- Suppose {et : t = −1, 0, 1, . . .} is a sequence of iid random variables with mean zero and variance 1. Define a stochastic process by xt = et − 0.5et−1 + 0.5et−2, t = 1, 2, . . . a. Is xt stationary? Show your work. b. Is xt weakly dependent? Again, show your work.
- Determine whether each of the following series has a unit root or not. If applicable, also determine whether each series has drift or trend. Use both the methods of Enders and Elder & Kennedy. I also want you to assess whether the series is growing or not growing, based on a visual inspection of the series. In other words, you should do this first.
The series to analyze are:
a. The U.S. industrial production index, starting from January 1980 (FRED mnemonic: INDPRO);
b. The U.S. trade balance in goods and services, balance of payments basis, starting from January 1992 (FRED mnemonic: BOPGSTB), and
c. The total business inventories to sales ratio, starting from January 1992 (FRED mnemonic: ISRATIO). You will need to go to the St. Louis Fed’s FRED website to download the series. In case you need them, the critical values from Enders are: • To test the null that trend = 0 given a unit root: 2.79 (for α = 0.05) and 3.53 (for α = 0.01); • To test the null that drift = 0 given a unit root: 2.54 (for α = 0.05) and 3.22 (for α = 0.01.)