Zero Mean Series - Time Series Analysis - Exam, Exams of Mathematical Statistics

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STAT 479 - Sample Exam II

1. (a) Define what it means for a series to be invertible.

(b) Is the zero-mean series defined by

Xt = Xt− 1 −. 5 Xt− 2 + wt

stationary? Why or why not?

2. Consider the following model for the movement of a certain type of particle. In the tth^ unit of time, the particle travels a distance wt, where w 1 , w 2 ,... are independent random variables with a mean of zero and a variance of σ^2 w. A positive value of w represents a forward movement, a negative value a backwards movement. Let Xt represent the position of the particle after the tth^ time period (X 0 = 0).

(a) Identify the series {Xt} as a particular ARIMA model. (b) Is the series weakly stationary? Why or why not?

3. (a) Identify the following as a particular ARIMA(p, d, q) × (P, D, Q)s model:

Xt = Xt− 1 + Xt− 12 − Xt− 13 + wt − θwt− 1 − Θwt− 12 + θΘwt− 13.

(b) Write out an expression, similar to that given in part (a), for an ARIMA(0, 1 , 1)× (1, 0 , 1) 4 model.

4. Consider the problem of predicting a random variable Y , on the basis of data repre- sented by a random variable X. Show that the function of X which minimizes the mean squared error MSE = E[{Y − g(X)}^2 ] is g(X) = E[Y |X], the conditional expectation of Y , given X.
5. Suppose one is given data {xt}Tt=1 from the stationary AR(1) model Xt = φXt− 1 + wt, with normally distributed white noise. Show that the MLE of φ is

φ^ ˆ =

PT − 1

P^ t=1^ xtxt+ T − 1 t=1 x (^2) t^.