Case U RT1: Unit Root Tests – More Examples
Case U RT1: Unit Root Tests – More Examples
Example 1: Case AR Revisited
Variable: Weekly Movement of TF Mayonnaise (32 oz) in # of Cases
Y = log transformed.
1. Timeplot
2. Unit Root Test Using τμ
(a) Regression output
|Dependent Variable: D(Y) |
|Method: Least Squares |
|Sample(adjusted): 2 108 |
|Included observations: 107 after adjusting endpoints |
| | | | | |
|Variable |Coefficient |Std. Error |t-Statistic |Prob. |
| | | | | |
|C |4.262106 |0.610344 |6.983121 |0.0000 |
| Y(-1) |-0.630127 |0.090107 |-6.993069 |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 |
|Durbin-Watson stat |2.091670 | Prob(F-statistic) |0.000000 |
| | | | | |
[pic]
(b) Unit Root Test output
|ADF Test Statistic |-6.993069 | 1% Critical Value* |-3.4922 |
| | | 5% Critical Value |-2.8884 |
| | | 10% Critical Value |-2.5809 |
| | | | | |
|*MacKinnon critical values for rejection of hypothesis of a unit root. |
| | | | | |
| | | | | |
|Augmented Dickey-Fuller Test Equation |
|Dependent Variable: D(Y) |
|Method: Least Squares |
|Sample(adjusted): 2 108 |
|Included observations: 107 after adjusting endpoints |
| | | | | |
|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 |
|Durbin-Watson stat |2.091670 | Prob(F-statistic) |0.000000 |
| | | | | |
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 ττ
(a) Regression [pic]
|Dependent Variable: D(Y) | | |
|Sample (adjusted): 1960Q2 2004Q4 | | |
|Included observations: 179 after adjustments | |
| | | | | |
| | | | | |
|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 |
| | | | | |
| | | | | |
|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 |
| | | | | |
| | | | | |
| | | | | |
(b) Unit Root Test
|Null Hypothesis: Y has a unit root | |
|Exogenous: Constant, Linear Trend | |
|Lag Length: 0 (Fixed) | | |
| | | | | |
| | | | | |
| | | |t-Statistic | Prob.* |
| | | | | |
| | | | | |
|Augmented Dickey-Fuller test statistic |-2.190338 | 0.4916 |
|Test critical values: |1% level | |-4.010143 | |
| |5% level | |-3.435125 | |
| |10% level | |-3.141565 | |
| | | | | |
| | | | | |
|*MacKinnon (1996) one-sided p-values. | |
| | | | | |
| | | | | |
|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 | |
| | | | | | |
| | | | | | |
| | | | | | |
Appendix
1. Three types of Unit Root Test
τ
τμ
ττ
For each test:
(a) Understand both H0 and H1.
(b) Learn how to implement:
To be able to specify the correct regression
To augment the regression when ε is not WN. (DW could tell)
Add ΔY(t-1) , ΔY(t-2), etc.
2. How To Select Proper Unit Root Test?
Personal opinion
a) Note the model of the alternative H1. -> τ test can be easily rejected.
b) If the first difference has clearly non-zero mean, then choose ττ test.
If the mean of the first difference is close to 0, then try both τμ and ττ tests.
The following is not complete
1) The nature of the variable precludes the existence of a unit root.
2) In the absence of a unit root, the most plausible process is AR(2) with non-zero mean.
Model: [pic]
Rewriting for test:
[pic]
= [pic]
= [pic]
[pic]
Hypotheses:
H0: [pic] (AR(1) in the first difference)
H1: [pic] (Stationary AR(2) process)
Test Using Regression:
|Dependent Variable: D(Y) |
|Method: Least Squares |
|Sample(adjusted): 3 108 |
|Included observations: 106 after adjusting endpoints |
|Variable |Coefficient |Std. Error |t-Statistic |Prob. |
|C |1.308143 |0.366499 |3.569297 |0.0005 |
|Y(-1) |-0.313898 |0.087968 |-3.568333 |0.0005 |
|D(Y(-1)) |-0.211258 |0.097010 |-2.177693 |0.0317 |
|R-squared |0.233052 | Mean dependent var |0.003104 |
|Adjusted R-squared |0.218160 | S.D. dependent var |0.362739 |
|S.E. of regression |0.320740 | Akaike info criterion |0.591520 |
|Sum squared resid |10.59602 | Schwarz criterion |0.666901 |
|Log likelihood |-28.35057 | F-statistic |15.64928 |
|Durbin-Watson stat |1.993541 | Prob(F-statistic) |0.000001 |
Test Using Unit Root Test Routine
|ADF Test Statistic |-3.568333 | 1% Critical Value* |-3.4928 |
| | | 5% Critical Value |-2.8887 |
| | | 10% Critical Value |-2.5811 |
|*MacKinnon critical values for rejection of hypothesis of a unit root. |
| | | | | |
| | | | | |
|Augmented Dickey-Fuller Test Equation |
|Dependent Variable: D(Y) |
|Method: Least Squares |
|Sample(adjusted): 3 108 |
|Included observations: 106 after adjusting endpoints |
|Variable |Coefficient |Std. Error |t-Statistic |Prob. |
|Y(-1) |-0.313898 |0.087968 |-3.568333 |0.0005 |
|D(Y(-1)) |-0.211258 |0.097010 |-2.177693 |0.0317 |
|C |1.308143 |0.366499 |3.569297 |0.0005 |
|R-squared |0.233052 | Mean dependent var |0.003104 |
|Adjusted R-squared |0.218160 | S.D. dependent var |0.362739 |
|S.E. of regression |0.320740 | Akaike info criterion |0.591520 |
|Sum squared resid |10.59602 | Schwarz criterion |0.666901 |
|Log likelihood |-28.35057 | F-statistic |15.64928 |
|Durbin-Watson stat |1.993541 | Prob(F-statistic) |0.000001 |
1)
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- test case template centers for disease control and
- test plan template centers for disease control and
- test plan unit and integration template
- test plan template strongqa
- case u rt1 unit root tests more examples
- validation verification and testing plan template
- alg2x unit 2
- test review unit 6 civil liberties civil rights
- unit test over the constitution
- sdm test plan template
Related searches
- business case examples pdf
- business case study examples pdf
- examples of unit vectors
- psychology case study examples pdf
- u s army unit patches
- examples of psychological case study
- examples of case management goals
- examples of business case study
- clinical case study examples psychology
- examples of clinical case studies
- case study examples in psychology
- thematic unit examples elementary