1. Hidden Markov Autoregressive Models

2. A Hidden Markov Model consists of
A sequence of states {Xt|t  T} = {X1, X2, ... , XT} , and
A sequence of observations {Yt |t  T} = {Y1, Y2, ... , YT}

3. The sequence of states {X1, X2,
The sequence of states {X1, X2, ... , XT} form a Markov chain moving amongst the M states {1, 2, …, M}.
The observation Yt comes from a distribution that is determined by the current state of the process Xt. (or possibly past observations and past states).
The states, {X1, X2, ... , XT}, are unobserved (hence hidden).

4. Given Xt = it, Yt-1 = yt-1, Xt-1 = it-1, Yt-2 = yt-2, Xt-2 = it-2, … , Yt-p = yt-p, Xt-p = it-p
The distribution of Yt is normal with mean
and variance

5. Parameters of the Model
P = (pij) = the MM transition matrix where pij = P[Xt+1 = j|Xt = i]
= the initial distribution over the states
where = P[X1 = i]

6. The state means
The state variances
The state autoregressive parameters

7. Simulation of Autoregressive HMM’s
HMM AR.xls

8. Computing Likelihood Assuming that it is known that
Y0 = y0, X0 = i0, Y-1 = y-1, X-1 = i-1, … , Y1-p = y1-p, X1-p = i1-p
Let u1, u2, ... ,uT denote T independent N(0,1) random variables.
Then the joint density of u1, u2, ... ,uT is:

9. Given the sequence of states X1 = i1, X2 = i2, X3 = i3, … , XT = iT
we have:
for t = 1, 2, 3, … , T:

10. The jacobian of this transformation is:
since
and
for s > t

11. Hence the density of y1, y2, ... ,yT given X1 = i1, X2 = i2, X3 = i3, … , XT = iT is
where
for t = 1, 2, 3, … , T:

12. Also
and

13. Efficient Methods for computing Likelihood
The Forward Method
Let
and
Consider

14. This will eventually be used to predict state (probability) from observations

15. Note:
where

16. Then

17. where
Finally

18. The Backward Procedure
Let
and
Define

19. Also
Note: