Scaling and Units of Measurement

1. VIJAYA BHARATHI
II MA ECONOMICS
MADRAS CHRISTIAN COLLEGE

2. CHANGING MEASUREMENT UNITS:
SCALING VARIABLES
We have data of import of crude oil
Source: Oil Companies & DGCIS
Having simple linear regression model
V = α + β Q + e
Q- Quantity is Independent Variable ( million tons )
V- Cost is Dependent Variable(Crores of Rupees)
? What happens when we change scale of dependent variable
? What will happen if independent variable’s scale is changed
? What kind of change will it bring in the interpretation

3. Running regression to the simple linear regression model.
We get-
α = -356119.731
β = 5.565
t value = 11.920
Sig value= .000
R Square= .905
V = α + βQ + e

4. Given: V = α + β Q + e
OLS Estimator:
𝛽 =
𝑞𝑣
𝑞2
𝛼 = 𝑉- β 𝑄
Scale the value of Q by factor y. [where y =
1
1000
]
And regress Value on (yQ) [ Billions of tons]
V = α* + β* (yQ) + e
Applying OLS:
β* =
yq v
(yq)2 =
𝑦 𝑞𝑣
𝑦2 𝑞2 =
1
𝑦
β =
β
𝑦
α* = 𝑉 - β* (y 𝑄) = 𝑉 -
β
𝑦
(y 𝑄) = α
Coefficient of slope variable change
Coefficient of intercept term DOES NOT change
Statistical Interpretation DOES NOT change
Scaling Independent Variable:
Running Regression we get:
V= α +
β
𝑦
Q + e
V= -356119.731 + 5564.583Q + e
α = -356119.731
β = 5564.583
t value = 11.920
Sig value= .000
R Square= .905

5. Now change V to V* by a factor x and leave Q unchanged.
[Where x = 1/100 & V* in billion rupees]
V* = (xV) = α* + β*Q + e
β* =
𝑞(𝑥𝑣)
𝑞2 = x
𝑞𝑣
𝑞2 = x β
α* = x 𝑉 - β* 𝑄 = x 𝑉 - (x β) 𝑄 = x( 𝑉- β 𝑄) = x α
Scaling dependent variable
Coefficient of slope variable change
Coefficient of intercept term change
Statistical Interpretation DOES NOT change
Running Regression we get:
xV = x α + (xβ)Q + e
V= -3561.197 + .056 Q + e
α = -3561.197
β = .056
t value = 11.920
Sig value= .000
R Square= .905

6. Now scale V by x (=1/100) and Q by y (=1/1000)
xV = α* + β*(yQ) + e
OLS Estimator of,
β* =
(𝑦𝑞)(𝑥𝑣)
𝑦𝑞2 =
𝑥𝑦 𝑞𝑣
𝑦2 𝑞2 =
𝑥
𝑦
β
α* = x 𝑉 - β*(y𝑄) = x 𝑉 -
𝑥
𝑦
β(y𝑄) = x( 𝑉- β 𝑄) = x α
Scaling Both the Independent and Dependent Variables:
Coefficient of slope variable change
Coefficient of intercept term change
Statistical Interpretation DOES NOT change
Running Regression we get:
xV = x α + (
𝑥
𝑦
β)Q + e
V= -3561.197 + 55.646Q + e
α = -3561.197
β = 55.646
t value = 11.920
Sig value= .000
R Square= .905

7. NO CHANGE SCALING
INDEPENDENT
VARIABLE
SCALING
DEPENDENT
VARIABLE
SCALING BOTH
DEPENDENT &
INDEPENDENT
VARIABLLE
Estimator of intercept 𝛼 α* = α α* = x α α* = x α
Estimator of slope 𝛽 β* =
β
𝑦
β* = x β β* =
𝑥
𝑦
β
Equation V = α + βQ + e V= α +
β
𝑦
Q + e xV = x α + (xβ)Q + e xV = x α + (
𝑥
𝑦
β)Q + e
V= -356119.731 +
5.565Q + e
V= -356119.731 +
5564.583Q + e
V= -3561.197 +
.056 Q + e
V= -3561.197 + 55.646Q +
e
Alpha (α) -356119.731 -356119.731 -3561.197 -3561.197
Beta (β) 5.565 5564.583 .056 55.646
T value 11.920 11.920 11.920 11.920
Sig value .000 .000 .000 .000
R square .905 .905 .905 .905
COMPARING THE CHANGES:

8. THANK YOU