## Description

Q1 (2 points): Consider the autoregression model

xt = −.9xt−2 + wt

, wt iid

∼

N (0, 1)

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 σ

2

w.

1. Show that the model can be written as xt = δt +

Pt

j=1 wj

.

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 xt =

1

3

(wt−1 + wt + wt+1), 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: xt = cos

2π(

t

12

+ ϕ)

, 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.

1