ARCH models represent the residual series as a moving average process. 

Consider the Markov chain whose one-step transition probability matrix is  Which of the following is its transition probability diagram?

Consider the following time series plot: Does this time series appear to be a candidate for a hidden Markov model?  Explain. 

A time series was modeled as a Poisson hidden Markov model with three states.  The R output is below.  An approximate 95% confidence interval for the Poisson mean when in state 3 is

Time series, cross-correlation, and ACF plots for series A and series B are below.  The lag between the two series is approximately two years.

Consider the Markov chain whose one-step transition probability matrix is  What is the period of this Markov chain?

Consider the ARMA(1,2) process  { X t : t ≥ 0 } {“version”:”1.1″,”math”:”\{X_t : t \ge 0\}”}given by  X t = 1 2 X t − 1 + e t + 1 4 e t − 1 − 1 4 e t − 2 . {“version”:”1.1″,”math”:”X_t = \frac{1}{2}X_{t-1}+e_t+\frac{1}{4}e_{t-1}-\frac{1}{4}e_{t-2}.”}Determine if this process is stationary by finding the roots of the AR characteristic polynomial.  Show your work as best you can below.

Chastity is a(n) ________ expression of an inward ________.

According to trends shown in class, cohabitation with children has done what in the last 30 years?

Short Answer Read each question carefully and write your answer on the Written Response Sheets provided with your exam. Be sure to write the number of the question with your answer, and to answer all parts of the question. Each question is worth 10 points; this section is worth 30 points total.