Problem Set 4

Question 1

Download the file guessTheModel1.txt from the course website and import the data into R.

a. (20 points)

What time series model best describes the data? Justify your answer both visually and with parameter estimates.

Solution:

b. (20 points)

What do you notice about the standard errors for the parameters of the model you estimated in part (a)? Estimate a new model that only includes terms that were significant in part (a). How does this model differ from what you previously selected?

Solution:

c. (15 points)

Repeat part (a) for only the first 100 observations and the first 1000 observations of the dataset.

Solution:

Question 2

Suppose that \(Y_1,Y_2,\ldots\) and \(W_1,W_2,\ldots\) are \(AR(1)\) and \(MA(1)\) processes, respectively, such that \[\begin{align*} Y_t & = c+\phi Y_{t-1} + \varepsilon_t \\ W_t & = \mu +\varepsilon_t + \theta \varepsilon_{t-1}, \end{align*}\]

where \(\varepsilon_t \stackrel{i.i.d.}{\sim} N(0, \sigma_e^2)\).

a. (15 points)

Solve for \(\text{Cov}( Y_t,W_t)\).

Solution:

b. (15 points)

Solve for \(\text{Cov}( Y_t,W_{t-1})\).

Solution:

c. (15 points)

Solve for \(\text{Var}( Y_t+W_{t})\).

Solution: