Chapter 2: Simple Linear Regression
[Pages:58]Chapter 2: Simple Linear Regression
1 The model
The simple linear regression model for n obser-
vations can be written as
yi = 0 + 1xi + ei, i = 1, 2, ? ? ? , n. (1)
The designation simple indicates that there is only
one predictor variable x, and linear means that the model is linear in 0 and 1. The intercept 0 and the slope 1 are unknown constants, and
they are both called regression coefficients; ei's
are random errors. For model (1), we have the following assumptions:
1. E(ei) = 0 for i = 1, 2, ? ? ? , n, or, equivalently E(yi) = 0 + 1xi.
2. var(ei) = 2 for i = 1, 2, ? ? ? , n, or, equivalently, var(yi)) = 2.
3. cov(ei, ej) = 0 for all i = j, or, equivalently, cov(yi, yj) = 0.
2 Ordinary Least Square Estimation
The method of least squares is to estimate 0 and 1 so that the sum of the squares of the difference between the observations yi and the straight
line is a minimum, i.e., minimize
n
S(0, 1) = (yi - 0 - 1xi)2.
i=1
4
3
2
E(Y|X=x)
1 = Slope 1
1
0
0 = Intercept
0
1
2
3
4
Predictor = X
Figure 1: Equation of a straight line E(Y |X = x) = 0 + 1x.
The least-squares estimators of 0 and 1, say ^0
and ^1, must satisfy
n
-2 (yi - ^0 - ^1xi) = 0 (2)
i=1 n
-2 (yi - ^0 - ^1xi)xi = 0 (3)
i=1
Simplifying these two equations yields
n
n
n^0 + ^1 xi = yi
n
i=1
i=1
n
n
(4)
^0 xi + ^1 x2i = yixi
i=1
i=1
i=1
Equations (4) are called the least-squares nor-
mal equations. The solution to the normal equa-
tions is
^1 =
n i=1
xiyi
-
nx?y?
n i=1
x2i
-
nx?2
=
=
Sxy Sxx
,
^0 = y? - ^1x?.
ni=1(xi - x?)(yi - y?) ni=1(xi - x?)2
The difference between the observed value yi
and the corresponding fitted value y^i is a residual,
i.e.,
ei = yi-y^i = yi-(^0+^1xi), i = 1, 2, ? ? ? , n
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