$$

\begin{aligned}

x_{t+1} & = A x_t + C w_{t+1}

X_{t+1} & = A X_t + C w_{t+1}

\\

y_t & = G x_t + H v_t

Y_t & = G X_t + H v_t

\end{aligned}

$$

where the shocks $w_t$ and $v_t$ are IID standard normals.

where $X_t$ and $Y_t$ denote random variables, and the shocks $w_t$ and $v_t$ are IID standard normals.

To connect this with the notation of this lecture we set

Suppose that

* all variables are scalars

* the hidden state $\{x_t\}$ is in fact constant, equal to some $\theta \in \mathbb{R}$ unknown to the modeler

* the hidden state $\{X_t\}$ is in fact constant, equal to some $\theta \in \mathbb{R}$ unknown to the modeler

State dynamics are therefore given by {eq}`kl_xdynam` with $A=1$, $Q=0$ and $x_0 = \theta$.

State dynamics are therefore given by {eq}`kl_xdynam` with $A=1$, $Q=0$ and $X_0 = \theta$.

The measurement equation is $y_t = \theta + v_t$ where $v_t$ is $N(0,1)$ and IID.

The measurement equation is $Y_t = \theta + v_t$ where $v_t$ is $N(0,1)$ and IID.

The task of this exercise to simulate the model and, using the code from `kalman.py`, plot the first five predictive densities $p_t(x) = N(\mu_t, \Sigma_t)$.

The task of this exercise is to simulate the model and, using the code from `kalman.py`, plot the first five predictive densities $p_t(x) = N(\mu_t, \Sigma_t)$ for $X_t$.

As shown in {cite}`Ljungqvist2012`, sections 2.9.1--2.9.2, these distributions asymptotically put all mass on the unknown value $\theta$.

```{code-cell} ipython3

# Parameters

θ = 10 # Constant value of state x_t

θ = 10 # Constant value of state X_t

A, C, G, H = 1, 0, 1, 1

ss = LinearStateSpace(A, C, G, H, mu_0=θ)

```{code-cell} ipython3

ϵ = 0.1

θ = 10 # Constant value of state x_t

θ = 10 # Constant value of state X_t

A, C, G, H = 1, 0, 1, 1

ss = LinearStateSpace(A, C, G, H, mu_0=θ)

:label: kalman_ex3

```

As discussed {ref}`above <kalman_convergence>`, if the shock sequence $\{W_t\}$ is not degenerate, then it is not in general possible to predict $x_t$ without error at time $t-1$ (and this would be the case even if we could observe $x_{t-1}$).

As discussed {ref}`above <kalman_convergence>`, if the shock sequence $\{W_t\}$ is not degenerate, then it is not in general possible to predict $X_t$ without error at time $t-1$ (and this would be the case even if we could observe $X_{t-1}$).

Let's now compare the prediction $\mu_t$ made by the Kalman filter

against a competitor who **is** allowed to observe $x_{t-1}$.

against a competitor who **is** allowed to observe $X_{t-1}$.

This competitor will use the conditional expectation $\mathbb E[ x_t

\,|\, x_{t-1}]$, which in this case is $A x_{t-1}$.

This competitor will use the conditional expectation $\mathbb E[ X_t

\,|\, X_{t-1}]$, which in this case is $A X_{t-1}$.

The conditional expectation is known to be the optimal prediction method in terms of minimizing mean squared error.

(More precisely, the minimizer of $\mathbb E \, \| x_t - g(x_{t-1}) \|^2$ with respect to $g$ is $g^*(x_{t-1}) := \mathbb E[ x_t \,|\, x_{t-1}]$)

(More precisely, the minimizer of $\mathbb E \, \| X_t - g(X_{t-1}) \|^2$ with respect to $g$ is $g^*(X_{t-1}) := \mathbb E[ X_t \,|\, X_{t-1}]$)

Thus we are comparing the Kalman filter against a competitor who has more

information (in the sense of being able to observe the latent state) and

behaves optimally in terms of minimizing squared error.

Our horse race will be assessed in terms of squared error.

Our horse race will be assessed in terms of realized squared error.

In particular, your task is to generate a graph plotting observations of both $\| x_t - A x_{t-1} \|^2$ and $\| x_t - \mu_t \|^2$ against $t$ for $t = 1, \ldots, 49$.

In particular, your task is to generate a graph plotting simulated realizations of both $\| X_t - A X_{t-1} \|^2$ and $\| X_t - \mu_t \|^2$ against $t$ for $t = 1, \ldots, 49$.

In the code below, `x[:, t]` is the realized value of $X_t$ along the simulated path.

For the parameters, set $G = I, R = 0.5 I$ and $Q = 0.3 I$, where $I$ is

the $2 \times 2$ identity.

and $\mu_0 = (8, 8)$.

Finally, set $x_0 = (0, 0)$.

Finally, set the realized initial state to $x_0 = (0, 0)$.

```{exercise-end}

Observe how the diagonal values in the stationary solution $\Sigma$ (see {eq}`kalman_dare`) increase and decrease in line with this coefficient.

The interpretation is that more randomness in the law of motion for $x_t$ causes more (permanent) uncertainty in prediction.

The interpretation is that more randomness in the law of motion for $X_t$ causes more (permanent) uncertainty in prediction.

```