question archive Question 1 (term structure model) The discrete time version of the generalized-CIR model for the term structure postulates that the short—term interest rate rt satis?es the following dynamic equa— tion: Tt = M1 — (/5) + 95735—1 + (050 4’ Til—limb with ut N NID(0, a2) and 040, a1 > 0 are additional parameters
Question 1 (term structure model)
The discrete time version of the generalized-CIR model for the term structure
postulates that the short—term interest rate rt satis?es the following dynamic equa—
Tt = M1 — (/5) + 95735—1 + (050 4’ Til—limb
with ut N NID(0, a2) and 040, a1 > 0 are additional parameters.
Write the code to estimate this model using MLE, deriving also the asymptotic
covariance matrix using the Gaussian loglikelihood:
[(6) = Zlog?rt | Tit—1:6)
7“ r_ 6 2—6 (“0+”-
?tl t 1’ ) 27r(a0+r3_11)2a2 ’
6 Z (M: $7 027 0507 a1),-
Compare the results of the estimation when leaving ¢ free from ?xing gt 2 0.5.
Also, how does its ?t go as compared with the Vasicek model
n = M(1 — ¢) + 92571—1 + at,
with at w NID(O, a2)?
0 G0 to the website of the St.Louis Federal Reserve Bank FRED (https://fred.
stlouisfed.0rg/) and download the data corresponding to the 3—M0nth Treasury
Bill: Secondary Market Rate (TB3MS), see https: //fred.stlouisfed.0rg/series/TB3MS,
at monthly frequency from 1960—m1 t0 2021—m5; use the most recent release
0 Evaluate the MLE, and its asymptotic covariance matrix, for the for the generalized—
CIR model, and the Vasicek model, using the data above.
0 Comment on the results.
Question 2 (Time Series)
Consider the Vector Autoregression (VAR) model:
yt = Ct + Alyt—l + A2yt—2 + ' ‘ ‘ + Ap’yt—p + um 75: 1: ~ ~ - 7T, (1)
where yt is a N X T vector of time series variables, Aj, (j = 1... p) are (N X N)
coef?cient matrices and is a white noise vector of time series with at N (0,21,).
Secondly, suppose that yt = [gdph inft, mt, it, hpt, lt, ipt], so N = 7, where: gdp is the
logarithm of the (nominal) Gross Domestic Product, inf is the in?ation rate, In is the
logarithm of money supply (in MZ—aggregate), i the logarithm nominal interest rate,
hp is the house price, 1 the number of hours worked and ip the industrial production
0 Go to the website of the St.Louis Federal Reserve Bank FRED (https://fred.
stlouisfed.0rg/) and download the data corresponding at the above indicated
variables at at quarterly frequency from 1980—Q1 to 2019—Q4; use the most
recent release available. (HINT: for the variable hpt use the series named:
Average Sales Price of Houses Sold for the United States and for It the Weekly
Hours Worked: Manufacturing for the United States).
0 For each time series, perform an Augmented Dickey—Fueller (ADF) test, for
lags going from 1 to 12; do the suitable transformation of the data according
to the ADF test before next step. Comment the results.
0 Estimate the VAR(p) model using a standard OLS7 variable by variable, and
select the best number of lags p using the adjusted R2 as a criterion. Comment
0 Now re—estimate the model setting 19 = 1 and evaluate the impulse response
([7 — A1L)_12 = 2.4ka
plotting the result for each of the 7 diagonal elements of the above matrix for
k: = 1, ..., 100 (that is consider the diagonal elements of Alf) What does this
0 Finally, use the model for forecastingzproduce the one—step forecasts, the four—
steps forecasts, and the 12—steps forecasts, and summarize using the mean
squared forecasting error. Comment the results.