Applied Exercises

This section provides some exercises that are meant to deepen your knowledge in the topics covered in this section and to gain experience solving real-world problems.‌

1. Forecast density

In this exercise you will simulate a density forecast from a simple AR(1) model.

Exercise

• Using an AR(1) model (see below) and assuming that $\rho=0.7$ and $y_{T}=1.5$, simulate 1000 paths for the process, 20 periods into the future, assuming that the error term is iid $N(0,0.5^2)$.

• Using the prctile function, find the median path for the process.

• Create a plot over time of the median path and the $16^{th}$ and $84^{th}$ percentiles of the forecast "distribution".

Theory

To forecast from an AR(1) model, in general we need to know the parameter values and the last value of the process, $y_{T}$. We can then construct the forecast for time $T+1$ as

$\hat{y}_{T+1} = \rho \; y_{T}$

We can then use the forecasted value for $T+1$ to create a forecast for $T+2$.

Note that above the error term is ignored as $E(\epsilon_{T+i}\vert \Omega_T)=0, \forall i>0$ where $\Omega_T$ is the information set at time $T$. We can however use Monte Carlo simulation to incorporate the uncertainty inherent in the model and get a "distribution" of forecasts.

2. Finding the probability of an event using Monte Carlo simulation

In this exercise you will write a Monte-Carlo simulation to answer a simple question on probabilities.

Exercise

• Write a script that performs a Monte Carlo simulation to find the probability that the sum of the numbers coming up on two (fair) dice is equal to 6.

• Perform the simulation 10, 100, 1000, 10,000 times and compare the results to the theoretical answer.

3. Large-sample distribution of the OLS estimator

In this exercise you will simulate the large-sample distribution of the OLS estimator.

Exercise

1. Write a Monte Carlo simulation to explore these large sample properties. Assume that the true model is $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon$, where $\beta_0=1, \beta_1= 2,$ and $\beta_2=5$and

   $\epsilon \sim N(0,1)$. Let the sample size be 50 and set the number of replications to 2000.

   • Create the data sets assuming that the regressors are iid $N(0,1)$

   • For each replication, estimate the OLS coefficients, the variance of the error term, and the variance of the OLS coefficients.

2. Run the following experiments

   • Check that the OLS estimate has the right mean, i.e. compare the mean of the estimated coefficients to the true coefficients.

   • Check that the variance of the OLS coefficients is correctly estimated, i.e., compare your estimate to the covariance of the coefficient estimates using the cov function.

   • Check that the OLS estimator has the right distribution: Compare the CDF of the normalized estimate of $\beta_2$ to the CDF of the standard normal distribution

Check