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Multiple Linear Regression Model
Y = ß
0
+ ß
1
x
1
+ ß
2
x
2
+ … + ß
k
x
k
+ ε
deterministic component
random
component
***Note: Linear w.r.t. Parameters ß
0,
ß
1
… ß
k
and not predictor variables.
Assumptions:
1.
The specified regression model has the correct form
. Therefore for given values of x
1
, x
2
, … x
k
, E(Y) = ß
0
+ ß
1
x
1
+ … + ß
k
x
k
and E(ε) = 0 This implies that when the least squares estimates are determined, the least squares equation
Ŷ = b
0
+ b
1
x
1
+ … +
b
k
x
k
estimates the average value of Y, given a set of values for the predictor variables x; and, the model correctly represents
the form of the association between the response variable and the predictor variables.
2.
The error variance is constant
. The σ
2
ε
is constant over all values of the predictor variables. Thus, the range of deviations of
the Y values from the regression model is the same regardless of the values of x
1
,
x
2
, …,x
k
.
3.
Random errors, ε’s, are independent and normally distributed
. Hence, the random errors associated with the Yvalues are
statistically independent of one another and normally distributed.
The residual variance S
2
e
is defined by:
S
2
e
= SSE / (nk1)
where SSE = ∑ (Y
i
– Ŷ
i
)
2
S
2
e
remains an absolute measure of how well the least squares equation fits the sample Y – values. If the fit were
perfect, all residuals would equal 0 and S
2
e
= 0.
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View Full Document Partitioning the Total Variation
SST = SSR + SSE
where
_
_
SST = ∑(Y
i
–Y)
2
SSR = ∑(Ŷ – Y)
2
SSE = ∑(  Ŷ)
Ү
2
The coefficient of determination has the same interpretation as in simple linear regression, that is, the fraction of the total
variation in the sample
Yvalues that has been explained by the predictor variables in the least squares equation.
R
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This note was uploaded on 12/07/2009 for the course MGMT 302 taught by Professor Canavos during the Spring '09 term at VCU.
 Spring '09
 CANAVOS

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