Reference no: EM131368049

TOPICS IN MONETARY ECONOMICS ASSIGNMENT

1. Consider the following model of the economy.

y_t = 0.7y_t-1 + 0.2 (π_t - π_t^e) + η_t                        (1)

where y_t is the output gap, π_t is the inflation rate, π_t^e are private agents' expectations of π_t, and η_t is a i.i.d. Normal shock, with mean zero and variance σ²_η.

The inflation rate is set according to:

π_t = x_t + v_t                                                      (2)

where x_t is the policy instrument and v_t is i.i.d. N(0, σ_v²), with E (η_tv_t) = 0.

Policymakers know that private agents form expectations of π_t based on the following rule:

π_t^e = 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: A₀S_t = A₁S_t-1 + Bx_t + ω_t; with State vector defined as: S_t = [y_t  π_t  π_t^e  1]'. Define the matrices A₀ and A₁, and the vector B: Define the vector of shocks !t; and write the covariance matrix E (ω_tω'_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^{^}₀Z_t = b A^{^}₁Z_t-1 + B^{^}x_t + u_t, with State vector defined as: Zt = [y_t  π_t  1]'. Define the matrices A^{^}₀, A^{^}₁, and C, and the vector B^{^}. Define the vector of shocks u_t; and write the covariance matrix ∑ = E (u_tu'_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: x_t = -FS_t-1, while using the State Space representation written in point b. you obtained: x_t =  -GZ_t-1.

i. Write the expressions for π_t and π_t^e under the policy rule x_t = -FS_t-1 and under the policy rule x_t = -GZ_t-1. The two policy rules will deliver the same representation for π_t and π_t^e, 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 FS_t-1 = GZ_t-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 x_t = -FS_t-1 and x_t = -GZ_t-1, the State Space representations become: A₀S_t = (A₁ - BF) S_t-1 + ω_t and A^{^}₀Z_t = (A^{^}₁ - B^{^}G)Z_t-1 + u_t. 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: x_t = fy_t-1. Policymakers' objective is to minimize the loss function:

L = λvar(y_t) + var(π_t)

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(y_t) = var(y_t-1).

Use the policy rule x_t = fy_t-1 in addition to (2) and (3) to rewrite the equation for y_t (assume that 0 < α < 1). Using this equation, show that var(y_t) = ((0.2)²σ²_v+σ²_η/[0.6+0.2(1-α)f]²). Since π_t = x_t + v_t, you know that var(π_t) = f²var(y_t) + σ²_v. Use this expression, together with the one for var(y_t), to obtain an expression for L. Show all your work.

e. Look at the expressions for var(y_t) and var(π_t) that you obtained in point d. For any value of the parameter   such that 0 < α < 1, how are var(y_t) and var(π_t) affected by the value of the policy parameter f? Will a higher value of f change var(y_t) and var(π_t) 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:

π_t = αx_t + (1 - α)π_t^e + ε_t                                                   (4)

where x_t is a policy instrument, π_t^e are private agents' expectations of inflation and ε_t ∼ N(0, σ_ε²). Private agents' expectations are unobservable to policymakers, but they are known to follow the process:

π_t^e = (1 - γ)π* + γπ_t-1^e + v_t                                             (5)

where π* is the long run inflation target, γ is a parameter satisfying 0 < γ < 1, v_t ∼ N (0, σ²_v) and E(ε_tv_t) = 0. Assume that policymakers know the values of α and σ_ε² (α = 0.9 and σ_ε² = 0:49), but do not know γ and σ_ε². Assume that everyone in the economy knows that π* = 2%:

Policymakers' optimal estimate of π_t^e given the information available at time t-1 is π_t|t-1^e ≡ E(π_t^e|Ω^t-1) = 3%, where Ω^t-1 = {π_t, x_t, π_t-1, x_t-1, ...}, and their estimate of P_t|t-1 = E(π_t^e - π_t|t-1^e)² = 0.5.

At time t, policymakers set x_t = 2%, and observe an inflation rate of 4%.

a. Using the Kalman filter, compute policymakers' updated beliefs of π_t^e, i.e. π_t|t^e ≡ E(π_t^e|Ω^t), after they have implemented x_t = 2%, and observed π_t = 4%.

b. Forecast π_t+1^e|Ω^t) and compute P_t+1|t = E(π_t+1^e - π_t+1|t^e)², the mean squared error of the forecast. Write them in terms of the unknown parameters γ and σ²_v. To what extent do γ and σ²_v  affect π_t+1|t^e and P_t+1|t^e? Explain your answer.

c. Write the forecast π_t+1|t ≡ E (πt+1|Ω^t) and its mean squared error Γ_t+1|t = E(π_t+1 - π_t+1|t)² as a function of the policy variable x_t+1 and of the unknown parameters γ and σ²_v. To what extent do γ and σ²_v affect π_t+1|t and Γ_t+1|t? Explain your answer.

d. Assume that policymakers could affect γ and σ²_v 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 σ²_v that they cannot attain by choice of x_t? Explain your answer.

3. Empirical Question: Consider the following model of the Phillips curve:

u_t = α₀ + α₁u_t-1 + β₀Δπ_t + ε_t                            (6)

where u_t is the unemployment rate, π_t is the inflation rate and Δπ_t = (π_t - π_t-1), and ε_t is a i.i.d Gaussian shocks with mean zero and variance σ²_ε. You know that E (π_tε_t) = 0; and that all the OLS assumptions are satisfied.

a. Find data on the unemployment rate u_t and the inflation rate π_t. 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 π_t = log(Index_t) - log(Index_t-1), and compute Δπ_t = (π_t - π_t-1). Make sure that all your data (for both u_t and π_t) 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 u_t and π_t).

State your selected Country, sample period, data frequency, and the specific variables that you used to obtain u_t and π_t.

b. Using your data, estimate equation (6) by OLS. Report the values of the estimated parameters α₀, α₁, and β₀. In addition, estimate σ²_ε using the formula: (1/T-4)_t=1∑^T(ε^{^}_t)², where T is the sample size and ε^{^}t = u_t - (α₀ + α₁u_t-1 + β₀Δπ_t), with α₀, α₁, and β₀ being your estimated parameters. Report the value of your estimate for σ²_ε.  

c. Assume that the inflation rate is controlled by the central bank, and that it is set it according to the feedback rule:

Δπ_t = f(u_t-1 - u*) + v_t                       (7)

where u* is the natural rate of unemployment and v_t is a i.i.d Gaussian shocks with mean zero, variance σ²_v, and such that E (v_tε_t) = 0. In (7), f represent a policy parameters that is chosen by policymakers to minimize the loss function:

L = λvar(u_t) + var(Δπ_t)

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 u_t and π_t are stationary, which implies that you can write: var(u_t) = var(u_t-1) and var(π_t) = var(π_t-1).

Using (6) and (7), rewrite the process for unemployment in the form:

u_t = a + bu_t-1 + ω_t                               (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 ω_t as a function of ε_t and v_t. Show your work.

d. Given (6), (7), and (8), we can write: var(u_t) = σ²/1-b², where σ²_ω = var(ω_t), and var(Δπ_t) = f²var(u_t). Use these expressions to write L as a function of the parameters α₀, α₁, β₀, f, σ²_ε and σ²_v. 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 σ²_ε 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 u_t will be explosive and var(u_t) 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 σ²_ε, 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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