Reporting Results, and choosing a functional form.

Hill et al chapter 6.

Explained and Unexplained Variation

1 2 ˆ

ˆ ˆ

ˆ ˆ( )

t t t

t t t

t t t

y b b x e

y y e

y y y y e

TotalExplained Unexplained

Decomposing the variation2 2

2 2

2 2

ˆ ˆ( ) [( ) ]

ˆ ˆ ˆ ˆ( ) 2 ( )

ˆ ˆ( )

t t t

t t t t

t t

y y y y e

y y e y y e

y y e

1. 2( ) ty y = total sum of squares = SST

2. 2ˆ( ) ty y = explained sum of squares = SSR

3. 2ˆ te = error sum of squares = SSE

link to proof

Coefficient of determination

2 1SSR SSE

RSST SST

Closer to 1, the better job we have done in explaining variation in y If 2R=1, SSE=0, and the model fits the data “perfectly.”

If y and x are show no linear association, the fitted line is

“horizontal,” and identical to y, SSR=0 and 2R=0.

When 0 < 2R < 1, it is “the percentage of the variation in y about its

mean that is explained by the regression model.”

Example Sum of Mean Source DF Squares Square Explained 1 25221.2229 25221.2229 Unexplained 38 54311.3314 1429.2455 Total 39 79532.5544

SST = 2( ) ty y = 79532. SSR = 2ˆ( ) ty y = 25221. SSE = 2ˆ te = 54311.

2 1SSR SSE

RSST SST

= 0.317

SSE/(T 2) = 2̂ = 1429.2455

Reporting Results2ˆ =40.7676 0.1283 0.317

(s.e.) (22.1387)(0.0305) t ty x R

2ˆ 40.7676 0.1283 0.317

( ) (1.84) (4.20) t ty x R

t

Choosing a functional form

• Model has been assumed to be linear in the parameters.

• Linear in parameters: parameters are not multiplied together, raised to a power etc.

• variables, however, can be transformed in any convenient way, as long as the resulting model satisfies assumptions SR1-SR5 of the simple linear regression model.

A functional form for food expenditure

It is expected that food expenditure will rise at a decreasing rate with income

Some common functional formsType

Statistical Model

Slope

Elasticity

1. Linear

1 2t t ty x e

2 2

t

t

x

y

2. Reciprocal

1 2

1t t

t

y ex

2 2

1

tx 2

1

t tx y

3. Log-Log

1 2ln( ) ln( )t t ty x e 2t

t

y

x

2

4. Log-Linear (Exponential)

1 2ln( )t t ty x e

2 ty

2 tx

5. Linear-Log (Semi-log)

1 2 ln( )t t ty x e 2

1

tx 2

1

ty

6. Log-inverse

1 2

1ln( )t t

t

y ex

2 2t

t

y

x 2

1

tx

Empirical issues

2ˆ 0.638 0.210 0.649

(0.064) (0.0022) ( . .)t ty x R

s e

An alternative functional form

3 2ˆ 0.874 9.68 0.751

(0.036) (0.824) ( . .)t ty x R

s e

Are the residuals normal (Jarque-Bera)

2

2

3

13

4

14

2

1

3

6 4

ˆ

ˆ

ˆ

T

ii

T

ii

T

ii

kTJB S

eS

ek

e

T

2

2

0

1

22

2.874 3400.3969 1.077

6 4

H : Residuals are normally distributed

H : Residuals are not normally distributed

5.99

JB

Appendix: proof

y y e y e y et t t t t b b x e y e b e b x e y et t t t t t t1 2 1 2

C o n s i d e r t h e t e r m s e t a n d x et t

e y b b x y T b b xt t t t t 1 2 1 2 0

x e x y b b x x y b x b xt t t t t t t t t 1 2 1 22 0

T h e l a s t e x p r e s s i o n s i n e a c h o f t h e s e e q u a t i o n s b e c o m e z e r o f r o m t h e n o r m a l e q u a t i o n s t h a t a r e u s e d t o s o l v e f o r t h e l e a s t s q u a r e s e s t i m a t o r s .

S u b s t i t u t i n g e t = 0 a n d x et t = 0 b a c k i n t o t h e o r i g i n a l e q u a t i o n , w e o b t a i n y y et t 0 .

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