TOPICS IN MONETARY ECONOMICS ASSIGNMENT
1. Consider the following model of the economy.
yt = 0.7yt-1 + 0.2 (πt - πte) + ηt (1)
where yt is the output gap, πt is the inflation rate, πte are private agents' expectations of πt, and ηt is a i.i.d. Normal shock, with mean zero and variance σ2η.
The inflation rate is set according to:
πt = xt + vt (2)
where xt is the policy instrument and vt is i.i.d. N(0, σv2), with E (ηtvt) = 0.
Policymakers know that private agents form expectations of πt based on the following rule:
πte = αxt + (1 - α)π* (3)
with 0 < α < 1. This rule implies that private agents set their expectations of inflation as a weighted average of this period's policy, which they are assumed to observe, and the inflation rate target π*.
a. Consider the model composed of (1), (2), and (3). Write the State Space representation of this model in the form: A0St = A1St-1 + Bxt + ωt; with State vector defined as: St = [yt πt πte 1]'. Define the matrices A0 and A1, and the vector B: Define the vector of shocks !t; and write the covariance matrix E (ωtω't) = Ω. Show all your work.
b. Consider again the model composed of (1), (2), and (3). Write the State Space representation of this model in the form: A^0Zt = b A^1Zt-1 + B^xt + ut, with State vector defined as: Zt = [yt πt 1]'. Define the matrices A^0, A^1, and C, and the vector B^. Define the vector of shocks ut; and write the covariance matrix ∑ = E (utu't). Show all your work.
c. Consider the fully optimal policy rule. Using the State Space representation written in point a., you computed that this optimal policy rule is: xt = -FSt-1, while using the State Space representation written in point b. you obtained: xt = -GZt-1.
i. Write the expressions for πt and πte under the policy rule xt = -FSt-1 and under the policy rule xt = -GZt-1. The two policy rules will deliver the same representation for πt and πte, even if it is difficult to show this result using simple algebra. However, it is easy to see that this result holds for the special case in which α = 0. So, assuming that α = 0, write FSt-1 = GZt-1 and find the relationships between the elements of the vectors F and G that will make this equality hold. Show all your work.
ii. Using the two policy rules xt = -FSt-1 and xt = -GZt-1, the State Space representations become: A0St = (A1 - BF) St-1 + ωt and A^0Zt = (A^1 - B^G)Zt-1 + ut. Write the matrices D and D^, where D = (A1 - BF) and D^ = (A^1 - B^G). Show all your work.
d. Assume now that instead of looking at the fully optimal policy rule, the policymaker decides to use the following simple rule: xt = fyt-1. Policymakers' objective is to minimize the loss function:
L = λvar(yt) + var(πt)
where λ ≥ 0. Assume that policymakers will always choose a value of the policy parameter f for which the model is stationary, which implies that you can write var(yt) = var(yt-1).
Use the policy rule xt = fyt-1 in addition to (2) and (3) to rewrite the equation for yt (assume that 0 < α < 1). Using this equation, show that var(yt) = ((0.2)2σ2v+σ2η/[0.6+0.2(1-α)f]2). Since πt = xt + vt, you know that var(πt) = f2var(yt) + σ2v. Use this expression, together with the one for var(yt), to obtain an expression for L. Show all your work.
e. Look at the expressions for var(yt) and var(πt) that you obtained in point d. For any value of the parameter such that 0 < α < 1, how are var(yt) and var(πt) affected by the value of the policy parameter f? Will a higher value of f change var(yt) and var(πt) in the same direction? What does this imply for the value of the optimal policy parameter f (i.e. the value of the policy parameter that minimizes losses L) How would your answer change if α = 0? How would your answer change if α = 0? Explain all your answers.
2. Consider a model of the economy in which inflation is determined by the following equation:
πt = αxt + (1 - α)πte + εt (4)
where xt is a policy instrument, πte are private agents' expectations of inflation and εt ∼ N(0, σε2). Private agents' expectations are unobservable to policymakers, but they are known to follow the process:
πte = (1 - γ)π* + γπt-1e + vt (5)
where π* is the long run inflation target, γ is a parameter satisfying 0 < γ < 1, vt ∼ N (0, σ2v) and E(εtvt) = 0. Assume that policymakers know the values of α and σε2 (α = 0.9 and σε2 = 0:49), but do not know γ and σε2. Assume that everyone in the economy knows that π* = 2%:
Policymakers' optimal estimate of πte given the information available at time t-1 is πt|t-1e ≡ E(πte|Ωt-1) = 3%, where Ωt-1 = {πt, xt, πt-1, xt-1, ...}, and their estimate of Pt|t-1 = E(πte - πt|t-1e)2 = 0.5.
At time t, policymakers set xt = 2%, and observe an inflation rate of 4%.
a. Using the Kalman filter, compute policymakers' updated beliefs of πte, i.e. πt|te ≡ E(πte|Ωt), after they have implemented xt = 2%, and observed πt = 4%.
b. Forecast πt+1e|Ωt) and compute Pt+1|t = E(πt+1e - πt+1|te)2, the mean squared error of the forecast. Write them in terms of the unknown parameters γ and σ2v. To what extent do γ and σ2v affect πt+1|te and Pt+1|te? Explain your answer.
c. Write the forecast πt+1|t ≡ E (πt+1|Ωt) and its mean squared error Γt+1|t = E(πt+1 - πt+1|t)2 as a function of the policy variable xt+1 and of the unknown parameters γ and σ2v. To what extent do γ and σ2v affect πt+1|t and Γt+1|t? Explain your answer.
d. Assume that policymakers could affect γ and σ2v to some extent by increasing the transparency of their policy decisions and their communication to the public. In view of your answer to c., what additional benefits could they obtain through γ and σ2v that they cannot attain by choice of xt? Explain your answer.
3. Empirical Question: Consider the following model of the Phillips curve:
ut = α0 + α1ut-1 + β0Δπt + εt (6)
where ut is the unemployment rate, πt is the inflation rate and Δπt = (πt - πt-1), and εt is a i.i.d Gaussian shocks with mean zero and variance σ2ε. You know that E (πtεt) = 0; and that all the OLS assumptions are satisfied.
a. Find data on the unemployment rate ut and the inflation rate πt. You can use data for your favorite country and time period, and you can choose any data frequency (but make sure that you have a large enough number of observations). You can pick your favorite unemployment variable (civilian unemployment rate, unemployment rate full time workers, male unemployment rate,...) and your favorite price index (CPI, core CPI, GDP deflator,...). To compute the inflation rate from your price index, define πt = log(Indext) - log(Indext-1), and compute Δπt = (πt - πt-1). Make sure that all your data (for both ut and πt) has the same frequency (month, quarter, year,...) and is expressed in the same unit (i.e., you can write 10% as either 10 or 0.1, but you need to use the same unit for both ut and πt).
State your selected Country, sample period, data frequency, and the specific variables that you used to obtain ut and πt.
b. Using your data, estimate equation (6) by OLS. Report the values of the estimated parameters α0, α1, and β0. In addition, estimate σ2ε using the formula: (1/T-4)t=1∑T(ε^t)2, where T is the sample size and ε^t = ut - (α0 + α1ut-1 + β0Δπt), with α0, α1, and β0 being your estimated parameters. Report the value of your estimate for σ2ε.
c. Assume that the inflation rate is controlled by the central bank, and that it is set it according to the feedback rule:
Δπt = f(ut-1 - u*) + vt (7)
where u* is the natural rate of unemployment and vt is a i.i.d Gaussian shocks with mean zero, variance σ2v, and such that E (vtεt) = 0. In (7), f represent a policy parameters that is chosen by policymakers to minimize the loss function:
L = λvar(ut) + var(Δπt)
where λ is the weight attached to unemployment stabilization relative to inflation stabilization.
Assume that policymakers will always set the inflation rate so that the processes for ut and πt are stationary, which implies that you can write: var(ut) = var(ut-1) and var(πt) = var(πt-1).
Using (6) and (7), rewrite the process for unemployment in the form:
ut = a + but-1 + ωt (8)
Find the expressions for a, b and c as a function of the parameters α0, α1, β0, β1, u* and f. In addition, find the expression for ωt as a function of εt and vt. Show your work.
d. Given (6), (7), and (8), we can write: var(ut) = σ2/1-b2, where σ2ω = var(ωt), and var(Δπt) = f2var(ut). Use these expressions to write L as a function of the parameters α0, α1, β0, f, σ2ε and σ2v. Show your work.
e. We are now going to do a simple grid search exercise. Set λ = 1: Use the expression for L that you obtained in point d. and the values of the parameters α0, α1, β0, β1, and σ2ε that you estimated in point b. Consider five possible values for the policy parameter: f = 0; f = 0:25; f = 0:5; f = 0:75; and f = 1. For each one of these values:
i. Compute the value of the parameter b in (8). If |b| ≥ 1; then the process for ut will be explosive and var(ut) will be infinite. Thus, if |b| ≥ 1 then L = ∞.
ii. If |b| < 1, you can compute the value of the loss function using the expression for L, the estimated parameters α0, α1, β0, β1, and σ2ε, and the value of f.
Which one of the five possible values of f minimizes policymakers' losses L? Explain your answer.
f. Now set λ = 0.01, and repeat the exercise that you performed in point e. What is the value of f that minimizes the loss function L in this case? Is the optimal response of inflation to the unemployment rate stronger or weaker when λ = 1 relative to the case in which λ = 0.01? Explain your answer.
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