12
We also tested three other aspects of the money-demand model in (5):
price homogeneity, the absence of any trends, and the normality of the

residuals. For the first of those, we obtained:

(6) (m-p-n)j = 1.01 + .68 (i-n)s + 213 py ~ 18.4 RNG - .04 G(p+i);
(.38) (.08) (.04) (3.8) (.26)

+ 1.1 Wy + 16.3 8
5) 2 (aia) 2

J = 36 6 = 5.03% dw= 1.85.
The coefficient on Pj is significant at the 99% level. Next, we obtained:

(1) (m~p-n)j = 2.32 + 414 (i-n)y - 16.0 BN; + .23 G(p+i);
(.52) (16) (2.7) (.24)

+ 1.8 Wy + 6.1 Sj + .0090 ty
(.5) (3.7) (0019)

J = 36 6 = 4.35% dw= 1.99 .

The coefficient on tj is statistically significant at any reasonable level,
so we reject the first two hypotheses. Normality of the disturbances is
‘also rejected, since Carlos Jarque and Anil Bera'ts (1980) statistic for
testing against skewness (SK) and excess kurtosis (EK) is —,(2) = 6.9 in
(5), with SK = 1.1 and EK = .6.!1 However, for the sub-sample estimates,
the value of €,(2) is negligible for the first half and 12.7 for the second.
(The 1% point of x2(2) is 9.2.) Thus, the distributions of the residuals
have "fat" tails and a skewed shape, inconsistent with normality. Further,
(7) fits better than (5) prior to either "removing the serial correlation"
(on which issue see Sargan (1980b) and Hendry and Grayham Mizon (1978)) or

adding the dummy variables W and S. Equation (5) is clearly neither an

11wWe modify Jarque and Bera's (1980, p. 257) statistic to be

&5(2) = [(T-k)/6]-[Sk? + EK2/4] ,
given that the number of regressors fitted is often large relative to the
sample size. £&,(2) is asymptotically distributed as x2(2) under the null
hypothesis of normality, in which case skewness and excess kurtosis are
both zero.


32 .
be co-integrable (see Appendix C on co-integrability and for a discussion
of testing for the existence of such a long-run relationship). Table IV
reports the estimates of the unrestricted autoregressive-distributed lag

model and the associated test statistics. Equation (22) records our

presently "preferred" error-correction simplification of that unrestricted

model.
° 2 2 .
(22) Ay (m-p), = +374) (mp) ¢-1 - +064) (m-p)¢-2 ~ .20(m-p-i),-y
; [.04] [.07] [.022]
+ 66(Oyig/4) - 470% p_ - .14a%pE-p ~ . TERS,
[.10] [.04] [.06] [.20]

~_3-3(bgR&E/2) + 1.9D) + 3.6D2 + .64D3 - .086
1] (1.1] {.6] [.80]° [.010]

T= 93 R? = 82 6 = 1.7118 0, (10,71) = 1.44 n2(6,75) = 3.5

&u(1) = .2 €5(2) = .03 SK = -.05 EK = -.005

(Figure V shows plots of the actual and predicted annual changes in real
money over the period 1878-1970.) The static equilibrium solution of (22)

yields (ignoring the Dj):
(23) M/(P*I) = exp(-.44 - 4.0RS) = .64e7-O4ORS*

where RS* is RS measured in Percentage points. Thus, a one percentage
Point increase in interest rates (e.g., 5% to 6%) reduces M relative to PI
by four percent in the "long run". That long-run solution (i.e.,
hypothetical equilibrium: see Aris Spanos (1981)) is within the framework
of the quantity theory.

Within the straitjacket of the information set FS use, equation (22)
is not unreasonable. Directly testing the significance of the additional
variables implicit in Table IV yields n3(21,60) = .97, so that the residual
of (22) appears to be an innovation on the information set generated by

(m, p, i, RS, RQ, Dy, Do, D3). Note that the unrestricted equation


28
series descriptions of {py} and {it} must be very different from that of
{mp}: otherwise aggregation over variables could not have caused such a

loss of information. That is indeed the case as shown in (16) and (17).

(16) appt = -640,pE-1 - -154;pE-2 + 63D] ~ 1.439 + 3.61D3 + .0106
[.20] (.14] (2.8] [1.2] [2.5]° [.0046]

T= 93 R? = 52 G = 3.96% n, (10,77) = .24 = €,(10) = 7.3

W

&.(1) = 7.1 &s(2) = 379. SK = -1.3) EK = 9.9 ng(&,79) = 5.4

(17) ait = 4264, it-1 + «23D, + .57D2 -  .18D3 + .0112
(.12] (.99] [(.82] [1.59] [.0052]
T= 93  R? = .07 @ = 3.50% 1, (10,78) = .15  €,(11) = 18.6

&.(1) = 8 €5(2) = 41. SK = -.8 EK = 2.9 6 (5,83) = .36

Since the values of 6? in these equations are very much larger than that
in (14), it is unsurprising that imposing unit coefficients on prices and
incomes at all lags should lead to a severe deterioration in fit.25
Returning to Figure I, three relatively distinet epochs cf behaviour
of v suggest themselves visually: a “cyclic around an average" period for
1875-1914, a "lower but more volatile" period during the inter-war years,
and a "long upward trend" from 1947 to 1970 (after which floating exchange
rates and competition and credit control were introduced). Any claim to
the constancy of a model for money demand would need both constant
parameters and a similar goodness-of-fit over each of these epochs; more
Stringently, it would need to stay constant since 1970. A possible
objection might be that a money-demand equation's "constancy" need not be
precise but only relatively better than (say) the consumption function's
constancy. Since FS assert that their model is indeed constant, such an

objection is not germane. However, in response to Mayer (1982, Pp. 1534),

25Note that the dummy variable D2, which FS identified as corresponding to a
"shift in liquidity preference" for 1921-1955, is not Significant in any of
equations (12)-(17).


34
(iv) 1950-1982 (which thereby adds post-1970 data to (iii)). That not only
investigates parameter constancy directly, but also implicitly allows all
of the parameters to shift in period (ii) when Do is non-zero throughout.
Table VI records the results. The coefficients are not so changeable
between epochs as to be unrecognizable. Given the simplicity of the
specification, these results provide a useful starting point for a more
detailed examination of such issues as functional form, omitted variables
(e.g., wealth and taxes), and so on, which may produce an improved model
with more constant values of 3 (see Imre Lakatos (1970) on progressive
research strategies). By way of comparison with (iv), estimating (22) over
the 105 observations through 1982 produces:

(24) ay(m-p)y = 410° (m=p)g-y + 100° (m-p)y-2 ~ .20(m-p-i) 5-1
07] [.10] ]

+ .64(AyiE/4) - .5004p_ - .070%pE-2 - .63RSy
[.16] (.07] [.08] (.13]

~ 49(OaR%/2) + 1.4D, + 4.2D2 + .7D3 - .097
(.88] [1.1] (.8] (.8] [£.015]

T= 105 R* = 66 & = 2.419% n, (12,81) = 8.75 n,(6,87) = 1.3

n3(21,72) = 2.41 €,(1) = 6.7 &.(2) = 10.2 SK = -.1 EK = 1.6

The most noticeable parameter changes from (22) are a fall in the
coefficient of AoR&, and a large increase in 6, so the orthogonal
parameterisation does prove useful for the most part (cf. Table y).e7

Nevertheless, constancy is clearly rejected (see n,(-)).

2Tve (1983, pp. 77-78) experimented with RS replaced by RN, FS's (p. 270)
measure of the marginal cost of money (RN = RS-H/M). The resulting
estimates are remarkably similar to those found using RS and reveal no
improvement in the constancy of the interest-rate coefficient. Following
suggestions by Chris Pissarides and recent work by Ross Starr (1983), we
also experimented with adding variables which measured interest rate
volatility, e.g., using .2 Veo (RS_-y-RS{)2 where RSt = .2 Theo RSp-4-
The effect was largest for the most recent period but did not produce
constant parameters or a constant fit.


8
where a super-script "bar" denotes phase-averaging and there are J phases.®
Thus, mys Pye and ij are the phase-average data on logs of the money stock,
price level, and constant price income such that (p+i) j is the phase
average of the log of measured nominal income. Figure IV shows the time
series for Vj and reveals a highly autoregressive variable, as confirmed by
inspecting the Durbin-Watson statistic (dw) when velocity is regressed on a

constant:

Throughout, estimated standard errors are shown in parentheses (except in
Table I, where we report results from FS) and @ is the standard

deviation of the residuals, adjusted for degrees of freedom.! One
possible statistical equivalent of a will-o'-the-wisp is a random walk.
Altnough the sample size is small due to aggregation over time, one cannot
reject the hypothesis that vj is a random walk using the table for the
Dur>in-Watson statistic prepared by Denis Sargan and Alok Bhargava

(1933) .8 Alternatively, using the approach proposed by David Dickey

and Wayne Fuller (1979, 1981), we obtain:

Svelocity is V = (P+I)/M, so the log of velocity is v = pti-m. Without loss
of clarity, we often refer to v as "velocity".

Tall our estimates using phase-average data are based on weighted least
squares, correcting for the different phase lengths. However, experiments
revealed that parameter estimates were not very different whether ordinary
or weighted least squares were used.

8sargan and Bhargava propose testing that the errors in a regression are a
random walk by seeing how close dw is to zero, and give upper and lower
bounds for dw under the hypothesis of a random walk. Note that, for a
static regression, dw = 2(1-p) where p is the first-order_autoregressive
coefficient of the disturbance. Adding FS's two dummies W and S to

(2) increases dw to .56, indicating some success by FS in selecting the

period over which S is non-zero.


61

Kiviet, J.F. (1982) "Size, Power and Interdependence of Tests in Sequential
Procedures for Modelling Dynamic Relationships", Discussion Paper
No. 8/82, University of Amsterdam, March.

Kloek, T. (1984) "Dynamic Adjustment when the Target: is Nonstationary",
International Economic Review, 25, 2, 315-326.

Laidler, D. (1984) "The "Buffer Stock' Notion in Monetary Economics",
Economic Journal, 94, supplement, 17-34.

Lakatos, I. (1970) "Falsification and the Methodology of Scientific
Research Programmes" in I. Lakatos and A. Musgrave (eds.) Cri
and the Growth of Knowledge, Cambridge, Cambridge University Press,
91-796.

Leamer, E.E. (1978) Specification Searches: Ad Hoc Inference with
Nonexperimental Data, New York, John Wiley and Sons.

Leamer, E.E. (1983) "Let's Take the Con Out of Econometrics", American
Economic Review, 73, 1, 31-43.

Longbottom, A. and S. Holly (1985) "Econometric Methodology and Monetarism:
Professor Friedman and Professor Hendry on the Demand for Money",
Discussion Paper No. 131, London Business School.

MacKinnon, J.G. (1983) "Model Specification Tests Against Non-nested
Alternatives", Econometric Reviews, 2, 1, 85-158 (with discussion).

Marshall, A. (1926) Official Papers, London, Macmillan and Co.
Mayer, Th. (1982) "Monetary Trends in the United States and the United

Kingdom: A Review Article", Journal of Economic Literature, 20, 4,
1526-1539.

McAleer, M., A.R. Pagan, and P.A. Volker (1985) "What Will Take the Con Out
of Econometrics?", American Economic Review, 75, 3, 293-307.

Mizon, G.E. (1984) "The Encompassing Approach in Econometrics", Chapter 6
in D.F. Hendry and K.F. Wallis (eds.) Econometrics and Quantitative
Economies, Oxford, Basil Blackwell, 135-172.

Mizon, G.E. and J.-F. Richard (1983) "The Encompassing Principle and Its
Application to Testing Non-nested Hypotheses", Econometrica,
forthcoming.

Moore, B.J. (1983) "Monetary Trends in the United States and in the United
Kingdom, A Review", The Financial Review, 18, 2, 146-166.

Nicholls, D.F. and A.R. Pagan (1983) "Heteroscedasticity in Models with
Lagged Dependent Variables", Econometrica, 51, 4, 1233-1242.

Pesaran, M.H. (1982) "A Critique of the Proposed Tests of the Natural
Rate-Rational Expectations Hypothesis", Economic Journal, 92, 367,
529-554.

Popper, K.R. (1959) The Logic of Scientific Discovery, London, Hutchinson.



43
o(+) denotes the standard deviation of the relevant variable. Other series
from FS show analogous results: roughly equal standard deviations in
levels, and large reductions in standard deviations by differencing, more
so for the annual data. The original series shows 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. The analogous case now is equivalent to

a,=0 in (B1) and (in mean deviation form) yields the baseline model:

(B16) ye = P¥tE-1 + Et t= 1,...,T
with

2 2 2
(B17) sy * o/(1-p) «

For n-period averaging,
(B18) yj = Ppyj-1 * ej J = lyeeegd

where pp is defined by [E(¥j-17)] 'E(¥j¥j-1)) i-e., E(Vj-1ej)<0- Thus,
= 2 2
(B19) 0°(¥) = o8/(1~p2) = oF En(1-9*)-201-p")I/EN=(1-p)°I

(B20), = o(1-0")®En(1-p?)-20(1-p") J

Moreover,
(B21) Ayye = eg + (Co-1)¥t-1 and
B22) Aqy; = ej + (p.-1)y¥q- so that
(B22) Ayyj j Py VY¥j-1

2 2
(B23) o (Ay) = 20,/(1 +6) and
(B24) o°(Ay7) = 20g/(148,) «

When n = 3 and p = .96, we have Pp = .92: Table B.IV gives relevant

standard deviations in terms of o¢,


41
higher orders dies off rapidly.
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:

(B10) E(zgug) = pdgae/E(1~p°) (1px) J Ozy (say)

which is zero only if pho = 0. E(zp-yez) = 0 for all i 2 0 (and for all p
and A>), so estimation of (B1)-(B2) by autoregressive least squares is

consistent. Turning to the phase-average data in (B5):
(BI) B(ZjHj) = 2p02(14p)/L4(1-p) (1-pe)]

when n=2 (for illustrative purposes). Thus, even if p = 0 (and so
Ozu = 0), E(z3u3) * O unless Ap = 0. Conversely, the strict exogeneity of
zp (i.e., Ap = 0) is sufficient for both (B10) and (B11) to be zero.

Next, we consider the effects of two-period differencing of the

phase-averages in (B5) to get (B6):

sie) - oe p(2~p")(1-p-")
B12 var(.5Aouj) = 1-
’ 2n(i-p)> n(1=p°)
and
(B13) BL(.5a2m5)(.5aguj—4)] = (1-p°)p(1~p") 02/41 -p°) (1-9) 7]

After rescaling (B12) 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 = .9), in which case the former will still exhibit substantial first-
order autocorrelation although the latter will not.

For the differenced data, the equation equivalent to (B11) is:
(BIN) E(Ap%jrAg0j) = [2 - (144) Iago2 74

when p=0 and n=2. So, in general, different biases will result from
fitting (B5) and (B6). Only if z_ is strictly exogenous (Az=0) will both

biases be zero.


NT
test of non-existence, since all lags can be re-parameterised as changes
relative to the current period, and hence must be of a lower order of
integrability than the levels. For example, Pp = Ayp_ is I(d-1) if pp is
I(d). Thus, omission of lagged variables will not vitiate the test: if the
levels are co-integrable, the changes must be also; and if the changes are
not co-integrable then the levels cannot be either. Both the Durbin-Watson
statistic (dw) used by Bhargava (1983) and the 't'-statistic which Dickey
and Fuller (1979) denote by t, (we use t) provide approximate statistics
for investigating the random walk hypothesis in regression residuals, and
we calculate both below.

The most general of FS's claims concerning money demand seems to be
that there exists a linear relationship linking m, p, i, RS and R&, which
we translate to mean that that set of variates must be co-integrable. On
the basis of the dw and t statistics for all of those series individually,
the hypothesis that they are I(1) cannot be rejected (although R& had an
estimated root larger than unity). Consequently, the putative existence of
hypothetical "long-run" relationships is of substantive interest. A
narrower version of FS's claim is that there exists a linear relationship
connecting v, Pp, RS and R&. We have investigated both the general and
narrower claims by estimating "staticised" relationships and report tests
of the hypothesis that the residuals are a random walk against the
alternative of a stationary process, i.e., we seek to reject the null that
the residuals from a regression of mg on pt, ig, RS_, and RM (or of Ve on
RSt and R&z) follow a random walk (i.e., are 1(1)). We obtained:

(C1) fig = 1.17p_ + 834g - 8.2RS_ + 1.11

T= 93 6 = 10.3% dw = 46 t= 73.4 6* = 6.15% dw* = .77

(C2) VE = 7-ORS_ + .308

T= 93 @= 11.1% dw-= .32 t= -2.8 6* = 6.608 dw* = .73


51
estimates on annual data for the inter-war and post-war periods separately
are reported. The standard error (3) in both periods is .5% of real non-
durable consumers' expenditure (cf), itself about 90% of total consumers!
expenditure during those periods; the sets of estimated parameters are
closely similar; and the errors are approximately white noise and

homoscedastic in both periods. A "representative" equation for 1922-38 is:

(B1) wet = .50a,it - .o80(ct-ity,  - 10530

Tt [.06}! t t-1

+
P
[.026] C.o27]! *

T= 17 R? = .81 G = .53% ni(4,10) = 1.1 n2(2,12) = .4

E,(1) = 630 &3(2) = 4.5 SK = -1.0 EK = 1.9 ng(4,10) = .6
where t is real personal disposable income in 1938 prices; et is the
implicit deflator of cts and if, Ph and ef are the logs of those
variables, That is a remarkably constant relationship, as the equivalent

post-war estimates (1954-1980) are:

(B2) acet = .uealit - .12(ct-it)

t
- .16A,p
Tt c.04]! t [.02] vt

eT [103]

T= 27 R% = .90 6 = .52% n,(4,20) = 1.0 n2(2,22) = .2

E4(1) = 3.6  &5(2)

1.41 SK = .5 EK = -.2  16(5,19) = .8.

See Hendry (1983, equations (13) and (9)).

We consider a "debate" about the relative constancy of these
relationships to be sterile: expenditure requires a transactions medium;
and as Alfred Marshall (1926, pp. 38-39) remarks, narrow money demand and
expenditure profiles are essentially mirror images. We see no immediate
implications for policy from the equations for consumers' expenditure and
money demand. For example, (E1) is compatible with large changes in the

percentage of income consumed, as actually occurred, with ett! falling

substantially in both periods. Thus, even if the equation itself does not

shift, demand management may be required to stabilise net output.


39

APPENDIX B. An analogue model for averaging and differencing

In this appendix, we consider the statistical effects that fixed
n-period phase-averaging has on time series generated by a simple dynamic
process. We focus on the variance and autocorrelation of the disturbance
term for both the levels and difference equations, as FS make claims about
what their properties ought to be. We also present certain features of the
observed series on velocity and examine how well our analytical model
captures them. While it is an extremely simplified characterisation of the
phase~averaging adopted by FS, the following model does offer several
insights into the likely consequences of their approach. 3!

Consider the time series {(yg:zy)'; t =1,...,T} such that:

(BI) ye = ag + ayZzp + UE
(BZ) ug = pug-7 + ef et te) ° 0

~ IN( ,| ).
(B32) 2 = AqZp-1 + Aave-1 + Vt Yt 0 o 6

Except when otherwise noted, we assume that (ygize)' is stationary; hence
all roots of (B1)-(B3) are stable, i.e., |p| <1 and |Ay+a,A2| <1. For
convenience, denote the second root (A;+a,A42) by x. If Ap = 0, zp is
strongly exogenous for (ag:a;); otherwise it is correlated with ut.

Letting the n-period phase average of an arbitrary series {xg} be:

n
cent - r
(B4) oxy en - X (jot) net J1,2,.0.0505) 5
then
(BS) Yj = a9 + a4 + uj
from (B1). Average two-(phase-)period differences result in:
(B6) — (.5dgyy) = a4 (.54225) + (.54205)
Our concern is how inference is affected by using (B5) (or (B6)) rather

than (B1)-(B3).

3lWe are indebted to Julia Campos for carefully checking an earlier version
of this appendix.


16
is likewise rejected by the data. More fundamentally, we cannot accept the
procedures FS adopt in pre-processing the data to obtain "phase averages",
These involve a data-based set of filters, but with no account being taken
of the statistical effects of such filtering on later inferences,
Moreover, the aggregation to phase averages entails a loss of information
on dynamics which we believe renders further modelling of such data a waste
of time, especially since phase-averaging failed to achieve the objectives
stated by FS. Thus we turn to an analysis of the annual observations in an
attempt. to develop a money~demand equation which encompasses (5) and explains
its failures. To do so, we first consider the statistical framework and the
theory of testing on which the analysis of FS's claims (above) was based, as

both will prove invaluable in modelling the annual data.

IV. A_statistical framework for analysing FS's data

In this section and the one following, we discuss the most relevant
aspects of our statistical approach for analysing FS's data. An
econometric model is viewed as a (reduced) re-parameterisation of the data
generation process achieved by marginalising and conditioning, the latter
operation being related to the economic notion of contingent plans based on
weakly exogenous variables. Such operations entail that the "error" is a
derived rather than an autonomous process, suggesting designing the model
to satisfy data-based and theory criteria. For a general exposition and
bibliographic perspective, see Hendry and Jean-Frangois Richard (1982,
1983), Hendry (1983), and Hendry and Kenneth Wallis (1984).

FS believe in the existence of a number of constant relationships
linking the variables under study and implicitly assume log-normality of
those variables (e.g., see pp. 75, 223, 236 combined). Thus, we seem to

share the hypothesis that there exists a joint data density for the


32a

Table IV. ‘A general autoregressive-distributed lag representation for money (mg),

conditional on incomes, prices, and interest rates

lag i
Variable
0 1 2 3 4 5
mi |-1.0 1.534 [.104]] -.890 [.179]} 500 [.206]| -.147 [.177]] -.059 [.093]
itt .075 [.063]] .019 [.079]} .025 [.066]] -.052 [.076]] .030 [.099]} -.017 [.052]
Peet 369 [.061]| -.418 [.086]]} .265 [.086]] ~.273 [.086]] .058 [.089]] 046 [.076]

RSp~4 ~.526 [.287]] 415 [.365]] -.208 £.370]} .148 [.355]} .098 [.390]} -.410 [.311]
Re; |-2.001 [.893]] .027 [1.370] } 2.484 [1.498] | ~1.433 [1.623] ] 2.258 [1.515] |) -.442 [1.138]

Dy 2.805 [.797] - - - - -
Dp +216 [1.614] - - - - -
D3 2.516 [1.116] - - - - -

Constant) -.186 [.105] - - - - -

T= R? = .99983 6 = 1.718% n,(10,50) = 1.20 n2(6,54) = 3.79 E,(1) = .06

&5(2) = .07 SK = .083 EK = ~.039



60

Granger, C.W.J. and A.A. Weiss (1983) "Time Series Analysis of
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Johnston, J. (1963) Econometric Methods, New York, McGraw-Hill Book
Company.


2
their claims either are in part directly testable from tneir data or have

testable implications. Nevertheless, FS do not actually present any tests

of those claims. By itself, that lacuna removes the credibility from their
claims although it does not establish that their assertions are incorrect.
However, we have tested those assertions, focussing on the specification of
their money-demand equation over the historical period they selected. We
have found that most of their assertions are in fact rejected by their
data, leaving their conclusions stranded as assertions devoid of empirical
support. Whether their theoretical models constitute a useful framework
for viewing "long-run" monetary history is logically a separate issue
(albeit an important one) from that of the credibility of their

assertions. Evidence pertinent to the former issue is also adduced below.
Nevertheless, our primary concern here is to demonstrate that one cannot
take at face value many of the inferences conducted by FS.

After considering the data and data transformations used by FS
(Section II), we evaluate their results for velocity and money-demand
relationships as fitted to phase-average data for the UK and reject almost
all their claims for those relationships (Section III). Our analysis of
such claims derives from a coherent statistical framework and theory of
testing which we exposit in Sections IV and V. Systematically applying
that methodology, we investigate demand-for-money functions using the
annual UK data series published by FS and attempt to reconcile our
estimates based on annual data with theirs on phase-average data (Section
VI and Appendix B). Section VII concludes the paper.

Although we reject many of FS's central empirical claims, this paper
is not an "anti-monetarist" critique. Rather, it is a "pro-econometrics"
tract which highlights the practical dangers of seeking to analyse complex

stochastic processes while eschewing modern econometric methods.


62

Salmon, M. (1982) "Error Correction Mechanisms", Economic Journal, 92, 367,
615-629.

Sargan, J.D. (1958) "The Estimation of Economic Relationships using
Instrumental Variables", Econometrica, 26, 3, 393-415.

Sargan, J.D. (1964) "Wages and Prices in the United Kingdom: A Study in
Econometric Methodology" in P.E. Hart, G. Mills, and J.K. Whitaker
(eds.) Econometric Analysis for National Economic Planning, Colston
Paper Vol. 16, London, Butterworths, 25-63 (with discussion);
reprinted in D.F. Hendry and K.F. Wallis (eds.) (1984) Econometrics
and Quantitative Economics, Oxford, Basil Blackwell, 275-314.

Sargan, J.D. (1980a) "Some Approximations to the Distribution of
Econometric Criteria which Are Asymptotically Distributed as
Chi-squared", Econometrica, 48, 5, 1107-1138.

Sargan, J.D. (1980b) "Some Tests of Dynamic Specification for a Single
Equation", Econometrica, 48, 4, 879-897.

Sargan, J.D. and A. Bhargava (1983) "Testing Residuals from Least Squares
Regression for Being Generated by the Gaussian Random Walk",
Econometrica, 51, 1, 153-174.

Sims, C.A. (1980) "Macroeconomics and Reality", Econometrica, 48, 1, 1-48.

Spanos, A. (1981) "Disequilibrium, Latent Variables and Identities",
unpublished paper, Birkbeck College, London.

Starr, R.M. (1983) "Variation in the Term Structure of Asset Holding and
Behavior of the Monetary Aggregates: The Maturity Shift Hypothesis",
Discussion Paper No. 82-9, University of California, San Diego.

Trundle, J.M. (1982) "The Demand for M1 in the U.K.", Bank of England
discussion paper.

White, H. (1980) "A Heteroskedasticity-consistent Covariance Matrix
Estimator and a Direct Test for Heteroskedasticity", Econometrica,
48, 4, 817-838.

Working, H. (1934) "A Random-difference Series for Use in the Analysis of
Time Series", Journal of the American Statistical Association, 29,
185, 11-24.

Working, H. (1960) "Note on the Correlation of First Differences of
Averages in a Random Chain", Econometrica, 28, 4, 916-918.

Yule, G.U. (1926) "Why Do We Sometimes Get Nonsense-correlations between
Time-series? -- A Study in Sampling and the Nature of Time-series",
Journal of the Royal Statistical Society, 89, 1, 1-69 (with
discussion).

Zellner, A. (1979) "Causality and Econometrics" in K. Brunner and
A.H. Meltzer (eds.) Three Aspects of Policy and Policymaking:
Knowledge, Data, and Institutions, Amsterdam, North-Holland Publishing
Co., 9-54.



26
respectively (see Appendix A); [-] denotes heteroscedast icity-consistent
estimated standard errors (see Halbert White (1980) and Desmond Nicholls
and Pagan (1983)). Note that (12) has approximately white-noise
homoscedastic (but non-normal) residuals. Equation (12) has no "long-run"
solution for the level of velocity, being in statistical terms a

generalisation of a random walk. Nevertheless, this will-o'-the-wisp model

of velocity substantially variance-dominates all of the monéy~demand models

reported by FS, so that their formulation cannot encompass even this naive
specification.

The notion that velocity is a near constant24 has a long history in
economics, probably because it seems natural to expect that the money stock
normalised by nominal income should manifest much less variability than
real money (which is not standardised for real income growth) or nominal
money (which ignores changes in both the price level and real income). It
is a non sequitur that statistical models of those three magnitudes (i.e.,
v, (m-p), m) have error variances following the same ranking. In fact,
equation (12) is a very poor model of the real money stock conditional on
real income (noting that vy = -(m-i-p),) since it imposes a unit elasticity
at all lags. Surprisingly, it is variance-dominated by a corresponding

time-series model for real money which excludes real income altogether:

(13) By (m-p)y = .3682(m-p)-y + .05D, + .89D2 + 2.403 + .0090
[.11] (1.64) [.83] [1.9] [.0037]

T= 93 R? = .22 6 = 3.42% n,(10,78) = .12 &(10) = 16.0

&4(1) = 12.6 €,(2) = 5.2 SK =-.2 EK = 1.1 ne (5,83) = 1.7.

The estimated error variance has dropped to just over 60% of that in (12).

24 concerning its "constancy" over time, note that, excluding war years, the
largest change in V_ over a two-year period is 26% and that after its
trough in 1947, V_ rose by 80% in the next 23 years. (All percentage
measures use the log-symmetric form.) Compare FS's Chart 5.5 (in which UK
velocity appears virtually constant) with Figure I.


13
adequate characterisation of the data nor consistent with a theory claiming
a constant money~demand equation, homogeneous of degree zero in prices.
What is perhaps the most important single claim of FS therefore has no
empirical basis, despite their assertion (p. 7) that "[t]his parallelism!2
is a manifestation of the stable demand curve for money plus the excellence
of the simple quantity theory approximation".

The other "equations" using phase-average data were not further
investigated because, as explained in our Appendix D, "re-normalising"
regressions (as FS do) does not provide additional inferences.

FS's primary justification for phase-averaging appears to te 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, FS argue phase-averaging should reduce serial correlation in the
data and lower the data variance. Secondly, FS claim phase-averaging
reduces the effects of measurement errors (p. 86): that also would imply a
reduction in the data variance. Contradicting those claims, we now show
that phase-averaging negligibly reduced the variance and (substantial
serial correlation in their series for velocity, a series from which they
claim to draw important inferences. Further, we demonstrate that a simple
first-order autoregressive process can explain those features of the data,
and we briefly address the issue of selecting the phase periods.

First, we record the following regression using FS's "raw" annual data:

(8) p= 153
(.017)
T = 100 6 = 16.91% dw = .08 T= 71.2

for T annual observations. The results in (8) are very close to those in

120 nominal income and the nominal quantity of money, and of the rate of
change of nominal income and the rate of change of the nominal quantity of
money.


19

have testable implications: where possible, those have been investigated
and relevant statistics are reported. For example, if % has been chosen
too small, e¢ will not be an innovation; and, if the {mj} are not constant,
predictive failure should be observable on sub-samples. Addit:onally,
Granger non-causality tests play the vital role of testing assertions about
strict exogeneity within the data set reported (i.e., conditional on xh;
see Robert Engle, Hendry, and Richard (1983)).16 In Appendix G, various
tests for strict exogeneity are presented, showing that the null of Granger
non-causality can be rejected at the 5% level in most cases. Estimation of
(11) also provides baseline innovation variances for the various series;
corresponding residual standard errors for m, p, and i are 1.9%, 2.6%, and
2.8%, where %=5 and xi = (mp hy it pt RS_ Ret et). 17

Given that (11) is not decisively rejected at the outset, one can turn
to modelling pt = Yh TiXt-i- There are many basic approaches to doing so,
including causal chains, simultaneous systems, block recursive models, and
various simplifications based directly on (11). All of these raise

problems in econometric modelling. Surprisingly, apart from their errors-

16The four distinct concepts of exogeneity, namely weak, strong, super and
strict, discussed by Engle et al (1983), correspond to different notions of
being "determined outside the model under consideration" according to the
purposes of the inferences being conducted (i.e., conditional estimation,
prediction, policy analysis, and forecasting, respectively). In no case is
it legitimate to "make variables exogenous" simply by not modelling them.
We can find no necessary connections between the "causality" of one
variable y for another z and their respective exogeneity status in that zt
being "exogenous" or endogenous is neither necessary nor sufficient for it
to influence y, (except for the trivial case that by definition strictly
exogenous variables cannot be Granger-caused by endogenous variables). For
a useful discussion of the concept of causality in econometrics, see Arnold
Zellner (1979).

17, constant term and the dummy variables Dj, Do, and D3 are also included,
We have chosen £=5 on the grounds that that implies 39 parameters estimated
per equation in Hendry and Ericsson (1983, Appendix C) and Appendix G
herein (vs. 93 observations total), in line with guidelines in Sargan
(1980b, p. 880). Also, &£=5 implies a maximum lag of approximately one
cycle (as measured by FS).


15
similar. Further, given an annual autoregressive coefficient of .96
(which is consistent with dw = .08 in (8)), the theoretical model in
Appendix B predicts an autoregressive coefficient of .92 for the
phase-average series, closely in line with dw = .31 in (2). While we do
not think velocity is simply a first-order autoregressive process (and
give evidence supporting our view below), these results nevertheless
demonstrate the ineffectiveness of phase-averaging in attaining its
supposed principal benefits.

Before concluding this section, we briefly return to the issue of
selecting phases (raised in Section II). 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, but a first approximation is given by examining the over-all
effects of phase-averaging on the data series and on econometric modelling
in light of the results for fixed-length phase-averaging reported in
Appendix B. For velocity, the model of phase-averaging with fixed phase
lengths explains several salient features of the data with little direct
evidence of the effects of selection. In terms of econometric models, the
standard error of FS's "money-demand equation" is around 5% (p. 5) (which
some reviewers, and FS, apparently regard as "good"); the equivalent 6
from our annual model is under 2% (see (22)). Thus the additional dynamic
information leads to a more than tenfold reduction in the residual variance.
That is a larger improvement than that predicted by the simple (fixed
phase-length) model, despite FS using carefully selected dates over which
to average.

FS's own phase-average data reveal the patent mis-specification and
parameter non-constancy of their money-demand equation, despite their

assertions to the contrary. Their claim that velocity is a near constant


27
Even retaining price homogeneity at all lags is a poor assumption since,
for the nominal money stock (i.e., excluding all information on both prices

and incomes), we have:

(14) Am = ~-37Aymp-2 + 4.5D, - .64D2 + 4.0D3 + .0105
[.08] (1.5) €.54]) [1.1]°  £.0031]

T= 93 R? = 40 @ = 2.21% (10,78)

+63 &2(10) = 26.0

w

E4(1) = 62 Es(2) = 2.8 SK = -.05 EK = .9  16(5,83) = 5.4.

The estimated error variance is now roughly a quarter of that in (12) and
under half that in (13). It is remarkable that 6? should fall so
dramatically from (12) to (14), precisely opposite to what one might have
anticipated from FS's theories. Since the residual sum of squares can only
fall from its value in (14) when pt and ig and their lags are added
unrestrictedly, the implicit constraints imposed to obtain (13) and (12)

are rejected at all reasonable levels of significance. Indeed:

(15) ye = S4AyvE-7 + .90Aq LE + 671A DE ~ -574,it—1 - -4T7ApPE-1
: [.08] [.06] (.06] [.10] (.07]

-_3.7D, + .53D2 - 4.0D3 - .0070
(1.2]) (€.48] [1.1]° [.0032]

T= 93 R* = 846 G= 1.94% n,(10,74) = .44 2 (6,78) = 3.8

n3(4,84) = 89.2 €,(1) = .4 &s(2) = .5 SK = .2 EK = .01.

Here, nj(+) tests the specialisation of (15) to (12), which is decisively
rejected.

This set of findings does not rule out the class of models which
imposes price homogeneity and/or unit income elasticities as being valid
characterisations of the equilibrium solution. The flaw in univariate
analyses of vy; or (m-p),_ of the type just presented is to impose such
restrictions at all lags.

Further, the three results recorded in (12)-(14) imply that the time-


10
demanc for money, the effects of money on income, and the effects of money
on prices. "Levels" and "rates of change" equations are analysed by FS,
but they see little to choose between them (e.g., see p. 286). For
brevity, we focus on the equations in levels. While full details can be
found in FS, we briefly summarise their interpretation of ‘the estimates as
follows: on the assumption that M is "exogenous", M does not cause I
(equation (I.4)), and M does cause P (1.3) and hence P-I (1.2).
Consequently, (I.1) is interpreted as a money-demand equation, but it is
determining P, not M (or RN). If FS are insistent that M is "exogenous",
we are in effect left with three models of P, all conditional on M and an
interest rate, but with different parameter values and standard errors.
While we have not yet presented a coherent statistical framework for
interpreting their estimates, the evidence seems strongly against the
restrictions imposed on (1.2) to generate (I.1) (@ increases by 40%),
and even more against those in (I.3). However, in keeping with FS's own
interpretation, we take (I.1) as being the money~demand function for which
a number of assertions are testable.

FS (p. 230) discuss the striking relationship between real money
balances and real income illustrated by Chart 6.2, even though residual
autocorrelation is apparent visually on noting that real income increases
over time. Relaxing the restriction of unit income elasticity in (2), we

obtain:

(4) (m-p-n) 5 = 611 + .86 (i-n);
(.38) (.08)

J = 36 R? = .96 6 = 16.49% dw = .33.
Given the value of dw, that regression should allow the reader to draw
his/her own conclusions on the validity of and "insights" that can be

gained from "consider[ing] variables one or two at a time" (p. 215;


49

(D1) FCx |X, ag) =f meg,

1% EPpy 2p lke)

= Beagle pe Xpan Fr) + Boley [Ae e) +
Either f;(-) and f2(+) or g;(+) and go(+) may be used; but f;(+) and gj(-)
do not comprise a legitimate representation, with both sets of "parameters"
being of independent interest. For example, when (10) holds (hence F(x, |)

is the bivariate normal density) with 2%=0 and zy non-singular and

constant, f;(+) and gj(+) yield:

2
(D2) x14 = 8 * aXe * Ere Elen) = 14
2
(D3) X54 =o * PaXie * oe Bleae) = 922

= = i = R2
with E(e )=0. Then ay 44 b1/%59 with a,b, RY x," Because the

€
1t®2t 1X2

density in (10) is "saturated" by five parameters and the equations in (D2)
and (D3) have six, an automatic restriction results. That finding

generalises so that for three variables, say:

2
(DAY i = Vig * Tar%ye * Thokst * Ste Bere) = iy

(i,j,8=1,2,3; j,s*i; s>j), then:

(D5) Y44/911 7 Yoq/%20 » Yyo/944 = /%33 » and Y.

31 22/%22 * 32/633 -
The regression coefficients are all functions of a smaller set of
parameters. Adding an unrestricted set of regressors to every equation
does not affect that result either. While the algebra becomes much more
complicated if different additional regressors are included in each
equation, it remains invalid in principle to mix the choice of
factorisations: running regressions with different normalisations cannot
establish separate "structural equations" or produce new inferences.

For FS's results reported in Table I, the equations in (D5) do not

hold precisely because of the other restrictions imposed and the different

sets of auxiliary variables in each regression. Nevertheless, they are


y

average lengths of 2.1 and 3.4 years respectively for contraction and
expansion. The phase averages, weighted by duration, are taken as the
basic units of analysis (pp. 75-79 and Table 5.8). FS analyse their phase-
average data both in levels and as "rates of change". The latter are based
on "differences" from one expansion (or contraction) to the next, so using
triplets of overlapping phases (again weighted appropriately) .2 The
data to which these moving-average filters are applied are the logarithms
of money, prices, incomes, and population but the original values of
interest rates.3

Throughout their book, FS emphasise primarily an "errors-in-variables"

paradigm. However, precisely what properties FS ascribe to those "errors

There are typographical errors in the reported values for "rates of change"
in Table 5.10 affecting observations 18 through 21. We corrected those
errors following the procedures described by FS (Section 3.2): our
amendments are reported in Appendix F.

3We have several caveats about their data; and while these are not germane
to an attempt to re-analyse their numbers, nevertheless we briefly note the
most important issues which would need clarification prior to drawing
substantive conclusions from the results reported below.

(i) Their choice of monetary measure seems too broad to represent
transactions demand and too narrow for an overall index of "liquidity",
especially given the rapid growth of the Building Society movement over the
last century. M is based on the UK monetary measure M2 (pp. 111-114), and
excludes shares and deposits in Building Societies. Note that FS centre
their money-stock figures on mid-years by averaging successive end-of-year
values.

(ii) The measurement of the price series (P) is "corrected" for
rationing and controls (pp. 115-120), with real income then derived by
deflating nominal income. It is difficult to understand why they should
wish to hold measured nominal income constant when they believe measured
prices are incorrect. Furthermore, over a century, one cannot but be
concerned about the effects on the measurement of P of the many dramatic
changes which have occurred in quality-adjusted real (i.e., relative)
prices (such as for computational power).

(iii) The rate of return on physical assets is measured as the rate of
change of nominal income G(p+i). Since changes in nominal income
cannot be excluded on a priori grounds as a determinant of current holdings
of money, such an approximation confounds dynamic reactions with
substitution effects. In any case, the evidence FS report in favour of
their proxy seems to us rather unfavourable (compare FS's discussion on
pp. 508, 510 with their Chart 10.12).

Proper treatment of those difficulties is outside our scope of
evaluating FS's results on the data they record.



24

agreement about and satisfaction of (C)).20 For single equations estimated
by least squares, a necessary condition for encompassing is variance
dominance where one equation variance-dominates another if the former has a
smaller variance.2! It seems natural that a poorly fitting equation cannot
account for why a well-fitting equation fits well. Below, we often use
variance dominance as a criterion for excluding unacceptable mcdels. For
comprehensive accounts of tests for encompassing and of related non-nested
hypothesis tests, see Mizon and Richard (1983), Mizon (1984), James
MacKinnon (1983), and Hashem Pesaran (1982).

We demonstrate below that FS's results on phase-average data are
variance-dominated by the simplest of univariate time-series models for

money fitted to their annual data and hence they cannot even encompass that

elementary hypothesis. An immediate implication is that their money -demand
equation is mis-specified (in part because of the time aggregation), so all
inferences based thereon are of dubious validity. Indeed, we rejected

(above) the constancy of their "best representative" money-demand equation
on phase data and will reject the constancy of the best that we can obtain
on annual data from 1878 to 1970; constancy fails dramatically when testing
over the period since competition and credit control was introduced in

1971.

20 encompassing can be understood intuitively from the following example.
Suppose Model 1 predicts a as the value for the parameter(s) a in Model 2,
whilst Model 2 actually | has estimate a. Then we test the closeness of a to
a, taking account of the uncertainty arising in estimation. Model 1
encompasses Model 2 if a is "statistically close" to a, so that Model 1
explains why Model 2 obtains the results it does.

21 Formally, variance dominance refers to the underlying (and unknown) error
variances. Without loss of clarity, we often will say a model variance-
dominates another if the estimated residual variance of the former is
smaller than that of the latter.


&-2
e%-3
ety
5

D2
Constant

T

R?

1006
ni (10,32)
£2(7)
&(1)
n3 (5,42)
n3 (30,42)

56

Table G.III. Estimates of equation (11) with h: observations for WWI and WWII excluded
(ives, 1914-1919, 1939-1946)
My hy it Pt RSt, Ru, et
1.29 ( .19)} 53 ( .28)| — .26 ( -.33) | -.04 ( .29)] 11 (11) | 031 (.029)] 1.52 ( .58)
~.83 ( .30)| -.67 ( .44)] -.52 ( .51)] -.33 ( 45) | -.30 ( .16)] -.082 (.045)} -1.79 ( .90)
+62 ( 631) | 54 ( 46) | 64 (54) 80 ( .47)] «37 (617) | «107 (.047) AH (94)
~.15 ( .28)] -.30 ( 41) } -.92 ( 48) } -.42 ( 242) ] -.23 ( .15)} -.070 (.043)] -.30 ( .85)
~.05 (14) ] 616 ( 621) | 5H (225) ] 03. ( 22) | 07 ( .08)| = .012 (022) 43 (43)
+16 ( .12)} «72 ( £18) | ~.39 ( .21)] 60 ( .18) | -.03 ( .07)] -.009 (.013)] -.95 ( .36)
+19 ( .17) +13 ( 6) +33 ( .29)| -.03 ( .25) -O4 ( .09) -002 (.026) 1.12 ( .51)
-.17 ( £18) | -.28 ( .26) +07 ( .30)| -.76 ( .26)}| -.08 ( .10)] ~-.008 (.026)] -.52 ( .52)
+18 ( .17)} 09. ( .25)} «14. ( 629) 56 ( 25) | 04 ( 09) | -.004 (.025) +16 ( .50)
203 ( .12)} -.19 ( .17)] -.16 ( .20)] -.04 ( .18)] -.02 ( .07)] .002 (.018)]}  -.47 ( .36)
-.04 ( .09)} 01 ( .13)} «78 ( .15)} = «08 ( .13) | 08 ( 05) ] = -.021 (013) +22 ( .26)
+05 (11) ] «16 ( .16)] -.06 ( .19)} -.08 ( .16)] -.03 ( .06)] -.003 (.017)] -.48 ( .33)
~.03 ( .10)| -.10 ( .15)} -.04 ( .18)] -.14 ( .15)] -.12 ( .06)} -.027 (.016) +27 (431)
01 (611) | 206 ( .15)} «17 ( .18) | -.14 ( .16)] 11 ( 06) ] 006 (.016) +12 ( .32)
04 ( .09)} 07 ( .13)} «18 ( 615) | 09 ( .13)] -.02 ( .05)] 014 (.013) |] -.07 ( .26)
-.26 ( .11) +08 ( .16)} ~.27 ( .19) +99 ( .16) -09 ( .06) 027 (.016)| -.24 ( .33)
+21 ( .15)| 22 ( .22)| «31 (26) | -.23 (23) | -.01 ( .08) | -.004 (.023) 16 ( 46)
~.43 (615) | -.07 ( .22)] ~.42 ( .26)} -.35 ( .23)] -.09 ( .08)] -.001 (.023) 18 ( 245)
-.03 ( .15) 07 ( .21) -2 ( .3) +05 ( .22) +08 ( .08)} -.005 (.022) -56 ( .44)
-.0 ( .09)} .16 ( .13)} 01 ( .15) | -.06 ( .13)} -.02 ( .05)] -018 (.014) | -.09 ( .27)
07 ( .36)} ~.51 ( .53)] -.71 ( .62)] -.21 ( .54) 56 ( .20)} -.055 (.055)} -.04 (1.08)
-.52 ( .40)| -.70 ( .59) | 1.64 ( .69) +09 ( .60)| -.10 ( .22) -093 (.061)] 4.01 (1.20)
-.13 ( .50) | -.08 ( .73)] -.55 ( .86)] -.30 ( .75)] «31 ( .27) | -.024 (.076) | 4.12 (1.50)
~.38 ( .53)| -.15 ( .77)] -.34 ( .90) +26 ( .78)| -.25 ( .29) 074 (.079)} 2.30 (1.57)
-06 ( .43)} -.84 ( .62)] -.72 ( .73)] -1.05 ( .64)} -.21 ( .23)] -.159 (.065) ~.61 (1.28)
~.11 (1.14) | -1.54 (1.66) 72 (1.94) | 2.91 (1.69) -51 ( .62)] 1.303 (.172)]  -.78 (3.41)
4.56 (1.97) | .96 (2.87) | 88 (3.35) | 78 (2.92) | -.54 (1.07) | -1.111 (.297) | 12.31 (5.89)
~.07 (2.05) | -1.72 (2.98) | 3.67 (3.49) | 3.36 (3.04) | -.06 (1.12) 678 (.309)} 8.26 (6.13)
4.00 (2.13) | 1.80 (3.11) | -2.49 (3.64) } 1.52 (3.17) | 1.50 (1.16) | -046 (.323)} -9.49 (6.39)
“1.98 (1.41) | -.36 (2.05) | 06 (2.40) | -2.51 (2.09) | -1.01 ( .77)] -.046 (.213)] 8.35 (4.22)
-.01 ( .05)} -.10 ( .08) +01 ( .09) 06 ( .08) 01 ( .03) | -.003 (.003) 87 ( .16)
02 ( .07) +03 ( .10)} -.02 ( .12)} -.08 ( .10) 01 ( .04) +002 (.010) ~.46 ( 21)
-.05 ( .07)} 612 ( .10)] 02 ( 612) ] 15 (011) | 02 (04) | 015 (.011) +21 ( .21)
~.11 ( .06)] ~-.07 ( .08)} -.09 ( .09)} -.18 ( .08)} .00 ( .03)} .004 (.008) “17 ( 16)
01 ( .05)|  .04 ( .07)] -.09 ( .08)] .06 ( .07)} -.01 ( .02)} -.007 (.007)] -.06 ( .14)
2.29 (1.91) | 2.48 (2.78) | 1.22 (3.25) | -7.13 (2.83) | -.61 (1.04) } -.049 (.288)]) 4.82 (5.71)
“1.19 ( .27)] «74 ( 239) ] 08 ( 45) | 1.16 ( .39)] «11 (214) | 029 (.040)} 1.86 ( .79)
93 93 93 93 93 93 93
-99991 -99981 -99848 99944 90007 -98709 -98273
1.535 2.237 2.617 2.280 8315 +2321 4,598
43 -15 +39 30 +30 433 1.77
11.4 8.0 6.1 9.7 8.8 24.0 7.7
.0 4,0 5.8 2.1 3 1.5 9
18.69 1.10 1.06 1.00 98 1.20 1.78
2.02 1.62 1.86 4,23 1.42 171 2.85



58

References

Akerlof, G.A. (1979) "The Case Against Conservative Macroeconomics: An
Inaugural Lecture", Economica, 46, 183, 219-237.

Artis, M.J. (1984) "Book Review", Economica, 51, 202, 205-207.
Bhargava, A. (1983) "On the Theory of Testing for Unit Roots in Observed
Time Series", Discussion Paper No. 83/67, International Centre for

Economics and Related Disciplines, London School of Economics.

Boland, L.A. (1982) The Foundations of Economic Method, London, George
Allen and Unwin,

Box, G.E.P. and D.A. Pierce (1970) "Distribution of Residual
Autocorrelations in Autoregressive~integrated Moving Average Time
Series Models", Journal of the American Statistical Association, 65,
332, 1509-1526.

Chalmers, A.F. (1976) What Is this Thing Called Science? An Assessment of
the Nature and Status of Science and Its Methods, St. Lucia,
Queensland, University of Queensland Press.

Chow, G.C. (1960) "Tests of Equality between Sets of Coefficients in Two
Linear Regressions", Econometrica, 28, 3, 591-605.

Congdon, T. (1983) "Has Friedman Got It Wrong?", The Banker, July, 117-125.

Davidson, J.E.H., D.F. Hendry, F. Srba, and S. Yeo (1978) "Econometric
Modelling of the Aggregate Time-series Relationship between Consumers'
Expenditure and Income in the United Kingdom", Economic Journal, 88,
352, 661-692.

Davidson, J.E.H. and M. Keil (1981) "An Econometric Model of the Money
Supply and Balance of Payments in the United Kingdom", Discussion
Paper No. 81/27, International Centre for Economics and Related
Disciplines, London School of Economics.

Desai, M.J. (1981) Testing Monetarism, London, Frances Pinter.

Dickey, D.A. and W.A. Fuller (1979) "Distribution of the Estimators for
Autoregressive Time Series with a Unit Root", Journal of the American
Statistical Association, 74, 366, 427-431.

Dickey, D.A. and W.A. Fuller (1981) "Likelihood Ratio Statistics for
Autoregressive Time Series with a Unit Root", Econometrica, 49, 4,
1057-1072.

Dyke, C.E. (1981) Philosophy of Economics, Englewood Cliffs, New Jersey,
Prentice-Hall.

Engle, R.F. (1982) "Autoregressive Conditional Heteroscedasticity with
Estimates of the Variance of United Kingdom Inflation", Econometrica,
50, 4, 987-1007. TO


59

Engle, R.F., D.F. Hendry, and J.-F. Richard (1983) "Exogeneity",
Econometrica, 51, 2, 277-304.

Evans, G.B.A. and N.E. Savin (1981) "Testing for Unit Roots: 1",
Econometrica, 49, 3, 753-779. :

Evans, G.B.A. and N.E. Savin (1984) "Testing for Unit Roots: ea",
Econometrica, 52, 5, 1241-1269.

Friedman, M. (1951) "Comment" in G. Haberler (ed.) Conference on Business
Cycles, New York, National Bureau of Economic Research, 107-114.

Friedman, M. (1953) Essays in Positive Economics, Chicago, University of
Chicago Press.

Friedman, M. and A.J. Schwartz (1963) A Monetary History of the United
States, 1867-1960, Princeton, Princeton University Press.

Friedman, M. and A.J. Schwartz (1982) Monetary Trends in the United States
and the United Kingdom: Their Relation to Income, Prices, and Interest
Rates, 1867-1975, Chicago, University of Chicago Press.

Fuller, W.A. (1976) Introduction to Statistical Time Series, New York, John
Wiley and Sons.

Godfrey, L.G. (1978) "Testing Against General Autoregressive and Moving
Average Error Models when the Regressors Include Lagged Dependent
Variables", Econometrica, 46, 6, 1293-1301.

Goodhart, C.A.E. (1982) "Monetary Trends in the United States and the
United Kingdom: A British Review", Journal of Economic Literature,
20,4, 1540-1551.

Goodhart, C.A.E. (1984) Monetary Theory and Practice: The UK Experience,
London, Macmillan Press.

Gould, J.P. and C.R. Nelson (1974) "The Stochastic Structure of the
Velocity of Money", American Economic Review, 64, 3, 405-418.

Granger, C.W.J. (1969) "Investigating Causal Relations by Econometric
Models and Cross-spectral Methods", Econometrica, 37, 3, 424-438.

Granger, C.W.J. (1981) "Some Properties of Time Series Data and Their Use
in Econometric Model Specification", Journal of Econometrics, 16, 1,
121-130.

Granger, C.W.J. (1983a) "Co-integrated Variables and Error-correcting
Models", Discussion Paper No. 83-13, University of California, San
Diego.

Granger, C.W.J. (1983b) "Discussion", International Statistical Review, 51,
2, 151-153.

Granger, C.W.J. and R.F. Engle (1984) "Dynamic Model Specification with
Equilibrium Constraints: Co-integration and Error-correction", mimeo,
University of California, San Diego, August.


22
Corresponding to the resulting sixfold partition of the information [i.e.,
(Ai), (Az), (As), (B), (C), (D)], we have the following six evaluation
criteria: innovation errors, weak exogeneity, parameter constancy, theory
consistency, data admissibility, and parameter encompassing. '8 Those
comprise a minimal set of criteria, the satisfaction of which is necessary,
but not sufficient, to sustain empirical inferences, and hence forecasting
and policy analysis. In statistical terms, each criterion yields a
testable null hypothesis (subject to minimal identification requirements),
and we have essentially stated those criteria in terms of their
corresponding nulls. In Section III (above), we investigated how well
certain of their equations perform on most of those criteria, and rejected
almost all of the claims FS make for their empirical model of money
demand.'!9 We cannot do better than cite Friedman (1951, p. 107) in

support of our approach:

It is one of our chief defects that we place all too much
emphasis on the derivation of hypotheses and all too little on
testing their validity. This distortion of emphasis is
frequently unavoidable, resulting from the absence of widely
accepted and objective criteria for testing the validity of
hypotheses in the social sciences. But this is not the whole
story. Because we cannot adequately test the validity of many
hypotheses, we have fallen into the habit of not trying to test
the validity of hypotheses even when we can do so. We examine
evidence, reach a conclusion, set it forth, and rest content,
neither asking ourselves what evidence might contradict our
hypothesis nor seeking to find out whether it does.

18(p) may be partitioned into (D,), (D,), and (D;), rather naturally inducing
encompassing on past data, on exogeneity conditions, and on forecasts.
E.g., see Mizon and Richard (1983) and Mizon (1984).

191 is unclear to which concept of exogeneity their notion corresponds.
Hence it is exceedingly difficult to formulate tests of their assertion
that) money is "exogenous" although, as noted in Section IV, we can reject
the null hypothesis that M is strictly exogenous.


3a

The log of the velocity of money (vi) and the yield on 3-month bills (RS,) (annual data).

Figure I.

[e) oO [e} [o} {2} oO {2} [o} [e)

62 @2F ©@ 8 + HY ALOE

ed [o) oO oO oO oO oO oO oO [o} oO

|

la q

E_ =

LL 4

be |
:
N

aa

Oe

55

re 4

Ze

2¢

zo

eu

ve i”

r |i

i

‘

| it Cae |

t

on

20

18

16
4
12.
10
8
6
4
2
i)

1980

1960

1940

1920

1900

1880


II. The data series and transformations

We consider immediately the annual and phase-average data series to be
used in our study, note certain reservations on the measurement and
interpretation of those series, and discuss some of the statistical effects
of transforming the annual data to phase averages.

In Table 4.9, FS report data series from 1871 to 1975 on the "annual"
values in the UK of the broad money stock (M), real net national income
(I), the price level (P), short-term and long-term nominal interest rates
(RS and R&), population (N), the exchange rate against the US dollar (E, in
$ per £), and high-powered money (H). We also include FS's (Table 4.8)
series for the price level in the USA (P*) in our data set.! Unless
otherwise noted, capital letters denote both the generic name and the
level; logs of scalars are in lower case, vectors and matrices in bold face.
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. Otherwise, the data are unaltered. Figures I-III show various
time series after the rescaling.

However, FS do not directly analyse this "raw" data, and instead
transform the annual series by averaging separately over contraction and
expansion phases of data-selected choices of "reference business eyeles",
Their objective in so doing is to extract the "longer-term movements" in
the data (the focus of their study; nb. pp. 13-14). Further, they claim
that phase-averaging reduces serial correlation arising from the business
cycle (p. 78) and attenuates measurement errors (p. 86). Attaining those
effects is important to their statistical analysis. Over some hundred

years of annual observations, FS identify 37 such phases for the UK, with

lFor details of construction and definition of all series, see FS (chapters
4 and 5) and our Appendix A.


11
contrast with J. Johnston (1963, pp. 179ff) on the effects of
autocorrelation, and see Christopher Sims (1980) for a forceful advocacy of
less restrictive modelling). On adding RN; to (4), dw increases to only
+59, but increases to 1.51 for the model of (m-p-n) recorded in Table I.

We could closely, but not precisely, reproduce those numbers:

(5) (m-p-n)y = .012 + .885 (i-n)j - 11.21 RN; - .22 G(p+i);
(419) (049) (3.3) (.29)

+ 1.37 Wy + 20.6 3;
(.58) (2.7)

J = 36 & = 5.66% dw= 1.51.

Much of the reduction in variance from (4) to (5) is due to 55, their data-
based dummy for "[a]n upward demand shift, produced by economic depression
and war" (p. 281), equal to a 21% shift during 1921-55.

A major issue in FS is the claimed (parameter) constancy of their
demand-for-money equation (e.g., see pp. 283, 624) although many
investigators would regard the need for a shift dummy for one-third of the
sample as casting severe doubt on the model's constancy. Interestingly, FS
nowhere actually test for constancy. They apparently believe that "high"
values of R? and/or "low" values of 3 are sufficient in themselves to
indicate constancy: such a conclusion is false. We tested the constancy of
their money-demand equation once only, on a "split half" basis, refitting
(5) to observations 1-18 and predicting 19-36, and testing for constancy
using Chow's (1960, pp. 594-5) statistic: that yielded n,(18,12) = 6.3
which exceeds the 1% point of the F-distribution. Even although the 95%
confidence interval based on @ is +11%, parameter non-constancy can be
detected. The values of the residual standard error for the two
corresponding sub-periods are dramatically different: 2.8% and 6.0%

respectively. Such evidence is far from supporting any claim to constancy.


3c

Figure I11. The log of the implicit price deflator (p,) and the log of the nominal money stock (m,) (annual data).

T T T T T T T T Toman 42

| ! ! i i i Pt 6
1880 1900 1920 1940 1960 1980



17
variables in FS's Table 4.9 with a meaningful parameterisation. We denote
that density by FOR | X03 8) where x_ is the vector of observations on
those variables at time t (t=1,...,T), x = (x1 «++ X_) so that x is the
complete sample of data, Xo denotes the matrix of all relevant initial
conditions, and @ is the vector of associated parameters. However, from
our theoretical perspective, the density F(+) over the variates in their
Table 4.9 has already been subjected to considerable marginalisation
involving variables which we believe to be important in accounting for the
behaviour of the reported series (e.g., fiscal variables, investment, etc.).
That could jeopardise the constancy of 8, which is potentially a function

of the omitted variables and would vary as they did.

Given the sequential nature of economic behaviour, F(+) is factorised as:
(9) Fc I
9) FO%p [X08 8) = ML POse | Xt-15 at)

where X¢-1 = (Xo X1 +++ Xt-1) and (Ay... At)" = A = £(@) is the
corresponding re-parameterisation. A critical claim to be tested is the
constancy over time of certain components and/or functions of {dt}. Once
the data are appropriately transformed to make normality reasonable, (9)
entails:
(10) xe [Xt-1 > Nuts Zt)
where ut = E(x¢|Xt-1) and so At comprises the non-redundant elements of Et
and fy. Note that if ey = xt-nte, then {e¢} is a sequence of martingale
differences (and so is an innovation with respect to X¢-1, and hence is
white noise).

We are already in a position to interpret some of the empirical
findings in Section III. For example, averaging over phases of business

cycles further reduces the density FIL) and the associated

J *#- - *
parameterisation, resulting in iL, F (x5 /Xj-15 Aj) (say). Thus, if FS's
je


International Finance Discussion Papers
Number 270

Revised: December 1985

ASSERTION WITHOUT EMPIRICAL BASIS: AN ECONOMETRIC APPRAISAL
OF MONETARY TRENDS IN ... THE UNITED KINGDOM
BY MILTON FRIEDMAN AND ANNA J. SCHWARTZ

by

David F. Hendry and Neil R. Ericsson

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.


35

The post-1970 evidence could be interpreted as support for our belief
that money is a "buffer" for short-run shocks, so that the error variance
is not’ constant for such equations, but varies with shocks to money supply,
e.g., when the authorities attempt to control that variable. However, in
models with nearly orthogonal regressors, omission of important explanatory
factors whose time-series properties alter also would be reflected
primarily in @ changing.

The large effect of Do in (22) is interesting in view of the estimates
in the earlier time-series models since its role appears to be to remove an
anomaiy between v and RS, probably due to the invalidity of unit-elasticity
restrictions on p and i. One reason for seeking to explain M without
needing Do is that we cannot accept that a "shift" of 20% persisting for
over 30 years in the relation of M to PI (see equation (22)) is just an
"anomaly". 28 Whether one believes that M is exogenous and "causes" P or
that it is endogenous and determined by a "stable" money-demand function,
the observable discrepancy is massive evidence against either belief .29
For 1920-1956, (m-p) is grossly out of line with i (even adjusted for RS),

a point seen graphically in Figure II. Yet the price level falls in almost

281¢ Do is in fact excluded from the regressor set when simplifying the
regression in Table IV, price homogeneity is rejected; or, if price
homogeneity is imposed, ¢ increases. Thus a radically different approach
seems required to sustain even the weaker view that agents plan to hold a
certa:n money stock via a constant behavioural model. Noting that nominal
money Mt almost never falls, the class of models discussed by George

Akerlof (1979) also merits consideration.

290ne possible explanation (suggested by Nicholas Dimsdale) for the behaviour
of a broad money aggregate like M is via a portfolio model in which the
ratio of M to other financial assets is a function of the opportunity costs
of ho.ding a non-interest bearing asset and the risks of other assets.
Andrew Threadgold has kindly provided his data on Central Government Debt
over our period and the similarity of its behaviour to that of M/(PI) is
Suggestive: we are currently investigating this approach to modelling M
without Do.


6

FS's phase-averaging. We take the first immediately whilst postponing
discussion of the latter two until Section III.

Phase-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 unwanted intermediate observations), thereby
entailing a (statistical) reduction in that re-parameterisation.> A
loss of information results from such aggregation. That would be
unimportant if all the parameters of interest could be recovered from the
aggregated series and clearly FS believe such parameters can be (e.g., the
long-run elasticity of interest rates in a money-demand equation). Many of
the parameters which we consider to be of interest cannot be obtained from
FS's phase 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", Further,
some of those parameters are fundamental in deriving "long-run" parameters
(cf, Davidson et al (1978, p. 681)). Hence, we will analyse primarily the
annual data and only briefly investigate the phase-average series. That
precipitates the issue of reconciling and comparing results from the two

data sets: Appendix B attempts an analysis, based on one possible, if

5To illustrate, consider fixed n-period phase-averaging of the annual series
{xpi t=..,1,2,...,T} with T a multiple of n. Letting L be the lag operator
such that Lxt=xp-1, the first filter is (1+L+L2+...+L"-1)/m, so the
averaged series is {x_*; t=..,1,2,...,T} where

Xp = (kptxE-7+X_-2+...+Xt-n+1)/n. The second filter selects every nth
observation of x~*, so the phase-average series is

Gxgnts Je-.51,2,-2.5(T/n)}, denoted {%j; J=..,1,2,...,d}. In fact, FS
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
(.5+L+L2+,,..+#L971+.5L")/n 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
thar. annual) data (cf. pp. 75, 84-85 and our Appendix B).


5
of measurement" we could not ascertain from their discussion. Several
hints are provided: for example, FS claim that the importance of errors in
variables is reduced by averaging, but enhanced by differencing (p. 86).
That suggests the model of the measurement errors being close to white
noise since averaging is of little help if errors are persistent (e.g.,
highly autoregressive; see Appendix B).

All the variables undoubtedly contain substantial measurement errors,
especially when interpreted as correspondences to economically meaningful
latent constructs. But, it seems unimaginable that such time series of
over a century would not contain large systematic errors. Differencing
would remove most effects of such errors, so rates of change could well be
absolutely more accurate than levels in that their errors were (say) 2% of
the level when the level was mismeasured by (say) 10%.4 While FS use
"differencing" as a filter, it is not used that way in our re-evaluation
below, but occurs only as a convenient way of imposing parameter
transformations (cf. James Davidson, Hendry, Frank Srba, and Stephen Yeo
(1978, pp. 673-674)). -Thus, the relative accuracy of estimation in levels
and differences is not directly relevant to this study, as all our
econometric models are effectively in levels. Where any assumption is
needed, we will take the measurement errors of the levels of variables to
be highly autoregressive with a small innovation variance (see Appendix B).

Use of phase-average data raises three distinct issues requiring
examination: the theoretical statistical effects of phase-averaging (qua
aggregation), the impact of selecting the intervals over which to average

by prior analysis of an interrelated data set, and the observed effects of

Using the earlier example, consider the enormous falls in the real price of
but increases in the volume of computing since World War II and the
consequential difference it makes as to whether 1947 or 1982 prices are
used in measuring real income. Yet the changes from year to year could be
reasonably accurately measured as a percentage of the level.


48
where 1 was computed from the estimated residuals, and dw* and 6* relate
to equivalent regressions including D,, Dz and D3. The large increase
between dw and dw* is due almost entirely to FS's shift dummy Do. Adding
R& had little effect on 6 or dw.

While the tests are inconclusive, the existence of those relationships
at their Waterloo is as much a close~run thing as Wellington's was at his:
when the dummies are excluded, the values of dw are close to the lower
bounds in Sargan and Bhargava (1983), especially given that an optimal
linear combination of stochastic regressors is being estimated. At best,
those tests point to an exceedingly tenuous relationship, salvaged only by
the inclusion of the shift dummy variable constructed by FS after they
noted discrepancies from a linear regression between real money and real
income (see FS (pp. 225-238) and our Figure II). Note also that (C1) and
(c2) fit much worse than the time-series models reported above, indicating

the important role of dynamic reactions in accounting for the data variances.

APPENDIX D. Re-normalisations of regression equations

This appendix briefly illustrates why we view FS's models for (pti),
P, and i (the second through fourth equations reported in Table I) as
approximate re-normalisations of their equation for (m-p-n) and why such
re-normalisations produce no new inferences.

Now, if the joint density F(x, |X ) in (9) involves q different

t-1 At
xXit's, it can be factorised in q! different ways. Any given factorisation
is statistically legitimate, although most need not yield either constant
or interesting parameters. However, in general, it is not legitimate to
pick elements from different factorisations and treat them as "independent"

summaries of F(x |X 5A). For example, if q = 2 with x' = (x x ),
~t~t at ~t 1t at

-1

there exist two valid factorisations:


33

automatically encompasses the models in equations (12)-(14) as well as
(22); that (22) variance-dominates FS's money-demand model in levels by a
factor of over tenfo1d26; that the parameters correspond to nearly
orthogonal variables (see Table V); that we did not need to run "hundreds
of regressions" (p. 266) to establish (22); but that it might be useful to
formulate a theory-model of agents' plans and expectations which could
yield insight into its dynamics and perhaps produce a more parsimonious
Pparameterisation. The parameters of the non-dummy variables in (22) are
precisely estimated; and most of the diagnostics are acceptable, although
n2(+) in Table IV and (22) is significant, with the partial sixth-order
residual autocorrelation coefficient for the latter being -.4, suggesting
that further lags may be necessary in Table IV. The adjustment lags are
long in (22), given that the error-correction term (m-p-i) enters at four
lags. Further, only changes in the rate of inflation (and not the level of
inflation) matter; otherwise, (22) is similar in form to the models
developed in Hendry and Mizon (1978), Hendry (1979), John Trundle (1982),
and Davidson and Manfred Keil (1981). Compared to (13), the time-series
component is retained almost unaltered, so (22) encompasses the univariate
model in (13) while highlighting the roles of both the feedback from
disequilibria via (m-p-i) and the additional economic variables.

Given that FS postulate a shift in liquidity preference over 1921-55,
it seemed sensible to estimate (22) separately for each of the main epochs

noted earlier, namely (i) 1878-1913, (ii) 1922-1938, (iii) 1950-1970, and

26gy appropriately (statistically) reducing the density F(yt¢|Zt»Xt-13 $1)
for our model, one could derive an estimate of the parameters in FS's model
corresponding to the density F* (xX |x jr1s Bj ). From that constructed
estimate and the estimate which Fe ad agtually obtain, one could test whether
our model using annual data encompasses theirs using phase-average data.
Note also that it is easy to construct examples for which the parameters in
FS's model would not be constant over time but those in ours would be
(e.g., in (21), , is constant, Ze is Granger-caused by Yee and 25 changes
over time).


23a

Table III. Criteria for evaluating econometric models

Null Alternative Statistic Sources

(A,) qt8-order residual &2(q)3 Box and Pierce (1970);
autocorrelation n2(q,T-k-q) Godfrey (1978), Harvey

(1981, p. 173)

(A,) q invalid parameter n3(q,T-k-q) Johnston (1963, p. 126)
restrictions

(A,) first-order ARCH &,(1) Engle (1982)

(A,) skewness (SK) and &5(2) Jarque and Bera (1980)
excess kurtosis (EK)

(A,) heteroscedasticity Ne (q,T-k-q) White (1980), Nicholls
quadratic in regressors and Pagan (1983)

(q quadratic terms)

(A,) q instrumental &7(q-k); Sargan (1958, 1964);
variables not n,(q-k,T-q) Sargan (1980a, p. 1136)
independent of errors

(A;) predictive failure over n,(q,T-k-q) Chow (1960, pp. 594-5)

a subset of q
observations

Notes: 1. We have labelled the Chow statistic n,(q,T-k-q) both to highlight
the pre-eminence of the issue of constancy in the substantive
debate on monetary behaviour and because of its crucial role as
an indirect test of weak exogeneity through testing the
conjunction of hypotheses embodied in super exogeneity.

2. The value of q may differ across Statistics, as may those of k
and T across models and samples.


36

every year from 1921 to 1935 and real income experiences its most drastic
historical fall in 1920-1922! Thus, the economy seems capable of operating
without inflation despite an "excessive" real money stock.

Our econometric model of the demand for money in the UK satisfies a
range of statistical criteria as well as incorporates economic theory.
Both aspects are essential for a model to characterise adequately the
underlying data generation process. However, even with our best effort,
our model is not fully satisfactory: estimated adjustment lags are rather
long; some parameter non-constancy is evident, mainly through the equation
standard error and the coefficients on interest rates; and the dummy
variable Do seems necessary . 30 That may be less than surprising as the data
span a century during which financial institutions altered dramatically:
witness the growth of building societies and, after 1970, the introduction
of competition and credit control and of floating exchange rates. In spite
of manifest deficiencies in our model, the evidence herein suggests that
the econometric techniques used to obtain that model are vastly superior in

practice to the maze of prior transformations in which FS indulge.

VII. Conclusions

A number of assertions in Friedman and Schwartz (1982) concerning the
empirical validity of their money-demand equation have been tested using
their data series for the United Kingdom and were found to be without
empirical support. Their procedure of averaging data over business-cycle
phases did not greatly reduce the serial correlation in the data series,
but did lose a great deal of information, thus leading to rather badly
fitting equations. We find nothing to recommend in that practice as

against analysing the annual data and modelling both "trend" and "cycle".

30For a follow-up to Section 6, see Andrew Longbottom and Sean Holly (1985).


23

While there are only six evaluation criteria described above,
departures from the null hypothesis could take many forms. Table III lists
the bulk of the test statistics reported below; the convention used is that
€4(q) and nj(q,r) denote statistics which have central x(q) and F(q,r)
distributions respectively under a common null and against the ith
alternative. Thus, &,(q) and n,(q,T-k-q) both test for qt-order residual
autocorrelation. There are T observations and k regressors in the model
under the null. Most of these tests are applied in the Lagrange Multiplier
Spirit, with all the data used in the estimates quoted and where (e.g.)
ni(q,*) acts as a post-estimation diagnostic for predictive failure over
the last q observations. Joint tests could be constructed as in Jarque and
Bera (1980); and, as Jan Kiviet (1982) also notes, many of the statistics
are asymptotically independent so that their x°-forms could be added
together to construct a "portmanteau" mis-specification statistic. Otherwise,
care should be taken to control for Type I errors over the set of tests.

: Even if FS had tested and had not rejected such necessary conditions
for valid inference entailed in (A), (B) and (C), as discussed in Hendry
(1983), those criteria are minimal in that they often can be satisfied
simply by designing empirical models appropriately. For example, a
theory-based model imposed on data and with any residual serial correlation
removed usually satisfies (A,) and (B); and so on. Consequently, we
additionally would require evidence on the ability of their models to
encompass rival hypotheses, that is, to demonstrate that the information in

(D) is irrelevant, conditional on (A) and (B) (here we assume common


dependent
variable

regressoP

M1
m-2
m-3
m4
M5
pr-1*
Pt-o*
Pt-3*
Pt-u*
Pt-5*
dt
it-2
it-3
it-y
its
Pt-1
Pt-2
Pt-3
Pt-4
PLS

Rt

RSt-2

Constant

T

R?

1008
7, (10,44)
&2(7)
&,(1)
n3 (5,54)
ng(30,54)

55

Table G.I. Estimates of equation (11) with p*: observations for WWI and WII included
M Pr* ig Pt RSy, Ruy, et
1.65 ( .16) 50 ( .33)} 48 (23) 43 ( .25)| ~.05 (.06)] -.027 (.019)] .51 ( .37)
-.€8 ( .29) -.26 ( .60)| -.85 ( .41) ~14 (44) | -.05 (211) | ~.001 (034) } -.04 ( .67)
+28 ( 31) +59 ( .63)] «70 ( 44) 221 ( 47) ] 20 (.12)]} — .061 (036) | -.75 ( .71)
04 ( .28) -.82 ( .58)} -.89 ( .40) ~.03 ( .43)} -.20 (.11) | -.067 (.033) 31 ( 65)
-.C6 ( .14) +48 ( .29)] 4B ( .20) 04 ( .22)] 09 (.05)] 023 (.017)] «11 (-.33)
C1 (609) ] 1.05 ( 618) } 14 ( 613) } -.03 ( .14) | .00 (.03)} 006 (.010)] 41 (21)
+10 ( .12) +08 ( .25)| -.13 ( .17) +06 ( .18)] 05 (.05)} 003 (.014) | -.27 ( .28)
~.01 ( .12) -.09 ( .25)| -.41 ( .18) -14-( 419) +06 (.05) 2017 (.014) | -.44 ( £29)
~CT ( .12)|  -.25 ( .23)] 20 ( .16) | -.20 ( .17) | -.10 (.04) | -.022 (.013)]  .18 ( .26)
-.C9 ( .10) -.10 ( .21) +20 ( .15) ~.05 ( .16)} -.01 (.04)} -.002 (.012) +06 ( 24)
CO. ( 11) ] 204 ( .22)) 64 ( £15) | -.03 ( 16) } 08 (04) | 024 (.012)} -.12 ( .24)
~07 (14) | 628 ( 628) | -.02 ( .20)} -.05 ( .21) | -.04 (.05) | -.008 (.016)| -.57 ( .32)
-.C8 ( .13) -.39 ( .26) 04 ( .18) -.29 ( .20)] -.12 (.05)} -.027 (.015) “4 (29)
CO ( .13) 202 ( .26) +18 ( 18) ~.08 ( .19) 13 (.05) +008 (.015) 31 ( .29)
-.03 ( .10) +26 ( .21) 23 ( .14) 202 ( .15)] -.03 (.04) 015 (.012) | -.20 ( .23)
~10 (14) | 14 (628) | -.37 (19) } 1.11 (221) | 08 (05) } 023 (.016) | -.51 ( .31)
201 (619) | 636 ( HO) «34 (27) | -.57 ( .29) | -.09 (.07)] -.017 (.023)]} -.19 ( .44)
-.28 ( .19) 00 ( 40) 04 ( .28) =.14 (30) | ~.07 (.07)} -.014 (.023) 69 ( .45)
01 ( 19) +10 ( .38) 08 ( .26) -.00 ( .28) «14 (.07) 016 (.022) | -.02 ( .43)
+10 ( .12) +04 ( 24) ] -.02 ( .17) +02 ( .18)] ~-.05 (.04) +003 (.014)] -.21 ( .27)
7.12 (45) | -.04 ( .92)] 13 ( 63) -.93 ( .68)] 46 (217) | -.094 (.052)} 1.18 (1.03)
-.16 ( .49)} 1.21 (1.01) | -1.65 ( 70) +50 ( .75)| -.18 (.19)] .060 (.057)} 2.58 (1.13)
636 (5H) } 1.04 (1.11) ] -.05 ( 277) | 1.14 ( .83) «47 (20) 042 (.063) | -3.17 (1.24)
1 ( 62) |  -.43 (1.28) | -1.12 ( .88) +30 ( .95)} ~.34 (.24)]  .034 (.073) | -.18 (1.43)
~631 (51) | 1.08 (1.04) ] 015 (672) | 1.17 ( 78) | -.24 (.19) | -.176 (.059) | 1.64 (1.17)
1.13 (1.34) | 1.40 (2.73) | 1.14 (1.89) } 4.85 (2.03) | «38 (.50) | 1.207 (.156) | 5.51 (3.07)
4.89 (2.17) | 1.47 (4.43) | 3.74 (3.07) | -.52 (3.30) | -.47 (.82)| -.872 (.252) | 2.11 (4.97)
+29 (2.27)| 8.94 (4.64) | -.89 (3.22) 6.26 (3.45) } -.12 (.86) 539 (.264) | 1.77 (5.21)
2.36 (2.35) | 3:70 (4.79) | -.70 (3.32)] 1.08 (3.56)} 1.53 (.89)] .068 (.273) | -.57 (5.38)
~.73 (1.72) | 2.40 (3.51) | -2.60 (2.43) | -2.32 (2.61) | -.09 (.65)]| .163 (.200)} 3.03 (3.94)
-.04 ( .06)| -.08 ( .13)] -.03 ( .09) +03 ( .10)] ~.03 (.02)} -.011 (.007)} .73 ( .15)
201 (608) | -.12 ( 616) ] 611. (211) | 206 (12) ] 01 (03) ] 002 (.009) | -.15 ( .18)
-.05 ( .08) -14(.16)] 03 ( .11) «05 ( .12)] 01 (.03)] 018 (.009)} .06 ( 17)
~.07 ( .07)| -.07 ( .13)] -.07 ( .09)] -.08 ( .10) 01 (.02)} .000 (.008)} .11 ( .15)
09 ( .05) +08 ( 411) ] -.15 ( .08) 04 ( .08)] -.01 (.02)} -.004 (.006)} -.16 ( .12)
4.53 (1.55) ] 6.56 (3.15) | 1.65 (2.19)] 2.57 (2.35)] «15 (.58)} «210 (.180)] 46. (3.54)
5.97 (2.10) | 12.93 (4.28) | 1.71 (2.97) | “14.31 (3.19) | -.60 (.79) | -.139 (.244) | 2.24 (4.80)
4.€5 (1.44) | 3.29 (2.93) } -.40 (2.03)| 2.98 (2.18) | -.55 (.54) | -.223 (.167) 61 (3.29)
~.34 (17) ~.3B ( .34) =30 ( 24) -.59 ( .26)| -.00 (.06)| -.019 (.020) +59 ( .39)
98 93 93 93 93 93 93
+99980 -99627 -99773 -99872 89592 -98339 -97801
1.997 4.076 2.826 3.034 -DB 2322 4.576
AT 43 32 1.19 a) 4.25 1.59
9.4 8.2 12.4 10.0 TA 17.5 11.4
1.2 3 1.4 .0 7 5 a
5.89 2.01 1.92 3.24 1.61 1.96 117
1.45 1.58 2.44 2.76 1.93 1.77 3.00



3b

Figure I1. The log of the real money stock [(m-p),] and the log of real income (i,) (annual data).

T T T T T T T T Tym 9.00

(m-p),, ~~ |

fo ttrcneeeee fg - 0-6 oa

— 8.50
— 8.00

4
— 7.50
— 7.00
L | i l i 1 l titi! 6.50

1880 1900 1920 1940 1960 1980


14

(2).13 Comparison with (2) immediately reveals that phase-averaging has
little impact on the standard deviation of "velocity" or on the significant
serial correlation present in the series. In Figure IV, the annual and
phase-average data for velocity are plotted, graphically illustrating the
effects of phase-averaging: there is only a small reduction in the (high-
frequency) variability at the cost of increasing the size of the "jump"
between successive observations. Consequently, phase-averaging has little
net effect on the variance of velocity (contrast with pp. 73ff).

Next, we present analytical evidence on the effects of phase-averaging.
In Appendix B, we derive the variance formulae based on a first-order
autoregressive process for "annual" and phase-average series, both in
levels and in differences. In Table II, values for a three-period phase
average (assuming an autoregressive coefficient of .96) are compared with

the observed statistics for velocity.

Table II. Standard deviations of velocity

Series Observed value Value for autoregressive model

v -169 16914
v +169 +166
Ay O47 +048
Ay? +071 067

As can be seen, the theoretical and actual standard deviations are quite

131f¢ we partial out war years from (8), we obtain:

. 14
Y= 655 + 2 ayWDy
(,018)  i=1

T = 100 @ = 16.6% dw = .17

where WDj denotes a dummy variable which is 1 for the th year of major war
and zero otherwise. These results are virtually identical to those in (8).

14the values in Appendix B (Table B.IV) have been normalised on the standard
deviation of v because the results in Appendix B are for relative, not
absolute, variances.


46

APPENDIX C. Testing for the existence of a relationship between variables

In this appendix, we summarise recent results on tests for the
existence of a relationship between variables and apply such tests to FS's
data for money, prices, incomes, and interest rates. A prior question to
what specification to choose in modelling money demand is whether there
exists any "long-run", or equilibrium, relationship between money, prices
and incomes. If a relationship is defined across evolutionary variables
(i.e., ones which need differencing to be stationary), then one
characterisation of its existence or the lack thereof is whether or not the
variance of deviations from the relationship is bounded over time (e.g.,
see Clive Granger and Andrew Weiss (1983)). If one cannot reject the
hypothesis that the deviations have unbounded variance over time, then the
postulated relationship is at best exceedingly tenuous, and at worst
meaningless. Since random walks have unbounded variability over time, a
natural hypothesis to seek to reject is that the deviations from the
reiationship constitute a random walk. That comprises the basis of Granger
and Engle (1984), drawing on Fuller (1976), Dickey and Fuller (1979, 1981),
G.B.A. Evans and N.E. Savin (1981, 1984), and Sargan and Bhargava (1983).

Granger (1981; 1983a, b) defines variables to be integrable of order d
(denoted I(d)) if differencing d times is needed to induce stationarity
(i.e., I1(0)). Variables which are I(d) for d 2 1 have variances which
increase rapidly over time whereas I(0) variables do not exhibit that
characteristic in large samples. A set of I(d) variables is co-integrable
if some linear combination of them is I1(0). Thus, we will test the
proposition that certain sets of variables which empirically behave as I(1)
are co-integrable (i.e., define a long-run relationship) by testing whether
the deviations from the relationship are I(1) against the alternative that

they are I(0). The exact dynamic specification does not matter for this


53

APPENDIX G. Estimation of equation (11)

Table G.I records the estimates pertinent to (11) for & = 5 for most
of the variables in FS's Table 4.9 (i.e., for those other than population).
For m, i, p, and e, the null hypothesis of Granger non-causality is
rejected at the 5% level. In no case is there significant autoregressive
conditional heteroscedasticity (i.e., ARCH; &,), or (with the exception of
the long-term interest rate R& and high-powered money H) parameter
non-constancy over the decade 1961-70 (n,) or residual autocorrelation (£2).
In several cases, however, there is evidence of significant outliers,
casting some doubt on the assumption of normality, although dummy variables
for each year of the two world wars would remove most such cases (see Table
G.III). Overall, therefore, the evidence is consistent with (11).

Table G.II reports related estimates when H is replaced by the US
price level P*. Some interesting differences result, unsurprisingly, as P*
has rather different time-series behaviour from H. Note that, whether or
not war years are included, P* is not Granger-caused by the UK variables
and so can be treated as strictly exogenous (cf. Tables G.II and G.IV).
Also, M is no longer Granger-caused on this information set, whereas the
exchange rate E is the variable most significantly affected by lags of the
other variables. Otherwise, results similar to those in Table G.I emerge.

In all four tables, n,(5,°) is the F statistic for testing for the
exclusion of Mt-1,+++,Mt-5 from the reported equation. n,(30,°) is the F
statistic for testing for the exclusion of all variables except lags of the
dependent variable, dummies, and the constant term from the equation (i.e.,
testing for Granger non-causality). The 5% significance points for
F(5,54), F(30,54), F(5,42), and F(30,42) are 2.38, 1.67, 2.44, and 1.73

respectively. The estimation period is 1878-1970 inclusive (T=93).


30
corresponding observable variables y_ and zt, a general autoregressive-
distributed lag relationship is postulated, with the parameters satisfying
the restriction entailed by the steady-state solution. For expositional
simplicity, we consider only current and one-period lags of yy, and zt

entering the dynamic relationship, as in the following equation:
(18) Ye = % + OYt-1 + Boze + Bize-7 + VE ve ~ IN(O, 0°)

In (18), long-run homogeneity between y and z requires a,+B,+fy=1. Re-

writing (18) with that restriction gives:
(19) Aqyg = a + BAYZE + Vlyeg-17-Zt-1) + ve Y#0

where a, 8, and Y are corresponding unrestricted parameters. That equation
is respresentative of a large class of models satisfying both the economic
theoretic restrictions and allowing for general dynamic responses.
Intuitively, the term BA;z_ reflects the immediate impact that a change in
Z, has on yg. The term Y(y¢-1-z~-1) (with Y negative for dynamic
stability) is statistically equivalent to having Y(yt-1 -K~-Zt-1) instead in
(19), and hence reflects the impact on Ajy¢ of having y¢-1 out of line with
K+Z_-1- Such discrepancies could arise from errors in agents' past
decisions, with the presence of Y(y¢-1-zy-1) reflecting their attempts to
correct such errors; so, (19) belongs to the class of error-correction
models. For a steady-state growth rate of Z, equal to g (i.e.,

B=A12¢74,¥_) and v_=0, then, solving (19), we have:
(20) Yq = Zeexp{[-atg(1-B)]/¥} ,

reproducing the assumption of proportionality between Y_ and Zy from the
non-stochastic steady-state theory. Note also that K = exp{[~atg(1-8)]/Y},
which is independent of g only if §=1 or a depends on g. See Teun Kloek

(1984) and Mark Salmon (1982).


. 31

Estimation of the "long-run parameter" K requires estimating all
parameters in (19), including those corresponding to short-run and dis-
equilibrium effects (i.e., B and Y). When using phase-average data, least-
squares estimation is inconsistent for long-run parameters of important
subclasses of (18) unless zy is strictly exogenous (see Appendix B). Even
if z_ is strictly exogenous, efficient estimation of those parameters and
consistent estimation of standard errors (necessary for valid inferences)
require properly modelling the dynamics present. Phase-averaging is
neither necessary nor sufficient to capture such dynamics or to permit
valid inferences from least-squares estimation. By way of contrast,
estimation of (19) by least squares allows valid inference on both
(a BY a) and K.

Error-correction models fall within the statistical framework of
Section IV, noting that (18) (and hence (19)) involves a further

factorisation of (9), namely:

(21) FOxe[Xe-15 At) = F(velzes Xe-15 1) + Flacl¥t-13 g2)

where x} = (ye Zt)» corresponding to the assertion that the variates in zt,
are weakly exogenous for the parameters $; in the conditional density (for
(19), o} = (a BY Oy)» xt (yt zt), and £=1). Such a factorisation of the
density is legitimate: whether it successfully isolates parameters of
interest will be considered later.

In the remainder of this section, we present a general autoregressive-
distributed lag representation of money conditional on prices, incomes, and
interest rates, estimated to establish the innovation variance. That
formulation is simplified to an error-correction model based on previous
money-demand models developed recently in the UK. The static-equilibrium

solution is presumed to be of a form ve=¥ot¥,RSz, thus taking vy and RS_ to


42
Tne variable-length weighting procedures in FS 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, (.5a2u;} should show negative second-order

autocorrelation of the form:

orth (4p?) (1 -p)2
(B15) -.5 + .
2 0(2-p%) (1-92)
2n(1-p°)[1 - —————____

n(1-*)

Some representative values of (B15) are given in Table B.II.

Table B.II. Values of second-order autocorrelation for Apu;

N 1 2 3 5

.0 -.50 -.50 ~.50 -.50
5 ~.38 ~.45 -.48 7.50
8 -.18 ~.25 ~.32 ~.W1
+95 -.05 ~.03 -.06 -.12

That effect may not carry over for data-selected phase averages.
Descriptively, FS's phase-average data series remain very highly
autoregressive (e.g., the Durbin-Watson statistic is -31 for velocity on a
constant and remains as low as .59 for real per capita money on a constant,
real per capita income, and interest rates). Thus the analysis ought to
account for the pure time-series phenomena of the data. Table B.IJII

details such phenomena, using velocity as an example.

Table B.III. Standard deviations of observed velocity

Annual Phase-average
Sample size 100 36

Levels o(v) = .169 o(¥) = .169

4

Differences o(A,v) = .O47 o(A1v)

.071


20
in-variables paradigm, FS ignore almost every other econometric issue, and
not just parameter non-constancy and autocorrelation and heteroscedasticity
in the residuals. There are no allowances for or tests of mis-
specification resulting from the simultaneous determination of the xt,
inappropriate functional forms, dynamic mis-specification, and/or omitted
variables.

In whatever approach is taken to modelling ¥t» aS shown above, the
associated disturbance is a derived process (being the unexplained
component of Xt) rather than an autonomous process. Hence it is relevant
to ask whether that disturbance (or any other aspect of the model, e.g., a
given parameter) has certain properties necessary for the model to be an
adequate representation of the underlying data generation process. The
nex section considers such properties and the corresponding model

evaluation criteria.

V. Model evaluation criteria

Statistical inference in multivariate time-series processes is a
hazardous and contentious issue. From the time of R. Hooker (1901) and
especially Udny Yule (1926) onwards, the enormous difficulties inherent in
conducting valid inference in such processes have gradually become
documented. Presently, econometricians are much more aware of the
pitfalls in analysing economic time series than of methods which ensure,
with any reasonable likelihood, that sensible and sustainable conclusions
can be reached. An empirical "conclusion" is deemed sustainable only if it
satisfies a range of criteria discussed in detail below. Most of those
criteria are well-known and widely accepted, we consider all of them to be
justifiable, and we contend that satisfying such criteria constitutes a

minimal necessary condition for judging an empirical model to be credible.


45

Table B.V. Standard deviations of various series (p=1.0)

Annual Phase-average
og (¥) = 7.00 o(y) = 7.04
de = 1.00 de = 1.40

That tatle is a much poorer match to Table B.III than is Table B.IV. Like
the significant error-correction coefficients established in Section VI,
the eviclence here favours a highly autoregressive but "just" stationary
process for velocity. Conversely, the analysis for that case explains why
the models using phase-average data fit much less well than those using
annual cata.

The effects of measurement errors (denoted by {ct¢}) on the above
analysis: critically depend on the time-series characteristics of such
errors. If the {z,} are white-noise errors in measuring A,yt, then they in
effect add to {e4} in (B16) when p is close to unity. However, if the {cy}
are white-noise errors in measuring Yt as ye = yt + St, then the error in
(B16) fitted to tye} becomes (e¢+e¢-pot-1). Thus, such errors would induce
substantial negative residual autocorrelation if on the variance of tt,
were comparable in magnitude to oy and that would be especially marked for

eros and o*(ayy°) = 2(0e + o2/(14p))«

the data {aye}. Also, o-(y°) =
Conversely, for ¥S = Vj + oj (say), (7°) = o*(y) + o/n and o-(A1¥°) =
o°(aqy) + 200 /n. Since the effect of measurement errors is much smaller
for y than for y, given (B16)-(B24) and the values observed in Table B.III,
oe woulc have to be negligible to match the variance ratios. While that is
depender.t on assuming (unrealistically) that the process is fully
characterised by (B16), nevertheless the evidence is more consonant with
white-ncise errors in measuring the changes, and relatively small errors

generally in comparison to the large standard deviations of the levels of

the variables (especially the trending series such as M, P, I, RS, or Rf).


dependent.
variable

ome

regressor
Me
M2
m-3
M4
M5
hey
h-2
hy~3
hey
hes
it-t
it-e
1t-3
ity
its
Pt-t
Pt-2
Pt-3
PL-y
Pt
RSt-1
RSt-2
RSp-3
RSt-y
FSp5
Re-1
R&-2
R&-3
Rey —y
Re 5

et
&t-2
3
ety
5

D
Da
D3

Constant

T

R?

1006
7, (10,44)
&2(7)
&4(1)
n3(5,54)
ns (30,54)

54

Table G.I. Estimates of equation (11) with h: observations for WWI and WWII included
Mm, he it Pt RSt, Ree, ee
1,38 ( .19) -61 ( .31) «42 ( .29)} -.19 ( .26) +02 (.08) | -.008 (.024) TH (46)
-.73 ( .30)| -.36 ( 48) 55 ( 45) } ~.22 ( .4O)] -.22 (.13) | -.035 (.038) -.31 ( .73)
+24 ( .32)} -.13 ( .53) “hg (.49)} 61 ( 44) 637 (614) | .095 (.041)]  -.87 ( .80)
209 ( .29)| 53 ( .48) | ~.98 ( 44) ] 215 ( 40) | -.25 (.13)] -.071 (.037) -64 ( .72)
~10 (615) ]} -.24 ( .24)] 663 ( .22)} -.04 ( .20)] .06 (.06)] —.010 (.019) 207 ( .37)
+18 ( .09) +73 ( .15)} -.21 ( .14) 49 (212) 02 (.04) +006 (.012) -.36 ( .22)
-02 ( .12) 05 ( .19)} .01 ( .18)] ~.01 ( .16)} 05 (.05) | .003 (.015) +32 ( .29)
+08 ( .12)} -.12 ( .20)] 05 ( .18)} -.13 ( .16)} -.09 (.05) | -.017 (.015) +08 ( .30)
+03 ( .12)] -.06 ( .20)]  .35 ( .18)|  .01 ( .17)] -.01 (.05)] -.010 (.016)] -.07 ( .30)
~.01 ( .10)] -.16 ( .16)} -.29 ( .14)] 210 ( .13)]  .03 (04) | .016 (.012)] -.34 ( .24)
+02 ( .09) +20 ( .14) +80 ( .13) O04 ( £12) -O7 (.04) 022 (.011) +24 (22)
09 ( .12) «04 ( 619) | -.06 ( .18) +05 ( .16)] ~-.02 (.05)] -.009 (.015) -.50 ( .29)
~.07 (11) | -.16 ( .18)] -.08 ( .17)} -.28 ( .15)] -.11 (.05) | -.023 (.014) 15 ( .28)
-105 ( 12) | -.11 (619) ] 17 (617) | -.13 (16) ] «10 (.05) | .004 (.015) 14 ( .28)
+00 ( .09)} -.00( .15)} 22 ( 14) ] 012 ( 213) | -.02 (.04) | .017 (.012) | -.07 ( .23)
~.22 ( 11) +09 ( .17)} ~-.14 ( .16) +96 ( .15) +05 (.05) +024 (014) -.06 ( .%)
+16 ( 15) +29 ( 3) 14 (423) | -.40 ( 221) +00 (.07)} -.008 (.019) -.38 ( .38)
7.30 ( .15)] -.22 ( .2)] -.34 ( .23) | -.12 ( .21)] -.06 (.07) | -.004 (.020) M1 (£38)
-.09 ( .15) | ~.04 ( .25)} 40 ( .23)] -.26 ( .21)] .07 (.07) | -.004 (.020) +23 ( .38)
04 ( .10)] 21 ( £16) | -.01 ( .15)] 02 ( .14) | ~.02 (.04)] 011 (.013)] -.09 ( .25)
o11 (42) | -.34 ( 68) | -.68 ( .63)] -.30 ( .57)]  .61 (.18)| -.050 (.053) 04 (1.03
-.25 ( 47) | -.42 ( 77) | -1.43 ( .71) +24 ( 64) | -.19 (.20) -059 (.060)] 3.80 (1.16)
-.14 (53) 19 ( .86)|  .43 ( 80) | -.88 ( .72)|  .37 (.23)|  .015 (.068) ] -2.70 (1.30)
+09 ( .57)| -.03 ( .93)| -.69 ( .86)]} .58 ( .78)| -.23 (.25) |  .046 (.073) +72 (1.41)
~.25 ( YT) | -.75 (77) | --56 (.71) | -1.14 ( 264) | -.20 (.20) | -.154 (.060) +24 (1.16)
-.94 (1.24) | -3.15 (2.02) } .76 (1.88) | 4.61 (1.69)]  .62 (.53) | 1.255 (.159) | -3.73 (3.06)
4.17 (1.98) | 3.35 (3.23) | 3.31 (3.00) | 1.04 (2.70) | -.48 (.85)} -.864 (.253)] -2.46 (4.88)
~.4O (2,12) | -1.92 (3.46) | -1.64 (3.21) | 6.07 (2.89) } -.22 (.91)] 541 (2771) +27 (5.23)
2.62 (2.21) | 2.65 (3.60) | -.85 (3.35) | 1.76 (3.01) ] 1.43 (.95)] .050 (.283) |  -.93 (5.45)
7-93 (1.51) | 1.10 (2.47) | 84 (2.29) | -3.10 (2.06) } -.92 (.65)| -.055 (.193)| 4.87 (3.73)
-.04 ( .06)] -.09 ( .09)] -.07 ( .09) -02 ( .08) +00 (.03) | -.003 (.007) 77 ( .14)
-.01 ( .07)} 05 ( .12)] 02 ( .11) | -.08 ( .10)] —.00 (.03)] .002 (.009)] -.27 ( .18)
~.02 ( .07)] 05 ( 612) | -.01 (611) ] «13. (10) } 04 (.03) | 020 (.009) -09 ( .18)
~.10 ( .06)| -.06 ( .10)] -.08 ( .09)] -.13 ( .08)] -.01 (.03)| -.004 (.008) 11 (215)
+05 ( .05)| 07 ( .08)] -.04 ( .07)] -.02 ( .07)] -.02 (.02)] -.007 (.006)] -.05 ( 12)
2.34 (1.50) | 16.03 (2.45) | 5.49 (2,28) | -2.89 (2.05) | -.79 (.65)} .037 (.192) -66 (3.71
1.03 (2.07) | -4.64 (3.38) -O4 (3.14) | 6.69 (2.82) | -.06 (.89)] -.067 (.265)] 3.48 (5.11)
4U6 (1.42) | 7.26 (2.31) | 2.16 (2.15) +70 (1.94) | -1.16 (.61) | -.334 (.182) +37 (3.50)
-.97 ( .27) TH (4) 46 (41) | 1.48 ( 237) +03 (.12)} -.008 (.034) 1.18 ( .66)
93 93 93 93 93 93 93
99982 -99954 -99772 -99910 -88081 +98234 97765
1.872 3.052 2.833 2.551 -8066 2393 4,613
234 +3 234 94 53 4.73 1.44
5.5 18.0 8.8 14.9 8.2 21.4 13.8
6 2.3 2.9 A 3 7 9
16.83 1.43 2.11 1.21 1.44 1.39 1.15
1.90 1.51 2.42 4.64 1.46 1.57 2.92



I. Introduction

«e+, the only relevant test of the validity of a hypothesis

is comparison of its predictions with experience. The
hypothesis is rejected if its predictions are contradictéd ..."
Milton Friedman (1953, pp. 8-9)

"In the social sciences there is no such thing as refutation;

only embarrassment." .
Converse's Law as formulated by John Converse in 1977
and quoted in Charles Dyke (1981, p. 71).

In their 1982 book Monetary Trends in the United States and the United

Kingdom: Their Relation to Income, Prices, and Interest Rates, 1867-1975,

Milton Friedman and Anna J. Schwartz present a wealth of empirical
statistical material claimed to substantiate a range of economic hypotheses.
Their approach involves transforming annual observations to averages over
phases of "NBER reference cycles", followed by detailed graphical and
regression studies of the resulting numbers. For excellent reviews and
useful summaries, see inter alia Charles Goodhart (1982), Thomas Mayer
(1982), Tim Congdon (1983), Michael J. Artis (1984) and Basil Moore (1983).
Their chapter summarising their "Principal Empirical Findings" is ten
pages long and contains many claims based on inferences from their
statistical analyses. There are too many claims to investigate in a single
Paper, but we show below that, despite their painstaking analyses, Friedman
and Schwartz (1982) (abbreviated to FS henceforth) have not credibly
established a significant number of their main empirical claims about
monetary behaviour in the United Kingdom. The claims in question concern
the "exogeneity" of money; the constancy over time of the parameters in
their money-demand model and the adequacy of its specification; the
validity of ignoring the mutual interdependence of money, income, prices,
and interest rates when using regression techniques; and the success of

their phase-averaging procedure in isolating long-run behaviour. Most of


34a

Table VI. Sub-sample estimates of (22): A, (arp) 4

period 1878-1970 1878-1913 1922-1938 1950-1970 1950-1982
regressorm (0) (i) (ii) (iii) (iv)
Aor), 3T C04] OT Cor] | 3 Cor) | 39 C06) | to C14
rp)» -.06 [ .07] “41 C12) | -.29 [ .20] w2 Cn) | 22 0.15)
Carpi), y ~.20 [ .022] ~17 C04] 04 [ .07] -.13 [ 03] | -.12 C .03]
yi -66 [ .10] -78 [ .15] ~.05 C .34] _ 03 £ .35] +2 [ 245]
Wo, -.uT £04] -.68 [ 06] | -.05 C17] | -.60 £08] | -.85 C .14]
re» 14 [ .06] 29 Cty | 4 C12) 01 0.15) | -.10 C .29)
RS, ~.78 [ .20) -.6 C 34] +11 .18] ~.69 [ .36] -.21 [ .19]
RU /2 3.3) 11.1] “4.4 [3.0] 7.4 [2.0] -.45 [ .83] 15 [ .51]
D 19° [1.1] - - - -

Dp 3.6 C .60] - - 2 0.68) | 1.7) 01.5)
D3 64 C .80] - - - -
Constant -.086 [ .010] -.075 C .027] -033 [ .032] ~.043 [ .013]} -.070 C .021]
T 93 36 17 a 3

R? 82 .19 89 93 62
1008 1.71 1.143 1.824 1.015 2.939
m.(a,T-k-Q) 44 1.4, 10 12,5 19, 2 8,5 15.1, 12
ne(q,T-K-q).q 3.5, 6 9, 6 (dw=2.7) (dw=2.9) 1.2, 6
E,4(1) 12 105 3 07 1.9

SK, EK, &5(2) | -.05, -.005, .03] -.3, -.4, .6 ~.5, 6, 4 5, ~H, 5 1,1, 2.8, 12.1



2
1 Arp),

33a

2
2. 4, rp) 9 -.04 {-.09}  -.13

3. (m-p-i)
4, Oi

5. APE

6. Php

7. RS,
8. ARR, /2

10. D,

1. =D,

Table V. Data correlation matrix for the variables in (22)
arp), 1 2 3 4 5 6 7 8 9
40 { .63}

~.15 {-.68} -.05 -.11

~.11 {55} -.12 -.06 .2

-.45 {-.76} 16 20 04.26

-.02 {-.21} .06 -.80 .08 .22 -.21

~.28 {-.43} -.07 04 -.62 -.06 -.02 .03

~.53 {-.33} -.32 -.04 -.21 .08 .01 13.62
-.06 { 24} -.07 -.05 -.16 04 16 12 13.16
18-{ 54} 01 00) .71 -.07 -.10 -.08 -.47 -.28 -.19
+25 { 09} .09 .01 .23) 10 -.02 04 -.33 -.12 -.07

(Figures in curly brackets are partial correlations.)

10


Pt-2
Pt-3
Pt-4
PS

Constant

R?
1006

7, (10,32)

(7)

&y(1)

ng (5,42)

n3 (30,42)

5T

Table G.IV. Estimates of equation (11) with p*: observations for WWI and WWII excluded
Cive., 1914-1919, 1939-1946)
me, pr* it Pt RS, Rey et
1.57 ( .17) 47 ( .38)] .26 ( 25) 48 ( .26)] .00 (.08)] .005 (.022)} .95 ( .46)
-.64 ( .29) -.18 ( .66)} -.54 ( .44) ~.13 ( .46)] -.11 (.13)] -.049 (.038)] -.83 ( .80)
+30 ( .30) +19 ( .69) +79 ( 46) -.11 (£49) +20 (14) +089 (.0H0)} -.12 ( .84)
-03 ( .27) -.17 ( .62) | -1.03 ( 241) 26 ( £43) | -.13 (.13)} -.065 (.036)} -.11 ( .75)
-.03 ( .13) +23 ( .30)} 47 ( .20)}  -.03 ( .21) | 04 (.06)] .013 (.017)] 20 ( .37)
~.07 ( .10) 1.13 ( .22) +28 ( 14) +02 ( .15) 02 (.04) +019 (.012) +72 ( .26)
+20 ( .12) +26 ( .27)| -.34 ( 18) 24 (419) 04 (.05) | -.001 (.015)} -.47 ( .33)
~.05 ( .13) | -.22 ( .31) | -.27 ( .20)]  -.05 ( .21)| 03 (.06)} -.002 (.018) | -.60 ( .37)
~03 (11) ]  -.35 ( .26)] 622 (217) | -.24 (£18) } -.08 (.05)] -.010 (.015)]  .25 (31)
7.10 ( .09)]}  -.03 ( .21)] 016 (14) ]  -.02 ( 615) | -.01 (.04)] -.004 (.012)} 210 ( .26)
~.02 (10) }  -.18 ( .23)} 666 (15) | -.08 ( .16)} 08 (.05) | 018 (.013) } -.25 ( .27)
-.11 (213) -.35 ( .30) -O4 ( .20) ~.06 ( .21) | -.05 (.06)] -.003 (.017) | -.39 ( .36)
~.03 ( .12)}  -.08 ( .27)} ~-.04 ( .18)} -.09 ( .19) | -.12 (.06)} -.021 (.016)| .56 ( .33)
203 ( .12) +03 ( .26)} .20( .17)}  -.10 ( .18) | 14 (05) } .008 (.015)] —.30 ( .32)
~.05 ( .10) 12 (23) +19 ( .15) ~.06 ( .16)} -.06 (.05) +005 (.013) | -.36 ( .28)
206 ( .15)|  -.37 ( .33) | -.57 ( .22) +98 ( .23)} .06 (.07) | — .003 (.019) | -1.06 ( .40)
-.09 ( .19) ~.34 (43) «71 ( .29) ~.64 ( .30)| -.07 (.09)| -.004 (.025) M1 (£53)
~.19 ( .20) 17 ( 46) | -.27 ( .30) +18 ( .32)| -.08 (.09)] -.001 (.026)] 71 ( .56)
~.03 ( .19) 12 (42) +03 ( .28) ~.03 ( .30) +12 (.09) | -.002 (.024) | -.07 ( .51)
205 (611) ] 610 ( 625) ] 06 ( .16)}  -.05 (617) | -.05 (.05)} 011 (014) | -.22 ( .30)
-.22 ( 40) -.29 ( .91)] -.16 ( .60) ~.74 ( 64) 54 (.19) | -.057 (.052) | 1.21 (1.10)
~.26 (43) 1.21 ( .99) | -1.82 ( .65) +46 ( .69)] -.15 (.20) -066 (.057)} 2.90 (1.20)
7.06 (451) 1.53 (1.17) | -.48 (77) ] 77 (82) 241 (.24) | 002 (068) | -3.74 (1.42)
~-84 (58) | 1.45 (1.33) } -.65 ( .88)]  -.21 ( .93)] -.38 (.27)] 060 (.077)| 63 (1.61)
201 ( .50)}  -.42 (1.15) | -.09 ( .76)] -.60 ( .80)} -.12 (.23) | -.124 (.066)] 2.30 (1.39)
-.T! (1,28) +T1 (2,90) | 2.87 (1.91) } 2.30 (2.04) | 05 (.59) | 1.184 (.167) | 4.01 (3.52)
4.74 (2,14) ~.88 (4.86) | 2.49 (3.21) +29 (3.41) -08 (.99)] -.936 (.280) | -3.58 (5.90)
~.69 (2.17) | 9.58 (4.94) | 2.52 (3.26)]} 3.72 (3.47) | -.14(1.01)] 674 (.285) | 5.85 (6.00)
4.29 (2.25) | -1.44 (5.11) | -2.16 (3.37) | 2.64 (3.59) | 1.90(1.04) | -.001 (.295) | 1.64 (6.20)
1.36 (1.63) 1.98 (3.70) | -3.18 (2.44) | -1.03 (2.60) | -.48 (.76) +100 (2214) | 4.45 (4.50)
~.05 ( .06) ~.06 ( 14) | -.01 ( .09)} -.03 ( .10)] -.03 (.03)} -.013 (.008) 69 ( .17)
OF ( .O7)}  -.07 ( .17)] «12. (11) | -.04 ( .12)] 201 (.03)] 002 (2010) | -.17 ( .20)
-.10 ( .07) +06 ( .17)} 03 ( .11) +00 ( .12)} -.00 (.03)] .011 (.010)] -.00 ( .20)
~.04 ( .06) 00 ( .14)} -.13 ( £09) +01 ( .10)] .02 (.03)] 007 (.008)] .20 ( .17)
+05 ( .05) «05 ( 611) | -.13 ( .07) 04 ( .08) | -.01 (.02) | -.008 (.007) | -.21 ( .14)
6.74 (1.93) | -16.81 (4.38) +67 (2.89) | -14.37 (3.07) | -1.02 (.90) | -.087 (.253) | 4.32 (5.32)
+39 (16) | -.66 ( .36)} £18 ( .24)] 652 ( .25)} 600 (.07)] 000 (.021)] 91 (244)
93 93 93 93 93 93 93
+99989 -99TH6 99860 99923 91331 98838 98247
1.678 3.817 2.518 2.678 -7800 2201 4.632
3 AT 3b -73 AB 5.37 1.59
12.7 13.5 9.5 10.7 4,3 22.9 12.4
9 5 7.2 1.2 6 5.4 0
53.55 1.64 1.47 2.55 58 1.16 1.20
1.47 1.69 2.12 2.68 1.85 2.06 2.79



8a

Figure IV. The log of the velocity of money: annual and phase-average data.
T J T T T T T T TP trrprremrrerprrnnrrerr
Annual data (v,)
wen ennee Phase-average data (¥;) /\
4

!

POT FOTTOYENE TOTTOTETY

1880

1900 1920 1940

1960

1980

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10
0.00


yy

Table B.IV. Standard deviations from models (B16) and (B18) (p=.96)

Annual Phase-average
oy = 3.57 o(y) = 3.51
o(ayy) = 1.01 o(ayy) = 1.42
Oe = 1.00 Ge = 1.39

Those values closely match the corresponding ratios in Table B.III. (See
Table II in Section III.)

The final alternative we consider is that the series are randon
walks: certainly the graphs of several of the variables look like those of
the random-walk series reported by Holbrook Working (1934), and p = .96 is
not "far" from unity 32 However, if p = 1 in (B16), then e(y2) = toe
(for Yo=0) and the mean of the standard estimator of the variance of yt is:

(B25) E(=—
t

J
wrag

2 2 = 2
ve = OL T(T+1)/[2(T-1)] = on (y)

(e.g., see Fuller (1976, p. 367)) whereas o(A1y) = oe. 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:
(B26) 02 = o@(2n+1)/(3n) «

However, the mean of the standard estimator of the variance of Yj is:

(B27) E ) = o2d(3un-n°+3ne1) /L6n(J-1)]

1 55
J-1 2

2.-
oy) s

gard 7

assuming yo=0. Table B.V gives various standard deviations for T = 96,

J = 35 and n = T/J (=2.74).

3245 noted in the text, 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 (1983)) and does not entail accepting the null.-


50
near enough satisfied to cast doubt on the notion that three "separate"
equations are being reported.

Interestingly, it is not only easy to show that the analysis above is
an exact sampling result, but in the process the formulae for any value of
q are obtained. The main insight required is simply the identity of
inversion and regression. If y is to be regressed on X, it is well-known
that all the relevant ingredients can be obtained by inverting the
augmented second-moment matrix A = (y:X)'(y:X). Then:

. (y'gy)! =) Ty xan |
(06) a! = i 4 a a 4
OOTY OO [Eee yy) Vt XD]
where Qy = I - xcept, Thus (y'Q,y) is the residual sum of squares,
etc. Normalising on the first diagonal term yields:

1 -3"

(or) a = (yt -

-B  (T-k)Var(B)LI+H]

for k regressors, where 8 = (xx) Txty and H = xyCytay) Bt Thus,
regression is just a normalised and appropriately interpreted inversion of
the augmented moment matrix A. But A involves all of the variables, and
the different normalised regressions simply correspond to dividing by
different diagonal terms in a and interpreting the associated columns.
Consequently, (D5) holds for all T; and the relevant generalisation for any
q = k+1 follows immediately. In the alternative possibility that the
parameters of three interdependent equations linking xj, X2_ and x3¢ are
of interest, then we have a simultaneous equations model and so least~

squares estimates are inconsistent.

APPENDIX E. Comparative results on the consumption function

The evidence contrasts interestingly with that on M (see Mayer (1982,

p. 1534)); detailed documentation is provided in Hendry (1983), where


9a

Table I. Final "levels" estimates with phase-average data
for the "main" equations in FS

Source
(1.1) (m-p-n) = .16 + .88(i-n) - 11.16RN - .22G(p+i) p. 282
(.08) (18.13) (3.42) (.74)
+ 1.4 + 21.8
(2.38) (7.56)
R? = .970 6 = 5.54%
(1.2) (pti) = -1,02 + .84m + .60(i-n) + 18.80RN - .08G(p+i) —p._ 349
- .9W - 10.8
R? = .9990 6 = 3.94%
(1.3) P= -5.99 - .015t + 1,02m + .Q4RS - 1.10G(p+i) pe 420
(31.1) (7.4) 17.4) .9) (2.8)
8 = 6.0%
(1.4) i = 6.50 + .O17t - .05m + 2.34RS + 2.20G(p+i) p. 420

(34.2) (8.3) (49) (2.4) (5.7)
6 = 5.9%

Notes: 1, Notation is as in Section II, but 't'-values are _in parentheses.
RN = RS*H/M; G(+) denotes a rate of change; and W and 5 are FS's
dummies for "post-war adjustment" and "demand shift", rescaled by
1/100 (see Appendix A).

2. Below, all values of @ for our equations with logarithmic
regressands are quoted relative to the level of the regressand
in its original units (i.e., if log(¥) is the regressand,

1006 is a percentage of Y).


ASSERTION WITHOUT EMPIRICAL BASIS: AN ECONOMETRIC APPRAISAL

OF MONETARY TRENDS IN ... THE UNITED KINGDOM

BY MILTON FRIEDMAN AND ANNA J. SCHWARTZ

David F. Hendry
Nuffield College
Oxford OX1 1NF England

and

iieil R. Ericsson*
International Finance Division
Federal Reserve Board
Washington, D.C. 20551

Revised: December 1985

*This research was supported in part by UK E.S.R.C. grants HR8789 and

B 0O 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, or other members of their staffs. Helpful discussions with and
comments from Chris Allsopp, Julia Campos, Rob Engle, John Flemming, David
Germany, John Geweke, Dale Henderson, Kevin Hoover, Bonnie Loopesko, Mary
Morgan, John Muellbauer, Charles Nelson, Ken Rogoff, Ken Wallis, and Arnold
Zellner are gratefully acknowledged. We are indebted to Jeroen Kremers and
Aileen Liu for their invaluable assistance with the many calculations
reported below and to Cathy Crosby for her preparation of the figures.

This is a revised and shortened version of a paper presented to the Bank of
England's Panel of Academic Consultants (Hendry and Ericsson (1983)).


29

Appendix E briefly documents the comparable constancy of the consumption
function using a simple error-correction model over some equivalent epochs,
and shows it to be remarkably constant both before and after World War II.

As many reviewers of FS noted, it is extremely difficult to accept in
the United Kingdom's institutional structure that money is an exogenous
variable (see especially Congdon (1983) and Goodhart (1982)). Rather,
given prices, income, and interest rates, and subject to adjustment lags,
the private sector determines the volume of money outstanding (for more
extensive discussions, see R. Hawtrey (1938, ch. 1-3) and Goodhart (1984)).
While agents may control the mean (i.e., planned) value of money, money
also acts as a "temporary buffer" for shocks, a buffer which agents cannot
(or do not find optimal to) control precisely in every period (see David
Laidler (1984)). Thus, we design our model of money demand to allow for
disequilibria in agents' holdings of money relative to their ex ante plans
(rather than postulating instantaneous adjustment). Such disequilibria are
removed gradually through "error correction"; the reaction lags of money to
changes in its various determinants are allowed to differ for every
variate, and are determined from the data. From that viewpoint, velocity
is a derived variate, which seems consistent with its time-series behaviour
in periods when any of money, prices, or income change substantially.

The properties of and intuition behind error-correction models are
important for understanding our approach (see Davidson et al (1978,
pp. 679-682) and Hendry, Pagan, and Sargan (1984)). Suppose that a
non-stochastic steady-state theory suggests proportionality between two
variables Y and Z (e.g., consumption and income, money and nominal income,
or wages and prices) so that Y=KZ where K is constant for a given growth
rate of Z (and so of Y). In logs, that theory becomes y=x+z with x=ln(K).

Without a solid economic theory of the dynamic relationship between the


18
claim that phase-averaging removes the serial correlation in the data
arising from the business cycle (p. 78) were true, then this would sustain
valid :nference without needing to condition on Xj-1 in P*(+).15 Certainly
they proceed as if conditioning is unnecessary by freely interpreting
coefficients divided by standard errors according to (possibly non-central)
't'-distributions in models which are not conditional on the past. It is
very regrettable that FS present no evidence in their book to support that
practice. Evidence in Section III demonstrated substantial residual serial
correlation in some of their models. Huge biases in estimated standard
errors can result from untreated residual autocorrelation even when all
regressors are strictly exogenous (nb. dw in (2) and (4) above). Those
biases are generally downwards for positive autocorrelation. Moreover,
biases in parameter estimates occur from serial correlation interacting
with regressors which are not all Strictly exogenous (see below).

Several assumptions additional to those implicit in (10) must be made
to allow an empirically testable model to be formulated. It must be
assumed that X¢-1 can be adequately approximated by tt (where & is the
fixed longest lag considered) without invalidating the innovation
properties of the fet}. Since we have assumed only conditional normality
for xz, we also need to assume that kt is well approximated by a linear

function of ae Thus, we postulate:

we

(11) xe = YL maxe-a + et et ~ IN(Q,Z), t#1,...,T.
ist

Many of the assumptions entail restrictions on the observables and hence

15We interpret phase-averaging as FS's attempt to eliminate X,-1 from
F(x¢[Xt-13 At)e, However, the conditions under which Xj-1 does

not appear in F°(;|Xj-13 45) and the parameters of interest can be obtained
from {a} are quite restrictive. Thus we will incorporate dynamics into
our model directly rather than rely on those conditions. Further, the data
reject a critical one of those conditions, the strict exogeneity of m, p,

and i (see Hendry and Ericsson (1983, Table C.I) and Appendix G below).


ABSTRACT

This paper critically re-evaluates some of the fundamental empirical
claims about monetary behaviour in the United Kingdom made by Milton

Friedman and Anna J. Schwartz in their 1982 book Monetary Trends in the

United States and the United Kingdom. We focus on six aspects of their

analysis: the exogeneity of money; their claims of the constancy and
correct specification of their money-demand equation; their interpretation
of a dummy variable in that equation as capturing a "shift in liquidity
preference" for 1921-55; their treatment of the interdependence of money,
income, prices, and interest rates; and their use of phase-average data.
They fail to support many of their empirical assertions with valid
econometric evidence: in particular, they leave untested many conditions
necessary to sustain their inferences. However, those conditions either
are in part directly testable from their data or have testable
implications: we test many of those hypotheses and reject virtually all of
them. We reject basic claims made for their empirical model of money
demand, e.g., those of parameter constancy, price homogeneity, and
normality of the disturbances. En route, we show that their model of
velocity as a constant performs poorly relative to the "will-o'-the-wisp"
model of velocity as a random walk. As constructive evidence against their
models, we develop a money-demand model superior to either model of
velocity, and which has an unexplained residual variance less than one
tenth that of their money-demand equation. This paper, however, is not an
"anti-monetarist" critique; rather, it is a pro-econometrics tract which
highlights the practical dangers of seeking to analyse complex stochastic

processes while eschewing modern econometric methods.


3T

The failure by FS to present statistical evidence pertinent to their
main assertions leaves these devoid of credibility. In fact, many of FS's
inferences from equations based on the phase-average data are invalidated
by residual autocorrelation. Moreover, their "final" money-demand equation
is not constant (contrary to their claim); and, on testing obvious
assumptions (such as price homogeneity and the absence of trends) implicit
in their model, rejection results. Such negative findings are consistent
with those reported by Meghnad Desai (1981, especially ch. 4).

Simply "corroborating" a subset of the implications of a theory is not
an adequate justification for deeming it useful (see inter alia Friedman
(1953), Karl Popper (1959, Section 82) and Lawrence Boland (1982, ch. 1)).
That is well illustrated by the contrast between FS's claims to have
empirically "corroborated" various aspects of their theories and our
evidence that those assertions are actually refutable from the same data.
Only well-tested theories which have successfully weathered tests outside
the control of their proponents and encompass the gestalt of existing
empirical evidence seem likely to provide a useful basis for applied
economic analysis and/or policy. Our econometric approach was briefly
exposited and illustrated above and it emphasised the crucial role of
testing and evaluating models with (at least) the six criteria outlined.
The failure to emphasise evaluation and testing is a major lacuna in
Leamer's (1983) analysis, and we would view this paper in part as a
counter-example to the view that data evidence does not. discriminate
between alternative hypotheses (cf. McAleer, Pagan, and Volker (1985)).
Despite the manifest insufficiency of even "sophisticated falsificationism"
as a methodology (e.g., see Alan Chalmers (1976)), rigorous evaluation of
empirical claims seems a necessary first step towards taking the con out of

econom:.cs.


25

VI. Econometric modelling of "money demand" using the annual data

As a positive critique to complement the analysis of Section III, we
model the demand for money in the UK using FS's annual data. Before doing
so, we first develop simple time-series models of the main variables,
namely, money, prices, and income: such models establish a useful baseline
against which to evaluate money-demand equations. In modelling the demand
for money itself, we start with a general autoregressive-distributed lag
representation of money conditional on incomes, prices, and interest rates,
and evaluate a simplification thereon in light of the model design criteria
described in Section V (cf. Hendry and Mizon (1978) and Michael McAleer,
Adrian Pagan, and Paul Volker (1985) on modelling from general to simple,
and Edward Leamer (1978) for an insightful analysis of specification
searches). We use series for 1878-1970 (T = 93) since the data from 1971
to 1982 appear to have a different stochastic structure (e.g., see Table VI
and equation (24) below).22

Starting from a fifth-order autoregression, 23 we obtained the

following simplified description of the time-series behaviour of Vet

(12) Oyve = HOO vE-4 + .54D) + .14D2 ~ 2.8D3 + .0037
[.09] (2.11) [1.16] £1.97" [.0044)]

T= 93 R? = 619° 6 = 4.34% n, (10,78) O07 €2(11) = 17.5

€4(1) = «2 &s(2) = 32.3. SK = -1.1 EK = 2.1 n¢(5,83) = 1.6.

(See Table III above for definitions of the statistics.) In (12), Dy, Do,

and D3 are zero-one dummies being unity for 1914-18, 1921-55, and 1939-45

22Most estimates have been computed with and without data for war years:
where interesting differences emerged, they will be noted and a format like
that in footnote 13 used (even though we removed the WDj by “prior
regression" which entailed setting the data for war years to zero after all
relevant lags, transformations, etc. were completed).

23wWe chose a fifth-order autoregression because %£=5 in the unrestricted
modeis estimated in Section IV.


32b

Equation (22): actual and fitted values for 4,(m-p),.

Figure V.

Ay (mp).

0.10

06

02

10


21

The main criteria which we have in mind relate to goodness-of-fit,
absence of residual autocorrelation and heteroscedasticity, valid
exogeneity, predictive ability, parameter constancy, the statistical and
economic interpretation of estimated coefficients, and the validity of a
priori restrictions. Rather than discuss each of those issues separately
and in an ad hoc manner, we propose the following taxonomy in which design
criteria are related to particular types of information available to the
modeller. Such information may be partitioned into four primary groups:

(A) sample data believed relevant by the given investigators (i.e., xb),

(B) theory information (n),

(C) the supposed structure of the measurement system (ML), and

(D) sample data not in x but believed relevant by other

investigators (i).

Different investigators may use different subsets of Ty as well as
different data. In (A)-(D), all the information sets are dated since, a
priori, the content of “ and om need not remain unaltered over time (and
generally changes as knowledge accrues).

Within (A) (and equivalently, for (D)), a further threefold
partition of xh into (xy Xpe 1) is useful as that corresponds to the
division of the data into the (relative) past, present, and future, denoted
(A,), (Az), and (A,). That the past is immutable and the future is
uncertain is among the basic tenets of economics, so it is unsurprising
that procedures for model evaluation should focus separately on those
various subsets. The evaluation issues which arise under (A) include:

(A,) goodness-of-fit and residual variance, residual serial
independence and homoscedasticity;

(A,) the legitimacy of conditioning on contemporaneous variables; and

(A;) parameter constancy and predictive performance.


7
simple, data process. That analysis implies properties for the
phase-average and annual data closely in line with those observed; and, it
Supports the notion that a substantial loss of information has indeed

resulted from phase-averaging.

III, Estimates and tests using the phase-average data

In this section, we address two fundamental claims of FS: the
constancy and the correct specification of their money-demand equation. En
route to testing those propositions, we find that their model of velocity
performs worse than the "will-o'-the-wisp" model of velocity as a »andom
walk. For their money-demand equation, the data reject their claims and
assumptions of parameter constancy, price homogeneity, absence of <rends,
and normality of the disturbances. Further, we find that their models of
nominal income and prices appear to add no new inferences since they are
approximate re-normalisations of their money-demand equation. Finally, we
show that phase-averaging does not achieve its principal claimed benefits.
The approach taken leads naturally into Sections IV and V on modelling and
testing.

The first empirical issue which we address is the assertion by FS that

++. a numerically constant velocity does not deserve the sneering

condescension that has become the conventional stance of economists.

It is an impressive first approximation that by almost any measure

accounts for a good deal more than half of the phase-to-phase

movements in money or income. Almost certainly, measurement errors
aside, it accounts for a far larger part of such movements than the
other extreme hypothesis —— that velocity is a will-o'-the-wisp

reflecting independent changes in money and income. (p. 215)

We begin by testing that proposition.

Consider the data on the so-called velocity of circulation of the

money stock:

GQ) ¥y = = GB-T; Js teed


52

APPENDIX F. Corrections and extensions to Friedman and Schwartz's data.

Table F.I. Data series from Table 5.10 for phase numbers 16 through 22, corrected

Table 5.10 Rates of Change Computed from Triplets of Reference Phase
Averages (Annual Percentage): United Kingdom, 1870-1975

Central Phase Per Capita Short-Term Long-Term High- Money Stock
of Triplet Real Money Interest Interest Powered per Unit of
Balances Rate Rate Money Output Weight
Number Midpoint gm DRg DR, 8H Guy Sw

16 1923.0 3.757 -0.296 0.162 2.925 -5.462 23.8
17 1925.5 1.028 0.296 0.003 -1.617 4.702 22.6
18 1927.0 2.255 0.080 0.031 -1.372 2.155 TH
19 1928.0 2.477 0.170 -0.010 -1.294 -2.179 4.0
20 1929.0 1.614 -0.407 0.050 -1.134 0.839 12.8
21 1931.0 3.146 -0.641 -0.256 1.817 0.494 67.1
22 1935.0 2.880 0.434 -0.189 3.293 0.155 69.3

Table F.II. Data for 1975-198233

Year Money Nominal Implicit Population Short-Term Long-Term Exchange
Stock Income Price Interest Interest Rate
Deflator Rate Rate
(£m) (£m) (1929=100.) (millions) (4% p.a.) (% p.a.) ($/£)
M I-P P N RS RQ E
1975 36,480. 83,573. 163.7 55.900 10.62 14.66 2.2200
1976 42,572. 97,447. 870.6 55.886 11.19 14.25 1.8049
1977 46,731. 111,915. 975.2 55.852 7.90 12.31 1.7455
1978 52,624. 129,106. 1085.9 55.836 8.97 11.92 1.9197
1979 59,861. 148,566. 1228.0 55.881 13.43 11.38 2.1223
1980 69,329. 169,352. 1454.8 55.945 15.87 11,86 2.3281
1981 80,306. 182,959. 1612.9 56.252 13.16 13.00 2.0254
1982 89,256. 197,003. 1746.3 n/a 11.57 11.91 1.7489

33We are indebted to John Trundle of the Bank of England for providing us
with these data.


38

APPENDIX A. Legend

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

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

D3 ~ A dummy variable for World War II (= 1 for 1939-45 inclusive,
zero elsewhere)

E ~ Exchange rate ($/£)

H ~- High-powered money (million £)

I ~ Real net national product (million 1929 £)

M - Money stock (million £)

N ~ Population (millions)

P ~ Deflator of I (1929 = 1.00)

p® - Deflator of net national product in the USA (1929 = 1.00)

RQ ~- Long-term interest rate (fraction)

RS - Short-term interest rate (fraction)

3 ~ 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 Dy}, Do, D3, 8,

and W are reported times 100 for readability.

These data are as in FS (Tables 4.8 and 4.9), but are rescaled using
the figures for 1920 to exclude a proportional effect of Southern Ireland,
and have been kindly extended through 1982 by John Trundle at the Bank of
England (see Appendix F). Also, P, P*¥, RS, and R& have been divided by 100
so that the values of P and P* in 1929 equal 1.00 (rather than 100) and RS

and R& are expressed as fractions (rather than as percentages).


40
As a baseline, we note that the va

(B1) are:
2 2
ol ep)

(BT) ty E(ugut-i)

i 2

proe/(1~p°)
€

Hence the autocorrelation function for

for uj in (B5), the levels equation wit

2
5 it
(B8) —s
n(1-p)*
tT = E(ujujy-4) = 4
vw n(i-1) ae
(B9) p °
n2(1-p")(1

The error variance of (B5) is less than that of (B1) for

both equations manifest residual autocorrelation unless p = 0.

riance and autocovariances of uy in

i= 1,2,...

ut is pt (i>0). Equivalent formulae
hh phase-average data, are:

2p(1-p")
: n(1-p")
0)*0e
-p)*
all n> 1, but

Some

representative values of the variance of uy and its first-order

autocorrelation r, (= B( G55 )/E(0;7) )

Table B.I. Variance and first-

are shown in Table B.I.

order autocorrelation of uy

Var(a3)/0® (from (B8))

x 1 2 3 5

0 1. 5 +33 22
5 1.33 1.0 81 +59
8 2.8 2.5 2.3 2.0
+95 10.3 10.0 9.8 9.5

The variance falls as n increases, but
effect as p increases. As an example,

(B8) yields 2.302 whereas oe from (B7)
u

a small decrease in variance. Moreover

ry (from (B8) and (B9))
1 2 3 5
+0 +0 +0 +0
5 +38 +28 17
8 +72 +64 +50
+95 +93 +90 84
with an ever smaller proportional

consider that for p = .8 and n = 3,

is 2.80", so averaging produces only
€

» for large p, there is only a small

reduction in first-order error autocorrelation, whereas autocorrelation at


(3) Oyvj = .023 - .028 vj-4
(.042) (.077)

J = 35 6 = 7.22% dw = 1.70 .
The t-ratio on Vj-1 (denoted 1) is only -.37, so again there is no
significant evidence against the random walk hypothesis.9

We note that John Gould and Charles Nelson (1974) sought to test
whether velocity was a random walk, following Friedman and Schwartz's
(1963) results for the USA. As Bhargava (1983) shows, the claim that
velocity is a constant can be tested as an assertion of existence through
testing whether the deviations from that constant are a random walk; that
is one interpretation of Gould and Nelson's approach to testing. !0 (For
an exposition, see Appendix C.) On that basis, the results in (2) and (3)
“do not allow us to reject the non-existence of the model that velocity is a
constant. Further, since the standard deviation of 4,¥j is under half
that of ¥j, the will-o'-the-wisp model appears to account for a far
larger percentage of the observed variability in velocity than the model
that velocity is a constant.

Turning to regression results, Table I reports the "main" estimates

gleaned from FS which they record as establishing evidence concerning the

9With the lag operator L defined as Lxy=xt-1, we let the difference
operator Ay be (1-L); hence A;x_=x_~-xt-1- More generally, Apxpe(1-LP)Ixz.
If q (or r) is undefined, it is taken to be unity. For phase-average data,
differencing is defined in terms of phases rather than years, so that
(e.g.) O4Xj=%j-Xj-1-

If ¥; has a unit root, then the coefficient of Vj-1 in (3) is zero.
Dickey and Fuller (1979; 1981, pp. 1065-1066) derive the distribution of
the t-ratio on ¥j-1 if that is so: Dickey in Fuller (1976, p. 373) gives
points on that distribution. Also, note that, while vj = Vj-1 is
equivalent to ¥; = c (a constant) in a deterministic world, there is a
massive difference in implications when random errors are added to the two
possidilities.

10Bhargava also demonstrates the invariance of the Durbin-Watson statistic to
the value of the constant and reports significance levels for the null
hypothesis that the errors are a random walk.