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2 value.
y t = β 1 + β 2 x t + e
t * * * * β 2 = β 2 /c
* * e t = e
t /c y t = y
t /c where * 6.23
30 Copyright 1996 Lawrence C. Marsh Effects of Scaling the Data Changing the scale of x and y y t /c = ( β 1 /c)
+ (c β 2 /c)x t /c
+ e t /c y t = β 1 + β 2 x t + e
t β 1 = β 1 /c
* and No change in the R
or the
t-statistics or in regression results for β 2 but all other stats change. y t
β 1 + β 2 x t + e
t * * * * x t = x
t /c
* * e t = e
t /c y t = y
t /c where * 6.24
Copyright 1996 Lawrence C. Marsh The term
in a simple regression model does not mean a linear relationship between variables, but a model in which the
parameters enter the model in a linear way. 6.25
Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 x t + e t Linear Statistical Models: Nonlinear Statistical Models: ln(y
t ) =
β 1 + β 2 x t + e
t y t = β 1 + β 2 ln(x t ) + e
t y t = β 1 + β 2 x t + e t 2 y t = β 1 + β 2 x t
+ e t β 3 y t = β 1 + β 2 x t + exp( β 3 x t )
+ e t y t =
β 1 + β 2 x t + e
t β
Linear vs. Nonlinear 6.27
Copyright 1996 Lawrence C. Marsh y x nonlinear relationship between food expenditure and income Linear vs. Nonlinear food expenditure income 0
Copyright 1996 Lawrence C. Marsh Useful Functional Forms 1. Linear 2. Reciprocal 3. Log-Log 4. Log-Linear 5. Linear-Log 6. Log-Inverse Look at each form and its slope and elasticity 6.28
Copyright 1996 Lawrence C. Marsh Linear
y t = β 1 + β 2 x t + e
t
slope: β 2 elasticity: β 2 y t Useful Functional Forms x t
31 Copyright 1996 Lawrence C. Marsh Reciprocal y t
β 1 + β 2
+ e
t
Useful Functional Forms 1 x t slope: elasticity: 1 x
2 − β
2 1 x t y t − β 2 6.30
Copyright 1996 Lawrence C. Marsh x t y t Log-Log ln(y t )= β 1 + β 2 ln(x t ) + e
t
slope: β 2
elasticity: β 2 Useful Functional Forms 6.31
Copyright 1996 Lawrence C. Marsh Log-Linear ln(y t
β 1 + β 2 x t + e
t
slope: β 2 y t
elasticity: β 2 x t Useful Functional Forms 6.32
Linear-Log y t
β 1 + β 2 ln(x t )
+ e t
_ slope:
β 2
elasticity: β 2 1 x t y t 1 _ Useful Functional Forms 6.33
Useful Functional Forms ln(y t
β 1 - β 2
+ e
t
1 x t Log-Inverse slope: β 2 elasticity: β 2 x 2 t y t 1 x t 6.34
Copyright 1996 Lawrence C. Marsh 1. E (e
t ) = 0
2. var (e t ) = σ 2 3. cov(e i , e
j ) = 0
4. e t ~ N(0, σ 2 ) Error Term Properties 6.35
32 Copyright 1996 Lawrence C. Marsh Economic Models 1. Demand Models 2. Supply Models 3. Production Functions 4. Cost Functions 5. Phillips Curve 6.36
Copyright 1996 Lawrence C. Marsh 1. Demand Models
* quality demanded (y d ) and price (x) * constant elasticity ln(y t
β 1 + β 2 ln(x) t + e
t
d 6.37 Copyright 1996 Lawrence C. Marsh 2. Supply Models
* quality supplied (y s ) and price (x) * constant elasticity Economic Models ln(y t
β 1 + β 2 ln(x t ) + e
t
s 6.38 Copyright 1996 Lawrence C. Marsh 3. Production Functions * output (y) and input (x) * constant elasticity Economic Models ln(y
t )=
β 1 + β 2 ln(x t ) + e
t
Cobb-Douglas Production Function: 6.39 Copyright 1996 Lawrence C. Marsh 4a. Cost Functions
* total cost (y) and output (x) Economic Models
y t =
β 1 + β 2 x 2 t + e t 6.40
Copyright 1996 Lawrence C. Marsh 4b. Cost Functions
* average cost (x/y) and output (x) Economic Models (y t /x t ) = β 1 /x t + β 2 x t + e t /x t 6.41
33 Copyright 1996 Lawrence C. Marsh
* wage rate (w t ) and time (t) Economic Models unemployment rate, u t
t-1 % ∆ w t = w t − w t-1
= γα + γη
u t 1 nonlinear in both variables and parameters 6.42
Copyright 1996 Lawrence C. Marsh The Multiple Regression Model Chapter 7 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. 7.1
Copyright 1996 Lawrence C. Marsh Two Explanatory Variables y t
β 1 + β 2 x t2 + β 3 x t3 + e t ∂ y t ∂ x t2 = β 2 ∂ x t3 ∂ y t = β 3 x t ‘s affect y t
separately But least squares estimation of β 2
t2 and x
t3 . 7.2 Copyright 1996 Lawrence C. Marsh Correlated Variables y t
x t2 = capital x t3 = labor Always 5 workers per machine. If number of workers per machine is never varied, it becomes impossible to tell if the machines or the workers are responsible for changes in output. y t = β 1 + β 2 x t2
+ β 3 x t3
+ e t 7.3 Copyright 1996 Lawrence C. Marsh The General Model y t
β 1 + β 2 x t2 + β 3 x t3 +. . .+ β K x tK
+ e t The parameter β 1 is the intercept (constant) term. The “variable” attached to β 1 is x t1 = 1. Usually, the number of explanatory variables is said to be K − 1 (ignoring x t1 = 1), while the number of parameters is K. (Namely: β 1 . . . β K ). 7.4
Copyright 1996 Lawrence C. Marsh 1. E(e
t ) = 0
2. var(e t ) = σ 2 3. cov ( e t , e
s ) = 0
for t
≠ s
4. e t ~ N(0, σ 2 ) Statistical Properties of e t
34 Copyright 1996 Lawrence C. Marsh 1. E (y
t ) =
β 1 + β 2 x t2 +. . .+
β K x tK
2. var(y t ) = var(e t ) =
σ 2 3. cov(y t ,y s ) = cov(e t , e
s ) = 0 t ≠ s
y t ~ N( β 1 + β 2 x t2 +. . .+
β K x tK , σ 2 ) Statistical Properties of y t
7.6 Copyright 1996 Lawrence C. Marsh Assumptions 1. y t
β 1 + β 2 x t2 +. . .+
β K x tK + e
t 2. E (y t ) =
β 1 + β 2 x t2 +. . .+
β K x tK
3. var(y t ) = var(e t ) =
σ 2 4. cov(y t ,y s ) = cov(e t ,e
) = 0 t ≠ s 5. The values of x tk are not random 6. y t ~ N( β 1 + β 2 x t2 +. . .+
β K x tK , σ 2 ) 7.7 Copyright 1996 Lawrence C. Marsh Least Squares Estimation y t
β 1 + β 2 x t2 + β 3 x t3 + e t S ≡ S(
β 1 , β 2 , β 3 ) = Σ ( y t − β
1 − β
2 x t2 − β
3 x t3 ) 2
= 1 T Define:
y t
= y t
− y
x t2
= x t2
− x 2 * x t3 = x t3
− x 3 * 7.8
Copyright 1996 Lawrence C. Marsh b 1 = y −
b 1 − b 2 x 2 − b 3 x 3 b 3 = (Σ y t x t3 )(Σ x t2
) − (Σ y t x t2 )(Σ
x t3 x t2 )
* * * * * 2 (Σ x t2 )(Σ
x t3
) − (Σ x t2 x t3 ) * * * * 2 2 2 b 2 = (Σ y t x t2 )(Σ x t3 ) − (Σ y t x t3 )(Σ
x t2 x t3 )
* * * * * 2 (Σ x t2 )(Σ
x t3
) − (Σ x t2 x t3 ) * * * * 2 2 2 Least Squares Estimators 7.9
Dangers of Extrapolation Statistical models generally are good only “within the relevant range”. This means that extending them to extreme data values outside the range of the original data often leads to poor and sometimes ridiculous results. If height is normally distributed and the normal ranges from minus infinity to plus infinity, pity the man minus three feet tall. 7.10
Error Variance Estimation σ
^ = Τ − Κ e t ^ 2 Σ Unbiased estimator of the error variance: σ
2 σ
2 ^ (Τ − Κ)
∼ Τ − Κ
χ Transform to a chi-square distribution: 7.11
35 Copyright 1996 Lawrence C. Marsh Gauss-Markov Theorem Under the assumptions of the multiple regression model, the ordinary least squares estimators have the smallest variance of all linear and unbiased estimators. This means that the least squares estimators are the B est Linear U
nbiased Estimators (BLUE). 7.12
Copyright 1996 Lawrence C. Marsh Variances y t
β 1 + β 2 x t2 + β 3 x t3 + e t var(b 3 ) =
( 1 − r 23 ) Σ (x t3 −
x 3 ) 2 2 σ 2 var(b 2 ) = ( 1 − r 23 ) Σ (x t2 −
x 2 ) 2 2 σ 2 Σ (x t2 −
x 2 ) 2 Σ (x t3 −
x 3 ) 2 where r
23 = Σ (x t2
−
x 2 )(x
t3 −
x 3 ) When r 23 = 0 these reduce to the simple regression formulas. 7.13
Copyright 1996 Lawrence C. Marsh Variance Decomposition The variance of an estimator is smaller when: 1. The error variance, σ
, is smaller: σ
2 0 .
2. The sample size, T, is larger:
Σ (x
−
x 2 ) 2 . 3. The variable’s values are more spread out:
(x t2 −
x 2 ) 2 . 4. The correlation is close to zero: r 23
2 t = 1
T 7.14
Copyright 1996 Lawrence C. Marsh Covariances y t
β 1 + β 2 x t2 + β 3 x t3 + e t where r 23
= Σ (x t2 −
x 2 ) 2 Σ (x t3 −
x 3 ) 2 Σ (x t2 −
x 2 )(x t3 −
x 3 ) ( 1 − r 23 ) Σ (x t2 −
x 2 ) 2 Σ (x t3 −
x 3 ) 2 cov(b
2 ,b 3 ) = 2 −
r 23
σ 2 7.15 Copyright 1996 Lawrence C. Marsh Covariance Decomposition 1. The error variance, σ
2 , is larger.
2. The sample size, T, is smaller. 3. The values of the variables are less spread out. 4. The correlation, r 23 , is high . The covariance between any two estimators is larger in absolute value when: 7.16
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