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Personal sector money demnad in the UK.

Non-parametric demand analysis is utilised to establish weakly separable subsets of monetary assets for the UK personal sector in the context of a utility function consisting of durable and non-durable consumption goods, services, leisure, and monetary assets. The admissible collections of assets are then combined using Divisia aggregation in order to produce monetary aggregates which are consistent with economic aggregation theory. We investigate the money demand properties of the aggregates and find relatively simple, stable long-run money-demand functions for the UK personal sector with well determined error correction dynamics.

1. Introduction

THE UK economy has suffered three serious recessions since the Bretton Woods monetary regime ended in the early 1970s. Few doubt that these episodes were made worse in the United Kingdom (relative to the experience in some other countries) by mis-timed and excessive monetary policy interventions on the part of the UK monetary authorities. The monetarist strategy of targeting growth rates of specific monetary aggregates was designed to avoid such damaging swings in monetary policy. The targeted monetary aggregate was supposed to be a leading indicator of impending inflation, so timely reactions were to have avoided the boom-bust scenarios which emerged in practice. Monetary targeting failed because chosen aggregates no longer appeared to be stably related to the economy and formal targeting was abandoned by a so-called monetarist government in 1986.

It is well known that the financial innovations of the 1980s introduced instability into estimated demand functions for broad money, and it was for this reason that monetary targeting was abandoned. It may well be, however, that this observed instability was attributable, at least in part, to the inadequate specification of traditional broad money demand functions. Hence, the present paper aims to improve our knowledge about the behaviour of monetary indicators. In particular, we seek to establish that there is a stable underlying personal sector demand function for the collection of assets traditionally called money. We choose to concentrate on the personal sector since previous research (Belongia and Chrystal, 1991; Drake and Chrystal, 1994) has indicated that there may be significant differences between the demand for money across sectors. The latter study, for example, indicated that the money-demand function estimated for the corporate sector differed markedly from the money-demand functions typically estimated using official monetary data which aggregates across sectors. Clearly, this type of aggregation across sectors is likely to be inappropriate if the sectors are characterised by quite distinct money-demand functions.

In addition to the potential problems associated with inappropriate aggregation across sectors, traditional money demand studies have typically utilised official monetary aggregates which incorporate simple sum aggregation across asset groupings which may or may not form legitimate aggregates in the context of weak separability. Our approach, therefore, involves applying microeconomic aggregation theory to both the choice of weakly separable asset groups and to the method of aggregation applied. We discover that two admissible Divisia aggregates have both plausible long run properties and well-specified dynamics.

In what follows, Section 2 tests for the existence of admissible collections of monetary assets by seeking weakly separable subsets in a framework which accommodates consumer choices between durable and non-durable consumer goods, services, and leisure, as well as financial assets. Section 3 discusses the construction of Divisia aggregates for the chosen groupings. Section 4 reports the empirical results testing for the existence of long-run money-demand functions and short-run adjustment dynamics. Section 5 concludes.

2. Non-parametric demand analysis

In order to construct admissible monetary aggregates, the asset components of the aggregate must be weakly separable from other goods in the utility function. A common approach in money-demand studies where separability tests are undertaken is to assume that monetary assets are weakly separable from goods, services and leisure and to proceed to conduct tests for weak separability across the set of monetary assets alone. This is the approach adopted by Belongia and Chrystal (1991) and Drake and Chrystal (1994).

This approach is questionable on theoretical grounds, however, as the maintained hypothesis of weak separability of monetary assets from other goods in the utility function requires that the marginal rate of substitution between any two monetary assets must be independent of any goods outside the monetary asset category. Clearly, if this necessary and sufficient condition is not satisfied then the non-parametric demand results may be invalid. Furthermore, if these results are used as a basis for constructing monetary aggregates, the results of money-demand analysis conducted using these aggregates may also be erroneous. It is also relevant to note that the `shopping time' model of McCallum and Goodfriend (1987) would suggest that leisure, goods, and monetary assets should all appear in the representative agent's utility function. Diewert (1974) can also be seen as an early contribution to the shopping time literature.

The subsequent analysis therefore uses non-parametric demand analysis to establish permissible monetary aggregates in the context of a utility function which contains consumption goods, leisure, and monetary assets.

2.1. Data

In order to conduct weak separability tests using non-parametric demand analysis, five general categories of goods were specified in the utility function: non-durable goods, services, durable goods, leisure, and monetary assets. For durables, non-durables, and services, aggregate data was utilised using quarterly data supplied, for the most part, by Datastream. As the main focus of this analysis is on monetary assets, however, the monetary category was disaggregated into all the individual assets held by the UK personal sector. These seven assets are notes and coins, non-interest-bearing (NIB) sight deposits, interest-bearing (IB) sight deposits, time deposits, building society deposits, National Savings investment accounts, and certificates of deposit (CDs). Due to the fact that CDs were not held by the UK personal sector until 1986 Q3, the sample period for the non-parametric demand analysis had to be restricted to the period 1986 Q3 to 1993 Q1.

With respect to consumption we make the reasonable assumption that the appropriate decision-making unit is likely to be the household rather than the individual (see Patterson, 1991). Hence, with the exception of leisure, all quantity series are expressed in real per household terms for the purpose of the non-parametric demand analysis.

In addition to quantity series, corresponding price data is also required for the non-parametric demand analysis and further detail on the construction of the appropriate price and quantity series is provided below.

2.2. Quantities

For non-durable goods and services, the appropriate quantity series are taken to be the real expenditure flows (constant prices) per quarter. For durable goods, however, the appropriate definition of quantity is not quite so straightforward. As Diewert (1974) initially demonstrated, the appropriate quantity for durable goods is the net stock and not the expenditure flow over a given period. Furthermore, the appropriate price for a durable good is its user cost and not the implicit deflator for new purchases as is the case with non-durables and services.

The net stock of durable goods is calculated using data on the real expenditures on durable goods and unpublished data obtained from the CSO on the average life length of the three main components of durable goods (10 years for cars and other vehicles, 23 years for furniture and floor coverings, and 10 years for other durable goods) with the average life-length data being used to derive appropriate depreciation rates. In order to ensure a representative net stock of durable goods at the start of the sample period (1986 Q3) the net stock computations commenced 10 years earlier in the case of cars, 23 years earlier for furniture, and 10 years earlier for other durable goods. The aggregate series for durable goods is simply the sum of these three components.

With respect to leisure, we follow the standard approach (see Patterson, 1991) in defining leisure hours per quarter as

[98 hours - (average hours worked per week during the quarter)]* 13

where 98 hours is assumed to be the maximum number of discretionary hours available per week. The series on average weekly hours worked are taken from The Employment Gazette--New Earnings Survey which is published annually each April. Linear interpolation was employed to construct a quarterly series.

Finally, with respect to the seven monetary assets, it is assumed that the services provided by the monetary assets in the utility function are proportional to the real stock of assets held. Data on the nominal stocks of each asset held by the UK personal sector were provided by the Bank of England(1) and these series were subsequently deflated by the GDP deflator in order to construct real quantities.(2)

2.3. Prices

As mentioned above, the appropriate prices for non-durable goods and services are the implicit price deflators while the appropriate price for durable goods and monetary assets is the calculated user cost or rental price. Following Diewert (1974), the user cost of durable good i is calculated as

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the spot price of good i, [R.sub.t], is the discount rate or benchmark rate of interest (see below), [[Delta].sub.i] is the computed rate of depreciation for good i and [[Pi].sub.i] is the period inflation rate for good i.

With respect to the monetary assets, the rental price or user cost of a particular financial asset is taken to be the one-period holding cost or opportunity cost of the asset in question. This is derived using the formula developed by Barnett (1978)

(2) [P.sub.it], = ([R.sub.t] - [r.sub.it])/(1 + [R.sub.t])

Where [r.sub.it], is the market yield on the ith monetary asset and [R.sub.t] is the yield available on a benchmark asset. The latter is generally taken to be the highest yield available on an asset held as a store of value rather than for liquidity purposes. Obvious candidates for the benchmark asset which have been used in previous studies are gilt yields, corporate bond yields, etc. In the UK over the sample period in question, however, it was often the case that the highest yield on offer was that on certain building society products such as notice accounts or term deposits. For this reason the maximum retail rate offered by the Halifax Building Society (the UK's largest building society) is used as the appropriate benchmark rate of interest. This is also the benchmark rate used by Patterson (1991).

The own yield on notes and coins and non-interest bearing sight deposits is taken to be zero, while the three-month CD rate and the National Savings investment account rate are used as the own yields for CDs and National Savings respectively. For personal sector holdings of building society deposits, the appropriate own yield is taken to be the building society average share rate.

Although data are available on bank seven-day deposit rates, this has long been unrepresentative of the rates available on the generality of bank deposit accounts. Hence, for the rates of return on bank IB sight deposits and bank time deposits we have used the representative interest rate series calculated internally by one of the largest UK clearing banks. These are calculated as weighted average rates of interest across all the various deposit products offered by the bank and should therefore represent a good proxy for the rates available over time to personal sector customers on these two categories of assets.

Finally, the appropriate price of leisure is taken to be the wage rate. Again, an annual series for average gross hourly earnings of full time employees (unaffected by absence and excluding overtime) is provided in the New Earnings Survey. In this case, in order to construct a quarterly series we used the movements in the average quarterly earnings index for all employees in the whole economy provided by The Employment Gazette. Although it would be desirable to use a net hourly wage rate as the appropriate price of leisure, it proved impossible to construct a consistent data series for this variable.

2.4. Methodology

Once a vector of prices and quantities is specified for all the components of the overall utility function, non-parametric demand analysis can be used to test whether the data are consistent with the existence of a stable underlying utility function and further to test for the existence of weakly separable subsets of goods. In this particular paper we are clearly interested to see whether any subset of monetary assets is weakly separable from non-durable and durable goods, services, leisure, and also other assets outside the monetary subset in question.

Although parametric techniques can also be employed to test hypotheses concerning an estimated utility function, indirect utility function, or cost function this requires the specification and estimation of a particular functional form. As Varian (1983) has pointed out, however, with this approach the hypothesis testing is always a joint hypothesis concerning both the restrictions, i.e. weak separability of preferences, and the particular choice of functional

Varian (1982) demonstrated that the consistency property requires that the data set must satisfy the generalised axiom of revealed preference (GARP) while weak separability requires that some subset of goods satisfies GARP and that the aggregate of this subset, together with the remainder of the goods, still satisfies the consistency property.(3)

2.5. Results

As was mentioned previously, the fact that personal sector holdings of CDs were zero prior to 1986 Q3 dictated that the sample period for the nonparametric demand analysis be restricted to 1986 Q3-1993 Q1, although the subsequent aggregation and cointegration analysis utilises the full sample period of 1976 Q2-1993 Q1.(4)

The results of the non-parametric demand analysis are given in Table 1, together with the number of violations in cases where GARP or weak separability is rejected. It can be seen from utility function 1 that the data set overall is consistent with the theory of utility maximisation in the sense that it exhibits no violations of GARP. Although not of central interest in this paper, it is interesting to note (in utility function 2) that consumption goods and leisure are weakly separable from monetary assets.

TABLE 1 Non-parametric tests for consumer goods, leisure and monetary assets: sample 1986 Q3-1993 Q1
Utility function                                          GARP

1-U[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]                   [check]
2-U[[V.sup.1](1, 2, 3, 4), 5, 6, 7, 8, 9, 10, 11]        [check]
3-U[1, 2, 3, 4, [V.sup.1](5, 6), 7, 8, 9, 10, 11]        [check]
4-U[1, 2, 3, 4, [V.sup.1](5, 6, 7), 8, 9, 10, 11]           x
5-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8), 9, 10, 11]        [check]
6-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9), 10, 11]        [check]
7-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9, 10), 11]        [check]
8-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9, 10, 11)]        [check]

                                                         weak
Utility function                                     separability

1-U[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]               not applicable
2-U[[V.sup.1](1, 2, 3, 4), 5, 6, 7, 8, 9, 10, 11]        [check]
3-U[1, 2, 3, 4, [V.sup.1](5, 6), 7, 8, 9, 10, 11]        [check]
4-U[1, 2, 3, 4, [V.sup.1](5, 6, 7), 8, 9, 10, 11]           x
5-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8), 9, 10, 11]        [check]
6-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9), 10, 11]           x
7-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9, 10), 11]           x
8-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9, 10, 11)]        [check]

                                                         No. of
Utility function                                       violations

1-U[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]                      0
2-U[[V.sup.1](1, 2, 3, 4), 5, 6, 7, 8, 9, 10, 11]           0
3-U[1, 2, 3, 4, [V.sup.1](5, 6), 7, 8, 9, 10, 11]           0
4-U[1, 2, 3, 4, [V.sup.1](5, 6, 7), 8, 9, 10, 11]          10
5-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8), 9, 10, 11]           0
6-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9), 10, 11]          15
7-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9, 10), 11]          15
8-U[1, 2, 3, 4, [V.sup.1](5, 6, 7, 8, 9, 10, 11)]           0


Notes:

U[ ]: utility functions

[V.sup.1](): sub-utility functions

Components of utility function: 1. Non-durable goods; 2. Services; 3. Durable goods; 4. Leisure; 5. Notes and coins; 6. NIB sight deposits; 7. IB sight deposits; 8. Time deposits; 9. Building society deposits; 10. National Savings investment accounts; 11. Certificates of deposit.

In respect of the sub-utility functions [V.sup.1](...), GARP is a necessary (but not sufficient) condition for weak separability. Hence, a tick under GARP in the table confirms that the sub-utility function passes this necessary condition. If this necessary condition fails, however, the subsequent sufficient test for weak separability cannot be undertaken (see footnote 2) and a rejection of weak separability is recorded by a cross in the appropriate column. If the necessary condition is passed, however, the sufficient condition test is undertaken to confirm or reject weak separability and is indicated by a tick or cross under weak separability.

With respect to monetary assets, however, it is clear that not all possible sub-groupings of assets are weakly separable from goods and leisure (and other assets).(5) Hence, not all possible sub-groupings of monetary assets can form potentially meaningful monetary aggregates in the sense of forming a valid weakly separable subset of the utility function. Specifically, while notes and coins and NIB sight deposits form a valid monetary grouping (utility function 3), it is not possible to add IB sight deposits (utility function 4) without also adding bank time deposits (utility function 5). Interestingly, this latter grouping corresponds broadly with M3 which was the government's favoured broad money aggregate until it was dropped in 1986. Significantly, however, the monetary grouping which corresponds to the official M4 aggregate by including building society deposits together with time deposits and IB sight deposits (utility function 6) does not form a weakly separable subset and therefore cannot be interpreted as a meaningful aggregate, at least in terms of the preferences of the UK personal sector.

It can also be seen from Table 1 that an even broader monetary aggregate including National Savings (utility function 7) is invalid in the context of the weak separability tests. Hence we are left with the result that the only meaningful aggregates relate to non-interest-bearing assets, the latter plus bank interest-bearing deposits, and finally a grouping which consists of all assets held by the personal sector, including CDs (utility function 8). This latter grouping is significant as the result contrasts with the findings of Belongia and Chrystal (1991). Using aggregate data with Varian's non-parametric technique they found clear evidence that wholesale deposits should not be aggregated with retail deposits, although it should be noted that this result was obtained in the context of aggregate (rather than personal sector) data and a specified utility function consisting only of monetary assets.

The non-parametric results in Table 1 have extremely important implications for the practical construction of monetary aggregates as the results suggest that it is not appropriate to move from narrow to broad aggregates simply by the ad hoc addition of `broader' assets, a result which echoes the findings of Drake and Chrystal (1994). More specifically, it is not appropriate to add interest-bearing sight deposits to NIB assets without also adding bank time deposits. In the context of the traditional monetary aggregates, this implies that personal sector NIB M1 and M3 are valid aggregates but personal sector M1 is not.

Similarly, moving from utility function 5, it is not possible to add what are sometimes referred to as near money assets on an individual basis. Rather all these near money assets must be included together to form a genuine broad money aggregate. The latter might be expected to be the case, a priori, in the sense that these assets are all relatively high-yielding and would be expected to exhibit relatively high cross elasticities of substitution. It follows from this, however, that it would be unwise to construct aggregates such as M4 (which includes building society deposits as the only near-money asset) as such an aggregate would be prone to destabilising substitutions between assets within and assets outside the aggregate. As Barnett (1980) has pointed out, this problem would not exist with an aggregate constructed on solid micro demand and aggregation theoretic foundations as these potentially destabilising substitutions would be internalised within the aggregate.

It is interesting to note, therefore, that with respect to UK money demand studies, much of the observed instability in estimated money demand functions has related to broad monetary aggregates. Hall et al. (1989), for example, found it was impossible to find a stable conventional money demand function for M4. In order to obtain a stable money demand function they had to resort to the incorporation of essentially ad hoc proxies for financial innovation. Based upon the non-parametric results outlined above, however, it seems a strong possibility that the observed broad money demand instability in the UK stems, at least in part, from the use of inappropriate monetary groupings and the simple sum aggregation of these assets rather than the use of weighting schemes consistent with economic aggregation theory such as Divisia aggregation. It is the purpose of the remainder of this paper, therefore, to use the non-parametric demand results to construct monetary aggregates which are consistent with respect to both demand and aggregation theory, and to establish whether it is possible to construct stable conventional money demand functions on the basis of these aggregates for the personal sector.

3. Divisia aggregation

The results of the non-parametric demand analysis indicated that only three monetary asset groupings were weakly separable from other assets, goods, services, and leisure in the utility function. Hence, only the subset of assets in utility functions 3, 5, and 8 can legitimately be combined to form meaningful monetary aggregates. In the interests of brevity, however, and given the fact that much of the observed money demand instability in the UK has centred on the broader aggregates, this paper will focus on the subset of assets in utility functions 5 and 8.

The next issue to be addressed, therefore, is how to combine these assets in the formation of monetary aggregates. Official monetary aggregates such as those produced by the Bank of England and other Central Banks are based upon simple sum aggregation. That is, the nominal quantities of each component asset are simply added together and each component is implicitly given an equal weighting in the construction of the aggregate. This approach can be criticised, however, as it implies that all of the component assets are viewed as perfect substitutes for one another by consumers/asset holders. In turn, this would imply infinite elasticities of substitution between each pair of component assets. However, empirical evidence produced, for example, by Drake (1992) for the UK, suggests that the observed elasticities of substitution between pairs of assets are actually quite low.

A more appropriate way to construct monetary aggregates, therefore, is to produce aggregates which are consistent with the underlying preferences of asset holders, i.e. to produce aggregates which are consistent with economic aggregation theory. Diewert (1976) demonstrated that index number theory can provide parameter-free approximations to economic aggregates which are consistent with the underlying preferences of consumers over the components of these aggregates. Furthermore, with respect to the choice of a particular index number, Diewert (1976) identified a class of index numbers which approximate arbitrary exact aggregator functions up to a third-order remainder term and termed these `superlative' index numbers. One such superlative index number is the Divisia index which has been widely applied to monetary aggregation, (Barnett et al., 1984; Belongia and Chrystal, 1991; Drake and Chrystal, 1994; Spencer, 1994; Drake, 1996) and which will be used to construct the two monetary aggregates utilised in the subsequent money demand analysis.(6) These Divisia aggregates, together with their component assets are detailed below.
Divisia Aggregate    Component Assets

D4                   Notes and coins, NIB sight deposits, IB sight
                     deposits, and bank time deposits

D7                   As above plus building society deposits,
                     National Savings investment accounts,
                     and CDs


In addition to the construction of the two Divisia monetary aggregate quantity indices, we compute the implicit Divisia rental price or user cost indices, P4 and P7, which are also utilised in the subsequent money demand analysis.

Although the discrepancy between the sample periods for the non-parametric demand analysis and the cointegration analysis provides an automatic caveat in respect of assessing the latter, the use (as far as is possible) of legitimate asset groupings and appropriate aggregation techniques should, at the very least, ensure that the money demand results are superior to similar studies conducted using official simple sum aggregates.

4. Money demand analysis

Having derived the two Divisia monetary aggregates (indices) together with the associated rental price indices, the next stage of the analysis is to investigate the money demand properties of these Divisia aggregates. The Johansen maximum likelihood technique (Johansen, 1988) is utilised in order to test for the presence of long-run cointegrating relationships. In addition to the Divisia monetary aggregates (in nominal terms) and the associated rental prices, the other variables included in the morley demand analysis are: Real GDP (RGDP), Real Personal Sector Disposable Income (RPSDI), Real Consumer Expenditure (REXP) (the latter two variables being alternative scale variables to Real GDP), and the GDP deflator (GDPDEF). All variables with the exception of the rental prices are expressed in log terms.

4.1. Order of integration tests

Prior to embarking upon the cointegration analysis, it is necessary to check the order of integration of the variables. In doing this we employ the Augmented Dickey Fuller (ADF) test statistic (Dickey and Fuller, 1981). This is used in preference to the more recent alternative non-parametric test proposed by Phillips and Perron (1988), as the latter has been found to have poor size properties (i.e. the tendency to over-reject the null when it is true) in Monte Carlo studies.

The results of the ADF tests of the null of a unit root on the levels of the variables are illustrated in Table 2. It is clear that for all variables the null hypothesis is accepted and hence the variables are not stationary in levels. This result is also confirmed using the appropriate Johansen test. This is a multivariate test and involves the testing of zero restrictions within the cointegrating VAR. In the case of this particular test the null is stationarity rather than a unit root and the null is clearly rejected based on the appropriate [chi square] critical value.
TABLE 2
Tests of the stationarity of the data in levels

                           ADF(1)
                     No trend   Trend    Johansen

LD4                   -0.953    -2.547    47.265
LD7                   -1.129    -2.545    52.671
P4                    -2.168    -3.324    30.417
P7                    -2.869    -2.866    39.581
LRGDP                 -0.955    -2.821    56.247
LGDPDEF               -1.638    -3.239    47.471
5% critical values    -2.930    -3.489     7.820


(1) The order of the lag is chosen to ensure that the errors are uncorrelated.

Having established that the data are not stationary in levels, the usual procedure adopted is to test for stationarity of the data in first difference terms and if this is also rejected to test for stationarity in second differences. Dickey and Pantula (1987) point out, however, that this is not an appropriate testing procedure as the alternative hypothesis is always assumed to be stationarity. This is clearly not appropriate when attempting to distinguish between I(1) and I(2) variables.

A more appropriate procedure, therefore, is to start with the maximum assumed order of integration (usually I(2)) and to test the null of two unit roots against the alternative of one unit root. In the case of a variable [Y.sub.t], for example, this test can be conducted by running the following regression

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Under the null hypothesis of two unit roots, [[Beta].sub.1] = [[Beta].sub.2] = 0 and this can be tested using an F test against the alternative of one unit root. Banerjee et al. (1993) and Dickey and Pantula (1987), however, argue that a more powerful test involves regressing [Delta][Delta][Y.sub.t] on [Delta][Y.sub.t-1] (together with the ADF correction involving [Delta][Delta][Y.sub.t-i])), computing the t ratio on [[Beta].sub.1] and comparing it to the appropriate ADF critical value. In this case, under the null of two unit roots [[Beta].sub.1] = 0 and the alternative is one unit root.

If the null hypothesis of two unit roots is rejected, Dickey and Pantula and Banerjee et al. advocate testing the null of one unit root against the alternative of stationarity. Under the null [[Beta].sub.1] < 0 and [[Beta].sub.2] = 0, while under the alternative [[Beta].sub.1] < [[Beta].sub.2] and [[Beta].sub.2] < 0. Hence, a test of the null simply involves computing the t ratio on [[Beta].sub.2] in (3) and comparing this with the appropriate ADF critical value. It is clear from Table 3 that the null of two unit roots is rejected for all variables regardless of the presence of a trend. It is also apparent that the null of one unit root is accepted for all variables indicating that they are I(1) and hence valid candidates for inclusion in the cointegration analysis. This evidence is consistent with the ADF and Johansen tests in Table 2 which accepted the null of a unit root over the alternative hypothesis of stationarity in levels.
TABLE 3
Tests of the null of two unit roots

                     No trend    Trend

LD4                    -8.256    -8.349
LD7                    -7.776    -8.045
P4                    -10.590   -10.582
P7                    -12.455   -12.369
LRGDP                  -8.678    -8.606
LGDPDEF                -5.017    -6.079
5% critical values     -2.930    -3.500

Tests of the null of one unit root

                     No trend    Trend

LD4                    -1.301    -1.451
LD7                    -1.732    -1.196
P4                     -2.154    -2.346
P7                     -2.830    -3.018
LRGDP                  -1.336    -2.022
LGDPDEF                -2.729    -2.926
5% critical values     -2.930    -3.500


4.2. Empirical results for D7

Although it is common when using the Johansen cointegration technique to adopt a VAR lag of 4 with quarterly data, recent studies have suggested that the results may be sensitive to the lag length specification. Hall (1991), for example, finds that while the resultant coefficient estimates are not very sensitive to the lag specification, the likelihood ratio tests used to determine the number of cointegrating vectors can be sensitive to the choice of VAR lag. In recognition of this potential problem, Table 4 summarises the Johansen results with alternative lag length specifications for the VAR. The maximal eigenvalue test was used to establish the number of cointegrating vectors and, in the case of more than one vector, only the vector which conforms to economic priors is reported.

TABLE 4 Summary of Johansen cointegration results for LD7 (normalised coefficients)
             No. of coint.
Lag length      vectors        P7     LRGDP   LRPSDI   LREXP

                   3         -0.246   1.085
    4              4         -0.068           2.034
                   3         -0.268                    1.198

                   3         -0.207   0.992
    5              3         -0.703           0.041
                   3         -0.114                    1.148

                   1         -0.136   1.178
    6              3         -0.447           0.432
                   4         -0.973                    0.763

Lag length    LGDPDEF               CON

              1.015(*)           -15.700
    4         0.396([dagger])    -23.600
              0.782([dagger])    -15.497

              1.010(*)           -14.690
    5         1.351([dagger])     -4.754
              0.780([dagger])    -15.138

              0.938(*)           -16.590
    6         1.212(*)            -8.806
              1.057([dagger])    -10.928


(*)([dagger]) Denotes that the [chi square] test of a unit coefficient on LGDPDEF was accepted (rejected).

It is clear from Table 4 that the best results are obtained when the log of Real GDP (LRGDP) is used as the scale variable.(7) The use of GDP rather than PSDI in the personal sector money demand function can be rationalised on the grounds that the entire national income ultimately accrues to individuals. The coefficient estimates are remarkably consistent across the varying lag specifications and accord strongly with economic priors. The coefficient estimates on P7 vary between -0.246 and -0.136, and both the income and price coefficients are consistently close to unity. With respect to the latter, the restriction tests proposed by Johansen (1988) suggest that the important theoretical price homogeneity restriction is satisfied (coefficient on LGDPDEF equal to 1) for all the VAR lag lengths specified.

In order to establish the appropriate lag length for the VAR we employed the Schwartz Bayesian Information Criteria (SBIC) which suggested that a lag length of six was the most appropriate. As can be seen from Table 4, this gives us a unique cointegrating vector which is extremely useful given that the interpretation of the coefficients in a cointegrating vector is somewhat controversial in the presence of multiple vectors. Table 5 reproduces the Johansen cointegration results for the VAR lag length of 6 and provides additional information. The value of the [chi square](1) test statistic of 1.1220 compared to the critical value of 3.84 (at the 5% level) indicates that the price homogeneity restriction is convincingly accepted. This is a significant result as price homogeneity must be satisfied in order for the vector to satisfy theoretical money-demand restrictions. It is a restriction, however, which is frequently imposed but rarely tested in empirical money-demand studies. Finally, the speed of adjustment coefficient of 0.6784 obtained from the Johansen estimation indicates that the personal sectors demand for the broad aggregate D7 adjusts remarkably quickly to any disequilibrium in money holdings

TABLE 5 Johansen cointegration results for LD7
VAR lag = 6
                                     Eigenvector

Eigenvalue         LD 7         P7      LRGDP      LGDPDEF      CON

   0.63925       -7.0028     -0.949     8.2494     6.566    -116.182

                                Normalised coefficients

                  -1.000     -0.136      1.178     0.938      -16.59

                             Max eigenvalue test

Null           Alternative    Likelihood ratio         95%
                               test statistic     critical value

r = 0             r = 1           60.1547             28.1380

r [is less
than or
equal to] 1       r = 2           19.6655             22.0000


Speed of adjustment = 0.6784 [chi square](1) = 1.1220.

TABLE 6 Johansen cointegration results for LD4
                              Specification A
                               (VAR LAG = 5)
                                Eigenvector

Eiqenvalue      LD4       P4     LRPSDI    LGDPDEF      CON

0.50248       -4.3882   -0.775   3.1571    4.4111     -50.49

                          Normalised coefficient

              -1.000    -0.176   0.7194    1.0052     -11.506

                            Max eigenvalue test

                               Likelihood ratio          95%
Null           Alternative      test statistic      critical value

r = 0            r = 1              39.7925            28.1380

r [is less
than or
equal to] 1      r = 2              20.1887            22.0000

[chi square](1) = 0.15285

                              Specification B
                               (VAR LAG = 6)
                                Eigenvector

Eiqenvalue      LD4      P4       LRGDP    LGDPDEF      CON

0.61850       6.1854    0.877    -6.279    -5.5192     90.29

                          Normalised coefficient

             -1.000    -0.141     1.0152    0.89229   -14.49

                            Max eiqenvalue test

                              Likelihood ratio          95%
Null           Alternative     test statistic      critical value

r = 0            r = 1              56.8548            25.559
r [is less
than or
equal to] 1      r = 2              23.0309            22.000

r [is lees
than or
equal to] 2      r = 3              14.1869            15.6720

[chi square](2) = 3.8578

Speed of adjustment = 0.72940


Although the Johansen procedure does produce estimates of the speed of adjustment of money holdings, it is useful to specify and estimate a second-stage error correction model in order to subject the short-run dynamic model to the full range of diagnostic tests, in particular, tests for parameter stability and mis-specification. In order to develop the error correction model, the lagged residuals from the cointegrating regression are incorporated in an OLS estimation incorporating the first difference of LD7 as the dependent variable and the first differences and lagged first differences of all the variables from the cointegrating vector as independent variables. The methodology employed in deriving the preferred short-run dynamic model is the general-to-specific approach. Initially, a highly general error correction model was specified which included lags up to the fourth order. This general model was then tested down in order to arrive at a parsimonious preferred short-run dynamic specification. The final preferred equation for D7 is given below(8)

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where RES = residuals from cointegrating regression and R2 = 0.5666, DW = 1.7353, [Sigma] = 0.0151, ARCH(4) = 2.6846, ARCH(8) = 3.8425, [LM.sub.1](1) = 1.2039, [LM.sub.1](4) = 4.8051, [LM.sub.1](8) = 9.0117, BJ(2) = 5.6251, [CHOW.sub.2](8) = 8.9607, [CHOW.sub.2](4) = 2.2659, and [CHOW.sub.2](8) = 3.8407.

Clearly, this model passes a wide range of diagnostic tests including the ARCH test for Autoregressive Conditional Heteroskedasticity and the Bera-Jarque (BJ) test for the normality of the regression residuals. Based upon Lagrange Multiplier ([chi square]) test statistics, there is no evidence of serial correlation ([LM.sub.1]). Initial estimates did suggest the presence of heteroskedasticity, however, and the standard errors shown are therefore White's heteroskedastic consistent estimates. The model convincingly passes a test of parameter stability ([CHOW.sub.1]) and both a four-period and eight-period [chi square] forecast test ([CHOW.sub.2]). Finally, both the CUSUM and CUSUMSQ statistics remain firmly within their 5% bands throughout the sample period indicating that the estimated equation is highly stable and shows no sign of underlying mis-specification.

Hence we have established the presence of a long-run cointegrating regression based upon an extremely simple money-demand specification and without the need to resort to arbitrary proxies for financial innovation. Furthermore, this long-run relationship has produced a sensible dynamic error-correction model that is structurally stable even through the 1980s and early 1990s, which was an extremely volatile period in terms of new product innovation, rising own rates of return on `near monies', and changing monetary regimes. This is an important result given the well-documented instability in broad money-demand functions in the UK since the 1970s and echoes the results obtained by Drake and Chrystal (1994) for corporate sector money demand. These results taken together strongly suggest that the observed broad money-demand instability stems from a combination of the following factors: inappropriate aggregation across sectors (personal and corporate); the use of inappropriate subsets of financial assets (assets which do not satisfy the appropriate aggregation conditions in terms of weak separability); and finally, the use of simple sum aggregation across these assets rather than a procedure consistent with economic aggregation theory such as the Divisia index number approach.

4.3. Empirical results for D4

Table 6 illustrates the Johansen cointegration results for D4. When the log of Real PSDI (LRPSDI) is used as the scale variable in Specification A we obtain a unique cointegrating vector which corresponds with economic priors in the sense of signs and magnitudes. Furthermore, the price homogeneity restriction is convincingly satisfied using the appropriate [chi square](1) test statistic. The log of Real GDP is used as the scale variable in Specification B, and although it is not possible to isolate a unique vector the first cointegrating vector (corresponding to the largest eigenvalue) out of a maximum of two appears to form a sensible money-demand function. In fact, the coefficients on the rental price and the scale variable are very close to those obtained on the broader aggregate, D7, and the price homogeneity restriction is again satisfied on the basis of the appropriate [chi square](2) test statistic.

It is interesting to note in comparing the results from Specifications A and B, that the income coefficient is somewhat lower for LRPSDI than for LRGDP. In fact, the former indicates modest economies of scale in money holdings, as would be predicted by the Inventory Theoretic approach of Baumol and Tobin, whereas the latter suggests income homogeneity in money demand. This is a common finding in money-demand studies and indeed is frequently imposed in estimations even though, unlike price homogeneity, it is not a theoretical requirement. It is also interesting to compare this result of an income coefficient of broadly unity (in Specification B) with the results obtained by Drake and Chrystal (1994)with respect to corporate sector money demand. The latter study also used LRGDP as the scale variable and found income coefficients in the range 2.5 to 3. Theoretical work by, for example, Miller and Orr (1966) suggests that these values are plausible for the corporate sector whereas theoretical work on household money demand, together with previous empirical money-demand results, suggests that a coefficient close to (or possibly below) unity is much more plausible for the personal sector. This conclusion is supported by the results obtained in this paper although they do indicate that the magnitude of the coefficient (and hence the theoretical interpretation) may be sensitive to the choice of scale variable. The contrast between the income coefficients obtained for the corporate and personal sectors provides further evidence against aggregation across sectors.

With respect to the speed of adjustment, the coefficient of 0.7294 derived from the Johansen procedure is very similar to that derived for D7 and suggests that, for broad-money aggregates in general, the personal sector's adjustment of their holdings of these aggregates is relatively rapid in response to any perceived disequilibrium.

In the interests of brevity and in order to be consistent with our analysis of D7, the preferred error correction model for Specification B only is illustrated below. As with the error correction model for D7, initial estimations suggested the presence of heteroskedasticity. The standard errors stated are therefore White's heteroskedastic consistent estimates.

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where RES = residuals from cointegrating regression, and R = 0.529, DW = 1.722, [Sigma]=0.01, ARCH(4)= 2.644, ARCH(8)=4.388, [LM.sub.1](1)= 1.328, [LM.sub.1](4)= 1.768, [LM.sub.1](8) = 14.009, BJ(2) = 1.776, [CHOW.sub.1](7) = 2.6104, [CHOW.sub.2](4)=0.398, and [CHOW.sub.2](8) = 2.621.

Again, this error correction model performs extremely well in terms of the diagnostics, especially the parameter stability ([CHOW.sub.1]) and post sample forecasting tests ([CHOW.sub.2]). These suggest that the dynamic model shows no signs of underlying instability or mis-specification and this is supported by the CUSUM and CUSUMSQ statistics which remain within their 5% bands throughout the sample period.

In short, we have found that there exist well-determined long-run money-demand functions for our chosen monetary aggregates which have highly plausible parameter estimates, broadly consistent with what economic theory might lead us to expect. We have also found well-determined error-correction dynamics indicating fairly rapid elimination of disequilibrium in personal sector money holdings.

5. Conclusions

The monetary authorities need to be able to judge when monetary conditions are too tight or too loose. For this purpose they need to have a stable model of underlying money demand. Monetary targeting was dropped because the demand function for M4 (and M3[pounds sterling] before it) was demonstrably unstable. Instability was widely attributed to financial innovation. We have shown that there was a stable underlying demand for monetary assets on the part of the UK personal sector, when Divisia aggregates are used, for a period stretching from the mid-1970s to the early 1990s--a period of substantial change in the UK financial system. The advantage of such aggregates is that they endogenise the substitution effects which result from yield changes on component assets. The strong support for both a unit income elasticity and a unit price elasticity in the long run is remarkable and reassuring. That these parameter estimates contrast with those reported elsewhere for the corporate sector (Drake and Chrystal, 1994), adds clear support to the strategy of sectoral disaggregation.

The message for the past is that monetary policy might have been better in the 1980s if the UK authorities had been able to use Divisia aggregates as indicators of impending inflation, rather than their simple sum counterparts. For the future, we would not suggest a return to any form of monetary targeting. However, there is a strong case for the authorities to pay more serious attention to the behaviour of Divisia money measures than to broad simple sum monetary aggregates. Rigid rules would be dangerous because it is likely that financial innovation will continue, and the financial innovations of the next ten years may be very different from those of the last ten. But better information makes for better policy, and there can be little doubt that, in the present state of knowledge, information about monetary policy stance is best achieved by a careful monitoring of appropriate Divisia indices, rather than traditional broad monetary aggregates.

ACKNOWLEDGEMENTS

The authors are very grateful to Sophie Haincourt for her excellent research assistance. Financial support for this research was provided by the ESRC (Research Grant No. R000234614) and is gratefully acknowledged. The authors are also grateful for helpful comments provided by three anonymous referees and by the editor.

(1) The authors are grateful to the Bank of England for their provision of data on the personal sector holdings of assets. Unfortunately, data are not formally collected on the split between personal sector non-interest-bearing and interest-bearing sight deposits. Data are available, however, for personal sector total sight deposits and personal sector non-interest-bearing bank deposits. Hence, the latter were taken as a proxy for non-interest-bearing sight deposits (as non-interest bank deposits will be almost wholly sight deposits), with interest-bearing sight deposits being derived as the residual.

(2) The GDP deflator is used in preference to the retail price index as the personal sector is broader than the household sector and also includes unincorporated businesses. Both the non-parametric demand analysis and the subsequent cointegration analysis were found to be essentially invariant to the choice of the price index.

(3) This is the necessary condition for weak separability. The software used to test for weak separability utilises the Afriat sufficient conditions (Afriat, 1967). This requires that the data satisfy GARP when the sub-utility functions are calculated using the Afriat inequalities. The use of this sufficient condition does imply, however, that the non-parametric test is biased toward rejection of weak separability. The authors are grateful to Hal Varian for making available the non-par software package necessary to undertake the non-parametric demand analysis.

(4) A referee suggested that the non-parametric demand analysis should be conducted over the same sample period as the money demand analysis. The reason the non-parametric demand analysis is restricted to a shorter period, however, is that this was the longest period for which a complete set of consistent data on all the assets was available. Prior to 1986 Q3 personal sector holdings of CDs are recorded as zero and it is not possible to have zero entries in the non-parametric tests. It would be possible to construct a new data set going back over the whole period either by pretending that CDs did not exist or by subsuming them with something else. These would not strictly be correct procedures, however, with the latter requiring the aggregation of potentially heterogeneous assets. While the aggregate money demand results are interesting and potentially valuable in their own right, we feel that the non-parametric demand results do provide important information and can be viewed as necessary, if not sufficient, for proceeding with the chosen aggregation and money-demand analysis. With the ideal longer data set more violations of GARP could be generated but not fewer, and hence any rejections evident in the smaller data set would still be evident in the longer data set.

(5) Following the suggestion of a referee, we also tested other groupings of monetary assets (including the specification of two monetary asset sub-utility functions) for weak separability in addition to the incremental groups shown in Table 1. No other groupings of monetary assets were found to be weakly separable, however, and for the sake of brevity these results are not detailed in Table 1.

(6) The Divisia index was contrasted with other aggregation procedures such as simple sum and currency equivalent aggregation and was found to be superior in respect of producing sensible cointegrating vectors.

(7) A dummy variable is included as an additional I(0) variable in the VAR in all the Johansen cointegration analysis. This reflects the data distortions caused by the conversion of the Abbey National Building Society to bank status during 1989. Hence, the dummy variable takes the value of zero prior to 1989 Q2 and unity thereafter.

(8) It was also found, not surprisingly, that the inclusion of the dummy referred to in footnote 7 above also improved the performance of the dynamic error correction models for both D7 and D4.

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LEIGH DRAKE, Department of Economics, Loughborough University, Loughborough, LE11 3TU

K. ALEC CHRYSTAL, Department of Banking and Finance, City University Business School
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Date:Apr 1, 1997
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