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[time_series_with_matrices] Address feedbacks #534

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[time_series_with_matrices] Address feedbacks #534

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88d4d21
address feedbacks from reading group
HumphreyYang Jul 23, 2024
1a0a299
update based on feedback
HumphreyYang Jul 26, 2024
0c33b84
remove "for"
HumphreyYang Jul 26, 2024
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66 changes: 36 additions & 30 deletions lectures/time_series_with_matrices.md
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Original file line number Diff line number Diff line change
Expand Up @@ -4,7 +4,7 @@ jupytext:
extension: .md
format_name: myst
format_version: 0.13
jupytext_version: 1.16.2
jupytext_version: 1.16.1
kernelspec:
display_name: Python 3 (ipykernel)
language: python
Expand Down Expand Up @@ -49,7 +49,12 @@ We will use the following imports:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
plt.rcParams["figure.figsize"] = (11, 5) #set default figure size

# Custom figsize for this lecture
plt.rcParams["figure.figsize"] = (11, 5)

# Set decimal printing to 3 decimal places
np.set_printoptions(precision=3, suppress=True)
```

## Samuelson's model
Expand Down Expand Up @@ -142,8 +147,8 @@ T = 80
α_1 = 1.53
α_2 = -.9

y_neg1 = 28. # y_{-1}
y_0 = 24.
y_neg1 = 28.0 # y_{-1}
y_0 = 24.0
```

Now we construct $A$ and $b$.
Expand Down Expand Up @@ -197,6 +202,13 @@ point precision:
np.allclose(y, y_second_method)
```

$A$ is invertible as it is lower triangular and [its diagonal entries are non-zero](https://www.statlect.com/matrix-algebra/triangular-matrix)

```{code-cell} ipython3
# Check if A is lower triangular
np.allclose(A, np.tril(A))
```

```{note}
In general, `np.linalg.solve` is more numerically stable than using
`np.linalg.inv` directly.
Expand Down Expand Up @@ -392,7 +404,13 @@ class population_moments:
Parameters:
α_0, α_1, α_2, T, y_neg1, y_0
"""
def __init__(self, α_0, α_1, α_2, T, y_neg1, y_0, σ_u):
def __init__(self, α_0=10.0,
α_1=1.53,
α_2=-.9,
T=80,
y_neg1=28.0,
y_0=24.0,
σ_u=1):

# compute A
A = np.identity(T)
Expand Down Expand Up @@ -437,8 +455,7 @@ class population_moments:
return self.μ_y, self.Σ_y


series_process = population_moments(
α_0=10.0, α_1=1.53, α_2=-.9, T=80, y_neg1=28., y_0=24., σ_u=1)
series_process = population_moments()

μ_y, Σ_y = series_process.get_moments()
A_inv = series_process.A_inv
Expand Down Expand Up @@ -483,12 +500,17 @@ Notice how the population variance increases and asymptotes.
Let's print out the covariance matrix $\Sigma_y$ for a time series $y$.

```{code-cell} ipython3
series_process = population_moments(
α_0=0, α_1=.8, α_2=0, T=6, y_neg1=0., y_0=0., σ_u=1)
series_process = population_moments(α_0=0,
α_1=.8,
α_2=0,
T=6,
y_neg1=0.,
y_0=0.,
σ_u=1)

μ_y, Σ_y = series_process.get_moments()
print("μ_y = ", μ_y)
print("Σ_y = ", Σ_y)
print("Σ_y = \n", Σ_y)
```

Notice that the covariance between $y_t$ and $y_{t-1}$ -- the elements on the superdiagonal -- are *not* identical.
Expand All @@ -502,9 +524,9 @@ We describe how to do that in [Linear State Space Models](https://python.quantec
But just to set the stage for that analysis, let's print out the bottom right corner of $\Sigma_y$.

```{code-cell} ipython3
series_process = population_moments(
α_0=10.0, α_1=1.53, α_2=-.9, T=80, y_neg1=28., y_0=24., σ_u=1)
series_process = population_moments()
μ_y, Σ_y = series_process.get_moments()

print("bottom right corner of Σ_y = \n", Σ_y[72:,72:])
```

Expand All @@ -529,26 +551,13 @@ To study the structure of $A^{-1}$, we shall print just up to $3$ decimals.
Let's begin by printing out just the upper left hand corner of $A^{-1}$.

```{code-cell} ipython3
with np.printoptions(precision=3, suppress=True):
print(A_inv[0:7,0:7])
print(A_inv[0:7,0:7])
```

Evidently, $A^{-1}$ is a lower triangular matrix.


Let's print out the lower right hand corner of $A^{-1}$ and stare at it.

```{code-cell} ipython3
with np.printoptions(precision=3, suppress=True):
print(A_inv[72:,72:])
```

Notice how every row ends with the previous row's pre-diagonal entries.





Since $A^{-1}$ is lower triangular, each row represents $ y_t$ for a particular $t$ as the sum of
- a time-dependent function $A^{-1} b$ of the initial conditions incorporated in $b$, and
- a weighted sum of current and past values of the IID shocks $\{u_t\}$.
Expand All @@ -566,9 +575,6 @@ This is a **moving average** representation with time-varying coefficients.
Just as system {eq}`eq:eqma` constitutes a
**moving average** representation for $y$, system {eq}`eq:eqar` constitutes an **autoregressive** representation for $y$.




## A forward looking model

Samuelson’s model is *backward looking* in the sense that we give it *initial conditions* and let it
Expand Down Expand Up @@ -638,7 +644,7 @@ for i in range(T):
```

```{code-cell} ipython3
B
print(B)
```

```{code-cell} ipython3
Expand Down