Skip to content

[additive_functionals] Style guide review #272

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Closed
github-actions wants to merge 1 commit into from
Closed

[additive_functionals] Style guide review #272

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
33 changes: 14 additions & 19 deletions lectures/additive_functionals.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -35,15 +35,15 @@ In addition to what's in Anaconda, this lecture will need the following librarie

## Overview

Many economic time series display persistent growth that prevents them from being asymptotically stationary and ergodic.
Many economic time series display persistent growth that prevents them from being asymptotically stationary and ergodic.

For example, outputs, prices, and dividends typically display irregular but persistent growth.
For example, outputs, prices, and dividends typically display irregular but persistent growth.

Asymptotic stationarity and ergodicity are key assumptions needed to make it possible to learn by applying statistical methods.

But there are good ways to model time series that have persistent growth that still enable statistical learning based on a law of large numbers for an asymptotically stationary and ergodic process.
But there are good ways to model time series that have persistent growth that still enable statistical learning based on a law of large numbers for an asymptotically stationary and ergodic process.

Thus, {cite}`Hansen_2012_Eca` described two classes of time series models that accommodate growth.
Thus, {cite}`Hansen_2012_Eca` described two classes of time series models that accommodate growth.

They are

Expand All @@ -63,9 +63,9 @@ We also describe and compute decompositions of additive and multiplicative proce
1. an asymptotically **stationary** component
1. a **martingale**

We describe how to construct, simulate, and interpret these components.
We describe how to construct, simulate, and interpret these components.

More details about these concepts and algorithms can be found in Hansen {cite}`Hansen_2012_Eca` and Hansen and Sargent {cite}`Hans_Sarg_book`.
More details about these concepts and algorithms can be found in Hansen {cite}`Hansen_2012_Eca` and Hansen and Sargent {cite}`Hans_Sarg_book`.

Let's start with some imports:

Expand All @@ -81,12 +81,12 @@ from scipy.stats import norm, lognorm

{cite}`Hansen_2012_Eca` describes a general class of additive functionals.

This lecture focuses on a subclass of these: a scalar process $\{y_t\}_{t=0}^\infty$ whose increments are driven by a Gaussian vector autoregression.
This lecture focuses on a subclass of these: a scalar process $\{y_t\}_{t=0}^\infty$ whose increments are driven by a gaussian vector autoregression.

Our special additive functional displays interesting time series behavior while also being easy to construct, simulate, and analyze
by using linear state-space tools.

We construct our additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)
We construct our additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)

```{math}
:label: old1_additive_functionals
Expand All @@ -107,7 +107,7 @@ of $\{y_t\}_{t=0}^\infty$ as linear functions of

* a scalar constant $\nu$,
* the vector $x_t$, and
* the same Gaussian vector $z_{t+1}$ that appears in the VAR {eq}`old1_additive_functionals`
* the same gaussian vector $z_{t+1}$ that appears in the VAR {eq}`old1_additive_functionals`

In particular,

Expand All @@ -117,8 +117,7 @@ In particular,
y_{t+1} - y_{t} = \nu + D x_{t} + F z_{t+1}
```

Here $y_0 \sim {\cal N}(\mu_{y0}, \Sigma_{y0})$ is a random
initial condition for $y$.
Here $y_0 \sim {\cal N}(\mu_{y0}, \Sigma_{y0})$ is a random initial condition for $y$.

The nonstationary random process $\{y_t\}_{t=0}^\infty$ displays
systematic but random *arithmetic growth*.
Expand Down Expand Up @@ -714,16 +713,14 @@ Notice the irregular but persistent growth in $y_t$.

### Decomposition

Hansen and Sargent {cite}`Hans_Sarg_book` describe how to construct a decomposition of
an additive functional into four parts:
Hansen and Sargent {cite}`Hans_Sarg_book` describe how to construct a decomposition of an additive functional into four parts:

- a constant inherited from initial values $x_0$ and $y_0$
- a linear trend
- a martingale
- an (asymptotically) stationary component

To attain this decomposition for the particular class of additive
functionals defined by {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals`, we first construct the matrices
To attain this decomposition for the additive functionals defined by {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals`, we first construct the matrices

$$
\begin{aligned}
Expand Down Expand Up @@ -759,8 +756,7 @@ A convenient way to do this is to construct an appropriate instance of a [linear

This will allow us to use the routines in [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) to study dynamics.

To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the
definitions just given,
To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the definitions just given,

$$
\begin{bmatrix}
Expand Down Expand Up @@ -844,8 +840,7 @@ interest.

The class `AMF_LSS_VAR` mentioned {ref}`above <amf_lss>` does all that we want to study our additive functional.

In fact, `AMF_LSS_VAR` does more
because it allows us to study an associated multiplicative functional as well.
In fact, `AMF_LSS_VAR` does more because it allows us to study an associated multiplicative functional as well.

(A hint that it does more is the name of the class -- here AMF stands for
"additive and multiplicative functional" -- the code computes and displays objects associated with
Expand Down