CALIFORNIA STATE UNIVERSITY DEPARTMENT OF STATISTICS

Statistics 6871: Seminar - Introduction to Times Series Analysis Summer 2003

Midterm

Instructions: This is a take-home midterm. You may use your book, notes, and you may consult teacher if you have questions. You may not collaborate with other students in the class.

1. (Stat 4866/6871) Consider a time series {x_{t}} consisting of a quadratic trend and additive independent white noise {w_{t}} with mean 0 and variance σ^{2}. That is

x_{t} = α_{0} + α_{1}t + α_{2}t^{2} + w_{t}  (1)

where α_{0}, α_{1}, and α_{2} are unknown parameters.

(a) Show that {x_{t}} is nonstationary.

(b) Suggest a transformation of {x_{t}} that is stationary and prove that it is in fact stationary.

2. (Stat 6871) Identify the following ARMA models and determine if they are causal and/or invertible.

(a) (1 − 1.05B + 0.4B^{2})x_{t} = w_{t}

(b) (1 − 1.05B)x_{t} = w_{t}

(c) (1 + 0.8B)x_{t} = (1 − 0.25B)w_{t}

3. (Stat 6871) Consider the ARIMA(0,1,1) model written as

x_{t} = x_{t−1} + w_{t} − θw_{t−1}  (2)

where |θ| < 1.

(a) Show that the model is invertible and solve for the {π_{j}} coefficients.

(b) Re-write the invertible representation of the model where x_{t} is equal to past values of time series and white noise.

(c) Determine the one-step ahead prediction formula by taking the conditional expectation. Note that

it can be written in the form

x̃_{n+1} = (1 − θ)x_{n} + θx̃_{n}  (3)

for , n > 1.

(d) Indicate how the prediction formula could be used in practice. In particular indicate where truncation will be need to make the calculations. (Remark: The prediction formula derived is a forecasting method known as exponentially weighted moving averages, (EWMA).)

4. (Stat 4866/6871) Consider the following pairs of ACF’s and PACF’s. In each case indicate whether the pairs of diagrams is associated with an AR or MA process, state the order of the process, and in each case write an appropriate model that you think would fit the data well.