Augmented Dickey Fuller Adf Test Finance Essay

Researchs on the causal relationship between equity monetary values and exchange rates have been conducted with assorted econometric methods. In this survey, I employ the vector autoregressive theoretical account and dynamic Granger ( 1969 ) causality trial to analyze the relationship between the variables under survey. Empirical surveies which are premised on clip series informations assume that the implicit in clip series is stationary. On the contrary, many empirical surveies have shown that this premise is non ever true and that a important figure of clip series variables are non-stationary ( Engle and Granger, 1987 ) . Consequently, using a non-stationary clip series informations in a arrested development analysis may ensue in specious consequences ( Granger and Newbold ( 1974 ) ) . Therefore, shiping on surveies affecting clip series informations necessitates that stationary trial is conducted to set up the implicit in procedure of the information series.

3.1 Stationary Trial

A information generating procedure is considered stationary if it has time-invariant first and 2nd minutes, and the covariance of two clip periods is changeless notwithstanding which clip periods are used and the distance between them, Gujarati ( 2003 ) . The procedure is said to be decrepit stationary if the two first conditions are fulfilled but the covariance between two clip periods depends on the distance between the clip periods, but non on when it is calculated. If the procedure is stationary around a tendency, it is said to be trend-stationary. There are a assortment of unit root trials used in the econometric literature chiefly Augmented Dickey-Fuller ( ADF ) , Dickey-Fuller, Phillip-Perron, Ng-Perron trials, etc to look into whether the clip series informations used in a survey are stationary or non. I employ the Augmented Dickey-Fuller to analyze the stationarity of the variables.

3.1.1 Augmented Dickey-Fuller ( ADF ) Trial

The ADF theoretical account tests the void hypothesis that there is unit root, against the alternate hypothesis that there is no unit root in the arrested development. The ADF trial is obtained by the undermentioned arrested development:

( 1 )

where represents the variable that we are analyzing its belongingss, is the difference operator, , and are the coefficients to be estimated, P is the chosen slowdown length, T is the clip tendency, and T is the white-noise mistake term. has a stochastic tendency under the void hypothesis but under the alternate hypothesis is stationary. By and large, the slowdown length for carry oning the ADF trial is unknown but can be estimated utilizing information standards such as the Akaike Information Criterion ( AIC ) and Bayesian Information Criterion ( BIC ) applied to the arrested developments of the signifier in equation ( 1 ) . If the Data Generating Process ( DGP ) is stationary in the information series at degrees, so it will be concluded to be integrated of order nothing, I ( 0 ) . On the contrary, it is non ever the instance and the implicit in procedure of the informations series may be non-stationary. In consequence, the original series demand to be transformed into a stationary province by taking difference ( vitamin D ) times. If after taking first difference of the series, it is found out that they are all stationary so we can reason that the DGP is integrated at order one, I ( 1 ) . Furthermore, if the original series used in the survey are found out to be integrated of the same order, it is utile to prove for cointegration relationship between the incorporate variables.

3.1.2 Cointegration Test

It is by and large accepted that arrested development which involves non-stationary clip series will take to specious consequences. However, Engle-Granger ( 1987 ) proposed that a additive combination of these non-stationary series may be stationary in which instance we can state that the series are cointegrated. To calculate the Engle-Granger trial, allow the vector denote the tth observation on N clip series, each of which is known to be I ( 1 ) . If these times series are cointegrated, there exists a vector such that the stochastic procedure with observation is I ( 0 ) . However, if they are non cointegrated, there will be no vector with this belongings, and any additive combination of y1 through yN and a invariable will still be I ( 1 ) . The cointegration arrested development is estimated as follows:

Yt1= +

for a sample of size T+1. With regard to this arrested development, it is assumed that all the variables are I ( 1 ) and might cointegrate to organize a stationary relationship, and therefore will ensue in a stationary residuary term, . The void hypothesis of non-cointegration is that is I ( 1 ) . Unit root trial is conducted on the remainders to happen out whether they are stationary or otherwise. To this terminal, the ADF trial is employed to carry on the unit root trial. If the remainders are stationary I ( 0 ) , so one rejects the void hypothesis of no-cointegration. However, if they are non-stationary I ( 1 ) , so one accepts the void hypothesis of no-cointegration.

3.2 Vector Autoregressive ( VAR ) Model

A vector autoreggresion is a set of n series of arrested developments in which the regressors are lagged values of all the n series. The implicit in premise of the theoretical account is that all variables are endogenous a priori, and allowance is made for rich kineticss. VAR theoretical accounts offer some degree of flexibleness and hence easy to utilize for analyzing multiple clip series.This is against the background that one needs non to stipulate which variables are exogenic or endogenous. But there are still some failings. First, it is difficult to see which variables have important consequence on the dependant variable. Second, VAR theoretical accounts require that all the information series in the system should be stationary. However, most fiscal series have a characteristic of the non-stationarity. In instance the variables are found non to be stationary at degrees, so harmonizing to Granger ( 1969 ) , it is more appropriate to gauge vector mistake rectification theoretical account if they become stationary after taking difference of the variables. The vector mistake rectification theoretical account is discussed in the subsequent subdivision. The simplest signifier of the VAR is the bivariate theoretical account which involves two variables. The bivariate theoretical account can by and large be estimated as follows:

Where A°A?’A?it is a white noise term with E ( A°A?’A?it ) = 0, E ( u1tu2t ) =0.

3.3 The Granger Causality Test

Harmonizing to Granger ( 1969 ) a variable Ten could be defined as causal to a clip series variable Yif the former aid to better the prognosis of the latter. Therefore, X does non Granger-cause Y if

Pr ( | ) = Pr ( | ) ( 3 )

where Pr ( . ) is the conditional chance, is the information set at clip T on past values of Y and is the information set incorporating values of both Xand Yup to clip point T

If the variables are found non to be cointegrated, so the following VAR will be estimated and the Granger causality trial is accordingly conducted:

( 4 ) ( 5 )

where SI is the stock market monetary value in footings ; ER is the exchange rate of the Ghana cedi to the US dollar andare uncorrelated white noise footings, ln represents the natural log, I ” difference operator and T denotes the clip period. If the lagged coefficient vector of equation ( 5 ) is non important so the consequences imply that there is unidirectional causality from exchange rate to stock monetary value returns. However, if the lagged coefficient vector in equation ( 5 ) is statistically important but the lagged coefficient vector in equation ( 4 ) is non statistically important so the consequences imply that there is unidirectional causality from stock monetary values returns to interchange rate returns. Furthermore, if the lagged coefficient vectors of both equations ( 4 and 5 ) are statistically important so the consequences imply that there is bidirectional causality from the stock returns and exchange rate returns. Finally, if both lagged coefficient vectors are statistically undistinguished, so this implies that there is no causality between these variables.

3.4 Vector Error Correction Model ( VECM )

Harmonizing to Engle and Granger ( 1987 ) , the VECM is a preferred theoretical account to the VAR ( equations 4 and 5 ) if it is found that there is no-cointegration relation between and or among the information series. The VECM discriminates between both the dynamic short-term and long-term Granger causality. The VECM equations are written as follows:

where SI is the stock monetary value in footings ; ER is the exchange rate, is the mistake rectification term lagged one period ; and are uncorrelated white noise footings. The mistake rectification term ( ) is derived from the long tally cointegration relationship between the variables. The estimations of the mistake rectification term of ( ) besides shows how much of the divergence from the equilibrium province is corrected in each short period. To happen out the presence of long tally causality between the two informations series, we will prove for the significance of the coefficient of the mistake rectification term in equations ( 6 and 7 ) by using the t-test. Finally the Wald or F-statistic is used to prove for the joint significance of both the mistake rectification term and the assorted synergistic footings in equations ( 6 and 7 ) . If the lagged coefficient vector of equation ( 6 ) is statistically important but the lagged coefficient of vector of equation ( 7 ) is non important so the consequences imply that there is unidirectional causality from exchange rate to stock monetary value returns. However, if the lagged coefficient vector in equation ( 7 ) is statistically important but the lagged coefficient vector in equation ( 6 ) is non statistically important so the consequences imply that there is unidirectional causality from stock monetary values returns to interchange rate returns. Furthermore, if the lagged coefficient vectors of both equations ( 6 and 7 ) are statistically important so the consequences imply that there is bidirectional causality from the stock returns and exchange rate returns. Finally, if both lagged coefficient vectors are statistically undistinguished, so this implies that there is no causality between these variables.

3.4 Lag Length choice Criteria

To gauge the VAR/VECM theoretical account requires taking the slowdown length that reduces the information loss. Therefore, taking the slowdown length involves neutralizing the tradeoff between adding more slowdowns against the fringy benefit of extra appraisal uncertainness. Therefore, excessively many slowdowns included in the theoretical account will take to extra appraisal mistakes and pieces excessively few slowdowns may go forth out potentially valuable information. To incorporate this job, there are so many theoretical accounts to utilize to choose the slowdown order, viz. the Akaike Information ( AIC ) and Bayes Information Criteria ( BIC ) . I used the BIC to find the slowdown order to the estimation theoretical account and therefore I will discourse it briefly. The BIC and AIC are expressed as follows:

( 8 )

( 9 )

where SSR ( P ) is the amount of squared remainders of the estimated AR ( P ) . The BIC calculator of, is the value that minimizes BIC ( P ) out of the scope of slowdowns available. The SSR decreases as more slowdowns are introduced, nevertheless the 2nd term additions as more slowdowns are introduced.

Furthermore, the sum of addition in the 2nd term of the AIC is comparatively smaller to that of the BIC. Thus, the BIC awards more punishment factor relation to the AIC. This implies that BIC gives a consistent estimation of the true slowdown length unlike the BIC. This makes the BIC preferred to AIC which tends to overrate the slowdown order with positive chance. Therefore, the 2nd term of the AIC is smaller compared to the BIC.

3.5 Test for Break Dates

To prove for structural interruption ( s ) in the arrested development coefficients, I estimate an autoregressive distributed slowdown ( ADL ) with silent person variables to stand for the periods before and after the redenomination of the cedi. Furthermore, to take the appropriate slowdown length for both the dependant and independent variables to include in the ADL, I estimate the arrested development equations with different slowdown lengths and compare the ensuing BICs. In consequence, the slowdown length that resulted in the lowest BIC is chosen to gauge the ADL and so the structural interruption trial is conducted.

The ADL was estimated as below:

where

SIt = stock monetary value returns

ERt = exchange rate returns

Dt = Dummy variable where Dt = 1 if t aaˆ°A? 3 July, 2007 ; Dt = 0 if t aaˆ°A¤ 3 July, 2007

aE†aˆ = difference operator

T = clip period ; vitamin D, are the coefficients of the parametric quantities

Chow ( 1960 ) theoretical account trials for structural interruption in which instance the interruption day of the months must be known a priori and the determination is made on the F-statistic that tests the void hypothesis of no interruption ; against the alternate hypothesis that at least one of vitamin D is nonzero. Therefore, in instance of the Chow ( 1960 ) test the research worker has to pick an arbitrary interruption day of the month or pick a known day of the month based on the characteristic of the information series. In consequence, the consequences can be extremely sensitive to these arbitrary picks and as the true interruption day of the month can be missed. However, in this survey, the interruption day of the month is identified by the redenomination of the cedi.

3.6 The CUSUM of Squares Test

In an effort to prove for the stability of the discrepancy, I employ the CUSUM of Square trial ( Brown, Durbin, and Evans, 1975 ) . This trial is chiefly based on the square of the remainders on the secret plan of the measures. This trial involves pulling a brace of critical lines on the diagram which is parallel to the average value line so that the chance that the sample way crosses one or both critical lines is the significance degree. If the sample way stays between the brace of lines without traversing any of the two lines, so one can reason that the discrepancy is changeless over the period. However, motion outside of the critical lines implies parameter or discrepancy instability. The CUSUM of squares test is based on the trial statistic:

( 11 )

The average value of under the hypothesis of parametric quantity stability is:

which goes from nothing at t = K to integrity at t = T.