Copyright 1996 Lawrence C. Marsh 0 PowerPoint Slides for Undergraduate Econometrics by
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unknown population constants . The formulas that produce the sample estimates b 1 and b 2 are
called the estimators of
β 1 and
β 2 . When b
0 and b
1 are used to represent the formulas rather than specific values, they are called estimators of β 1
and
β 2 which are random variables because
they are different from sample to sample. 4.4
Copyright 1996 Lawrence C. Marsh • If the least squares estimators b 0 and b 1 are random variables , then what are their their means, variances, covariances and probability distributions? • Compare the properties of alternative estimators to the properties of the least squares estimators. Estimators are Random Variables ( estimates are not ) 4.5
Copyright 1996 Lawrence C. Marsh The Expected Values of b1 and b2 The least squares formulas (estimators) in the simple regression case: b 2
T Σ x t y t - Σ x t
Σ y t T Σ x t -( Σ x t )
2 b 1 = y - b 2 x where y =
Σ y t / T and
x = Σ x t / T
(3.3.8a) (3.3.8b)
4.6 Copyright 1996 Lawrence C. Marsh Substitute in y t
β 1 + β 2 x t + ε t to get:
b 2 = β 2
+ T Σ x t ε t -
Σ x t Σε t T Σ x t -(
Σ x t ) 2 2 The mean of b 2
Eb 2 = β 2
+ T Σ x t E ε t - Σ x t Σ E ε t T Σ x t -( Σ x t )
2 Since
E ε t = 0
, then Eb 2 = β 2 . 4.7
19 Copyright 1996 Lawrence C. Marsh The result Eb 2
β 2 means that the distribution of b 2 is centered at β 2 . Since the distribution of b 2
is centered at β 2 ,we say that b 2 is an unbiased estimator of β 2
An Unbiased Estimator 4.8
Copyright 1996 Lawrence C. Marsh The unbiasedness result on the previous slide assumes that we are using the correct model . If the model is of the wrong form or is missing important variables, then
E ε t ≠ 0 , then Eb 2 ≠
β 2 . Wrong Model Specification 4.9
Copyright 1996 Lawrence C. Marsh Unbiased Estimator of the Intercept In a similar manner, the estimator b
of the intercept or constant term can be shown to be an
estimator of β 1
when the model is correctly specified. Eb 1 = β 1 4.10 Copyright 1996 Lawrence C. Marsh b 2 = T Σ x t y t
− Σ x t
Σ y t T Σ x t − ( Σ x t ) 2 2 (3.3.8a)
(4.2.6) Equivalent expressions for b 2 :
b 2 = Σ (x t − x ) ( y t − y ) Σ( x t − x )
4.11
Variance of b 2 Given that both y t and
ε t have variance σ
, the
variance of the estimator b 2 is:
b 2 is a function of the y t values but var(b 2
t directly. Σ( x
t
− x ) σ
2 var(b
2 ) =
4.12 Copyright 1996 Lawrence C. Marsh Variance of b 1 Τ Σ(
x
t −
x )
var(b
) =
σ
Σ x
t 2 the
variance of the estimator b 1 is:
b 1 = y − b
2 x Given 4.13 20 Copyright 1996 Lawrence C. Marsh Covariance of b 1 and b
2 Σ(
x
t −
x )
cov(b
,b
) = σ
− x
If x = 0, slope can change without affecting the variance. 4.14
What factors determine variance and covariance ?
σ
: uncertainty about y t values uncertainty about b 1 , b 2 and their relationship. 2. The more spread out the x t values are then the more confidence we have in b 1 , b 2 , etc. 3. The larger the sample size, T, the smaller the variances and covariances. 4. The variance b 1 is large when the (squared) x t values are far from zero (in either direction). 5. Changing the slope, b 2 , has no effect on the intercept, b 1 , when the sample mean is zero. But if sample mean is positive, the covariance between b 1 and b 2 will be negative, and vice versa. 4.15
Copyright 1996 Lawrence C. Marsh Gauss-Markov Theorm Under the first five assumptions of the simple, linear regression model, the ordinary least squares estimators b 1
and b 2 have the smallest variance of all linear and unbiased estimators of β 1 and β 2 . This means that b 1 and b
2
are the Best Linear Unbiased Estimators (BLUE) of β 1 and β 2 . 4.16
Copyright 1996 Lawrence C. Marsh implications of Gauss-Markov 1. b
1 and b
2 are “best” within the class of linear and unbiased estimators. 2. “Best” means smallest variance within the class of linear/unbiased. 3. All of the first five assumptions must hold to satisfy Gauss-Markov. 4. Gauss-Markov does not require assumption six: normality. 5. G-Markov is not based on the least squares principle but on b 1 and b
2 . 4.17 Copyright 1996 Lawrence C. Marsh G-Markov implications (continued) 6. If we are not satisfied with restricting our estimation to the class of linear and unbiased estimators, we should ignore the Gauss-Markov Theorem and use some nonlinear and/or biased estimator instead. (Note: a biased or nonlinear estimator could have smaller variance than those satisfying Gauss-Markov.) 7. Gauss-Markov applies to the b 1 and b 2 estimators and not to particular sample values (estimates) of b 1 and b 2 . 4.18 Copyright 1996 Lawrence C. Marsh Probability Distribution of Least Squares Estimators b
~ N β
, Σ(
t
−
x ) σ
2 2 b
~ N β
, Τ Σ(
x
t −
x )
σ
Σ x
t 2 4.19
21 Copyright 1996 Lawrence C. Marsh y t and ε t normally distributed The least squares estimator of β 2 can be expressed as a linear combination of y t ’s:
b 2 = Σ w t y t
b 1 = y − b
2 x Σ( x
t −
x )
where w
= ( x
t −
x ) This means that b 1 and b
2 are normal since linear combinations of normals are normal. 4.20
Copyright 1996 Lawrence C. Marsh normally distributed under The Central Limit Theorem If the first five Gauss-Markov assumptions hold, and sample size, T, is sufficiently large, then the least squares estimators, b 1 and b
2 , have a distribution that approximates the normal distribution with greater accuracy the larger the value of sample size, T. 4.21
Consistency We would like our estimators, b 1 and b 2 , to collapse onto the true population values, β 1 and β 2 , as sample size, T, goes to infinity. One way to achieve this
property is for the variances of b 1 and b 2 to go to zero as T goes to infinity. Since the formulas for the variances of the least squares estimators b 1 and b 2 show that their variances do, in fact, go to zero, then b 1 and b 2 , are consistent estimators of β 1
β 2 . 4.22 Copyright 1996 Lawrence C. Marsh Estimating the variance of the error term, σ
e t
t
− b 1
− b 2 x t ^ Σ e t ^ t =1 T 2 T − 2 σ
2
= σ
2
is an unbiased estimator of σ
^
4.23 Copyright 1996 Lawrence C. Marsh The Least Squares Predictor, y o
^ Given a value of the explanatory variable, X o , we would like to predict a value of the dependent variable, y o . The least squares predictor is: y o = b 1 + b 2 x o (4.7.2)
^ 4.24
Copyright 1996 Lawrence C. Marsh Inference in the Simple Regression Model Chapter 5 Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. 5.1
22 Copyright 1996 Lawrence C. Marsh 1.
y t = β 1 + β 2 x t
+ ε t 2. E( ε t ) = 0 <=> E( y t ) = β 1 + β 2 x t 3. var( ε t ) =
σ
2 = var(
y t ) 4. cov( ε i , ε j ) = cov( y i , y j ) = 0
5. x t ≠ c
for every observation 6.
ε t ~N(0, σ
2 ) <=> y t ~ N( β 1 + β 2 x t , σ
2 )
Assumptions of the Simple Linear Regression Model 5.2
Copyright 1996 Lawrence C. Marsh Probability Distribution of Least Squares Estimators b
~ N β
, Τ Σ(
x
t −
x )
σ
Σ x t
b
~ N β
, Σ( x
t −
x ) σ 2 2 5.3
Copyright 1996 Lawrence C. Marsh σ
2 ^ = Τ − 2 e t ^ 2 Σ Unbiased estimator of the error variance: σ
2 σ
2 ^ (Τ − 2) ∼ Τ − 2
χ Transform to a chi-square distribution: Error Variance Estimation 5.4
Copyright 1996 Lawrence C. Marsh We make a correct decision if: • The null hypothesis is false and we decide to reject it. • The null hypothesis is true and we decide not to reject it. Our decision is
if:
• The null hypothesis is true and we decide to reject it. This is a type I error . • The null hypothesis is false and we decide not to reject it. This is a type II error . 5.5 Copyright 1996 Lawrence C. Marsh b
~ N β
, Σ(
t
−
x ) σ
2 Create a standardized normal random variable, Z, by subtracting the mean of b
and dividing by its standard deviation: b
− β
2
var(b 2 ) Ζ = ∼ Ν(0,1) 5.6
y t = β 1 + β 2 x t
+ ε t where E
ε t = 0 y t
~ N( β 1 + β 2 x t , σ
2 ) since Ey t =
β 1 + β 2 x t ε t =
y t
− β 1 −
β 2 x t Therefore, ε
~ N(0, σ
2 ) .
5.7 23 Copyright 1996 Lawrence C. Marsh Create a Chi-Square ε
t ~ N(0, σ
2 ) but want N(0, 1 ) .
( ε
t / σ)
~ N(0, 1 ) Standard Normal . ( ε
t / σ) 2 ~ χ 2 (1)
Chi-Square . 5.8
Copyright 1996 Lawrence C. Marsh Sum of Chi-Squares Σ t =1 ( ε t
σ) 2
=
( ε
/ σ) 2 + ( ε 2 / σ) 2 +. . .+ ( ε T / σ) 2
χ 2
+ χ 2 (1) +. . .+ χ 2 (1)
=
χ 2 (Τ)
Therefore, Σ t =1 ( ε t
σ) 2
∼ χ 2 (Τ) 5.9
Copyright 1996 Lawrence C. Marsh Since the errors
ε t =
y t −
β 1
−
β 2 x t are not observable, we estimate them with the
sample residuals e t =
y t
− b 1 −
b 2 x t . Unlike the errors, the Download 0.54 Mb. Do'stlaringiz bilan baham: |
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