Econ 423 – Assignment 2
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Econ 423 – Assignment 2
(Due on OCT. 31)
1. Derive the autocovariance functions for AR(2), MA(2) and AMRA(2,2) processes. Briefly comment on their relationships.
2. (1) Verify whether the following process is stationary or not. If so, cal- culate its autocovariance. (2) Verify whether the following process is Ergodic or not.
yt = (1 + 2.4L +0.8L )e2t et ∼ WN (0, 1)
3. Consider the following bilinear time series model,
yt = et + βet − 1 et −2
where e ∼ SWN (0,σ2 ). Explain whether yt is stationary or not. (SWN stands for the strict WN.)
4. Consider the following bivariate probability distribution
|
|
X=0 |
X=1 |
|
Y=0 |
0.1 |
0.3 |
|
Y=1 |
0.5 |
0.1 |
(Note: The table can be read as prob(X=0, Y=0)=0.1 ,etc.)
(i) Find the marginal distribution of X and Y; Prove that X and Y are not statistically independent.
(ii) Find E(X|Y = 0) and Var(X|Y = 1);
(iii) Find Cov(X, Y) and Corr (X, Y).
5. Plot the Autocorrelation Function (ACF) for the ARMA (2,1) process yt = φ 1yt − 1 + φ2yt −2 + et + θ1 et − 1
for the lags j=0, 1, 2, ..., 20 with the parameter values, φ 1 = 0.6, φ2 = −0.2 and θ 1 = 0.1. Repeat the exercise with θ 1 = 0, the other two parameters being unchanged, in order to see how the moving-average component affects the ACF in this case.
6. Do the following simulation:
(1) Assume et ∼ N.I .D(0, 1). The simulate 1000 data points for et (t=1, 2, ..., 1000).
(2) Let φ 1 = 0.2022. With the initial condition y0 = 0, construct the time series yt for t=1, 2, ..., 1000, from the AR(1) process,
yt = φ 1yt − 1 + et
(3) With the constructed yt , empirically calculate the mean, variance. Empirically calculate autocovariance and the ACF for the first 5 lags.
(4) Theoretically calculate the mean, variance, autocovariance and the ACF correspondingly and compare the results from (3).
7. Repeat the whole process specified in 6 again for MA(1),
yt = et + θ1 et − 1
Note: Let θ 1 = 0.2022 and e0 = 0.
8. Apply AR models to the real return data. (note: Choose any two time series (out of five) from your assignment 1.)
(i) Fit the return data with AR process with order 1 up to 5. Report your results along with the standard errors.
(ii) Compute the first four moments from your estimated models (AR(1) to AR(5)) and compare them with the empirical corresponding moments. Construct a table for comparison. In your opinion, which model is the “best” for your data?
(iii) Forecast the stock prices of the next five days (out of your sample pe- riod) using the AR(1)-AR(5) . Compare your forecasts with the actual prices.
9. Suppose you are investing $ 1M in a portfolio consisting of your five time series. How do you optimally allocate your investment among these five? (You can assume constant risk across time.) Based on your investment strat- egy, what are the expected returns and risk levels? Draw the reward-risk plot.
2022-12-01