Econ 3790: Business and Economics Statistics

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Econ 3790 Business and Economics Statistics

• Instructor Yogesh Uppal
• yuppal_at_ysu.edu

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Chapter 15 Multiple Regression Model

• The equation that describes how the dependent
  variable y is related to the independent
  variables x1, x2, . . . xp and an error term is
  called the multiple regression model.

y b0 b1x1 b2x2 . . . bpxp e
where b0, b1, b2, . . . , bp are the
parameters, and e is a random variable called
the error term
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Estimated Multiple Regression Equation
A simple random sample is used to compute
sample statistics b0, b1, b2, . . . , bp that are
used as the point estimators of the parameters
b0, b1, b2, . . . , bp.
The estimated multiple regression equation is
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Interpreting the Coefficients
In multiple regression analysis, we
interpret each regression coefficient as
follows
bi represents an estimate of the change in y
corresponding to a 1-unit increase in xi when
all other independent variables are held
constant.
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Multiple Regression Model
Example Car Sales Suppose we believe that
number of cars sold (y) is not only related to
the number of ads (x1), but also to the minimum
down payment required at the (x2). The
regression model can be given by
y ?0 ?1x1 ?2x2 ?
where y number of cars sold x1 number
of ads x2 minimum down payment required
(000)
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Estimated Regression Equation
y 14.4 3.7 x1 25 x2

• Interpretation?
• Estimated values of y?
• Error?
• Prediction?

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Multiple Coefficient of Determination

• Relationship Among SST, SSR, SSE

SST SSR SSE
where SST total sum of squares SSR
sum of squares due to regression SSE
sum of squares due to error
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Multiple Coefficient of Determination
R2 SSR/SST
R2 84.63/89.2 .949
Adjusted Multiple Coefficient of Determination
Standard Error of Estimate
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Testing for Significance t Test
Hypotheses
Test Statistics
Rejection Rule
Reject H0 if p-value lt a or if t lt -t????or t gt
t???? where t??? is based on a t
distribution with n - p - 1 degrees of freedom.
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Example Testing for significance of coefficients
Hypotheses
For ? .05 and d.f. ?, t.025
Rejection Rule
Test Statistics
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Testing for Significance of Regression F Test
H0 ?1 ?2 . . . ?p 0 Ha One or more
of the parameters is not equal to zero.
Hypotheses
F MSR/MSE
Test Statistics
Rejection Rule
Reject H0 if p-value lt a or if F gt F?, where F?
is based on an F distribution with p d.f. in the
numerator and n - p - 1 d.f. in the denominator.
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Multiple Regression Model

• Example 2 Programmer Salary Survey

A software firm collected data for a
sample of 20 computer programmers. A
suggestion was made that regression analysis
could be used to determine if salary was
related to the years of experience and the
score on the firms programmer aptitude test.
The years of experience, score on the
aptitude test, and corresponding annual salary
(1000s) for a sample of 20 programmers is
shown on the next slide.
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Multiple Regression Model
Exper.
Score
Score
Exper.
Salary
Salary
4 7 1 5 8 10 0 1 6 6
9 2 10 5 6 8 4 6 3 3
78 100 86 82 86 84 75 80 83 91
88 73 75 81 74 87 79 94 70 89
38 26.6 36.2 31.6 29 34 30.1 33.9 28.2 30
24 43 23.7 34.3 35.8 38 22.2 23.1 30 33
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Multiple Regression Model
Suppose we believe that salary (y) is related
to the years of experience (x1) and the score
on the programmer aptitude test (x2) by the
following regression model
y ?0 ?1x1 ?2x2 ?
where y annual salary (1000) x1 years
of experience x2 score on programmer
aptitude test
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Solving for b0, b1 and b2
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Anova Table
Source of Variation Sum of Squares Degrees of Freedom Mean Square F-statistic
Regression 500.34 .. .
Error .. . .
Total 599.8 ..
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Estimated Regression Equation
SALARY 3.174 1.404(EXPER) 0.251(SCORE)
b1 1.404 implies that salary is expected to
increase by 1,404 for each additional year of
experience (when the variable score on programmer
attitude test is held constant).
b2 0.251 implies that salary is expected to
increase by 251 for each additional point scored
on the programmer aptitude test (when the
variable years of experience is held constant).
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Prediction

• Suppose Bob had an experience of 4 years and had
  a score of 78 on the aptitude test. What would
  you estimate (or expect) his score to be?
• 3.174 1.404(4) 0.251(78)
• 28.358
• Bobs estimated salary is 28,358.

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Error

• Bobs actual salary is 24000. How much error we
  made in estimating his salary based on his
  experience and score?
• So, we shall overestimate Bobs salary.

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Multiple Coefficient of Determination

• Relationship Among SST, SSR, SSE

SST SSR SSE
where SST total sum of squares SSR
sum of squares due to regression SSE
sum of squares due to error
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Multiple Coefficient of Determination
R2 SSR/SST
R2 500.3285/599.7855 .83418
Adjusted Multiple Coefficient of Determination
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Testing for Significance t Test
Hypotheses
Test Statistics
Rejection Rule
Reject H0 if p-value lt a or if t lt -t????or t gt
t???? where t??? is based on a t
distribution with n - p - 1 degrees of freedom.
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Example
Hypotheses
For ? .05 and d.f. 17, t.025 2.11 Reject H0
if p-value lt .05 or if t gt 2.11
Rejection Rule
Test Statistics
Since t7.07 gt t0.025 2.11, we reject H0.
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Testing for Significance of Regression F Test
H0 ?1 ?2 . . . ?p 0 Ha One or more
of the parameters is not equal to zero.
Hypotheses
F MSR/MSE
Test Statistics
Rejection Rule
Reject H0 if p-value lt a or if F gt F?, where F?
is based on an F distribution with p d.f. in the
numerator and n - p - 1 d.f. in the denominator.
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Example
H0 ?1 ?2 0 Ha One or both of the
parameters is not equal to zero.
Hypotheses
For ? .05 and d.f. 2, 17 F.05 3.59 Reject
H0 if p-value lt .05 or F gt 3.59
Rejection Rule
Test Statistics
F MSR/MSE 250.17/5.86 42.8
F 42.8 gt F0.05 3.59, so we can reject H0.