Correlation and Regression

CFA Level II Quant
#1

Could anyone explain/provide a mathematical proof for the following statement?

‘If the units of the independent variable are tons instead of pounds, the estimated slope coefficient will be 2000 times larger’

#2

2,000 lbs. = 1 ton

#3

Thanks. I get that. I don’t understand why the slope coefficient needs to be any larger.

Isn’t

y = a + 1 ( 1 pound) , the exact same as,

y = a + 1 (0.0005tons)

Wouldn’t the change in unit of the independent variable be corresponded by a change in unit for the dependent variable, as opposed to an increase in the slope coefficient, which just serves as a multiplier?

#4

Suppose that the price of asparagus is USD1.99/lb. If we measure the weight of asparagus in pounds, then:

p = 1.99_q_

and the slope is

m = USD1.99/lb.

If we measure the weight of asparagus in tons, then:

p = 3,980_q_

and the slope is

m = USD3,980/ton

Note that 3,980 = 1.99 × 2,000.

#5

You will get from:

y = a + 1 (0.0005tons)

y = a + (1 / 0.0005) (1 tons)

y = a + 2,000 (1 tons)

The effect of units will be seen in the calculated coefficients as long as the unit measure of the dependent variable and the other independent variables stay the same.

#6

Makes sense now. thanks