Journal of Econometrics 43 (1990) 275-292. North-Holland

AN ANALOGUE MODEL OF PHASE-AVERAGING PROCEDURES*

Julia CAMPOS

Bunco Cerrtrul de Venezuekr, Curucus 1010, Venezuelu

Neil R. ERICSSON

Federul Reserve Bad, Wushingtotl, DC 205.71. USA

David F. HENDRY

NuSJield College, Oxford OX1 1 NF, England

Received May 1987, final version received January 1989

This paper analyses the statistical and econometric effects that fixed n-period phase-averaging has on time series generated by some simple dynamic processes. We focus on the variance and autocorrelation of the data series and of the disturbance term for levels and difference equations involving the phase-average data. Further, we consider the effect of phase-averaging on the exogeneity of variables in those equations and the implications phase-averaging has for conduct- ing statistical inference.

To illustrate these analvtical results, we examine features of the observed annual and phase- average series on money, &come, prices, and interest rates from Friedman and Schwartz (1982). While the analogue model developed above is a simplified characterisation of the NBER phase-averaging techniques employed by Friedman and Schwartz, it does offer several insights into the likely consequences of this approach.

1. Introduction

The nature of business cycles has evoked extensive research in economics over the last several decades, including such diverse studies as Beveridge

*This research was supported in part by U.K. E.S.R.C. grants HR8789 and B00220012. We arc grateful for the financial assistance from the E.S.R.C. although the views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting those of the E.S.R.C.. the Board of Governors of the Federal Reserve System, the Banco Central de Venezuela, or other members of their staffs. Helpful discussions with and/or comments from Rob Engle, David Howard. Bob Marshall, David Pierce, Ken Wallis. and an anonymous referee are gratefully acknowledged. We are indebted to Aileen Liu for excellent research assistance in preparing the figures. This paper is based in part on Hendry and Ericsson (1985, app. B), itself developed from appendix B for Hendry and Ericsson (1983).

0304-4076/90/$3.501c‘ 1990. Elsevier Science Publishers B.V. (North-Holland)

276 J. Cumpos er 01.. An unalogue model of phuse-ctverugingprocedures

(1921) Slutzky (1937) Schumpeter (1939) Burns and Mitchell (1946) Friedman and Schwartz (1963, 1982) Sims (1977) and recent investigations on ‘real business cycles’ by (e.g.) Lucas (1977) Long and Plosser (1983). and King, Plosser, Stock, and Watson (1987). Simplifying (and in some cases, overly so), economic series might be imagined to be composed of a trend and a cycle.’ Burns and Mitchell (1946). for example, focus on the cycle, abstracting from noncyclical temporal differences. To do so, they first establish the NBER reference business cycles, dating the contraction and expansion phases of the cycles over the sample period, and then average the data across cycles for various points within the cycle. Conversely, Friedman and Schwartz (1982) focus on the longer-run features of the data. To filter out short-run (i.e., cyclical) properties of the data, they transform (raw) annual series by averag- ing separately over contraction and expansion phases of the reference business cycles.* Our paper analytically and empirically evaluates statistical and econo- metric effects of phase-averaging on economic time series.

Use of phase-average data raises three distinct issues: (i) the theoretical effects of phase-averaging (qua time aggregation), (ii) the observed effects of phase-averaging, and (iii) the effects of selecting the intervals over which to average (e.g., by prior analysis of an interrelated data set such as that underlying reference business cycles). In section 2, we abstract from (iii) and address (i), using as a model fixed n-period phase-averaging of time series generated by a simple dynamic process.3 We focus on the effects that phase-averaging has on the variance and autocorrelation of the data series and of the disturbance term for levels and difference equations involving phase-average data. Further, we examine the effect of phase-averaging on the exogeneity of variables in those equations and the implications phase-averag- ing has for conducting statistical inference. To illustrate those analytical results, in section 3 we examine features of the observed annual and phase- average series on money, income, prices, and interest rates from Friedman and Schwartz (1982) i.e., (ii). While the analogue model from section 2 is a simplified characterisation of the NBER phase-averaging techniques employed by Friedman and Schwartz, it does point to several likely consequences of this approach.

‘Allais (1966). Neftqi (1984, 1986). and Stock (1987) inrer alra present various reasons for the existence of business cycles and give empirical evidence thereon.

‘See Maverick (1933) for early advocacy of phase-averaging in econometrics.

‘Without undertaking a Monte Carlo study, it is difficult to deduce the statistical effects of selecting the phases over which to average when using information from related time series [i.e., (iii)]. However. a first approximation is given by examining the effects of fixed-length phase-aver- aging. The model of phase-averaging with fixed phase lengths below explains salient features of the data for velocity, leaving a small role for the effects of selection of phase lengths.

J. Cumpos et ul., An unalogue model of phase-averaging procedures 271

Original series {x,} x0 x1 X2 x3 x4 X5 X6 x,

Filter

- ii, - -

Fig. 1. A schema for fixed three-period phase-averaging.

2. Analytical effects of phase-averaging

This section describes a model of phase-averaging and considers the effects that phase-averaging has on time series generated by some simple dynamic

processes. Phase-averaging sequentially applies two filters to the annual data: the first

averages that data (as with a moving average) and thus implies a reparameteri- sation of the data-generation process; the second selects the phase-average data from the averaged series (i.e., marginal&es the data density with respect to the intermediate observations), thereby entailing a (statistical) reduction in that reparameterisation. By analogy with seasonal adjustment of quarterly data, the first filter is like the X-11 procedure; the second discards all but the

fourth-quarter data points. For analytical tractability, consider fixed n-period phase-averaging of the

annual series {x!; t= . . . . 1,2 ,..., T} with T a multiple of n. Letting L be

the lag operator such that Lx, = x,_~, the first filter is (1 + L + L2 + . . . + L” _ ‘)/n, so the averaged series is

xt*= (X,+X,pl+X,_Z+ ... +x,-,+,)/n, (1)

t= . . . . 1,2 ,..., T.

The second filter selects every n th observation of x7, so the phase-average seriesis{x,T,; j= . . . . 1,2 ,..., (T/n)},denoted{x,; j= . . . . 1,2 ,..., J}where J = T/n. Fig. 1 shows the steps involved in going from { xt} to {XI” } to {X,} for fixed three-period phase-averaging.4

41n fact. Friedman and Schwartz include turning points in both preceding and following phases (but weight them by half the normal weight in each), so the first filter is actually (0.5 + L + L2 + + L”- ’ + OSL”)/tt and the second is unchanged. However, the statistical effects of using phases-averages with (rather than without) overlap appear minor in comparison to those of using phase-average (rather than annual) data (cf. pp. 75, 84-85). Here and below, unreferenced page numbers refer to Friedman and Schwartz (1982).

278 J. Cumpos et al., An analogue model of phase-uveraging procedures

There is little loss of information information results from the selection process.

That loss would be if (a) all the parameters of interest could be recovered from the phase-average series and (b) there were no loss of power in testing the resulting model. However, many parameters which often are of interest cannot be obtained from phase-average data, including parameters relevant to tests of Granger noncausality, short-run in the postu- lated relationships,

exogeneity: as will be seen below, variables weakly exogenous at annual observations

information is also apparent if phase-averaging

frequencies. While it would be possible to derive the following results by spectral we work in the time domain because the concepts which arise are ones of the time domain (contemporaneous exogeneity), phase-averaging time-domain filter, and the models are most parsi- monious in the time domain (as is the process generating the data).

To analyse the effects of phase-averaging, T} generated by the simple dynamic bivariate

process:

y, = a0 + (YlZ, + u,, (2)

Except when otherwise noted, we assume that ( yt : z,)’ is stationary; hence all roots of (2)-(4) are stable, i.e., IpI < 1 and 1~1 < 1 where K = A, + a,h,. If X z = 0, z, is strongly exogenous for ( a0 : al)‘; otherwise it may be correlated

‘For instance, cf. Banerjee. Dolado, Hendry, and Smith (1986) on cointegration tests with static and dynamic models and note the similarity between those models and the phase-average and annual models herein.

J. Cumpos et ut., An onafogue model ofphuse-aueraging procedures 279

with u r.6 Although (2)-(4) do not include a deterministic cycle, for suitable parameters, they easily can generate data with cyclical behaviour.7 The n-period phase-averaging of the data in (2) gives

y, = (YO + cur?, + uj, j=1,2 ,...,(T/n), (5)

where (e.g.)

5, = n-l f zc,-l),,+,.

/=1

Average two-(phase-)period differences result in

(A,Y,/2) =q(A2F,/2) + (A,u,/2), j=l,T...,(T/n), (6) where differencing for phase-average data is defined in terms of phases rather than years so that (e.g.)

A2X,=X,-X j-2’

Our concern is how inference is affected by using (5) [or (6)] rather than (2)-(4). Unless otherwise noted, discussion is limited to the effects on

‘If (_r; : z,)’ is normally distributed, conditional upon its own past, and the conditional mean of ( .v, : z, )’ is linear in its past and only includes y,_ , and z,_ t, then the corresponding conditional density of .,; (conditional upon z,) is yr = 8, + S,z, + a,~,_, + &z,_i + E, and the corresponding marginal density for z, is (4). If the conditional density of y, includes a common factor (1 - pL), that density can be written as (2)-(3). Cf. Hendry, Pagan, and Sargan (1984) for further discussion on common factors,

Technically speaking, if X, + 0 and so Z, is not strongly exogenous for (a,, : q)‘, then z, is not wecrklv exogenous for (a(, : a,j’ either, because stationarity requires (X, + a,X,( i 1 and so (h, : A,) and (a(, : a, )’ cannot be variation free; cf. Engle, Hendry, and Richard (1983, p. 282). Weak exogeneity could be ‘restored’ either by assuming that (X, : A,)’ and (a,,: a,)’ lie within a subspace of the region for stationarity where they are variation free or by relaxing the condition of joint stationarity to that of the conditional stationarity of JJ, (requiring only that IpI < 1). In the latter situation, if 2, is integrable of order 1, then so will be y,, and y, and Z, will be cointegrated. For further details on cointegration, see Engle and Granger (1987) Phillips and Durlauf (1986). and articles in the August 1986 special issue of the Oxford Bulletin of Economics und Statistics on cointegration.

‘For instance, Slutsky (1937) demonstrates that moving uuerage processes can exhibit cyclical behaviour, and (2)-(4) can be well approximated by a moving average process.

‘In the context of Friedman and Schwartz (1982), it is natural to interpret ‘periods’ and ‘phases’ as years and half-cycles. However, the analytical properties are quite general and could be applied to other time intervals, e.g., quarterly data aggregated to annual data. In fact, a substantial literature exists on the effects of time aggregation; cf. Theil (1954), Telser (1967) Amemiya and Wu (1972). Sargan (1976) Tiao and Wei (1976) Wei (1978), and Weiss (1984). Those studies show that, inter ah. sizeable inefficiencies result from using least squares even if the regressors are strongly exogenous. Further, if they are only weakly exogenous (e.g., include lagged dependent variables), least squares is generally inconsistent; cf. Wei (1982). Our results below are in line with those general results. However, the motivation for our study is fundamentally different from those of time aggregation. In the latter, the data are generated at unit intervals but observation occurs at multiples of that unit: hence relevant questions include the effects on inference, forecasting, and policy analysis of not having available observations at natural time intervals. In the case of phase-averaging. observations at the unit interval are available, but the investigator chooses to use the aggregated (phase-average) data.

280 J. Campos et ul., An analogue model of phase-averaging procedures

phase-average data of positive p because economic data tend to be highly positively autocorrelated. The effects of negative p are qualitatively different, particularly for n even; cf. (9) and (10) below.9 However, the formulae derived apply for al/ IpI -c 1.

Levels of phase averages. As a baseline, we note that the variance and auto-covariances of U, in (2) arelo

T, = E( u,u,_;) = $ir,,~~z, i = 0,

E 3 i=1,2 )... .

(7) (8)

Hence the autocorrelation function for U, is p’ (i > 0). Equivalent formulae for tij in (5) are

?, = E( tc,U,_,) =

0,” I [ n(l -P)’

1_ 2P0-p”)

41 -P”)

P .W,(, - pya,’

n2(1 - p2)(1- p)2 ’ i=1,2 )... . (10)

The error variance of (5) is less than that of (2) for all n > 1, but both equations manifest residual autocorrelation unless p = 0. The logarithm of the variance of ii, (normalised by u,‘) and its first-order autocorrelation rl

( = E( ii,U,_ ,)/E( 6:)) are shown in figs. 2 and 3, respectively. The variance falls as n increases, but with an ever smaller proportional effect as p increases. As an example, for p = 0.8 and n = 3, (9) yields F. = 2.3~~~ whereas IJ,’ from (7) is 2.8u,*, so averaging produces only a small decrease in variance. More- over, as is apparent from fig. 3, phase-averaging only marginally reduces the first-order error autocorrelation when p is large. By contrast, autocorrelation at higher orders dies off rapidly as n increases [nb., the exponents of p in (lo)].

Least-squares estimation of structural parameters in the presence of auto-

‘Figs. 2-5 are only for positive p and (where applicable) first-order autocorrelation; Campos. Ericsson, and Hendry (1987) contains comparable graphs which include negative values of p and higher orders of autocorrelation.

“Telser (1967. table 1) obtains parallel results for n = 3 (i.e., his m = 2). noting that his table 1 is for a typical ‘summed process’. Thus, his c(k) is our ?(k/n) (1 - p’) .n , where k is a multiple of n and his parameter c1 is our p. Ahsanullah and Wei (1984) obtain equivalent formulae for the AR(l) process: Stram and Wei (1986) generalise those results to ARIMA processes.

J. Cumpos et ul., An anulogue model of phase-averaging procedures 281

n=l 3- ----_- n=2

-----_-- n=3 ----

2-

-, ____-----

/- /-

-/-

-2 “““““““““‘I 0 .I .2 .3 .4 5 .6 .7 .a .9 1.0

P

Fig. 2. The logarithm of the variance of U, (normalised by u,‘) as a function of p and n.

1.0 -

- ----_- n=2 __-_-_-- n=3

o,6 _ ----n=5

0.6 -

0.4 -

0 .I .2 .3 .4 .5 .6 .7 .a .9 1.0 P

Fig. 3. The first-order autocorrelation coefficient of ii, as a function of p and II.

282 J. Cumpos et ul., An unalogue model ojphuse-averaging procedures

correlated disturbances is inefficient. Further, if right-hand-side variables are correlated with the disturbance, such estimation is inconsistent:

E(z,u,> = ph,~,~/[(l- p2)(1 - PK)] = Q, (say), 01)

which is zero only if ph 2 = 0. E( zt_ ;e,) = 0 for all i 2 0 (and for all p and A 2), so estimation of (2)-(3) by autoregressive least squares is always consistent. Turning to the phase-average data in (5):

E( z,U,) = A,$(1 + ~)/I40 - P)(I - PK)] 3 (14

when n = 2 (for illustrative purposes). Thus, even if p = 0 (and so a,, = 0), E( z/G,) # 0 unless X2 = 0. Conversely, the strict exogeneity of z, (i.e., A, = 0) is sufficient for both (11) and (12) to be zero.

Differences of phase averages. Next, we consider the effects of two-period differencing of-the phase averages in (5) to get (6):

var( A 2U,/2) = 0,’

2n(l - p)’ l-

and

p(2 - P”)(l - P2n) fi(l- P2) 1 (13) (1 - pZ”)p(l - p”)2u,2

E[(A,u,/2)(A,u,-,/2)1 = 4n2(1 _ p2)(1 _ p)’

After resealing (13) by 4, the error variances for levels and ‘rates of change’ models will be similar only for quite large values of p (e.g., p = 0.9) in which case the former will still exhibit substantial first-order autocorrelation al- though the latter will not; cf. figs. 4 and 5.” Further, and in contrast to the results for levels, the differenced series { A,U,/2} exhibits (typically negutiue) second-order autocorrelation which is near zero only for large values of p. For equations with the differenced data, the formula paralleling (12) is

E[(G,)(A,u,)] = [2 - (1 + K)2K2] .X,0,2/4, 05)

when p = 0 and n = 2. So, in general, different biases will result from fitting (5) and (6). Only if z, is strictly exogenous (h, = 0), will both biases be zero.

Phase average of stationary AR(I) data. One particularly interesting special case of (2)-(4) is the stationary mean-zero AR(l) process, i.e., LYE = CX~ = 0 in

“The variable-length weighting procedures in Friedman and Schwartz (1982) should not alter the substance of this last result; but being data-based, it could seriously alter the nature of the residual autocorrelation. For example, the negative second-order autocorrelation of differenced data may not carry over for data-selected phase averages.

J. Cumpos et 01.. An analogue model of phase-uveragingprocedures 283

1 -n=l 0.5 -_----n=2 /’

--__-_-_” =3 / _---_n=5 / .

-2.5 11 11 11 11 11 11 11 11 11 11 0 .l .2 .3 .4 .5 .6 .7 .a .9 1.0

P

Fig. 4. The logarithm of the variance of A,U,/2 (normal&d by 0,‘) as a function of p and n.

0.6

0.5

0.4

r; 0.3

n=l ------ n=2 -_-_-___ n=3 ----

Fig. 5. The first-order autocorrelation coefficient of A,“,/2 (rI*) as a function of p and n.

284 J. Cumpos et ul.. An unalogue model

(2). That yields

Y, = PY,-1 + Et3 t=l,...,T,

with uJ? = a,‘/(1 - p2). For n-period phase-averaging,

.P, = P,Y,-~ + e,, j=l 7.e.7 J,

where pp is defined by [E(j,2_1)]-‘E(jjjj_1), i.e., E(j,_,e,) = 0. Thus,

(16)

(17)

C?(y) = 0;/(1 - p;) =$[n(l - p’) - 241 - p”)]/[n2(1 - P)‘]

08)

(which is equivalent to To) and

Pp = PC1 - P”)2/qn(l - P2> - 2P(l - P”>l (19)

(which is ?i/?,). Moreover, Ay,= E*+ (p - l)y,_, and AjJ,= ej+ (pp - l)j,_, so that a’(Ay) = 2ue2/(1 + p) and u2(Aj) = 2ue2/(l + pp). These formulae will be used in section 3 for analysing the empirical characteristics of velocity and will be contrasted with similar formulae for the random walk model (developed below).

Phase averages of a random walk. The final alternative we consider is that the series is a random walk. If p = 1 in (16) then E(y:) = tu,’ (for y, = 0) and the mean of the standard estimator of the variance of yt is

E[&+) =+T(T+l)/[2(T-I)] =&r) (20)

[see Fuller (1976, p. 367)] whereas u2(A_v) = u,‘. Working (1960) considers an n-period average of a random walk and shows that, for e, defined by j, =

jj_l + e,, its variance is

u,’ = uE2(2n2 + 1)/(3n). (21)

However, the mean of the standard estimator of the variance of yj is

~2. J(3Jn2 - n2 + 3n + 1)/[6n( J - 1)] = u,‘(J),

(22)

J. Cunlpos et ul., An unalogue model of phase-averaging procedures 285

assuming y,, = 0. The variance u,‘( jj) is virtually linear in T and nearly invariant to n, given T. That follows from (22) which gives u:(y) = a;(y) + O(T’) = aF2[( T/2) + O(T’)], where each of the terms O(T”) depends upon n only, to 0( T- ‘).12

3. Interpreting Friedman and Schwartz’s application of phase-averaging

To illustrate the value of the analytical results above, we examine the annual and phase-average series on U.K. velocity from Friedman and Schwartz (1982) and compare the actual data properties with the predictions of the analogue model and those by Friedman and Schwartz. We have chosen velocity, given

its prominence in discussions of money demand (cf. their chapters 2,6-7). The analogue model for an AR(l) process implies properties for the phase-average and annual data closely in line with those observed. The evidence also suggests a substantial loss of information from phase-averaging. Finally, we compare the properties of money-demand equations using annual and phase-average

data. Friedman and Schwartz justify phase-averaging on the grounds that it

reduces cyclical effects (their pp. 13-14, 78, 81) thereby allowing them to focus on their primary concern, monetary trends. In filtering out the cycle, they argue that phase-averaging reduces serial correlation in the data (p. 78) and attenuates measurement errors (p. 86): each implies a reduction in the data variance. Attaining those effects is important to sustain the validity of their statistical analysis of money, income, and prices since no explicit account is made for biases in coefficient estimates and estimated standard errors. Thus, variances and autocorrelations are measures with more than usual importance.

Time-series evidence. To evaluate some basic time-series properties of veloc- ity u, consider the following first-order autoregressions with annual and

phase-average data over the last century for the United Kingdom:13

U^,= 0.019 + 0.968 u,_r, (23) (0.016) (0.029)

T= 95, a^=4.798%, dw= 1.13, dw* =0.08, dw** = 1.15,

“Due to the nonstationarity of the series generated, in/erence requires special attention; cf. Dickey and Fuller (1979, 1981), Evans and Savin (1981, 1984), Sargan and Bhargava (1983). Phillips and Durlauf (1986) and Engle and Granger (1987).

“The (logarithm of) velocity is v = ln( I’. I/M), where P. I is nominal income and M is the money stock. The appendix briefly describes these data.

Our estimates and summary statistics using phase-average data are based on weighted least squares, correcting for the different phase lengths. However, parameter estimates are not very different whether ordinary or weighted least squares are used.

One phase observation is lost in the calculation of lags, so the sample sizes are J = 35 (phase observations 2-36) and T= 95 (annual observations 1879-1973), with the span of the annual data matching that for the phase-average data.

286 J. Cumpos et ul., An unulogue model of phase-averaging procedures

Series

Table 1

Standard deviations of velocity (percentages).

Predicted values

Observed values AR(l) model Random walk model

V 17.35 17.14 33.43 u 17.31 16.84 33.60 ALI 4.80 4.85 4.80 A6 7.19 6.82 6.67 E 4.80 4.80 4.80 e 7.28 6.68 6.67

and

:, = 0.023 + 0.972 (0.042) (0.077)

uJ_t, (24)

J= 35, 0^=7.279%, dw= 1.70, dw* =0.31, dw** = 1.74.

Values in parentheses (.) are conventionally calculated standard errors, dw

denotes Durbin and Watson’s (1950, 1951) statistic, dw* is its value when the dependent variable is regressed on a constant, and dw** is its value when the first difference of the dependent variable is regressed on a constant.t4 The series in levels (both annual and phase-average) show first-order autocorrela- tion close to unity, with the resulting differenced series being far less positively autocorrelated than the levels, and close to ‘white noise’ for the phase-average data. Those features influence the standard deviations of velocity, listed in the second column of table 1. For the annual and phase-average series, the standard deviations for levels are roughly equal; large reductions in standard deviations are obtained by differencing, more so for the annual data. Other series from Friedman and Schwartz (1982) have similar properties.

Assuming that velocity is a stationary AR(l) process and that n = 3 and p = 0.96, the analogue model in section 2 predicts standard deviations for the annual and phase-average data in terms of a,, the standard deviation of the underlying innovation process. Those values appear in the third column of table 1 and closely match the corresponding observed values.” Further, the implied first-order autocorrelation coefficient of the phase-average data is pP = 0.92, close to that in (24) above. While we think velocity is more

r4The Durbin-Watson statistic dw* and the r-ratio for (B - 1) may be interpreted as statistics for testing for the nonexistence of a relationship between money, prices, and income: cf. Sargan and Bhargava (1983), Dickey and Fuller (1979, 1981) and Engle and Granger (1987).

“The values in columns 3 and 4 of table 1 have been normalised on the annual AR(l) regression residual standard error because the results in section 2 are for relative, not absolute, variances.

J. Cumpos et al., AU anulogue model ofphuse-averaging procedures 287

1.0

E -Annual data (v,) - - - - - - Phase-average data (Vi)

0.8

0.6

01 I I I I I I I I I I I 1880 1900 1920 1940 1960 1980

Fig. 6. The logarithm of the velocity of money for the United Kingdom: annual and phase-aver- age data.

complicated than a univariate AR(l) process [and give evidence supporting that view in Hendry and Ericsson (1990)], these results nevertheless show the ineffectiveness of phase-averaging in eliminating ‘ . . . the bulk of the systematic cyclical fluctuation. . . ’ (p. 81) and hence reducing the serial correlation arising from cyclical components and the data variance. These results are seen more informally in fig. 6 which jointly plots the phase-average and annual data for velocity in the United Kingdom. As is visible, phase-averaging negligibly reduces either the variance or serial correlation in this time series. These results also explain why the models in Friedman and Schwartz using phase- average data fit less well than those in Hendry and Ericsson (1990) using annual data.

Alternatively, the series for velocity might be generated by a random walk. Certainly, the graphs of velocity and of several other variables of interest to Friedman and Schwartz (1982) look like those of the random-walk series in Working (1934) and (e.g.) p = 0.96 is not ‘far’ from unity.16 Also, that hypothesis for velocity has received considerable attention in the literature [cf. Gould and Nelson (1974) Nelson and Plosser (1982) and Bhargava (1986)]. The last column of table 1 lists various standard deviations for T = 95, .I = 35 (and so n = T/J = 2.71), assuming p = 1. Those results match the observed

IhAs noted in Hendry and Ericsson (1985), it is difficult to reject the null that p = 1; but that could be due to the low power of unit-root tests against alternatives like p = 0.95 [e.g., see Bhargava (1986)] and does not entail accepting the null.

288 J. Cumpos et ul., An anrrlogue model of phase-averaging procedures

standard deviations less well than the predictions from the stationary AR(l) model. Like the significant error-correction coefficient established in Hendry and Ericsson (1990), the evidence here favours a highly autoregressive but ‘just barely’ stationary process for velocity.

Econometric evidence. In terms of estimating econometric models, the stan- dard error of Friedman and Schwartz’s (1982, p. 282) money-demand equation for the United Kingdom is around 5%. We could closely reproduce their equation:

(m_)j= 0.012 + 0.885 (i-fi)j- 11.21RIVj (0.19) (0.049) (3.3)

0.22 G(i,+i),+ 1.37 w,+ 20.6 q, - (0.29) (0.58) (2.7)

(25)

J = 36, II = 2.75, c?= 5.66%.

By contrast, the equivalent a^ for many dynamic annual models is under 2%. For example, Hendry and Ericsson (1990, eq. (10)) obtain the following:i7

A(z),= 0.45 A( [0.06]

m -p),pl - 0.10 A2(m -P)~-~ [0.04]

- 0.60 Ap,+ 0.39 AP,_~ - 0.021 Ars, [0.04] [0.05] [0.006]

0.062 A,rl,+ 0.51 (1 - 59,_,)i& - [0.021] [0.12]

-t 0.005 + 3.7 (01 +I&, [0.002] [0.6]

(26)

T= 93 (1878-1970), R* = 0.87, o^= 1.424%.

The variable ii, is the residual from the Engle-Granger static cointegration regression of (m - i - p), on a constant and the short-term interest rate RS,.

Thus the additional information leads to a jifteenfozd reduction in the residual variance relative to that for the equation using variable-length phase averages. This is larger than predicted by section 2 and may arise in part from the data-selected choices of reference business cycles.

“White (1980) heteroscedasticity-consistent standard errors appear in square brackets [.I; cf. Nicholls and Pagan (1983). The appendix briefly describes the data in eq. (23) and (24) for which J = 36 (phase observations l-36) and T= 93 (annual observations 1878-1970), respectively. Hendry and Ericsson (1990) further discuss both regressions.

J. Cun~pos Ed ul., An unalogue model of phase-uveruging procedures 289

Exogeneify. In their study, Friedman and Schwartz implicitly assume that phase-averaging does not affect the exogeneity or otherwise of the variables of interest (pp. 35-36, 206). Hendry and Ericsson (1985, app. G) obtain substan- tial evidence against Granger noncausality for annual money, income, prices, and interest rates. Thus, the weak exogeneity of prices, income, and interest rates [as in (26)] may well be lost when a money-demand equation is estimated as in (25) using phase-average data. l8 That happens because some feedback from (e.g.) money to interest rates occurs within a phase, so the dynamics of the annual data introduce simultaneity in the phase-average data.

4. Summary

Business cycles have been the focus of much economic analysis throughout this century: phase-averaging has been proposed as one empirical method of separating cyclical and secular components. This paper analyses the statistical and econometric effects such filtering has on data from some simple dynamic processes and illustrates those effects by comparison of annual and phase- average data from Friedman and Schwartz (1982). For Friedman and Schwartz in particular, phase-averaging empirically does not obtain two results impor- tant to their analysis, namely, reductions of serial correlation and of the data variance. In general, phase-averaging involves a loss of information and thereby can result in the inconsistency of previously consistent estimation procedures and in the endogeneity of previously exogenous variables.

Appendix: The data

D, = a dummy variable for World War I (= 1 for 1914-18 inclusive, = 0 elsewhere),

D, = a dummy variable for World War II (= 1 for 1939-45 inclusive, = 0 elsewhere),

G( p + I) = growth rate of phase-average nominal income,

H = high-powered money (million &),

I = real net national product (million 1929 &), M = money stock (million .&)), N = population (millions), P = deflator of I (1929 = l.OO), RI = long-term interest rate (fraction),

RN = RS - H/M,

RS = short-term interest rate (fraction),

“Nb.. Engle et al. (1983) on the different concepts of exogeneity; cf. Wei (1982).

290 J. Cumpos et ul., An unalogue model of phase-aoeraging procedures

s = a dummy variable for phase observations 16-28 (1921-1955; = 1 for observations 16-28 inclusive, = 0 elsewhere),

W = a dummy variable for phase observations 13-15 and 26-28 (1918-1921 and 1946-1955; = -4, -3, -2, 8, 5, and 3 for phase observations 13, 14, 15, 26, 27, and 28, respectively; = 0 elsewhere).

Coefficients and estimated standard errors of the dummies D,, D,, g, and I@ are reported times 100 for readability.

These data are as in Friedman and Schwartz (1982, tables 4.8, 4.9) but relevant series are resealed proportionately from 1871 to 1920 to remove the break in 1920 when Southern Ireland ceased to be part of the United Kingdom. A capital letter denotes the level of a variable, lower case denotes the logarithm. Also, P, RS, and RI have been divided by 100 so that the value of P in 1929 equals 1.00 (rather than 100) and the interest rates are expressed as fractions (rather than as percentages).

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