Instructions: Show your answers in details.
Q1 (2 points): Consider the autoregression model
1. Generate n = 200 observations from this process.
2. Apply the moving average filter νt = (xt + xt−1 + xt−2 + xt−3)/4 to the data you generated.
3. Plot xt as a line and superimpose νt as a dashed line. Comment! Q2 (2 points): Consider the random walk with drift model
xt = δ + xt−1 + wt
for t = 1,2,···, with x0 = 0, where wt is white noise with variance .
1. Show that the model can be written as .
2. Find the mean function and the autocovariance function of xt.
3. Show that xt is not stationary.
4. Show that ρx(t − 1,t) = p(t − 1)/t as t → ∞. What is the implication of this result?
5. Suggest a transformation to make the series stationary, and prove that the transformed series is stationary.
Q3 (2 points): For the following yearly time series (10 years).
(t,xt) : (1,24), (2,20), (3,25), (4,31), (5,30), (6,32), (7,37), (8,33), (9,40), (10,38)
1. Compute the sample autocorrelation function, ˆρ(h), at lags h = 0,1,2, and 3.
2. Test the null hypothesis that the theoretical autocorrelation at lag h = 1 equals zero.
3. Use R and redo parts (1) and (2).
Q4 (2 points):
1. Simulate a series of n = 500 observations from the process ), where wt is Gaussian white noise series and compute the sample ACF, ˆρ(h), to lag 20. Compare the sample ACF you obtain to the actual ACF, ρ(h).
2. Repeat part (1) using only n = 50. How does changing n affect the results?
Q5 (2 points): Consider the process: , where ϕ is a random variable with a
Uniform distribution (0,1), and t = 0,±1,±2. Show that this process is weak stationary and find its autocorrelation function.
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