(To approximate the probability of event $E$, we can simulate many times and count the fraction of times that $E$ occurs).

Nice functionality for simulating Markov chains exists in [QuantEcon.py](http://quantecon.org/quantecon-py).

Nice functionality for simulating Markov chains exists in [QuantEcon.py](https://quantecon.org/quantecon-py).

* Efficient, bundled with lots of other useful routines for handling Markov chains.

However, it's also a good exercise to roll our own routines --- let's do that first and then come back to the methods in [QuantEcon.py](http://quantecon.org/quantecon-py).

However, it's also a good exercise to roll our own routines --- let's do that first and then come back to the methods in [QuantEcon.py](https://quantecon.org/quantecon-py).

In these exercises, we'll take the state space to be $S = 0,\ldots, n-1$.

To implement this simulation procedure, we need a method for generating draws from a discrete distribution.

For this task, we'll use `random.draw` from [QuantEcon](http://quantecon.org/quantecon-py), which works as follows:

For this task, we'll use `random.draw` from [QuantEcon](https://quantecon.org/quantecon-py), which works as follows:

```{code-cell} python3

ψ = (0.3, 0.7) # probabilities over {0, 1}

### Using QuantEcon's Routines

As discussed above, [QuantEcon.py](http://quantecon.org/quantecon-py) has routines for handling Markov chains, including simulation.

As discussed above, [QuantEcon.py](https://quantecon.org/quantecon-py) has routines for handling Markov chains, including simulation.

Here's an illustration using the same P as the preceding example

np.mean(X == 0)

```

The [QuantEcon.py](http://quantecon.org/quantecon-py) routine is [JIT compiled](https://python-programming.quantecon.org/numba.html#numba-link) and much faster.

The [QuantEcon.py](https://quantecon.org/quantecon-py) routine is [JIT compiled](https://python-programming.quantecon.org/numba.html#numba-link) and much faster.

```{code-cell} ipython

%time mc_sample_path(P, sample_size=1_000_000) # Our homemade code version

It's clear from the graph that this stochastic matrix is irreducible: we can eventually

reach any state from any other state.

We can also test this using [QuantEcon.py](http://quantecon.org/quantecon-py)'s MarkovChain class

We can also test this using [QuantEcon.py](https://quantecon.org/quantecon-py)'s MarkovChain class

```{code-cell} python3

P = [[0.9, 0.1, 0.0],

$\psi$ such that $\psi = \psi P$ is a left eigenvector associated

with the unit eigenvalue $\lambda = 1$.

A stable and sophisticated algorithm specialized for stochastic matrices is implemented in [QuantEcon.py](http://quantecon.org/quantecon-py).

A stable and sophisticated algorithm specialized for stochastic matrices is implemented in [QuantEcon.py](https://quantecon.org/quantecon-py).

This is the one we recommend:

(new_interp_sd)=

This gives us another way to interpret the stationary distribution --- provided that the convergence result in {eq}`llnfmc0` is valid.

The convergence asserted in {eq}`llnfmc0` is a special case of a law of large numbers result for Markov chains --- see [EDTC](http://johnstachurski.net/edtc.html), section 4.3.4 for some additional information.

The convergence asserted in {eq}`llnfmc0` is a special case of a law of large numbers result for Markov chains --- see [EDTC](https://johnstachurski.net/edtc.html), section 4.3.4 for some additional information.

(mc_eg1-2)=

### Example

Tauchen's method {cite}`Tauchen1986` is the most common method for approximating this continuous state process with a finite state Markov chain.

A routine for this already exists in [QuantEcon.py](http://quantecon.org/quantecon-py) but let's write our own version as an exercise.

A routine for this already exists in [QuantEcon.py](https://quantecon.org/quantecon-py) but let's write our own version as an exercise.

As a first step, we choose

$\{x_0, \ldots, x_{n-1}\} \subset \mathbb R$ and $n \times n$ matrix

$P$ as described above.

* Even better, write a function that returns an instance of [QuantEcon.py's](http://quantecon.org/quantecon-py) MarkovChain class.

* Even better, write a function that returns an instance of [QuantEcon.py's](https://quantecon.org/quantecon-py) MarkovChain class.

```

```{solution} fm_ex3

:class: dropdown

A solution from the [QuantEcon.py](http://quantecon.org/quantecon-py) library

A solution from the [QuantEcon.py](https://quantecon.org/quantecon-py) library

can be found [here](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/markov/approximation.py).

```