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fix syntax for example
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mmcky committed Jul 15, 2024
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Expand Up @@ -636,17 +636,15 @@ The probability of being in recession (either mild or severe) in 6 months' time
$$
(\psi_t P^6)(1) + (\psi_t P^6)(2)
$$
```{index} single: Markov Chains; Cross-Sectional Distributions
```

```{index} single: Markov Chains; Cross-Sectional Distributions
```

```{prf:example} Cross-Sectional Distributions
````{prf:example} Cross-Sectional Distributions
:label: cross-sectional-distributions
### Example 2: Cross-Sectional Distributions
The distributions we have been studying can be viewed either
1. as probabilities or
Expand Down Expand Up @@ -674,16 +672,15 @@ So for a very large (tending to infinite) population, $\psi_t P^{10}$ also repre
This is exactly the cross-sectional distribution.
```{note}
```{note}
A cross-sectional frequency measures how a particular variable (e.g., employment status) is distributed across a population at a specific time, providing information on the proportions of individuals in each possible state of that variable.
```

````

(stationary)=
## Stationary distributions


As seen in {eq}`fin_mc_fr`, we can shift a distribution forward one
unit of time via postmultiplication by $P$.

Expand All @@ -698,8 +695,6 @@ P = np.array([[0.4, 0.6],

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



Such distributions are called **stationary** or **invariant**.

(mc_stat_dd)=
Expand Down Expand Up @@ -753,7 +748,7 @@ corresponds to unemployment (state 0).
Using $\psi^* = \psi^* P$ and a bit of algebra yields

$$
p = \frac{\beta}{\alpha + \beta}
p = \frac{\beta}{\alpha + \beta}
$$

This is, in some sense, a steady state probability of unemployment.
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