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Log-level and Log-log transformations in Linear Regression Models A. Joseph Guse Washington and Lee University Fall 2012, Econ 398 Public Finance Seminar

Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

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Page 1: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-level and Log-log transformations inLinear Regression Models

A. Joseph Guse

Washington and Lee University

Fall 2012, Econ 398 Public Finance Seminar

Page 2: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Level-Level

A “Level-level” Regression Specification.

y = β0 + β1x1 + ǫ

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

Page 3: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Level-Level

A “Level-level” Regression Specification.

y = β0 + β1x1 + ǫ

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

How do we interpret β1?

Page 4: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Level-Level

A “Level-level” Regression Specification.

y = β0 + β1x1 + ǫ

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

How do we interpret β1?

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

dydx

= β

Page 5: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Level

A “Log-level” Regression Specification.

log(y) = β0 + β1x1 + ǫ

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

Page 6: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Level

A “Log-level” Regression Specification.

log(y) = β0 + β1x1 + ǫ

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?)

Page 7: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Level Continued

How do we interpret β1?

Page 8: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Level Continued

How do we interpret β1?First solve for y .

log(y) = β0 + β1x1 + ǫ

⇒y = eβ0+β1x1+ǫ

Page 9: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Level Continued

How do we interpret β1?First solve for y .

log(y) = β0 + β1x1 + ǫ

⇒y = eβ0+β1x1+ǫ

Then differentiate to get the marginal effect:

dydx1

= βeβ0+β1x1+ǫ = β1y

So the marginal effect depends on the value of y , while β

itself represents the growth rate.

β1 =dydx1

1y

For example, if we estimated that β1 is .04, we would saythat another year increases the volume of lumber by 4%.

Page 10: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Log

A “Log-Log” Regression Specification.

log(y) = β0 + β1 log(x1) + ǫ

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).

Page 11: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Log

A “Log-Log” Regression Specification.

log(y) = β0 + β1 log(x1) + ǫ

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) = β0 + β1 log(x1) + ǫy = eβ0+β1 log(x1)+ǫ

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

dydx1

=β1

x1eβ0+β1 log(x1)+ǫ = β1

yx1

Page 12: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Log Continued.

‘Log-Log” Regression Specification Continued...

log(y) = β0 + β1 log(x1) + ǫ

From previous slide the marginal effect is

dydx1

= β1yx1

Page 13: Log-level and Log-log transformations in Linear Regression ...home.wlu.edu/~gusej/econ398/notes/logRegressions.pdf · Log-level and Log-log transformations in Linear Regression Models

Log-Log Continued.

‘Log-Log” Regression Specification Continued...

log(y) = β0 + β1 log(x1) + ǫ

From previous slide the marginal effect is

dydx1

= β1yx1

Solving for β1 we get

β1 =dydx1

x1

y

Hence β1 is an elasticity. If x1 is price and y is demand andwe estimate β1 = −.6, it means that a 10% increase in theprice of the good would lead to a 6% decrease in demand.