Advanced Macroeconomics Take-Home Mid-Term Exam
Due: 11:59PM (Japan Standard Time), November 13, 2023
Graduate School of Economics Keio University
Fall 2023
• Consider the following DSGE model, and complete the exercises (a), (b), (c), and (d).
• Report your results and answers in PDF, and submit a PDF file along with MAT- LAB codes (m-file) used in the exercises via K-LMS.
• I strongly recommend submitting your answers sufficiently ahead of the deadline. Last minutes submission can result in unsuccessful submission because of internet connection problems. Any make-up exam will NOT be held for any reasons.
• You must NOT collaborate or share your answers with your classmates. If someone copies your answers, you will also receive a failing grade.
• To be fair, I will not take any questions during the exam.
yˆ= 1 Eyˆ + b yˆ −1−b(rˆ−Eπˆ )+zd, t 1+b t t+1 1+b t−1 1+b t t t+1 t
πˆt = β Etπˆt+1 + ι πˆt−1 1+ιβ 1+ιβ
(1)
(1 − ξ) (1 − ξβ) ?? 1 ? b ?
+ ξ(1+ιβ)
?
(2) +φyˆ+z, (3)
p η+1−b yˆt−1−byˆt−1 +zt,
?πˆt +πˆt−1 +πˆt−2 +πˆt−3? ? trt−1 rπ 4 ytt
rˆ=φrˆ +(1−φ)φ
where yˆ , πˆ , and rˆ denote the percentage deviation of output, inflation, and the nominal
r
ttt
interest rate from their long-run trend or steady-state values. Et is the expectation operator conditional on information available in period t, b ∈ [0, 1] is the degree of external habit persistence in consumption preferences, β ∈ (0, 1) is the subjective discount factor, ι ∈ (0, 1) denotes the weight of price indexation to past inflation, ξ ∈ (0, 1) measures the price stickiness , η > 0 is the inverse of the elasticity of labor supply, φr ∈ [0, 1) is the degree of interest rate smoothing, and φπ,φy ≥ 0 are the degrees of policy responses to inflation and output. ztd represents a demand shock, ztp is a cost-push shock, and ztr is a monetary policy shock.
We assume that the shocks ztx, x ∈ {d, p, r} are all governed by univariate stationary AR(1) processes
zx =ρ zx +εx, t xt−1 t
where ρx ∈ [0, 1) is an autoregressive coefficient and εxt is an i.i.d. disturbance with mean zero and standard deviation σx.
Parameter values are summarized in Table 1. Table 1: Parameters
β η b ι ξ φr φπ φy ρd ρp ρr σd σp σr 0.995 1.0 0.5 0.5 0.75 0.75 1.5 0.125 0.5 0.5 0.5 0.2 0.2 0.2
(a) Compute the impulse responses of the yˆ , πˆ , and rˆ to one-percent shocks about ttt
demand, cost-push, and monetary policy, respectively. Then, draw graphs to show all the responses. Provide an intuitive economic explanation on each response.
(b) The excel file “us data.xlsx” contains the data of output, the inflation rate, and
the nominal interest rate (output is detrended whereas the inflation and nominal interest
rates are demeaned so that they are comparable to the model variables yˆ , πˆ , and rˆ .) ttt
in the U.S. over the sample from 1985Q1 to 2008Q4 (before the Fed conducted the zero interest rate policy). Import the data to your MATLAB work space, and then calculate the variance-covariance matrix of the data using a MATLAB command.
(c) Compute the variance-covariance matrix of yˆ , πˆ , and rˆ in the model economy, ttt
and compare it with the one calculated in the exercise (b). Point out which properties of the data are not well replicated by the model.
(d) Modify the model so that the model can better replicate the variance-covariance matrix of the data. You may introduce new parameters and parameterize them as you like. However, you must NOT change the existing parameter values. The modified model may not be necessarily micro-founded but should be economically reasonable. Which part of the data properties is well replicated after changing the model? Intuitively explain why your modification can improve the empirical performance of the model. (Extra points will be given if no other students make similar modification to yours.)
