(Requires Internet access for the test question) The following question requires you to download data from the internet and to load it into a statistical package such as STATA or EViews

Question

a. Your textbook suggests using two test statistics to test for stationarity: DF and ADF. Test the null hypothesis that inflation has a stochastic trend against the alternative that it is stationary by performing the DF and ADF test for a unit autoregressive root. That is, use the equation (14.34) in your textbook with four lags and without a lag of the change in the inflation rate as a regressor for sample period 1962:I — 2004:IV. Go to the Stock and Watson companion website for the textbook and download the data "Macroeconomic Data Used in Chapters 14 and 16." Enter the data for consumer price index, calculate the inflation rate and the acceleration of the inflation rate, and replicate the result on page 526 of your textbook. Make sure not to use the heteroskedasticity-robust standard error option for the estimation.

b. Next find a website with more recent data, such as the Federal Reserve Economic Data (FRED) site at the Federal Reserve Bank of St. Louis. Locate the data for the CPI, which will be monthly, and convert the data in quarterly averages. Then, using a sample from 1962:I — 2009:IV, re-estimate the above specification and comment on the changes that have occurred.

c. For the new sample period, find the DF statistic.

d. Finally, calculate the ADF statistic, allowing for the lag length of the inflation acceleration term to be determined by either the AIC or the BIC.

What will be an ideal response?

Answer 1

Answer:
a. For the sample period 1962:I — 2004:IV, the result is as follows:

Dependent Variable: D2LP
Method: Least Squares
Date: 12/31/10 Time: 10:44
Sample: 1962Q1 2004Q4
Included observations: 172

Coefficient Std. Error t-Statistic Prob.

C 0.51 0.21 2.37 0.02
DLP(-1) -0.11 0.04 -2.69 0.01
D2LP(-1) -0.19 0.08 -2.32 0.02
D2LP(-2) -0.26 0.08 -3.15 0.00
D2LP(-3) 0.20 0.08 2.51 0.01
D2LP(-4) 0.01 0.08 0.13 0.90

R-squared 0.24 Mean dependent var 0.02
Adjusted R-squared 0.21 S.D. dependent var 1.71
S.E. of regression 1.51 Akaike info criterion 3.70
Sum squared resid 380.61 Schwarz criterion 3.81
Log likelihood -312.37 Hannan-Quinn criter. 3.75
F-statistic 10.31 Durbin-Watson stat 1.99
Prob(F-statistic) 0.00

Hence the ADF statistic is -2.69. You cannot reject the null hypothesis of non-stationarity at the 5% level (critical value -2.86), but you could at the 10% level (critical value -2.57).

b. Not much has changed. The intercept became smaller, but was statistically insignificant anyway. The slope coefficient increase somewhat (as did the Regression R2 with it) and its t-statistic also became stronger. Some of this is the result of data revisions (even for the old sample period the slope coefficient increased somewhat) while part of it has changed because of the longer sample period.

Dependent Variable: D2LP
Method: Least Squares
Date: 12/31/10 Time: 11:20
Sample: 1962Q1 2009Q4
Included observations: 192

Coefficient Std. Error t-Statistic Prob.

C 0.62 0.26 2.36 0.02
DLP(-1) -0.15 0.05 -2.75 0.01
D2LP(-1) -0.29 0.08 -3.54 0.00
D2LP(-2) -0.30 0.09 -3.45 0.00
D2LP(-3) 0.03 0.08 0.31 0.76
D2LP(-4) -0.05 0.08 -0.62 0.54

R-squared 0.24 Mean dependent var 0.01
Adjusted R-squared 0.22 S.D. dependent var 2.20
S.E. of regression 1.95 Akaike info criterion 4.20
Sum squared resid 707.46 Schwarz criterion 4.31
Log likelihood -397.64 Hannan-Quinn criter. 4.25
F-statistic 11.54 Durbin-Watson stat 2.00
Prob(F-statistic) 0.00

c. The DF statistic is obtained by simply regressing the change in the inflation rate on the lagged level of the inflation rate. The t-statistic on the lagged inflation level is the ADF statistic, which is -5.28, rejecting the null hypothesis of non-stationarity.

d. Both the AIC and the BIC have a minimum for two lags. For that case, the ADF statistic is -2.94 and the null hypothesis of non-stationarity can therefore be rejected at the 5% level, but not at the 1% level.