granger(1981)_properties of time series data

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Journal of Econometrics 16 (1981) 121-130. North-Holland Publishing Company

SOME PROPERTIES OF TIME SERIES DATA AND THEIR

USE IN ECONOMETRI C MODEL SPECIFICATION

C.W.J. GRANGER

Uni versity of Cali fornia at San Diego, La Joll a, CA 92093, USA

1 Introduction

It is well known that time-series analysts have a rather different approachto the analysis of economic data than does the remainder of the econometricprofession. One aspect of this difference is that we admit more readily tolooking at the data before finally specifying a model, in fact we greatlyencourage looking at the data. Although econometricians trained in a moretraditional manner are still very much inhibited in the use of summarystatistics derived from the data to help model selection, or identification, itcould be to their advantage to change some of these attitudes. In fact, I haveheard rumors that econometricians do data-mine in the privacy of their ownoffices and I am merely suggesting that some aspects, at least, of this practiceshould be brought out into the open.

The type of equations to be considered are generating equations, so that asimulation of the explanatory side should produce the major properties ofthe variable being explained. If an equation has this property, it will be saidto be consistent, reverting to the original meaning of this term. As a simpleexample of a generally non-consistent model, suppose that one has

yt = a + bx, + e,

where y, is positive, but x, is unbounded in both directions. A more specificexample is when y, is exponentially distributed and x, normally distributed.The only case when such a model is consistent is when b is zero. Although itwould be ridiculous to suggest that econometricians would actually proposesuch models, it might be noted that two models that appear in the financeliterature have

D,=a+bD,_,+cE,+e,,

and

P,=d+eD,+fE,+e,,

0165-7410/81/0000-0000/$02.50 0 North-Holland Publishing Company


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22 C.WJ. Gr anger, Ti me seri es dat a and econometri c model specr cati on

where D , represents dividends, E, is earnings and P, is share price. Note thatP, and D , are necessarily positive, but that E, can be both positive andnegative, as Chrysler and other companies can testify.

A further example arises from consideration of the question of whether ornot a series is seasonal. For the purposes of this discussion, a time series willbe said to be seasonal if its spectrum contains prominent peaks round theseasonal frequencies, which are 2rrjJ12, j = 1,2,. . ., 6, if the data are recordedmonthly. In practice, this will just mean that a plot of the series through timewill display the presence of a fairly regular twelve-month repeating shape.Without looking at the data, one may not know if a given series is seasonalor not and economic theory by itself may well not be up to the task ofdeciding. If now we look at a group of variables which are to be modelled,

how does the presence, or lack, of seasonality help with model specification?Considering just single-equation models, which are suitable for the simplepoint to be made, of the form

y,=a+bx,+cz,+e t, (1.1)

then it would clearly be inconsistent to require

(i) if y, were seasonal, xt, z, not seasonal that e, be white noise (or non-seasonal), or

(ii) generally, if y, were not seasonal, but just one of x, or z, was seasonaland that e, be white noise or AR(l) or any non-seasonal model.

Clearly, we may have information about the time-series properties of thedata, in terms of spectral shapes, that will put constraints on the form ofmodels that can be built or proposed. As the point is a very simple one, andways of dealing with the seasonality are well understood, or are at least cur-rently thought to be so, this case will not be pursued further. There is, however,a special case that is worth mentioning at this point. Suppose that y , is notseasonal, but that both x, and z, are seasonal, then is model (1.1) a possibleone, with e, not seasonal? In general, the answer is no, but if it shouldhappen that the term bx, +cz, is non-seasonal, then the model (1.1) is notruled out. This could only happen if there is a constraint f=c/b such thatthe seasonal component in x, is exactly the reverse of f times the seasonalcomponent in z,. A simple case where this would occur is if the seasonalcomponents is x, and zt were identical and f takes the value minus one. Thisdoes at first sight appear to be a highly unlikely occurrence, but an examplewill be given later, in a very different context, where such cancellations couldoccur.

It is obvious that the spectrum of one side of a generating equation, suchas (l.l), must be identical to the spectrum of the other side. If the spectrumof one side has a distinctive feature, it must be reproduced in the spectrum of


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C.WJ. Gr anger, Ti me seri es dat a and econometri c model specr cati on 123

the other side; obvious examples being periodic components, such as theseasonal and trend terms. The majority of this paper will be concerned withdiscussions of this point in connection with a generalized version of the trend

term.

2. Integrated series and filters

To proceed further, it is necessary to introduce a class of time seriesmodels that have been popular in parts of electrical and hydraulicengineering for some years, but which have so far had virtually no impact ineconometrics.

Suppose that x, is a zero-mean time series generated from a white noise

series a, by use of the linear filter a(B), where B is the backward operator, sothat

xt=4 , (2.1)

with

Further suppose that

u(B)= (1 -B)-%(B), (2.2)

where a(z) has no poles or roots at z =O. Then x, will be said to beintegrated of order d and denoted

x,-Z(d).

Further, defining

x;= (1 -B)dX,=.(B)Et,

then

x:-z(o)

from the assumed properties of a(B). a(B) will be called an integrating filterof order d. If a(B) is the ratio of a polynomial in B of order m divided by apolynomial of order I, then x, will be ARIMA (I, d, m) in the usual Box and

Jenkins (1970) notation. However, unlike the vast majority of the literatureon ARIMA models, the class of models here considered allow the order ofintegration, d, to be possibly non-integer. Clearly, not constraining d to be an


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124 C.WJ. Granger Ti me seri es data and economet ri c model specr cati on

integer generalizes the class of models before considered, but to be relevant,the generalization has to be shown to be potentially important. Some earlieraccounts of similar models may be found in Hipel and McLeod (1978),

Lawrence and Kottegoda (1977), Mandelbrot and Van Ness (1968) andMandelbrot and Taqqu (1979), although some details in the form of themodels are different than those here considered, which were first introducedin Granger and Joyeux (1981).

Some of the main properties of these models may be summarized asfollows: The spectrum of x,, generated by (2.1) and (2.2), may be thought ofas

.Uo)= (l/II -z/2d)la(412,

if var (Ed)= 1, from analogy with the usual results from filtering theory. It isparticularly important to note that for small w,

fx(o)=CW-2d. (2.3)

It was shown in Granger and Joyeux (1981) that the variance of x, increasesas d increases, and that this variance is infinite for d 23, but is finite for d ~4.Further, writing

x,= f bjc -jj=O

and denoting

pj = correlation (x,, x,_ j),

then, for j large,

pj=AljZdvl, d-c+, d O,

andbj=A2jde1, d51 df0,

where A, and A, are appropriate constraints. When d =O, both pj and bjdecrease exponentially in magnitude as j increases, but with d PO, it is seenthat these quantities decline much slower. Because of this property theintegrated series, when d O, have been called long-memory. For long-termforecasting, the low frequency component is of paramount importance and

(2.3) shows that if d is not an integer, this component cannot be wellapproximated by an ARIMA (1, d, m) model with integer d and low order for1 and m.


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It is not clear at this time if integrated models with non-integer d occur inpractice and only extensive empirical research can resolve this issue.However, some aggregation results presented in Granger (1980) do suggestthat these models may be expected to be relevant for actual economicvariables. It is proved there, for example, that if xjt, j= 1,. . .,n, are set ofindependent series, each generated by an AR(l) model, so that

Xj =ajXj,t- 1 +Ejt, j=l 9 . ., N,

where the sjt are independent, zero-mean white noise and if the ajs arevalues independently drawn from a beta distribution on (0, l), where

dF(cr)= (2/B(p,q))azP-(1 -GC*)~-~ da, OSaSl, (2.4)P>O, 4>0,

then, if Xt= 2 xj,(, for N largej=l

X,-1(1 -q/2). (2.5)

The shape of the distribution from which the as are drawn is only criticalnear 1 for this result to hold.

A more general result arises from considering xjt is generated by

xj,=a.x. J J t-1 +Yj,t+bjK+ jt> (2.6)

where the series yj,,, & and sjt are all independent of each other for all j,sjt are white noise with variances oj2, yj,, has spectrum &(cu, Qj) and is at leastpotentially observable for each micro-component. It is assumed that there isno feedback in the system and the various parameters a, Bj, /l and 0 are allassumed to be drawn from independent populations and the distributionfunction for the LYS s (2.4). Thus, the xj,( are generated by an AR(l) model,plus a independent causal series Y,,~ and a common factor causal series W;.With these assumptions, it is shown in Granger (1980) that (i)

where d, is the largest of the three terms (1 -q/2 + d,), 1 q + d, and(1-q)/2, where &-I@,), kK~l(d,), and (ii) if a transfer function model ofthe form

is fitted, then both a,(B) and a,(B) are integrating filters of order 1-q.

JOE-E


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It should be noted from (2.6) that, if c(~< 1 then the spectrum of Xj,t is

so that if one assumes that x~,~ Z(O) it necessarily follows that yj,, and Kare both Z(0).

In Granger (1980) it was shown that integrated models may arise frommicro-feedback models and also from large-scale dynamic econometricmodels that are not too sparse. Thus, at the very least, it seems thatintegrated series can occur from realistic aggregation situations, and so dodeserve further consideration.

3. The algebra of integrated series and its implications

The algebra of integrated series is quite simple. If x,-Z(&) and a(B) is anintegrating filter of order d, then a(B)x, will be I(d,+d). Thus, d, isunchanged if x, is operated on by a filter of order zero. Further, if x, -Z(d,),y,-Z(d,) then z, =bx,+cy,-Z(max(d,,d,)) in general. This result is provedby noting that the spectrum of z, is

(3.1)

where CT(O) is the cross-spectrum between x, and y, and has the propertythat Icr(o)l ~J,(o&(,(o). For small o,

fX(w)=A3W-2dx and f,(w) = AoPZdy,

and clearly the term with the largest d value will dominate at lowfrequencies. There is, however, one special case where this result does nothold, and this will be discussed in the following section.

Suppose now that one is considering the relationship between a pair ofseries x, and y,, and where d, and d, are known, or at least have beenestimated from the data. For the moment, it will be assumed that d, and d,are both non-integer. If a model of the form

wqY,=cwx,+w ~, (3.2)

is considered, where all of the polynomials are of finite order, and willusually be of low order, and E, is white noise, independent of xt, then thismodel is consistent, from consideration of sRectra1 shapes at low frequencies,

only if d, =d,. If one knows that d, cd, then to make the model consistent,either c(B) must be an integrating filter of order d,-d, or h(B) is anintegrating filter of order d,, or both. In either case, the polynomials cannot


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where d,>O, ME, is Z(d,) and var(c) = 1. The spectrum of the right-handside will be

(4.2)

where now cr(w) is the cross-spectrum between X, and y,. The special case ofinterest has :

(i) fX(o)=a2fi(o), w small, so d,=d,,(ii) cr(w)= ct.,(w), w small, so that the coherence C(o)= 1 and the phase

4(0)=0 for w small.

A pair of series obeying (i) and (ii) will be called co-integrated.

If further, g= -cca, the part of the spectrum (4.2) inside the main bracketswill vanish at low frequencies and so a model of the form (4.1) will beappropriate even when d, 0. Thus, if unanticipated money supply, x,+ 1 -fn,,l, is used in a

model, this can be appropriate if the variable being explamed has d,>O.


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It should be emphasized that for this result to hold fn,h must be onoptimal forecast and, if d, is not an integer, then this means that intheory the forecast has to be based on an infinity of lagged xs. If 1 >d,> 0, but an ARIMA (1, d, m) model is used to form forecasts, with integerd, the forecasts will not be optimal and the series and its forecast willnot be co-integrated.

There obviously are pairs of economic series, such as prices and wages,which may or may not be co-integrated and a decision on this has to bedetermined by an appropriate theory or an empirical investigation. It mightbe interesting to undertake a wide-spread study to find out which pairs ofeconomic variables are co-integrated.

In the frequency domain, the conditions for co-integration of two seriesstate that the two series move in a similar way, ignoring lags, over the longswings of the economy and in trend, although the idea of trend is rarelycarefully defined and will here mean just the very low frequency component.Although the two series may be unequal in the short term, they are tiedtogether in the long run.

The use of the difference between two series to explain the change in aseries has been suggested by Sargan (1964) and Hendry (1978) andimplemented in a number of models, particularly in Britain. An example is amodel of the form

and the use of the term p( y,_ 1 -x,_ 1) has been found in some cases toproduce a better model, in terms of goodness of lit. The form of the modelhas an important property. The difference equation without the innovations

e,,

is such that if x, and y, each tend to equilibrium, so that Ax,+0 and dy,+Othen x, and y, tend to the same equilibrium level. When the stochasticelements e, are present, equilibrium becomes much less meaningful, and isreplaced by x, and y, tending to having identical means, assuming the meansexist. However, if the d values of x, is greater than d,, this generating modelensures that x, and yt will be co-integrated. They, therefore, will move closelytogether in the long run, which is possible the property that most naturallyreplaces the concept of equilibrium for stochastic processes. It is important

to note that this property does not hold if d,=d, =d,, as then the coherencebetween x, and y, need not be high at low frequencies, depending on therelative variances of e, and J,.


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5. Conclusion

Having, I hope, made a case for the prior analysis of time series data

before model specification inter-relating the variables, it has now to beadmitted that the practical implementation of the rules suggested above isnot simple. Obviously, one can obtain satisfactory estimates of the spectrumof a series, but it is not clear at this time how d values should be estimated.In the references given earlier, a variety of ways of estimating d aresuggested, and a number of sensible modifications to these can easily beproposed, but the statistical properties of these d estimates need to beestablished. It is possible that too much data is required for practical use ofthe specification rules or that d values for real economic variables are allintegers. Only further analysis, both theoretical and empirical can answerthese questions.

References

Box, G.E.P. and G.M. Jenkins, 1970, Time series analysis, forecasting and control (Holden-Day,San Francisco, CA).

Davidson, J.E.H., D.F. Hendry, F. Srba and S. Yeo, 1978, Econometric modelling of theaggregate time-series relationship between consumers expenditure and income in the UnitedKingdom, Economic Journal 88, 661692.

Granger, C.W.J., 1980, Long-memory relationships and the aggregation of dynamic models,Journal of Econometrics 14, 227-238.

Granger, C.W.J. and R. Joyeux, 1981, An introduction to lone-memorv time series andfractional differencing, Journal of Time Series Analysis V.1. - _

Hioel, W.H. and A.I. McLeod. 1978, Preservation of the resealed adiusted range, Part 1, WaterResources Research 14, 491-518.

Lawrence, A.J. and N.T. Kottegoda, 1977, Stochastic modelling of river-flow time series, Journalof the Royal Statistical Society A140, 147.

Mandelbrot, B.B. and J.W. Van Ness, 1968, Fractional Brownian motions, fractional noises andapplications, Siam Review 10, 422437.

Mandelbrot, B.B. and MS. Taqqu, 1979, Robust R/S analysis of long-run serial correlation,Research report RC 7936 (IBM, Yorktown Heights, NY).

Sargan, J.D., 1964, Wages and prices in the United Kingdom: A stu y in econometricmethodology, in: P.E. Hart, G. Mills and J.K. Whitacker, eds., Econometric analysis for

national economic planning (Butterworths, London).

granger(1981)_properties of time series data

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