series modeling procedures. This paper represents a preliminary attempt to analyze multivariate time series data. The empirical result may or may not stand further scrutiny. However the topic in my view is of sufficient importance and interest to justify further work in this area rather than to be brushed aside as ad hoc or not serious. Appendix: Proof of various characterizations
of causal eve&’
Let the notation H,(r) stand for the completion with respect to the meansquare norm of the linear space of random variables spanned by y, for ss t. Let u, be the difference between yI and the projection of y, on H,,x,n(t- 1). Let & be the difference between x, and the projection of x, on H,,,,$1). Let u, be that part of 5, which is orthogonal to u,. Similarly, let w, be that part of the difference between z, and the projection of z1 on Hy,x,z(t- 1) which is orthogonal to u, and u,. By definition, u,, u,, and w, are contemporaneously uncorrelated and are uncorrelated with past values of each other. Also H,,,,,(r) is identical to HySx,Jt), and {y,,x,,z,} has a moving average representation of the form (3) [Rozanov (1967)]. We similarly define u: as the difference between y, and the projection of y, on H&t1) and u: as that part of the difference between x, and the projection of x, on HY,.Jt- 1) which is orthogonal to u:; hence H,,,,(t) is identical to H,,,(t). We also let u:* denote the difference between y, and the “For an illustration, see Hsiao (1982). ‘Hosoya (1977) has proved the Granger non-causality in non-stationary cases. The result does not seem easily generalizable to the proof here because in the non-stationary case the processis no longer a Cauchy sequence in Hilbert space, thus making it very dilkult to operate be&en different dimensions.
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projection of y, on HY,&- 1) and w,** denote that part of the difference between q and the projection of z, on H,,.(t - 1) which is orthogonal to u:*. Thus, H,,,(t) is identical to HP,+(t). Lemma I. Zf c$l21# 0 bnd $1 31# 0 for some 1 in (3), then the {y, x} process has an MA representation of the form
0(
Yf 4:1(L) Xl = dJt,w
4:2(L) u: ML) >( 0: )
(10)
with $rz,#O for some 1. Proof. We denote by D(t) the orthogonal subspa= H,, x.A%
complement
of H,,,(t)
in the
Then, by construction, u: ED@-- l)@u,,
(12)
where @ denotes the direct sum. ~,,&)~&,&) and o(t)=H,,,,(t)BH,,,.(t). = H,(t) . Therefore,’ H,*(t)Gzf,(t)8H,(t-
By construction we know that Hence o(t)~H,,.,,(t)OH,,,(t)
I).
(13)
By Weld’s decomposition theorem [Rozanov (1967)], {y,x} from moving averages of uncorrelated processes as follows:
Given
are obtained
=~:,@Ju:+4:,(Lb:,
(14)
= 4;,w:
(15)
that H,(t)lH,,,(f)
+ 4M4u:. and (13), we know
that
{y, x} may not be
sWe note that if Cp,3,=0 for all I and &,#O for some I, then we may switch the roles of 1) and H,(t-1) and construct an MA representation with 4T2,=0 for all I under certain conditions. H,(t-
C. Hsiao, Autoregressive modeling and causal ordering
255
represented as
x, = 62,w: Thus,
+ 622(~h.
(17)
. H”(t)EH:(t).
W
However, &I#0 for some 1 means that the projection of y, on H,(t - 1) is non-zero. Consequently, the projection of y, on H:(t- 1) is non-zero as well. Hence $&#O for some 1. Proof of Corollary 2. Expressing (10). Since a2(y 14 = a2(y 1A -Z), H,,.&1) lies in H&-l), and and u,lH,(t); hence, 4:,(~5)=4,,(L) then 4r2,=0 and $r3,= 0 for all y. If 4:21 #O for some 1, by (3) and
Letting C(L) = @2(L)/+f2(L),
{yt,x,} in terms of the @, $, we have we know that the projection of y, on therefore u:=u~. Furthermore u,lHV,,(t) and @i(L)=&(L). If &=O for all I, 1, and it is trivial that z does not cause (10) we have [Ansley et al. (1977)]
we have
Theorem 1 says that a2(y 1Y,Z)=a’(y 1Q if and only if there exists a moving average representation for {yl,z,} of the form
J+= 4Gw) 0 u:* 0Zf ( 4:w av) )( w:*>* By Lemma +131=0 for all From Theorem does not cause
(22)
1, a necessary condition to construct (22) from (3) is that 1. However, (21) says that 4ra1=0 for all 1, so are +231=O. 1 we know that 4 rsl=O and $J~~~=Ofor all 1 if and only if z y.
Proof of Theorem 2. We note that a2(y 1zZ)=a’(y 1p) implies that the moving average representation for {y,,z,} has the form (22) with #i=O for all 1. If both 4121=0 and r$131=0 for all 1, the projection of y, on H,,x,z(t- 1) lies in H,(t - 1) alone. That is, a2(y 1A) = 02(y I A -X-Z), and neither z nor x
256
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ordering
causes y. On the other hand, if both +I21 #O and rjr 31# 0 for some 1, then by Lemma 1, cbt$ # 0 for some 1. Therefore we assume 4121 #O for some I and 41sI=0 for all 1.’ With 4111#0, +121#0 for some I, the only way for (22) to hold is that
Such a representation exists if and only if there exists a C(L) such that
If 4231also equal zero for all 1, then Q(L) is lower block triangular, and hence a2(y 1A) = c2(y 1A -2). This is a contradiction and, therefore, 4231 # 0 for some 1. Together with bIzl#O for some 1, this implies that $r3r#0 for some 1, i.e., c2(y 1A) < fl”(y ) A -Z). Condition (ii) can be derived by taking the inverse of the Q(L) matrix. Proof of Corollary 3.
Condition
(i) of Theorem 2 is automatically
satisfied
with C(L) 3 0. Proof of Theorem 3. We first prove condition 02(y 17, Z) < cr2(y1Y) if and only if
(i). Theorem
y, = 4::(L) 9:w u,** 0Zt (W(L) MIL)>(w:*) ’
1 says that
(26)
with @zl#O for some 1. However, condition (ii) of no causality from z to y implies 4rS1=0 and 4231= 0 for all 1 in (3). Therefore, {y,,z,} has the following representation: Y, =
41
lw4
+
4J12uh
(27)
- From a2(z 1ii) < c2(z 1A-X) and g2(z I X, Z) < a2(z ( Z) we know that 4b321# 0 for some 1. Given (27) and (28), as the proof of Corollary 2 shows that a necessary condition for the existence of (26) is +t2, #0 for some 1, and no ‘Equivalently, we (6) should 13 .
yf$on
can assume be replaced
d, Izr=O for all I and 4131#0, &#O by the equivalent condition ol [4,,(L)
for some I; then q&,(L)]=C(L)[4,,(L)
C. Hsiao,
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257
existence of C(L) such that
Condition (i) implies condition (ii) follows straightforwardly from taking the inverse of Q(L). To show that condition (ii) implies condition (i), we note that h31=0, ~~31=0 for all 1 and $121#0 for some 1 jointly imply a’(yIa =a2(yp-Z)Io2(yp-X) and a2(xIA~=a2(xIA-Z). That is c&,=0, 4231 = 0 for all I and 4i2r #O for some 1. $a2, #O for some I implies that there does not exist a C(L) such that (29) holds. Proof of Theorem 4. We first prove condition (i). We note that a2(y 1A) = 02(y1A-Z) if a n d only if (21) holds. If C(L) = 0, then this implies that +121= 0, r$isl =O for all 1. That is, y can be viewed as generated by an innovation process that is uncorrelated with the innovations in the x and z processes. Therefore C(L) must be non-zero for a2(y 1r z) < a2(y 1q to hold. On the other hand if dz3,= 0 for all 1, by (21) +131=0 for all 1. Then by Theorem 1, (r2(x 1A) = 02(x 1A -Z). (Neither can 4 i 21--0 for all I, since this will imply 4221=0 for all 1.) When 4121#0, 4131#0 for some 1, by Lemma 1 the {j&z,} process will have 473: # 0 for some 1 in (26), that is a2(y I z z) < a20, I q. Condition (ii) can be proved by taking the inverse of @(L).
Proof of Coro&ry 4. Definition 1 says that if z does not cause y directly either 02(y I a=a2(yl A-Z) or a2(y I ~Z)=02(yl 7) or both. We first show that 02(y I a=a2(y I A-Z) and a’(y I EZ)=a’b I 7) cannot hold simultaneously under the assumption that z-x and x-y. We note that a2(yp)=a2(yp-z) implies (21) holds. Therefore, 4i2r $0 for some 1 unless C(L) is identically equal to zero --which is ruled out by the assumption that x=y. Thus by Lemma 1, a2(y ) Y,Z) = a2(y I ?) implies that 4i3[ =0 for all 1. By (21), 4231=0 for all 1. This contradicts the assumption that z*x. Therefore, either a2(yIA)=aZ(yIA-Z) and o’(y( Y,Z)<02(y) Y) or and a2(y 1KZ) = a2(y ( n. The former is indicated as a2(yIAT
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ordering
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