FIN5330-Homework 2 Solved

Problem 1
Simulate T = 500 observations from an AR(1) process for φ = {0.25,0.5,0.75,0.8,0.9}.
yt = φyt−1 + t

Treat the artificial from the simulations above as observed data by an econometrician.
Estimate each model via OLS.
Test the standard null hypothesis of φ = 0 with a standard t-test for significance levels {0.01,0.05,0.10} for each one and report the results in a table. Provide test statistics, standard errors, critical values, p-values, etc.
Pick one of the parameter values for the models above and do the following:Use the Central Limit Theorem to derivive a sampling distribution for φˆ. Present the parameter values of the sampling distribution. Produce a graph of the distribution.
Use parametric Monte Carlo to simulate the sampling distribution. Use M = 10,000 repititions. Use the sample mean and standard deviation to estimate the parameter values of the distribution. Produce a histogram.
Use the IID Bootstrap to simulate the sampling distribution. Use B = 10,000 repititions. Use the sample mean and standard deviation to estimate the parameter values of the distribution. Produce a histogram.
Compare all three methods.
Can you interpret the last two distributions as predictive densities?
Return to the problem in 5 above and redo the simulation from step one, but replace the error distribution with a Student-T distribution with df = 5 (degrees of freedom parameter). Even though we know at the generation stage that the errors come from the Student-T distribution, the econometrician assumes a normal distribution when using the CLT and parametric Monte Carlo. The bootstrap obviously does not need to make such assumptions. Compare to the results above.
Problem 2
Simulate an AR(1) process with parameter φ = 0.8 by using the MA(∞) representation. Hint: you will have to truncate the MA(∞) representation, yielding an approximation to the AR(1). Recall that the AR(1) can be represented by the following (i.e. the MA(∞) representation):

                                                   xt t−j                       where x0 is the initial condition.

j=0

Plot the simulated time series.
Estimate φˆ via OLS. Report the usual suspects.
Problem 3
Take the AR(1) model above:

                                                                            yt = φyt−1 + εt                     with φ = 0.8

Run the following simulation:
Set ε0 = 1.0 – Set y0 = 0.0 – Set all εt = 0.0 for t > 0
Plot the simulated process {y}Tt=0 as a function of time. This is called the impluse response function. Interpret it in terms of the MA (θ) coefficients for the AR(1) representation.
Simulate T = 50 time steps in the process.
Problem 5
Simulate T = 500 observations from the following ARMA model:

yt = φ1yt − 1 + φ2yt−2 + θ1εt−1 + θ2εt−2 + εt

Choose appropriate values for the AR and MA coefficients, as well as for σε.

Plot the simulated time series.
Calculate γ0, γ1 and γ2.
Problem 6
Simulate T = 500 time steps for the following two equations:
yt = yt−1 + u1,t xt = xt−1 + u2,t

where uj,t j = 1,2 are independent standard white noise processes.
Next regress yt on xt and estimate β (slope coefficient) via OLS in the following regression
yt = α + βxt + t

Test the null hypothesis H0 : β = 0 against the alternative Ha : β = 06 . Use the standard t-test with standard significance levels (0.01, 0.05, and 0.10). What should you find? What do you find?
Repeat the process M = 50,000 times and store the β coefficients for each run of the simulation.
Summarize the simulated sampling distribution for β.
Make a histogram plot of the simulated coefficients.
Problem 7
Repeat the exercise in Problem 1 above for φ = 1.
Comment on your findings.
Problem 8
Simulate T = 500 time steps from the random walk model
xt = xt−1 + u1,t • Next simulate T = 500 time steps from the model

yt = α + βxt + t

Where α = 0.22 and β = 2.
∼ N(0,1) (white noise process)
Use the Augmented Dickey-Fuller Test to check for the presence of a unit-root in both yt and xt. What do you find? What should you find?
Implement the Engle-Granger two-step method by:First, test for cointegration by submitting ˆt to the ADF Test. What do you find?
Obtain βˆ via OLS.
Estimate the error-correction model with p = 1 and include contemporaneous xt.