Time series, cross-correlation, and ACF plots for series A and series B are below.  Both series exhibit possible periodicity, with period of about six months.

Consider the Markov chain whose one-step transition probability matrix is  Write/type the limiting distribution, if it exists.

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

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

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 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 ________.