The Narrow and Expanded Money Supply and Its Impact on Interest Rate and Product of the Private Sector in Jordan during the Period (1990–2019)

The study requires testing the various time series of the variables included in the models to ensure that they are stable at the level and for the unit root test (the static test). Therefore, the common test is Augmented Dickey-Fuller test (ADF) which is considered as development of the Dickey-Fuller test (DF) that takes into account the possibility of autocorrelation between random errors in unit root test equations.

The time series that has at least a unit root equal to one is considered as non-stationary time series. In case that the time series are not stationary at their levels, the first and the second differences are taken, and so on until the time series becomes stable. If the first difference series is stable, then the original series is co-integration of order one I(1). If the series is stable after obtaining the second difference, then the original series is co-integrated of order two I(2). If the original series is stable, it is integrated from the order zero I(0) meaning it is stationary at the level.

When using the Augmented Dickey-Fuller test to find out whether the study variables have a unit root (non-stationary), the following hypotheses are tested: The series has unit rootH0:α=0The series has no unit rootH1:α≠0 \matrix{ {{\rm{The}}\;{\rm{series}}\;{\rm{has}}\;{\rm{unit}}\;{\rm{root}}} \hfill & {{{\rm{H}}_0}:\alpha = 0} \hfill \cr {{\rm{The}}\;{\rm{series}}\;{\rm{has}}\;{\rm{no}}\;{\rm{unit}}\;{\rm{root}}} \hfill & {{{\rm{H}}_1}:\alpha \ne 0} \hfill \cr }

If the absolute value of test statistic is less than the critical value, then the null hypothesis (H₀) is accepted which means that there is a unit root (the time series is unstable) which requires removing the static in the series by treating the non-stationary of series variance as well as removing the general trend by using the difference method (Atiya, 2005).

The time series stationary was tested using Augmented Dickey-Fuller test on the basis of the level and on the basis of the first difference which takes the logarithmic form. If the trend (T) is not included, it is estimated according to the following Dickey-Fuller formula for the level (Maddala and Kim, 1990): Xt=μ+γXt−1+εt {X_t} = \mu + \gamma {X_{t - 1}} + {\varepsilon _t}

For differences, the mathematical formula is as follows: ΔXt=μ+γXt−1+∑i=1nϕΔXt−i+εt \Delta {X_t} = \mu + \gamma {X_{t - 1}} + \sum\limits_{i = 1}^n {\phi \Delta {X_{t - i}} + {\varepsilon _t}}

If the trend is included, the mathematical formula for the level is as follows: Xt=μ++βT+γXt−1+εt {X_t} = \mu + + \beta T + \gamma {X_{t - 1}} + {\varepsilon _t}

When taking the differences, the formula becomes ΔXt=μ++βT+γXt−1+∑i=1nϕΔXt−i+εt \Delta {X_t} = \mu + + \beta T + \gamma {X_{t - 1}} + \sum\limits_{i = 1}^n {\phi \Delta {X_{t - i}} + {\varepsilon _t}}

Table 1 shows the results of this test for all-time series of study variables.

According to Augmented Dickey Fuller test, the overview of different time series by level shows that all of their coefficients have a unit root, and they are not static at level 5% and level 10% level.

After taking the first difference for the time series, all the time series (with a trend and a segment) became static at a significance level of 5%.

The causality test is one of the important statistical tests in order to determine the direction of the relationship between economic variables. To verify the direction of the relationship between the variables of time series models, Granger has introduced Granger's causality which is used to test the direction of the relationship between variables and determine whether the causal relationship goes from X to Y or from Y to X.

The idea of Granger causality is based on the assumption that the past can cause the present, but the future cannot affect the present or the past. Granger believes that the problem of autocorrelation is one of the problems inherent in the analysis of time series which makes the process of determining the direction of causation difficult (Gujarati, 2003).

Moreover, if the hysteresis values of the independent variable X play an important role in explaining the change in the dependent variable Y, then in this case it can be said that there is a unidirectional causal which means that there is a causal relationship goes from X to Y (X Granger Causes Y) and also there can be a one-way causation from Y to X (Y Granger Causes X). But, in case that the hysteresis values of each independent and dependent variables affect the other, there is a bilateral causality and none of the causal relationships between the two variables may exist (Greene, 1993).

After conducting Grainger's causal test for the variables included in the analysis for each of product significance, narrow money supply significance and product significance in the presence of expanded money supply, the results are as shown in Tables 2 and 3 as follows.

After verifying that the time series are unstable at their levels, the co-integration test has been conducted between the study variables. The theoretical framework of this test showed that if the two series (X_t, Y_t) are unstable at the level, it is not necessary, when using them in the estimation of a relationship, to obtain a misleading regression, especially if they are co-integral.

Co-integration is defined in economic terms that any two variables have joint integration if they have a long-term equilibrium relationship (Gujarati, 2003). In other words, the fluctuations of one of these two variables lead to canceling the fluctuations in the second variable in a way that makes the ratio between their values stable over time, thus the time series becomes stable if they are taken as one group.

The occurrence of co-integration requires that the two series (X_t, Y_t) be complementary of first order of each. This in turn will make the residuals resulting from the estimation of relationship between them integrated from the order zero I(0). The comparison is made between the test statistic and the critical values.

If the test statistic is higher than the critical value, we reject the null hypothesis, which states that there is no co-integration. Therefore, the residual series is stable at the level. So, it can be said that the series X_t and Y_t are characterized by the co-integration property, and the estimated regression is not misleading (Atiya, 2005).

The results of co-integration test for the variables included in each of the significance of private sector product and narrow money supply and the significance of private product and expanded money supply indicate that there is a co-integration between these variables.

And by studying in Table 4, it shows that there are two complementary vectors of the first significance at significance level 5%, which means accepting the alternative hypothesis r = 1 and r = 2 and rejecting the null hypothesis r = 0, knowing that r expresses the number of integral vectors.

Table 5 shows that there are four integrative vectors in the second significance as the test statistic values are higher than the critical values at levels of significance 5% and 10%, and thus accepting the alternative hypothesis r = 1, r = 2, r = 3 and r = 4 versus rejecting the null hypotheses r = 0 and r < 1.

If there is a co-integration between the model variables, two methods can be used to estimate the economic significance: the first is the error correction model (ECM), which provides a methodology capable of examining the issue of non-stationarity of time series and misleading correlation, implicitly assuming the existence of a short-term relationship between the variables of the model. The second method is the fully modified ordinary least squares (FM-OLS) method of Phillips and Hansen (1990), which will be used in this study.

This method is used to obtain a better estimate than the estimation of the ordinary least squares method; in other words, this method is used to eliminate parameter errors that affect the approximate distribution of the ordinary least squares estimator by correcting that estimator to get rid of the problem of autocorrelation between random errors (serial correlation) and the problem of endogeneity between time series. To apply the fully modified ordinary least squares method to estimate long-term parameters, there is a condition that must be met, which is the existence of co-integration between the study variables of the first order I(1).

The general idea of the fully modified ordinary least squares method (FM-OLS) is based on estimating the relationship between an integrated dependent variable of the first order I(1) and a set of independent and integral variables of the first order I(1) according to the following formula (Maddala and Kim, 1998): Y1t=β′Y2t+u1tΔY2t=u2t \matrix{ {{Y_{1t}} = \beta '{Y_{2t}} + {u_{1t}}} \hfill \cr {\Delta {Y_{2t}} = {u_{2t}}} \hfill \cr } where:

Y_2t: a vector representing the explanatory variables of first-order co-integration;

We also assume that each element of Y_2t has only one unit root with no co-integration relationships between variables. And we assume that they are stationary with an arithmetic mean equal to zero, and the covariance matrix is equal to ∑=[σ11σ21′σ21∑22], ∑≻0 \sum = \left[ {\matrix{ {{\sigma _{11}}} & {\sigma _{21}^\prime} \cr {{\sigma _{21}}} & {{\sum _{22}}} \cr } } \right],\,\sum \succ 0

This matrix is called the long-term covariance matrix which is indicated by Ω and expressed as follows: Ω=limT→∞1T∑t=1T∑s=1TE(utus′) \Omega = \mathop {\lim }\limits_{T \to \infty } {1 \over T}\sum\limits_{t = 1}^T {\sum\limits_{s = 1}^T {E\left( {{u_t}u_s^\prime} \right)} }

It is a set of forward and backward covariances and joint covariances; in other words, it contains the covariance and joint covariance between u_t and u_s, so that Ω=Σ+Λ+Λ′ \Omega = \Sigma + \Lambda + \Lambda ' since

Σ: covariance matrix Σ=E(u0u0′) \Sigma = E\left( {{u_0}u_0^\prime} \right)

The fully modified ordinary least squares method is one of the methods used to address the two problems of overlapping dependence of the time series of variables and the autocorrelation of errors. This method corrects the parameter β^ \hat \beta estimated by the ordinary least squares method, as follows: 1)

Correct Y_1t so that it becomes y^1t+=y1t−ω^12Ω^11Δy2t {\hat y_{1t}}^ + = {y_{1t}} - {\hat \omega _{12}}{\hat \Omega _{11}}\Delta {y_{2t}} And correct the random error u_t so that it becomes u^1t+=u1t−ω^12Ω^11Δy2t {\hat u_{1t}}^ + = {u_{1t}} - {\hat \omega _{12}}{\hat \Omega _{11}}\Delta {y_{2t}} This is to solve the problem of overlapping dependence of time series for independent variables.

Therefore, the fully modified ordinary least squares method combines these two corrections to estimate the ordinary least squares estimator β^ \hat \beta according to the following formula: β^=(Y2′Y2)−1(Y2′y^1+−Tδ^+) \hat \beta = {\left( {Y_2^\prime{Y_2}} \right)^{ - 1}}\left( {Y_2^\prime\hat y_1^ + - T{{\hat \delta }^ + }} \right)

Thus, the fully modified ordinary least squares method is similar to the unit root test according to the Philips-Peron (PP) method, as it starts with the estimation according to the ordinary least squares method, and then does the correction process for the estimated parameters in order to address the problem of interdependence and the problem of autocorrelation of errors (Maddala and Kim, 1998).

It should be noted that there are studies conducted using the Monte Carlo method of researcher Johansen. These studies have proven that the FM-OLS method is superior to it because it uses Single Equation Estimation Technique while Johansen's method uses Multivariate Co-integration Technique (Athukorala and Riedel, 1996).

The fully modified ordinary squares method was used, which is one of the methods of co-integration analysis to estimate the parameters of the product and money supply significance as it works to correct the two problems of interdependence of data and bias of parameters, and the following results were concluded.