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 |

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