Dickey-Fuller Unit Root Test
First, download the excel formatted data file named "US_cpi_data" from the "Sample Data" of Econ3600 homepage.
Second, open the EVIEWS program and click "file", "new", and "workfile", then the "Workfile Range" window will be appeared as following. Choose to highlight "Monthly" (because the series is a monthly sample data) and types the dates for "Start date" and "End date" in the dialogue box to specific the starting and ending of the sample.
After click "OK" and get this "Workfile" window.
Next, click the "Process", "Import", "Read Text-Lotus-Excel", then get the following output. And you need to type the names of series and the necessary information of the upper-left cell in the dialogue boxes as follow. Notice the "B9" in the dialogue box represents the sample data starting at the upper-left cell of "B9" in the Excel file.
Then click "OK" again and get this output in "workfile" window:
Now, you are ready to carry out the Dickey-Fuller (DF) Unit Root Test for any type of time series data. Pick one sample series among the other nine series in this "Workfile", simply choose "cpi" to test whether it is a stationary series or not. (Students are encouraged to test the other eight time series on their own.)
Let cpit = Yt , the DF Unit Root Test are based on the following three regression forms:
I. Without Constant and Trend
2. With Constant
3. With Constant and Trend
The hypothesis is:
If t* > ADF crtitical value, ==> not reject null hypothesis, i.e., unit root exists.
If t* < ADF critical value, ==> reject null hypothesis, i.e., unit root does not exist.
Run each regression equation separately:
I. For testing the first regression equation , the steps are as follows:
Step 1. Double click the item "cpi" in the workfile and get
Important: To get a rough idea of a time series whether it is stationary or not, simply click "View", "Line Graph" and plots the series as follow. The series seems as a non-stationary data since it is increased upward as time changes.
Step 2. Click "View", "Unit Root Test" and get the following windows, and then choose "Augmented Dickey-Fuller", "Level", "None" and type "0" in the "Unit Root Test" dialogue box as following:
After click the "OK" and get the regression result:
Step 3. Since the computed ADF test-statistics (17.62461) is greater than the critical values --"tau"( -2.5742, -1.9410 and -1.6164 at 1%, 5% and 10% significant level, respectively), we cannot conclude to reject Ho. That means the CPI series has an unit root problem and the CPI series is a non-stationary series. (However, this result is not reliable because the the Durbin-Watson statistics is very small that means the CPI series may has autocorrelation problem.)
II. For testing the second regression equation , the steps are similar as previous to click for the "Unit Root Test" and choose "Augmented Dickey-Fuller", "Level", "0" and "Intercept" in the dialogue box as following:
After click "OK" and get the result:
The computed ADF test-statistic (-3.515) is smaller than the critical values - "tau" ( -2.5742, -2.873, -3.4592 at 10%, 5%, 1% significant level, respectively), therefore we can reject Ho. It means the CPI series doesn't has an unit root problem and the CPI series is a stationary series at 1%, 10% and 5% significant level. (Again, this result is also not reliable because the the Durbin-Watson statistics is still very small that means the CPI series may has autocorrelation problem.)
III. For testing the third regression equation , again, the steps are similar as previous to click for the "Unit Root Test" and choose "Augmented Dickey-Fuller", "Level", "0" and "Trend and Intercept" in the dialogue box as following: (Notice: now changing from "Intercept" to "Trend and Intercept"):
After click "OK" and get the following result:
Again, the computed ADF test-statistic (-2.569347) is greater than the critical values - "tau" (-3.9996, -3.4298, -3.1381 at 1%, 5% and 10% significant level, respectively), thus we cannot conclude to reject the Ho. That means the CPI series is a non-stationary series. (Again, because the Durbin-Watson statistics is not significantly to reject the autocorrelation, so we still cannot rely on the simple DF unit root test).
According to the above three separate regressions, there is confuse to determine whether the CPI is stationarity or not? In order to confirm, we need further to adopt the
Augmented Dickey Fuller (ADF) Test, the regression equation is based:
Same steps as above except change the "lag difference" from "0" to "1", such as follow (Notice: of course you can try and add more lag terms)
By clicking "OK", the result appear as
Now, it is clearly we have passed the Durbin-Waston Test and we can trust the regression result. Since the computed absolute t-statistic is smaller than the absolute critical "tau" value, thus we cannot reject the Ho. That means the "cpi" is a non-stationary time series which consistent with our priori expectation from the line graph.
How can we transform the time series data from non-stationary to stationary?
For data with deterministic trend, we can use either Trend-Stationary Process (TSP) or Difference-Stationary Process (DSP).
For data with stochastic trend, we can use DSP rather than TSP (i.e. just as our demonstration case here). Since we have done the TSP before, but it does no succeed. So, we can try to use DSP to get a stationary time series. The regression equation is:
Same step as above except changing the chosen items of "1st difference" as following:
By clicking "OK", the result is:
Now the absolute computed ADF test-statistic (-9.940211) is smaller than the critical "tau", thus we can reject the Ho. That means the 1st-difference of "cpi" becomes stationarity.
We also can double check the same conclusion from plotting the line graph of the 1st-difference of "cpi". After running the above regression, we choose the "GENR" command to generate the first difference series of "cpi", named "dcpi", after type the "dcpi = d(cpi)" and click "OK", we will have the item "dcpi" will be added into the workfile. By double click the "dcpi", we can see the series data. Next, choose "View", "Line Graph" and finally get the time plot graph as the following:
This graph shows the series has a constant mean and constant variance which implies the first difference series of "cpi" achieves stationarity.
What will happen if we added the trend variable into the DSP regression model? Can it improve the regression result? Let's try it as below.
In the "Unit Root Test", choose "Augmented Dickey-Fuller", "1st differenc", "0" and "Trend and Intercept" in the dialogue box as following:
After clicking "OK", the result is as following:
As you can see, the regression result is even better than before which implies that the "cpi" series has a time trend, so if we detrend the series, we can also get the stationary series. Therefore, we can conclude that the "cpi" series is a non-stationary series, but either took the 1st-difference or detrend would generate the stationary.