Topic 2b .pptx

RMIT Classification: Trusted
Richard Tay
VC Senior Research Fellow
School of Business IT & Logistics
Demand Estimation for Transport Services
OMGT1058, OMGT2102, OMGT2227, OMGT2227, OMGT2303
Transport Economics
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Outline of Topic
 Review Basics of Demand Analysis
 Estimation and Forecasting of Demand
• Review of Basic Statistics
o Measures of Central Tendency
o Measures of Spread
o Simple Hypothesis Test
• Correlation
• Basic Trend Analysis
• Basic Regression Analysis
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What is Statistics?
 “There are lies, damned lies and statistics”
(former British Prime Minister)
 “Statistics is like a bikini; it reveals a lot but also covers some of the
most important parts”
(student in Singapore)
 “If I had one day left to live, I would live it in my statistics class, it
would seem so much longer”
(American student)
 Systematic way of processing data
to help us make decisions.
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What is Statistics?
 Statistical Inference:
• In general, we are interested in some parameters of a population.
• Often, we are not able to measure the parameters directly. Why?
• Thus, we take a sample and use its sample statistics to infer about
the population parameters.
 Main steps in statistics:
• Problem definition
• Data collection
• Data processing
• Analysis of results
• Make recommendation
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Descriptive Statistics
 Measure of Central Tendency or Location:
• Mean
• Mode
• Median
 Measures of Spread or Dispersion:
• Range
• Standard Deviation
 Questions:
• Examples when each is more appropriate
• What are the strengths and weaknesses of the three measures?
n
x
x i


 
1
2
2




n
x
x
s i
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Hypothesis Testing
 Null Hypothesis: A population parameter equals to some constant.
Examples: 𝐻𝑜: μ = 5; 𝐻𝑜: 𝜇1= 𝜇2; 𝐻𝑜: 𝜎2
≥ 2
 Alternative Hypothesis: The competing hypothesis to be "accepted" if
null hypothesis is rejected.
Examples: 𝐻𝑜: μ ≠ 5; 𝐻0: 𝜇1 − 𝜇2 ≠ 0 𝐻𝑜: 𝜎2< 2
 Note the equal sign is always part of the null hypothesis. It is what we
use to do the test.
 The alternative hypothesis is not tested, only the null hypothesis.
 We do not show that the alternative is correct, only reject or do not
reject the null hypothesis.
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Hypothesis Testing
 Remember that only the null hypothesis is
tested for rejection.
 Is there any evidence to reject the null
hypothesis?
 Assuming that the null hypothesis is true,
what is the probability of getting the sample
we have?
 If the probability is small, we reject the null
hypothesis. We are confident that the null
hypothesis is false.
 If probability is large, we cannot reject the
null. The sample may have come from a
population stated in the null hypothesis.
 Traditionally, we use threshold of a = 0.05
for the p-value to reject or not reject the null
hypothesis
Assume Ho is true
Take a random sample
Calculate probability (p) of
drawing this sample from
the population if Ho is true
Reject Ho
p-value
is small
p-value
is big
Don’t Reject Ho
Rejection is “Good”
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Example: Testing Mean
 𝐻0: 𝜇 = 𝜇0 {e.g. μ = 4}
𝐻𝑎: 𝜇 ≠ 𝜇0 {e.g. μ ≠ 4}
 Decision Rule: Reject the null if estimated
t-statistic is greater than the critical value.
 Reject Ho if
 When the alternate hypothesis is not equal
to a constant, we have a two-tail test and
a/2 is used as the error.
 When the alternate hypothesis is greater or
less than a constant, we have a one-tail
test and a is used as the error.
   
a

1
0
/



 n
t
n
s
x
t
≠
< >
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Example: Testing Mean
 Sample: {3, 4, 5, 7, 6, 5}
 =
(3+4+5+7+6+5)
6
= 5
 =
(3−5)2+(4−5)2+(5−5)2+(7−5)2+(6−5)2+(5−5)2
(6−1)
= 2
 
1
2
2




n
x
x
s i
 = 1.73
 
6
/
2
4
5
/




n
s
x
t o

 Since t < 2.57 (df = n-1 = 6-1 = 5, a/2 = 0.05/2 = 0.025), we cannot
reject the null.
 Alternatively, p-value for a t-statistics of 1.73 with 5 degrees of
freedom is 0.145 which is larger than the threshold of 0.05, thus we
cannot reject the null hypothesis.
 Note: for a large sample (n>30), the threshold value for t-statistic is
1.96.
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Hypothesis Testing
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Hypothesis Testing
11
Finding the p-value
Click on the fx button
Select Statistical
Select T.DIST.2T
Input 1.73 for X
Input 5 for deg-freedom
Finding the critical t-value
Click on the fx button
Select Statistical
Select T.INV.2T
Input 0.05 for probability
Input 5 for deg-freedom

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 Note that we cannot accept the null hypothesis, we can only reject the
null hypothesis.
 To illustrate, suppose we conducted another test to see if the mean is
equal to 5.0, 4.5 or 5.5, the t-tests would also not reject the null.
 Is the mean equal to 4.0, 4.5, 5.0 or 5.5?
 If we reject the null hypothesis, the default choice is to accept the
alternate hypothesis.
 Thus, hypothesis testing is valid only if we have only two mutually
exclusive choices and one is the null.
Hypothesis Testing
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Hypothesis Testing
 Type I Error:
• Reject the null when the null is true.
• Usually denoted by the symbol a.
• Confidence Level = 1 - a.
 Type II Error:
• Do Not Reject the null when it is false.
• Usually denoted by the symbol b.
• Power of the test = 1 - b.
• Quite difficult to compute, seldom reported.
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Scatter Plot
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 The correlation between two random variables is a measure of the
linear relationship between them.
 The correlation coefficient, , gives a quantitative measure of how well
two variables move together
 The correlation coefficient ranges between -1 and 1.
Correlation
0
No
Relationship
-1 1
Perfect
Negative
Relationship
Perfect
Positive
Relationship
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Correlation
Y
X
𝑌
𝑋
𝑌𝑖 > 𝑌
𝑋𝑖 > 𝑋
(𝑌𝑖 − 𝑌)(𝑋𝑖 − 𝑋) > 0
𝑌𝑖 < 𝑌
𝑋𝑖 > 𝑋
(𝑌𝑖 − 𝑌)(𝑋𝑖 − 𝑋) < 0
(𝑌𝑖 − 𝑌)(𝑋𝑖 − 𝑋) < 0
(𝑌𝑖 − 𝑌)(𝑋𝑖 − 𝑋) > 0
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Correlation
Exercise: Match
the correlation
coefficients to
the scatter plots
r1 = +0.87
r2 = +0.73
r3 = -0.42
r4 = -0.77
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 Forecasting is the process of predicting the direction of future
trends based on historic and present data.
 Necessary conditions for forecasting validity:
• There is good information about the past.
• The information can be quantified in the form of data.
• Some aspects of the past pattern will continue into the future.
Simple Time Series Forecasting
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 Transport and logistics data may be collected by different methods.
• Time series data are simply data collected over time on a hourly,
daily, weekly, monthly or yearly basis.
• Cross sectional data are data collected over different decision
makers (consumers, household, train lines, cities, states,
countries, etc.) during the same time period
• Panel data contains both time series and cross sectional data.
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Linear Trend Line
 The most basic type of time series analysis is uncovering the
underlying long term time trend and using it for forecast future
demand.
 We assume that the long term trend line can be represented by a simple
linear equation or straight line.
 𝑦𝑡 = 𝛽0 + 𝛽1𝑡
Year Train Boarding
(million)
Year Train Boarding
(million)
1983-84 92 1997-98 113.5
1984-85 99 1998-99 118.1
1985-86 100 1999-2000 124.2
1986-87 112.7 2000-01 130.3
1987-88 101 2001-02 131.8
1988-99 107 2002-03 133.8
1989-90 108 2003-04 134.9
1990-91 108.5 2004-05 145.1
1991-92 109.5 2005-06 159.1
1992-93 105.9 2006-07 178.6
1993-94 101.1 2007-08 201.2
1994-95 105.5 2008-09 213.9
1995-96 109.2 2009-10 219.3
1996-97 112.5 20

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Linear Trend Line
 We can eye-ball the data and try to fit the “best” line through the data.
 However, we may end up with different lines
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Ordinary Least Square Regression
errors
0
50
100
150
200
250
1980 1985 1990 1995 2000 2005 2010 2015
Train Boardings
errors
 Best line to represent the data is the one that minimises the error
or the difference between the actual data and the estimated line.
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 𝑌 = 𝛽0 + 𝛽1𝑋 + 𝜀
• where Y is dependent variable (variable you like to predict, such
as quantity demanded)
• X is the independent variable or explanatory variable (e.g., time,
income, price, population, etc.)
• b0 and b1 are unknown parameters or constants to be estimated.
• e is the error term
• The error term is assumed to be normally distributed
 Error  e = (𝑦𝑡 − 𝛽0 − 𝛽1𝑋)
 Note that some of the points will be above the line while others will be
below the line, which means that some errors will be positive while
others will be negative.
 We can try to minimise the absolute value but this is difficult to do
analytically. Hence, we minimise the square of the errors.
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 Choose the values of b0 & b1 to minimise sum of squares of error
 Minimise 𝑛=1
𝑁
e2 = (𝑦𝑡 − 𝛽0 − 𝛽1𝑡)2
 Solving the above minimisation problem gives:
𝛽0 = 𝑌 − 𝛽1𝑋







 n
i
i
n
i
i
i
X
X
Y
Y
X
X
1
2
1
)
(
)
)(
(
𝛽1
Y = mx + c
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 The value of y can be separated into two parts:
• Value estimated or predicted by the regression line
• Error
𝛽0
Actual value
Estimated value
value of Y using
regression line
Error or residual
𝑦 = 𝛽0+𝛽1X
Y
X
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 Changes in the value of y can also be separated into two parts:
• Changes predicted by the regression line
• Error
 Total Sum of Squares (SST) = Regression Sum Squares (SSR) +
Error Sum of Squares (SSE)
 Coefficient of Determination: R2 measures how well the estimated
regression line fit the data. It is the most commonly used indicator for
the goodness-of-fit in regression.
 R2 =
𝑆𝑆𝑅
𝑆𝑆𝑇
= 1 −
𝑆𝑆𝐸
𝑆𝑆𝑇
 R2 measures the amount of variation in Y that can be explained by
the variation in X
 R-squared is always between 0 and 1 (0% and 100%)
 Low R2 value indicates that the model is not very useful – knowing
the value of X does not help us to predict the value of Y
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1. Click the Microsoft Office Button , and then click Excel to open the
excel program
2. On the top left corner, click File, then click Options on left bottom. The
excel option panel will open up.
3. On the left column, Click Add-Ins to open the add-ins panel.
4. Select Analysis Toolpak, and in the Manage box, select Excel Add-ins,
click on Go. The Add-in panel will open up.
4. Select Analysis ToolPak, and then click OK.
5. If you get prompted that the Analysis ToolPak is not currently installed
on your computer, click Yes to install it.
6. After you load the Analysis ToolPak, the Data Analysis command is
available in the Analysis group on the Data tab.
 You will need the analysis toolpack to run regression in excel. If
you do not have it installed yet, follow the instructions below.
Data Analysis Using Excel
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 You will need the analysis toolpack to run regression in excel. If
you do not have it installed yet, follow the instructions below.
Data Analysis Using Excel
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Correlation Analysis Using Excel
 We start by finding the correlation between two variables using the train
boarding data in the excel file named “Linear Trend”.
• Download the file from Canvas into your laptop.
• Click on it to open.
• The file contains data in two columns: year, train boarding.
• Click on Data Analysis on the top right to open the data analysis box
• Scroll down and select Correlation and click on OK to open the
correlation analysis box
• Check Group by: Column and Labels in First Row
• Go to Input Range box and then select the both columns of data
• Click OK
• Excel will calculate the correlation requested.
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Year Train Boarding
Year 1
Train Boarding 0.8536 1
 The correlation between “year” and “train boarding” is 0.8536 which
positive and quite high (closer to +1 than 0).
 This implies that there is a strong linear relationship between the two
variables
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Linear Trend Line Using Excel
 We will now fit a simple linear time trend using the train boarding data
in the excel file named “Linear Trend”.
• Open the excel file and select the data
• Click on Insert on the top and the insert menu will show up
• Click on the scatter plot icon (x and y-axis and several points)
• Excel will product a graph of train boarding over time
• Click on one of the data points
• Right click to open the options box
• Select Add Trendline and another option box will open.
• Select Linear (it is the default choice), Display Equation on
Chart” and Display R-squared value on Chart.
• Excel will add the estimated line to your graph.
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y = 3.8349x - 7525.8
R² = 0.7286
0
50
100
150
200
250
1980 1985 1990 1995 2000 2005 2010 2015
Train Boarding
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 How good is the linear trend line?
• R-square is 0.7286 (r = 0.8536 – same as correlation estimate)
• 72.86% of the variations in boarding can be explained by the
model; that is, 72.86% of the variations can be explained by time.
• From both the graph and the R-squared value (0.7286 is much
closer to1 than 0), we can say that the model is reasonably useful.
 Estimated regression equation
• 𝑦 = −7525.8 + 3.8349𝑋
• The intercept (𝛽0 = −7525.8) gives the value of y when X (time in
years) is zero which is not relevant in this example since there is
no train service in the year 0 B.C. or A.D.
• The slope estimate (𝛽1 = 3.8349) indicates the change in the
dependent variable (number of train boarding) with a one unit
increase in the independent variable (time in years).
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 The model tells us that train boarding is increasing at a rate of 3.8349
units (million passenger) per year.
 This result is expected and consistent with economic theory
because population and income has been increasing over the years,
which means that demand will be increasing over the years.
 Exercise: What is the expected number of boarding in 2019?
 To do this, we need to substitute the value of 2019 into the estimated
equation:
 𝑦 = −7525.8 + 3.8349𝑋 = −7525.8 + 3.8349 2019 = 216.8
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 We will now re-run the linear trend line estimation using the regression
analysis option in excel.
• Open the excel file LinearTrend again.
• Click on Data at the top to open up the data panel
• Click on Data Analysis on the top right to open the data analysis box
• Scroll down and select Regression and click on OK to open the
regression analysis box
• Tick Labels to include the name of the variables
• Go to Input Y Range box, then select the Y data (2nd column – train
boarding)
• Go to Input X Range box, then select the X data (1st column – year)
• Click OK
• Excel will run the regression requested.
Simple Regression Analysis Using Excel
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SUMMARY OUTPUT
Regression Statistics
Multiple R 0.8536
R Square 0.7286
Adjusted R Square 0.7178
Standard Error 18.9443
Observations 27
ANOVA
df SS MS F Significance F
Regression 1 24089.44 24089.44 67.12 1.51782E-08
Residual 25 8972.139 358.89
Total 26 33061.58
Coefficients
Standard
Error t Stat P-value Lower 95% Upper 95%
Intercept -7525.7720 934.2968 -8.0550 2.073E-08 -9449.9923 -5601.5516
Year 3.8349 0.4681 8.1929 1.517E-08 2.8709 4.7990
E-08 means move
the decimal point 8
places to the left
(0.0000000151782)
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 How good is the regression model?
• R-square is 0.7286 which means that 72.86% of the variations in
boarding can be explained by the model.
• From the R-squared value (0.7286 is much closer to1 than 0), we
can say that the model is reasonably useful.
• Next, we look at the F and Significance F. The F-test is used to test
the null hypothesis that ALL the slope estimates are equal to zero.
• If all the slope estimates are zero, then none of X is useful in
explaining the behaviour of Y.
• The significance F value is the probability that all the slope estimates
are equal to zero.
• Significance F is very small (1.5E-08 = 0.000000015 – much
smaller than 0.05), which means that not all the slope estimates are
equal to zero.
• There is a very small chance that the model is useless or simply
that the model is useful.
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• 𝑦 = −7525.8 + 3.8349𝑋
• The slope estimate (𝛽1 = 3.8349) indicates train boarding is
increasing at a rate of 3.8349 units (million passenger) per year.
• This result is expected and consistent with economic theory
because population and income have been increasing over the years,
which means that demand will be increasing over the years.
• Next, we need to look at the p-values of the estimates. The t-test is
used to test the null hypothesis that the slope estimate for each
individual X is equal to zero.
• If the slope estimate is zero, then that particular X is NOT useful in
explaining the behaviour of Y.
• The p-value is the probability that the slope is equal to zero.
• p-value is very small (1.5E-08 = 0.000000015 – much smaller than
0.05), which means that the slope is not likely to be zero.
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• Next, we look at the 95% confidence interval
• We are 95% confident that the slope is between 2.9709 and 4.7990.
• Note that our best estimate is 3.8349 and we are 95% confidence it is
between 2.7909 and 4.7990.
• Note that for one independent variable model, the t-test and F-test
are the “same” (8.19292 = 67.12; p-value = Significance F).
 Exercise: What is the expected number of boarding in 2019?
• To do this, we need to substitute the value of 2019 into the estimated
equation:
• 𝑦 = −7525.8 + 3.8349 ∗ 𝑦𝑒𝑎𝑟 = −7.525.8 + 3.8349 ∗ 2019 = 216.8
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Multiple Regression Analysis Using Excel
 In economics, it is often necessary to use more than one independent
variable to explain the behaviour of the dependent variable.
 For example, the demand and quantity demanded of product or service
will depend on the price of the product/service, as well as other factors
like population, income, taste, price of related goods, etc.
 Conceptually, multiple linear regression is a straightforward extension
of the simple linear regression.
 For this exercise, we will use the excel file named “Multiple Regression”
 We will run the model for train boarding using one new variable (price
or fare) and then estimate another model using two independent
variables (price and population).
 Follow the previous described steps in estimating a simple regression
model with price as the independent variable.
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SUMMARY OUTPUT
Multiple R 0.7463
R Square 0.5570
Observations 27
ANOVA
Regression 1 18421.61 18421.61 31.43 7.8339E-06
Residual 25 14652.48 586.10
Total 26 33074.09
Coefficients
Standard
Intercept -65.8935 35.0249 -1.8813 0.0716 -138.0286 6.2416
Price 20.2436 3.6108 5.6063 7.8E-06 12.8069 27.6802
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• From the R-squared value (0.55 is “middle of the road” between 0
and 1), we can say that the model is slightly useful.
 Significance F is very small, which means that there is a very
small chance that the model is useless.
• 𝑦 = −65.8935 + 20.2436𝑋
• The slope estimate (𝛽1 = 20.2436) indicates the change in the
dependent variable (number of train boarding) with a one unit
increase in the independent variable (price or fare).
• p-value is very small, which means that the slope is not likely to
be zero.
• In fact, we are 95% confident that the slope is between 12.8069 and
27.6802.
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 This result is not expected and inconsistent with the economic
theory because it contradicts the Law of Demand.
• The demand curve is drawn holding all other factors constant.
• By ignoring the other important factors in the model, we are not
holding them constant.
• When we collect data on fare (price) and train boarding (quantity),
they are the equilibrium price and quantity.
• When we only have price and quantity data, we could be estimating
the demand curve, the supply curve or a bit of both.
• The importance of understanding the underlying economic
theory cannot be over emphasised.
• We will run the model for train boarding using two independent
variables (price and population).
• Note that population is one of the key determinants of demand.
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SUMMARY OUTPUT
Multiple R 0.9280
R Square 0.8611
Observations 27
ANOVA
Regression 2 28480.30 14240.15 74.40 5.1546E-11
Residual 24 4593.79 191.41
Total 26 33074.09
Coefficients
Standard
Intercept -225.2197 29.7267 -7.5763 8.155E-08 -286.5727 -163.8667
Price -9.1155 4.5453 -2.0054 0.0563 -18.4966 0.2657
Population 0.1313 0.0181 7.2492 1.725E-07 0.0939 0.1686
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• From the R-squared value (0.86 is much higher and closer 1), we can
say that the model is slightly useful.
 Significance F is very small, which means that there is a very small
chance that the model is useless or simply that the model is useful.
• 𝑦 = −225.2 − 9.11 ∗ 𝑃𝑟𝑖𝑐𝑒 + 0.13 ∗ 𝑃𝑜𝑝
• Each unit increase in price ($1) is associated with a 9.11 unit
decrease in train boarding or a decrease of 9.11 million passengers
• p-value is small (0.056 – close to traditional cut-off of 0.05), which
means that the slope is not likely to be zero.
• The negative relationship between price and quantity demanded is
expected due to the law of demand.
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• 𝑦 = −225.2 − 9.11 ∗ 𝑃𝑟𝑖𝑐𝑒 + 0.13 ∗ 𝑃𝑜𝑝
• Each unit increase in population is associated with a 0.13 units
increase in train boarding or an increase of 0.13 million passengers
• p-value is very small (1.7E-07), which means that the slope is not
likely to be zero.
• The positive relationship between population and quantity demanded
is expected and consistent with economic theory because an
increase in population is expected to increase the number of potential
buyers, resulting in an increase in demand.
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Regression Residual Analysis
 Regression residuals are the estimated errors of the data points.
 If the model fits the data well, the residuals should be randomly distributed.
No clear pattern - Good Non-linearity
Outlier Variance not constant
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Regression Residual Analysis Using Excel
 We will run the model for train boarding using two independent variables (price
and population).
 When running the regression model, you need to check the boxes Residuals
(or Standardized Residuals) and Residual Plots
-20
-15
-10
-5
0
5
10
15
20
25
30
6 7 8 9 10 11 12
Residuals Price
Price Residual Plot
Clear Non-Linearity Problem Not as clear – can assume to be OK
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Transformation and Elasticity
 When you have non-linearity problem, you need to use a non-linear
transformation of the X, such as log(X), X2, etc.
• 𝑌 = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + 𝛽3𝑋2
2
+ 𝜀
• Since X2 are known numbers, 𝑋2
2
are also known numbers.
• We can simply rename 𝑋2
2
as X3
• 𝑌 = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + 𝛽3𝑋3 + 𝜀
• The regression equation is still a linear equation, in terms of the
unknown parameters b0, b1,b2 & b3.
• In excel data sheet, create a new variable call Pop2 which is the
square of population.
• Run regression with price, population and Pop2.
54

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55

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SUMMARY OUTPUT
Multiple R 0.9916
R Square 0.9833
Observations 27
ANOVA
Regression 3 32522.1 10840.7 451.67 1.41806E-20
Residual 23 552.0 24.0
Total 26 33074.1
Coefficients
Standard
Intercept 1199.8677 110.3218 10.8761 1.53E-10 971.64977 1428.08564
Price -5.9430 1.6280 -3.6505 0.001334 -9.31085 -2.57523
Population -0.7158 0.0656 -10.9134 1.43E-10 -0.85152 -0.58015
Pop2 0.0001 9.41E-06 12.9768 4.57E-12 0.00010 0.00014
increased
56

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-15
-10
-5
0
5
10
15
6 7 8 9 10 11 12
Residuals
Price Residual Plot
-15
-10
-5
0
5
10
15
2550 3050 3550 4050 4550
Residuals
Population (000s) Residual Plot
-15
-10
-5
0
5
10
15
7000000 9000000 11000000 13000000 15000000 17000000
Residuals
Pop2 Residual Plot
 The residual plots now
look more random
 No obvious problem
57

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 The own price elasticity of demand is ε =
%∆𝑄
%∆𝑃
=
(∆𝑄)(𝑷)
(∆𝑃)(𝐐)
 (ΔQ)/(ΔP) is the slope and it is constant along a straight line.
 For the regression equation: 𝑌 = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + 𝛽3𝑋2
2
+ 𝜀,
b1 is the slope of the quantity demanded with respect to price.
 The elasticity of any point along the linear demand curve is
then given by:
• 𝜀 = 𝛽1(
𝑃
𝑄
)
• For example: the mean is (𝑃 = 9.61, 𝑄 = 128.72)
• Elasticity at the mean = (-5.943)(9.61/128.72) = 0.44
 Note that the change in demand due to a change in population
depends on both b2 & b3 (b2 + 2b3).
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 Similar estimation can be performed for income elasticity and cross
price elasticity if these variables are used in the linear regression.
 One of the most common type of transformation in economics is the
natural log or ln function.
 ln 𝑦 = 𝛽0 + 𝛽1 ln 𝑃𝑟ice + 𝛽2 ln 𝐼𝑛𝑐𝑜𝑚𝑒 + 𝛽3 ln 𝑅𝑒𝑙𝑎𝑡𝑒𝑑𝑃𝑟𝑖𝑐𝑒 + 𝜀
• own price elasticity is b1
• income elasticity is b2
• cross price elasticity is b3
 Note that in the ln-ln functional form (a.k.a log-linear demand curve),
the elasticity is constant along whole the demand curve and does not
depend on where it is evaluated.
59

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