An Analogue Model of Phase-Averaging Procedures

International Finance Discussion Papers Number 303

February 1987

AN ANALOGUE MODEL OF PHASE-AVERAGING PROCEDURES by

Julia Campos, Neil R. Ericsson and David F. Hendry

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

This paper considers 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 examine the effect of phase-averaging on the exogeneity of variables in those equations and the implications phase-averaging has for conducting statistical inference.

To illustrate our analytical results, we investigate claims by Friedman and Schwartz in their 1982 book Monetary Trends in the United States and the United Kingdom about what the properties of phase-average data and the relationships between those data ought to be. We present certain features of the observed series on velocity, examine how well our analytical model captures them, and contrast them with Friedman and Schwartz's predictions. While our model is an extremely simplified charact:erisation of the phase-averaging adopted by Friedman and Schwartz, it does offer several insights into the likely consequences of their

approach.

Keywords: business cycles. conditional models. conditioning. dynamics. econometrics. exogeneity. marginalising. methodology. money demand. phase averages. quantity theory. statistical inference. temporal aggregation. time series. velocity of circulation.

An Analogue Model of Phase-averaging Procedures by

Julia Campos, Neil R. Ericsson and David F. Hendry”

1. Introduction

The nature of business cycles has evoked extensive research in economics over the last several decades, including such diverse studies as Beveridge (1921), Slutzky (1937), Schumpeter (1939), Burns and Mitchell (1946), Tinbergen and Polak (1950), Friedman and Schwartz (1963, 1982), Sims (1977), Lucas (1977), and recent investigations on "real business cycles" by (e.g.) 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.t Burns and Mitchell (1946), for example, focus on the cycle, abstracting from non-cyclical temporal differences. To do so, they establish the NBER reference business cycles, dating the contraction and expansion phases of the cycles over their sample period, and across cycles average the data at

various points in the cycle. Conversely, Friedman and Schwartz in their

1982 book Monetary Trends in the United States and the United Kingdom focus

‘“The authors' affiliations are: Banco Central de Venezuela, Carmelitas, Caracas 1010 Venezuela; International Finance Division, Federal Reserve Board, Washington, D.C. 20551; and Nuffield College, Oxford OX1 INF England. This research was supported in part by UK E.S.R.C. grants HR8789 and

B 0G 220012. We are 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 comments from Rob Engle, David Howard, Bob Marshall, David Pierce, and Ken Wallis 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, Appendix B), itself developed from Appendix B for Hendry and Ericsson (1983).

lallais (1966, 1972), Neftci (1984, 1986), Stock (1985), and Falk (1986) inter alia present various reasons for the existence of business cycles and give empirical evidence thereon.

on the longer-run features of the data. In an effort to filter out short-run (i.e., cyclical) properties of the data, they transform their (raw) annual series by averaging separately over contraction and expansion phases of the reference business cycles, afterwards presenting detailed graphical and regression studies of the resulting series, both in levels and in differences. They claim that such phase-averaging reduces serial correlation arising from the business cycle (p. 78) and attenuates measurement errors (p. 86), and implicitly claim that phase-averaging does not affect the exogeneity or otherwise of their variables of interest

(pp. 35-36, 206): attaining those effects is important to sustain the validity of their statistical analyses.2 Our paper analytically evaluates statistical and econometric effects that phase-averaging has on time series and considers whether it obtains empirically the results claimed by Friedman and Schwartz.

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 our model fixed n-period phase-averaging of time series generated by a simple dynamic process.2

We focus on the effects that phase-averaging has on the variance and

2See Maverick (1933) for early advocacy of phase-averaging in econometrics. Unreferenced tables and page numbers refer to Friedman and Schwartz (1982).

3without 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-averaging. 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.

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-averaging has for conducting statistical inference. To illustrate those analytical results, in Section 3 we examine claims by Friedman and Schwartz (1982) about what the properties of phase-average data and the relationships between those data ought to be. We present certain features of the observed series on velocity (i.e., (ii)), examine how well our analytical model captures them, and contrast them with Friedman and Schwartz's predictions. While our model is an extremely simplified characterisation of the phase-averaging adopted by Friedman and Schwartz, it does offer several insights into the

likely consequences of their approach.

2. Analytical effects of phase-averaging

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

Fhase-averaging sequentially applies two filters to the annual data: the first averages that data (as with a moving average) and thus implies a re-parameterisation of the data generation process; the second selects the phase--average data from the averaged series (i.e., marginalises the data density with respect to the intermediate observations), thereby entailing a (statistical) reduction in that re-parameterisation. By analogy with seasonal adjustment of quarterly data, the first filter is like the X-1l procedure; the second discards all but the fourth-quarter data points.

To illustrate, consider fixed n-period phase-averaging of the annual

series (xp3 t™..,1,2,...,T} with T a multiple of n. Letting L be the lag

4

Operator such that Lxy=x;_1, the first filter is (1+L+L2+...+L9-1) sn, Yo)

the averaged series is (xi t=..,1,2,...,T) where

% =

*t

(Xp tXe_y tXp_gt. -+Xe_n gy) /n. . (1) The second filter selects every nth observation of x so the phase~average series is (x3 ju..,1,2,...,(7/,)}, denoted (x45 jnm..,1,2,...,J}. To illustrate, Figure 1 shows the steps involved in going from {x,} to (xt) to {xj} for fixed 3-period phase-averaging.4

There is little loss of information from the use of the moving-average data per se but a distinct loss of information results from the selec‘:ion process. That loss would be unimportant if (a) all the parameters of interest could be recovered from the phase-average series (and Friedman and Schwartz seem to believe such parameters can be, including the long-run elasticity of interest rates in a money-demand equation) and (b) there was no loss of power in testing the resulting model. However, many of the ‘parameters which we consider to be of interest cannot be obtained from phase-average data, including parameters relevant to tests of Granger (1969) non-causality, short-run variability in the postulated relationships, and the dynamic mechanisms whereby the economy adjusts to "shocks". Point (a) also ties in closely with issues of exogeneity: as will be seen below, variables exogenous at annual observations may no longer be so when phase-averaged. Even for parameters which can be

retrieved from the phase-average data, tests about them may be low in

4In 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 (.S5+Lt+L2+...4L9-14, 519) sn and the second is unchanged. However, the statistical effects of using phase-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).

Figure 1. A schema for fixed 3-period phase-averaging.

Original series (x, } rs AS Filter 1: Averaging & *& *& Fe & Averaged series (x, } tee Ky Xp XS, x, x Xt bald (X_tX_-1+%_~2)/3 Filter 2: Sampling Phase-average series {x,) eee Xo - - x) ~

power relative to tests based upon the annual data.° To illustrate the statistical effects of phase-averaging, we begin with the time series ((yz:z_)'; t =l,...,T} generated by the simple dynamic

bivariate process:

Ye = Ag + OZ + Ut (2)

Ut = PUey + Et Et 0 0 0 (3) ~ IN( ’ o|)°

Ze = Anze-1 + Azye-1 + Ve Vt 0 0 64, (4)

Except when otherwise noted, we assume that (y;:z;,)' is stationary; hence all roots of (2)-(4) are stable, i.e., |p| < 1 and |Aj+oyA9| <1. For convenience, we denote the second root (Aj+ajA7) by xk. If Ap = 0, z¢ is strongly exogenous for (ag:a,)'; otherwise it is correlated with uz -6

Although (2)-(4) do not include a deterministic cycle, for suitable

5For instance, cf. Banerjee, Dolado, Hendry, and Smith (1986) on co-integration tests with static and dynamic models and note the similarity between those models and the phase-average and annual models herein.

61¢ (ye22¢)' is normally distributed, conditional upon its own past, then (2)-(3) is the corresponding conditional density of yz (conditional upon _ zt) and (4) is the corresponding marginal density for z_, on the assumption that the conditional mean of (yz:z_)' is linear in its past and only includes yy.) and z¢_,, and that the resulting equation associated with the conditional density of y_ (e-g-, Ye = Sgtdy2e+doy¢_1+632z¢_1+€¢) includes a common factor (l-pL). Cf£. Hendry and Mizon (1978) and Hendry, Pagan, and Sargan (1984) for further discussion on models with common factors.

Technically speaking, if Az * 0 and so Zt is not strongly exogencus for (ag:aj)', then z_ is not weakly exogenous for (agta,)' either because stationarity requires [AytayAg | < 1 and so (Ay:A9)' and (agta,)' cannet be variation free; cf. Engle, Hendry, and Richard (1983, p. 282). Weak exogeneity could be "restored" either by assuming that (A,:A7)' and (ag:a))' lie within a sub-space 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 yy (requiring only that || <1). In the latter situation, if z_ is integrable of order 1, then so will be yz and yt and z_ will be co-integrated. For further details on co-integration, see Granger and Engle (1987), Phillips and Durlauf (1686), and articles in the August 1986 special issue of the Oxford Bulletin of Economics and Statistics on co-integration, especially Granger (1986) and Hendry (1986).

parameter values, it easily can generate data with cyclical behaviour.’

The n-period phase average of an arbitrary series {xz} is:

T =), (5)

_ n Xj 7 nly x i=l

(j-1) nti jum,2,--.,(

sO

yj = ao + 0424 + uj (6) from (2). Average two-(phase-)period differences result in:

(.5A9yj) = ay (.5A92,) + (.5Au,) (7)

where differencing for phase-average data is defined in terms of phases rather than years so that (e.g.) AlX} ™ Xj-Xj-y- Our

concern is how inference is affected by using (6) (or (7)) rather than (2)-(4) .8 Unless otherwise noted, we limit our discussion to the

effects on phase-average data of positive p because economic data tend to

be highly positively autocorrelated. The effects of negative p are

7S5lutzky (1937) demonstrates that moving average processes can exhibit cyclical behaviour, and (2)-(4) can be well approximated by a moving average process.

8In the context of Friedman and Schwartz (1982), it is natural to interpret “periocs" and "phases" as years and half-cycles. However, the analytical properties are quite general and could be applied to other time intervals, e.g-, cuarterly data aggregated to annual data. In fact, a substantial literature exists on the effects of time aggregation; cf. Theil (1954), Durbin (1960), Mundlak (1961), Telser (1967), Moriguchi (1970), Zellner and Montmarquette (1971), Amemiya and Wu (1972), Sargan (1976), Tiao and Wei (1976), Wei (1978), and Weiss (1984). Those studies show that, inter alia, sizeable inefficiencies result from using least squares even if the regressors are strongly exogenous. Further, if they are only weakly exogenous (or include lagged dependent variables), least squares is general.ly 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 aggregzted (phase-average) data.

qualitatively different, particularly for n even; cf. (10) and (11) below.? However, the formulae derived apply for all |p| <1.

Levels of phase averages. As a baseline, we note that the variance and autocovariances of Ut in (2) are:10

07/(1-p") i= 0 (8) tT, = E(uguy-y) = i2

Hence the autocorrelation function for uy is pi (i>0). Equivalent formulae

for uj in (6), the levels equation with phase-average data, are:

0 2p(1-p") 5 [1 _ — i=0 (10) n(1-p) n(1-p”) ty = E(ujuj_;) n(i-l) ,, .n\2.2 p p(i-p ) O.. i=-1,2,.... (11)

n?(1-p”)(1-p)” The error variance of (6) 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 uy (normalised by 02) and its first-order autocorrelation r; (= B(tyuy_1)/E(Ut)) are shown in Figures 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 = .8 and n = 3, (10) yields To = 2.302 whereas 02 from (8)

is 2.802, SO averaging produces only a small decrease in variance. E

9Figures 2-7 below are for positive p only; Appendix B contains comparable graphs which include negative values of p.

l0Telser (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 t(k/n)*(1-p?)°n? where k a multiple of n and his parameter a is

our p. Ahsanullah and Wei (1984) obtain equivalent formulae for the AR(1) process; Stram and Wei (1986) generalise those results to ARIMA processes.

Figure 2. The logarithm of the variance of u, (normalised by 0°) as a function of p and n. - 2 In(t)/o.)

3.5

n=l n=2 3 n= 3 n=5 2.5 2 Oe fed the 1 77 eee 3° - we “” ~~ co" a“ + = =. -- ° 0 wes ao a or oo” a“ - rae _ " 0.5 eon _a ° - > a — 1 _—— _—— 1.5 2 3 4 5 6 7 8 9 l

16

Figure 3. The first-order autocorrelation coefficient of uy (x) as a function of p and n.

ll

Figure 4. The second-order autocorrelation coefficient of a, (r,) as a function of p and n.

12

Moreover, as is apparent from Figure 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. (11), and compare Figure 3 with Figure 4, the latter plotting the Second-order autocorrelation of uj, denoted ro).

Least-squares estimation of structural parameters in the Presence of autocorrelated disturbances is inefficient. Further, if right-hand side variables are correlated with the disturbance, such estimation is inconsistent:

E(zeug) = prgo2/[(1-p2)(1-pk)] = py (say) (12) which is zero only if pla = 0. E(ze_ye¢) = 0 for all i 2 O (and for all p and Ay), so estimation of (2)-(3) by autoregressive least squares is always consistent. Turning to the phase-average data in (6):

E(zyuy) = Ago2(1+p)/[4(1-p)(1-px) ] (13) when n = 2 (for illustrative purposes). Thus, even if p = O (and so Szy = 0), E(zjuj) * O unless Ay = 0. Conversely, the strict exogeneity of z_ (i.e., Az = 0) is sufficient for both (12) and (13) to be zero.

Differences of phase averages. Next, we consider the effects of

two-period differencing of the phase-averages in (6) to get (7):

0 p(2-p")(1-p7") var(.5Aqu,) = —F [1 - a (14) 2n(1-p) n(1-p")

and

EL(.SAgiiy)(.5Agiy-1)] = (1-p7™)p(1-p")702/f4n2(1-p2)(1-p)7] . (15)

After rescaling (14) by 4, the error variances for levels and "rates cof change" models will be similar only for quite large values of p (e.g.,

p = .9), in which case the former will still exhibit substantial first-

13

order autocorrelation although the latter will not; cf. Figure 5 and 6.11 Further, and in contrast to the results for levels, the differenced series - {.5AQu,} exhibits (typically negative) second-order autocorrelation of the form: nt+1 2 p ~(1-p 9) (1-p")? r = -.5 + , (16) 2 2 5 p(2-p")(1-p*") 2n(1-p*) [1 - rr

n(1-p”)

as portrayed in Figure 7 for some representative values of n. Only for

very large values of p is (16) near zero.

For the differenced data, the equation equivalent to (13) is: - - 2.2 2 E(d9zj°Aguy) = [2 - (1tk)“K JAQ00/4 (17)

when p = 0 and n= 2. So, in general, different biases will result from fitting (6) and (7). Only if z_ is strictly exogenous (Ag = 0) will both

biases be zero.

Phase averages of stationary AR(1) data. One particularly interesting

special case of (2)-(4) is the stationary AR(1) process, i.e., with a; = 0 in (2). In mean deviation form (i.e., ag = 0), that yields: Yt = P¥t-1 + Et t =1,...,T (18) with 2 mw g2/(1-p2 %y oe/(1 p+). (19)

For n-period averaging,

11 the variable-length weighting procedures in Friedman and Schwartz (1982) should not alter the substance of this last result; but being data-based to eliminate "cyclical behaviour" (which, e.g., a positively autocorrelated series would appear to manifest), it could seriously alter the nature of the residual autocorrelation. For example, the negative second-order autocorrelation of {.5Aguy} in (16) may not carry over for

data-selected phase averages.

14

Figure 5. The logarithm of the variance of -5h53, (normalised by 0°) as_a function of p and a.

0.75

3p 35 5 tl : WW nN —

ae Gnas eee eS ome

o fo mn 2 as)

2 05 > n=l a 0.75 < 5 1 E ees 1.25 ao 1.5 “7 U- 1.75 a “7 2 oT a 2.25 2.5 0 wl 2

15

- k Figure 6. The first-order autocorrelation coefficient of Shou, (e,) as a function of p and n.

0.6" 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1

0.050

16

- * Figure 7. The second-order autocorrelation coefficient of Shou, (ry) as a function of p and n.

0.15 0.2 0.25 ; 0.3

0.35

0.4

0.45 0.5

0.55

17 where fp is defined by [E(yj_17)]-lE(yj¥j-1), i-e-, E(¥j-1ej)=0. Thus, 04(y) = 02/(1-p2) - 02 [n(1-p?)-2p(1-pM)1/[n2(1-p)?] (21) (which is equivalent to tg) and Pp = p(1-p®)2/[n(1-p2)-29(1-p™)] (22)

(which is 1)/tg). Moreover,

Aye = Ex + Cp-L)ye-1 and (23) A1Y; =e; + (Pp-L)¥4-1 so that (24) 62(Ayy) = 202/(1+p) . and (25) 62(A,y) = 202/(1+pp) . (26)

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). Fhase averages of a random walk. The final alternative we consider is that the series is a random walk. If p = 1 in (18), then E(y2) = to2 (for

Yo = 0) and the mean of the standard estimator of the variance of y; is:

T 1 2 2 - 2 E> Ze = o- T(THL)/[2(T-1)] = 05(y) (27)

(e.g., see Fuller (1976, p. 367)) whereas 62(Ayy) = 02. Working (1960) considers the case of an n-period average of a random walk and shows that, for ej defined by yj = Yj-1 + ej , its variance is:

62 = 62(2n241)/(3n) . (28) However, the mean of the standard estimator of the variance of Yj is:

J E(s4> 3 ¥4) = o25(3Jn7-n7+3nt1)/[6n(J-1)] = 05(9) , (29) Toy" € J

assuming yq = 0. Figure 8 plots 62(y) for different values of T 0 J =

18

Figure 8. The mean of the standard estimator of the variance of Ys as a function of T and n.

a)

19

and n with 6, = 1.12 ft strikingly portrays the linearity of 02(y) in T and the near invariance of 02(y) with

respect to n, given T. Both follow from (29) which gives 02(y) = o2(y) + 0(T9) = 02[(1/2) + 0(T°)] where each of the terms 0(T9)

depends upon n only, to o(t-1) .13

3. Interpreting Friedman and Schwartz's phase-averaging

To illustrate the value of our analytical results from Section 2 above, we investigate claims by Friedman and Schwartz (1982) about what the properties of phase-average data and the relationships between those data ought to be. We present certain features of the observed series on velocity, examine how well our analytical model captures them, and contrast them with predictions by Friedman and Schwartz. While our model is an extremely simplified characterisation of the phase-averaging adopted by Friedman and Schwartz, it does offer several insights into the likely consequences of their approach.

Friedman and Schwartz's primary justification for phase-averaging appears to be the claim that it reduces cyclical effects (pp. 13-14, 78), thereby allowing them to focus on their primary concern, monetary trends. By filtering out the cycle, they argue that phase-averaging should reduce serial correlation in the data and lower the data variance. Secondly, they claim phase-averaging reduces the effects of measurement errors (p. 86): that also would imply a reduction in the data variance. Attaining those

effects is important to their statistical analysis. In practice, as is

12Note that p (=1) is no longer a determinant of the variances whereas T now is. Also, 02(y) = o2(y) for n= 1.

l3Due to the non-stationarity of the series generated, inference in the presence of a unit root requires special attention; cf. Dickey and Fuller (1979, L981), Evans and Savin (1981, 1984), Sargan and Bhargava (1983), Phillips and Durlauf (1986), and Granger and Engle (1987).

20

clearly visible in Figure 9 which jointly plots the phase-average and annual data for velocity in the United Kingdom, phase-averaging negligibly reduces either the variance or serial correlation in this time series, a series from which they claim to draw important inferences.14 The remainder of this section shows that the simple AR(1) model in Section 2 | implies properties for the phase-average and annual data closely in line with those observed. The evidence also supports the notion that a substantial loss of information has resulted from phase-averaging. Time-series evidence. To evaluate some basic time-series properties of velocity v, consider the following first-order autoregressions with

annual and phase-average data over the last century for the United Kingdom.15

a

(.016) (.029) T = 95 8 = 4.798% dw= 1.13) dw* = .08 dwt* = 1.15. Vj = .023 + .972 Vj-1 (31) (.042) (.077) J= 35 8 = 7.279% dw= 1.70 dw* = .31 dwt* = 1.74.

Values in parentheses (*) are conventionally calculated standard errors.

dw denotes Durbin and Watson's (1950, 1951) statistic, dw* is its value

14From Table 1 below, the estimated standard error of the annual data is 17.35% whereas that for the phase-average data is 17.31%. The extent of serial correlation in both series is evident from (30) and (31).

15The (logarithm of) velocity is v = In(P°I/M) where P*I is nominal income and M is the money stock. Appendix A briefly describes these data.

All our estimates 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. The summary statistics for phase-average data also are calculated with weights accounting for the differences in phase lengths.

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.

21

Figure 9. The logarithm of the velocity of money for the United Kingdom: annual and phase-average data.

J | | ~] T T 1.00

———-—— Annual data (v4)

; 0.90 woe e---- Phase-average data (v5) 0.80 0.70 0.60 0.50 0.40 0.30

0.20

0.10

0.00

1880 1900 1920 1940 1960 1980

22

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.16 The series in levels (both annual and phase-average) show first-order autocorrelation 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.

Table Ll. Standard deviations of velocity (percentages)17

Predicted values

Series Observed values ae AR(1) model random walk model

Vv 17.35 17.14 33.43 Vv 17.31 16.84 33.60 Av 4.80 4.85 4.80 AlV 7.19 6.82 6.67 € 4.80 4.80 4.80

e 7.28 6.68 6.67

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 analogous properties.

‘Assuming that velocity is a stationary AR(1) process and that n = 3 and p = .96, the analogue model in Section 2 predicts relevant standard deviations for the annual and phase-average data in terms of O-, the

Standard deviation of the underlying innovation process. Those values are

eo

16The Durbin-Watson Statistic dw* and the t-ratio for (f-1) may be interpreted as statistics for testing for the non-existence of a relationship between money, prices, and income; cf. Sargan and Bhargava (1983), Dickey and Fuller (1979, 1981), and Granger and Engle (1987).

17The values in columns three and four of Table 1 have been normalised on the annual AR(1) regression residual standard error because the results in Section 2 are for relative, not absolute, variances.

23

given 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 = -92, close to that in (31) above. While we do not think velocity is simply a first-order autoregressive process (and give evidence supporting that view in Hendry and Ericsson (1985)), these results nevertheless illustrate the ineffectiveness of phase-averaging in attaining its supposed principal benefits, namely, reduction in the data variance and elimination of serial correlation arising from cyclical components. These results also explain why the models using phase-average data fit much less well than those 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 reported by Working (1934); and (e.g.) p = .96 is not "far" from unity.1é Also, that hypothesis for velocity has received considerable attention in the literature (cf. Gould and Nelson (1974), Nelson and Plosser (1982), and Bharghava (1986)). The last column of Table 1 gives various standard deviations for T = 95, J = 35 (and son = T/J « 2.71). Those results are a much poorer match to the observed standard deviations than the predictions from the Stationary AR(1) model. Like the significant error-correction coefficients established in Hendry and Ericsson (1985), the evidence here favours a highly autoregressive but "just barely"

stationary process for velocity.

184s 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 = .95 (e.g., see Bhargava (1986)) and does not entail accepting the null. ~~

24

- Econometric evidence. In terms of estimating econometric models, the Standard error of Friedman and Schwartz's (1982, p. 282) money-demand equation for the United Kingdom is around 5%. We were able to reproduce

closely, but not precisely, their numbers:

(m-p-h) j = .012 + .885 (i-h), - 11.21 RNy - .22 G(pti)

(.19) (.049) (3.3) (.29) + 1.37 Wi, + 20.6 Sj (32) (.58) (2.7) J = 36 n= 2.75 6@= 5.66% .

By contrast, the equivalent 6 from many dynamic annual models is under 2%. For example, Hendry and Ericsson (1985, equation (22)) obtain the

following.19

A, (mp) = .37Ai(m-p)¢_] - -06A?(m-p)¢_2 - -20(m-p-i)¢_y [ .04] [.07] [.022]

+ .66(A,i¢/4) - .47A!py - .14A!pe_p - .78RSy

[.10] [04] [.06] [.20] ~ 3.3(A,R&_/2) + 1.9D, + 3.6D, + .64D, - .086 (33) [1.1] [1.1] [.6] [.80} [.010]}

T= 93 R* = .82) 6 = 1.711%

Thus the additional information leads to a more than tenfold reduction in the residual variance relative to that for their model using variable-length phase averages. This is a larger improvement than predicted by the analysis of fixed-length phase-averaging and may arise in

part from the data-selected choices of reference business cycles.

'19White (1980) heteroscedasticity-consistent standard errors appear in square brackets [*]; cf. Nicholls and Pagan (1983). Appendix A briefly describes the data in equations (32) and (33) for which J = 36 (phase observations 1-36) and T = 93 (annual observations 1878-1970) respectively. Hend:ry and Ericsson (1985) extensively discuss the properties of both regressions.

25

Exogeneity. Hendry and Ericsson (1985, Appendix G) obtain substantial evidence against Granger non-causality for money, income, prices, and interest: rates. Thus, the weak exogeneity of (e.g.) prices, income, and interest: rates in (33) may well be lost when a money-demand equation is estimated as in (32) using phase-average data.29 That happens because some feedback from (e.g.) interest rates to money occurs within a phase, so those 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 stat:istical 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 important 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.

20Nb. Engle et al. (1983) on the different concepts of exogeneity; cf. Wei (1982), Hendry and Ericsson (1985, footnote 15), and Longbottom and Holly (1985).

26

APPENDIX A. The data

Legend

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

D, - A dummy variable for 1921-55 (= 1 for 1921-55 inclusive, zero elsewhere)

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

G(pti) - Growth rate of phase-average nominal income

H - Population (millions)

I - Real net national product (million 1929 £)

M ~ Money stock (million £)

P - Deflator of I (1929 = 1.00)

RL - Long-term interest rate (fraction)

RN - RS‘H/M

RS ~ Short-term interest rate (fraction)

S - A dummy variable for phase observations 16-28 (1921-1955; = 1 for observations 16-28 inclusive, zero 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; zero elsewhere)

Coefficients and estimated standard errors of the dummies D,, D,, D,,

S, and W are reported times 100 for readability.

These data are as in Friedman and Schwartz (1982, Tables 4.8 and 4.9), but relevant series are rescaled 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 R& have been divided by 100 so that the values of P in 1929 equal 1.00 (rather than 100) and the

interest rates are expressed as fractions (rather than as percentages).

27

APPENDIX B. Phase-averaging of negatively autocorrelated data

In the text, the effects of phase-averaging are depicted for positive p in (3) via variances and autocovariances in Figures 2-7. As noted in Section 2, the effects are qualitatively different for negative p. This appendix contains corresponding figures (numbered B.2 through B.7) with p both positive and negative. The properties for even n (i.e., n=2 in the

figures) are particularly asymmetric.

28

Figure B.2. The logarithm of the variance of Bs (normalised by 07) as a function of 9 and n.

2.8

In (7/02)

1O+

2.8

29

Figure B.3. The first-order autocorrelation coefficient of u, (r) as a function of 9 and n.

1 a=] ii 0.9 oe n= 2 ve 0.8 cm emo —- naz 3 vf ° To n= of / 0.7

a aA 0.4 yf 0.3 a 0.2

2 —_—— 0.4

ee Ou Zo 0.2

Y 0.3 / 4 0.4 0.5 0.6 0.7 0.8 0.9

30

Figure B.4. The second-order autocorrelation coefficient of u; (x5) as a function of p and n.

=] i!

Wwry re

| ' I ‘ I ‘ I ' I 5.5 nou

-——oo -- -_-—oo ee oe ee ee

+ 0

0.1 -.5 0 .

_

31

Figure B.5. “he logarithm of the variance of 5454, (normalised by 3°) as a function of p and n.

]

2 €

In[var(.54)u,)/o

32

- * Figure B.6. The first-order autocorrelation coefficient of .5A,u (x) as a function of p anc. n.

Figure B.6. The first-order autocorre:

1 2 meme n = 3 5

0.3

0.2

0.1

tO+

0.1

0.2

0.3

0.4

0.5

33

- * Figure B.7. The second-order autocorrelation coefficient of 15Ayu, (x4) as a function of op and no.

34

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IF DP NUMBER

303

302 301

300

299

298 297

296

295

294

293

292

291

290

38

International Finance Discussion Papers

TITLES 1987

An Analogue Model of Phase4Averaging Procedures

The Outrof=Sample Forecasting Performance of Exchange Rate Models When Coefficients are Allowed to Change

Financial Concentration arid Development: An Empirical Analysis of the Venezuelan Case

1986 The International Debt Situation

The Cost Competitiveness of the Europaper Mar ket

Germany and the European Disease

The United States International Asset and Liability Position: A Comparison of Flow of Funds and Commerce Department Presentation

An International Arbitrage Pricing Model with PPP Deviations

The Structure and Properties of the FRB Multicountry Model

Short8term and Longterm Expectations of the Yen/Dollar Exchange Rate: Evidence from Survey Data

Anticipated Fiscal Contraction: The Economic Consequences of the Announcement of Gramm*Rudman#Hollings

Tests of the Foreign Exchange Risk Premium Using the Expected Second Moments Implied by Option Pricing

AUTHOR( 3)

Julia Campos Neil R. Ericsson David. F. Hendry

Garry J. Schinasi P.A.V.B. Swamy

Jaime Marques Janice ShackMar quez

Edwin M. Truman Rodney H. Mills John Davis

Patrick Minford

Guido E. van der Ven John E. Wilson

Ross Levine

Hali J. Edison Jaime R. Merquez Ralph W. Tryon Jeffrey A. Frankel Kenneth A. Froot

Robert A. Johnson

Richard K. Lyons

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal Reserve System, Washington, D.C. 20551.

39

International Finance Discussion Papers

IF DP

NUMBER TITLES AUTHOR(s)

289 [eposit Risk Pooling, Irreversible Robert A. Johnson Investment, and Financial Intermediation

288 The YenADollar Relationship: A Recent Manuel H. Johnson Historical Perspective Bonnie E. Loopesko

287 Should Fixed Coefficients be Reestimated P.A.V.B. Swamy Every Period for Extrapolation? Garry J. Scehinasi

286 An Empirical Analysis of Policy Hali J. Edison Coordination in the U.S., Japan and Ralph Tryon Europe

285 Comovements in Aggregate and Relative B. Dianne Pauls Prices: Some Evidence on Neutrality

284 Labor Market Rigidities and Unemployment: Michael K. Gavin “he Case of Severance Costs

283 A Framework for Analyzing the Process Allen B. Frankel of Financial Innovation Catherine L. Mann

282 Is the ECU an Optimal Currency Basket? Hali J. Edison

281 4Sre Foreign Exchange Forecasts Rational? Kathryn M. Dominguez New Evidence from Survey Data

280 Taxation of Capital Gains on Foreign Garry J. Schinasi 2xchange Transactions and the Non=neutrality of Changes in Anticipated Inflation

279 The Prospect of a Depreciating Dollar Jacques Melitz and Possible Tension Inside the EMS

278 The Stock Market and Exchange Rate Dynamics Michael K. Gavin

2TT Can Debtor Countries Service Their Debts? Jaime Mar quez Income and Price Elasticities for Exports Caryl McNeilly of Developing Countries

276 Post#simulation Analysis of Monte Carlo Neil R. Ericsson Experiments: Interpreting Pesaran's (1974) Study of Non#nested Hypothesis Test Statistics

275 A Method for Solving Systems of First Order Robert A. Johnson Linear Homogeneous Differential Equations When the Elements of the Forcing Vector are Modelled as Step Functions

274 International Comparisons of Fiscal Policy: Garry Schinasi

The OECD and the IMF Measures of Fiscal Impulse