6. 200865341278,59917,962,00030,383,000,0003,390,000,00024,7
46,000,00043,500$3.45395,000,000354Q-
200863239776,00015,183,00026,448,000,0003,046,000,00020,8
25,000,00042,490$2.93339,000,000
Regression Analysis Assignment
The objective of this assignment is to provide you with
experience applying practical tools to analyzing and interpreting
actual business data. Secondary objectives include providing
you with experiences working within a group as well as with
business writing. I will assess your assignment in three parts
including (1) your responses to case questions one through four,
(2) your memo to the CEO summarizing your conclusions (case
question five), and (3) your team effort. The assessment
breakdown is as follows:
Responses to case questions 1 – 4 20 Memo to the CEO 20
Team effort 10 Total 50
To the extent that you respond to the case questions, you will
receive full credit for this portion of the assignment. Therefore,
for these questions, I suggest that you focus your efforts more
on analysis than writing.
Much of your grade for this assignment will derive from the
summary of your analysis and conclusions in your memo to the
CEO. Here, I will assess the validity of your assumptions and
conclusions, and how accurately you interpret your results. I
will base your memorandum grade primarily on these factors.
However, I will also assess your ability to convey the intended
information, including your ability to summarize the
information concisely and to use reasonable grammar.
Admittedly, this portion of the grade can be very subjective, but
I believe it will be useful for you in your careers. The grammar
and writing style portion of the grade should not comprise more
than 5 points in the total assignment.
For grading purposes, the team effort portion of this assignment
will be treated as a separate assessment and will not be reported
in direct conjunction with this assignment. Each individual in a
7. group will be assessed by the other members of the group using
a brief survey that will be conducted after the regression
assignment due date.
This assignment is due Monday, November 2 by 10:00am. To
submit your assignment, please have a member of your group
email me a zipped file containing your assignment deliverables.
Best
Continental Airlines Case Study
THE DECISION CONTEXT
In 2008, the senior management team at Continental Airlines,
commanded by Lawrence Kellner, the Chairman and Chief
Executive Officer, convened a special meeting to discuss the
firm’s latest quarterly financial results. A bleak situation lay
before them. Continental had incurred an operating loss of $71
million dollars—its second consecutive quarterly earnings
decline that year. Likewise, passenger volume was significantly
down, dropping by nearly 5 percent from the prior year’s
quarter. Continental’s senior management needed to act swiftly
to reverse this trend and return to profitability.
Being the fourth largest airline in the U.S. and eighth largest in
the world, Continental was perceived as one of the most
efficiently run companies in the airline industry. Nonetheless,
2008 brought unprecedented challenges for Continental and the
entire industry as the United States and much of the world was
heading into a severe economic recession. Companies cutting
deeply into their budgets for business travel, the highest
yielding component of Continental’s total revenue, together
with a similar downward trend from the leisure and casual
sector, combined to sharply reduce total revenue.
Concurrent with this revenue decline, the price of jet fuel
soared to record levels during 2008.1 Thus, while revenue was
decreasing, Continental was paying almost twice as much in
fuel costs. Interestingly, fuel costs surpassed the firm’s salaries
and wages as the highest cost in Continental’s cost structure.
This obviously had a negative impact on the bottom line,
8. squeezing even further the already strained profit margins.
The outlook for a quick recovery in the U.S. economy and,
consequently, an upturn in the demand for air travel in the short
term did not seem likely. Continental’s internal forecasts
indicated that a further decline in passenger volume should be
anticipated throughout 2009, with a recovery in travel possibly
occurring by the middle of 2010.
To summarize, adverse economic conditions in the U.S.,
coupled with the rise in fuel costs, were dragging down
Continental’s profits and relief was unlikely through the
foreseeable future.
Given the situation described above, management needed to act
swiftly to restore profitability. Several strategic options were
evaluated. Since the U.S. and much of the world was facing a
severe recession, the prospect for growing revenues by either
raising airfares or passenger volume seemed futile. Contrary to
raising revenue, Continental’s managers believed that raising
fares could potentially erode future revenues beyond the present
level. Discounting fares did not seem a plausible solution
either, because given the severity of the economic situation a
fare cut could fall short in stimulating additional passenger
demand and lead to lowering revenues.
1 To illustrate, jet fuel is tied
to the price of oil and, over the past year, oil prices surged from
about $70 to $135 per barrel. Consequently, the price of jet fuel
increased markedly, from an average of $1.77 per gallon to
$4.20 by the mid-summer of 2008.
Thus, because management anticipated that revenues would
remain flat for most of the year, the only viable short-term
solution to restoring profits was a substantial and swift
reduction in operating costs. This could most effectively be
accomplished in two ways. First, through a reduction in flying
capacity adjusted to match projected passenger demand. With
this in mind, Continental’s management agreed to reduce flying
capacity by 11 percent on domestic and international routes.2
As a result of this action, Continental would eliminate the least
9. profitable (or unprofitable) flights and, accordingly, would
ground several planes in the fleet. Management anticipated that
this decision would reduce several of the firm’s operating costs.
Apart from this, Continental could achieve further reductions in
costs by implementing several cost-cutting initiatives and
through operational efficiencies. For example, management
projected that it could achieve reductions in Passenger Services
expenses by consolidating several tasks during passenger check-
in and by reducing food and beverage waste served during
flights. Additionally, the firm could reduce various
miscellaneous expenses through targeted cuts in discretionary
spending.
In sum, to close the gap in profitability, Continental’s strategy
was geared toward slashing operating costs by cutting capacity
and through aggressive identification and implementation of
cost-cutting initiatives.
The next step would be for management to know precisely how
their decision to downsize capacity would impact the firm’s
future operating costs, and also identify specific areas in which
the firm could achieve additional cost reductions. Additionally,
the cost analysis would help forecast the firm’s operating costs
and projected profits (or losses) for the upcoming fiscal year.
However, before we can proceed with such analysis, an
examination of how the various categories of Continental’s
costs behave is in order.
Before we begin, let us prepare with an overview of the airline
industry and its competitive landscape, and an understanding of
why cost behavior bears particular relevance in this case.
Relative to other industries, airlines are a very difficult business
to manage. In particular, they are exposed to tremendous risks
brought by volatility inherent in their business model, as they
deal with high fixed costs, labor unions, instability in fuel
prices, weather and natural disasters, passenger safety, and
security regulations. These aspects bring a large burden to
airlines’ cost structures. Moreover, competition within the
industry is fierce; the proliferation of discount carriers, such as
10. Southwest Airlines and, most recently, Jet Blue, and the end of
fare regulation in 1978, has hindered airlines’ pricing power
and their ability to spur revenues. For these reasons, cost
containment is a critically important aspect of profitability in
this industry.
In order for Continental to restore profitability in this harsh
environment of weak demand for air travel, it must be able to
contain its operating costs, especially its massive fixed costs,
which are visible in several ways. For example, salaries for
pilots, flight attendants, and mechanics, as well as aircraft
leasing costs, are typically fixed, varying little with shifts in
passenger volume. Because fixed costs typically embody the
amount of operating capacity of a firm, they are
2 Specifically, on June 13, 2008, Continental Airlines
announced that it planned to reduce its flight capacity by 11
percent. By shrinking capacity, Continental expected to reduce
the number of domestic and international flights from its three
major hubs in Houston, Cleveland, and Newark (Maynard 2008).
commonly referred as “capacity” costs. Since fixed costs do not
self-adjust to fluctuations in passenger volume, the only way in
which they can be decreased (or increased) is if management
adjusts them in accordance to the level of operating capacity. In
contrast, other costs, such as passenger services and reservation
and distribution costs, behave as variable and would selfadjust
with variations in volume or operating activity.
Hence, to assess the impact of this strategic decision to alter
Continental’s cost structure, and identify the areas that could
achieve the greatest reduction in costs, we must resolve how
Continental’s operating costs behave and what drives them. In
what follows, we learn how to apply regression analyses to
examine cost behavior and forecast future costs, and then use
that knowledge to assess how the reduction in flying capacity
would affect Continental’s operating costs and profitability in
the near term.
ESTIMATING COSTS USING REGRESSION ANALYSES
The previous discussion highlighted the importance of
11. examining the behavior of Continental’s operating costs to pave
the way for a cost and profitability analysis using regression
analysis. Regression analysis is a powerful statistical tool that
is frequently used by firms to examine cost behavior and predict
future costs. The idea behind regression analysis is
straightforward: historical data for costs, and the various
activities that could potentially drive operating costs, are
inserted into a mathematical calculation which yields the
average amount of change in that particular cost that has
occurred over time. Average values provided by regression
calculations may then be applied to estimate future change that
will occur in that cost given a one-unit change in one or more of
the business activities which drive that cost.3 More precisely, in
a regression model, cost is a function of one or more business
activities (or factors) underlying a business operation. Simply
put, the business activities are the drivers of operating costs.
Therefore, since activities drive costs, our first step in the
estimation of a cost function is to identify the underlying
activities or other potential factors that drive the cost in
question—the cost drivers. This requires extensive knowledge
of the business operation. In the case of Continental Airlines,
the potential drivers of operating costs vary greatly. For
instance, as previously noted, the number of passengers that
Continental flies may drive the costs related to Passenger
Services. Likewise, Aircraft Maintenance and Repairs costs
could be driven by the number of aircraft in the fleet and by the
level of flying capacity set by Continental (i.e., available seat
miles).
In synthesis, to predict how Continental’s operating costs would
be affected by the decision to reduce capacity, and to identify
those areas in which additional room is available for cost
cutting, we need to identify which costs in this firm’s cost
structure behave as variable, fixed, or mixed (in which elements
of both variable and fixed are observable). Equally important,
we should also identify the specific drivers (if any) of each
cost.
12. Your job is to assist management in their quest to restore
profitability at Continental Airlines. Specifically, you must
conduct regression analyses to examine cost behavior and then
use this information to forecast operating costs and profitability
for the upcoming year. As part of your cost analysis, you should
investigate how the decision to cut flying capacity would impact
the 3 For ease in exposition,
cost functions and regression analyses are discussed briefly
here. For further insight on cost functions and the mechanics of
regression analyses, refer to the Appendix.
firm’s future operating costs and, equally important, identify
those specific expense categories (or operating areas) in which
this firm could attain additional cost saving by implementing
costcutting initiatives. Your conclusions should be outlined in a
memorandum directed to Continental’s Executive Management
team.
You are provided next with a description of Continental’s
operating costs and the potential drivers of costs so you can
conduct regression analysis to estimate the corresponding cost
functions. To help you in estimating the regressions, a
comprehensive set of instructions for performing regression
analysis using Microsoft Excel is provided in the Appendix.
Immediately following the description of costs, a series of
questions is provided that should help guide your analysis.
Additionally, to help you estimate your regressions, the Excel
Spreadsheet accompanying this case presents past quarterly data
for all of the above expenditures for the period of April 2000
through December 2008, and also quarterly operations data over
the same period.
CONTINENTAL’S OPERATING COSTS AND POTENTIAL
COST DRIVERS
The spreadsheet provides ten categories of operating costs,
including salaries and wages, aircraft fuel and related taxes,
aircraft rentals, airport fees, aircraft maintenance and repairs,
depreciation and amortization, distribution costs, passenger
services, regional capacity purchases, and other expenses. Of
13. these, some represent a single expense item. For example, the
cost of aircraft rentals and airport fees together comprise a
single cost item. Other costs represent cost pools comprising
several cost items. Such is the case of passenger services and
other expenses. The following provides a detailed description of
each cost, along with the potential cost drivers.
Salaries and wages This account represents costs related to
salaries and wages, as well as fringe benefits, of Continental’s
workers. These include salaries for pilots and wages for flight
attendants and ground crew, as well as wages for Continental’s
mechanics. Additionally, a significant portion of this salary
pool represents wages of reservation specialists, customer
service representatives at airports, and the salaries for
administrative and support personnel (e.g., flight schedulers,
technology personnel, accountants, and division managers_. A
possible cost driver of salaries is the available seat miles.4
Aircraft fuel and related taxes This represents the cost of jet
fuel and related fuel taxes. Jet fuel cost tends to be driven by
the current price of jet fuel and gallons of jet fuel consumed.
Aircraft rentals These are expenses for operating leases of
aircraft. The main driver is the number of leased planes in
Continental’s fleet, including regional jets operated on behalf of
Continental by four regional airlines under various capacity
purchase agreements.
4 Available seat miles is
calculated as the number of seats available for passengers
multiplied by the number of scheduled miles those seats are
flown.
Airport fees Represents landing fees and passenger security fees
paid to the various domestic and international airports where
Continental flies. Landing fees are driven by the number of
passengers.
Aircraft maintenance and repairs These are expenses associated
with the service and maintenance of planes. These include
expenses related to scheduled maintenance, spare parts and
materials, and airframe and engine overhauls. The main drivers
14. of these costs are the total aircraft in the fleet and the number
of miles flown.
Depreciation and amortization This represents depreciation and
amortization expenses of aircraft, ground equipment, buildings,
and other property. It must be emphasized that the largest
portion of depreciation expense relates to the depreciation of
aircraft. Although depreciation expenses are driven by the
acquisition cost of Continental’s capital assets, depreciation is
greatly influenced by both company policy and accounting
principles, such as the depreciation method, that a firm adopts.
Distribution costs These expenses represent credit card discount
fees, booking fees, and travel agency commissions, all of which
are affected by passenger revenue. Therefore, the driver of these
costs is total revenue.
Passenger services This is also a cost pool that includes
expenses related to processing and servicing passengers prior to
take-off, during flight, and after arrival at their destination. A
significant portion of these costs is generated by Continental’s
Field Services Division, the main function of which is to
provide service to planes prior to take-off. Some of these
expenses relate to checking in passengers, handling luggage on
and off planes, cleaning planes, stocking planes with beverage
and food, and refueling the aircraft prior to take-off. The
potential cost driver of these costs is the number of passengers.
Regional capacity purchases These are costs related to the
purchase of regional routes served by several regional airlines
on behalf of Continental (ExpressJet, Chautauqua, CommutAir,
and Cogan). These costs are driven by the combined flying
capacity of the four airlines: available regional seat miles.
Other expenses This is a cost pool that comprises many
ancillary and discretionary expenditures, including technology
expenses, security and outside services, general supplies, and
advertising and promotional expenses. Further, this cost pool
contains various special charges for gains and losses from the
sale of retired aircraft and costs of future leases. Given the
large variety of miscellaneous items, there is no clear driver of
15. these expenses; however, a large portion of them, such as
advertising and promotional expenses, are driven by total
revenue.
CASE QUESTIONS
1. Using the quarterly data for operating costs and the various
cost drivers of costs provided in the spreadsheet, estimate
regressions for each of the ten cost categories listed above.
Then, write the appropriate cost function for each category of
cost and interpret your regression results.
2. Based on your regression results and your interpretation of
those results, where do you see the largest reductions in costs if
flying capacity is lowered by 11 percent? Also, in which areas
do you see opportunities to achieve further cost reductions?
Why?
3. The table below provides a quarterly forecast of revenues, jet
fuel prices,5 and the projected operating activity for 2009.
Using the information from your regressions and the forecast
information provided, estimate Continental’s operating costs
and expected profit for the upcoming fiscal year.
4. Based on the results of your profitability analysis, what can
you say about the firm’s financial outlook? Would Continental
be earning an operating profit in 2009? If not, what should
Continental’s management do to restore profitability in 2009?
5. Summarize your conclusions in a memorandum addressed to
Continental’s CEO. In the memo, you must clearly communicate
your main findings, emphasizing specific areas in which you see
the greatest potential to achieve further reductions in costs and,
based on your profitability analysis, sum up the financial
outlook for 2009.
5 You should note that
Continental has entered into several future contracts to hedge
the exposed risks of rising fuel prices. The projected costs for
jet fuel on exhibit reflects the value of the various future
contracts which guarantee Continental a fixed price for jet fuel
at various maturity dates in 2009, as well the estimated gallons
16. of fuel that Continental plans to use during the year.
Projected Revenues and Operating Activity for 2009 Quarter
Variable 1 2 3 4 Revenues $2,962,000,000 $2,767,000,000
$2,947,000,000 $2,462,000,000 Total aircraft 634 617 604 601
Leased aircraft 398 394 380 379 Flights 77,778 80,675 78,948
83,008 Passengers 14,408,000 16,348,000 16,795,000
15,258,000 Available seat miles 26,323,000,000 28,007,000,000
28,933,000,000 26,291,000,000 Available regional miles
2,971,000,000 3,044,000,000 3,130,000,000 3,002,000,000
Passenger miles flown 24,562,000,000 25,186,000,000
25,529,000,000 24,550,000,000 Employees 39,600 42,300
43,000 42,000 Fuel price $1.82 $2.07 $1.99 $1.98 Fuel
consumed 403,000,000 430,000,000 369,000,000 479,000,000
Definitions of Operations Variables: Total aircraft =
number of planes in the fleet, including regional routes aircraft
and leased aircraft; Number of leased planes = number of leased
planes; Flights = number of flights from a single point of
departure to a single destination; Passengers = number of
paying passengers; Available seat miles = the number of seats
available multiplied by the number of miles flown; Available
regional miles = available seat miles on regional routes;
Passenger miles flown = number of paying passengers
multiplied by the number of miles flown; Employees = number
of employees, including pilots, flight attendants, and ground
crew. Fuel price = average price per gallon of jet fuel in the
respective quarter; and Fuel consumed = number of gallons of
fuel consumed in the respective quarter.
APPENDIX
FUNDAMENTALS OF REGRESSION ANALYSIS
Regression analysis is a powerful statistical technique that is
commonly used to predict a future value of a variable of
interest, such as costs, revenues, etc., based on data from the
past. To perform regression analysis, we must specify a
regression model of the relationship between the variable of
interest, the dependent variable, and one or several explanatory
17. variables, the independent variables. One simple and frequently
used way to describe the underlying relationship between the
dependent variable and the independent variable is with a linear
regression model. A linear model assumes that the relationship
between the variables of interest is strictly linear and is
described in the following way:
� = � +�� +�.
To better illustrate, suppose that you want to estimate Aircraft
Maintenance and Repair costs at Continental Airlines. In this
case, the dependent variable is the underlying cost that we are
trying to predict and the independent variable is whatever factor
causes that cost to rise or drop. For example, a common factor
that tends to affect Aircraft Maintenance and Repair costs for
airlines is the number of aircraft in their fleet. Therefore, this
would be the independent or predictor variable of Aircraft
Maintenance and Repair costs.
In its most simple term, what the above model indicates is
whether the variable Y is related to X. More formally, the
regression model represents the mean of Y for a given change in
X. That is, whether the mean of Y is linearly related to X plus
some error term. Where Y represents the dependent variable
(Aircraft Maintenance and Repair costs), X is the independent
variable (number of aircraft), a and b are the estimated
coefficients, the constant and the slope of the regression model,
respectively, which will be explained next, and e is the residual
or estimated error of the model. The “a” coefficient is a value at
which the line intercepts the Y-axis. It is the value of the mean
of Y when � = 0. With respect to a cost function, it represents
the fixed costs in the cost function. The “b” coefficient is called
the slope because it measures the slope of the regression line. In
our example, it represents the amount by which the mean of Y
(aircraft maintenance costs) changes if X (number of the
aircraft) changes by one unit.
The next question is how do we perform regression analysis in
Excel? To explain the procedure, consider the following
quarterly data for Continental Airlines for Aircraft Maintenance
18. and Repair costs and the potential cost driver of such costs, the
total number of aircraft in Continental’s fleet:
Observation Aircraft Maintenance Total Aircraft 1 $340,000 10
2 $400,000 40 3 $440,000 50 4 $480,000 80 5 $530,000 110
The first step in regression analysis is to see whether a linear
relationship exists between the dependent variable and the
independent (predictor) variable. This may be accomplished by
plotting the data on a graph. Data for aircraft maintenance
costs, the dependent variable, is plotted on the Y (vertical) axis,
and data for the number of aircraft, the independent variable, on
the X (horizontal) axis.
To create this graph in Excel, follow these steps: 1. Copy the
data into an Excel spreadsheet, copying the data pertaining to
the cost driver in the first column and the cost data in the
second column. 2. Click on the “Insert” tab and select the
Scatter Chart option having only markers. 3. Once the chart
space has been created, select it. From the “Design” tab, click
the Select Data option. 4. Click the button next to the “Chart
data range” field. Select the two columns you copied into Excel,
including the headers, and then hit “Enter”. The chart should
populate similar to below.
As shown above, though the relationship scatter plots of Y and
X will not outline a perfectly straight line, we could observe
that the scatter graph shows that this relationship is indeed
linear and, thus, indicates that the total number of planes is a
good predictor of maintenance costs. Moreover, by plotting the
data, we could identify potential outliers (data points that do
not represent normal Activity) and eliminate them from the
analysis. In this particular case, it can be observed that no
outliers are present.
Next, we’ll proceed to estimate the regression model, where
aircraft maintenance would be a function of the number of
airplanes serviced. In this particular case, since we have a
single predictor variable, we refer to it as a univariate
regression. The regression model is expressed as follows:
$0
19. $100,000
$200,000
$300,000
$400,000
$500,000
$600,000
0 20 40 60 80 100 120 Total Aircraft
Aircraft Maintenance
�������� ����������� = � +� (�����
��������)+�,
where aircraft maintenance substitutes for the Y variable and
total aircraft for the X variable described previously; e is called
the residual or error term and is defined as the difference
between an actual observation of the dependent variable (cost)
and its estimated or forecasted value from the regression
estimation that we will run next.
The idea of regression analysis is to calculate the values of the
dependent variables that minimize the sum of square of these
residuals. The mechanics of regression analysis work as
follows. Using the data in your sample, regression would
calculate a mean and would use this mean as a benchmark to
compare as a central value in the calculations. Simply put, the
regression equation in this example will provide an estimate of
the relationship between aircraft maintenance and the number of
flights that Continental Airlines offered, on average, in the
quarter. Let us now perform the regression in Excel.
To estimate a regression in Excel (MS Office 97 thru 2005
versions), follow these steps:
1. On the “Tools” menu bar, select the “Data Analysis”
command, and then select “regression.” Note that if the “Data
Analysis” command is not available, you need to install the
“Analysis Toolpak add-in.” Here is how to add it: on the
“Tools” menu bar, select the “Add-in” command. Then, select
“Analysis Toolpak” and press “OK.”
2. Enter the cell reference for the dependent variable (i.e.,
aircraft maintenance); the range selected must consist of a
20. single column of data; then proceed to select the cell reference
for the independent variable (i.e., number of airplanes).
3. Select whether or not the first row or column of the input
ranges contain labels (or headings). Excel generates appropriate
data labels for the regression output table.
4. Click to create a new worksheet containing the regression
output.
5. Press “OK” to generate the Regression Output Table.
To estimate a regression in Excel (MS Office 2007 version),
follow these steps:
1. On the “Data” menu bar, select the “Data Analysis” command
located on the right hand corner, and then select “Regression
Analysis.” Note that if the “Data Analysis” command is not
available, you need to install the “Analysis Toolpak add-in.”
Here is how to add it: press the “Office” icon located on the
left-hand side of the Excel menu bar, select the “Excel Option,”
then select “Add-ins” command. Then, select “Analysis
Toolpak” and press “OK.”
2. Enter the cell reference for the dependent variable (i.e.,
aircraft maintenance); the range selected must consist of a
single column of data; then proceed to select the cell reference
for the independent variable (i.e., number of airplanes).
3. Select whether or not the first row or column of the input
ranges contain labels (or headings). Excel generates appropriate
data labels for the regression output table.
4. Click to create a new worksheet containing the regression
output.
5. Press “OK” to generate the Regression Output Table.
Using the above data, the Regression Output Table should
appear as in Exhibit A1, below. Let us now proceed to analyze
each main statistic of the regression output. The intercept, with
a value of $328,707.48, is commonly referred to as the alpha
coefficient and is a constant value in the regression function. In
the specific case of cost estimation, this value represents the
amount of fixed costs present in the cost function. In our
example, this value indicates that $328,707 of maintenance
21. costs is fixed and would not change at all given any change in
the number of planes in Continental’s fleet. Further, the second
statistic of interest is the coefficient estimate for the total
number of Aircraft, which has a value of 1,884.35. This is the
slope of the regression function and it is referred as the beta
coefficient. In cost estimation, this value represents the portion
of variable costs in the cost function; that is, the portion of
maintenance costs that would vary given any changes in the
number of planes. In our example, the slope coefficient is
interpreted as follows: for each plane in which Continental
provides scheduled maintenance, the amount of maintenance
costs is expected to increase, on average, by $1,884.35 dollars.
An important issue in the examination of both coefficients is
that neither the intercept nor the slope is ever examined in
isolation. Each coefficient must be analyzed in combination
with its corresponding t-statistic, or the respective p-value. The
t-statistic is the ratio of the value of the coefficient to its
standard error. The standard error of the coefficient represents
the amount of variation in costs that is unexplained by the cost
driver; in our example, total aircaft. The lower (higher) the
standard error, the better (worse) each coefficient is. Going
back to the t-statistic, this value indicates whether each of the
two coefficients is different from zero. A rule of thumb is that
if the t-statistic is at least 1.96 or greater, then we are 95
percent confident that the value of the coefficient is
significantly greater than zero. This implies that the coefficient
is a valid estimate and, therefore, could be used in the cost
function as a way to predict future costs. If the t-value is less
than 1.96, then we cannot rule out the possibility that the
coefficient is zero and, therefore, the coefficient should not be
used in the prediction of costs. If we take a look at the
Regression Output Table, the t-statistics for the intercept and
the slope coefficients are 32.34 and 12.49, respectively.
Therefore, we can conclude that both coefficients are not zero
and, thereby, the estimates of fixed costs and variable costs per
unit are valid estimates.
22. The second statistic of importance in the examination of each
coefficient is the probability value, or the p-value. Each t-
statistic has a corresponding p-value. The p-value is the
reciprocal of the tstatistic in that it tells us the probability that
the coefficient estimate is significantly different from zero. If
the p-value is less than or at least equal to 0.05, we are 95
percent confident that the coefficient estimate for the intercept
or the slope coefficient is statistically significantly different
from zero. A smaller p-value indicates a larger t-statistic. For
example, the p-value for the tstatistic in the intercept
coefficient equals 0.0000649, and 0.0011 for the slope
coefficient.
Exhibit A1 SUMMARY OUTPUT
Regression Statistics Multiple R 0.990524645 R Square
0.981139072 Adjusted R Square 0.974852096 Standard Error
11566.62648 Observations 5
ANOVA
df SS MS F
Significance F Regression 1 20878639456 20878639456
156.0589831 0.001105628 Residual 3 401360544.2
133786848.1 Total 4 21280000000
Coefficients
Standard Error t Stat P-value Lower 95% Upper 95%
Lower 95.0%
Upper 95.0% Intercept 328707.483 10163.56279 32.3417575
6.49662E-05 296362.4901 361052.4758 296362.4901
361052.4758 No. of Aircraft 1884.353741 150.8405294
12.49235699 0.001105628 1404.311856 2364.395627
1404.311856 2364.395627
This means that both coefficients are not zero. More formally,
this means that fixed costs have a probability of being zero
about six in 100,000 times, and the variable cost about one in
1,000 times.
The last statistic of interest in our analysis is the R2, or the
Adjusted R2. The R2 represents the goodness of fit of the
23. model, or the explanatory power of the model. In the case of the
Adjusted R2, it measures the same thing, but after adjusting for
degrees of freedom in the regression model. Simply put, the R2
(Adjusted R2) represents the percentage of variation in the
dependent variable (aircraft maintenance) that is explained by
the independent variable (total aircraft). This value ranges
between 0 and 1, with the larger value representing a higher
explained variation in costs. In our example, the R2 equals 0.98,
which indicates that approximately 98 percent of the variation
in aircraft maintenance costs is explained by the total number of
planes which were serviced. A follow-up question is “which
cutoff value is acceptable?” This is subjective and will
definitely depend on the underlying analysis. In the case of cost
estimation, perhaps we may want to have at least 30 percent of
the variation in costs explained by the cost driver.
Multivariate regression analyses
Thus far, we have explored regression analyses with one
independent variable. Then, the question becomes what happens
if we have two or more independent variables (in our case, two
cost drivers) as predictors of the dependent variable (Aircraft
Maintenance and Repair costs). For example, it is feasible that
besides the number of planes in Continental’s fleet, aircraft
maintenance and repair costs may also be driven by other
factors, such as the average utilization of each plane in miles or
hours.
In this case, the regression model varies subtly from the case of
a single predictor variable. The mathematical notation for a
multivariate regression model is expressed as follows:
� = � +�1�1 +�2�2 +⋯+���� +�,
where Y represents the dependent variable (Aircraft
Maintenance and Repair costs), �1 is independent variable one
(number of aircraft in the fleet), and �2 is independent variable
two (average daily miles flown on each aircraft). Just as in the
case of a univariate regression, a and b are the estimated
coefficients of the constants and the slopes of the regression
model; that is, the fixed costs and the slopes of the cost
24. function, respectively, and e is the residual or estimated error of
the regression model.
Assume the following quarterly data for Aircraft Maintenance
and Repair costs and for each of the potential cost drivers
discussed above:
Observation
Aircraft Maintenance Total Aircraft Daily Miles Flown 1
$340,000 10 1,166 2 $400,000 40 1,270 3 $440,000 50 1,457 4
$480,000 80 1,450
5 $530,000 110 1,433
To estimate a multivariate regression in Excel, we must follow
Step 1, described previously, plus the following additional
steps:
1. Enter the cell reference for the dependent variable (i.e.,
aircraft maintenance); the range selected must consist of a
single column of data.
2. Then, proceed to select the cell reference for all of the
independent variables (number of airplanes, average age, and
the average number of hours flown daily); the range selected
must consist of two or more columns. Note that all independent
variables must be listed in sequential order; that is, next to each
other.
3. Select if the first row or column of the input ranges contain
labels (or headings); clear if your input has no labels; Excel
generates appropriate data labels for the regression output table.
4. Click to create a new worksheet containing the regression
output.
5. Press “OK” to generate the Regression Output Table.
Using the above data, the Regression Output Table for the
multivariate regression appears as shown in Exhibit A2.
The statistics of interest are the same from the case of the
univariate regression. The only difference is that now we have
two slope coefficients. Let us summarize the main statistics of
interest. The Intercept represents the alpha coefficient or the
constant value in the regression function and, thus, indicates
that there is $181,834.40 of fixed costs. The coefficient
25. estimate for the Number of Aircraft has a value of $1,552.18
and indicates that for every aircraft in Continental’s fleet,
Maintenance and Repair costs rise by this amount. And, the
coefficient estimate for Daily Miles Flown indicates that for
every mile that each aircraft in the fleet is flown daily,
maintenance costs increase, on average, by $122.59. Note that
the t-statistic for each coefficient is greater than 2 and, thus,
indicates that both coefficients are different from zero. The
Adjusted R2 indicates that 99% of the variation in aircraft
maintenance is explained by both cost drivers. An additional
statistic of interest in multivariate regression is the F-statistic,
which indicates the fitness of the model; that is, how well all
predictors as a whole explain changes in the dependent variable.
An F-statistic of two or greater indicates that the independent
variables as a whole make a good prediction of changes in
aircraft maintenance costs.
Let us now proceed to discuss how we can develop the cost
functions as a way to estimate Aircraft Maintenance and Repair
costs.
Exhibit A2 SUMMARY OUTPUT
Regression Statistics Multiple R 0.999500305 R Square
0.999000859 Adjusted R Square 0.998001718 Standard Error
3260.500131 Observations 5
ANOVA
df SS MS F
Significance F Regression 2 21258738278 10629369139
999.8596569 0.000999141 Residual 2 21261722.21
10630861.11 Total 4 21280000000
Coefficients
Standard Error t Stat P-value Lower 95% Upper 95%
Lower 95.0%
Upper 95.0% Intercept 181834.4026 24729.33755 7.352983161
0.01799797 75432.65089 288236.1544 75432.65089
288236.1544 No. of Aircraft 1552.183215 69.95682026
22.18773251 0.002025132 1251.183311 1853.183119
26. 1251.183311 1853.183119 Daily Miles Flown 122.5936916
20.50237078 5.979488566 0.026847454 34.37911005
210.8082732 34.37911005 210.8082732
Development of a cost function to estimate future costs
A cost function is simply an algebraic representation of how a
given cost would change given a change in the cost driver. For
example, in our example, the cost function would represent how
Aircraft Maintenance and Repair costs would change given a
one-unit change in the number of aircraft. Using our prior
results from the univariate regression, the cost function is
written as follows:
� = $328,707.48+$1,324.94(�).
The above cost function indicates that $328,707.48 of Aircraft
Maintenance and Repair costs is fixed and would not change
irrespective of changes in the number of aircraft in the fleet,
and Maintenance and Repair costs would increase by $ 1,884.35
for every aircraft in Continental’s fleet.
After setting up the cost function, we can now proceed to
estimate Maintenance and Repair costs. Let us assume that
Continental Airlines plans to have 142 planes in its fleet next
year. Plugging this figure into the cost function, we obtain the
following result:
� = $328,707.48+$1,324.94(142) = $516,848.96.
Therefore, given the projection with respect to the number of
airplanes, we can conclude that Continental Airlines would
incur approximately $516,848.96 in Aircraft Maintenance and
Repair Costs for next year.
The same procedure is applied to the case in which there are
multiple cost drivers. Referring to our prior results from the
multivariate regression, the cost function is written as follows:
� = $181,834.40+$1,552.18(�1)+$122.59(�2).
Using the same projection as before of 142 planes for next year,
plus assuming that the average utilization of miles for each
plane is 1,550 miles, then total Aircraft Maintenance costs for
next year is calculated as follows:
� = $181,834.40+$1,552.18(142)+$122.59(1,550) =
27. $592,258.50.
Hence, given the projection of the number of airplanes and the
average utilization of miles that Continental plans to fly each
plane, we can conclude that Continental Airlines would incur
approximately $592,259 in Aircraft Maintenance and Repair
costs for next year.
Guidelines to Writing a Memorandum
1. Begin your memorandum with a general statement about the
core issue you are discussing.
2. The second paragraph should include a short explanation as
to how you arrive at your conclusion. Specifically, it must
describe how the analysis was conducted (i.e., regression
analysis) and the assumptions taken.
3. The memorandum should then incorporate a more detailed
discussion of the findings supporting your conclusion.
Specifically, indicate which areas would experience the largest
reduction in costs and how that can be achieved. Also discuss
the profit outlook for 2009 and provide a list of
recommendations to further reduce operating costs to reverse
losses and return the company to profitability.
4. Pay special attention to grammar, including spelling,
fragmented, and convoluted sentences. Focus on parsimony over
length, as long as clarity and understanding are not jeopardized.
The memo should not exceed 1-2 pages double-spaced. (Note:
memos are typically one page, single-spaced, but double-spaced
between paragraphs.)
References
Maynard, M. 2008. Big airlines in a rush go small. The New
York Times (June 6).
