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140 changes: 61 additions & 79 deletions lectures/additive_functionals.md
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Expand Up @@ -41,13 +41,13 @@ For example, outputs, prices, and dividends typically display irregular but per

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 with persistent growth while enabling 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

1. **additive functionals** that display random "arithmetic growth"
1. additive functionals that display random "arithmetic growth"
1. **multiplicative functionals** that display random "geometric growth"

These two classes of processes are closely connected.
Expand All @@ -58,14 +58,14 @@ In this lecture, we describe both additive functionals and multiplicative funct

We also describe and compute decompositions of additive and multiplicative processes into four components:

1. a **constant**
1. a **trend** component
1. an asymptotically **stationary** component
1. a **martingale**
1. a *constant*
1. a *trend* component
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 @@ -83,10 +83,9 @@ from scipy.stats import norm, lognorm

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.
Our special additive functional displays interesting time series behavior and is easy to construct, simulate, and analyze 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 @@ -98,7 +97,7 @@ Here

* $x_t$ is an $n \times 1$ vector,
* $A$ is an $n \times n$ stable matrix (all eigenvalues lie within the open unit circle),
* $z_{t+1} \sim {\cal N}(0,I)$ is an $m \times 1$ IID shock,
* $z_{t+1} \sim {\cal N}(0,I)$ is an $m \times 1$ iid shock,
* $B$ is an $n \times m$ matrix, and
* $x_0 \sim {\cal N}(\mu_0, \Sigma_0)$ is a random initial condition for $x$

Expand All @@ -125,7 +124,7 @@ systematic but random *arithmetic growth*.

### Linear state-space representation

A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
We can conveniently represent our additive functional using a [linear state space system](https://python-intro.quantecon.org/linear_models.html).

To do this, we set up state and observation vectors

Expand Down Expand Up @@ -211,7 +210,7 @@ $$

are strictly greater than unity in absolute value.

(Being a zero of $\phi(z)$ means that $\phi(z) = 0$)
Being a zero of $\phi(z)$ means that $\phi(z) = 0$.

Let the increment in $\{y_t\}$ obey

Expand All @@ -223,17 +222,17 @@ with an initial condition for $y_0$.

While {eq}`ftaf` is not a first order system like {eq}`old1_additive_functionals`, we know that it can be mapped into a first order system.

* For an example of such a mapping, see [this example](https://python.quantecon.org/linear_models.html#second-order-difference-equation).
For an example of such a mapping, see [this example](https://python.quantecon.org/linear_models.html#second-order-difference-equation).

In fact, this whole model can be mapped into the additive functional system definition in {eq}`old1_additive_functionals` -- {eq}`old2_additive_functionals` by appropriate selection of the matrices $A, B, D, F$.

You can try writing these matrices down now as an exercise --- correct expressions appear in the code below.

### Simulation

When simulating we embed our variables into a bigger system.
We embed our variables into a larger system when simulating.

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

All of these objects are computed using the code below

Expand Down Expand Up @@ -674,7 +673,7 @@ def plot_martingales(amf, T, npaths=25):
return mart_figs
```

For now, we just plot $y_t$ and $x_t$, postponing until later a description of exactly how we compute them.
We plot $y_t$ and $x_t$, postponing until later a description of how we compute them.

(addfunc_egcode)=

Expand Down Expand Up @@ -714,16 +713,9 @@ 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 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, and an (asymptotically) stationary component.

- 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 obtain this decomposition for additive functionals defined by {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals`, we first construct the matrices

$$
\begin{aligned}
Expand Down Expand Up @@ -755,7 +747,7 @@ It is convenient for us to introduce the following notation:

We want to characterize and simulate components $\tau_t, m_t, s_t$ of the decomposition.

A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) by using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
We can do this by constructing an instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).

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

Expand Down Expand Up @@ -844,8 +836,7 @@ 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 All @@ -865,12 +856,11 @@ When we plot multiple realizations of a component in the 2nd, 3rd, and 4th panel

We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$ (you can tell this from the shape of the 95% probability coverage shaded areas).

Notice tell-tale signs of these probability coverage shaded areas
Notice tell-tale signs of these probability coverage shaded areas.

The purple shaded area for the martingale component $m_t$ grows with $\sqrt{t}$.

* the purple one for the martingale component $m_t$ grows with
$\sqrt{t}$
* the green one for the stationary component $s_t$ converges to a
constant band
The green shaded area for the stationary component $s_t$ converges to a constant band.

### Associated multiplicative functional

Expand Down Expand Up @@ -917,7 +907,7 @@ plot_multiplicative(amf, T)
plt.show()
```

As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LinearStateSpace class.
As before, the 2nd, 3rd, and 4th panels show multiple realizations of a component along with population 95% confidence bands computed using the LinearStateSpace class.

Comparing this figure and the last also helps show how geometric growth differs from
arithmetic growth.
Expand All @@ -931,18 +921,14 @@ Let's see what happens when we set $T = 12000$ instead of $150$.

### Peculiar large sample property

Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the martingale component
$\widetilde M_t$ of the multiplicative decomposition
Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe two properties of the martingale component $\widetilde M_t$ of the multiplicative decomposition.

* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$,
nevertheless $\ldots$
* as $t \rightarrow +\infty$, $\widetilde M_t$ converges to
zero almost surely
While $E_0 \widetilde M_t = 1$ for all $t \geq 0$, nevertheless as $t \rightarrow +\infty$, $\widetilde M_t$ converges to zero almost surely.

The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition
$\widetilde M_0 = 1$.

The second is a **peculiar property** noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.
The second is a *peculiar property* noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.

The following simulation of many paths of $\widetilde M_t$ illustrates both properties

Expand All @@ -960,8 +946,7 @@ The purple 95 percent frequency coverage interval collapses around zero, illustr

## More about the multiplicative martingale

Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in
more detail.
Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in more detail.

As we have seen, it has representation

Expand All @@ -971,14 +956,13 @@ $$

where $H = [F + D(I-A)^{-1} B]$.

It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$ and that consequently ${\widetilde M}_t$ is log normal.
It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$, and consequently ${\widetilde M}_t$ is log normal.

### Simulating a multiplicative martingale again

Next, we want a program to simulate the likelihood ratio process $\{ \tilde{M}_t \}_{t=0}^\infty$.

In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and
$[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and $[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.

After accomplishing this, we want to display and study histograms of $\tilde{M}_T^i$ for various values of $T$.

Expand Down Expand Up @@ -1022,23 +1006,23 @@ class AMF_LSS_VAR:

# Build A matrix for LSS
# Order of states is: [1, t, xt, yt, mt]
A1 = np.hstack([1, 0, 0, 0, 0]) # Transition for 1
A2 = np.hstack([1, 1, 0, 0, 0]) # Transition for t
A3 = np.hstack([0, 0, A, 0, 0]) # Transition for x_{t+1}
A4 = np.hstack([ν, 0, D, 1, 0]) # Transition for y_{t+1}
A5 = np.hstack([0, 0, 0, 0, 1]) # Transition for m_{t+1}
A1 = np.hstack([1, 0, 0, 0, 0]) # transition for 1
A2 = np.hstack([1, 1, 0, 0, 0]) # transition for t
A3 = np.hstack([0, 0, A, 0, 0]) # transition for x_{t+1}
A4 = np.hstack([ν, 0, D, 1, 0]) # transition for y_{t+1}
A5 = np.hstack([0, 0, 0, 0, 1]) # transition for m_{t+1}
Abar = np.vstack([A1, A2, A3, A4, A5])

# Build B matrix for LSS
Bbar = np.vstack([0, 0, B, F, H])

# Build G matrix for LSS
# Order of observation is: [xt, yt, mt, st, tt]
G1 = np.hstack([0, 0, 1, 0, 0]) # Selector for x_{t}
G2 = np.hstack([0, 0, 0, 1, 0]) # Selector for y_{t}
G3 = np.hstack([0, 0, 0, 0, 1]) # Selector for martingale
G4 = np.hstack([0, 0, -g, 0, 0]) # Selector for stationary
G5 = np.hstack([0, ν, 0, 0, 0]) # Selector for trend
G1 = np.hstack([0, 0, 1, 0, 0]) # selector for x_{t}
G2 = np.hstack([0, 0, 0, 1, 0]) # selector for y_{t}
G3 = np.hstack([0, 0, 0, 0, 1]) # selector for martingale
G4 = np.hstack([0, 0, -g, 0, 0]) # selector for stationary
G5 = np.hstack([0, ν, 0, 0, 0]) # selector for trend
Gbar = np.vstack([G1, G2, G3, G4, G5])

# Build H matrix for LSS
Expand Down Expand Up @@ -1133,7 +1117,7 @@ def population_means(amf, T=150):
# Pull out moment generator
moment_generator = amf.lss.moment_sequence()

for tt in range (T):
for tt in range(T):
tmoms = next(moment_generator)
ymeans = tmoms[1]
xmean[tt] = ymeans[0]
Expand All @@ -1142,8 +1126,7 @@ def population_means(amf, T=150):
return xmean, ymean
```

Now that we have these functions in our toolkit, let's apply them to run some
simulations.
Now that we have these functions in our toolkit, let's apply them to run some simulations.

```{code-cell} ipython3
def simulate_martingale_components(amf, T=1000, I=5000):
Expand All @@ -1157,7 +1140,7 @@ def simulate_martingale_components(amf, T=1000, I=5000):
for i in range(I):
foo, bar = amf.lss.simulate(T)

# Martingale component is third component
# martingale component is third component
add_mart_comp[i, :] = bar[2, :]

mul_mart_comp = np.exp(add_mart_comp - (np.arange(T) * H**2)/2)
Expand All @@ -1173,16 +1156,15 @@ amc, mmc = simulate_martingale_components(amf_2, 1000, 5000)
amcT = amc[:, -1]
mmcT = mmc[:, -1]

print("The (min, mean, max) of additive Martingale component in period T is")
print("The (min, mean, max) of additive martingale component in period T is")
print(f"\t ({np.min(amcT)}, {np.mean(amcT)}, {np.max(amcT)})")

print("The (min, mean, max) of multiplicative Martingale component \
print("The (min, mean, max) of multiplicative martingale component \
in period T is")
print(f"\t ({np.min(mmcT)}, {np.mean(mmcT)}, {np.max(mmcT)})")
```

Let's plot the probability density functions for $\log {\widetilde M}_t$ for
$t=100, 500, 1000, 10000, 100000$.
Let's plot the probability density functions for $\log {\widetilde M}_t$ for $t=100, 500, 1000, 10000, 100000$.

Then let's use the plots to investigate how these densities evolve through time.

Expand Down Expand Up @@ -1248,21 +1230,21 @@ plt.tight_layout()
plt.show()
```

These probability density functions help us understand mechanics underlying the **peculiar property** of our multiplicative martingale
These probability density functions help us understand mechanics underlying the **peculiar property** of our multiplicative martingale.

As $T$ grows, most of the probability mass shifts leftward toward zero.

For example, note that most mass is near $1$ for $T =10$ or $T = 100$ but most of it is near $0$ for $T = 5000$.

As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.

* As $T$ grows, most of the probability mass shifts leftward toward zero.
* For example, note that most mass is near $1$ for $T =10$ or $T = 100$ but
most of it is near $0$ for $T = 5000$.
* As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.
* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$
even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.
Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$ even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.

### Multiplicative martingale as likelihood ratio process

[This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies **likelihood processes**
and **likelihood ratio processes**.
[This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies likelihood processes
and likelihood ratio processes.

A **likelihood ratio process** is a multiplicative martingale with mean unity.
A likelihood ratio process is a multiplicative martingale with mean unity.

Likelihood ratio processes exhibit the peculiar property that naturally also appears
[here](https://python.quantecon.org/likelihood_ratio_process.html).
Likelihood ratio processes exhibit the peculiar property that also appears [here](https://python.quantecon.org/likelihood_ratio_process.html).