Testing for the presence of non-linearity, long-run relationships and short-run dynamics in error correction model

Transcription

1 Testing for the presence of non-linearity, long-run relationships and short-run dynamics in error correction model Hassan M.A. Hussein Department of Statistics, Faculty of Commerce, Zagazig University, Egypt. Hussein, H. M.A., Abstract Recent research has increasingly suggested that the demand deposit may be characterized by non-linear behavior. This paper examines whether such non-linear behavior is evident, not in the demand deposit themselves,but in the adjustment of the demand deposit back to fundamental equilibrium. Thus, we examine whether a series of the demand deposit and currency outside banks exhibit non-linear error-correction dynamic behavior. In order to test the presence of non-linearity in the errorcorrection models an artificial regression error correction model can be developed, the model has been regarded as a generalization of a number of linear and non-linear models. Two types of non- linear error correction models can be captured as nested cases depending upon the shape of the transition function, the first is referred to as the Logistic error-correction (LEC) model, the second model is referred to as the exponential error-correction (EEC) model. Moreover,we analyze some asymptotic properties such as long-run relationships and short-run dynamics in linear error correction models in small samples(t =5; 50), by using simulation study which designed to shed some light on these properties. The empirical results showed that non-linear behavior is present specially in the currency outside banks series.the latter findings are,the existence of short run and long run bi-directional causality between the demand deposit and currency outside in the sector of banks in Egypt.. Introduction: The recent literature has been an explosion of studies of non-stationarity time series. Non-stationarity (with a unit-root) mandates a revision of the standard inference tools. Most economic variables that exhibit strong trends, such as, consumption, money demand, the price level and exchange rate are not stationary. In many cases, stationarity can be achieved by simple differencing or some other - -

2 transformation (such as seasonal adjustment). The underlying series is said to be integrated of order one, denoted I(), because taking a first difference produces a stationary process,etc. Since Fuller (976), Dickey & Fuller (979, 98), Nelson & Plosser (98), Long & Plosser (983) and Phillips & Perron (988) devised a univarite tests that the null-hypothesis is a non-stationary process with a unit root (difference stationary process) and an opposing hypothesis is a trend stationary process. Perron (989), Perron & Vogelsang (99), Zivot & Andrews (99), Perron (997), and Lee & Strazicich (003), developed univariate tests allowing for structural changes occurring at unknown dates. Only more recently the theoretical literature has provided tests to assess the existence of instabilities in the cointegrating or long-run relation-ships between the variables. Two or more individual series may be non stationary (with a unit root), but a linear combination between these individual series may be stationary. If such a stationary linear combination exists, then the non- stationary time series are said to be cointegrated. The stationary linear combination is called a cointegrating equation. Sometimes, the results from the cointegration tests may suggest that there is no consistent evidence of a long-run relationship the underlying series, this still leaves the possibility for the existence of short-run relationships, so, we may employ Granger causality tests to determine how much of a current variable, y t, can be explained by past values of y t and whether adding lagged values of another variable, x t, can improve the explanation. In this case, y t is said to be Granger-caused by x t if x t helps explain to predict y t, what one is looking for is the coefficients on the lagged x s to see if they are statistically significant based on an F-test or χ -test. The well-known Engle Granger representation theorem (Engle & Granger,987), states that if two variables y t and x t are cointegrated, then the relation ship between the two variables can be expressed as the so-called error correction model (ECM ), this model provides a one-to-one relationship between the framework of cointegration. Boswijk (995b) advocates the conditional ECM, specifying the joint data-generating process (DGP)of - Note: the null hypothesis in the augmented Dickey Fuller (ADF) test is that there exists a unit root in the time series, that is, the time series is non-stationary. The null hypothesis is rejected if the ADF statistic is greater than the Mackinnon critical values. -There is this difference between tests for unit root and tests for cointegration; tests for unit roots are preformed on univariate [i.e., single] time series. In contrast, cointegration tests deal with the relationship among a group of variables, where (unconditionally) each has a unit root process. - -

3 endogenous variables conditional on a set of contemporaneous variables being weakly exogenous as defined in Engle et al. (983). A particular advantage of the single equation version of this model is to allow efficient inference which is very close to ordinary least-squares (OLS) procedures (see e.g. Boswijk, 993; Kremers et al., 99).The presence of cointegration leads the stage for using the error-correction model (ECM). If a set of non-stationary variables is cointegrated, then an unrestricted vector autoregressive (VAR) model comprised of the first differences of these variables will be misspecified. The reason is that the first differences of non- stationary impose too many unit roots and thus information on long-run equilibrium relationships among the variables will be lost. In this case, the error-correction model must be used. This model includes a vector of error terms that represent deviations from the long-run equilibrium and lagged short-term. These deviations may be due to the non- linear factors affecting the relation-ships among the underlying variables. Non-linearities in error-correction have been examined in macroeconomic data. Granger et al.(993), for example, focused on U.S. sales, production and inventory data, and Escribano & Pfann (998) reported non-linear error-correction in a system of UK employment, wages and capital stock. In a separate, though related, context Taylor & Peel (000) examine non-linear adjustment to equilibrium in a monetary model of exchange rate determination. An important question regarding the demand deposit in the sector of banks concerns non-linearities in their behavior. This paper examines whether there is non-linear threshold adjustment back to equilibrium in a system of the demand deposit and currency outside banks. We examine whether the parameters of an errorcorrection model differ according to some switching rule. To this end we consider a non-linear error-correction model, which is able to capture two types of asymmetric adjustment. First, the parameters of the model are allowed to change depending on whether the error-correction term takes a value above or below the threshold parameter, i.e., depending on the direction of disequilibrium. Second, the parameters of the model change depending on whether the error- -Alternative approaches considered by Granger et al.(993), Escribano & Pfann (998) and Escribano & Granger (998) include the error-correction term in a standard error-correction model in a non-linear fashion. - Lau(008). -3 -

4 correction term takes a large or small value, i.e., depending on the size of disequilibrium. The paper is organized as follows. In Section we discuss in detail the presence of non linearity in error-correction models. Section 3 outlines the data used in the empirical analysis. Section 4 explains the empirical evidences and reports summary statistics, including preliminary unit-root and cointegration tests and simulation study which has been designed for investigating the properties of the error-correction models in the small samples. Finally, results and conclusions of the analysis are given in Section 5. -Alternative bivariate system linear and non-linear error-correction models: The empirical procedure of this paper is developed for a bivariate error correction model in which the demand deposit and the currency outside banks are I() and cointegrated with cointegrated vector (, - ). In this section we consider alternative bivariate linear and non-linear error-correction models for the demand deposit and currency outside banks..- Linear error-correction model: The standard linear bivariate error-correction model (Engle & Granger, 987) is given by: p t 0, i t i t t i = Δy = α + α Δy + γz + ε () where y t represents a vector containing the I() demand deposit and currency outside banks series, z t represents the error-correction (equilibrium error) term obtained from our co-integrating regression, p refers to the lag length and t is the time measured chronologically. The α i parameters represent the short-run dynamics and indicate possible causality between the demand deposit and currency outside banks, that is, whether the lags of the currency outside banks are significant in the equation for these variables and vice versa. The parameter γ measures the speed of adjustment to equilibrium, with a negative (positive) value for the demand deposit and currency outside banks indicating mean- reversion. Statistical significance indicates whether the series exhibit (long-run) causality or exogeneity. -4 -

5 .: Non-linear error-correction models: In order to test for the presence of non-linearity in the error-correction model we construct an artificial regression error correction model, this model has been regarded as a generalization of a number of linear and non-linear error correction models( in statistical analysis this idea is known as the general - to simple approach), the model can be expressed as : x p p p p = β + β x + β x x + β x x + β t 0 j t j j t j t d 3j t j t d 4j j= j= j = j = x t j x 3, t d where the vector x t includes both differenced I() demand deposit,currency outside banks and the stationary error-correction term and d represents the delay parameter, the lag length p is determined when specifying the linear error-correction model. The appropriate lags are chosen by using Akaike information (AIC) and Bayesain information (BIC) criteria, one could select a lag structure by increasing the number of lags up to the point where the AIC or BIC reaches a minimum value. In order to determine the appropriate error-correction model, we use an Lagrange Multiplier (LM-type) test with the null hypothesis being expressed as : H 0 : β j = β 3j = β 4j =0, j =,..., p. The test statistic is given by: T(SSR SSR o) LM =, (3) SSR l where T is the number of observations, SSRl is the sum of squared residuals from the linear model(), and SSRis o the sum of squared residuals from the LM-test (), McMillan (005) mentioned that, the test statistic is asymptotically distributed as χ (3p). Two types of non- linear error correction models can be captured as nested cases depending upon the shape of the transition function,f(z t-d ).The first is the logistic model, in which the estimated parameters differ between positive and l () - The AIC and BIC are based upon multiplied by the average log-likelihood function adjusted by a penalty function, where the penalty is much larger for the BIC. The equations are given by: AIC: (l /T) + (k/t), and BIC : (l /T) + log k(t)/ T. where l is the value of the log-likelihood, T is the number of observations and k is the number of parameters. - McMillan (005) mentioned that,f(z t d ) function allows the parameters to change monotonically with z t d.as γ, F(z t d ) becomes a Heaviside function: F(z t d )=0, z t d c, F(z t d )=, z t d c, and reduces to a transition autoregressive model of order(p). When γ 0, () becomes a linear AR(p) model. As noted, the LEC model allows non-linearity between the differing dynamics of positive and negative deviations from equilibrium. -5 -

6 negative values of the error- correction term, with the model thus referred to as the Logistic (LEC) model: Δy ( p p t = π0 + π x + θ + θ x + F( z ) + ε, i t i 0 i t i t d t i = i = t d ) = ( + exp( γ(z t d c))), γ > 0, ) F(z (4) where the vector x t includes both differenced I() demand deposit,currency outside banks and the stationary error-correction term, γ the smoothing parameter and c the transition parameter. The second is the exponential model, where the parameters of the model change depending upon whether there are large or small values of the error- correction term. The resulting model is referred to as the Exponential (EEC) model: Δy t p p 0 + π x + θ + θ x + F( z ) + ε, i t i 0 i t i t d t i = i = t d ) = exp( γ(z t d c) ), γ > 0, = π ( ) F(z (5) Prior to estimate the presence of non-linearity in the error-correction models, we extend the specification strategy for smooth transition autoregressive models of Granger & Ter asvirta (993) and Ter asvirta (994) to the error-correction models considered here. To set up the non-linearity in the error-correction model, first specify the appropriate linear model (); second, test the null hypothesis of linearity against the nonlinearity in the error-correction model over several values of the delay parameter, d, thus simultaneously determining the presence of non-linearity and the value of d in (); and, third, choose between EEC and LEC models using a sequence of nested tests. Following Granger & Ter asvirta (993) and Ter asvirta (994), (under a range of d values consider appropriate ) we could choose whether the series are more appropriately modeled by EEC or LEC models, by conducting a series of nested tests within (eq()) based on LM test. The sequence of hypotheses to be tested is as follows: H 04 : β4j = 0, j =,...,p (a) H 03 : β3j = 0 β4j = 0, j =,...,p (b) - The d selected is the one which minimizes the p-value of (eq(3)) from all d considered. The rationale for such a selection method is as follows. When d is correctly specified, (eq()) is the appropriate auxiliary regression against the non-linear alternative. If another d is selected, then (eq()) is mis-specified. The power of the corresponding test against the miss-specified non-linear model will thus be weaker than the power of the test based upon the correctly specified auxiliary regression. -6 -

7 H 0 : βj = 0 β 3j = β 4j = 0, j =,...,p. (c) The rationale behind this sequence is based on interpreting the coefficients β ij as functions of the parameters of the artificial regression error correction model (), with either (4) or (5). If the model is EEC, then (a) should not be rejected, but (b) should be rejected. However, if (a) is rejected, then the LEC model is chosen instead. In practice, however, rejection of all three nested tests is possible. Moreover, to analyze some asymptotic properties such as long run relationships and short run dynamics in a bivariate linear error correction models, following (Herwartz & Neumann (005)), single equation test statistics are adopted for testing these hypotheses in small samples, again we assume that two non-stationary series for the demand deposit and currency outside banks variables denoted by y t and x t are observed, in this case the error correction model can be expressed as : Δyt = δ + α ( yt + βx t ) + γδxt + ut, (6) where - < α < 0. The marginal process is formalized by: Δx t = δ + α ( yt + βxt ) + νt (7) Since the model (6) evaluates the expectation of Δy t conditional on an information set containing Δx t, it is natural to impose Cov(u t,ν t )=0 (Boswijk,995b). (i) For testing the long-run equilibrium relations (i.e. To test the cointegrated parameter β) we assume α = 0: The null hypothesis is: H 05 : y t and x t are cointegrated and β = -. The equilibrium errors under H 05 can be defined as y t0 = α 0 + β 0 x t corresponding to a stationary linear combination of the integrated variables Within single equation models we investigate Likelihood Ratio (LR-type) tests. (ii) Finally, for testing the short-run dynamics in a bivariate linear error correction models, we consider testing the significance of parameter estimates of stationary explanatory variables augmenting the conditional model in (6), Δy t- say. To be precise, we assume instead of (6) the following model: Δy t = δ + γ Δ x t +γ Δy t- + u t. (8) and the null hypothesis of interest is H 06 : γ = 0. - when two hypotheses are nested and a maximum likelihood approach to estimate is adopted, the ratio L(R)/L(U) forms the basis for a test of the two hypotheses, where L(R) and L(U) are the ML values for the model with and without restrictions respectively. -7 -

8 One usually expects that the LR test is asymptotically χ (q) distributed, where q is the number of restrictions imposed under the null hypothesis. In our case, the number of excess parameters is. 3. Data: As noted above, all previous specifications of error correction models are tested using real data about Egypt. The data set used in this paper consists of monthly time series observations covers the period (98: 00:) ( ) for estimation purpose and over the period //00 0/9/003 for the forecasting purpose for both linear and non-linear error correction models. This data represents the demand deposit in local currency (y t ) in the sector of banks in Egypt and its more related independent variable. The statistical results obtained from the previous studies for money demand concluded that, the most important independent variables that are affecting the demand deposit in the sector banks is the currency outside banks plus quasi-money (x t ). Monthly data for the underlying variables which have been collected from different issues, of International Financial Statistics both inclusive (Total 40 observations) are used. In the empirical study, all variables are transformed to the nature logs and denoted by Ly t and Lx t respectively. 4. Empirical Evidences: Prior to the estimation, a number of statistical tests are performed to reveal the data properties. For each individual variable, the autocorrelation structures of the variables are examined. Since the time lag structure is different for each data series, it is desirable to estimate the optimal time lag p. AIC and BIC criteria are examined for each variable up to 6 lags, the appropriate lags vary between and 3 for the two variables. For analysis purposes, firstly, we conduct the more popular unit root tests, namely, ADF tests in order to search the stationarity in the time series of the underlying variables. Secondly, the cointegration tests which use to analyze the relationships between these non-stationary time series are the Engle-Granger cointegration tests (Engle and Granger, 987). Thirdly, we examines whether such ( ) - Source: I.M.F. International Financial Statistics, Various Issues - Mubarak., (998). -8 -

9 non-linear behavior is evident not in the underlying variables themselves but in the adjustments of these variables back to fundamental equilibrium, moreover single equation test statistics are used for testing the hypotheses of long run relation-ships and short run dynamics in error correction models in small samples. 4.: Unit root and cointegration tests: We first conduct a t-value type test of the levels of the underlying time series in nature logs. The presence of a unit root in these data is well-established, hence our work focuses on Ly t and Lx t as the first difference of the logarithm series, Table presents summary statistics for the demand deposit and currency outside banks series. The economic theory suggests that the demand deposit and the currency outside banks should be cointegrated with the error correction term given by z t = y t βx t and cointegrating vector (, -), i.e., β=. Table : Descriptive statistics Measures / variables demand deposit (y t ) currency outside banks (x t ) Error correction term (z t ) Mean Standard deviation Skew (0.74) (0.7) 0.08(0.6) Kurt (0.33) 00.38(0.57) 80.54(0.) Q * 7.334*.7876* ADF * * -33.4* β Wald (Standard deviation) (0.0309) (0.000) - Notes :The Ljung-Box Q-statistic is for first-order autocorrelation (The Q-statistic is distributed χ (q)). The ADF test lag length was determined by AIC, BIC and parameter significance. The Wald tests are use for testing the weak exogeneity of the variables. The numbers in parentheses under the coefficient restriction Wald tests are p-values. * Test statistic significant in rejecting the null at the 5% level. As Table shows, the two individual series (Ln y t and Ln x t ) in first differences of logarithms are trend stationary at 5% significance level. In conclusion, the series Ln y t and Ln x t have a single unit root or are integrated of degree one, I(). Moreover unit-root tests and summary statistics for z t are also reported in Table. These show that z t is indeed stationary and that, y t and x t are thus cointegrated, while coefficient restriction tests support the hypothesis that the cointegrating vector is (, -). Moreover, the summary statistics suggest that the error-correction term exhibits -9 -

10 many of the same characteristics as the individual series, that is, a small mean value which is dominated by a larger standard deviation, evidence of non-normality through (negative and positive) skwness and excess kurtosis, and significant first-order autocorrelation. It should be noted, however, that the values of these latter two statistics are much stronger in magnitude for the error-correction term than for the individual series. The empirical results of estimating the linear error-correction models outlined in section. are presented in Table. The lag lengths for the difference terms in the linear error-correction models were chosen on the basis of individual significance tests and information criteria AIC and BIC. The lag lengths chosen (The demand deposit lag length, The currency outside banks lag length) were (,) this indicates short-run bi-directional causality between demand deposit and currency outside banks. The error-correction term is a statistically significant, suggesting long-run bidirectional causality, and has a positive sign for demand deposit and negative sign for currency outside banks. This suggests that for positive deviations from equilibrium, where the demand deposit currency outside banks exceeds, z t- y t- βx t- > 0, the demand currency outside banks deposit falls and rises, implying meanreversion in both demand deposit and currency outside banks series. Table : Linear error-correction model demand deposit currency outside banks constant -4.48E-5*(0.0003) *(0.000) Demand deposit(-) -.349*(0.005) *(0.0309) currency outside banks(-) *(0.005) -0.59*(0.0435) currency outside banks (-) *(0.009) z t *(0.0303) -0.90*(0.045) Notes: For equation specification see Eq. (). The numbers in parentheses are standard errors. *Significant at the 5% level. We next test for the presence of non-linearity threshold effects using the procedure outlined in Section.. Preliminary tests to discriminate between LEC and EEC models based on LM-type test, as discussed above, were also conducted. both LEC and EEC models were estimated for y t and x t series. However, successful estimation, i.e., significant parameters and reasonable parameter magnitudes, was only achieved with the LEC model. -0 -

11 These test statistics (and associated probability values assuming d is unknown) are presented in Table 3. The results suggest that at all delay lags linearity is rejected (except for d = 4 for the currency outside banks). Using the lowest p-value rule, a delay parameter of one is selected for the two series.) Table 3: Non linear tests values of the delay parameter(d) demand deposit currency outside banks d= 3.69(.56e-5) 4.04(.37e-6) d= 4.0(0.005) 8.97(0.003) d=3.(0.030) 4.85(0.0453) d=4 7.53(0.059) 3.84 (0.4947) Notes: For test specification see Eqs. (4) and (5). Numbers in parentheses are p-values. However, while the findings of significant non-linearity and estimation of the appropriate non-linear form are in themselves important, a further issue concerns whether estimating a non-linear model improves the in-sample fit or the out-ofsample forecasting performance. Table 4 presents model-selection tests in particular, a serial correlation test and the AIC and BIC information criteria, and standard rootmean-squared-error statistics for the out- of-sample forecasts. The Q-test for serial correlation suggests no remaining serial correlation for any estimated model. The information criteria select the preferred model by minimizing the estimated variance, subject to penalty criteria for the number of parameters, where the penalty function is heavier for the BIC, and thus tends to select a more parsimonious model as discussed in footnote (pp.6).the results of this exercise show that AIC and BIC largely agree on the preferred model, with the linear model typically selected for demand deposit and the non-linear model preferred for currency outside banks. Table 4 also presents a standard forecasting performance, namely, the root-meansquared error, for the models estimated above, forecast over the period //00 0/9/003. For the two series, the forecasting performance of the non-linear model dominates that of the linear model. Notably, the linear forecast has the lower RMSE only for the demand deposit, while the currency outside banks series reports the lowest RMSE. Thus, both the demand deposit and currency outside banks are forecast more accurately by the LEC model. - -

12 Table 4:Model selection results Model / variables demand deposit currency outside banks Linear ECM - model: Q AIC * BIC -9.39* RMSE * 0.0 Non-LinearECM-model: Q AIC * BIC * RMSE 0.005* * Note: * Lowest test statistic. 4-. Testing some asymptotic properties in error correction models in small samples ( Monte-Carlo Simulation) : In order to test the asymptotic properties such as long-run relationships and short-run dynamics in linear error correction models in small samples (T =5; 50), the following simulation study is mainly,designed to shed some light on these properties based on the LR-type tests. Critical values for LR-type statistics are taken from the χ - distribution. Within single equation models, we investigate LR-type tests for the following restrictions: (i) The long-run parameter in the conditional model (6) ( This hold if H 05 : y t and x t are cointegrated and the cointegrating parameter β =-). (ii) The short-run dynamics within the conditional model (8)( This hold if H 06 : γ = 0) The simulated processes : In order to conduct this analysis, we consider the following stochastic processes: y x t = φ y t + φ x t + ε t, = x + ε t t t (0) It is well known that (9) specifying the (DGP) as defined in Engle et = is the identity matrix. Obviously al.(983).where εt ( ε t,εt ) ~N( 0,I ) and I x t is generated as a random walk without drift. Moreover, y t and x t are cointegrated with (9) - Herwartz & Neumann (005). - -

13 cointegrating parameter β = φ ( φ ) / terms of a conditional ECM like (6), with δ = 0; α = φ. By construction. The DGP in (9) may also be respecified in β and γ = φ -, =φ ( φ ) / Δ x does not respond to violations of the long-run equilibrium t relation implying that x t is weakly exogenous for inference on β or α.to characterize the dynamics of the DGP in (9) and(0) the following equivalent bivariate representation is useful : yt = xt 0 φ φ 0 0 yt xt ε + ε t t, () where + φ ε t ~ N( 0, ε ) and ε = φ φ. For values of φ smaller than but close to y t will show only weak error correcting effects. We simulated alternative processes allowing different degrees of error correcting dynamics by varying α between -0.0 and or, equivalently, by choosing the following values for φ = in (9): α + φ = 0.99, 0.95, 0.90, 0.85,0.80, 0.70, 0.60, 0.50, 0.0, 0.05, 0.03, 0.0 Since β = φ /( φ ) cointegrating parameter β = -., this mean that : y t and x t series are cointegrated and the Also for the issue of testing the null hypothesis of the short run dynamic (H 06 :γ = 0),the following augmented ECM is used : Δy t = δ + α ( y t + βxt ) + γ Δxt + γ Δy t + u t. () Monte-Carlo experiments are mostly performed with small samples (T = 5 and T =50). We generate each process Q = 000 times. Empirical size estimates are regarded to differ significantly at the 5%. -For testing the long-run relationship; consider the first testing H 05 : y t and x t are cointegrated and β =-. Based on P-values associated with the LR - test,we find that -3 -

14 LR statistics has slightly less than the χ (). This finding may be addressed to the fact that long-run bi- directional causality is observed for the relationship between the demand deposit and the currency outside banks series. - For testing the short-run dynamics (H 06 : γ = 0) ; we turn to hypothesis tests on parameters governing the short-run dynamics as H 06 : γ = 0 in the augmented regression model (). Opposite to testing H 05 inference on short-run dinamics appears to be unaffected by the error correction coefficient underlying the true DGP, in other words we note, the existence of short-run bi-directional causality between the demand deposit and the currency outside banks series (It should be noted that these latter findings insure the linear error correction model results discussed in section (4-)). 5. Summary and conclusion : This paper has investigated whether non-linear error-correction adjustment exists in the demand deposit and currency outside banks series. The non- linearity threshold error-correction model may be appropriate to analyzing the relationship between the demand deposit and the currency outside banks series, where deviation from fundamental equilibrium implies non-linear adjustment. Small deviations may be considered of little importance, whereas large deviations result in active relationship. Starting with linear error-correction models, which indicate both shortrun and long- run bi-directional causality and mean-reversion, we conduct nonlinearity tests on these models, which suggest that non-linear behavior is present specially in the currency outside banks series. Hence, we proceed to estimate whether non-linearity threshold models with both logistic (LEC) and exponential (EEC) models. Estimation of the latter, however, did not generate reasonable parameter magnitudes and significance. The estimation results for the non-linear, -4 -

15 logistic error- correction models (LEC) suggest non-linear adjustment to long-run equilibrium with behavior patterns depending upon the sign of the deviation from equilibrium. Finally the later findings obtained from analyzing some asymptotic properties such as long run relation-ships and short run dynamics in a bivariate linear error correction models based on simulated data assure that first, long-run bidirectional causality is observed for the relationship between the demand deposit and the currency outside banks series. Moreover the existence of short-run bi-directional causality between the demand deposit and the currency outside banks series. -5 -

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