Reference no: EM132356379

Question 1

You are given a random walk with a drift process defined to be the following:

Y_t = Φ₀ + Φ₁Y_t-1 + ε_t is white noise and Φ₀ < 0 and Φ₁ = 1.

a. What would the line plot look like for the Y_t data series?
b. Solve for the five period-ahead forecast, Y^{^}_t+5.
c. Solve for the variance of the five period-ahead forecast error, Var(e_t+5).
d. Use your answers to parts (b) and (c) to comment on your ability to forecast a data series that has a unit root (and, therefore, can be modeled as a random walk with a drift process).

Question 2

A researcher obtains data for the X and Y variables and models simple linear regression estimation as follows:

Y_t = β₁ + β₂X_t + ε_t

Suppose both the X and Y variables are non-stationary. In particular, each is a random walk process such that,

(2) Y_t = Y_t-1 + ε_t

(3) X_t = X_t-1 + ε_t

where ε_t is a white noise error term.

a. If the researcher estimates model (1) as it is, what is the main problem that he will encounter? Explain clearly.

b. You have advised the researcher that he needs to run model (1) using the first-differences of the variables. In order to explain why this technique works, first obtain the one-period lag of model (1). Second, re-write model (1) in first-differences. Now apply what you know about the X and Y processes. Then explain clearly why the first-differenced model solves the problem encountered in part (a).

Question 3
A researcher models the German inflation rate series, Y_t, as an AR(1) process in the first-differences and obtains the following results. The standard errors are given in parentheses.

ΔY_t = 0.628 + 0.417ΔY_t-1
   (0.033)          (0.024)

R-squared = 0.73; n = 262
Serial Correlation LM test statistic _{lag 1} = 2.75
Serial Correlation LM test statistic _{lag 2} = 1.99
Serial Correlation LM test statistic _{lag 3} = 4.87

a. Is there evidence of serial correlation of order three in the model? Briefly explain.

b. Is ΔY_t a stationary process? Conduct a relevant hypothesis test to answer this question.

c. Is Y_t a stationary process? Conduct a relevant hypothesis test to answer this question.

d. Based on your answer to part (c), what does the line plot look like for the German inflation rate series?

e. Next, the researcher formally tests for the existence of a unit root in the German inflation rate series. He implements an Augmented Dickey Fuller (ADF) test and obtains the following results. Can this researcher formally conclude that the Japanese inflation rate series has a unit root?

Note: The relevant Dickey Fuller critical value at the 5 percent significance level is 90.34.

Unrestricted model: ΔY_t = 0.002 - 0.183_t + 0.764Y_t-1 + 0.035ΔY_t-1 + 0.174ΔY_t-2
                                   (0.000)  (0.000)   (0.038)     (0.049)        (0.002)

Sum of Squares for Error = 48.69

Restricted model:

ΔY_t = 0.004 + 0.166ΔY_t + 0.278ΔY_t-2
         (0.000)   (0.046)     (0.032)

Sum of Squares for Error = 87.57