Problem 1

1. Simulate _T = 500 observations from an _AR(1) process for = {0_.25_,0_.5_,0_.75_,0_.8_,0_.9}.

yt = yt1 + t

2. Treat the artificial from the simulations above as observed data by an econometrician.
3. Estimate each model via OLS.
4. Test the standard null hypothesis of = 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.
5. 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 . 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?
6. 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 = 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):

x_{t tj} where _x0 is the initial condition.

j=0

• Plot the simulated time series.
• Estimate via OLS. Report the usual suspects.

Problem 3

Take the _AR(1) model above:

y_t = y_t1 + _t with = 0.8

• Run the following simulation:
• Set ₀ = 1.0 Set y₀ = 0.0 Set all _t = 0.0 for t > 0
• Plot the simulated process {_y}^T_t=0 as a function of time. This is called the impluse response function. Interpret it in terms of the _MA () 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 = 1yt 1 + 2yt2 + 1t1 + 2t2 + t

Choose appropriate values for the _AR and _MA coefficients, as well as for .

• Plot the simulated time series.
• Calculate ₀, ₁ and ₂.

Problem 6

• Simulate _T = 500 time steps for the following two equations:

yt = yt1 + u1,t xt = xt1 + u2,t

• where _{uj,t j} = 1_,2 are independent standard white noise processes.
• Next regress _yt on _xt and estimate (slope coefficient) via OLS in the following regression

y_t = + x_t + _t

• Test the null hypothesis _H0 : = 0 against the alternative _Ha : = 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 coefficients for each run of the simulation.
• Summarize the simulated sampling distribution for .
• Make a histogram plot of the simulated coefficients.

Problem 7

• Repeat the exercise in Problem 1 above for = 1_.
• Comment on your findings.

Problem 8

• Simulate _T = 500 time steps from the random walk model

xt = xt1 + u1,t Next simulate _T = 500 time steps from the model

y_t = + x_t + _t

• Where = 0_.22 and = 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 _t to the ADF Test. What do you find?
  • Obtain via OLS.
  • Estimate the error-correction model with _p = 1 and include contemporaneous _xt.