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

Subject:MathPrice: Bought3

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.

Write the code to estimate this model using MLE, deriving also the asymptotic

covariance matrix using the Gaussian loglikelihood:

T

[(6) = Zlog?rt | Tit—1:6)

where

1 ——%i1)—%2(Tt—M(1—¢)—¢Tt—i)2

7“ r_ 6 2—6 (“0+”-

?tl t 1’ ) 27r(a0+r3_11)2a2 ’

and

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)?

Summarizing:

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

available.

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

index. Then:

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

the results.

0 Now re—estimate the model setting 19 = 1 and evaluate the impulse response

function

([7 — A1L)_12 = 2.4ka

k—O

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

mean economically?

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.