* have heard of the Kalman filter but don't know how it works, or

* know the Kalman filter equations, but don't know where they come from

Subsequent lectures use the same recursive logic in more applied and more econometric settings.

See {doc}`kalman_2` for an economic application in which a firm infers a worker's hidden human capital and effort.

See {doc}`kalman_filter_var` for a derivation of the innovations representation and its connection to vector autoregressions.

For additional (more advanced) reading on the Kalman filter, see

* {cite}`Ljungqvist2012`, section 2.7

```

```{note}

A proof can be found in {cite}`Bishop2006`. To get from his expressions to the ones used above, you will also need to apply the [Woodbury matrix identity](https://en.wikipedia.org/wiki/Woodbury_matrix_identity).

A proof can be found in {cite}`Bishop2006`.

To get from his expressions to the ones used above, you will also need to apply the [Woodbury matrix identity](https://en.wikipedia.org/wiki/Woodbury_matrix_identity).

```

Here $\Sigma G^\top (G \Sigma G^\top + R)^{-1}$ is the matrix of population

These are the standard dynamic equations for the Kalman filter (see, for example, {cite}`Ljungqvist2012`, page 58).

```{note}

Here $\mu_t$ is the filter's prediction of the hidden state $X_t$. In much of the Kalman filter literature it is written $\hat x_t$, emphasizing that it is an estimate of $X_t$.

Here $\mu_t$ is the filter's prediction of the hidden state $X_t$.

In much of the Kalman filter literature it is written $\hat x_t$, emphasizing that it is an estimate of $X_t$.

```

(kalman_convergence)=

Conditions under which a fixed point exists and the sequence $\{\Sigma_t\}$ converges to it are discussed in {cite}`AHMS1996` and {cite}`AndersonMoore2005`, chapter 4.

A sufficient (but not necessary) condition is that all the eigenvalues $\lambda_i$ of $A$ satisfy $|\lambda_i| < 1$ (cf. e.g., {cite}`AndersonMoore2005`, p. 77).

A sufficient (but not necessary) condition is that all the eigenvalues $\lambda_i$ of $A$ satisfy $|\lambda_i| < 1$.

See, for example, {cite}`AndersonMoore2005`, p. 77.

(This strong condition assures that the unconditional distribution of $X_t$ converges as $t \to \infty$.)

:class: dropdown

```

Here is one solution:

```{code-cell} ipython3

# Parameters

θ = 10 # Constant value of state X_t

:class: dropdown

```

Here is one solution:

```{code-cell} ipython3

ϵ = 0.1

θ = 10 # Constant value of state X_t

:class: dropdown

```

Here is one solution:

```{code-cell} ipython3

# Define A, C, G, H

G = np.identity(2)