Copyright 1996 Lawrence C. Marsh 0 PowerPoint Slides for Undergraduate Econometrics by
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sample residuals are
not independent since they use up two degrees of freedom by using
b
and b 2 to estimate β 1 and β 2 . We get only T − 2 degrees of freedom instead of T. Chi-Square degrees of freedom 5.10
Copyright 1996 Lawrence C. Marsh Student-t Distribution t = ~ t (m)
V / m
where Z ~ N(0,1) and V ~ χ (m)
2 5.11
Copyright 1996 Lawrence C. Marsh t = ~ t (m)
V /
(
T − 2) where Z = (b 2 − β 2 ) var(b 2 ) and var(b 2 ) =
σ
2 Σ ( x
i − x ) 2 5.12
Copyright 1996 Lawrence C. Marsh t =
Z V / (T-2) (b 2
2 ) var(b 2 ) t = (T − 2) σ
2 σ
2 ^ ( T − 2) V =
(T − 2) σ
2 σ
2 ^ 5.13 24 Copyright 1996 Lawrence C. Marsh var(b
2 ) =
σ
2 Σ ( x
i − x ) 2 (b 2 − β 2 ) σ
2 Σ ( x
i − x ) 2 t = = (T −
σ
2 σ
2 ^ ( T − 2) (b 2 − β 2 ) σ
2 Σ ( x
i − x ) 2 ^ notice the cancellations 5.14
Copyright 1996 Lawrence C. Marsh (b 2 − β 2 ) σ
Σ ( x
i − x ) 2 ^ t = = (b 2 − β 2 ) var(b 2 ) ^ t = (b 2 − β 2 ) se(b 2 ) 5.15 Copyright 1996 Lawrence C. Marsh Student’s t - statistic t = ~ t (T − 2) (b 2 − β 2 ) se(b 2 ) t has a Student-t Distribution with T
2 degrees of freedom. 5.16
Figure 5.1 Student-t Distribution ( 1−α
) t 0 f(t) -t c t c α / 2 α / 2 red area = rejection region for 2-sided test 5.17
Copyright 1996 Lawrence C. Marsh probability statements P(-t c
≤ t
≤ t c ) = 1 −
α P( t < -t c ) = P( t > t c ) =
α/ 2 P(-t c
≤ ≤ t c ) = 1 −
(b 2 − β 2 ) se(b 2 ) 5.18 Copyright 1996 Lawrence C. Marsh Confidence Intervals Two-sided (1 −α )x100% C.I. for β 1
b 1
− t α /2 [se(b
1 )], b
1 + t
α /2 [se(b 1 )] b 2
− t α /2 [se(b 2 )], b 2 + t
α /2 [se(b 2 )] Two-sided (1 −α )x100% C.I. for β 2
5.19 25 Copyright 1996 Lawrence C. Marsh Student-t vs. Normal Distribution 1. Both are symmetric bell-shaped distributions. 2. Student-t distribution has fatter tails than the normal. 3. Student-t converges to the normal for infinite sample. 4. Student-t conditional on degrees of freedom (df). 5. Normal is a good approximation of Student-t for the first few decimal places when df > 30 or so. 5.20
Hypothesis Tests 1. A null hypothesis, H 0 . 2. An alternative hypothesis, H 1 . 3. A test statistic. 4. A rejection region. 5.21
Rejection Rules 1.
If the value of the test statistic falls in the critical region in either tail
of the t-distribution, then we reject the null hypothesis in favor of the alternative. 2.
If the value of the test statistic falls in the critical region which lies in the left tail of the t-distribution, then we reject the null hypothesis in favor of the alternative. 2.
If the value of the test statistic falls in the critical region which lies in the right tail of the t-distribution, then we reject the null hypothesis in favor of the alternative. 5.22
Format for Hypothesis Testing 1. Determine null and alternative hypotheses. 2. Specify the test statistic and its distribution as if the null hypothesis were true. 3. Select α and determine the rejection region. 4. Calculate the sample value of test statistic. 5. State your conclusion. 5.23
practical vs. statistical significance in economics
but not statistically significant: When sample size is very small, a large average gap between the salaries of men and women might not be statistically significant.
but not practically significant: When sample size is very large, a small correlation (say, ρ = 0.00000001) between the winning numbers in the PowerBall Lottery and the Dow-Jones Stock Market Index might be statistically significant. 5.24
Copyright 1996 Lawrence C. Marsh Type I and Type II errors Type I error:
We make the mistake of rejecting the null hypothesis when it is true. α = P(rejecting H 0 when it is true). Type II error:
We make the mistake of failing to reject the null hypothesis when it is false. β
0 when it is false). 5.25
26 Copyright 1996 Lawrence C. Marsh Prediction Intervals A (1 −α
o is: y o
c se( f ) ^ se( f ) = var( f ) ^ f = y o
− y o ^ Σ( x t
−
x ) 2 var( f ) = σ
1 + + ^ 1 Τ ( x o
−
x ) 2 ^ 5.26 Copyright 1996 Lawrence C. Marsh The Simple Linear Regression Model Chapter 6 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. 6.1
Copyright 1996 Lawrence C. Marsh Explaining Variation in y t
t without any explanatory variables: y t
β 1 + e t Σ e t =
Σ (y t −
β 1 ) 2 2
t = 1 T T = −2 Σ (y t
− b 1 ) = 0 ∂Σ e t 2 t = 1 t = 1 T T ∂ β 1 Σ (y t
− b 1 ) = 0 t = 1 T Σ y t
− Tb 1 = 0 t = 1 T b 1 = y Why not y? 6.2
Explaining Variation in y t
t = b
1 + b
2 x t + e t ^ Unexplained variation: y t = b 1 + b 2 x t ^ Explained variation: e t
t − y t = y
t
− b 1 − b 2 x t ^ ^ 6.3 Copyright 1996 Lawrence C. Marsh Explaining Variation in y t
t = y
t + e
t ^ Why not y? ^ y t − y = y t − y + e t ^ ^ using y as baseline SST = SSR + SSE Σ (y
− y) 2 = Σ (y t − y) 2 + Σ e t
T ^ ^ T T t = 1 t = 1 2 cross product term
drops out
6.4 Copyright 1996 Lawrence C. Marsh Total Variation in y t
SST measures variation of y t around y Σ (y t − y) 2
T SST
= 6.5
27 Copyright 1996 Lawrence C. Marsh Explained Variation in y t
y t = b 1 + b
2 x t ^ Fitted y
t values:
^ y t
y ^ Σ (y t − y) 2
T SSR
= ^ 6.6 Copyright 1996 Lawrence C. Marsh Unexplained Variation in y t
SSE measures variation of y t around y t ^ e t = y t − y t = y t
− b 1 − b 2 x t ^ ^ Σ (y t − y t ) 2
= Σ e t 2
T SSE
= ^
T ^ 6.7 Copyright 1996 Lawrence C. Marsh Analysis of Variance Table ^ Table 6.1 Analysis of Variance Table Source of Sum of Mean Variation DF Squares Square Explained 1 SSR SSR/1 Unexplained T-2 SSE SSE/(T-2) [ = σ
] Total T-1 SST 6.8
Coefficient of Determination 0 ≤
2
≤ 1 What proportion of the variation in y t
SSR SST
R 2 = 6.9 Copyright 1996 Lawrence C. Marsh Coefficient of Determination SST = SSR + SSE SST SSR SSE SST SST SST = + SSR SSE SST SST 1 = + Dividing
by SST SSR
SST R 2 = = 1 − SSE SST 6.10
Copyright 1996 Lawrence C. Marsh R 2 is only a descriptive measure. R 2 does not measure the quality of the regression model. Focusing solely on maximizing R 2 is not a good idea. Coefficient of Determination 6.11
28 Copyright 1996 Lawrence C. Marsh cov(X,Y)
ρ = var(X) var(Y) Correlation Analysis cov(X,Y)
r = var(X) var(Y) Population: ^ ^ ^ Sample: 6.12
Correlation Analysis var(X) = ^ Σ (x t − x) 2 / ( T − 1 )
T var(Y) =
^ Σ (y t − y) 2 / ( T − 1 ) t = 1 T cov(X,Y) = ^ Σ
t − x)(y t − y) / ( T − 1 ) t = 1 T 6.13
Copyright 1996 Lawrence C. Marsh Correlation Analysis Σ (x
− x) 2 Σ (y t − y) 2 t = 1 T Σ (x t − x)(y t − y) t = 1 T r =
t = 1 T Sample Correlation Coefficient 6.14
Correlation Analysis and R 2 For simple linear regression analysis: r
= R 2
R 2
is also the correlation between y t
y t measuring “goodness of fit”. ^ 6.15
Copyright 1996 Lawrence C. Marsh Regression Computer Output Table 6.2 Computer Generated Least Squares Results (1) (2) (3) (4) (5) Parameter Standard T for H0: Variable Estimate Error Parameter=0 Prob>|T| INTERCEPT 40.7676 22.1387 1.841 0.0734 X 0.1283 0.0305 4.201 0.0002 Typical computer output of regression estimates: 6.16
Regression Computer Output se(b 1
1 ) = 490.12 = 22.1287 ^ se(b
2 ) = var(b 2 ) = 0.0009326 = 0.0305 ^ b 1 = 40.7676 b 2 = 0.1283 se(b
1 ) t = = = 1.84 b 1 40.7676 22.1287
se(b 2 ) b 2 t = = =
4.20 0.1283
0.0305 6.17
29 Copyright 1996 Lawrence C. Marsh Regression Computer Output Table 6.3 Analysis of Variance Table Sum of Mean Source DF Squares Square Explained 1 25221.2229 25221.2229 Unexplained 38 54311.3314 1429.2455 Total 39 79532.5544 R-square: 0.3171 Sources of variation in the dependent variable: 6.18
Regression Computer Output SSR SST
R 2 = = 1 − = 0.317 SSE SST
SSE /(T-2) =
σ 2
= 1429.2455 ^ SSE =
Σ e t 2 = 54311
^ SST =
Σ (y t − y) 2 = 79532 SSR =
Σ (y t − y) 2 = 25221 ^ 6.19 Copyright 1996 Lawrence C. Marsh y t = 40.7676 + 0.1283x t (s.e.) (22.1387) (0.0305) y t = 40.7676 + 0.1283x t (t) (1.84) (4.20) Reporting Regression Results 6.20
Copyright 1996 Lawrence C. Marsh R 2 = 0.317
Reporting Regression Results This R
2 value may seem low but it is typical in studies involving
data analyzed at the individual or micro level.
A considerably higher R 2 value would be expected in studies involving time-series data
analyzed at an aggregate or macro level.
6.21 Copyright 1996 Lawrence C. Marsh
Effects of Scaling the Data Changing the scale of x y t = β 1 + (c β 2 )(x t /c) + e
t y t = β 1 + β 2 x t + e t y t = β 1 + β 2 x t + e t * *
β 2 = c β 2 * x t = x t /c
* where
and The estimated coefficient and standard error change but the other statistics are unchanged. 6.22
Copyright 1996 Lawrence C. Marsh Effects of Scaling the Data Changing the scale of y y t /c = ( β 1 /c)
+ ( β 2 /c)x t + e
t /c
y t = β 1 + β 2 x t + e
t β 1 = β 1 /c
* and All statistics are changed except for the t-statistics and R
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