Econ 33220
Midterm Exam
If necessary, recall that the solutions to a quadratic polynomial λ2 + pλ + q = 0 are given by
Problem 1 (Total: [ 20 pt ]) Consider the AR(2)
yt = ayt-1 — byt-2 + ∈t (1)
1. State the characteristic polynomial and its roots.
2. Suppose that a = 3/2 and that b takes one of the three values
For each of these values, calculate the roots explicitly and make an educated guess about the impulse response (try to draw it by hand) and its key properties.
Problem 2 (Total: [ 25 pt ]) Consider the VAR for yt ∈ RI2 given by
where
1. State the characteristic polynomial and calculate its roots.(Hint: re- member that the determinant of a 2 × 2 matrix is given by det = ad — bc and that the inverse is given by /det. )
2. Find the eigenvectors for the two eigenvalues. (Hint: remember that the eigenvectors solve Bv = λv, where λ is an eigen- value. Try with setting the second entry vy of the eigenvectors to vy = 1 in each case, and solve for vx.)
3. Decompose B = VDV-1, normalizing so that D(1, 1) is the smallest eigenvalue, V (2, 1) = 1 and V (2, 2) = 1 .
4. Provide the error-correction representation resulting from this decom-position, i. e. state Q, β as well as QβI .
5. Calculate the Cholesky decomposition for Σ .
6. Find the Blanchard-Quah decomposition of Σ .
Problem 3 (Total: [ 10 pt ]) Which one of these statements is true or false? If you provide the correct answer, you get 2 points. If you provide no answer, you get 0 points. If you provide the wrong answer, you will get a deduction of 3 points.
1. The Lettau-Ludvigson cointegration relationship between consumption, assets and income is helpful for improving the forecast for any of these three variables, beyond the simple random walk forecast.
2. Assuming a concave production function y = f(n) with f(0) = 0, the procyclicality of average labor productivity is evidence in favor of technology-shock-driven business cycles.
3. The AR(1) yt = 1.1yt-1 + Et has damped impulse responses.
4. Consumption, hours worked and investment are strongly procyclical.
5. Government spending is strongly procylical, demonstrating that govern- ment spending is a key driver of business cycles.
Problem 4 (Total: [ 25 pt ]) Consider the following whiskey-drinking problem. You have received a nice bottle of whiskey, but it would be a waste to consume it all at once: you therefore decide to drink some now and store the rest, to drink the remainder in the future. So, if you have a certain amount of whiskey wt ≥ 0 at the beginning of the period, you can decide, how much of it to drink, ct ≥ 0, and how much of it to store st ≥ 0,
ct + st = wt
The whiskey evaporates a bit with storage: you will start the next period with wt+1 = Rst , where 0 < R < 1. Your utility function for drinking whiskey is given by
which you seek to maximize.
1. Write this as a dynamic programming problem for V (w) .
2. State the Lagrangian L.
3. Find the first-order conditions and the envelope condition.
4. Guess that the solution takes the form.
c(w) = ψw
for some parameter 0 < ψ ≤ 1. Plug this guess into your equations, and use that to calculate ψ in terms of the parameters above.
5. Suppose that η = 1/2, β = 0.6 and R = 0.5. Calculate ψ explicitly.
6. (For extra points:) Explicitly calculate V (w) in general and for these
parameters. ( Hint: you know the derivative already. 2/
√
0.82 ≈ 2.21.)