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Question 1

Consider the AR(1) process given below.

Y_t = Φ₀ + Φ₁Y_t-1 + ε_t

Where Φ₀ = 0, |Φ₁| < 1 and ε_t is a white noise error term.

a. What is the mean-reverting level μ for this data process?

b. Suppose the process is currently at μ. Then, in time period t a shock occurs to the system. In other words, ε_t ≠ 0. For simplicity, assume that there are no other shocks prior to or after this shock. Using relevant equations, trace the impact of this shock from period (t-1) to (t+k). Show that, as k approaches infinity, the above AR(1) process will eventually get back to its mean-reverting level μ that you solved for in part (a).

Question 2

A researcher models the Chinese stock returns series, Y_t, as an AR(1) process and obtains the following results. The standard errors are given in parentheses.

Y^{^}_t = 0.826 –  0.569 Yt-1
        (0.595)  (0.113)

R² = 0.79; n = 185

a. Is the intercept term significant to the model? Conduct a relevant hypothesis test to answer this question.

b. Is the slope significant to the model? Conduct a relevant hypothesis test to answer this question.

c. What is the necessary condition that needs to be satisfied to achieve covariance stationarity? Is that condition satisfied in this model? Conduct a relevant hypothesis test to answer this question.

d. Compute the mean-reverting level for the Chinese stock returns series and interpret what this number means.

Suppose the true underlying data generating process for the stock returns series is in fact the AR(1) process shown above.

e. What does the correlogram for the AutoCorrelation Function (ACF) of Yt look like up to 8 lags?

f. Suppose the process is currently at μ. Then, in time period t a shock occurs to the system. In other words, ε_t ≠ 0. For simplicity, assume that there are no other shocks prior to or after this shock. Use relevant equations to trace the impact of this shock from period (t-1) to (t+k).

g. Using your equations in part (f), show that the above AR(1) process will eventually get back to its mean-reverting level μ as k approaches infinity.