:depth: 2

```

In this quantecon lecture {doc}`A First Look at the Kalman filter <kalman>`, we used

In this QuantEcon lecture {doc}`kalman`, we used

a Kalman filter to estimate locations of a rocket.

In this lecture, we'll use the Kalman filter to

The firm learns about those things only by observing a history of the output that the worker generates for the firm, and from understanding how that output depends on the worker's human capital and how human capital evolves as a function of the worker's effort.

We'll posit a rule that expresses how the much firm pays the worker each period as a function of the firm's information each period.

We'll posit a rule that expresses how much the firm pays the worker each period as a function of the firm's information each period.

In addition to what's in Anaconda, this lecture will need the following libraries:

!pip install quantecon

```

To conduct simulations, we bring in these imports, as in {doc}`A First Look at the Kalman filter <kalman>`.

To conduct simulations, we bring in these imports, as in {doc}`kalman`.

```{code-cell} ipython3

import matplotlib.pyplot as plt

from scipy.stats import multivariate_normal

import matplotlib as mpl

mpl.rcParams['text.usetex'] = True

mpl.rcParams['text.latex.preamble'] = r'\usepackage{{amsmath}}'

mpl.rcParams['text.latex.preamble'] = r'\usepackage{amsmath,amsfonts}'

```

## A worker's output

A representative worker is permanently employed at a firm.

The workers' output is described by the following dynamic process:

A worker's output is described by the following dynamic process:

```{math}

:label: worker_model

\begin{aligned}

h_{t+1} &= \alpha h_t + \beta u_t + c w_{t+1}, \quad c_{t+1} \sim {\mathcal N}(0,1) \\

h_{t+1} &= \alpha h_t + \beta u_t + c \epsilon_{t+1}, \quad \epsilon_{t+1} \sim N(0,1) \\

u_{t+1} & = u_t \\

y_t & = g h_t + v_t , \quad v_t \sim {\mathcal N} (0, R)

y_t & = g h_t + v_t , \quad v_t \sim N(0, R)

\end{aligned}

```

* $h_t$ is the logarithm of human capital at time $t$

* $u_t$ is the logarithm of the worker's effort at accumulating human capital at $t$

* $y_t$ is the logarithm of the worker's output at time $t$

* $h_0 \sim {\mathcal N}(\hat h_0, \sigma_{h,0})$

* $u_0 \sim {\mathcal N}(\hat u_0, \sigma_{u,0})$

* $\epsilon_{t+1}$ is an IID standard normal shock to human capital

* $h_0 \sim N(\hat h_0, \sigma_{h,0})$

* $u_0 \sim N(\hat u_0, \sigma_{u,0})$

Parameters of the model are $\alpha, \beta, c, R, g, \hat h_0, \hat u_0, \sigma_h, \sigma_u$.

Parameters of the model are $\alpha, \beta, c, R, g, \hat h_0, \hat u_0, \sigma_{h,0}, \sigma_{u,0}$.

At time $0$, a firm has hired the worker.

The worker is permanently attached to the firm and so works for the same firm at all dates $t =0, 1, 2, \ldots$.

At the beginning of time $0$, the firm observes neither the worker's innate initial human capital $h_0$ nor its hard-wired permanent effort level $u_0$.

The firm believes that $u_0$ for a particular worker is drawn from a Gaussian probability distribution, and so is described by $u_0 \sim {\mathcal N}(\hat u_0, \sigma_{u,0})$.

The firm believes that $u_0$ for a particular worker is drawn from a Gaussian probability distribution, and so is described by $u_0 \sim N(\hat u_0, \sigma_{u,0})$.

The $h_t$ part of a worker's "type" moves over time, but the effort component of the worker's type is $u_t = u_0$.

The $h_t$ part of a worker's "type" moves over time, while the equation $u_{t+1} = u_t$ implies $u_t = u_0$ for all $t$.

This means that from the firm's point of view, the worker's effort is effectively an unknown fixed "parameter".

Thus, from the firm's point of view, effort is a fixed, unobserved component of the worker's type that must be inferred from output observations.

At time $t\geq 1$, for a particular worker the firm observed $y^{t-1} = [y_{t-1}, y_{t-2}, \ldots, y_0]$.

## A firm's wage-setting policy

Based on information about the worker that the firm has at time $t \geq 1$, the firm pays the worker log wage

At time $t \geq 1$, before observing current output $y_t$, the firm sets the worker's log wage using the past output history $y^{t-1}$:

$$

w_t = g E [ h_t | y^{t-1} ], \quad t \geq 1

w_t = g \mathbb{E}[h_t | y^{t-1}], \quad t \geq 1

$$

and at time $0$ pays the worker a log wage equal to the unconditional mean of $y_0$:

```{math}

\begin{aligned}

\begin{bmatrix} h_{t+1} \cr u_{t+1} \end{bmatrix} &= \begin{bmatrix} \alpha & \beta \cr 0 & 1 \end{bmatrix}\begin{bmatrix} h_{t} \cr u_{t} \end{bmatrix} + \begin{bmatrix} c \cr 0 \end{bmatrix} w_{t+1} \cr

\begin{bmatrix} h_{t+1} \cr u_{t+1} \end{bmatrix} &= \begin{bmatrix} \alpha & \beta \cr 0 & 1 \end{bmatrix}\begin{bmatrix} h_{t} \cr u_{t} \end{bmatrix} + \begin{bmatrix} c \cr 0 \end{bmatrix} \epsilon_{t+1} \cr

y_t & = \begin{bmatrix} g & 0 \end{bmatrix} \begin{bmatrix} h_{t} \cr u_{t} \end{bmatrix} + v_t

\end{aligned}

```

```{math}

:label: ssrepresent

\begin{aligned}

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

x_{t+1} & = A x_t + C \epsilon_{t+1} \cr

y_t & = G x_t + v_t \cr

x_0 & \sim {\mathcal N}(\hat x_0, \Sigma_0)

x_0 & \sim N(\hat x_0, \Sigma_0)

\end{aligned}

```

```

Next, to compute the firm's policy for setting the log wage based on the information it has about the worker,

we use the Kalman filter described in this quantecon lecture {doc}`A First Look at the Kalman filter <kalman>`.

we use the Kalman filter described in this QuantEcon lecture {doc}`kalman`.

In particular, we want to compute all of the objects in an "innovation representation".

u_hat_t = x_hat_t[1, :]

```

For a draw of $h_0, u_0$, we plot $E y_t = G \hat x_t $ where $\hat x_t = E [x_t | y^{t-1}]$.

For a draw of $h_0, u_0$, we plot $\mathbb{E}[y_t | y^{t-1}] = G \hat x_t$ where $\hat x_t = \mathbb{E}[x_t | y^{t-1}]$.

We also plot $E [u_0 | y^{t-1}]$, which is the firm inference about a worker's hard-wired "work ethic" $u_0$, conditioned on information $y^{t-1}$ that it has about him or her coming into period $t$.

We also plot $\mathbb{E}[u_0 | y^{t-1}]$, which is the firm inference about a worker's hard-wired "work ethic" $u_0$, conditioned on information $y^{t-1}$ that it has about him or her coming into period $t$.

We can watch as the firm's inference $E [u_0 | y^{t-1}]$ of the worker's work ethic converges toward the hidden $u_0$, which is not directly observed by the firm.

We can watch as the firm's inference $\mathbb{E}[u_0 | y^{t-1}]$ of the worker's work ethic converges toward the hidden $u_0$, which is not directly observed by the firm.

```{code-cell} ipython3

fig, ax = plt.subplots(1, 2)

ax[0].plot(y_hat_t, label=r'$E[y_t| y^{t-1}]$')

ax[0].plot(y_hat_t, label=r'$\mathbb{E}[y_t| y^{t-1}]$')

ax[0].set_xlabel('Time')

ax[0].set_ylabel(r'$E[y_t]$')

ax[0].set_title(r'$E[y_t]$ over time')

ax[0].set_ylabel(r'$\mathbb{E}[y_t]$')

ax[0].set_title(r'$\mathbb{E}[y_t]$ over time')

ax[0].legend()

ax[1].plot(u_hat_t, label=r'$E[u_t|y^{t-1}]$')

ax[1].plot(u_hat_t, label=r'$\mathbb{E}[u_t|y^{t-1}]$')

ax[1].axhline(y=u_0, color='grey',

linestyle='dashed', label=fr'$u_0={u_0:.2f}$')

ax[1].set_xlabel('Time')

ax[1].set_ylabel(r'$E[u_t|y^{t-1}]$')

ax[1].set_ylabel(r'$\mathbb{E}[u_t|y^{t-1}]$')

ax[1].set_title('Inferred work ethic over time')

ax[1].legend()

Evidently, entries in the conditional covariance matrix become smaller over time.

It is enlightening to portray how conditional covariance matrices $\Sigma_t$ evolve by plotting confidence ellipsoides around $E [x_t |y^{t-1}] $ at various $t$'s.

It is enlightening to portray how conditional covariance matrices $\Sigma_t$ evolve by plotting confidence ellipsoides around $\mathbb{E}[x_t | y^{t-1}]$ at various $t$'s.

```{code-cell} ipython3

# Create a grid of points for contour plotting

# Generate plots for y_hat_t and u_hat_t

fig, ax = plt.subplots(1, 2)

ax[0].plot(y_hat_t, label=r'$E[y_t| y^{t-1}]$')

ax[0].plot(y_hat_t, label=r'$\mathbb{E}[y_t| y^{t-1}]$')

ax[0].set_xlabel('Time')

ax[0].set_ylabel(r'$E[y_t]$')

ax[0].set_title(r'$E[y_t]$ over time')

ax[0].set_ylabel(r'$\mathbb{E}[y_t]$')

ax[0].set_title(r'$\mathbb{E}[y_t]$ over time')

ax[0].legend()

ax[1].plot(u_hat_t, label=r'$E[u_t|y^{t-1}]$')

ax[1].plot(u_hat_t, label=r'$\mathbb{E}[u_t|y^{t-1}]$')

ax[1].axhline(y=u_0, color='grey',

linestyle='dashed', label=fr'$u_0={u_0:.2f}$')

ax[1].set_xlabel('Time')

ax[1].set_ylabel(r'$E[u_t|y^{t-1}]$')

ax[1].set_ylabel(r'$\mathbb{E}[u_t|y^{t-1}]$')

ax[1].set_title('Inferred work ethic over time')

ax[1].legend()

ax.plot(u_hat_t - u_0, alpha=.5)

ax.axhline(y=0, color='grey', linestyle='dashed')

ax.set_xlabel('Time')

ax.set_ylabel(r'$E[u_t|y^{t-1}] - u_0$')

ax.set_ylabel(r'$\mathbb{E}[u_t|y^{t-1}] - u_0$')

ax.set_title(title)

else:

label_line = (r'$E[u_t|y^{t-1}]$' if name is None

label_line = (r'$\mathbb{E}[u_t|y^{t-1}]$' if name is None

else name)

title = ('Inferred work ethic over time'

if title is None else title)

ax.axhline(y=u_0, color=u_hat_plot[0].get_color(),

linestyle='dashed', alpha=0.5)

ax.set_xlabel('Time')

ax.set_ylabel(r'$E[u_t|y^{t-1}]$')

ax.set_ylabel(r'$\mathbb{E}[u_t|y^{t-1}]$')

ax.set_title(title)

```