INDR 372 - Spring 2024
Fikri Karaesmen
Review Exercises for the Midterm Exam
1. Let us assume that Dt = µ + ϵt (for t=1,2,...) where ϵt are independent
and identically distributed normal random variables with mean 0 and
variance σ 2 . Consider the following three forecasts: i) Ft = Dt−1 ii)
Gt = 0.8Dt−1 + 0.2Dt−2 , iii) Ht = Dt−1 + (Dt−2 − Dt−3 ). For all three
forecasts,
(a) Determine whether the forecast is unbiased.
(b) Find the variance of the error et = Ft − Dt .
(c) Are Ft and Dt−1 correlated? If so find the auto-covariance Cov(Ft , Dt−1 ).
(d) Are Gt and Dt−1 correlated? If so find the auto-covariance Cov(Gt , Dt−1 ).
2. Let us assume that Dt = a + bt + ϵt (for t=1,2,...) where ϵt are independent and identically distributed normal random variables with mean 0
and variance σ 2 .
(a) Assume that b is unknown (but constant). Show that 0.5(Dt−1 −
Dt−2 ) + 0.5(Dt−2 − Dt−3 ) is an unbiased estimator for b. What is
the variance of this estimator?
(b) Assume a is known, consider the following forecast: Ft = a +
(Dt−1 − Dt−2 )t. Is this unbiased?
3. The Australian Beer Production Data (see the blackboard page) for
1991 starts with the following monthly production quantities: y1 = 164,
y2 = 148, y3 = 152, y4 = 144,y5 = 155, y6 = 125.
(a) Compute F4 using a moving average with a 3 period window.
(b) Compute F4 using exponential smoothing with α = 0.8.
(c) Compute F4 using double exponential smoothing with α = 0.8
and β = 0.5. You can use an initial estimator of slope and level
â = 150 and b̂ = 10.
(d) Compute F4 using the following ARIMA model: Dt = 80 +
0.4Dt−1 + 0.1Dt−2 + ϵt .
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4. John Kittle, an independent insurance agent, uses a five-year moving
average to forecast the number of claims made in a single year for one of
the large insurance companies he sells for. He has just discovered that
a clerk in his employ incorrectly entered the number of claims made
four years ago as 1,400 when it should have been 1,200.
(a) What adjustment should Mr. Kittle make in next year’s forecast
to take into account the corrected value of the number of claims
four years ago?
(b) Suppose that Mr. Kittle used simple exponential smoothing with
α = 0.2 instead of moving averages to determine his forecast.
What adjustment is now required in next year’s forecast? (Note
that you do not need to know the value of the forecast for next
year in order to solve this problem.)
5. The owner of a small brewery in Milwaukee, Wisconsin, is using Winter’s method to forecast his quarterly beer sales. He has been using
smoothing constants of α = 0.2, β = 0.2, and γ = 0.2. He has currently obtained the following values of the various slope, intercept, and
seasonal factors: S10 = 120, G10 = 14, c10 = 1.2, c9 = 1.1, c8 = 0.8,
c7 = 0.9.
(a) Determine the forecast for beer sales in quarter 11.
(b) Suppose that the actual sales turn out to be 128 in quarter 11.
Find S11 and G11 , and find the updated values of the seasonal
factors. Also determine the forecast made at the end of quarter
11 for quarter 13.
6. Consider a monthly sales data that exhibits increasing trend with a 12
month seasonality.
(a) What would be an appropriate ARIMA model based on this information only? (i.e. Dt = α1 Dt−1 + α2 et−1 + ... ?)
(b) How would you analyze the data for a complete ARIMA study?
7. Consider the autocorrelation and the partial autocorrelation plots for
two different time series in Figure 1.
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(a) What is an appropriate ARIMA model for the series in Figure
1A?
(b) What is an appropriate ARIMA model for the series in Figure 1B?
(c) Assume that D1 = 20, D2 = 15, D3 = 12, and that you are using
an ARIMA(1,0,0) model with a0 = 10 and a1 = −0.5, what is
your forecast for period 4 and period 5 (made in period 3)?
8. Consider the bike share (the daily number of bikes that are checked
out from the Paris municipality bike share system) demand time series
whose plot and autocorrelation plot is in Figure 2.
(a) Explain what is observed in the ACF.
(b) What steps should be taken to fit an ARIMA model to this data?
9. Consider the bike share demand time series (the daily number of bikes
that are checked out from the Paris municipality bike share system).
The data also includes weather related measurements for each day.
Thinking that the number of bikes shared may be dependent on weather
factors, we perform a linear regression taking three weather-related
measurements. Note that the weather related measurements such as
wind speed, temperature and humidity are normalized to values between 0 and 1. The results are summarized in the Table in Figure
3.
(a) Which predictors (factors) are significant? Why?
(b) Is the regression satisfactory? What can be done to improve it?
(c) Assume that tomorrow is predicted to be an average day (i.e.
x1 = x2 = x3 = 0.5). Find a point estimate and a 90% confidence
interval for the number of bikes demanded?
10. Consider the refrigerator sales data from the lectures (the monthly
number of refrigerators sold over a number of years). Because there
appears to be both trend and seasonality in the data. We use the
following model to predict the monthly sales:
yt = b0 + b1 x1,t + b2 x2,t + . . . + b12 x12,t + ϵt
where x1,t models the trend: i.e x1,t = t and and xi,t (for i = 2, 3, ..12)
are the monthly binary (dummy) variables i.e : x2,t = 1 if month t is
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January and is zero otherwise. Similarly, x3,t = 1 if month t is February
and is zero otherwise. Note that we only need only 11 such variables
since: x13,t = 1 − (x2,t + x3,t + . . . + x12,t ) (the binary variable for
December is linearly dependent on the binaries for the other months).
The regression results are in the Table in Figure 4.
(a) Is the regression statistically significant? Explain.
(b) Which predictors are statistically significant? Explain.
(c) Predict the average sales of refrigerators for month 25 (January
of year 3).
(d) What is the expected change in sales of refrigerators from month
25 to month 27 (January to March of year 3)?
(e) What is the expected change in sales of refrigerators from month
25 to month 36 (January to December of year 3)?
(f) Would the accuracy of the regression (measured by R2 ) improve if
we added a new term wt = t2 ? for quadratic trend? What about
the adjusted R2 ?
(g) If we were to shrink (reduce) the model by a lasso regression,
which predictor is likely to vanish first as we increase the penalty
parameter λ? Which predictor is likely to survive until the end
as λ increases further? (Note that we can only make a reasonable
guess but cannot give a definitive answer since the optimization
problem is complicated. That’s why we need computational tools).
11. Mr. Meadows Cookie Company makes a variety of chocolate chip cookies in the plant in Albion, Michigan. Based on orders received and forecasts of buying habits, it is estimated that the demand for the next four
months is 850, 1,260, 510, and 980, expressed in thousands of cookies.
During a 46-day period when there were 120 workers, the company
produced 1.7 million cookies. Assume that the number of workdays
over the four months are respectively 26, 24, 20, and 16. There are
currently 100 workers employed, and there is no starting inventory of
cookies.
(a) What is the minimum constant workforce required to meet demand over the next four months?
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(b) Assume that cI = 10 cents per cookie per month, cH = $100, and
cF = $ 200. Evaluate the cost of the plan derived in part (a).
(c) Formulate as a linear program. Be sure to define all variables and
include the required constraints.
(d) Suppose now that the cost of hiring workers each period is $100 for
each worker until 20 workers are hired, $400 for each worker when
between 21 and 50 workers are hired, and $700 for each worker
hired beyond 50. Write down the complete linear programming
formulation of the problem for these hiring costs.
12. Consider the following extension to the standard production modeling
framework. The workforce has a learning curve and becomes more
efficient each month they work. Assume that each month of experience
leads to an additional r% productivity with respect to the previous
month i.e. if a newly starting worker makes K units in a day, the next
month he makes K(1 + r) units in day and at month τ + 1, he makes
K(1 + r)τ units in a day.
(a) Let Vi,j be the number of workers hired in month i who are still
working in month j (j > i). (Assume that workers that are fired
and later rehired start from the beginning of the learning curve.)
Write the constraints for a production plan that determines how
many workers to employ in each period depending on their experience (i.e. hiring and firing decisions depend on the time that a
worker started). Explain which workers to fire in priority under
such a structure.
(b) Assume that the wages of the workers also depend on their experience (i.e. wages are increasing in the number of months worked).
Write the objective function to take into account a q% increase in
the wage for each month worked.
Online Exam Questions
13. (3 points) Assume that we have observed the following demand over the
first five periods: d(1) = 12, d(2) = 18, d(3) = 12, d(4) = 15, d(5) = 17.
What is the forecast for the expected demand in period 6 if a Moving
Average forecast with a 5-period window is used?
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14. (3 points) Assume that we have observed the following demand over the
first five periods: d(1) = 12, d(2) = 18, d(3) = 9, d(4) = 15, d(5) = 11.
What is the forecast for the expected demand in period 6 if an Exponential Smoothing forecast with smoothing constant α=0.1 is used?.
Assume that the ES forecast for period 5 is: F5 = 19.
15. (3 points) Let Dt = µ + ϵt where ϵt are independent random variables
that are normally distributed with mean zero and standard deviation
7.
Consider the following forecast: Ft = (0.7)Dt−1 + (1 − 0.7)Dt−2 .
What is the variance of the forecast error et ? Note that ( et = Ft −Dt )?
16. (4 points) Let Dt = 30 + (5)t + ϵt where ϵt are independent random
variables with mean zero and standard deviation σ.
Consider the following forecast Ft = Dt−2 + (2)(5). Find the expected
value of the forecast F12 : E(F12 ).
17. (4 points) Let Dt = a+bt+ϵt where ϵt are independent random variables
with mean zero and standard deviation σ.
Consider the following forecast Ft = Dt−1 + (Dt−1 − Dt−2 ). Note that
this is an unbiased forecast. Assume that the observed demand in the
first three periods are d(1) = 10, d(2) = 13 and d(3) = 9.
Find a forecast for the demand in period 8 made after observing the
demand in the first three periods: F3,8 .
18. (3 points) Let Dt = 5 + (40)t + ϵt where ϵt are independent random
variables with mean zero and standard deviation σ.
Consider the following transformation: Zt = Dt − Dt−1 . Assume that
we obtain an unbiased forecast Gt for Zt .
Find the expected value of Gt : E(Gt ).
19. (3 points) Let Dt = 12 + (15)t + ϵt where ϵt are independent random
variables with mean zero and standard deviation σ.
Consider the following transformation: Zt = Dt − Dt−1 .
Find the variance of Zt : V ar(Zt ).
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20. (4 points) Consider the following Box-Jenkins process: Dt = 100 +
(0.1)Dt−1 + (0.3)Dt−2 + ϵt . Assume that the observed demand in the
first two perids are: d(1) = 75, d(2) = 70.
Find the forecast for the expected demand in period 3.
21. (4 points) Consider the following Box-Jenkins process: Wt = 100 +
(−0.5)Wt−1 + ϵt .
Now assume that Dt = (11)t + Wt . Note that Dt has a known trend.
Assume that we observed the outcomes of Wt in the first two perids as:
w(1) = 71, w(2) = 60.
Find the forecast for the expected demand in period 4 made after observing demands (and outcomes of Wt ) in the first two periods: F2,4 .
22. (4 points) Assume that we have quarterly demand data and we use the
following time series regression as a forecasting model: dt = b0 + b1 t +
b2 y2t + b3 y3t + b4 y4t + ϵt .
where yit (i = 2, 3, 4) are binary (dummy) variables for seasons 2, 3
and 4 (yit = 1, if period t is quarter i of a year and yit = 0 otherwise).
We fit a regular least squares regression and estimate the coefficients as
follows b0 = 80, b1 = 6.5, b2 = 10, b3 = 70 and b4 = −20. Let us assume
that all coefficients are statistically significant.
Find a forecast for the demand in period 17. (Assume that t = 1
corresponds to Q1, t = 2 corresponds to Q2 and so on.)
23. (16 points) Let Dt = a + bt + ϵt where ϵt are independent random
variables with mean zero and standard deviation σ.
Consider the following forecast Ft = Dt−1 + (0.1)(Dt−1 − Dt−2 ) + (1 −
0.1)(Dt−2 − Dt−3 )
(a) (3 points) Assume that the observed demand in the first three
periods are d(1) = 10, d(2) = 11 and d(3) = 16. Find the forecast
for the expected demand in period 4: F4 .
(b) (4 points) Establish whether Ft is a an unbiased forecast or not
for all t > 3.
(c) (5 points) Find the variance of the forecast F t: V ar(Ft ).
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(d) (4 points) Consider the forecast Gt = Dt−2 +2(Dt−1 −Dt−2 ). Find
V ar(Gt ) and compare with part d. Expain the difference.
24. (14 points) Assume that we have quarterly demand data and we use
the following time series regression as a forecasting model:
dt = b0 + b1 t + b2 y2t + b3 y3t + b4 y4t + ϵt
where yit (i = 2, 3, 4) are binary (dummy) variables for seasons 2, 3
and 4 (yit = 1, if period t is quarter i of a year and yit = 0 otherwise).
We fit a regular least squares regression to the first 6 years (24 quarters)
of demand observations and obtain the following results (assume that
t=1 corresponds to first quarter of the year):
Estimated Coefficients:
Estimate
SE
tStat
(Intercept)
104.29
9.0209 11.561
x1
4.7794
0.51429 9.2931
x2
17.39
9.9503 1.7477
x3
-28.776
9.9901 -2.8804
x4
39.867
10.056 3.9645
pValue
4.8499e-10
1.6914e-08
0.096663
0.0095816
0.00083098
Number of observations: 24, Error degrees of freedom: 19
Root Mean Squared Error: 17.2
R-squared: 0.885, Adjusted R-Squared: 0.861
F-statistic vs. constant model: 36.7, p-value = 1.09e-08
(a) (2 points) Which predictors are statistically significant? Explain.
(b) (2 points) What is the forecast for expected value of demand in
quarter 28?
(c) (2 points) Find a 90% prediction interval for the forecast in part
b. Note that z0.95 = 1.64.
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(d) (2 points) What is the expected change in demand between quarter 20 and quarter 28?
(e) (2 points) What is the expected change in demand between quarter 21 and quarter 28?
(f) (2 points) Assume that we also have data on the demand of the
competitor’s product which is correlated with our demand. How
would the R2 value change if we add the competitor’s demand as
a new predictor in the above model. Explain.
(g) (2 points) Assume that we have additional 16 quarters of data
that we can use as a test set. We can then test the above regression model on the test set. How would you expect the Root Mean
Squared Error to change in the test set. Explain.
25. (10) points Consider an online retailer that ships healthy meals to customers. Their veggie menu is becoming popular. The demand for this
menu for the next 4-days is forecasted as follows:
Day
1
2
3
4
Demand
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8
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Assume that each veggie menu requires two hours of preparation and
packaging time. Assume that the cooks for veggie meals work for eight
hours a day (and only cook veggie menus) and the menus can be prepared in advance and kept in inventory. Assume also that the workforce
(number of cooks) must be an integer.
(a) (2 points) What is the number of cooks needed to satisfy the demand on day 1 (disregarding the other days)?
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(b) (8 points) What is the minimum constant workforce level (number
of cooks employed) if there is no inventory available in the beginning and it is targeted to have no inventories of veggie meals at
the end of day 4? You can assume that production can be slowed
down to avoid excess in inventories in period 4 if required. Show
all your work and explain all steps.
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