伍德里奇计量经济学讲义

y y is the totalsum of squares (SST ) ˆ y is theexplainedsum of squares (SSE) y ˆ is theresidual sum of squares (SSR) u
2 2 i i 2 i
T henSST SSE SSR
The Simple Regression Model
y = b0 + b1x + u
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Some Terminology
In the simple linear regression model, where y = b0 + b1x + u, we typically refer to y as the
n n
1
y
n i 1 n i 1
i
ˆ b ˆ x 0 b 0 1 i

1
ˆ b ˆ x 0 x y b i i 0 1 i
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More Derivation of OLS
Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows
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Deriving OLS using M.O.M.
The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments
What does this mean? Recall that for E(X), the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean of the sample
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E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x)
y
f(y)
.
x1 x2
. E(y|x) = b + b x
0 1
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Ordinary Least Squares
ˆ ˆ ˆ ui yi b0 b1xi
2 i 1 i 1
n
n


2
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Alternate approach, continued
If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by n
To derive the OLS estimates we need to realize that our main assumption of E(u|x) = E(u) = 0 also implies that
Cov(x,u) = E(xu) = 0 Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y)
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More OLS
Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible, hence the term least squares The residual, û , is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point
i 1 n i 1 2 ˆ xi x yi y b1 xi x i 1 i 1
14
n
n
So the OLS estimated slope is
ˆ b 1
x x y y
i 1 i i
n
x x
i 1 i n i 1
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More Derivation of OLS
We want to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true The sample versions are as follows:
Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Covariate, or Control Variables

3
A Simple Assumption
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Sample regression line, sample data points and the associated estimated error terms
y y4
û 4{
.
ˆ b ˆx ˆb y 0 1
y3 y2
û } . 1 x1
. û { 2
.} û3
y1
x2
x3
x4
x
18
n
2
providedthat xi x 0
2
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Summary of OLS slope estimate
The slope estimate is the sample covariance between x and y divided by the sample variance of x If x and y are positively correlated, the slope will be positive If x and y are negatively correlated, the slope will be negative Only need x to vary in our sample
ˆ b ˆ x, yb 0 1 or ˆ yb ˆx b 0 1
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More Derivation of OLS
ˆ x b ˆ x 0 x y y b i i 1 1 i
i 1 n n



ˆ x y y b i i 1 xi xi x
7
Population regression line, sample data points and the associated error terms
y y4 E(y|x) = b0 + b1x . u4 {
y3 y2
u2 {.
.} u3
y1
.
} u1
x1
x2
x3
x4
x
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Deriving OLS Estimates
ˆ y b
n i 1 n i
ˆ x 0 b 0 1 i

ˆ b ˆ x 0 x y b i i 0 1i
i 1
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Algebraic Properties of OLS
The sum of the OLS residuals is zero Thus, the sample average of the OLS residuals is zero as well The sample covariance between the regressors and the OLS residuals is zero The OLS regression line always goes through the mean of the sample
Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Regressand

2
Some Terminology, cont.
In the simple linear regression of y on x, we typically refer to x as the
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Proof that SST = SSE + SSR
ˆ y ˆ y y y y y ˆ y ˆ y u ˆ 2 u ˆ y ˆ y y ˆ y u ˆ y ˆ y SSE SSR 2 u ˆ y ˆ y 0 and we know that u
Alternate approach to derivation
Given the intuitive idea of fitting a line, we can set up a formal minimization problem That is, we want to choose our parameters such that we minimize the following:
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