Finance 30210: Managerial Economics

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Finance 30210 Managerial Economics

• Demand Estimation and Forecasting

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The Truth is Out There...
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There is a true probability distribution that
governs the outcome of a coin toss (assuming a
fair coin)
The Truth
Suppose that we were to flip a coin over and over
again and after each flip, we calculate the
percentage of heads tails
(Sample Statistic)
(True Probability)
That is, if we collect enough data, we can
eventually learn the truth!
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Continuous distributions
Probability distributions identify the chance of
each possible event occurring
Probability
Event
Mean
1 SD
2 SD
3 SD
-1 SD
-2 SD
-3 SD
65
95
99
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Statistical Tests
Suppose that you wanted to learn about the
temperature in South Bend
The Truth
Temperature
We could find this distribution by collecting
temperature data for south bend
Sample Mean (Average)
Sample Variance
Note Standard Deviation is the square root of
the variance.
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Conditional Distributions
Obviously, the temperature in South Bend is
different in the winter and the summer. That is,
temperature has a conditional distribution
Temp (Summer)
The Truth
Temp (Winter)
Regression is based on the estimation of
conditional distributions
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Some useful properties of probability
distributions
Probability distributions are scaleable

3 X
Mean 1 Variance 4 Std. Dev. 2
Mean 3 Variance 36 (334) Std. Dev. 6
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Probability distributions are additive


Mean 1 Variance 1 Std. Dev. 1
Mean 2 Variance 9 Std. Dev. 3
Mean 3 Variance 14 (1 9 22) Std. Dev.
3.7
COV 2
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Suppose we know that your salary is based on your
shoe size
The Truth
Salary 20,000 2,000 (Shoe Size)
Salary
Shoe Size
Mean 6 Variance 4 Std. Dev. 2
Mean 32,000 Variance 16,000,000 Std. Dev.
4,000
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We could also use this to forecast
The Truth
Salary 20,000 2,000 (Shoe Size)
If Bigfoot had a jobhow much would he make?
Salary 20,000 2,000 (50) 120,000
Note There is NO uncertainty in this prediction.
Size 50!!!
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Surely, there are other important determinants of
income other than shoe size. Therefore, the
truth might look something like this
Salary 20,000 2,000 (Shoe Size) Error
This error term incorporates all other
information that explains shoe size, but is
unknown. All we require is that it is mean zero
and uncorrelated with shoe size
Now, our forecast of shoe size will have an error
associated to it
Salary 20,000 2,000 (50) 120,000
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Searching for the truth.
You believe that there is a relationship between
shoe size and salary, but you dont know what it
is.

• Collect data on salaries and shoe sizes
• Estimate the relationship between them

Note that while the true distribution of shoe
size is N(6,4), our collected sample will not be
N(6,4). This sampling error will create errors
in our estimates!!
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Slope b
a
Salary a b (Shoe Size) error
We want to choose a and b to minimize the
error!
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We have our estimate of the truth
T-Stats bigger than 2 are considered
statistically significant!
Salary 45,415 1,014 (Shoe Size) error
Intercept (a) Mean 45,415 Std. Dev. 1,650
Shoe (b) Mean 1,014 Std. Dev. 257
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Percentage of income variance explained by shoe
size
Error Term Mean 0 Std, Dev 11,673
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Using regressions to forecast (Remember, Bigfoot
wears a size 50).
50
Salary 45,415 1,014 (Shoe Size) error
Mean 45,415 Std. Dev. 1,650
Mean 1,014 Std. Dev. 257
Mean 0 Std. Dev. 11,673
(Recall, Shoe size has a mean of 6)
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Income
95
Forecast
-95
Shoe Size
Note that your forecast error will always be
smallest at the sample mean! Also, your forecast
gets worse at an increasing rate as you depart
from the mean
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Cross Sectional estimation holds the time period
constant and estimates the variation in demand
resulting from variation in the demand factors
Demand Factors
Time
t1
t-1
t
For example can we predict demand for Pepsi in
South Bend by looking at selected statistics for
South bend
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Estimating Cross Sectional Demand Curves
Lets begin by estimating a basic demand curve
quantity demanded is a function of price.
Next, we need to assume a functional form. For
simplicity, lets start with a linear model
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Next, Collect Data on Prices and Sales
Price
Quantity
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That is, we have estimated the following equation
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(3.004)
(0.774)
(10.02)
Our forecast of demand is normally distributed
with a mean of 23 and a standard deviation of
9.90.
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If we want to calculate the elasticity of our
estimated demand curve, we need to specify a
specific point.
2.50
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Given our model of demand as a function of
income, and prices, we could specify a variety of
functional forms
Linear Demand Curves
Here, quantity demanded responds to dollar
changes in price (i.e. a 1 increase in price
lowers demand by 4 units.
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Given our model of demand as a function of
income, and prices, we could specify a variety of
functional forms
Semi Log Demand Curves
Here, quantity demanded responds to percentage
changes in price (i.e. a 1 increase in price
lowers demand by 4 units.
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Given our model of demand as a function of
income, and prices, we could specify a variety of
functional forms
Semi Log Demand Curves
Here, percentage change in quantity demanded
responds to a dollar change in price (i.e. a 1
increase in price lowers demand by 4.
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Given our model of demand as a function of
income, and prices, we could specify a variety of
functional forms
Log Demand Curves
Here, percentage change in quantity demanded
responds to a percentage change in price (i.e. a
1 increase in price lowers demand by 4.
Log Linear demands have constant elasticities!!
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One Problem
Suppose you observed the following data points.
Could you estimate a demand curve?
D
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Estimating demand curves
A problem with estimating demand curves is the
simultaneity problem.
S
Market prices are the result of the interaction
between demand and supply!!
D
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Estimating demand curves
Case 1 Both supply and demand shifts!!
Case 2 All the points are due to supply shifts
S
S
S
S
S
S
D
D
D
D
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An example
Suppose you get a random shock to demand
Demand
The shock effects quantity demanded which (due to
the equilibrium condition influences price!
Supply
Therefore, price and the error term are
correlated! A big problem !!
Equilibrium
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Suppose we solved for price and quantity by using
the equilibrium condition
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We could estimate the following equations
The original parameters are related as follows
We can solve for the supply parameters, but not
demand. Why?
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By including a demand shifter (Income), we are
able to identify demand shifts and, hence, trace
out the supply curve!!
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D
D
D
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Time Series estimation holds the demand factors
constant and estimates the variation in demand
over time
Demand Factors
Time
t1
t-1
t
For example can we predict demand for Pepsi in
South Bend next year by looking at how demand
varies across time
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Time series estimation leaves the demand factors
constant and looks at variations in demand over
time. Essentially, we want to separate demand
changes into various frequencies
Trend Long term movements in demand (i.e. demand
for movie tickets grows by an average of 6 per
year)
Business Cycle Movements in demand related to
the state of the economy (i.e. demand for movie
tickets grows by more than 6 during economic
expansions and less than 6 during recessions)
Seasonal Movements in demand related to time of
year. (i.e. demand for movie tickets is highest
in the summer and around Christmas
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Suppose that you work for a local power company.
You have been asked to forecast energy demand for
the upcoming year. You have data over the
previous 4 years
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First, lets plot the datawhat do you see?
This data seems to have a linear trend
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A linear trend takes the following form
Estimated value for time zero
Estimated quarterly growth (in kilowatt hours)
Forecasted value at time t (note time periods
are quarters and time zero is 20031)
Time period t 0 is 20031 and periods are
quarters
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Lets forecast electricity usage at the mean time
period (t 8)
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Heres a plot of our regression line with our
error bandsagain, note that the forecast error
will be lowest at the mean time period
T 8
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We can use this linear trend model to predict as
far out as we want, but note that the error
involved gets worse!
Sample
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One method of evaluating a forecast is to
calculate the root mean squared error
Sum of squared forecast errors
Number of Observations
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Lets take another look at the datait seems that
there is a regular pattern
Q2
Q2
Q2
Q2
We are systematically under predicting usage in
the second quarter
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• Average Ratios
• Q1 .87
• Q2 1.16
• Q3 .91
• Q4 1.04

We can adjust for this seasonal component
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Now, we have a pretty good fit!!
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Recall our prediction for period 76 ( Year 2022
Q4)
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We could also account for seasonal variation by
using dummy variables
Note we only need three quarter dummies. If the
observation is from quarter 4, then
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Note the much better fit!!
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Ratio Method
Dummy Variables
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A plot confirms the similarity of the methods
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Recall our prediction for period 76 ( Year 2022
Q4)
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Recall, our trend line took the form
This parameter is measuring quarterly change in
electricity demand in millions of kilowatt hours.
Often times, its more realistic to assume that
demand grows by a constant percentage rather that
a constant quantity. For example, if we knew
that electricity demand grew by g per quarter,
then our forecasting equation would take the form
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If we wish to estimate this equation, we have a
little work to do
Note this growth rate is in decimal form
If we convert our data to natural logs, we get
the following linear relationship that can be
estimated
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Lets forecast electricity usage at the mean time
period (t 8)
BE CAREFUL.THESE NUMBERS ARE LOGS !!!
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The natural log of forecasted demand is 2.698.
Therefore, to get the actual demand forecast, use
the exponential function
Likewise, with the error bandsa 95 confidence
interval is /- 2 SD
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Again, here is a plot of our forecasts with the
error bands
T 8
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When plotted in logs, our period 76 ( year 2022
Q4) looks similar to the linear trend
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Again, we need to convert back to levels for the
forecast to be relevant!!
Errors is growth rates compound quickly!!
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There doesnt seem to be any discernable trend
here
Consider a new forecasting problem. You are asked
to forecast a companys market share for the 13th
quarter.
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Smoothing techniques are often used when data
exhibits no trend or seasonal/cyclical component.
They are used to filter out short term noise in
the data.
A moving average of length N is equal to the
average value over the previous N periods
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The longer the moving average, the smoother the
forecasts are
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Calculating forecasts is straightforward
MA(3)
MA(5)
So, how do we choose N??
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Total 78.3534
Total 62.48
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Exponential smoothing involves a forecast
equation that takes the following form
Forecast for time t
Forecast for time t1
Actual value at time t
Smoothing parameter
Note when w 1, your forecast is equal to the
previous value. When w 0, your forecast is a
constant.
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For exponential smoothing, we need to choose a
value for the weighting formula as well as an
initial forecast
Usually, the initial forecast is chosen to equal
the sample average
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As was mentioned earlier, the smaller w will
produce a smoother forecast
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Calculating forecasts is straightforward
W.3
W.5
So, how do we choose W??
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Total 87.19
Total 101.5