ECON321Economic Analysis

Tutorial 7

Transformations

Introduction• The lecture on functional forms explains that you

may want to transform your data and estimate non-linear relationships.

• This tutorial will show you how to do those transformations, but for more about the interpretations please refer to the chapter in Gujarati and the lecture PowerPoint slides.

• We use the model of house price (measured in dollars) as a function of:o ROOMS: The average number of rooms in houses.o NOX: The amount of nitrogen oxide in the air in parts per

million.o DIST: The distance of the community from employment

centres in miles.o STRATIO: The average student-teacher ratio of schools in

the community.

Using logs

• It maybe that the relationship is not this linear in parameters.o For example, as pollution gets higher and

higher, the associated decrease in the price become larger.

• You can check this by using a log-level model.

Using logs

Model Dependent variable

Independent variable

Interpretation of β1

Level-level y x ∆y = β1∆x

Level-log y log(x) ∆y = (β1/100)%∆x

Log-level log(y) x %∆y = (100*β1)∆x

Log-log log(y) log(x) %∆y = β1%∆x

Using quadratics

• In addition to logs, you may want to use quadratics to examine increasing or decreasing returns.o For example, there may not ne a straight-line

positive relationship between the number of rooms and PRICE.

• You can check this by adding a quadratic term.

Using quadratics

• You are interested in the effect of ROOMS on log(PRICE):

• Note, the coefficient on ROOMS is negative, and on SQROOMS it is positive.

• This means that at low values of ROOMS an additional room has a negative effect on log(PRICE).

• At some point the effect becomes positive and the semi-elasticity of PRICE with respect to ROOMS is increasing as ROOMS increase.

• You can determine that turning point by the formula• In this case it is

xxyxxy 2121

ˆ2ˆˆ so ,ˆ2ˆˆ

)ˆ2/(ˆ21

* x

4.4)]062(.2/[545.* ROOMS

You now know how to estimate models and interpret the results. The rest of the course will be about tests to check whether the CLRM assumptions hold and the corrective steps that you need to take when they do not.

Save your workfile and continue…