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fix arellano
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mmcky committed Mar 4, 2024
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23 changes: 8 additions & 15 deletions lectures/arellano.md
Original file line number Diff line number Diff line change
Expand Up @@ -709,8 +709,8 @@ The first figure shows the bond price schedule and replicates Figure 3 of
Arellano, where $ y_L $ and $ Y_H $ are particular below average and above average
values of output $ y $.
![https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_bond_prices.png](https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_bond_prices.png)
```{figure} _static/lecture_specific/arellano/arellano_bond_prices.png
```
- $ y_L $ is 5% below the mean of the $ y $ grid values
- $ y_H $ is 5% above the mean of the $ y $ grid values
Expand All @@ -729,16 +729,17 @@ The figure shows that
The next figure plots value functions and replicates the right hand panel of Figure 4 of {cite}`Are08`.
![https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_value_funcs.png](https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_value_funcs.png)
```{figure} _static/lecture_specific/arellano/arellano_value_funcs.png
```
We can use the results of the computation to study the default probability $ \delta(B', y) $
defined in {eq}`equation13_4`.
The next plot shows these default probabilities over $ (B', y) $ as a heat map.
![https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_default_probs.png](https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_default_probs.png)
```{figure} _static/lecture_specific/arellano/arellano_default_probs.png
```
As anticipated, the probability that the government chooses to default in the following period
increases with indebtedness and falls with income.
Expand All @@ -747,19 +748,15 @@ Next let’s run a time series simulation of $ \{y_t\} $, $ \{B_t\} $ and $ q(B_
The grey vertical bars correspond to periods when the economy is excluded from financial markets because of a past default.
![https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_time_series.png](https://python-advanced.quantecon.org/_static/lecture_specific/arellano/arellano_time_series.png)
```{figure} _static/lecture_specific/arellano/arellano_time_series.png
```
One notable feature of the simulated data is the nonlinear response of interest rates.
Periods of relative stability are followed by sharp spikes in the discount rate on government debt.
+++
## Exercises
+++
```{exercise-start}
:label: arellano_ex1
```
Expand All @@ -772,14 +769,10 @@ To the extent that you can, replicate the figures shown above
```{exercise-end}
```
+++
```{solution-start} arellano_ex1
:class: dropdown
```
Solution to this [exercise](https://python-advanced.quantecon.org/arellano.html#arella_ex1).
Compute the value function, policy and equilibrium prices
```{code-cell} ipython3
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