REGRESSION

1. Prediction Equation

2. Sample Slope

SSx= ∑ x2- (∑ x)2/n

SSxy= ∑ xy- ∑ x*∑ y/n

3. Sample Y Intercept

4. Coeff. Of Determination

5. Std. Error of Estimate

6. Std Error of b0 and b1

7. Test Statistic

8. Confidence Interval of b0 and b1

9. Confidence interval for mean value of Y given x

10. Prediction interval for a randomly chosen value of Y given x

11. Coeff. of Correlation

12. Adjusted R2

13. Variance Inflation Factor

14. Beta Weights

15. Partial F Test

SSER - sum of squares of error of reduced model SSEF - sum of squares of error of full model

r – no. of variables dropped from full model.

16. Outliers

Measure

Potential Outliers

Standardized residual, Studentized

> 3 (3 sigma level)

y i= β0+ β1 x i

β1=SSxySSxx

=∑ (x i− x ) ( y i− y )

∑ ( x i− x )2

β0= y− β1 x

R2=SSRSST

=1−SSESST

Se=√∑ (Y i−Y¿ )2

n−2

Se=√∑Y i2−β0∑Y i−β1∑ X iY i

n−2

S( β0 )=Se×√∑ x2

√nSSxxS( β1 )=

Se

√SS xx

t(n−2 )=Estimate−ParameterEst . std . error of estimate

¿β1

¿−β1

Se( β1 )¿

t(n−2 )=β1

¿

√SS xSe

β1±t(α /2, n−2)×Se( β1)β0±t(α /2 ,n−2)×Se( β0 )

A (1- α )100% confidence interval for E (Y|X ) :

Y i¿

±tα /2Se√1n

+( X i−X

−

)2

SS X

Here Y¿

is the E(Y |X ).

A (1- α )100% prediction interval for Y is:

Y i¿±tα /s Se√1+1

n+(X i−X

−

)2

SSxwhere Xs are observed values of independent variable .

Y¿

is the estimate of Y, n is the sample size and Se is thestandard error of Y

r=√R2=SS XY

√SS XX SSYY

RA2 =1−

SSE /( n−k−1)SST /(n−1 )

RA2 =1−(1−R2 )×n−1

n−( k+1)RA

2 =The adjusted coefficient of determinationR2= Unadjusted coefficient of determinationn = number of observationsk = no. of explanatory variables

VIF(X j)=1

1−R j2

R j2 is the coefficient of

determination for the regression of X j as dependent variable

Beta=βi×SxS y

Sx=Std dev of XS y=Std dev of Y

F r , n−(k+1)=( SSER−SSEF )/r

MSEF

residual

Mahalanobis distance

> Critical chi-square value with df = number of explanatory variables(Outliers in independent variable)

Cook’s distance > 1 implies potential outlier

Leverage values

> 2(k+1)/n, then the point is influential (k is the number of independent variables and n is the sample size)

DFBeta > 2/Ön

DFFit

17. Mahalanobis Distance

Mi = ((Xi – X)/ Sx)2

18. Cook’s Distance

Di =

∑j (Yj – Yj(i))2/k x MSE

19. Durbin Watson Test

Durbin Watson value close to 2 implies no auto-correlation

Durbin Watson value close to 0 implies positive auto-correlation

Durbin Watson value close to 4 implies negative auto-correlation

20. Relationship between F and R2

F = (R2/1- R2) x ((n-(k+1))/k)

21. Standard ErrorStandard Error =SQRT(MSE)

22. Leverage23.

FORECASTING

1. Exponential Smoothing

2. Double Exponential Smoothing

3. Theil’s Coeff

U1 is bounded between 0 and 1, with values closer to zero indicating greater accuracy.

If U2 = 1, there is no difference between naïve forecast and the forecasting technique

If U2 < 1, the technique is better than naïve forecast

If U2 > 1, the technique is no better than the naïve forecast.

¿2√(k+1 )/nLt=α∗Y t−1+(1−α )∗Lt−1

F t¿

=Lt

( i) Lt=α×Y t+(1−α )×(Lt−1+T t−1 )( ii ) T t=β×(Lt−Lt−1 )+(1−β )×T t−1

( iii )F t+1

¿

=Lt+T t

( iv )F¿

t+m=Lt+mT t

U 1=√∑t=1

n

(Y t−Ft )2

√∑t=1

n

Y t2+√∑

t=1

n

Ft2

,

U2 =√∑t=1

n-1

(F t+1−Y t+1

Y t )2

∑t=1

n-1

(Y t+1−Y tY t )

2