## ECMT 674: Economic Forecasting

### Homework 4; Due Date: April 1, 2019 (beginning of class)

This is a team assignment that could be conducted in teams of at most two. Please mark every person’s name on the document that is being returned. I would also like to get the signatures of every team member under a statement: “I have contributed to this assignment to a sufficient degree to get equal credit with my teammate and all other collaborations are properly acknowledged.” Please submit the files through eCampus. Everyone in the team will receive the same grade.

You will need Excel and R/RStudio to complete this assignment. I would like you to turn in one file with figures and answers to questions, i.e. an R Markdown file, and an Excel spreadsheet (if used) and an R script file documenting your work and the results. Please name your files in an informative manner: an example would be the course name/number + the first initials of the team members + the assignment number + a file identifier.

General notes: When plotting figures, make sure you use line specifications that can be easily distinguished even if one prints your figure with a black and white printer. In addition, for time series plots have the horizontal axes represent dates.

### Problem I: Downward bias

In this exercise you will conduct a Monte Carlo experiment that studies the phenomenon of downward  bias in time series regressions.  In a Monte Carlo study,  artificial data are generated  using a computer,  and then these artificial data are used to calculate the statistics being studied.      In this particular case the object of interest is the bias in the parameter estimates of autoregressive models.

Consider an AR(1) process with an autoregressive coefficient of θ = 0.9 and i.i.d. standard normal errors.

1. (0.25 points) Generate a sequence of T = 100 observations from this AR(1) process.

2. (0.25 points) Estimate the autoregressive coefficient using an AR(1) process. Store the pa- rameter estimate.

3. (0.5 points) Repeat the previous 2 steps (step 1 and 2) 5000 times (we will call this Monte Carlo iterations) and store the estimated autoregressive coefficient each time.

4.(1 point) Plot the sampling distribution of the autoregressive coefficient. Calculate the bias, i.e.    1   Σ5000(θˆi) − θ.

5. (1 point) Repeat steps 3 and 4, for a larger sample T = 500. How do the sampling distribution and the bias change?

6. (1 point) Consider the exercise above for various values of θ. In other words, in addition to θ = 0.9, also consider θ = 0.5, θ = 0.2 and θ = 0. Does the bias change?

### Problem II: Forecasting

Now we would like to revisit a modified version of the exercise for your final project. We will be forecasting employment with the spread series. Employment will be measured with seasonally adjusted Payroll Employment, while the spread will be measured by the difference between the interest rate on 10-year Treasury bonds and 3-month Treasury bills. Consider data at monthly frequency. Even though you would be asked for employment forecasts in levels (i.e. rounded to the nearest thousand person) as it is done in FREDcast, consider the forecasting regressions by using the growth rates of employment. You can either choose to connect to FRED directly to download the data or download it in Excel and further import it into R (If you pick the latter scenario, please also submit the Excel file as indicated above). Conduct the exercises below for the overlapping sample period only.

1. (0.5 points) Consider the best autoregressive model to use for forecasting employment growth. In other words, use an AR(p) model for employment growth, where p is picked optimally.

2. (0.5 points) Estimate the model with spread, where employment growth is explained by the spread.

3. (0.5 points) Estimate an ADL(p,1) model for employment growth with spread, where p attains the same value as that in the first sub-question of this problem.

4. (2.5 points) Use the three models above to forecast the one-month-ahead employment growth using fixed, rolling and recursive forecasting schemes. Use a rolling window that captures 10 years of data. This should also be the window size considered for the fixed estimation scheme, as well the initial window for the recursive estimation scheme. Plot the out-of- sample predictions as well as the realization of employment growth on the same plot. You would end up with three forecast series and one realization series for each forecasting scheme. Which one of the models does better in terms of the MSFE (mean squared forecast error)?

5. (1.5 points) Take the best model from before and consider the implied forecast for employment (in levels). Further consider a no-change forecast for employment (this is the forecast when assuming that employment is random walk). Which model does better in terms of MSFE (mean squared forecast error)?

6. (0.5 points) If you were to use what you have learned in this forecasting exercise to provide forecasts for Payroll Employment in FREDcast, how would your forecast for the next period   of FREDcast look like?