This week we will look at how to simulate Markov chains, and in
particular, how to gain an empirical understanding of their
stationary distributions.
First read carefully through the following example, trying it out as you go along, then tackle the
exercises below.
Try simulating AR(1) processes with different parameter
values. Note that you will have problems with parameter values outside
the range [-1,1]! Try starting off with parameters inside this range,
and see what happens as you get close to the boundaries. eg. try 0.9,
0.95, 0.99 on the positive side, and -0.9, -0.95, -0.99 on the
negative side. A parameter value of -1 often produces an interesting
plot! Notice how the chain
converges much more quickly for some parameter values than
others. Note also how the location, scale and shape of the stationary
distribution changes as the parameter values change.
2.
What is the theoretical asymptotic mean and variance of the
process, in terms of the parameter, a? Make sure you have the
right answer by comparing with your empirical findings for
various values of a.
3.
What is the (approximate) standard error of the sample mean
as a function of n and a? Hint: remember that the
observations are not independent! Use the information in your lecture
notes.
4.
For a particular sample of 10,000 that you generate for
a=0.995, construct a confidence interval for the mean. Pretend
that you don't know a, and use the sample lag 1 autocorrelation
to estimate it. Compare that to the confidence interval you would have
if the observations were independent.
5.
Calculate the sample lag 2 autocorrelation in the chain
from the previous question, and use this to verify that the assumption
of an underlying AR(1) process is reasonable.
6. (for postgrads)
Try out the C code for chain
generation. See if you can use the GSL for variate generation. Try out
the simple method for calling executables from R. Then try compiling
your C code as a shared object, and dynamically loading it into R, and
calling it directly.