ADF Test Statistic
|
-6.993069
|
1% Critical Value*
|
-3.4922
|
|
|
5% Critical Value
|
-2.8884
|
|
|
10% Critical Value
|
-2.5809
|
|
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|
*MacKinnon critical values for rejection of hypothesis of a unit root.
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Augmented Dickey-Fuller Test Equation
|
Dependent Variable: D(Y)
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Method: Least Squares
|
Sample(adjusted): 2 108
|
Included observations: 107 after adjusting endpoints
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|
Variable
|
Coefficient
|
Std. Error
|
t-Statistic
|
Prob.
|
|
|
|
|
|
Y(-1)
|
-0.630127
|
0.090107
|
-6.993069
|
0.0000
|
C
|
4.262106
|
0.610344
|
6.983121
|
0.0000
|
|
|
|
|
|
R-squared
|
0.317752
|
Mean dependent var
|
-0.000982
|
Adjusted R-squared
|
0.311255
|
S.D. dependent var
|
0.371408
|
S.E. of regression
|
0.308234
|
Akaike info criterion
|
0.502601
|
Sum squared resid
|
9.975877
|
Schwarz criterion
|
0.552561
|
Log likelihood
|
-24.88917
|
F-statistic
|
48.90301
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Durbin-Watson stat
|
2.091670
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Prob(F-statistic)
|
0.000000
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Example 2: Case ST Revisited
Variable: Gross Domestic Product in billions of dollars, seasonally adjusted annual rate, quarterly. Divided by GDP Implicit Price Deflator, Index 2000=100
Range: 1960:1 to 2004:4
1. Ttimeplot of log
2. Unit Root Test Using
(a) Regression 
Dependent Variable: D(Y)
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Sample (adjusted): 1960Q2 2004Q4
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Included observations: 179 after adjustments
|
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Variable
|
Coefficient
|
Std. Error
|
t-Statistic
|
Prob.
|
|
|
|
|
|
|
|
|
|
|
C
|
0.321734
|
0.142301
|
2.260937
|
0.0250
|
Y(-1)
|
-0.041774
|
0.019072
|
-2.190338
|
0.0298
|
QUARTER
|
0.000319
|
0.000152
|
2.098769
|
0.0373
|
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|
|
|
|
|
|
|
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R-squared
|
0.032387
|
Mean dependent var
|
0.008236
|
Adjusted R-squared
|
0.021392
|
S.D. dependent var
|
0.008588
|
S.E. of regression
|
0.008495
|
Akaike info criterion
|
-6.681989
|
Sum squared resid
|
0.012702
|
Schwarz criterion
|
-6.628569
|
Log likelihood
|
601.0380
|
F-statistic
|
2.945480
|
Durbin-Watson stat
|
1.386973
|
Prob(F-statistic)
|
0.055174
|
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(b) Unit Root Test
-
Null Hypothesis: Y has a unit root
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Exogenous: Constant, Linear Trend
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Lag Length: 0 (Fixed)
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t-Statistic
|
Prob.*
|
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|
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Augmented Dickey-Fuller test statistic
|
-2.190338
|
0.4916
|
Test critical values:
|
1% level
|
|
-4.010143
|
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5% level
|
|
-3.435125
|
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10% level
|
|
-3.141565
|
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*MacKinnon (1996) one-sided p-values.
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|
Augmented Dickey-Fuller Test Equation
|
|
Dependent Variable: D(Y)
|
|
|
Method: Least Squares
|
|
|
Date: 05/06/05 Time: 11:27
|
|
|
Sample (adjusted): 1960Q2 2004Q4
|
|
Included observations: 179 after adjustments
|
|
|
|
|
|
|
|
|
|
|
|
Variable
|
Coefficient
|
Std. Error
|
t-Statistic
|
Prob.
|
|
|
|
|
|
|
|
|
|
|
Y(-1)
|
-0.041774
|
0.019072
|
-2.190338
|
0.0298
|
C
|
0.338664
|
0.150335
|
2.252734
|
0.0255
|
@TREND(1960Q1)
|
0.000319
|
0.000152
|
2.098769
|
0.0373
|
|
|
|
|
|
|
|
|
|
|
R-squared
|
0.032387
|
Mean dependent var
|
0.008236
|
Adjusted R-squared
|
0.021392
|
S.D. dependent var
|
0.008588
|
S.E. of regression
|
0.008495
|
Akaike info criterion
|
-6.681989
|
Sum squared resid
|
0.012702
|
Schwarz criterion
|
-6.628569
|
Log likelihood
|
601.0380
|
F-statistic
|
2.945480
|
Durbin-Watson stat
|
1.386973
|
Prob(F-statistic)
|
0.055174
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
DW indicates autocorrelation in the residual. This could invalidate the test.
(c) Augmented Dickey – Fuller Test
For making the residual of the regression WN.
-
Null Hypothesis: Y has a unit root
|
|
|
Exogenous: Constant, Linear Trend
|
|
|
Lag Length: 1 (Fixed)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
t-Statistic
|
Prob.*
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Augmented Dickey-Fuller test statistic
|
-2.991965
|
0.1374
|
|
Test critical values:
|
1% level
|
|
-4.010440
|
|
|
|
5% level
|
|
-3.435269
|
|
|
|
10% level
|
|
-3.141649
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
*MacKinnon (1996) one-sided p-values.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Augmented Dickey-Fuller Test Equation
|
|
|
Dependent Variable: D(Y)
|
|
|
|
Method: Least Squares
|
|
|
|
Date: 05/06/05 Time: 11:31
|
|
|
|
Sample (adjusted): 1960Q3 2004Q4
|
|
|
Included observations: 178 after adjustments
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Variable
|
Coefficient
|
Std. Error
|
t-Statistic
|
Prob.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Y(-1)
|
-0.054580
|
0.018242
|
-2.991965
|
0.0032
|
|
D(Y(-1))
|
0.308088
|
0.070503
|
4.369858
|
0.0000
|
|
C
|
0.437102
|
0.143758
|
3.040543
|
0.0027
|
|
@TREND(1960Q1)
|
0.000422
|
0.000145
|
2.901395
|
0.0042
|
|
|
|
|
|
|
|
|
|
|
|
|
|
R-squared
|
0.136809
|
Mean dependent var
|
0.008310
|
|
Adjusted R-squared
|
0.121926
|
S.D. dependent var
|
0.008554
|
|
S.E. of regression
|
0.008015
|
Akaike info criterion
|
-6.792672
|
|
Sum squared resid
|
0.011179
|
Schwarz criterion
|
-6.721171
|
|
Log likelihood
|
608.5478
|
F-statistic
|
9.192547
|
|
Durbin-Watson stat
|
2.146510
|
Prob(F-statistic)
|
0.000011
|
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