Key Formulas of Chapter 10

1
Key Formulas of Chapter 10

• General Multiple-regression Model
• Y a b1X1 b2X2 . . . bnXn et
  (10-1)
• Big City Bookstore
• Y -18.80 2.525X1 - .545X2 (10-4)
• (-4.879) (19.50) (-4.432)
• _
• SYX 2.978 R2 .973 n 12
• F 199.21 DW 1.7705

2

• 
  _
• Multiple Adjusted Coefficient Of Determination
  (R2)
• _ Explained Variance Unexplained
• R2 1 -
• Total Variance Total
  Variance
• S2YX
• 1 -
• S2Y

3

• Partial Regression Coefficients, Given X1 30
• Yt -18.80 2.525(30) - .545X2 et
  (10-5)
• Yt -18.80 75.75 - .545X2 et
• Yt 56.86 - .545X2 et
• Standard Error Of Regression Coefficient
• SYX
• Sb1 (10-6)
• S x2 (1 - r122)

4

• Partial F-test Restricted And Unrestricted
  Models
• (SSER - SSEU ) / m
• Fcalculated (10-7)
• SSEU / (n - k)
• If Fcalc ? Ftable , then SSER SSEU no sign
  improvement.
• First-Order Serial Correlation
• Yt a bXt ret-1 et (10-8)

5

• Cochrane-Orcutt Iterative Least Squares (COILS)
• ?
• ret-1 r (Yt-1 - Yt-1 ) (10-10)
• r Yt-1 - (a bXt-1 ret-2)
• Yt a bXt rYt-1 - ra - rbXt-1 nt
• Yt - rYt-1 a bXt - rbXt-1 nt
  (10-11)
• (1 - rB)Yt a b(1 - rB)Xt nt (10-12)

6

• Forecasting With Serially Correlated Errors
• Yt 115.86 .476Xt .65et-1
• (one period ahead)
• Yt1 115.86 .476Xt1 .65(0) (10-15)
• ( gt one period ahead)

7

• Constant Elasticity For Logarithmic
  Relationships
• LnYt a bLnXt et (10-16)
• DLnYt LnYt - LnYt-1
• b
• DLnXt LnXt - LnXt-1

8

• Variable Elasticity For Linear Relationships
  Like Equation 10-4
• X1 DYt X1
• Elasticity b
• Y1 DXt Y1
• DYt / Y1
• (10-17)
• DXt / X1

9

• Heteroscedasticity Goldfeild - Quandt Test
• ESSL 104685.85 ESSH 5164.41
• ESSH / (n - d)/ 2 - k)
• Fcal
• ESSL / (n - d)/ 2 - k)
• 104685.85(106-2)
• 20.27
• 5164.41(106-2)
• F106-2, 106-2, a .05 1.39
• Fcalculated gtgt Ftable
  reject homoscedasticity
• 
  hypothesis

10

• Weighted Least Squares (e.g. Variance
  Proportional to X1)
• Y a b1X1 b2X2 e
• (10-19a)
• X1 X1 X1 X1 X1

11

• Heteroscedasticity Goldfeild - Quandt Test
• DY /SY DY /SY
• b1 , b2 (10-22)
• DX1/SX1 DX2 / SX2
• SX1 SX2
• b1 b1 , b2 b2 (10-23)
• SY SY

12

• Dichotomous (dummy) Variables For Modeling
  Events Changes in The Constant And Slope, Using
  Dummy Variables
• Yt a bXt ret-1 et
• T time
• It improvement in product at t 11
• It 0 for t 1 to 10
• It 1 for t 11 to 20
• Xt It t Xt 0 for t 1 to 10 Xt ?
  0 for t 11 to 20