Transcript No Slide Title

Econ 140
Classical Regression
Lecture 8
Lecture 9
1
Today’s Plan
Econ 140
• For the next few lectures we’ll be talking about the
classical regression model
– Looking at both estimators for a and b
– Inferences on what a and b actually tell us
• Today: how to operationalize the model
– Looking at BLUE for bi-variate model
– Inference and hypothesis tests using the t, F, and c2
– Examples of linear regressions using Excel
Lecture 9
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Estimating coefficients
Econ 140
• Our model: Y = a + bX + e
• Two things to keep in mind about this model:
1) It is linear in both variables and parameters
• Examples of non-linearity in variables:
Y = a + bX2 or Y = a + bex
• Example of non-linearity in parameters:
Y = a + b2X
• OLS can cope with non-linearity in variables but not
in parameters
Lecture 9
3
Estimating coefficients (3)
Econ 140
2) Notation: we’re not estimating a and b anymore
• We are estimating coefficients which are estimates
of the parameters of a and b
• We will denote the coefficients as  or aˆ and  or bˆ
• We are dealing with a sample size of n
– For each sample we will get a different  and  pair
Lecture 9
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Estimating coefficients (4)
Econ 140
• In the same way that you can take a sample to get an
estimate of µy you can take a sample to get an estimate of
the regression line, of  and 
Y
Sample 1
Regression
(Population)
Line
Sample 2
Lecture 9
X
5
The independent variable
Econ 140
• We also have a given variable X, its values are known
– This is called the independent variable
• Again, the expectation of Y given X is
E(Y | X) = a + bX
• With constant variance
V(Y) = 2
Lecture 9
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A graph of the model
Econ 140
Y
(Y1, X1)
Yˆ
e  Y  Yˆ
Y
Yˆ    X
X
Lecture 9
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What does the error term do?
Econ 140
• The error term e  Y  Yˆ gives us the test statistics and
tells us how well the model Y = a+bX+e fits the data
• The error term represents:
1) Given that Y is a random variable, e is also random,
since e is a function of Y
2) Variables not included in the model
3) Random behavior of people
4) Measurement error
5) Enables a model to remain parsimonious - you don’t
want all possible variables in the model if some have
little or no influence
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Rewriting beta
Econ 140
• Our complete model is Y = a + bX + e
• We will never know the true value of the error e so we will
estimate the following equation:
Y    X  
• For our known values of X we have estimates of , , and 
• So how do we know that our OLS estimators give us the
BLUE estimate?
– To determine this we want to know the expected value
of  as an estimator of b, which is the population
parameter
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Rewriting beta(2)
Econ 140
• To operationalize, we want to think of what we know
• We know from lecture 2 that there should be no correlation
between the errors and the independent variables
  i  0, E  i   0
  i X i  0, E  i X i   0

2
• We also know V ( X )   2 V (Y )  

• Now we have that E(Y|X) = a + bX + E(|X)
• The variance of Y given X is V(Y) = 2 so V(|X)= 2
Lecture 9
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Rewriting beta(3)
Econ 140
• Rewriting 
– In lecture 2 we found the following estimator for 
XY  nXY


2
2
X

n
X

• Using some definitions we can show:
E() = b
2
V ( ) 
2
x

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Rewriting beta (4)
Econ 140
• We have definitions that we can use:
yi  (Yi  Y )
2
2
xi  ( X i  X ) So that  xi   ( X i  X )
• Using the definitions for yi and xi we can rewrite  as
xy

 x2
• We can also write  xy 
Lecture 9

xY
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Rewriting beta (5)
   ciYi
• We can rewrite  as
Econ 140
where ci 
xi
2
x
 i
• The properties of ci :
 ci  0,  ( X i  X )  0,
2.  ci xi  1
1.
Lecture 9
3.
2 1
c
 i
4.
 ci X i  1 
2
x
 i
 xi X i
 xi 2
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Showing unbiasedness
Econ 140
• What do we know about the expected value of beta?
   ciYi
E (  )   ci E (Yi )
• We can rewrite this as E (  )   ci (a  bX i )
• Multiplying the brackets out we get:
E (  )   ci (a)   ci (bX i )
• Since b is constant,
E (  )   ci (a)  b ci X i
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Showing unbiasedness (2)
Econ 140
• Looking back at the properties for ci we know that
 ci  0,  ci X i  1
• Now we can write this as
E (  )  0  b(1)  b
• We can conclude that the expected value of  is b and that
 is an unbiased estimator of b
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Gauss Markov Theorem
Econ 140
• We can now ask: is  an efficient estimator?
• The variance of  is
2
2
V (  )   ci V (Y ) 
 xi2
Where    ciYi
• How do we know that OLS is the most efficient estimator?
– The Gauss-Markov Theorem
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Gauss Markov Theorem (2)
Econ 140
• Similar to our proof on the estimator for my.
• Suppose we use a new weight ci  ci  di
• We can take the expected value of E()
   ci E (Yi )
  ci (a  bX i )
 a( ci   di )  b( ci X i   di X i )
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Gauss Markov Theorem (3)
Econ 140
• We know that
E (  )  a (0   di )  b(1   di X i )
 a  di  b  b di X i
– For  to be unbiased, the following must be true:
 di  0
 di X i  0
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Gauss Markov Theorem (4)
Econ 140
• Efficiency (best)?
• We have    ciYi where ci  ci  di
• Therefore the variance of this new  is
•
2
2
V (  )   ci V (Y ) 
2+
 xi
Sdi2 +2Scidi
• If each di  0 such that ci c’i then
2
d
 i 0
• So when we use weights c’I we have an inefficient
estimator
Lecture 9
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Gauss Markov Theorem (5)
Econ 140
• We can conclude that
•
Lecture 9
XY  nXY


 X 2  nX 2
is BLUE
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Wrap up
Econ 140
• What did we cover today?
• Introduced the classical linear regression model (CLRM)
• Assumptions under the CLRM
1) Xi is nonrandom (it’s given)
2) E(ei) = E(ei|Xi) = 0
3) V(ei)= V(ei|Xi) = 2
4) Covariance (eiej) = 0
• Talked about estimating coefficients
• Defined the properties of the error term
• Proof by contradiction for the Gauss Markov Theorem
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