Seasonal Cointegration Analysis
of German Money Demand
using Simple–Sum and Divisia
Monetary Aggregates
Klaus Eberl
Diskussionsbeiträge der Katholischen Universität Eichstätt
Wirtschaftswissenschaftliche Fakultät Ingolstadt
Nr. 107
(ISSN 0938–2712)
September 1998
Klaus Eberl
Katholische Universität Eichstätt
Wirtschaftswissenschaftliche Fakultät Ingolstadt
Auf der Schanz 49
85049 Ingolstadt
Germany
Tel: ++49–841–937–1857
Fax: ++49–841–931–950
E–mail: [email protected]
1
Abstract
In a multivariate seasonal cointegration framework four different money demand systems for Germany are investigated, using simple–sum and Divisia M3 as
alternative measures of money. For each monetary aggregate both a nominal and
a real money demand system is examined and from each system a real money demand relation is derived. In order to exclude diverging seasonal trends the Kunst
and Franses (1998) approach is adopted. System stability is investigated by use of
the Hansen and Johansen (1996) recursive estimation analysis. The Divisia aggregate systems yield consistent and stable real money demand relations, whereas the
simple–sum M3 specifications show inconsistencies.
Acknowledgements
I gratefully acknowledge helpful comments and support from Josef Falkinger,
Ulrich Küsters, Robert Kunst, Johannes Schneider, Franz Seitz, Winfried Vogt,
and seminar participants in Ingolstadt, Regensburg, and Linz.
2
1 Introduction
Since the introduction of Divisia monetary index numbers by Barnett (1980) there
has been a debate if these indices could be serious alternatives to simple–sum monetary aggregates as measures for money in an economy.1 Some central banks, such
as the German Bundesbank, still use a simple–sum monetary aggregate, namely
M3, as an intermediate target in their conduct of monetary policy. Others, like the
Federal Reserve or the Bank of England, have abandoned following a targeting
strategy.
The question whether a monetary aggregate is a useful target or not is clearly
an object of empirical investigation. One major prerequisite is the existence and
stability of a money demand relation
where , , and denote the natural logarithm of a monetary aggregate, a transactions variable, and the price level, respectively, and denotes an opportunity cost
measure. A sufficient condition for the existence of a money demand relation that
has this special log–linear form is given if a quantity equation of money
holds, where
denotes the aggregate’s velocity of circulation which is assumed
to be a stable nondecreasing function of opportunity costs only. Furthermore, the
quantity equation concept imposes a nonpositive opportunity cost semi–elasticity
, price homogeneity
, and income homogeneity
.
In this paper we consider money demand relations using simple–sum and Divisia M3 as alternative monetary aggregates . We investigate four different
money demand systems in a multivariate seasonal cointegration framework: for
each aggregate one system in nominal terms with unrestricted, and one in real
terms where price homogeneity is imposed. Most of the empirical literature on
money demand considers real money, so the aim of this paper is to check if nominal vs. real system specifications give consistent results and can therefore be interpreted as evidence for the existence of a real money demand relation.
The observation sample contains quarterly data from 1975 to 1997. Because
of the evident seasonality in the data involved,2 all systems are investigated using seasonal cointegration analysis which was first introduced by Hylleberg et al.
(1990) and was extended to a full information maximum likelihood method by Lee
(1993). Here, the Kunst and Franses (1998) approach to seasonal cointegration is
adopted which takes account of the fact that the data show non–diverging seasonal trends. Money demand relations are extracted from the non–seasonal long
!"
!$#
%&#
1
2
For a more recent survey of the discussion, see Barnett et al. (1992).
Graphical illustrations of the data are given in the appendix.
3
run part of each system, i.e. they are looked for in the cointegration space at the
non–seasonal frequency.
Furthermore, the observation period covers German reunification in 1990. Besides the fact that from July 1990 on the data concern the reunified Germany which
manifests itself in an obvious jump in transactions and monetary data,3 this event
could possibly have led to a structural break in the data generation process and,
therefore, in the cointegrating relations. This issue is, for example, discussed in
Hansen and Kim (1995), or in Wolters and Lütkepohl (1996). In both articles the
authors find a stable real M3 demand relation in a single–equation non–seasonal
cointegration framework. Here, a multivariate approach is chosen which gives a
completer picture of the dynamics involved.
In order to investigate system stability the recursive estimation procedure proposed by Hansen and Johansen (1996) is applied. This approach focuses on the
evolution of long run relations while short run dynamics are held fixed. Here,
we consider constancy of cointegration ranks and cointegration spaces at the non–
seasonal frequency, and examine how estimated transactions and opportunity cost
(semi–)elasticities evolve over time.
The paper is organized as follows: In section 2 the monetary aggregate M3, the
monetary index number Divisia M3 and two corresponding opportunity cost measures are introduced, section 3 reviews multivariate seasonal cointegration analysis, section 4 contains the empirical results, and section 5 concludes.
2 Monetary Aggregates and Opportunity
Cost Measures
The German Bundesbank uses the monetary aggregate M3 to monitor the demand
for liquidity and transactions services within the German economy. M3 is defined
as
')(
*,+.0/ 13- 254 0 6 *
26
4 6 ** == currency,
deposits,
487 6 * = demand
deposits with up to 4 years time to maturity,
489 6 * = time
savings deposits at three months notice,
4 held by domestic
non-banks during period : .
where
The simple–sum aggregation procedure implies perfect substitutability between
the components of M3, which is implausible since some of its components –
namely time and savings deposits – bear interest and therefore are suited to con3
Again, see the appendix.
4
tribute to the accumulation of wealth. The motivation to hold these assets goes
beyond the pure transactions purpose.
In order to “concentrate out“ the liquidity–relevant part of money components,
Barnett (1980) proposes the use of Divisia monetary index numbers4 instead. Divisia M3 is defined as
Q
Q
Q
Q
; <=5> ?A@ B5C)D E FG; <H=
> ?I@ B
CJD E K
L FMOPQ N L%TS U V W E K
LX V W E Y > ; <=Z W E G; <H=%Z W E K5L F
R
with the liquidity cost shares
Q Q
Q
E M\[ L W E Z E W E E
V W ] ^N R [ ^ W Z ^ W
Q
E
where [ W equals the user cost of money component _ .
In an intertemporal representative consumerQ model with perfect foresight, this
Q
user cost is5
E Ma` E X Gb E W E
[ W
Q
`
S
b
where is the own rate of interest on component _ and ` is the interest rate on
a default–free reference asset which yields no transactions services and therefore
reflects the opportunity costs of holding money.
In a nutshell, Barnett’s argument is as follows: In an intertemporal model where
the liquidity services of money components contribute to the utility of the representative agent in terms of a linear homogeneous subutility function, the (continous)
Divisia index tracks this subutility exactly. This implies that the Divisia index mirrors liquidity preferences for a wider range of models than the very special and
unlikely case of perfect substitutes.
Of course, the formation of Divisia M3 as a measure of transactions utility
presupposes that the components of M3 can indeed be aggregated in a subutility
function and are therefore weakly separable from other monetary components or
from consumption goods. This assumption does not seem to cause any problems,
since nonparametric tests for the validity of GARP do not reject weak separability.6
However, even if one does not ”believe” in the microfoundation framework
the Divisia index nevertheless seems to be a useful measure for money, simply
because of the way it is constructed. The growth rate of Divisia M3 is a weighted
average of the growth rates of M3 components with time varying weights that are
determined by opportunity costs. If the own rate of a component is close to the
reference rate, it is likely to be a close substitute to longer term investments and
4
The expression “Divisia index“ is used throughout this paper to refer to what more precisely should be called
the “Törnqvist–Theil discrete approximation of the continous time Divisia index“.
5
See Barnett (1978).
6
The nonparametric testing strategy is developed in Varian (1982) and Varian (1983). For an application of
nonparametric tests to M3 components, see Scharnagl (1996).
5
therefore does contribute only a small amount to liquidity demand. If the own rate
is low compared to the reference rate, the component is likely to contribute a large
amount to liquidity, since it is likely to be held only contemporaneously. Simply,
one could say that Divisia M3 measures the liquidity content of M3.
For empirical considerations, I choose the following interest rates as own rates
and reference rate, respectively:
= 0%,
= 0.5%,
= rate on three months time deposits,
from 100,000 DM to 1,000,000 DM,
= rate on savings deposits at three months notice,
= yield on bonds with at least 4 years time to maturity
(Umlaufsrendite),
during period .7
In times of an inverted term structure, which for instance was the case in some
parts of 1992 and 1993, the calculation of Divisia user costs has to be modified in
order to yield empirically useful results. If own rate
exceeds the reference rate
, user cost
and user cost share
would become negative. In order to avoid
this undesirable feature it is assumed that in this case the money component does
not contribute to transactions services. In order to achieve this the user costs are
redefined to
if
if
c d ef
c g ef
c h ef
c i ef
j f
k
j f
m l ef
c l ef
n l ef
dx
m l e foqprs t{ ru v s w s
j fzyc l e f
j fz| c l e f
The intuition behind this modification is that a money component cannot be “less
illiquid“ than the reference asset.8 In the subsequent empirical analysis the modified costs will be used.
For money demand considerations appropriate opportunity cost measures are
needed for both monetary aggregates. As an opportunity cost measure for simple–
sum M3, the difference between reference rate and a weighted arithmetic average
of own rates
i l ef

j
c f} o f3~ J
c l ef
l €
d‚ ƒ f
is used.
7
There seems to be consensus about the choice of these interest rates for the analysis of German (Divisia)
money demand, e.g. see Wolters and Lütkepohl (1997), Gaab (1996).
8
A more consistent formulation of user costs could possibly be obtained by extending the intertemporal consumer model. On the one hand, it is not clear what has to be taken as reference rate. One could think of combining
stock indices and longer term bonds in a reference group of assets to give a better measure of opportunity costs of
holding money. On the other hand, an inverted term structure implies declining interest rates in the nearer future,
with short rates declining faster than long rates. Taking account of this inherent uncertainty could lead to a more
appropriate definition of user costs. Barnett, Liu, Xu, and Jensen (1996) derive user costs under uncertainty in a
CAPM setting, which is a first attempt of research in this direction, but it seems that a lot more has to be done.
6
From the Divisia quantity index an opportunity cost measure for Divisia M3
can be derived by forming the dual price index
„ …H†
‡ ˆI‰ Š
‹JŒ …Ž  Ž  ‘’„ …H†
‡ ˆI‰ Š
‹JŒ …Ž  Ž  “5” ‘•
˜ž
˜ž
˜ž
˜ž
•.—˜ ™ – ”%›š œ   “
”Ÿ   ‡ „ …†¡  ’„ …H†z¡  “5” ‘ ¢
˜ ž
¡
•a£ ) causes
Unfortunately, the possibility of an inverted term structure (with
consistency problems, so the following user cost aggregate will be used instead:
˜ ž
˜ž
˜ž
˜ž

’
¤
„ …†¤ ¥ ¦ •.˜— ™ – ”   „ …†¡  •.—˜ ™ – ”   „ …H†A§ Ÿ 
§
˜
ž
š
z¨ ¤ 
where the summands with §
are set to zero.9
3 Seasonal Cointegration
©
Consider a –dimensional vector of quarterly time series
sonal first differences
«
ª

with stationary sea-
 • ‡ ’¬ – ‘  •  ’  “
ª– q
ª ª ª –
š
‡ ­3‘
­J®¯ ,
that follows a VAR –process,
 Ÿ²± ”  “
” Ÿ ¢ ¢ ¢ Ÿ²±z³  “ ³%Ÿ´ 
ª •°
ª
ª
(1)
´
°
where is Gaussian white noise and contains all deterministic components including seasonal trends. Reparametrization gives the seasonal error correction
model10
«
with
 •µ° Ÿ ³¶A™ “ ” ·3” ž  “
« ”,Ÿ¶I¸ ·
¸ ž  “
”,Ÿ¶I¹ ·
¹ ž  “5¸Ÿ¶ ·
¹ ž  “
” Ÿ
Ÿ8º » –3” ¼ »  “ » Ÿ´  ½
–ª
–
ª–
ž
· ” ž  •¾‡ Ÿ ¬%‘ ‡ Ÿ ¬ ¸¸ ‘  •  Ÿ  “
” Ÿ  “¸ Ÿ  “5¹
·3¸ ž  •¾‡ š ’¬%¸‘ ‡ š Ÿ ¬ ‘ ª  • ª  ’ ª  “
”,Ÿ ª  “5¸ ’ ª  “¹
ª
·3¹  •¾‡ š ’¬ ‘ š  ª • ª  ’ ª  “5¸ ª
ª
ª
ª
š
(2)
ž
·
¹  “
” •
In order to be able to estimate the system by canonical correlation analysis
one has to impose the restriction that the so–called asynchronous cycles
Ô.
9
10
Note that
¿À3ÁÂ Ã Ä À implies ÅHÃ Ä À
ÆJÇ
and
È Ã Ä À
ÆJÇ . In this case É Ê Ë!Ì Í Î Ï ÐzÑ5È Ã Ä À É Ò Ó
ÅHÃ Ä À
ÆJÇ
See Hylleberg, Engle, Granger, Yoo (1990).
7
is taken as summand
ÛÜ ÕÖ
ÕÖ ×
ØIÙÕÖ ×5Ú
!Ý ÜÞàß
á Ø Þ
Ý!æIÞ&
ç ß
èæzÞ$é êë5ì
í Ýîæ ï
á,æ
ßð$èæzñ&ò
Ý!æ
do not have any influence on
, i.e.
.11 Under this
assumption the time series are said to be seasonally cointegrated at frequency
,
, or
, if the corresponding matrix
. Let
.
In the case of seasonal cointegration at frequency we have
, and
can be decomposed as
ß áâIÞ&ã
á Ú Þ&ãäå
ò8öè æ
óæ
ôæ
Ý æ Þó æ ô3æ õ
èæ
óæ
ôæ
with
–matrices and , both of rank .12
is called loading matrix,
matrix of cointegrating relations. Note that the main object of our interest is the
vector of moving annual averages,
, because it is the only –variable which
both has a simple economic meaning and whose dynamics is reduced to (at most)
a single unit root process.
Kunst and Franses13 have pointed out that without any restrictions on the deterministic seasonality contained in , model (1) implies diverging seasonal trends in
general. In order to avoid this unlikely feature of seasonal time series they propose
to restrict seasonal trend variables to values that cannot lead to common stochastic
trends with different drift terms. Under this restriction the seasonal error correction
model has to be written as
÷ Ø øÖ
÷
ù
Û Ü Õ Ö ÞµúAûÚ ó Ø Ú ô Øõ Ú÷ ø Ö Ø × ø Ö ×5Ø ûóÖâ í ô âõ ÷3â ø ÖÖ ×5Ø ûüýÖþ ï û
ûIó í ô× ØÜõ ÷ â Õ ûÖ ÿ × ý ûÖ ý ï
û ÛÜ û
where
ú
(3)
ýÖ þÖ Þ í ã ï ý Þ í â ï Ù
ý Ö Þ í â í ï ï and denotes remaining deterministic components.14
Efficient estimators for all parameters can be obtained from canonical correlation analysis between
and
plus relevant
,15 given
(
),
lagged fourth differences of
and all other deterministic terms. From the associated eigenvalue problems estimates of ranks and seasonal cointegration matrices
are derived by use of trace and maximum eigenvalue test statistics. In a second–
step OLS–regression estimates of ,
and are calculated.
Û ÜÕÕÖÖ
÷
æ ø Ö ×
Ø
ýÖ èæ
ú
ô5æ
11
÷ ø Ö ×5Ø Þ
ç óæ This restriction is necessary because otherwise #! $
"%
cannot be decomposed in a cointegration and a loading
matrix. For a detailed discussion of the complications in the general case
, see Johansen and Schaumburg
(1997).
12
See Hylleberg, Engle, Granger, and Yoo (1990), Lee (1992).
13
See Franses and Kunst (1996), Kunst and Franses (1998).
14
See Franses and Kunst (1996).
15
None if
,
if
, and
if
.
& #(' ) #* & #%+
) #%, - . & #%/
8
4 Empirical Results
4.1
Data and Model Specifications
The sample period considered is 1975 through 1997. The starting year 1975 is
often justified by two historical facts: the end of Bretton Woods in early 1973,
and the first announcement of a monetary target by the German Bundesbank at the
end of 1974. Both events could possibly have led to a structural break in German
money demand. For this reason most of the literature on German money demand
choose 1975 as sample start.16
M3 components and interest rates are taken from the monthly report of the German Bundesbank, which contains end–of–month series for money and monthly averages for interest rates. The Divisia M3 index and the opportunity cost measures
were constructed from these series on a monthly basis, Divisia M3 with starting
value
0
0
13254 687 95:%;3< =?>AB @ C H < 31 2JI C H <
C DEGF
in period 0, which is January 1975.17 Monetary and opportunity cost measures
were then transformed into quarterly time series by calculating arithmetic averages
over the corresponding quarter.
German GDP in prices of 1991 and the GDP–deflator, which are used as transactions variable and price index, respectively, are available on a quarterly basis and
are taken from the database of the Federal Statistical Office (Statistisches Bundesamt).
From July 1990 on the series include data concerning the reunified Germany,
which leads to a level shift in money, transactions and price variables. Levels and
first differences of the data are shown in the appendix, Figures A1 and A2.
and
are considered in absolute values, monOpportunity cost measures
etary aggregates, GDP and the deflator are considered in logarithms, with the abbreviations
,
,
, and
GDP–deflator .
Four different money demand systems are investigated in what follows:
0
K ML
3K 0 M N L
0
>
3
1
5
2
4
%
:
;
=
>
1
G
2
4
R
6
7
G
9
S
:
;
=
>
M PQO M
M T M 13254 U86RV M =
O M =
M
0
12G4
Y M >Z4 O M T M W M K ML
X Model 2 (nominal Divisia M3): Y M >\4 PQO M T M
X Model 3 (real M3): Y M >Z4 O M ] W M T M K ML = [
X Model 4 (real Divisia M3): Y M >Z4 PQO M ] W M T M
X
Model 1 (nominal M3):
16
=[
WM >
W M 3K M N L = [
K MN L = [
Scharnagl (1996) gives an overview of recent publications on German money demand.
This choice, which is nothing else but a weighted geometric average of M3 components, enables us to interpret the value of
directly as the (log of the) transactions–relevant “part“ of M3. This implies that
represents a monetary aggregate rather than merely denoting an index number.
17
^ _ `3a b c d eRf g h
^ _ `3a bc d eRf g h
9
For each system a seasonal error correction model (3) with restricted seasonal
dummies is estimated, including a constant and an impulse dummy as deterministic
terms. The impulse dummy has a value of 1 in 1990–3 and 0 otherwise in order to
account for the level shift due to German reunification.
TABLE 1
A KAIKE I NFORMATION C RITERION
i
j k l m noj
j k l m sAj
j u l m noj
j u l m poj
4
M ODEL 1
M ODEL 2
M ODEL 3
M ODEL 4
k l m poj
k l m toj
u l m sAj
u l m toj
5
k l m kAjk q m nojk q m r
k l m uAjk q m nojk q m s
u l m roju l m uoju l m q
u l m uAju l m lvju q m n
6
7
8
Table 1 gives the values of an AIC for each system.18 From these results, for
each model a VAR–degree of
was chosen.19
Model misspecification test statistics are given in Table 2. The Lagrange multiplier (LM) test for residual autocorrelation, the normality test and the vector heteroscedasticity test using squared residuals are taken from Doornik and Hendry
(1997). The test statistics were calculated using Ox 1.20a developed by Jurgen
Doornik.20
wxzy
TABLE 2
M ODEL M ISSPECIFICATION T ESTS21
M ODEL 1
autocorrelation
LM(1)
LM(4)
LM(8)
LM(1–4)
normality
heteroscedasticity
M ODEL 2
{"| l p } l n k ~5
l mt s 
0.84
2.08*
l mt s 
1.13
0.92
{J| p r3} l € q ~G
1.21
1.26
‚Gƒ | n ~
29.7**
29.9**
{"| k u q } u p p ~5
0.55
0.53
M ODEL 3
M ODEL 4
l mn t 
1.41
1.42
0.62
0.37
0.72
{J| € } l p q ~G
{"| k p } l p € ~5
1.18
‚Gƒ | p ~5
20.3**
17.8**
{J| l r r3} u r s ~G
0.90
0.82
0.74
There are only minor problems with autocorrelated or heteroscedastic residuals.
Vector normality is clearly rejected in each case, but this fact seems to be of less
importance since the full information maximum likelihood method applied here
for seasonal cointegration analysis is robust against deviations from normality.
18
See Lütkepohl (1993).
Note that a seasonal error correction model requires a VAR–degree of at least 4.
20
See Doornik (1996).
21
Throughout this paper, , *, and ** denote significance at the 10%, 5%, and 1% level, respectively.
19

10
TABLE 3
S EASONAL C OINTEGRATION T EST S TATISTICS
„J…?†
rank
M ODEL 1
M ODEL 2
M ODEL 3
M ODEL 4
4.2
‡
frequency
0
trace
60.2**
max ev
30.6*
23.6*
5.8
0.3
42.9**
Š8‹Œ
Š8‹S Ž   ‘ ’
Š8‹Ž 6.0
Š8‹“ 0.3
Š8‹Œ 70.3**
Š8‹S 27.4 Ž   ” ’
Š8‹Ž 6.0 5.7
Š8‹“ 0.3 0.3
Š8‹Œ 26.9 22.6*
Š8‹S 4.3 3.7
Š8‹Ž 0.6 0.6
Š8‹Œ 35.0* 29.1**
Š8‹S 6.0 5.8
Š8‹Ž 0.2 0.2
trace
144.6**
96.1**
48.2**
19.5**
140.9**
81.6**
39.4**
14.4**
110.9**
62.2**
23.0**
103.3**
55.5**
16.9**
max ev
48.5**
47.9**
28.6**
19.5**
59.2**
42.2**
25.0**
14.4**
48.7**
39.2**
23.0**
47.8**
38.5**
16.9**
trace
127.8**
82.8**
46.9**
21.8**
126.8**
70.7**
36.9**
14.1*
105.4**
54.8**
22.5**
96.7**
39.6**
13.7*
ˆ‰
max ev
44.9**
35.9**
25.0**
21.8**
56.1**
33.8**
22.8*
14.1*
50.6**
32.3**
22.5**
57.1**
26.0**
13.7*
Multivariate Seasonal Cointegration Analysis
Table 3 contains trace and maximum eigenvalue test statistics for each model at
each frequency. Critical values are taken from Franses and Kunst (1996) who
tabulate critical values for a sample with size
.22
For each model a maximum number of cointegrating relations at the seasonal
frequencies and
is accepted at a 5% significance level or higher. This means
that the matrices
and
have full rank and there is no stochastic seasonality
within neither system. The absence of multivariate stochastic seasonality implies
that none of the univariate time series has a seasonal unit root at
or .
For the non–seasonal frequency 0 in both nominal money demand models a
is accepted, whereas in the real money demand
cointegration rank of
models one cointegrating relation is found. Standardized cointegration vectors
are given in Table 4, with coefficients of the monetary aggregates normed to
.
Since the cointegration space is spanned by these vectors a money demand
relation has to be looked for in the set of all linear combinations of cointegrating
relations. In order to select an appropriate relation for models 1 and 2 which is
consistent with monetary theory, the real money demand relation that lies in the
respective cointegration space is depicted as money demand relation. Note that for
two cointegrating relations this real money demand relation is uniquely defined.
The (implied) real money demand relations are given in Table 4.
•z–\— ˜3˜
™
›™ š3œ
Ÿž
Ÿ ¡R—
¢R£
¤5¥–¦œ
¡R—
22
§¨‹© ©
ª«‹”
Although these values are only approximations – the observation period 1975–1 through 1997–4 covers a
sample of size
only (note that
) – it can be expected that the error is negligible because there is
only little difference between the critical values Franses and Kunst simulated for sample sizes of
and
. See Franses and Kunst (1996), Table 1.
§S‹%Ž Œ Œ
§‹¨ Œ Œ
11
TABLE 4
S TANDARDIZED C OINTEGRATING R ELATIONS
AT FREQUENCY 0
M ODEL 1
0.20 2.06
1.02 1.31
implied real money demand
1.35
1
M ODEL 2
1.61 0.51
1.25 0.91
implied real money demand
1.17
1
M ODEL 3
1.10
M ODEL 4
1.18
ݫ ¨
µ ­°¯²
´¶ ±· ¸ ³ ¹
´›µ
´¶ · ¶ º
´›µ
0.22
» ´›¬®
°
­
¼
¯
±½³
µ
0.16
´›µ
´¶ · µ ¾
´›µ
´¶ · ¿ ¿
´
¬ ´›¯À
µ ­
´¶ ±· Á ³ ¿
» ¬ ´ ´›¯À
µ ­
´¶ ± ·½ Â ³ ¾
The real money demand relation obtained from the nominal M3 model 1 shows
an income elasticity which is slightly higher than one. This finding is quite in accordance with theory since M3 holdings may partly contribute to the accumulation
of wealth. A serious problem arises from the positive semi–elasticity of the opportunity cost measure which stands in stark contrast to any theoretical conception of
a quantity equation of money.
The implied real money demand in model 2 also yields an income elasticity
exceeding unity, but to a smaller amount than the value in model 1, which confirms
that Divisia M3 expresses more closely the transactions relevant part of M3 money.
In addition, the opportunity cost semi–elasticity shows the right sign.
The real money demand obtained from model 3 is seemingly contradictory to
the findings in model 1. The parameter values fit better to economic theory than in
the nominal model. Obviously, the nominal and real M3 systems yield inconsistent
results, which shades doubt on the usefulness of M3 with regard to the formulation
of money demand relations.
Results are much more homogeneous for Divisia M3. The real money demand
relation estimated in model 4 shows only minor differences to the implied real
money demand obtained from model 2. The inconsistency concerning M3 does
not appear here.
In order to be able to interpret the estimation results as money demand relations we yet have to examine the role of the cointegration matrix within the full
system. For this issue the two–step estimation approach described in section 3 is
applied to each model. First all cointegration matrices are estimated by canonical correlation analysis. Because the objects of interest are the real money demand
relations already considered, for models 1 and 2 the first column of the cointegration matrix is replaced by the respective implied real money demand relation.
Note that this operation does not change the cointegration space. In a second step
Ã5Ä
ÃQÅ
12
TABLE 5
L OADING V ECTORS FOR R EAL M ONEY D EMAND R ELATIONS
(t–values in parentheses)
M ODEL 1
0.02154
0.01351
(
)
(2.55)
(3.21)
(
)
M ODEL 2
0.01422
0.80115
(0.46)
(
)
(
)
(3.56)
M ODEL 3
0.00391
0.08839
(1.17)
(
)
(4.05)
M ODEL 4
0.02505
0.15888
(2.67)
(
)
(2.66)
ÆJÇ É
ÆJÇ ÊË
Î Ï Ð Ï JÆ Ï Ï Ç Ñ È Ò
ÎÏ Ð Æ?Ñ Ó Ç Ô Ì Ô Í Õ
ÎÏ Ð Ï Õ
ÎÓ3Ð Ï Ò
Æ?Ç Ö È Î Ï Ð Ï Ï ÆJÒ ÇÙ É ÑAÎ Ï Ð Ï Ù ÆJÚ ÇÙ Ê×
Ú Æ?Ç Ì Ø Í
Î Ï Ð Ñ Ï
Î Ò ÐÔ Ó
Î
ÆJÇ Û È Ê3Ü Î Ï Ð Ï Ï JÆ Ô ÇÒ É Ù
Æ?Ç Ì Í
ÎÑ Ð Ú Ï
Î
Æ?Ç Û Ö È Ê3Ü Î Ï Ð Ï Ú ÆJÏ ÇÝ É Þ
Æ?Ç Ì Ø Í
Î?Ú Ð Ó Ô
ßà
estimators for the loading matrices and all other parameters are obtained from
an OLS–regression.
Table 5 gives the estimate of the first column of
(in row form) for each
model. The estimated coefficients reflect the short run adjustment of each variable
to a deviation from the “long run equilibrium“ defined by the cointegrating real
money demand relation.
The model 1 adjustment coefficients hardly fit to a money demand interpretation of the corresponding cointegrating relation. If a positive (negative) monetary
shock occurs, income and prices significantly adjust downwards (upwards) and opportunity costs adjust upwards (downwards), which is in contradiction to economic
theory.23 The coefficient on the monetary aggregate is found to be insignificant.
Model 2, however, is in accordance with theoretical considerations. Both coefficients on money and income growth are insignificant. All adjustment to a positive (negative) monetary schock takes place through a rise (fall) in prices and a fall
(rise) in opportunity costs.24
In models 3 and 4 all coefficients show the expected sign. In model 3 adjustment to a deviation from long run equilibrium takes place through income and
interest rates, whereas in model 4 only the coefficients of money and opportunity
costs are significantly different from zero (at a 5% level). This means that in the
M3 model a monetary shock implies real effects, whereas Divisia money is neutral.
Furthermore, if we compare the coefficients of each M3 model with its Divisia
counterpart we find that absolute values of (significant) coefficients in the Divisia
models are about twice to five times as large. Therefore, the Divisia systems adjust
to deviations from equilibrium at a faster rate than the M3 models.
ߛá
ß
ß
23
Note that money enters the cointegration relations with negative sign.
The opportunity cost measures are constructed basically from interest rate spreads and are therefore real
variables which should not be affected by inflation.
24
13
4.3
Recursive Estimation of Money Demand Relations
One of the main prerequisites for a money demand relation is its stability which
means that in an empirical analysis the relevant parameter values should remain
constant over time. Hansen and Johansen (1996) give a guideline along which stability analysis for the long run part of a multivariate cointegrated system can be
conducted. The idea is as follows: Take a subsample as a starting point, successively extend the sample back to its full size and observe the evolution of long run
characteristics of the system – eigenvalues, statistics and cointegrating vectors. As
Hansen and Johansen suggest, this can be achieved either by successive reestimation of the full system, or by keeping the short run parameters at their full sample
values and only calculate eigenvalues and cointegrating vectors recursively.25
Since for short sample periods the loss of degrees of freedom might become a
severe problem the latter method will be applied in this study. Furthermore, the
Hansen and Johansen approach is adopted in a modified way which accounts for
seasonality. Not only short run dynamics but also long run dynamics at the seasonal frequencies are considered constant, and only the system’s long run properties at the non–seasonal frequency are investigated by recursive canonical correlations analysis.
The sample is extended from 1975–1 through 1984–4 to 1975–1 through 1997–
4 by successively adding four quarters. This choice takes account of German reunification in 1990 which could possibly have led to a structural break in the considered money demand relations. Two sets of questions will be addressed: First,
we examine if the cointegration rank is constant over time. Second, the constancy
of the cointegration space is considered. We are especially interested in how the
estimated real money demand relations evolve.
Cointegration rank constancy is examined by consideration of recursive trace
statistics.26 In Figure 1 rescaled trace statistics are plotted, where a value of zero
means significance at the 10% level.27
First, from all models a phase of system instability during the late eighties can
be detected when no cointegrating relation is empirically found, whereas evidence
towards cointegration becomes stronger in the period after 1991. Second, the nominal money demand models 1 and 2 show similar features with regard to recursive
cointegration rank diagnostics, whereas the real models are more distinctive. At
the 10% level no cointegrating relation is detected for any subsample specification
in the real M3 model, but there is strong evidence towards cointegration in the real
Divisia M3 model, especially when the subsample ends after German reunification.
Third and perhaps most importantly, we observe that trace statistics according to
25
For details, see Hansen and Johansen (1996).
Maximum eigenvalue statistics do not yield further insights and are therefore omitted.
27
Again, critical values are taken from the small sample simulations by Franses and Kunst (1996). There is a
systematic error from the fact that these critical values were simulated for a sample size of 100 observations, but
this error does not seem to be severe as was argued before.
26
14
âãŸäåçæzè
âãŸäåçæ\é
the null hypotheses
and
are higher in average in the Divisia
models than in their simple–sum counterparts. This means that the cointegration
relationship between variables is more intense in the Divisia specifications.
Figure 2 shows recursive –statistics of a likelihood ratio test for restricted
cointegration where the full sample cointegration matrix is taken as restrictions
matrix.28 This test checks whether the full sample cointegration space equals the
considered subsample cointegration space and therefore serves as an indicator for
system stability. For recursive analysis, in the subsample specifications for models
1 and 2 a constant cointegration rank of 2 and for models 3 and 4 one cointegrating
relation was assumed. The statistics were rescaled by their respective 5% critical
values, so a value less than zero means insignificance. For each model the null hypotheses of a constant cointegration space is accepted, i.e. the cointegration space
does not change significantly.
In order to investigate the stability of cointegrating relations at the parameter
level we finally consider the recursive shape of (implied) real money demand relations. Figure 3 shows income elasticities and opportunity cost semi–elasticities
obtained from the recursive estimation procedure. Apparently, the estimates derived from the Divisia M3 models are much less volatile than the simple–sum M3
estimates (except for an outlier at sample end dates 1988–4 and 1989–4 in model
2). Furthermore, nominal and real Divisia money demand systems yield approximately equal elasticities whereas the simple–sum M3 demand systems show large
discrepancies. Estimated opportunity cost semi–elasticities for M3 range from
to
. Especially, the finding of opportunity cost semi–elasticities
about
with opposite signs in models 1 and 3 for subsample ends since 1990–4 confirms
the inconsistency result we already found for the full sample. Hence, the formulation of a nominal M3 system does not seem to be an appropriate approach to
modelling money demand.
êë
ìRé
í8èQî ï
28
For a description of the likelihood ratio test in a multivariate cointegration framework, e.g. see Johansen
(1995).
15
16
1995
-1
-1
(c) M ODEL 3
-.75
-.75
1995
-.5
-.5
1990
-.25
-.25
1985
0
0
.5
.25
r<=0
r<=1
r<=2
.25
.5
(a) M ODEL 1
-1
-.75
-.75
1990
-.5
-.5
1985
-.25
-.25
-1
0
0
.5
.25
r<=0
r<=1
r<=2
r<=3
.25
.5
1985
r<=0
r<=1
r<=2
1985
r<=0
r<=1
r<=2
r<=3
F IGURE 1
R ECURSIVE T RACE S TATISTICS AT F REQUENCY 0
(rescaled by 10% significance levels)
(d) M ODEL 4
1990
(b) M ODEL 2
1990
1995
1995
17
-.8
-1
-.8
-1
(c) M ODEL 3
-.6
-.6
1995
-.4
-.4
1990
-.2
-.2
1985
0
0
(a) M ODEL 1
-.8
-.8
1995
-.6
-.6
1990
-.4
-.4
-1
-.2
-.2
-1
0
0
1985
1985
1985
= COINTEGRATION MATRIX OF
(rescaled by 5% significance levels)
(d) M ODEL 4
1990
(b) M ODEL 2
1990
FULL SAMPLE
F IGURE 2
STATISTICS OF RESTRICTED COINTEGRATION TEST
RESTRICTIONS MATRIX
R ECURSIVE
ñð
1995
1995
18
1985
1990
1995
(c) M ODEL 3
-1
1995
-1
1990
0
0
1985
1
2
1
2
r^m
1995
2
1
1990
1995
1
1985
(a) M ODEL 1
1990
1.5
y
1985
-1
0
1
1.5
2
-1
0
1
1
1
r^m
1.2
1.2
r^dm
y
r^dm
y
1985
1985
1985
1985
IN REAL MONEY DEMAND RELATIONS
1.4
y
F IGURE 3
(d) M ODEL 4
1990
1990
(b) M ODEL 2
1990
1990
INCOME AND OPPORTUNITY COST ( SEMI –) ELASTICITIES
1.4
R ECURSIVE
1995
1995
1995
1995
5 Conclusion
In a multivariate seasonal cointegration framework we investigated four different
money demand systems for Germany, covering the period from 1975 to 1997. For
both a simple–sum and a Divisia M3 aggregate a nominal and a real money demand system was specified and from each system a real money demand relation
was derived. Stability analysis was performed by recursive canonical correlation
analysis at the non–seasonal frequency.
There is a discrepancy between the nominal and real simple–sum M3 systems.
Especially the nominal M3 system seems to be inappropriate for modelling money
demand because the derived real money demand relation implies a positive opportunity cost semi–elasticity. Recursive analysis confirms this feature which is
present for varying sample sizes. Furthermore, adjustment to deviations from long
run equilibrium takes place in a way that is contradictory to theoretical considerations.
On the other side the Divisia M3 model specifications give results according to
each other. Both models yield an income elasticity of about unity and a moderate
and negative opportunity cost semi–elasticity. In these models adjustment to deviations from long run equilibrium takes place through prices and interest rates only,
as would be expected from monetary theory. Parameter values are approximately
constant over time which means that both Divisia money demand systems picture
a stable relationship.
Despite several theoretical objections against Divisia monetary indices, concerning e.g. the choice of money components or of an adequate reference rate,
empirical evidence seems to confirm that Divisia M3 is a serious alternative to the
simple–sum aggregate for modelling money demand within the German economy.
19
References
William A. Barnett (1978). The User Cost of Money. Economics Letters 1, pp.
145–149.
William A. Barnett (1980). Economic Monetary Aggregates. Journal of Econometrics 14, pp. 11–48.
William A. Barnett, Douglas Fisher, and Apostolos Serletis (1992). Consumer
Theory and the Demand for Money. Journal of Economic Literature 30,
no. 4 (December), pp. 2086–2119.
William A. Barnett, Yi Liu, Haiyang Xu, and Mark Jensen (1996). The CAPM
Risk Adjustment Needed for Exact Aggregation over Financial Assets. Working Paper, Washington State University St. Louis.
Jurgen A. Doornik (1996). Object–oriented Matrix Programming using Ox. International Thomson Business Press, London.
Jurgen A. Doornik and David F. Hendry (1997). Modelling Dynamic Systems Using PcFiml 9.0 for Windows. International Thomson Business Press, London.
Philip H. Franses and Robert M. Kunst (1996). On the Role of Seasonal Intercepts in Seasonal Cointegration. Discussion Paper TI 96–175/7, Tinbergen
Institute, Rotterdam.
Werner Gaab. On the demand for Divisia and simple–sum M3 in Germany, 1960–
93. In: Andrew Mullineux (ed.), Financial Innovation, Banking and Monetary Aggregates, pp. 160–186. Elgar Publishing, Cheltenham, 1996.
Gerd Hansen and Jeong-Ryeol Kim (1995). The Stability of German Money Demand: Tests of the Cointegration Relation. Weltwirtschaftliches Archiv 131,
pp. 286–301.
Henrik Hansen and Søren Johansen (1996). Recursive Analysis of Eigenvalues in
Cointegrated VAR–Models. Working Paper, Institute of Mathematical Statistics, University of Kopenhagen.
Helmut Herwartz and Hans-Eggert Reimers (1996). Seasonal Cointegration Analysis for German M3 Money Demand. Diskussionspapier, Sonderforschungsbereich 373, Humboldt Universität Berlin.
Sven Hylleberg, Robert F. Engle, Clive W. J. Granger, and B. S. Yoo (1990). Seasonal Integration and Cointegration. Journal of Econometrics 44, pp. 215–
238.
Ottmar Issing, Karl–Heinz Tödter, Heinz Herrmann, and Hans–Eggert Reimers
(1993). Zinsgewichtete Geldmengenaggregate und M3 – ein Vergleich.
Kredit und Kapital 26, no. 1, pp. 1–21.
Søren Johansen (1995). Likelihood–Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford.
20
Søren Johansen and Ernst Schaumburg (1997). Likelihood Analysis of Seasonal
Cointegration. Working Paper ECO No. 97/16, European University Institute.
Robert M. Kunst and Philip H. Franses (1998). The Impact of Seasonal Constants
on Forecasting Seasonally Cointegrated Time Series. Journal of Forecasting
17, pp. 109–124.
Hahn S. Lee (1992). Maximum Likelihood Inference on Cointegration and Seasonal Cointegration. Journal of Econometrics 54, pp. 1–47.
òó
Helmut Lütkepohl (1993). Introduction to Multiple Time Series Analysis. 2
edition. Springer, Berlin.
Michael Scharnagl (1996). Geldmengenaggregate unter Berücksichtigung struktureller Veränderungen an den Finanzmärkten. Diskussionspapier 2/96, Volkswirtschaftliche Forschungsgruppe der Deutschen Bundesbank, Frankfurt am
Main.
Hal R. Varian (1982). The Nonparametric Approach to Demand Analysis. Econometrica 50, pp. 945–973.
Hal R. Varian (1983). Nonparametric Tests of Consumer Behaviour. Review of
Economic Studies 50, pp. 99–110.
Jürgen Wolters and Helmut Lütkepohl (1997). Die Geldnachfrage für M3: Neue
Ergebnisse für das vereinigte Deutschland. IFO Studien 43, pp. 35–54.
21
22
1975
6.25
6.5
1975
6.75
5
5.5
6
1975
6.5
7
7.5
y
dm
m
1980
1980
1980
1985
1985
1985
1990
1990
1990
1995
1995
1995
1975
4.25
4.5
4.75
1975
1
1.5
1975
2
1
1.5
F IGURE A1
DATA , LEVELS
p
r^dm
r^m
1980
1980
1980
1985
1985
1985
1990
1990
1990
1995
1995
1995
Appendix
23
1975
0
.1
1975
0
.05
.1
1975
0
.05
.1
Dy
Ddm
Dm
1980
1980
1980
1985
1985
1985
1990
1990
1990
1995
1995
1995
1975
-.025
0
.025
1975
.05
-.2
0
.2
1975
-.2
0
.2
Dp
1980
1980
Dr^dm
1980
Dr^m
F IGURE A2
DATA , FIRST DIFFERENCES
1985
1985
1985
1990
1990
1990
1995
1995
1995