Managerial Economics
in a Global Economy
Chapter 4
DEMAND ESTIMATION
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Regression Analysis
Year
X
Y
1
10
44
2
9
40
3
11
42
4
12
46
5
11
48
6
12
52
7
13
54
8
13
58
9
14
56
10
15
60
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Scatter Diagram
Prof. M. El-Sakka
CBA. Kuwait University
 Regression Line: Line of
Best Fit
 Regression Line:
Minimizes the sum of the
squared vertical
deviations (et) of each
point from the regression
line.
 Ordinary Least Squares
(OLS) Method
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University

REGRESSION ANALYSIS

Given the following demand function:

Y = A + B1 X + B2 P + B3 I + B4 Pr;

X = selling expenses (advertising) Pr = price of
substitutes

What we want are estimates of the values of A, B1, B2,
B3, & B4.

Regression analysis describes the way in which one
variable is related to another. It derives an equation that
can be used to estimate the unknown value of one
variable on the basis of the known value of the other
variable(s).
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University

Simple Regression Model.

the simple regression model takes the following form:
Yi = A + B Xi + ei;

Regression analysis assumes that the mean value of Y,
given the value of X, is a linear function of X. In other
words, the mean value of the dependent variable is
assumed to be a linear function of the independent
variable.

Yi is the ith observed value of the dependent variable
and Xi is the ith observed value of the independent
variable. Essentially ei is an error term, that is, a random
amount that is added to A+BXi (or subtracted if ei is
negative).
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University

Because of the presence of the error term, the observed
values of Yi fall around the population regression line
(A+BXi), not on it

Regression analysis assumes that the values of ei are
independent and that their mean value equals zero.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Sample Regression Line (based on a sample) .

sample regression line (estimated regression line)
describes the average relationship between the
dependent variable and independent variable. The general
expression of the sample regression line is:
ˆ
Yˆt  aˆ  bX
t




i.e, the value of the dependent variable predicted by the
regression line,
a & b = estimators of A and B.
a = the intercept of the regression line
b = the slope of the line, measure the change in the
predicted value of Y associated with a one unit increase in
X.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University

Method of Least Squares.

Used to determine the values of a and b. Since the
deviation of the ith observed value of Y from the
regression line equals Yi Yˆi , the sum of the squared
deviations equals:
2 n
2
n
 (Yi Yˆi )   (Yi abX i )
i1
i1

Where n is the sample size. Using minimization
technique we can find the values of a and b that
minimize this expression, by differentiating these
expression with respect to a and b and by setting
these partial derivatives equal to zero.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University



n
  (Yi Yˆi )2
n
i1
  2  (Yi abX i )  0
a
i1
n
  (Yi Yˆi )2
n
i1
  2  X i (Yi abX i )  0
b
i1
(1)
(2)
solving equations (1) and (2) simultaneously,
and letting equal the mean value of X in the
sample and equal the mean of Y, we find that;
n
n  ( X i  X ) (Yi  Y )
b  i 1
n
2
 ( Xi  Xi )
i 1
and
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a  Y  bX .
Prof. M. El-Sakka
CBA. Kuwait University
given the following data for company X. Given the following results of the table
below
Yˆ = 2.533 + 1.504X;
if Y = the observed value of sales
Yˆ = the computed (estimated) value of sales based on the regression line.
from the table Y = 4 when X = 1.
But using the regression line:
Yˆ = 2.533 + 1.504(1) = 4.037 (Note there is a difference between the observed
sales (4) and the estimated sales (4.037).
Selling Expenses
(X)
1
2
4
8
6
5
8
9
7
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Sales
(Y)
4
6
8
14
12
10
16
16
12
Prof. M. El-Sakka
CBA. Kuwait University

if X = 0 then Y = 2.533 +1.504(0) = 2.533 (the intercept:
the value of Y that intersects the vertical axis)

Interpretation: if the firm’s selling expenses = 0, sales
would be 2.533 million of units, and estimated sales go
up 1.504 million units when selling expenses increase
by 1m.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Ordinary Least Squares Estimation Using Excel
Yt  a  bX t  et
The model:
ˆ
Yˆt  aˆ  bX
t
et  Yt  Yˆt
Objective: Determine the slope and intercept that
minimize the sum of the squared errors.
n
n
n
t 1
t 1
t 1
2
2
ˆ )2
ˆ
ˆ
e

(
Y

Y
)

(
Y

a

bX
t  t t  t
t
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Estimation Procedure
n
bˆ 
(X
t 1
t
 X )(Yt  Y )
n
(X
t 1
t
 X )2
Time
Xt
1
2
3
4
5
6
7
8
9
10
10
9
11
12
11
12
13
13
14
15
120
n  10
n
44
40
42
46
48
52
54
58
56
60
500
n
t 1
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(X
t 1
n
-6
-10
-8
-4
-2
2
4
8
6
10
n
 Yt  500
t 1
Yt 500

 50
10
t 1 n
Y 
Yt  Y
-2
-3
-1
0
-1
0
1
1
2
3
n
 X t  120
X t 120

 12
n
10
Xt  X
Yt
t 1
X 
ˆ
â  Y  bX
t 1
( X t  X )2
12
30
8
0
2
0
4
8
12
30
106
4
9
1
0
1
0
1
1
4
9
30
t
 X ) 2  30
106
bˆ 
 3.533
30
t
 X )(Yt  Y )  106
aˆ  50  (3.533)(12)  7.60
n
(X
( X t  X )(Yt  Y )
Prof. M. El-Sakka
CBA. Kuwait University
n  10
n
X
t 1
t
n
X t 120
X 

 12
10
t 1 n
 120
n
 Yt  500
n
Yt 500
Y  
 50
10
t 1 n
t 1
n
(X
n
2
(
X

X
)
 30
 t
t 1
t 1
aˆ  50  (3.533)(12)  7.60
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t
 X )(Yt  Y )  106
106
ˆ
b
 3.533
30
Prof. M. El-Sakka
CBA. Kuwait University
Tests of Significance
Standard Error of the Slope Estimate
sbˆ 
2
ˆ
(
Y

Y
)
 t
(n  k ) ( X t  X )
2

2
e
 t
(n  k ) ( X t  X ) 2
• A measure of the amount of scatter of individual observations
about the regression line.
• It is useful in constructing prediction intervals - that is,
intervals within which there is a specified probability that the
dependent variable will lie.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University

if probability is set at 0.95, a very approximate
prediction interval is:
 Yˆ  2se;
since se = 0.3702
 if the predicted value of Y is 11, there is a probability
that the firm’s sales will be between:


10.26 (11 – (2 × 0.37))
and
11.74 (11 + (2 × 0.37))

Example Calculation
n
n
n
(X
 e   (Yt  Yˆt )2  65.4830
t 1
2
t
t 1
t 1
 (Y  Yˆ )
( n  k ) ( X  X )
2
sbˆ 
t
t
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2

t
 X ) 2  30
65.4830
 0.52
(10  2)(30)
Prof. M. El-Sakka
CBA. Kuwait University

The t-statistic (significance of individual variables).

Managers need to know whether a particular
independent variable influences the dependent variable.
The least square estimates of B’s by chance may be
positive even if their true values are zero. e.g., B1 =
1.76 i.e., selling expenses have an effect on sales
(t=0.0001).

To test whether the true value of B1 is zero we must
look at the t-statistic of B1. The t-statistic has a
distribution called t-distribution.

All things equal, the bigger the value of t-statistic (in
absolute terms), the smaller the probability that the true
value of the regression coefficient in question is zero.


In our case, there is only 1 in 10 000 that chance alone
would have resulted in a large t-statistic.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Calculation of the t Statistic
bˆ
3.53
t

 6.79
sbˆ
0.52
Degrees of Freedom = (n-k) = (10 - 2) = 8
Critical Value at 5% level =2.306 ( b^ is significant)
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Decomposition of Sum of Squares
Total Variation = Explained Variation + Unexplained Variation
2
2
2
ˆ
ˆ
(
Y

Y
)

(
Y

Y
)

(
Y

Y
)
 t

 t t

Coefficient of Determination
2
Explained Variation  (Yˆ  Y )
R 

2
TotalVariation
(
Y

Y
)
 t
2
R2 

373.84
 0.85
440.00
Coefficient of Correlation
r  R 2 withthe sign of bˆ
1  r  1
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r  0.85  0.92
Prof. M. El-Sakka
CBA. Kuwait University
Decomposition of Sum of Squares
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Multiple Regression Analysis
Model:
Y  a  b1 X1  b2 X 2 
 bk ' X k '
Adjusted Coefficient of Determination
(n  1)
R  1  (1  R )
(n  k )
2
2
Analysis of Variance and F Statistic
Explained Variation /(k  1)
F
Unexplained Variation /(n  k )
R 2 /(k  1)
F 
(1  R 2 ) /(n  k )
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Problems in Regression Analysis
Multicollinearity.


A situation in which two or more independent variables are
very highly correlated.
Under perfect linear correlation it is impossible to estimate
the regression coefficients.
e.g., (perfect linear correlation)
Y = A + B1 X1i + B2 X2i; where X1i = 3X2i-1
or X1i = 6 + X2i
or X1i = 2 + 4X2i;
imperfect linear correlation.

Y = A + B1 X1i + B2 X2i; where; X1 = price, X2 = nominal
income (p.Q)
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University

If two independent variables move together in a rigid fashion,
there is no way to tell how much effect each has separately,
all what we can observe is the effect of both combined.
Consequences of mutlicollinearity


- High R2 with no significant t-scores
- High simple correlation coefficients (cross correlation
matrix)
How to deal with multicollinearity



- Drop one or more of the multicollinear variables
- Transform the multicollinear variables (e.g. first difference)
- Increase sample size
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Serial Correlation (or Autocorrelation)

Error terms are not independent, if this year’s error term is
positive, next year is always positive ( positive serial
correlation ), and if this year’s error term is negative, next
year’s is always negative.

This is a violation of the assumptions underlying regression
analysis. [ should be E(reiej) = 0, if not, the simple correlation
between two observations of the error term is not equal to
zero]
Consequences of Serial Correlation


- Increases of the variances of the distributions
- Leads to underestimate the standard errors of the
coefficients.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University


Detecting Serial correlation.
- Durbin Watson Test
n
d
2
(
e

e
)
 t t 1
t 2
n
2
e
t
t 1




Compare the computed DW with the DW tables to show whether d is so high
or so low, that the hypothesis that there is no serial correlation should be
rejected.
if d < dL reject the hypothesis of no serial correlation.
if d > du accept the hypothesis of no serial correlation
if dL  d  du, the test is inconclusive.
e.g.; if the hypothesis is that there is a negative serial correlation, we should

- reject the hypothesis of no serial correlation if d<4-dL

- accept the hypothesis of no serial correlation if d<4-du

- if 4-du  d  4-dL the test is inconclusive.
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
How to deal with serial correlation


- take the difference of the variables
- use generalized least squares
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Steps in Demand Estimation
1. Model Specification: Identify Variables
- Identify the independent variables (in reality an empirical issue)
2. Specify Functional Form
- Specify the mathematical form of the equation relating the mean
value of the dependent variable to those of the independent
variables.
e.g.,
Y = f(X,P).
This can take the following forms:
 Y = A + B1 Xi + B2 Pi + ei;
or:
B1>0,
B2<0
B B
Y  AXi 1 Pi 2 ei ;
 log Y = log A + B1 log Xi + B2 log Xi + log ei;
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
3. Collect your data. Data can be:
- time series
- cross section
- cross section/time series (panel)
4. Estimate The Function
5. Test the Results
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University
Managerial Economics
Prof. M. El-Sakka
CBA. Kuwait University