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ECO 315: Econometrics Formula sheet Population and sample statistics Population

Sample _

E(X) = ∑Xf(x)

X

MEAN

X

i

/n _

2

2

( X  X) 2 S   i n 1 2 x

σ x = ∑(X-μ) f(x)

VARIANCE

σx

STANDARD DEVIATION

(X cov(X, Y)  

cov(X,Y) = ∑∑(X-μx)(Y-μy)f(x,y)

COVARIANCE

ρ = cov(X,Y)/σxσy

CORRELATION COEFFICIENT

3

S = E(X-μx) / σ

3

x

4

2 2

K = E(X-μx) / [E(X-μx) ]

Sx _

i

_

 X)(Yi  Y) n 1

r = cov(X,Y)/SxyS _

 X) 3

SKEWNESS

 (X S

KURTOSIS

 (X K

 X)4

i

n 1 _

i

n 1

Normal distribution If Xi ~ N(μ,σ2), then Xi can be converted into Zi ~ N(0,1) using Zi = (Xi – μ)/σ OLS formulas: Simple regression _

_

b1  Y b 2 X _

b2

_

 ( X  X )( Y  Y )   (X  X) i

i

_

2

i

_

var(b 1)   X i2  2 /[n  (X i  X ) 2 ] _

2 var(b 2 )   2 /  (X i  X )

se(b 1 )  var(b 1 ) se(b 2 )  var(b 2 )

^

Since σ2 is unknown, use  2   u i2 /( n  2)

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Hypothesis testing in simple regression

b1  1 ~ t df n 2 se(b 1 )

b 2  2 ~ t df n 2 se(b 2 )

Confidence intervals Limits:

b1 ± tαdf=n-2((se(b 1))

Limits:

b2 ± tαdf=n-2((se(b 2))

Analysis of variance TSS = ESS + RSS _

_

^

^

 (Y  Y)   (Y  Y)   (Y  Y ) 2

i

i

2

i

i

2

^

[Note that

 (Y  Y ) =  u i

i

2

2 i ]

Coefficient of determination in context of simple regression ^

r

_

(Y  Y )   (Y  Y )

2

i

2

_

 2

ESS TSS

i

Multiple regression t-ratios and confidence intervals are the same as in the simple regression case, but note that df = n-k, where k is the number of parameters. ^

^

ESS  2  yi x 2i   3  y ix 3i = 1 – (RSS/TSS) R   TSS  y i2 2

^

_

R

2

 u /(n  k ) 1   y /(n  1) 2 i

2 i

OLS formulas: Multiple regression ^

^

^

^

minu2i  (Yi  1 2 X2i 3 X3i) 2 ^

_

^

_

^

_

1  Y  2 X 2 3 X3 ^

2 

( yi x2 i )( x32i )  ( y ix 3i )( x 2i x 3i ) ( x22i )( x23i )  ( x 2ix 3i )2

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^

3 

( yi x3i )(  x 22i )  ( yi x 2 i )(  x 2i x 3i ) ( x22i )( x23i )  ( x 2i x 3i ) 2

_  2 1 X2  var(  1 )    n  



^

x

^

var( 2 ) 

2 3i

(  x )(  x )  (  x 2 i x 3i ) 2 2i

2 3i

2

where  2 

2 2i

(  x )(  x )  (  x 2 i x 3i ) 2 2i

2 3i

2 ^

x

^

var(  3 ) 

_ _ _  2 x3 i  X 23  x 22 i  2 X 2 X 3  x 2 i x 3 i  2  .  x 22 i  x 23i  (  x 2i x 3i ) 2  

2





^

u 2i

n3

2

Multiple regression: Hypothesis testing for overall significance ANOVA approach: F 

In terms of R2: F 

ESS /( k  1) RSS /( n  k )

R2 /(k  1) (1 R2 ) /(n  k )

Marginal contribution of an explanatory variable F test: F 

(ESSnew - ESSold ) / number of new regres sors RSSnew / df (  n  number of parameters in the new model)

Using R2: F 

(R2new - R2old ) / number of new regres sors (1 - R2new ) / df (  n  number of parameters in the new model)

Multiple regression: Hypothesis testing for equality of two coefficients ^

t

^

3 4 ^

^

^

^

var(  3 )  var(  4 )  2 cov(  3 ,  4 )

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Multiple regression: Hypothesis testing of linear equality restrictions ^

^

(2  3 )  (2  3 )

The t-test approach: t 

^

^

^

^

var( 2 )  var(3 )  2 cov(2 , 3)

F-test approach: F 

( RSS R  RSS UR ) / m RSS UR /( n  k )

or in terms of R 2: F 

(R 2UR  R 2R ) / m (1 R2UR ) /(n  k )

The Chow test

F

( RSS R  RSS UR ) / k ~ F( k ,( n1  n2  2 k ) ( RSS UR ) /( n1  n 2  2 k )

Testing whether error variances of two sub-sample regressions are the same:

RSS1 And ^2 RSS 2 2    n1  k n2  k ^

2 1

^

Then: F 

 12 ^



2 2

Multicollinearity: Variance Inflation Factor: 1 VIF  (1  r 223 ) ^

var( j) 

2 VIF  x2j

Auxiliary Regressions:

Fi 

R2xi .x2 x3 ...xk /(k  2) (1  R 2xi .x2 x3 ...xk ) /(n  k  1)

Heteroscedasticity: Spearman Rank Correlation Test:

  d 2i  rs  1  6   2  n ( n  1)  t

rs n  2 1  rs2

Page 9 of 17

Goldfelt-Quandt test:

 F

RSS2 / df RSS 1 / df

where df = [(n-c)/2] – k in the numerator and denominator

Breusch-Pagan-Godfrey test: ^ 2

~ 2

  ui / n ^

~ 2

pi  u i /  2



1 (ESS ) 2

Autocorrelation: Durbin-Watson statistic: t n

d

^

^

t n

^ 2 t

 ( u t  u t 1 ) 2 t 2

u t 1

^

d  2 (1  ) Model Specification and Diagnostic Testing: Ramsey RESET test:

(R 2new - R 2old ) / number of new regressors F 2 (1 - Rnew ) / df (  n  number of parameters in the new model)

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Eco315 June 2019

Eco315 June 2019

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