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

From standard asymptotic theory we know that the OLS estimator is normally distributed with mean equal to the true coefficients and covariance equal to $\sigma^2(X'X)^{-1}$.