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Dec 20, 2023
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Typos in markov_chains_I.md #327

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Update markov_chains_I.md
Jingni0117 Dec 14, 2023
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fix markup issues
mmcky Dec 20, 2023
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15 changes: 4 additions & 11 deletions lectures/markov_chains_I.md
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Original file line number Diff line number Diff line change
Expand Up @@ -107,8 +107,6 @@ From US unemployment data, Hamilton {cite}`Hamilton2005` estimated the followin

```

+++

Here there are three **states**

* "ng" represents normal growth
Expand Down Expand Up @@ -294,7 +292,7 @@ Looking at the data, we see that democracies tend to have longer-lasting growth
regimes compared to autocracies (as indicated by the lower probability of
transitioning from growth to growth in autocracies).

We can also find a higher probability from collapse to growth in democratic regimes
We can also find a higher probability from collapse to growth in democratic regimes.


### Defining Markov chains
Expand Down Expand Up @@ -411,7 +409,6 @@ def mc_sample_path(P, ψ_0=None, ts_length=1_000):
X = np.empty(ts_length, dtype=int)

# Convert each row of P into a cdf
n = len(P)
P_dist = np.cumsum(P, axis=1) # Convert rows into cdfs

# draw initial state, defaulting to 0
Expand Down Expand Up @@ -683,7 +680,7 @@ P = np.array([[0.4, 0.6],
ψ @ P
```

Notice that `ψ @ P` is the same as `ψ`
Notice that `ψ @ P` is the same as `ψ`.



Expand Down Expand Up @@ -772,11 +769,11 @@ For example, we have the following result
(strict_stationary)=
```{prf:theorem}
Theorem: If there exists an integer $m$ such that all entries of $P^m$ are
strictly positive, with unique stationary distribution $\psi^*$, and
strictly positive, with unique stationary distribution $\psi^*$, then

$$
\psi_0 P^t \to \psi^*
\quad \text{as } t \to \infty
\quad \text{ as } t \to \infty
$$
```

Expand Down Expand Up @@ -837,8 +834,6 @@ ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='k', s=60)
plt.show()
```

+++ {"user_expressions": [], "tags": []}

Here

* $P$ is the stochastic matrix for recession and growth {ref}`considered above <mc_eg2>`.
Expand Down Expand Up @@ -1083,8 +1078,6 @@ Solution 1:

```

+++

Since the matrix is everywhere positive, there is a unique stationary distribution.

Solution 2:
Expand Down