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Log-level and Log-log transformations inLinear Regression Models

A. Joseph Guse

Washington and Lee University

Fall 2012, Econ 398 Public Finance Seminar

Level-Level

A Level-level Regression Specification.

y = 0 + 1x1 +

This is called a level-level specification because rawvalues (levels) of y are being regressed on raw values of x .

Level-Level

A Level-level Regression Specification.

y = 0 + 1x1 +

This is called a level-level specification because rawvalues (levels) of y are being regressed on raw values of x .

How do we interpret 1?

Level-Level

A Level-level Regression Specification.

y = 0 + 1x1 +

This is called a level-level specification because rawvalues (levels) of y are being regressed on raw values of x .

How do we interpret 1?

Differentiate w.r.t. x1 to find the marginal effect of x on y . Inthis case, IS the marginal effect.

dydx

=

Log-Level

A Log-level Regression Specification.

log(y) = 0 + 1x1 +

This is called a log-level specification because the naturallog transformed values of y are being regressed on rawvalues of x .

Log-Level

A Log-level Regression Specification.

log(y) = 0 + 1x1 +

This is called a log-level specification because the naturallog transformed values of y are being regressed on rawvalues of x .

You might want to run this specification if you think thatincreases in x lead to a constant percentage increase in y .(wage on education? forest lumber volume on years?)

Log-Level Continued

How do we interpret 1?

Log-Level Continued

How do we interpret 1?First solve for y .

log(y) = 0 + 1x1 +

y = e0+1x1+

Log-Level Continued

How do we interpret 1?First solve for y .

log(y) = 0 + 1x1 +

y = e0+1x1+

Then differentiate to get the marginal effect:

dydx1

= e0+1x1+ = 1y

So the marginal effect depends on the value of y , while itself represents the growth rate.

1 =dydx1

1y

For example, if we estimated that 1 is .04, we would saythat another year increases the volume of lumber by 4%.

Log-Log

A Log-Log Regression Specification.

log(y) = 0 + 1 log(x1) +

You might want to run this specification if you think thatpercentage increases in x lead to constant percentagechanges in y . (e.g. constant demand elasiticity).

Log-Log

A Log-Log Regression Specification.

log(y) = 0 + 1 log(x1) +

You might want to run this specification if you think thatpercentage increases in x lead to constant percentagechanges in y . (e.g. constant demand elasiticity).To caculate marginal effects. Solve for y ...

log(y) = 0 + 1 log(x1) + y = e0+1 log(x1)+

... and differentiate w.r.t. x

dydx1

=1

x1e0+1 log(x1)+ = 1

yx1

Log-Log Continued.

Log-Log Regression Specification Continued...

log(y) = 0 + 1 log(x1) +

From previous slide the marginal effect is

dydx1

= 1yx1

Log-Log Continued.

Log-Log Regression Specification Continued...

log(y) = 0 + 1 log(x1) +

From previous slide the marginal effect is

dydx1

= 1yx1

Solving for 1 we get

1 =dydx1

x1y

Hence 1 is an elasticity. If x1 is price and y is demand andwe estimate 1 = .6, it means that a 10% increase in theprice of the good would lead to a 6% decrease in demand.