Econ 33220
Homework 1
Please correct any obvious typos. Explain your reasoning.
Problem 1. This homework requires e.g. matrix calculations, using a com-puter. Familiarize yourself with, say, MATLAB to do that.
Consider the AR(2)’s on slide 48 of “topic 1”,
yt = ayt−1 − byt−2 +
where a takes one of the values
a ∈ {1.85, 1.89, 1.898, 1.9, 1.9002}
and where b = 0.90.
1. For general a and b, state the companion matrix B.
2. For general a and b, state the characteristic polynomial.
3. For each value of a, calculate the roots up to two digits after the decimal point.
4. For each value of a, plot the impulse response over the horizon k = 0, . . . , 200.
5. Relate your plots to the roots. In particular, some impulse response will have cyclical features. Calculate the number of periods until a full cycle is completed. ( Hint: answer this by examining the roots. Remember that a complex number z can be written as z =|z|eiα, making it easy to calculate zk. Remember that eiα = cos(α) + isin(α).)
6. Suppose that the variance of is 1. For all values a above calculate the variance as well as the first three autocovariances and autocorrelations of yt.