Econ 325: Exam 2 Study Guide
1. Calculations you are expected to be able to make include:
a. Any calculation made in a homework assignment since the last exam.
b. Confidence interval
c. Autocorrelation
d. t statistics for Xbar and r
e. Perform hypothesis tests on the mean and autocorrelation (using the standard method and the
rule of thumb)
f. First difference
g. Second difference
h. Make a table of values relating to simple exponential smoothing or Holt’s exponential
smoothing.
2. Be prepared to answer questions similar to the following homework problems:
a. Chapter 2:
i. Case Questions: 1, 2 (and use excel to calculate the autocorrelation for k = 1), 3.
ii. Exercises: 1, 2, 3 (set up a two sided test in part b, not a one sided test as suggested
by the question), 4, 6 a & c, 7 a – d (only 95% CI), 8 – 11.
b. Chapter 3:
i. Case Questions: 1 (only answer the first two sub-questions, and replace "2004" by
"2008"), 2.
ii. Note: be able to discuss the graph on page 148 which is included in the solution to
case question 1.
iii. Exercises: 1, 3, 4, 7, 8, 9, 11, 12,
3. Confidence Intervals & Hypothesis Tests
a. When we talk about Xbar, to what are we referring?
b. If Xbar is a number, then why is there a distribution on Xbar (a sampling distribution)?
c. Where is the distribution of Xbar centered?
d. What is the standard deviation of this distribution? (Use an equation that relates the standard
deviation of the distribution of Xbar to the standard deviation to the underlying distribution of
X.)
e. What is the shape of this distribution for a reasonably large sample?
f. How do we know this?
g. Draw a distribution of Xbar for medium sized sample.
h. Indicate on your graph where we expect 95% of the values of Xbar calculated from a
randomly selected sample of X from the population to lie?
i. Please label your graph appropriately and include the mean of the distribution of
Xbar.
i. What’s the relationship of this interval to a 95% confidence interval?
j. Approximately how many standard deviations wide is a 95% confidence interval?
k. How do we know this?
l. What happens to the above mentioned confidence interval as the sample size increases?
m. How do we know this?
i. Show it with the aid of a graph.
ii. Show this using an equation.
n. Explain, without using the word “confident,” what a C% confidence interval (relating to μ)
tells us.
i. C% of the time we construct an interval of this size around a sample mean Xbar, the
population mean μ will fall within this interval.
o. Explain why someone would ever want to center a sampling distribution at a hypothetical
value and then analyze the location of a particular sample statistic in that distribution. In your
answer, use an example where you are trying to estimate a population mean.
i. If a null hypothesis about the mean μ of a random variable were true, then the
distribution of Xbar would be centered at the hypothesized value. Therefore, one
way of determining the quality of the hypothesis is to center a distribution of Xbar on
the null hypothesis, and then observe where the Xbar calculated from a sample falls
in that distribution. The further out it falls in the distribution, the less likely the null
hypothesis is true because the lower is the probability of getting an observed value
for Xbar that is further from the null hypothesis.
p. What is the general form of a t statistic?
i. (“sample statistic” – “hypothesized population value of the sample statistic”) /
“standard deviation of sample statistic”
ii. Note, you can interpret “population” in the above equation as “true.”
q. What exactly does a t statistic tell us?
i. How many standard deviations our sample statistic is from the hypothesized
population value.
r. Quickly (concisely), how is the above discussion relevant to statistics like autocorrelation?
i. The autocorrelation (rk) that you calculate is a sample statistic. This sample statistic
has a distribution, just as Xbar (a different sample statistic) has a distribution. Under
certain assumptions, rk has a t distribution, just as Xbar has a t distribution.
Therefore, just as we can do hypothesis tests on µ being a certain value, we can do
hypothesis tests on ρk being a certain value.
ii. Note, that a statistic is merely a function (mathematical manipulation) performed on a
sample.
4. Answer the following questions pertaining to correlation/autocorrelation.
a. What does correlation measure?
b. Draw three scatterplots (x variable on one axis, corresponding y variable on the other axis):
i. one which shows a high positive ρ
ii. one which shows a weak negative ρ
iii. and one which shows a ρ close to zero.
c. Write out a hypothetical GDP time series for years 2001 – 2007.
i. Write out a corresponding series for GDPt-1 and GDPt-2.
d. Explain how one could use a scatterplot to get a feeling for the ρ2 (use an actual graph if it
helps you).
e. What is the equation for the test statistic for rk?
f. Why do we do hypothesis tests on rk?
g. What is the rejection criterion for a hypothesis test that the autocorrelation equals zero?
h. Rewrite this inequality to get rejection criterion for rk.
i. Show how you go from the previous inequality to the rule of thumb.
5. Autocorrelation & Correlograms.
a. What is a correlogram?
b. Why are correlograms useful?
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
p.
q.
How might one determine whether a time series displayed a trend using a correlogram?
How might one determine whether a time series displayed seasonality using a correlogram?
Why might one want to remove a trend and or seasonality in a time series?
How might one remove a trend in time series data?
How would one confirm that the process had indeed removed the trend?
How does one first difference data?
How does one second difference data?
What is the point of first differencing data?
What is the point of second differencing data?
How would one go from a first difference of a data series to a forecast for that data series?
How would one go from a second difference of a data series to a forecast for that data series?
What is stationary data?
Why would you want stationary data?
Why are we ever interested in a hypothesis test on the autocorrelation?
What is the relationship of the correlograms you produced in your homework to a hypothesis
test on the autocorrelation? In your answer, refer to the horizontal lines in the correlograms
you produced in your homework.
r. What is the rule of thumb for determining whether there is evidence that the autocorrelation
for any particular k is not zero?
6. Moving Average
a. What are the appropriate conditions the use of the MA model?
b. Write an equation for a 4 period MA model.
c. Explain in words what your model says making reference to the relative size of the weights
across periods.
d. How might one evaluate MA models of different lengths?
7. Exponential Smoothing
a. What are the appropriate conditions for the use of the simple exponential smoothing model?
b. How would you go about determining if those conditions were met other than casual
observation of either the raw data or a graph of the data?
c. Give an equation for the simple exponential smoothing model.
d. Define each of your terms, including the smoothing constant.
e. Give an interpretation of this equation. (Don’t just say in words the mathematical expression
in the equation. Use the phrase “weighted average” in your answer.)
f. Rewrite your equation to display a second equation with a second interpretation of
exponential smoothing that shows how the model “learns” from past values.
g. Give an interpretation of this second equation. (Don’t just say in words the mathematical
expression in the equation.)
h. Rewrite the first equation another way to display a third equation with a third interpretation of
exponential smoothing.
i. You should expand the second equation until it contains at least 4 terms.
i. Give an interpretation of this equation if you were to expand it indefinitely.
j. What is the similarity between a moving average and simple exponential smoothing?
k. What is the difference in the pattern of how the moving average and simple exponential
smoothing model weight past data?
l. What is the allowable range of the smoothing constant in your equation?
m. Use two tables, one for a high level of alpha (0.9) and one for a low level of alpha (0.1), that
displays the weights on past values of X in your third interpretation of the exponential
smoothing model for periods t, t-1, t-2, & t-3.
n. What would be the sum of the weights if you were to expand your table indefinitely?
o. What does a high versus a low value for the smoothing constant imply about the simple
exponential smoothing equation?
p. If you were to choose the value of alpha based simply on a casual observation of your time
series, give an example of when would you choose a high value
q. If you were to choose the value of alpha based simply on a casual observation of your time
series, give an example of when would you choose a low value?
r. What criterion is often used to determine the most appropriate smoothing constant?
s. Is this always the best method for finding the appropriate smoothing constant?
t. How would one appropriately use the exponential smoothing model to forecast if the data
exhibited both seasonality and a trend?
u. What is the relationship between the third interpretation of the exponential smoothing model
and the trend estimate in Holt’s exponential smoothing model? (Note, both have a lagged
version of the left hand side variable on the right hand side of the equation.)
v. Explain what “initializing” or “warming up” the model means.
w. What are the advantages of the exponential smoothing model?
i. See p. 112.
x. What are the disadvantages of the exponential smoothing model?
i. See p. 112.
8. Holt’s Exponential Smoothing
a. What are the appropriate conditions for the use of Holt’s exponential smoothing model?
b. Give the three equations used in Holt’s exponential smoothing model.
i. Define each of your terms including the two smoothing constants.
c. What terms must be initialized in this model?
d. Give an interpretation of the first equation.
e. Give an interpretation of the second equation consistent with an interpretation of the right
hand side of the equation as an infinite sum of past terms.
f. What is the relationship between the second equation, and the second way of rewriting the
simple exponential smoothing model?
g. Give an interpretation of the third equation.
h. What terms must be initialized in this model?
9. Winter’s Exponential Smoothing
a. What are the appropriate conditions for the use of Winter’s exponential smoothing model?
b. Give the four equations used in Winter’s exponential smoothing model.
i. Define each of your terms including the three smoothing constants.
c. What terms must be initialized in this model?
d. Give an interpretation of the first equation.
e. Give an interpretation of the second equation consistent with an interpretation of the right
hand side of the equation as an infinite sum of past terms.
f.
g.
h.
i.
j.
k.
l.
m.
Give an interpretation of the third equation consistent with an interpretation of the right hand
side of the equation as an infinite sum of past terms.
What is the relationship between the third equation, and the second way of rewriting the
simple exponential smoothing model?
Give an interpretation of the fourth equation.
What terms must be initialized in this model?
What does a seasonal index tell us?
How does one deseasonalize data?
How does one reseasonalize data?
Is there an alternative exponential smoothing model to the Winter’s model when the data
exhibit seasonality but no trend? Clearly explain the alternative approach to forecasting the
data.