Introduction to econometrics
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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

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Introduction to econometrics

Introduction to Econometrics

Lecture 5

Extensions to the multiple regression model


Introduction to econometrics

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)


Introduction to econometrics

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.


Introduction to econometrics

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.


Introduction to econometrics

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


Introduction to econometrics

Log-linear regression models (4)


Introduction to econometrics

Log-linear regression models (5)


Introduction to econometrics

Log-linear regression models (6)

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


Introduction to econometrics

Log-linear regression models (7)


Introduction to econometrics

Log-linear regression models (8)


Introduction to econometrics

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.


Introduction to econometrics

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.


Introduction to econometrics

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.

Y

Men

wage

rate

Women

For women: Di = 0.

Yi = b1 + b2 Xi + ui

b1+ b3

b1

0

years of experience

X


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 slope

Yi = b1 + b2 Xi + b3Di + b4Di Xi + ui

Y

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

Men

wage

rate

b2 + b4

Women

Yi = b1 + b2 Xi + ui

b2

b1

b1 + b3

X

0

years of experience


Introduction to econometrics

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”


Introduction to econometrics

Partial adjustment mechanisms (1)


Introduction to econometrics

Partial adjustment mechanisms (2)


Introduction to econometrics

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


Introduction to econometrics

Stavrinos results


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