Are there 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?

The answer provided by Hansen and Scheinkman {cite}`hansen2009long` is yes.

The answer provided by Hansen {cite}`Hansen_2012_Eca` is yes.

They described two classes of time series models that accommodate growth.

He described two classes of time series models that accommodate growth.

They are

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

More details about these concepts and algorithms can be found in Hansen and Sargent {cite}`hansen2008robustness`.

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:

## A Particular Additive Functional

{cite}`hansen2009long` describe a general class of additive functionals.

{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.

When simulating we embed our variables into a bigger system.

This system also constructs the components of the decompositions of $y_t$ and of $\exp(y_t)$ proposed by Hansen and Scheinkman {cite}`hansen2009long`.

This system also constructs the components of the decompositions of $y_t$ and of $\exp(y_t)$ proposed by Hansen {cite}`Hansen_2012_Eca`.

All of these objects are computed using the code below

\end{aligned}

$$

Then the Hansen-Scheinkman {cite}`hansen2009long`, {cite}`Hans_Sarg_book` decomposition is

Then the Hansen {cite}`Hansen_2012_Eca`, {cite}`Hans_Sarg_book` decomposition is

$$

\begin{aligned}