return β_hat, σ_hat

```

### Exercise 1

```{exercise-start}

:label: apl_ex1

```

Look at the equation,

Verify that this equation is a regression equation.

### Exercise 2

```{exercise-end}

```

```{solution-start} apl_ex1

:class: dropdown

```

To verify that it is a **regression equation** we must show that the residual is orthogonal to the regressor.

Our assumptions about mutual orthogonality imply that

$$

E\left[\epsilon_{i,t}\right]=0,\quad E\left[\epsilon_{i,t}u_{t}\right]=0

$$

It follows that

$$

\begin{aligned}

E\left[\sigma_{i}\epsilon_{i,t}\left(R_{t}^{m}-R^{f}\right)\right]&=E\left[\sigma_{i}\epsilon_{i,t}\left(\xi+\lambda u_{t}\right)\right] \\

&=\sigma_{i}\xi E\left[\epsilon_{i,t}\right]+\sigma_{i}\lambda E\left[\epsilon_{i,t}u_{t}\right] \\

&=0

\end{aligned}

$$

```{solution-end}

```

```{exercise-start}

:label: apl_ex2

```

Give a formula for the regression coefficient $\beta_{i, R^m}$.

### Exercise 3

```{exercise-end}

```

```{solution-start} apl_ex2

:class: dropdown

```

The regression coefficient $\beta_{i, R^m}$ is

$$

\beta_{i,R^{m}}=\frac{Cov\left(R_{t}^{i}-R^{f},R_{t}^{m}-R^{f}\right)}{Var\left(R_{t}^{m}-R^{f}\right)}

$$

```{solution-end}

```

```{exercise-start}

:label: apl_ex3

```

As in many sciences, it is useful to distinguish a **direct problem** from an **inverse problem**.

E[(a + b R_t^m) R^f_t)] &= 1

\end{align*}

### Exercise 4

Using the equations above, find a system of two **linear** equations that you can solve for $a$ and $b$ as functions of the parameters $(\lambda, \xi, E[R_f])$.

Write a function that can solve these equations.

Please check the **condition number** of a key matrix that must be inverted to determine a, b

### Exercise 5

Using the estimates of the parameters that you generated above, compute the implied stochastic discount factor.

## Solutions

### Solution to Exercise 1

To verify that it is a **regression equation** we must show that the residual is orthogonal to the regressor.

Our assumptions about mutual orthogonality imply that

$$

E\left[\epsilon_{i,t}\right]=0,\quad E\left[\epsilon_{i,t}u_{t}\right]=0

$$

It follows that

$$

\begin{aligned}

E\left[\sigma_{i}\epsilon_{i,t}\left(R_{t}^{m}-R^{f}\right)\right]&=E\left[\sigma_{i}\epsilon_{i,t}\left(\xi+\lambda u_{t}\right)\right] \\

&=\sigma_{i}\xi E\left[\epsilon_{i,t}\right]+\sigma_{i}\lambda E\left[\epsilon_{i,t}u_{t}\right] \\

&=0

\end{aligned}

$$

### Solution to Exercise 2

The regression coefficient $\beta_{i, R^m}$ is

$$

\beta_{i,R^{m}}=\frac{Cov\left(R_{t}^{i}-R^{f},R_{t}^{m}-R^{f}\right)}{Var\left(R_{t}^{m}-R^{f}\right)}

$$

```{exercise-end}

```

### Solution to Exercise 3

```{solution-start} apl_ex3

:class: dropdown

```

**Direct Problem:**

Q: How close did your estimates come to the parameters we specified?

```{solution-end}

```

```{exercise-start}

:label: apl_ex4

```

Using the equations above, find a system of two **linear** equations that you can solve for $a$ and $b$ as functions of the parameters $(\lambda, \xi, E[R_f])$.

Write a function that can solve these equations.

Please check the **condition number** of a key matrix that must be inverted to determine a, b

```{exercise-end}

```

```{solution-start} apl_ex4

:class: dropdown

```

The system of two linear equations is shown below:

### Solution to Exercise 4

\begin{align}

$$

\begin{aligned}

a ((E(R^f) + \xi) + b ((E(R^f) + \xi)^2 + \lambda^2 + \sigma_f^2) & =1 \cr

a E(R^f) + b (E(R^f)^2 + \xi E(R^f) + \sigma_f ^ 2) & = 1

\end{align}

\end{aligned}

$$

```{code-cell} ipython3

# Code here

a, b, condM

```

### Solution to Exercise 5

```{solution-end}

```

```{exercise-start}

:label: apl_ex5

```

Using the estimates of the parameters that you generated above, compute the implied stochastic discount factor.

```{exercise-end}

```

```{solution-start} apl_ex5

:class: dropdown

```

Now let's pass $\hat{E}(R^f), \hat{\sigma}^f, \hat{\lambda}, \hat{\xi}$ to the function `solve_ab`.

```{code-cell} ipython3

a_hat, b_hat, M_hat

```

```{solution-end}

```