* productivity, etc.

We are going to study AR(1) processes partly because they are useful and

partly because they help us understand important concepts.

partly because they help us understand important concepts.

Let's start with some imports:

X_{t+1} = a X_t + b + c W_{t+1}

```

where $a, b, c$ are scalar-valued parameters

where $a, b, c$ are scalar-valued parameters

(Equation {eq}`can_ar1` is sometimes called a **stochastic difference equation**.)

```{prf:example}

:label: ar1_ex_ar

For example, $X_t$ might be

For example, $X_t$ might be

* the log of labor income for a given household, or

* the log of money demand in a given economy.

And we can use a theoretical AR(1) model to calculate the right hand side.

If $\frac{1}{m} \sum_{t = 1}^m X_t$ is not close to $\psi^*(x)$, even for many

observations, then our theory seems to be incorrect and we will need to revise

it.

If $\frac{1}{m} \sum_{t = 1}^m h(X_t)$ is not close to $\int h(x)\psi^*(x) dx$, even for many observations, then our theory seems to be incorrect and we will need to revise it.

## Exercises

return 0

k_vals = np.arange(6) + 1

sample_moments = np.empty_like(k_vals)

true_moments = np.empty_like(k_vals)

sample_moments = np.empty(len(k_vals), dtype=float)

true_moments = np.empty(len(k_vals), dtype=float)

for k_idx, k in enumerate(k_vals):

sample_moments[k_idx] = sample_moments_ar1(k)

For $K$ use the Gaussian kernel ($K$ is the standard normal

density).

Write the class so that the bandwidth defaults to Silverman’s rule (see

the “rule of thumb” discussion on [this

Write the class so that the bandwidth defaults to Silverman's rule (see

the "rule of thumb" discussion on [this

page](https://en.wikipedia.org/wiki/Kernel_density_estimation)). Test

the class you have written by going through the steps