results support each other. After the integration level of time series is found, the cointegration analysis must be
applied.
The basis of cointegration idea is that a linear combination of time series might be stationary even if the series
themselves contain unit root. If the time series are cointegrated, that means there is a long run equilibrium
relationship between series. The Johansen (1988) procedure is one of the most common tools in the estimation of
cointegrated systems. Johansen procedure has some advantages over the single equation approaches. In the system
estimation of Johansen method, the normalization problem does not appear and also the number of cointegrating
vectors is not fixed a priori but is determined in the course of estimation.
Johansen procedure applies maximum likelihood to the VAR (Vector autoregressive) model, assuming the errors
are Gaussian. The procedure leads to two test statistics for cointegration. The first is the Trace test, which tests the
hypothesis that there are at most r cointegrating vectors. The second called the Maximum Eigenvalue Test, which
tests the hypothesis that there are r+1 cointegrating vectors versus the hypothesis that there are r cointegrating
vectors. The results of the ADF, KPSS and Johansen test are reported in the Results section.
2.2. Results
The results of ADF test on our variables real GNP, real income tax revenue and the real taxable income are
reported in Table 2.
Table 2: Augmented Dickey Fuller Unit Root Test Results
|
Variables_____________ |
Level / First Difference____________ |
Constant________________ |
Constant & Trend______________ |
|
Ln (realT)____________ |
Level___________________________ |
-1,01957 [0]_____________ |
-2,1253[0]________________________ |
|
First Difference____________________ |
-4,8978[0]*_____________ |
-4,7957[0]*_____________________ | |
|
ln(realTI)______________ |
Level___________________________ |
-0,5850[1]_______________ |
-1,2540[1]________________________ |
|
First Difference____________________ |
-8,1190[0]*_____________ |
-8,0003 [0]*______________________ | |
|
Ln(realY)_____________ |
Level___________________________ |
-0,0177[1]_______________ |
-2,8090 [0]________________________ |
|
First Difference____________________ |
-7,2666 [0]*______________ |
-7,1447[0]*_____________________ |
Note: In the table, the test statistics are shown from the model including constant term and from the model including constant
term and trend. * denotes the rejection of null hypothesis at %1 significance level. That means the null hypothesis about the
series have a unit root is rejected. The numbers in parenthesis near estimated coefficients are the optimal lag numbers
determined by the SIC (Schwarz Information Criteria). Test statistics are compared with the critical values of
MacKinnon(1991)
Table 3: KPSS* Unit Root Test Results
|
Variables_____________ |
Level / First Difference____________ |
Constant________________ |
Constant&Trend________________ |
|
Ln (realT)____________ |
Level___________________________ |
0,6679 [4]**____________ |
0,12055[4]**____________________ |
|
First Difference____________________ |
0,0604[1]_______________ |
-0,0632[1]_________________________ | |
|
ln(realTI)______________ |
Level___________________________ |
0,7137[4]**_____________ |
0,886164[1]*____________________ |
|
First Difference____________________ |
0,1477[1]_______________ |
-0,07742 [1]________________________ | |
|
Ln(realY)_____________ |
Level___________________________ |
0,7242 [4]**_____________ |
0,2982 [0]*_______________________ |
|
First Difference____________________ |
0,04138[0]______________ |
0,041[0]_____________________________ |
7
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