STAT question

STAT 4363: Applied Time Series Analysis

HW 1

Written assignment (due 09/09 in class)

1. (10 points) Let X ∼ N(0, 2) and Y ∼ Exp(λ = 3) be two uncorrelated random variables. Find the covariances Cov(X, Y ) and Cov(Y, X + Y ).

2. (10 points) Let X1, X2, . . . , X500 be a white noise time series of length T = 500 following N(0, 1). Find

the covariances Cov(X10, X11), Cov(X10, X9) and Cov ( X10,

1 2 (X9 + X10)

) .

Data analysis assignment (due 09/09 on Canvas)

1. (10 points) Generate a Normally distributed white noise time series, following N(0, 2), of length T = 500. Apply a moving average filter length 3 and plot the filtered series. Increase the filter length from 3 to 5 to 15. Comment on what happens as the filter length is increased.

2. (10 points) Simulate and plot 500 observations based on the following two autoregressions:

Xt = 0.95Xt−1 + ϵt,

Xt = −0.95Xt−1 + ϵt,

where the error term ϵt is a white noise series ϵt ∼ N(0, 1). Comment on the differences you see between the two autoregressions.

3. (10 points) Textbook Exercise 1.2.

4. (10 points) Textbook Exercise 1.4 (a).

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