cointegration, error correction, and the demand for money in mexico

Cointegration, Error Correction, and the Demand for Money in Mexico

By

John Thornton

C o n t e n t s: I. Introduction. - II. Methodology, Data, and Results. - III. Conclu- sion.

I. Introduction

F inancial reforms in Mexico since 1988 have facilitated a shift f rom a system o f mone t a ry cont ro l based on interest rate ceil- ings, quant i ta t ive credit quotas, and reserve requirements to

one based mainly on open marke t operat ions. 1 Fol lowing the shift, the p r imary policy objective o f the Bank o f Mexico was redefined as tha t o f at ta ining price stability. 2 Moreover , as the 1995 peso crisis forced the a b a n d o n m e n t o f the exchange rate-based stabil ization policy, the Bank o f Mexico has focused on the cont ro l o f the m o n e t a ry aggregates as the intermediate variable most likely to bear a stable relat ionship with prices. Such an opera t ing p rocedure assumes that there is a stable long-run equil ibr ium relat ionship between real m o n ey balances, real income, and interest rates in Mexico. This paper exam- ines the issue by presenting estimates o f the long-run demand for na r row and b road definit ions o f the Mexican m o n ey supply over the per iod 1980 Q1 - 1994 Q4 utilizing the cointegra t ion test procedures o f Johansen and Juselius (1990) and the related not ion o f e r ror correction.

II. Methodology, Data, and Results

The e r ror correct ion me thodo logy was popular ized by H en d ry and his col laborators , no tab ly in Davidson et al. (1978). Subse-

Remark: The views expressed in this paper should not be attributed to the institutions with which I am affiliated. I am grateful for helpful suggestions from an anonymous referee and for comments from Sebastian Paris-Horvitz on an earlier version. i The reforms are discussed in Aspe (1993) and Coorey (1992). 2 The objective of price stability was formalized in August 1993 when the Mexican Congress approved a constitutional amendment granting autonomy to the Bank of Mexico and giving it a mandate to maintain price stability. The autonomy and the mandate became effective on April 1, 1994 (Bank of Mexico 1994).

Thornton: Demand for Money 691

quently, Engle and Granger (1987) showed that, if there exists an equilibrium or cointegrating relationship between nonstationary variables, there must exist an error correction representation of the data. Engle and Granger show that if a cointegrating relationship exists, then a simple ordinary least squares static regression affords consis- tent estimates of the long-run, equilibrium parameters. It is then straightforward to embed the parameters from the first-stage estimates in the full, dynamic, second-stage estimates of the error correction parameterization. A precondition for the existence of cointegration is that all the relevant variables are integrated of the same order. If this is established then the residuals from the first-stage (long-run) estimates can be used as the error correction to explain the short-run dynamics.

The data for the study consist of quarterly observations for the narrow (M1) and broad (M2) measures of money, each deflated by the gross domestic product deflator (P). Two variables are utilized for the scale variable (S V) in the demand function: real gross domestic product (RGDP) and real private consumption expenditure (PCON). This distinction follows, for example, Arestis and Demetriades (1991) and Mankiw and Summers (1986) who adopt consumption expenditure in demand for money equations to demonstrate the relative supe- riority of this variable. The argument, based on Friedman and Schwartz (1982) and Hall (1978), is that permanent income is a better proxy for the volume of transactions and that consumption is closely related to unobservable permanent income; Mankiw and Summers also argue that consumption is the most important component of income-generating demand for money. The interest rate on 91-day treasury bills (ITB) is used to represent the opportunity cost of hold- ing money. The source for all the data is the Bank of Mexico.

To ensure the use of stationary time series, augmented Dickey- Fuller (ADF) test statistics were computed for the presence of unit roots against the alternative hypothesis that the series are stationary round a linear time trend. The results of these tests are reported in Table 1.3 From the results the null hypothesis that the series contain unit roots (i.e., are nonstationary) cannot be rejected in all cases;

3 Trend stationarity was considered as the appropriate alternative hypothesis given that income velocity exhibited a marked trend over much of the sample period; suffi- cient lags were included in the regressions to eliminate serial correlation as a statistical problem as measured by the Lagrange Multiplier test statistic up to order 4.

692 W e l t w i r t s c h a f t l i c h e s A r c h i v 1996, Vol. 132(4)

Table I - Unit Root Test Results

Variable I A D F statistic

(M1/P) -1.86o6 (M2/P) - 0 . 8 8 1 6 RGDP - 2.4527 PCON - 1.9742 ITB - 1.7579

Variable I A D F statistic

A(M1/P) - -3 .8904* A(M2/P) - 3 . 5 6 1 0 " A RGDP - 3.9074 * A PCON - 3.8842 *

A ITB - 6.7669 *

Note: Critical values are f rom Fuller (1976: Table 8.5.2). * denotes significance at the 5 percent level.

however, the results from first-differenced data reject the hypothesis of a unit root for all series. 4

The cointegration equations include the log levels of real money balances, the scale variable (real GDP or real private consumption), and the treasury bill interest rate. In addition, an attempt is made to capture the effects of financial reform in Mexico after 1988 by the inclusion of a shift term by using a dummy variable that takes the value of one for 1988-1994, the key reform period, and zero else- where. The cointegration test results are presented in panel (a) of Table 2. A likelihood ratio test was used to select the number of lags required for the cointegration test. The maximum eigenvalue test statistics show that the hypothesis of a single cointegrating vector among the variables is not rejected for either demand function of real money balances, whether RGDP or PCON is used as the scale variable. An econometric interpretation of the results can be facilitated by normalizing the cointegrating vector on real money balances. The resulting values are the long-run elasticities and can be used to test the standard hypotheses of whether the long-run income elasticity is unity and whether the long-run interest elasticity is zero. The estimated long-run elasticities and hypothesis test results are reported in panel (b) of Table 2. For both (M1/P) and (M2/P) the signs on the income and interest rate elasticities are as one would expect (i.e., positive on RGDP and PCON and negative on the interest rate); the calculated test statistics (distributed as X2(1)) reject the zero interest elasticity hypothesis in all cases but reject the unitary income elasticity hypo-

t No te that this result is found even though Perron (1989) has shown that A D F tests are biased towards no t rejecting the null hypothesis o f a unit root when there is a structural break in a series. The financial reforms in the late 1980s induced a dramat ic change in M1.


Table 2 - Results of Johansen-Juselius Estimation Procedure

(a) Cointegration test result"

r = 0

In(M1/P) In(M2/P)

In(M1/P) In(M2/P)

r < l r < 2 r < 3

Scale variable: RGDP

29.4750" 16.1054 6.6623 2.3059 29.8904" 13.7101 8.6093 2.4090

Scale variable: PCON

32.6341 * 12.2516 8.0091 2.7161 32.4486 * 17.6639 11.4353 3.6130

(b) Implied long-run elasticities and hypothesis tests b

ln(M1/P) In(M2/P)

In(M1/P) In(M2/P)

Elasticity

Real income Interest rate

Test

~/,~p = 1 r/it b = 0

Scale variable: RGDP

1.0971 -0 .7017 0.0060 12.0286"* 2.0455 -0 .2585 1.9383 15.7884"*

Scale variable: PCON

2.3696 -0 .6228 3.6959 19.1959 ** 2.3931 -0 .2468 9.8452** 12.5467"*

"The cointegration tests are conducted using the maximum likelihood procedure with a maximum lag of 2. The number of significant cointegrating vectors is denoted by r. The Johansen-Juselius statistics test the hypothesis of at most one and zero cointegrating vectors, respectively. A trend was included in the vector autoregres- sions. * denotes significance at the 5 percent level. - b To test the restrictions, a model incorporating the test restriction is compared to an unrestricted model using a likelihood ratio test. The test statistic is distributed as X2(1) under the null hypothesis. The 5 percent critical value is 3.84. ** denotes rejection o f tbe null hypothesis.

thesis in the case of (M2/P) when PCON is the scale variable. The size of the income elasticities is relatively large (1.1 and 2.0 for (M1/P) and (M2/P), respectively, when RGDP is the scale variable and 2.4 for both (M 1/P) and (M2/P) when PCON is the scale variable), although not out of line with results from studies on developing economies. The size of the interest elasticities ( -0 .7 when RGDP is the scale variable and -0 .6 when PCON is the scale variable in the case of (M1/P), respectively, and about -0 .3 for (M2/P) whichever of the two scale variables is used) accords with expectations from theory as well as

694 Weltwirtschaftliches Archiv 1996, Vol. 132(4)

with the results of other studies. 5 Not shown in the results, the coefficient of the dummy variable to capture the effects of the financial reform is negative indicating that financial liberalization has reduced the demand for money at a given level of income. In addition, the coefficient related to the trend term is positive, probably reflecting the increase in income velocity that took place over much of the sample period.

Since (M1/P), SV and ITB, and (M2/P), SV and ITB arc cointe- grated the dynamic relationship between real money balances, the scale variable and interest rates can be expressed in terms of an error correction model (ECM). Following Engle and Granger (1987) the ECM for all variables in the cointegrating regression can be formu- lated as:

AIn(M/P)=o~o + AI EC,_I +Z,81 Aln (M/P),_ i (1)

+ Z ~ 2 Aln ( S V ) , + Z ~ 3 Aln (ITB)z-i+~t,

where (M/P) represents real money balances (either M1/P or M2/P), EC is the vector of stationary residuals from the cointegration regressions, and the other variables are as previously defined; in addition, the estimates included quarterly seasonal dummies (QD). In estimating the ECMs the number of lags has to be determined for each first- differenced independent variable. A common technique is Hendry's general-to-specific modeling strategy of eliminating lags with insignif- icant parameter estimates (Gilbert 1986). Accordingly, ECMs with four lags of each dependent variable were estimated, the lags whose coefficients were not significant were eliminated, and the resulting restricted models were estimated.

The results of the ECMs are presented in Table 3. The results from the (M2/P) equations generally are more satisfactory: the diagnostic statistics for serial correlation (LM4) and heteroscedasticity (ARCH1) are acceptable and the Chow statistic (CHPF) (from splitting the sample at 1988Q1) does not indicate predictive failure or general specification error. In addition, changes in the scale variable and the treasury bill rate are important factors that influence significantly the short-run dynamics, with the signs on the coefficient estimates consis- tent with theoretical expectations. Results for the (M1/P) equations are less satisfactory: the Chow statistics indicate specification errors

5 Laidler (1982) suggests as reasonable quantitative ranges 0.5 to 1.0 for the real income elasticity and -0.1 to -0.5 for the interest elasticity. However, Hafer and Kutan (1994) find an income elasticity of 1.1 for the monetary base and 1.3 for M2 in the case of China.

Thornton: Demand for Money

Table 3 - Dynamic Demand for Money Functions

695

Constant

AIn(RGDP)

A In (PCON)

AIn(ITB)

AIn(M2/P)t- 1

ECt- I

QDI

QD2

QD3

R2 DW SE LM4 ARCH1 CHPF(21,20)

Dependent variable

Aln(M1/P) AIn(M1/P) AIn(M2/P) AIn(M2/P)

Explanatory variables

-2.7910" -1.8862" -4.1167" 0.9717 (4.3966) (5.0655) (2.8569) (1.9905) 0.5767 0.7699"

(1.4997) (7.5881) 0.5847 0.7164"

(1.3505) (2.7107) -0.1494" -0.1339" -0.0568* -0.0541"

(3.6530) (3.5539) (2.6558) (2.3185) -0.2156" --0.3473*

(4.5381) (3.1023) -0.2156" -0.2657* -0.1473" -0.1301 *

(4.5381) (5.3439) (2.8668) (2.0495) -0.1346" -0.1447" 0.0494* -0.633*

(3.3351) (4.2835) (4.1923) (2.7690) -0.1181 * -0.1088"

(4.7979) (4.9268) -0.1164" -0.1521 * --0.0506*

(2.5486) (6.7845) (3.7357)

Summary statistics

0.7122 0.7565 0.6918 0.6332 1.5961 1.9631 2.1050 2.1895 0.0586 0.0539 0.0306 0.0341 2.9760 1.1878 4.0772 8.4739 0.0457 0.0961 0.0315 2.3099 6.4764 9.2382 2.0338 1.9841

Note: t-statistics are in parentheses below the coefficient estimates; * indicates significance at the 5 percent level. LM4 is the Lagrange Multiplier statistic for serial correlation of up to order 4 asymptotically distributed as X 2 with 4 degrees of freedom. ARCH1 is the test for first-order autoregressive conditional heteroscedas- tieity distributed asymptotically as X 2 with one degree of freedom under the null hypothesis. CHPF is the predictive failure F-test statistic.

a n d the coeff ic ients o f the scale var iables are n o t s tat is t ical ly significant . T h e relat ively p o o r p e r f o r m a n c e o f the M 1 e q u a t i o n s p r o b a b l y reflects the pos t -1988 f inancia l l iberal izat ion: in par t icu la r , there was a t e m p o r a r y surge in the g r o w t h ra te o f M1 be tween 1989 a n d 1992 fo l lowing the i n t r o d u c t i o n o f in te res t -bear ing check ing accoun t s . T h e e r r o r c o r r e c t i o n term, wh ich is nega t ive a n d s ta t is t ical ly s igni f icant in

696 Weltwirtsehaftliehes Archiv 1996, Vol. 132 (4)

each estimate, indicates the speed with which deviat ions f rom long- run equil ibr ium (the cointegra t ion equat ions) will be corrected. In all cases this would appear to take place slowly: only 2 2 - 2 7 percent o f the deviat ion is el iminated after one quar te r in the case o f the (M1/P) equat ions, and 1 3 - 1 5 percent is e l iminated after one quar te r in the case o f the (M2/P) equations.

As noted, an issue in the demand for m o n ey l i terature is whether a demand for m o n e y funct ion with consumpt ion as the scale variable per forms bet ter than one with income. A compar i son o f the s tandard statistical criteria and the diagnostics o f the estimates in Table 3 suggests that the equat ions with RGDP as the scale variable pe r fo rm bet ter than those in which PCON is the scale variable. To address this issue more formally, five non-nested tests are presented in Table 4: the N-test is a t t r ibuted to Cox (1962) and Pesaran (1974); the NT- tes t and W-test f rom Godf r ey and Pesaran (1983); the J-test f rom Davidson and M a c K i n n o n (1981); and the JA-test f rom Fisher and McAleer (1981). These statistics are designed to test the null hypothesis tha t Mode l 1 is t rue against the al ternative Mode l 2, and vice versa; in relatively small samples the NT-tes t and the W-test are preferable to the others shown in the table. Table 4 also reports two model selection criteria: the Akaike In fo rma t ion Cri ter ion (Akaike 1973) and the Schwartz Bayesian In fo rma t ion Cri ter ion (Schwartz 1978), bo th o f which use statistics which incorpora te measures o f the precision and pars imony in the parameter iza t ion o f the two models. All the test

Table 4 - Non-Nested Tests

Test statistic �9

N-test NT-test W - t e s t

J-test JA-test

H0: Model 1 H0: Model 2 HI: Model 2 HI: Model 1

(a) AIn(M1/P) b

-9.9148 - 1.3532 -7.7116 -0.9618 - 5.8659 -0.9266

5.1063 1.1884 3.8016 0.8173

H0: Model 1 HI: Model 2

H0: Model 2 HI: Model I

(b) Aln (M2/P) c

--3.0881 -0.6258 -- 2.5929 - 0.2609 - 2.2850 - 0.2572

2.4609 0.5766 2.4602 0.5324

�9 See text for definitions of test statistics. AIC and SBIC are Akaike's Information Criterion and Schwartz's Bayesian Information Criterion, respectively. - b Model selection criteria: AIC (Model 1 versus Model 2) -- 10.7737, favors model 2, SBIC (Model 1 versus Model 2) -- 10.7737, favors model 2. - c Model selection criteria: AIC (Model I versus Model 2) = -4.1166, favors model 2, SBIC (Model 1 versus Model 2) = - 5.0825, favors model 2.


statistics reject Model 1 (with private consumption as the scale variable) in favor of Model 2 (with real GDP as the scale variable). At the same time, in no case can the null hypothesis that Model 2 is true be rejected against Model 1. In addition, the two information criteria reported clearly favor Model 2. Thus, in contrast to the findings of Arestis and Demetriades (1991) and Mankiw and Summers (1986), the traditional income variable would appear to be a better scale variable than private consumption in the demand for money in Mexico.

IlL Conclusion

This paper has examined whether there exists a long-run equilibrium money demand relation for Mexico using the Johansen-Juselius cointegration test procedures. The results are directly relevant to con- cerns about which monetary aggregate best determines the long-run effects of monetary policy actions in Mexico. The results suggest that a single cointegrating relationship exists for real money balances (M1 and M2), a scale variable (real GDP and real private consumption), and the 91-day treasury bill rate. This would appear to establish the potential for the Bank of Mexico to achieve its objective of price level stability by controlling the growth rates of either M1 or M2. The results of the short-run dynamic equations favor M2 as the appropriate monetary aggregate to target, probably because it has been less affected by financial liberalization. Finally, the traditional income variable, real GDP, would appear to be the more appropriate scale variable in the demand for money function for Mexico.

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A b s t r a c t : Cointegration, Error Correction and the Demand for Money in Mexico. - Estimates of the long-run demand for narrow and broad definitions of the Mexican money supply over the period 1980 Q1-1994 Q1 suggest that a single cointegrating relationship exists for real money balances (M 1 and M2), a scale variable (real GDP or real consumption expenditure), and the 91-day treasury bill rate. The results


from short-run dynamic equations favor M 2 as the monetary aggregate to target and suggest that real GDP rather than real private consumption is a more appropriate scale variable in the demand for money function for Mexico. JEL no. E41

Z u s a m m e n fa s s u n g : Kointegration, Fehlerkorrektur und Geldnachfrage in Mexiko. - Sch/itzungen der langfristigen Nachfrage nach der eng und weit definierten Geldversorgung Mexikos fiir die Periode vom 1. Quartal 1980 bis zum 1. Quartal 1994 deuten darauf hin, dab es eine einzige Kointegrationsbeziehung gibt zwischen realen Geldbest/inden (M 1 und M 2), einer Aktivit/itsvariablen (reales BIP oder reale Konsum- ausgaben) und der Rendite fiir 3-Monats-Schatzwechsel. Die Ergebnisse aus kurzfristi- gen dynamischen Gleichungen empfehlen M2 als monet/ire Zielgr6Br und legen es nahe, das reale BIP anstelle des realen privaten Verbrauchs als gecignetere Aktivit/its- variable in der Geldnachfragefunktion fiir Mexiko zu verwenden.

cointegration, error correction, and the demand for money in mexico

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