metadata: authors: 0: author: Arrau, Patricio
1: author: de Gregorio, Jose
count: World
docna: 0: docna: Financial innovation and money demand :
            theory and empirical implementation
docty: Policy Research Working Paper
subtopic: Macro-Fiscal Policy,Taxation & Subsidies,Economic Adjustment and Lending,Public Sector Economics,Inflation,Economic Stabilization,Macroeconomic Management,Public Finance Decentralization and
            Poverty Reduction
teratopic: Macroeconomics and Economic Growth,Finance and Financial Sector Development,Private Sector Development
abstracts: cdata!: Empirically, traditional money demand
            equations are frequently characterized by periods of
            "missing money", unstable parameters, and
            autocorrelated errors.  The common practice to solve these
            problems consists of changing the specification of the
            regressions once the shifts (which are usually associated to
            financial innovation) are identified.  This paper provides
            an alternative approach to dealing with the unobservable
            process of financial innovation.  It consists of modelling
            financial innovation as shocks that have permanent effects
            on the money demand, analogous to productivity shocks in
            production functions.  This paper describes the theoretical
            model used and shows the failure of traditional money demand
            equations using cointegration techiques.  It describes a
            simple GLS-iterative econometric model which allows the
            authors to recover the path of financial innnovation and
            obtain sensible estimates of the relevant elasticities.  It
            also shows Monte Carlo simulations to evaluate the behavior
            of the estimation procedure for particular samples and data
            generating processes, and to study how robust the procedure
            to some deviations from the basic assumptions is.
repnme: repnme: Financial innovation and money demand :
            theory and empirical implementation
display_title: Financial innovation and money
            demand : theory and empirical implementation
listing_relative_url: /research/1991/01/23866007/financial-innovation-money-demand-theory-empirical-implementation
topicv3: Public Finance m1326315 713,Inflation m1326709 0870F4727ACED1CF8525815C00471347,Macroeconomic Stability and Growth
            m1326923 1088,Monetary Policy m1327609 444,Taxation m1327615 1121
publishtoextweb_dt: 1991-01-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Finance and Financial Sector Development,Private Sector Development
no_of_pages: 48
model_extractions: WPS585 <!-- Start of picture text --> Debt and International Finance<br><!-- End of picture text -->
<!-- Start of picture text --> Policy, Research, and External Affairs<br>Debt and International Finance<br><!-- End of picture text --> WPS 585 This paper - a product of the Debt and International Finance Division, International Economics Department-is part of a larger effort in PRE to look for better structural models for developing countries. Copies are available free from the World Bank, 1818 H Street NW, Washington DC 204 33. Please contact Sheilah King-Watson, room S8-025, extension 31047 (42 pages).

Traditional estimates of money demand are often characterized by periods of "missing money," unstable parameters, and autocorrelated errors.

Typically, these problems are resolved by changing specifications for the regressions once the shifts are identified. The shifts are usually associated with financial innovation.

Arrau and de Gregorio provide an alternative approach for dealing with the unobservable process of financial innovation. They derive from first principles a money demand that is consistent with many traditional models but that explicitly includes financial innovation. In their model, financial innovation is treated as shocks that have a permanent effect on demand for money.

They estimate the model using data for Chile (1975-89) and Mexico (1980-89).

The PRE Working Paper Series disseminates the findings of work under way in the Bank's Policy, Research, and External Affairs Complex. An objective of the series is to get these findings out quickly, even if presentations are less than fully polished. The findings, interpretations, and conclusions in these papers do not necessarily represent official Bank policy.

Produced by PRE Editorial Services
\.,' HG3~.s­ · l,_h~ _P6J_ fJD·S-~ c / 2-

# Financial Innovation and Money Demand: Theory and Empirical Implementation*

by Patricio Arrau and Jo8e de Gregorio Table of Contents

|1.|Introduction|1|
|---|---|---|
|2.|TheModel|3|
|.3.|TheLack of Cointegration of Traditional MoneyDemandEquations|9|
||3.1Data<br>3.2Estimations|11<br>12|
|4.|··EstimatingFinancialInnovation|13|
|5.|Monte Carlo Exi>ertments|22|
|6.|Conclusions|24|
|Ref|erences|27|
|Tab|les|30|
|Fig|ures<br>SECTORAL LIBRARY<br>INTERNATIONAL BANK<br>FOR|36|
|App|endix A Individual Optimiiation<br> <br>RECONSTRUCTION AND DEVELOPMENT|38|
|App|endix BDataSourcesandUnit Root Tests<br>FEB071991|40|
|App|endix CVAREstimations for ChileandMexico|42|

* The first version of this paper was written while the second author was a consultant at the Debt and International Finance Division, World Bank. We are grateful to Ricardo Caballero, Jim Hanson, Raul Labaan, Leo Leiderman, and Sweder van Wijnbergen for helpful comments and discussions. Special thanks to Carmen Reinhart for detailed comments to a previous version.
# 1. INTRODUCTION

Empirically, traditional money demand equations are frequently characterized by periods of "missing money", unstable parameters, and autocorrelated errors. The common practice to solve these problems consists in changing the specification of the regressions once the shifts are identified. These shifts are usually associated to financial innovation<sup>1</sup> • To solve the estimation problems, it is common to find the inclusion of dummy variables, lagged variables, proxies for expectations, etc. Often, these amendments improve the fit of the models, but provide no guarantee that the estimated equation is the true long-run relationship. In other words, spurious inclusion of variables may result in biased estimates of the interest rate and income elasticities of the money demand.

In this paper we provide an alternative approach to deal with the unobservable process of financial innovation. It consists in modelling financial innovation as shocks that have permanent effects on the money demand, analogous to productivity shocks in production functions. The entire path of financial innovation is estimated simultaneously with the other parameters, using data for Chile (1975-89) and Mexico (1980-89), by applying a procedure due to Cooley and Prescott (1973a,b,1976)<sup>2</sup> .

Our definition of financial innovation is very broad. It involves pure

> 1 Discussions of instability of the demand for money in the U.S. can be found in Goldfeld (1976) and Friedman (1988). A comprehensive survey on the state of the art of money demand equations in developed countries can be found in Goldfeld and Sichel (1990). For developing countries see Fasano-Filho (1986), Marquez (1987), Melnick (1989), Viren (1989) and Laban (1990).

> 2 In developing countries, not only "pure financial innovation", but volatility in rates of inflation and currency substitution (or "dollarization") have been also invoked as provoking shifts in money demand relationships. They will be included in our definition of financial innovation as long as they have permanent effects on money demand equations.

1
technological progress in transactions, which we usually may interpret as financial innovation, as well as policy changes such as financial regulation or deregulation. Therefore, negative financial innovations may be related to policy restrictions in the payment's system, rather than technological reversal. The key element of our definition.is that a financial innovation shock will permanently shift the demand for money. Empirically we identify financial innovation with all permanent shifts in money demand not explained by changes ih the scale variable (income or consumption) and the interest rate.

The theoretical model is a representative infinite-lived consumer with a transactions technology for money. Money is demanded because it produces transaction services. We assume a specific transactions technology with a productivity parameter whose innovations are identified as financial innovation. We show that the reduced form of the model is capable of including as particular cases a wide variety of monetary models; e.g. a class of money in the utility function, the fixed-velocity cash in advance model, and other,specifications ·such as Baumol-Tobin<sup>3</sup> , Cagan (1956), and Miller and Orr (1963). Thus, we provide microfoundations to traditional money demand equations and generalize other monetary models. The model follows the insights provided by Feenstra (1985), who shows the equivalence between the transactions technology and the money in the utility function approaches.

The paper is organized as follows. Section 2 describes the theoretical model. Section 3 shows the failure of traditional money demand equations using cointegration techniques. Both the equations fail to cointegrate and provide implausible estimates of the relevant elasticities. Section 4 describes a simple GLS-iterative econometric model (Cooley-Prescott procedure) which allows us to

> 3 See Baumol and Tobin (1989) for a recent reference.

2
\' recover the path of financial innovation and obtain sensible estimates of the relevant elasticities. Section 5 shows Monte Carlo simulations to evaluate the behavior· of the estimation procedure for our particular samples and data generating process, and to study how robust is the procedure to some deviations from the basic assumptions. Section 6 concludes.

# **2. THE MODEL**

The representative infinite-lived consumer maximizes the expected value of the utility function where the subscript s denotes time, c is consumption of the only perishable good, _fJ_ the discount factor and u(•) is a twice differentiable increasing and concave utility function. E5 is the expectations operator conditional on all the information at time s.

There are two assets in this economy: bonds and money. The nominal return on bonds is it and on money is zero. However, people demand money because it facilitates transactions. The larger the money holdings, the less expensive it is to transact, as less resources have to be devoted to shopping. The shopping technology describes the amount of resources required to buy one unit of the consumption good as a function of the ratio of money holdings to''corisumption spending (the inverse of velocity). The amount of resources spent in transactions is inversely related to this ratio.

The consumer maximizes (1) subject to the flow budget constraint 3
<!-- Start of picture text --> (2)<br><!-- End of picture text --> where capital letters represent nominal variables. At the beginning of period t the individual starts with bond holdings of Bt-l which yield the nominal return it- 1 ; Mt-l are nominal cash balances, and Yt is labor income in units of the consumption good. At time t, after observing capital and labor income the individual chooses consumption, money and bonds to be carried to period t+l. In real terms the budget constraint (2) can be written as where 1rt the inflation rate ( [Pt+i/Pt]-1) and rt is the real interest rate ([Pt<l+itl/Pt+iJ-1). The transactions technology is such that<sup>4</sup> h 1 (•) < 0 and h 11 (•) > 0, so a decrease in velocity (increase in m/c) decreases the unit cost of transactions, but at a decreasing rate. _8_ is a technological parameter in the transactions technology, by convention we assume h 2 (•) > 0<sup>5</sup> • Financial innovation is represented by changes of _8_ through time . A low value of _8_ could be thought as characterizing a well developed credit market where transactions can be conducted carrying low real balances. A fall in Ot could be caused by technological change as well as policy-induced changes in financial markets.

Maximization of (1) subject to (2'), and assuming that both it and _Ot_ are

> 4 hi denotes partial derivative with respect to i=c/m and _8._ Similar notation. is used for second partial derivatives.

> 5 This formulation to analyze financial innovation has been proposed in De Gregorio (1990). He also extends the analysis to allow for substitutes of money. Thus, _8_ .could represent government regulation (e.g. reserve requirements). The formulation is popular also in the literature on inflation taxation (Kimbrough, **1.986·;, ve·gh, 1989).** '.· ·

4
known at time t, yields one Euler equation for the consumption path and one intratemporal relation for money and'consuinption, which can be expressed as (see Appendix A for a formal derivation) where Equation (3) says that the cost of the last unit of consumption forgone in period t must be equal to its marginal benefit in terms of higher consumption next period. The marginal cost includes the marginal utility of consumption and the impact of the last unit of consumption forgone on the resources devoted to transactions. One less unit of consumption today reduces the unitary cost of transaction, so the marginal utility of consumption today must be discounted by a number bigger than one (the term Ht in the LHS of equation 3). Analogously the .- marginal increase in utility next period (times the real return) must also be discounted as the unit of consumption forgone increases the unitary cost of transactions next period.

Equation (4) is the equilibrium condition for money. The consumer allocates resources to money until the marginal cost of the last unit of money (interest lost as money is not an interest.bearing asset) is equal to the marginal benefit associated to the reduction of the cost of transactions today. The relevant cost of holding money is the nominal interest as holding money not only implies losing 5
the real return on interest bearing· assets but also the depreciation of value due to inflation. The interest rate is discounted back as this atemporal equilibrium condition requires to measure both the marginal cost and benefit at the same point in time.

Equation (4) is the. reduced form estimated in this paper. Under some specifications for h(•), we can write (4) in the form of a linear regression which represents a money demand.

Now we can consider a few particular specifications for h(•).

Money in the Utility Function (CES) For the specification where K is an arbitrarily large constant, such that h > 0, the reduced.form (4) for the. money demand equation is: ..

which is equivalent to first order conditions arising from a money in the utility function model for the case of a general C. E. S subutility be·:::ween consumption and money.

The parameter _p_ represents the intratemporal elasticity of substitution between money and consumption in the subutility (Dornbusch and Mussa, 1975). As discussed in Feenstra (1986) this formulation has, as a particular case, the fixed-velocity cash-in-advance model (e.g. Lucas, 1982) when the intratemporal elasticity of substitution between money and consumption is equal to zero.

6
Cagan money demand

> A traditional specification for the money demand function is Cagan' s (1956) form. Unlike the previous case, the interest rate is not in log form, but linear. The following form for the function h(•) generates the Cagan money demand:

where K is large enough to guarantee h 2>0, and h > 0. We require that _x/O_ be less than one for h 1 to be less than zero and the semielasticity of money demand with respect to the interest rate to be negative. This assumption is empirically verified below. This specification for the transactions technology yields the following reduced form: and l/a corresponds to the semielasticity of the interest rate<sup>6</sup> .

Again, the fixed-velocity cash-in-advance model can be obtained by setting l/a equal to zero. In other words, making the money demand equation insensitive to the nominal interest rate<sup>7</sup> . An interesting difference between obtaining the fix-velocity model by restricting l/a in (4b) or _p_ in (4a) to be equal to zero is that the later restriction also eliminates the financial innovation parameter. Finally, it is important to note that _()_ is not introduced multipicatively

> 6 Calvo and Leiderman (1990) derive a Cagan money demand directly from money in the utility function by assuming a separable utility function between consumption and money. In their case, however, the semielasticity of the interest rate depends on the marginal utility of consumption and the elasticity of consumption has no direct interpretation.

> 7 This point has been noted in Mccallum (1983). Other versions of cash-inadvance, like Svensson's (1985), do exhibit variable velocity and they do not have a direct interpretation from our transaction technology model. Englund and Svensson (1988) and Englund (1989) present models of financial innovation in a cash-in-advance economy with variable velocity.

7
as in th.e CES case. O:therwhe, the interest rate elasticity would,, also be affected ~yJinancial innovation .. As a starting point we prefer .to concentrate in financial innovation affecting money demand independently from the other variables.

Scale economies in transactions Note that until now equation (4) a],.lows only for a scale (consumption) elastkity equal to one. Of course this is a r.estrictive .;issumption. We know from traditional models a la Baumol-Tobin or Miller-Orr that transaction. technologies may involve increasing returns to scale, and therefore, the scale elasticity could be less than one. In fact, the Baumol-Tobin model suggests that this elasticity is equal to 1/2, apd the Miller-Orr model ,implies an elasticity equal to 1/3.

To obtain a scale elasticity less than one in the context of, our framework, it is enough to assume that the transactions tech?ology not only depends on velocity, but also on the level of consumption. Hence, high levels .of consumption would require. a lower ratio m/c for the same unitary cost of transactions .. A technology involving this assumption would be: where in this case the scale elasticity of the money demand is eql,lal t 0 _¢>._ Frain the above discussion, we can summarize the general form of the money demand equation which will be the focus of our empirical application. The equation, in a regression form, can be written as lit is a white noise distributed normally w:itli. variance _av2_ The error _llt_ can 8
I \ be considered a measurement error<sup>8</sup> , some form of e~pectational or· aggregational e'r:tor, or· other departures from·our model, e.g. iiquidity constraints<sup>9</sup> • St is some measure of the interest rate~ As·· discussed ··above, the interest rate could be in log form as in (4a), in which case a is an elasticity; or in linear form as in (4b), in which case a is the semielasticity. It c:iiso could be it or' it/(l+it) depending . ori whether we co~sider money accumulated today affecting today's or tomorrow;s transaction technology (in this paper we'use the latter convention as budget constraint (2) indicates). Finally, the scale elasticity must be less or equal to one.

# 3. THE LACK OF COINTEGRATION OF TRADITIONAL MONEY DEMAND EQUATIONS

By traditional money demand we mean the relation like (6) with a fixed intercept. Vari~tions of this regression equation constitutes most of the literature·on the ·demand for money'.

Typically, estimations· of money demand equations like (6) with a fix.ed intercept have found several anomalies. The existence of protracted periods of missing money, highly autocorrelated errors, and implausible parameter estimates, suggest that there is a misspecification in the model. The problems of traditional estimations are compounded by the time series properties of the 8 Our theory is a specification for household money demand: The' series m (Ml) includes other sectors (corporate, government, etc.) so we .can assume that household demand is being measured with error by Ml. Implicitly we assume household demand then, when taking log, we obtain the white noise error in the regression.

> 9 The presence of liquidity constraints aff~cts the intratemporal decision of consumption versus money holdings by changing the measure of the opportunity cost, which in this case would include the lagrange multiplier for the liquidity constraint.

9
variables themselves. As we. will see below, money, income (consumption), and nominal interest rates are often non-stationary processes (more specifically integrated of order one or I(l) processes). Hence, the "true" long-run relationship among these variables can only be assessed if cointegration holds. Most linear combinations among the level of I (1) variables are I (1), except for the cointegrating vector(s) (if it exists), in which case the linear combination is I (0) (Engle and Granger, 1987). This cointegrating vector captures a long-run relationship among the. variables. Loosely speaking, although the variahles may be moving everywhere as time goes on, there is (are) a relationship(s) among the variables that remains between certain limits. In our case real balances, consumption (or income) and some measure of the opportunity cost of money could be a stationary process. The coefficients of this linear combination describe the demand for money. Money demand relationships have been typiGal example in the development of tests and estimation procedures on cointegration (Engle and Granger, 1987, Stock and Watson, 1989, and Johansen and Juselius, 1990). They have also been recently used for estimation in developing countries (Melnick, 1989 and Laban, 1990).

Another important issue we want to explore is what is the right scale variable: private consumption or GDP. Our model is obtained -from household optimization, in which case the relevant scale variable is private spendinglO. Using the wrong scale variable will produce estimators without clear interpretation. It also has different policy implications. For example, Mankiw and Summers (1986) show in an IS-LM framework that when consumption is the relevant scale variable, a tax cut may be contractionary because of a large

> 10 If money is an input in the production process, it could be the case that firm's demand for money would be a function of output.

10
increase in interest rate in response to the increase in money demand. Although evidence for developed countries suggests that about two third. of real balances are held by households, the extrapolation of this conclusion to developing countries could be controversial as firms may have much less sophisticated cash management procedures. The resolution of this question, however, should be empirical. We address this issue by using both income and consumption as the scale variable.

Next we describe the data and investigate the time series properties of the variables. The section is closed with an assessment of the fixed-intercept money demand equation in the light of recent cointegration literature. 3.1 Data We estimate the model with quarterly dat.a from Chile during the period 1975-1989 and Mexico 1980-1989<sup>11</sup> • The data are described in Appendix B. For each country we used data on Ml, short-term interest rate, consumption and GDP. The series of money, consumption and GDP are normalized by total population, as suggested by the theory.

Using standard Dickey-Fuller and Augmented-Dickey-Fuller tests it can be seen that there is strong evidence that most of the series are well characterized by processes with unit roots. The results are presented in Table Bl, Appendix B. All the series are the log of the respective variable. The tests in levels do not reject the null hypothesis that the series have a unit root. The tests in first differences reject the null hypothesis of unit root. Therefore the evidence supports the hypothesis that the variables are difference stationary or I(l). The only exception is GDP in Mexico where stationarity around a deterministic trend

> 11 In Chile market interest rates start in 1975, and in Mexico quarterly consumption starts in 1980.

11
# appears _as a likely specification .

. Sitlc~:the variables are·well·char.;tcterized:oya I(l)'stochastic process a coint~grat:i,ng ·vector among. the . :variables entering . the ·money. demand- could be -· estiJ11ated, ._ As we claim below, t:he :variables in general fail to co integrate. ·,. ·• i

## 3. 2 ·-Estimations

·In Tables i arid,2-we report the results of the money-demand-·equatfon (6) with a fixed. intercept for . Chile and Mexico, respectively~ Dickey-Fuller ·and Augmented-Dickey .... Fuller s ta tis tics are· computed ·for. the regress ion error. We. use !>oth consumption and output as .the scale variabre in ·the regressions. The column Il:aJlle.9 interest. rate in th$ .. l'ables indicates whether the interest rate is in log (as in equati:ono4a) or l-inear (as in equation 4b)~. In a11· regressions we detect evidence of strong-seasonality,· so· seasonal dummies were· included . . The fit of the regressions is relatively-.-good for both countries.· A closer look, however, shows serious anomalies .. The estimated values' for the income elasticity iI\ Me~ico -~re surprisingly high (between 5 and 6). This 'happens mainly beca\,lse of th_e small vari~mce -of. income and consumption during our sample period as opposed .to that of M~ .. In the case .of. Chile, as·.·most previous estimations conclude, the income elasticity is larger than one which has neither theoretical nor practiCal appeal. However, it is worth noticing that t:he value for this parame_t;:er is ·i.n both countries smaller when consumption is employed as the scale variable Jnsteaci of- GDP·.

The results in both cases strongly reject cointegrat:lon: That is, ·we cannot rejec.t the _hyp_othesJs t~at .the error, term in the regressions is non-stationary. Therefore, the results.from these regressions,cannot be :interpreted as long;_run

> , .··

·;: 12
# money demand estimations<sup>1</sup> ?.

Recently,. Perron • (198.9)1 ·has ·suggested .that "dis'crete· b:["eaks." ;·in the .ec~n!'.>mic environment may induce unit ro_ots .in stationary series;~ Therefore, 'after filtering the. serie_s from ,:th~se large breaks'., the· resul,t1ng ·process ~may be stationary. Perron (1989) suggests that the crash of 1929 and the-.·c>.il .:shock of 1973 are e~~mp;J.es of such discr.ete .. breaks which. induce· GDP to "look-· Like" ·an I(l) process .. Fo,r a· similar, reason, .we· introduce·: dl.inimy variables· in Cfrile after 1984. III. and in Mexico:after 1987 .IV•• In both cases· casual evidence. suggests .a structural change.· Although the dummies appear;.to· .be' Significant.·· and· the absolutes values ,:of the: Dickey-Fuller•.and ·Augmented-'Dickey-Fuller tests· increase, they are sti).l below the. criti~·al value!! to :accept. coiritegratiori<sup>13</sup> • · The main conclusion from, these· results is that standa·rd money' "demand ~quations, no .matter whether consumption.or income is used as the: scale variable, do not provide, a· long-'-run relationship. ·. _·:_ .,. ·.·'· . r _.:.:: .._ , We int;erpret the failure of. the money qemand-equation to coiritegrate as· due to· the omission of an· important .unobservable ¢omponentwhich we call financial in:qovaticm. The next section shows_ the results '.of our -i>'roposed approach~

# 4. ESTIMATING FINANCIAL .INNOVATION

In this section we' describe the ass\.tmptions regarding financial innovation, the econometric technique to be used, and the motivations' .f<>r· thi·s choi'c:e;··

> 12 We also run the Stock and Watson (1989) regression to have non degenerated distributions for· the• standard errors, using four l'e.ads B,nd lag·s of the first differences of the RHS variables. The values of the scale variable elasticity raised even further .away from one·.

> 13 See Laban (1990) for a summary of the evidence in Chile and a discussi~n of cointegration when a dummy is incorporated. He does not reject cointegration when the dummy is added from 1984.III. His estimation is for a shorter period, output is the scale variable, and the data are not in per-capita terms.

\I I
The most important departure in this section from a standard money demand is that the intercept in the regression is no longer a constant. It may reflect technical progress in the transactions technology as well as institutional and policy conditions that may affect directly the necessity of money as a medium of exchange. For example, allowing "dollarization" and the use of foreign currency to make transactions and improvements in credit markets ability to provide alternative medium of exchange (e.g. credit cards) will shift the intercept of the money demand.

In modelling this financial innovation variable, we should bear in mind the results from the previous section. The failure to cointegrate of the traditional money demand equation might indicates that an unobservable non-stationary process is omitted from the regression. Consequently, we model financial innovation as a non-stationary process. This assumption is also consistent with the idea that financial innovation is like technical progress. Once financial innovation takes place, it permanently shifts the money demand.

We propose to model log(8t) in (6) as a difference stationary process, which for simplicity is assumed to be a random walk. Therefore, a realization of a financial innovation shock will have a permanent effect on the money demand. We also assume that the innovations in the random walk are orthogonal to the error in the regression.

These assumptions can be written as 14
where Therefore the money demand (6) becomes This is a standard regression equation besides the fact that the intercept evolves as a random walk.

Ye use a procedure first applied by Cooley and Prescott (1973a,b), which for the sake of completeness is described below<sup>14</sup> • For a sample size T and by recursive substitution, we can substitute ~t by ~T in (7) to obtain All parameters in equation (8) are time-invariant and therefore can be estimated by least squares. The only correction is that the error term is no longer i. i. d. Defining μt as lit - l::J=t+l E.,., it is easy to verify that where le•> is an indicator function that takes a value of one when the condition in parenthesis holds and zero otherwise. The Cooley-Prescott procedure consists of estimating equation (8) by GLS correcting by the variance-covariance matrix of the transformed residuals. This variance-covariance matrix does not need to be estimated because it is known.

Since we are interested in estimating the whole path {~tl, we can write,

> 14 Greene (1990) provides also a simpie description. For a detailed treatment and the discussion of the asymptotic properties of the estimators, see Cooley and Prescott (1976).

15
for any given· r e tl,: .. Tl, a similar equation to (8) as where μ(r) is just the following ·~traiglitforward generalization of _μ_ defined previously With this generalization and redefining the variances of the i. i. d. random variables as a~ - _-ya_<sup>_2_</sup> and a~ = (l--y)a<sup>2</sup> , it can be verified that the variancecovariance::matrix of μ(I'),· named here O(r) ,. is given ·by the expression where Therefore, given the matrix O(r) we can estimate equation (9) using GLS· for ail values of r':from~l to T~ This· procedure allows us -to recover the whole path . ; ~. _:_ Finally we need to estimate the value of 'Y· This parameter ·reflects the relative imp-ortance of the permanent shocks to the money demand with· respect to the transitory shocks. When 'Y - 1, only the permanent shock appears in the equation. 'In this · case· all · changes 'in money demand are due to financial innovatiOn. The estimatiOn' of this case is equivalent to estimating eqtiatio-n (9) in first differences. On the other extreme, 'Y ... 0 implies that _'I_ is a constant and OLS applied to the equation in levels would be the appropriate m~~hod. We can 'estimate 'the- maximum likelihood estimator of .. 'Y. The ·concentrated 16
likelihood function for.- .. ~ given -y .. (a·fter replac.i:ng. the. es"timat.or.s. pf• th~,,,other. parameters in the regression) is "''.-} . /. . ~ ' _:_ . , ..

where s<sup>2</sup> is the estimated variance of the regres~iori residuals. A grid search for ) . .., between 0 and 1 give us.th~ .-maximum·likelfho~d estimator.

Summing up, the st~ps of the estimaddn are' the following. We first estimate (9) by GLS and do _·a:_ grid search o~e·r 1'. for r-T<sup>15</sup> • We then choose .., th~t· maximizes tpe likelihc;>od .fu.n.ction,,a~q;prc;>~e~d to compu~~~ t,l;le path of., _'I_ :i...:terating O(r) . for· all. r=l. ,-T. : . .._, The procedure .has -t:wo adva!ltages when .compared to. other-al·~e-r:nativ:es. tq estimating time-varying parameters (e.g. Kalman fi~tering). First, it is possible to formally show that this method generates the same numerical estimates for all the parameters assumed to be constant· 'in our equad.on (a and 4'). Second, it uses the- whole Sl;lmp!e period tc;> esti_mate each variabl~ pa,i::amete:r·. It Js mo:re. than an updating rule because. it uses future i_nf.ormation .. TJ.i,er~ ;!.s no; reasqn .in O\,lr mod,~l to use only the information from 1 to t to estimate _'It_ instea~ of using the whole. s_ample .from J to T ..

We .. first; discuss the, results for Chile (Table ~) .!1-nd then for Mexico .(Table. 4). • I~ • . · In Chile the maxim\1111 likelihood estimat,or of 'L.is O_, 63 whep .the int.er~st·· rate is .. in a log fo:r111,· and. O·. 7 whe_n.it is. in~ linear- fqrm ,(Cagan), .·we also. show .

. ,.

> 1 ~ Tl>.is . s.t.~p could b~ .don~ for . a:ny r ,: The,., result~. how~yer . do riot. change significantly.

17.
the estimation for a low value of 7~0.2 and the maximum 7=1<sup>16</sup> . The likelihood function is relatively flat, so there is a problem of precision in the estimate of 7. The likelihood function increases very fast for 7 below 0 .1 or 0. 2, becoming then flatter for higher values. The scale elasticities (consumption as scale) of money demand are very close to one.

The interest rate elasticities also fall with respect to the OLS estimation, being the latter close to zero in the .log form and close to -0.4 in the linear case.

If we were to plot the actual and fitted series from our regression, we could hardly distinguish the two lines .. In general the R<sup>2</sup> of the estimated regression are very high, over 99%<sup>17</sup> • A useful way to present ·the results is to isolate the explanatory power of consumption and interest rate on one hand, and the financial innovation. shocks on the other·. We do that by plotting in Figure 1 the actual series of (the log of) money and the fitted value implied by the parameter estimates in Table 3 and a constant intercept. For the latter we use the estimated sample mean of the intercept. In Figure 2 we plot the path of the intercept. In both cases we plot the series obtained from regression 5 in Table 3, that is the linear (Cagan) form for the interest rate with the log· of consumption as the scale variable.

From Figure 1 we see that the effect of financial innovationAs important before 1983 causing alternative periods of under and over estimation of the money demand. By 1984 the money demand bec.omes relatively more stable and there is no "missing money". As we. mentioned above, previous evidence suggests an

> 16 The case 7=0 is the OLS regression from the section 4.1

> 17 As expected,. the regression residuals (estimated value of 11.) are very close to a white noise.

18
overestimation of money demand after 1984. It is important to recall that 1984 is the year where a strong recovery of the Chilean economy started. Output and consumption kept (_ growing for the next 5 years. Obviously, an upward-biased elasticity of the scale variable (Table 1) would produce an overestimation of the money demand during the period of recovery. From Figure 1 we can see that once the omitted "financial innovation" variable is included and the correct elasticity is recovered (close to 1), no apparent "missing money" is there. The missing money therefore is largely due to the misspecificat'ion of the money demand before 1984 which biases the relevant elasticities, as Figure 1 shows surprising stability after 1984.

During the financial liberalization that occurred in Chile in the late 70's, Figure 2 shows negative financial innovation, which may seems contradictory.· However financial innovation and financial intermediation are not necessarily related in a direct way. The former affects the saving opportunity (technology) available to households, whereas the latter affects the transactions technology. According to Figure 2, the increase in financial intermediation in the late seventies was· not accompanied by progress in transactions. Moreover, this period was also characterized by increasing macroeconomic stability, which may have cau·sed a return to money as a medium of exchange.

The estimations for Mexico are presented in Table 4 and Figures 3 and 4; again for regression 5 in Table 4.

In contrast to Chile, the likelihood function increases monotonically as we approach _-y_ equal to 1. Table 4. also presents _-y=O._ 2 ·and _-y=O._ 7 as a way to assess the estimation. Unlike Chile, the likelihood function is not flat and the value of _-y_ makes a significant difference on the values of the coefficients for consumption and the rate of interest. Surprisingly, we obtained much more 19
sensible results for both the income and interest rate elasticities as _-y_ increases. Take for instance the Cagan form and consumption as the scale variable. For _-y=O_ (third row in Table 2) the estimates for the consumption elasticity were about 6 and the interest semielasticity about -4.2. As we move _-y_ to 1, the consumption elasticity goes to 1.0 and the interest semielasticity to -0.54, both very reasonable figures<sup>18</sup> In Figure 3 we plot the actual and fitted money demand using again the sample mean of the intercept in order to disentangle the role of financial innovation (for regression 5 in Table 4). In Figure 4 we plot the path of the intercept. Now we can easily understand the origin of the unreasonable estimates in Table 2.

Looking at Figures 3 and 4 we see a decreasing path of the intercept during the period of increasing inflation. Th~ high elasticities of Table 2 are due to .. the bias introduced when trying to accommodate the evolution of financial innovation with the series of consumption and interest rate alone, both trending over the sample period. We also can conclude from Figures 3 and 4 that virtually all money demand variation for this sample period in Mexico is due to financial innovation. This observation ca~ explain the lack of response to v~locity in Mexico after the stabilization policy of 1988. If, as suggested by Figure 3, most of the variation of real balances in Mexico in the eighties corresponds to permanent shifts, then there is no reason to expect a substantial reduction in

> 18 The interest semi-elasticity result, however, is very sensitive to alternatives definitions of the quarterly interest rate and it should not be taken as very robust. For instance is we compute the quarterly interest rate from middle of the quarter to middle of the next quarter instead of end-of-period definitions, the semi-elasticity become bigger than one in absolute value. The scale elasticity however is very stable around one.

20
# velocity when the nominal interest rate is sharply reduced<sup>19</sup> •

We can now conclude that the higher elasticities obtained in both countries from the OLS regressions in the previous section are the result of the omission of financial innovation.

It is also very interesting to notice that the estimations are extraordinarily consistent across the two countries, with a scale elasticity close to one, a semielasticity between -0.4 and -0.6 for the linear form, and an · intratemporal elasticity _(p_ in 4b) between 0.03 and'0.14 for the log form<sup>20</sup> • Another interes·ting .implication of our model is related to the empirical instability of the relation inflation-seigniorage (Blanchard and Fischer, 1989, Table 10.1). -Our explanation is that the relation is unstable· due to financial innovation shocks. For instance, given the evolution of financial innovation shown in Figure 4 for Mexico; at the same inflation rate and consumption per capit~, Mexico would be able to collect seigniorage in 1989 60% below that in 1980.

One final.econometric concern remains. What is the performance of the Cooley-Prescott estimation in small samples, .and particularly, given the data generating process for our two countries? ,What are the estimation problems if the process assumed for the random shocks are · different to · those assum·ed in our estimations? We. answer these questions in the next section .

.<sup>19</sup> This monetary puzzle in Mexico is noted and discussed in Arrau (l990), where the hypothesis of currency substitution is explored to explain the puzzle. 20 The latter results differ substantially from the estimates for the intratemporal elasticity of substitution, p, between 0. 3 and 0. 7 provided by Arrau (1990) using the Euler approach for the same two countries.

21
# 5. MONTE CARLO EXPERIMENTS

It is not obvious what is the small sample performance of the econometric procedure employed here. It could be possible that given the data generating process of our variables, the estimator had some small sample bias._ For instance, looking at the results .for Mexico, one wonders whether the ML estimator of 1 for l could be due to a small sample problem.

In order to assess the above concerns, we perform Monte Carlo experiments. We generate data and simulate the model of Section 4, and then see whether there is evidence of any small sample bias. Specifically, we are interested in detecting any possible bias in the estimation of 'Y and the relevant elasticities. For both countries we use the parameter estimated in the linear (Cagan) ·model of the previous section for both the elasticities and variance of the errors. The data generating process of consumption and interest rate is assumed to be a VAR in first difference, which is consistent with the evidence reported in Appendix B. We estimate the VAR with four lags (Appendix C). For every iteration we compute the left-hand-Side variable by generating the data from the VAR representation and adding the financial innovation variable (random walk). In this way the results of the Monte Carlo simulations are not conditional to any particular realizations of these variables .

.. Table 5 summarizes the results. We provide both the median and mean of the empirical distribution of 'Y and the elasticities. Given the results in ·the previous section, we simulate the model for a range of·., between 0.4 and 1.0 for both countries. As we can see, the results are quite satisfactory. The scaie and interest rate elasticities are virtually identical· . to the true population parameters, both using the mean or the median criterion.· Th~ interest elasticity for Mexico, however, is somewhat imprecise.

22
Regarding _-y,_ we should only. consider the median estimator of the replicat.ions due to the character of the procedure, which restricts the search of _-y_ between zero .and one. Consequently, the mean will necessary be biased when the true _-y_ deviates from 0.5. For example if the true _-y_ is distributed around one the mean will be strictly less than one (because we search between zero and one only). Looking at the median of _-y,_ we can see a small bias downward for lower values of-y. For Chile, the median is equal to the true value for all _-y_ except 0.4. For Mexico, the median is lower than the true value of _-y_ for 0.4 and 0.6. From these Monte Carlo simulations we can conclude that, assuming our model is well specified, there is no small sample biases.

The second question we want to address is to assess any possible bias when the process assumed for· the errors in the -regressions are diff.erent to those assumed by the estimation procedure. This seems particularly important in the case of Mexico, where according to Figure 4, financial innovation appears to have a downward drift. We do tpe·next Monte Carlo simulations for Mexican data only. We depart from our· assumptions in.three ways.-First we assume that _'It_ has ' : a negative drift, so there is permanent financial innovation. Second we explore the case where _'It_ is a more complic_ated. integrated process, an ARIMA(l, 1, 0), instead of the random walk (ARIMA(0,1,0)). Finally we assume that the stationary shock of equation (7) is AR(l) instead. of a white noise. In the last two cases we use.a"coefficient ·of .autocorrelation of 0.8.

The results are presented in Table 6 for. true _-y_ equal to 0. 4, . 0. 7 and 1. 0. It is clear from this table that .when the . true _-y_ is different from one the estimator of _-y_ is biased upwards. Looking at the median estimate, we see that for a true _-y_ equal to 0.7, themedian of the· estimates of _-y_ is one in two out of the three cases , and O·. 9 5 in- the· third. When the true _-y_ is 0. 4, the median is be tween 23
0. 8 and. 1 in the_ three cases.· However it is·_ interesting to notice that the other elasticities . seem. to · b.e · very well .estimated around. -::0. 6 ... for the interest semielasticity and: 1 for the scale elasticity. This simulation allows us to conclude that deviations from the assumptions of the estimation along the lines exp.lqred here,. will, bias only the estimate of -y.

The results of this Monte Carlo study implies that for the Mexican case, if the true process of the errors were like the ones examined here, we shoul.d lo.ok at .. an equation with a lower -y than the ML estimator. For instance·, for the processes assumed in.these .experiments, -y=0.7 s~ems tobe. a reasonable estimat;e (equatioI). 5, Table 4). In this case, the interest rate. ~lasticity in Mexico would be -0. 8 and the. consumption elasticity 1. 5, which is greater than the theoretical bounds ..

..; .

# 6. CONCLUSIONS

Attempting to disentangle the shifts in money demand from moyemertts along the money demand sched,ule is worthwhile pursuing. Episodes of velocity shifts are abundant in the empirical literature from inflationary countries. Whether oi; not velocity. shifts permanently after a stabilization plan is importan:t ·to understand. the .events a·fter such a plan takes place. In Mexico we ·e>bserved that velocity stays up after the stabilization of 1987 (Ratchet effect). The same result is observed in Israel after the stabilization of 1986, and in Argentina after the Austral Plan (a result recently challenged by Melnick1 1990). In contrast, in Chile .~fter the .1975 stabilization, a- recovery in velocity is observed, which occurs also with a process of financial deepening.

We . provide a framework which allows us to disentangle those shifts and estimate the other parameters (elasticities) simultaneously. Perhaps the.best 24
argument in favor' of our approach' is the reasonable and consistent results we find· for Chile and Mexico. In both cases we show that the ·high elastici. ty estimates (inteiest' rate' arid the scale variable) provided by 'traditional estimations are appare'ntly due to an omitted and unobseriable variable which we call financial innovation. After controlling for this ·variable, under the simple asswilptiori that· those shocks (innovations) have 'permanent effects, sensible parameter<sup>1</sup> estimates are obtained in both countries. · We have been able to understand the source of the "missing" money in Chile after·· 1984 .. ·We have also. been able _to-_ obtain "economically sensible" money demand parameters for ·Mexico,as the ·estimates from traditional approaches do not make sense. In the latter case we found -that most of the movements of inohey demand during the eighties' are due to financial innovation shocks and the results appears to be sensitive to misspecifications.

The theoretical model presented in section 2 accommodates a w1de- variety of models of househoids demand for money, providing microeconomic foundations to traditional money demand estimations. For instance, the model replicates exactly the famous Cagan money 'demand· as the result of the c·onsumer i.ntertemporal optimization.

Our model also provides an explanation for the lack of a stablei .. relation between inflation and seigniorage. For instance, at the same inflation rate and consumption pe:t·capita, Mexico would be able to collect seigniorage in 1989 60% below that in 1980. This is the direct implication of the estimated path of financfal ·innovation· during the period of inflation. Because the shocks are permanent, velOcitY did not' fall after the· stabilization of 1987. · It would' be interesting to extend the framework of this study to tighten the definition of financial innovation.' Our definition of financial innovation 25
as "everything persistent different from conswnption and interest rate" is too broad. Distinguishing policy induced shifts from pure technological progress could be a fruitful agenda to explore in the future. For example, it could be important to disentangle a shift away from money due to macroeconomic instability (currency substitution and capital flights), from the introduction of "credit cards". If part of the estimated path of "financial innovation" for Mexico is actually due to macroeconomic instability, we should expect to see velocity falling over the next quarters as macroeconomic stability takes place.

26