VIJAYA BHARATHIII MA ECONOMICSMADRAS CHRISTIAN COLLEGE

CHANGING MEASUREMENT UNITS:SCALING VARIABLES

We have data of import of crude oilSource: Oil Companies & DGCIS Having simple linear regression model

V = α + β Q + eQ- 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

Running regression to the simple linear regression model.

We get-

α = -356119.731β = 5.565

t value = 11.920Sig value= .000R Square= .905

V = α + βQ + e

Given: V = α + β Q + e

OLS Estimator:

=

= - β

Scale the value of Q by factor y. [where y = ]

And regress Value on (yQ) [ Billions of tons]

V = α* + β* (yQ) + e

Applying OLS:

β* = = = β =

α* = - β* (y) = - (y) = αCoefficient 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 + Q + e

V= -356119.731 + 5564.583Q + e

α = -356119.731 β = 5564.583

t value = 11.920Sig value= .000R Square= .905

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* = = x = x β

α* = x - β* = x - (x β) = x(- β ) = 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 + eV= -3561.197 + .056 Q + e

α = -3561.197 β = .056

t value = 11.920Sig value= .000R Square= .905

Now scale V by x (=1/100) and Q by y (=1/1000)

xV = α* + β*(yQ) + e

OLS Estimator of,

β* = = = β

α* = x - β*(y = x - β(y = x(- β ) = 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 + ( β)Q + e V= -3561.197 + 55.646Q + e

α = -3561.197 β = 55.646

t value = 11.920Sig value= .000R Square= .905

NO CHANGE SCALING INDEPENDENT

VARIABLE

SCALING DEPENDENT VARIABLE

SCALING BOTH DEPENDENT & INDEPENDENT

VARIABLLEEstimator of intercept α* = α α* = x α α* = x α

Estimator of slope β* = β* = x β β* = β

Equation V = α + βQ + e V= α + Q + e xV = x α + (xβ)Q + e xV = x α + ( 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 55.646T value 11.920 11.920 11.920 11.920

Sig value .000 .000 .000 .000

R square .905 .905 .905 .905

COMPARING THE CHANGES:

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