4310 : Intertemporal macroeconomics
Preview Markov chains 10/22
Example: Weather transitions
R O
S
1
4
1
4
1
4
1
4
1
2
1
2
1
2
1
2
where
R is rain,
O is overcast, and
S is sunshine.
4310 : Intertemporal macroeconomics
Preview Markov chains 11/22
Represented as a transition matrix
t + 1
R O S
R 0.50 0.25 0.25
t O 0.25 0.50 0.25
S 0.50 0.50 0.00
Such a square array is called the matrix of transition probabilities,
or the transition matrix.
We denote the probability that, given the chain is in state i today,
it will be in state j n days from now p
(n)
ij .
What is the probability that it will be overcast in two days if it is
overcast today?
4310 : Intertemporal macroeconomics
Preview Markov chains 12/22
Represented as a transition matrix
The weather today is known to be overcast. This can represented
by the following vector:
x(0)
= 0 1 0
The weather tomorrow (one day from now) can be predicted by
x(1)
= x(0)
Π = 0 1 0


0.50 0.25 0.25
0.25 0.50 0.25
0.50 0.50 0.00

 = 0.25 0.50 0.25
The weather two days from now can be predicted by
x(2)
= x(1)
Π = 0.25 0.50 0.25


0.50 0.25 0.25
0.25 0.50 0.25
0.50 0.50 0.00

 =


0.3750
0.4375
0.1875


′
4310 : Intertemporal macroeconomics
Preview Markov chains 13/22
cont’d
The weather n days from now can be predicted by
x(n)
= x(0)
Πn
= 0 1 0


0.50 0.25 0.25
0.25 0.50 0.25
0.50 0.50 0.00


n
and in the limit
lim
n→∞
x(n)
= lim
n→∞
x(0)
Πn
= lim
n→∞
0 1 0


0.50 0.25 0.25
0.25 0.50 0.25
0.50 0.50 0.00


n
= 0.4 0.4 0.2