Economics 470: Economic Fluctuations and Forecasts
Take Home Midterm
Directions
This test will be posted online in the afternoon on Thursday, February 21st. Your answers will be due at 11 AM on Monday, February 26th in my office. Late submissions will not be accepted. There will be class held, as usual, on the 27th.
You are welcome to use any print or internet resources available. You are not to discuss this exam with others (and that means all other sentient beings). If you are in a position where you are unsure about breaking these simple rules, think of the worst case scenario—if you break these rules I will give you a zero on this exam.
I will be happy to help you with clarifying questions but will not answer questions that elucidate the material. Please address questions via e-mail (); I will post answers to clarifying questions at
Your answers may be typed or legibly handwritten. Feel free to submit Stata output if it helps explain your answer (it is easiest to “copy and paste” the Stata output into a Word file that contains your answers). Please answer on blank, white copy paper and only include your student id number as an identifier. Do not include your name on the exam. Please start a new page for each problem; though you may combine parts of a problem onto a single page.
Points possible are in parenthesis after each question.
1.A number of instances occur in economic data where the people measuring an event do so at frequencies other than those that cause that event. For instance, production happens daily (or hourly?) but we measure GDP only once per quarter. For another example, the Federal Reserve alters the money base minute-by-minute as it buys and sells securities but it measures the quantity of money once per week. The end result may, or may not, be innocuous for econometricians attempting to understanding the underlying process that generates this data.
To explore this, consider the following AR(1) relationship:
yt = B0 + φyt-1 + εt
whereεt is mean zero, homoskedastic and without autocorrelation. Further, assume this is a covariance stationary process. Though it is not necessary, you may find it easier to start in period zero where the value of yt is y0. Imagine yt occurs monthly, so t indexes months. However, like GDP, it is observed only once per quarter. Specifically, what is observed is:
Don’t let the notation bother you; YT is simply the sum of three yt’s for quarter T. Thus, for quarter 1, Y1 = y1 + y2 + y3 and for quarter 2, Y2 = y4 + y5 + y6. To be clear, the econometrician never observes the yt but instead only observes the YT.
Imagine using OLS (or, identically the ARIMA command) to estimate:
YT = α0 + α1YT-1
What properties do your estimates of α0 and α1 have? Specifically, are they unbiased and efficient estimates of the parameters B0 and φ that generate the yt? If not, how are α0 and α1 biased and/or inefficient? (10)
2.In class we have modeled all of our ARMA procedures as having error terms that were uncorrelated with each other and, less frequently stated, homoskedastic. However, a number of econometricians have claimed that the variance of error terms in a number of series may be a function of time. For instance, the Dow Jones Industrial Average appears to have had very mild shocks up until the late 1990s and thereafter experienced much greater volatility.
Dow Jones Industrial Average
For the purposes of this problem, assume εt is distributed normally with mean zero and variance σt2. The important thing to notice here is that σ2 varies with t. Imagine further that you identify a series as being an AR(2) process and, ignoring the variation in σ2 over time, you estimate it with OLS. Discuss the implications of using OLS to forecast this series. (8)
3.Consider theARMA(2,1) process: yt = φ1yt-1 + φ2yt-2 + εt + θεt-1.
a. What are ρ(1), ρ(2), ρ(3)? (12)
b. If you have 1000 observations of yt and estimate this process with an ARIMA model, what are your forecasts of y1001 and y1002? You may express these in terms of the y’s,φ’s, θ’s and anything else necessary. (10)
c. What are your forecast variances for y1001 and y1002? In expressing these, you may assume your forecasted parameters equal the actual parameters of the model. (10)
4.I have posted MLB Runs Per Game on my website. This is a stata dataset that consists of the average number of runs per game scored by both teams in major league baseball per year between 1872 and 2012. The data comes from:
Forecast the number of runs per game expected in both the 2013 and 2014 seasons. Provide 95% confidence intervals for these forecasts. (20)
For this problem, most of your grade will be determined based upon your explanation of how you arrived at your forecast rather than the forecast itself. In other words, I want you to explain what you did and why you did it.