#### 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. 