Exercise

# The standard error of the variances

The variable `return_matrix`

is already loaded into your environment. This variable contains the continuously compounded returns of each `VBLTX_prices`

, `FMAGX_prices`

, and `SBUX_prices`

.

Consider the constant expected return model (CER):

$$R_{it} = \mu_i + \epsilon_{it},$$ $$t = 1,\ldots,T$$ $$\epsilon_{it} \sim \text{ iid } N(0,\sigma_i^2)$$ $$cov(\epsilon_{it},\epsilon_{jt}) = \sigma_{ij},$$

where \(R_{it}\) denotes the continuously compounded return on asset \(i\), with \(i\) equal to the Vanguard long term bond index fund (VBLTX), Fidelity Magellan stock mutual fund (FMAGX), Starbucks stock (SBUX). The model thus assumes that the returns of these assets are normally distributed and that the time series of these continuously compounded returns are covariance stationary.

The parameters of the above CER model are of course unknown to us. It is your task now to estimate the model parameters \(\sigma_i^2\) for the different assets. The data that you constructed in the previous exercise is given. Remember from the course that an estimator for \(\sigma_i^2\) is just the sample variance.

Once you have these estimates, your second task is to investigate the precision of the estimates. More specifically, you should estimate the standard errors in order to get \(\hat{SE}(\hat{\sigma_i^2})\).

The formula is given by:

$$\hat{SE}(\hat{\sigma_i^2}) = \hat{\sigma_i^2} / (\sqrt{T / 2}).$$

Instructions

**100 XP**

- Assign to
`n_obs`

the number of observations in the matrix`return_matrix`

. - Assign to
`sigma2hat_vals`

the estimates of \(\sigma_i^2\) for the three different assets. - Assign to
`se_sigma2hat`

the estimates of \(SE(\hat{\sigma_i^2})\) and print the result to the console.