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Introduction to Econometrics. Lecture 5 Extensions to the multiple regression model. Lecture plan. logarithmic transformations - log-linear (constant elasticity) models dummy variables for qualitative factors

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introduction to econometrics

Introduction to Econometrics

Lecture 5

Extensions to the multiple regression model

Lecture plan
  • logarithmic transformations - log-linear (constant elasticity) models
  • dummy variables for qualitative factors
  • simple dynamic models with lagged variables - the partial adjustment mechanism
  • an application to illustrate the above - A study of cigarette consumption in Greece by Vasilios Stavrinos (Applied Economics, 1987 pp 323-329)
Log-linear regression models (1)

In many cases relationships between economic variables may be non-linear. However we can distinguish between functional forms that are intrinsically non-linear (and will need to be estimated by some kind of iterative non-linear least squares method) and those that can be transformed into an equation to which we can apply ordinary least squares techniques.

Log-linear regression models (2)

Of those non-linear equations that can be transformed, the best known is the multiplicative power function form (sometimes called the Cobb-Douglas functional form), which is transformed into a linear format by taking logarithms.

Log-linear regression models (3)

Production functions

For example, suppose we have cross-section data on firms in a particular industry with observations both on the output (Q) of each firm and on the inputs of labour (L) and capital (K).

Consider the following functional form

Log-linear regression models (6)

The parameters  and  can be estimated directly from a regression of the variable lnQ on lnL and lnK

Dummy variables (1)

Dummy variables (sometimes called dichotomous variables) are variables that are created to allow for qualitative effects in a regression model.

A dummy variable will take the value 1 or 0 according to whether or not the condition is present or absent for a particular observation.

For example suppose we are investigating the relationship between the wage (Y) and the number of years of experience (X) of workers in a particular industry.

Our initial model is

Y = a + b X + u

However we are concerned that the wages of female workers may be below that of male workers with similar experience. To test for this we can introduce a dummy variable to distinguish between the observations for male and female workers in the regression.

Dummy variables (2)

Define D = 1 for male workers and 0 for female workers.

The overall equation becomes

Y = a + b X + cD + u

where c will measure the differential between male and female workers, having taken account of differences in experience. We can run a normal multiple regression with X and D as explanatory variables. Assuming that c is positive it means that the regression line for male workers lies above that for female workers - c measures the extent of the upward shift. We can use its t value to test whether these differences are statistically significant.

Dummy variables (3)

Ramu Ramanathan (1998) includes a data set compiled by Susan Wong relating to

49 professionals in an industry (23 are for females and 26 for males).

The results show a large and significant difference in wages (which range between

981 and 3833 with a mean of 1820).

y i b 1 b 2 x i b 3 d i u i
Dummy variables (4) Testing for differences in intercept.Yi = b1 + b2 Xi+ b3 Di+ ui

Yi = (b1+ b3) + b2 Xi + ut

For men:Di= 1.






For women: Di = 0.

Yi = b1 + b2 Xi + ui

b1+ b3



years of experience


y i b 1 b 2 x i b 3 d i b 4 d i x i u i
Interactive dummies: Testing for differences in intercept and slopeYi = b1 + b2 Xi + b3Di + b4Di Xi + ui


Yi = (b1 + b3) + (b2 + b4) Xi + ui




b2 + b4


Yi = b1 + b2 Xi + ui



b1 + b3



years of experience

Dummy variables and time series data
  • With time series data we can have
  • impulsedummies – just affecting a particular period
  • stepdummies – affect remains on for a number of periods

We might also have seasonal dummies

e.g. lnQt = b0 + b1 lnYt + b2lnPt + d1D1t + d2D2t + d3 D3t + ut

D1 = 1 for quarter 1 observations, 0 otherwise

D2 = 1 for quarter 2 observations, 0 otherwise

D3 = 1 for quarter 3 observations, 0 otherwise

Beware of the “dummy variable trap”

Illustration: cigarette consumption in Greece (see Stavrinos, Applied Economics, 1987 19, pp323-329)