EPOS exam, 7 June 2022 Please address all questions

EPOS exam, 7 June 2022 Please address all questions. Write a short report (no more than 6 pages plus appendix with the code). You have to develop your own code in any language you prefer (R, MatLab, etc). 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 satisfies the following dynamic equation: rt = µ(1 − φ) + φrt−1 + (α0 + r α1 t−1 )ut , with ut ∼ NID(0, σ2 ) and α0, α1 > 0 are additional parameters. Write the code to estimate this model using MLE, deriving also the asymptotic covariance matrix using the Gaussian loglikelihood: l(θ) = X T t=2 logf(rt | rt−1, θ) where f(rt | rt−1, θ) = 1 p 2π(α0 + r α1 t−1 ) 2σ 2 e − 0.5 (α0+r α1 t−1 )2σ2 (rt−µ(1−φ)−φrt−1) 2 , and θ = (µ, φ, σ2 , α0, α1) 0 . Compare the results of the estimation when leaving φ free from fixing φ = 0.5. Also, how does its fit go as compared with the Vasicek model rt = µ(1 − φ) + φrt−1 + ut , with ut ∼ NID(0, σ2 )? Summarizing: • Go to the website of the St.Louis Federal Reserve Bank FRED (https://fred. stlouisfed.org/) and download the data corresponding to the 3-Month Treasury Bill: Secondary Market Rate (TB3MS), see https://fred.stlouisfed.org/series/TB3MS, at monthly frequency from 1960-m1 to 2021-m5; use the most recent release available. • Evaluate the MLE, and its asymptotic covariance matrix, for the for the generalizedCIR model, and the Vasicek model, using the data above. • Comment on the results. Question 2 (Time Series) Consider the Vector Autoregression (VAR) model: yt = ct + A1yt−1 + A2yt−2 + · · · + Apyt−p + ut , t = 1, . . . , T, (1) where yt is a N × T vector of time series variables, Aj ,(j = 1 . . . p) are (N × N) coefficient matrices and is a white noise vector of time series with ut ∼ (0, Σu). Secondly, suppose that yt = [gdpt , inft , mt , it , hpt , lt , ipt ], so N = 7, where: gdp is the logarithm of the (nominal) Gross Domestic Product, inf is the inflation rate, m is the logarithm of money supply (in M2-aggregate), i the logarithm nominal interest rate, hp is the house price, l the number of hours worked and ip the industrial production index. Then: • Go to the website of the St.Louis Federal Reserve Bank FRED (https://fred. stlouisfed.org/) 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 lt the Weekly Hours Worked: Manufacturing for the United States). • 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. • Estimate the VAR(p) model using a standard OLS, variable by variable, and select the best number of lags p using the adjusted R2 as a criterion. Comment the results. • Now re-estimate the model setting p = 1 and evaluate the impulse response function (I7 − A1L) −1Σ = X∞ k=0 A k 1L k 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 Ak 1 ). What does this mean economically? • Finally, use the model for forecasting:produce the one-step forecasts, the foursteps forecasts, and the 12-steps forecasts, and summarize using the mean squared forecasting error. Comment the results. END OF PAPER