Introduction to Econometrics

1
Introduction to Econometrics

• Lecture 5
• Extensions to the multiple regression model

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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)

3
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.

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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.

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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

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Log-linear regression models (4)

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Log-linear regression models (5)

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Log-linear regression models (6)

The parameters ? and ? can be estimated directly
from a regression of the variable lnQ on lnL and
lnK
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Log-linear regression models (7)

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Log-linear regression models (8)

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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.
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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.
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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).
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Yi b1 b2 Xi b3 Di ui
Dummy variables (4) Testing for differences in
intercept.
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
X
years of experience
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Yi b1 b2 Xi b3 Di b4 Di Xi ui
Interactive dummies Testing for differences in
intercept and slope
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
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Dummy variables and time series data

• With time series data we can have
• impulse dummies just affecting a particular
  period
• step dummies 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

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Partial adjustment mechanisms (1)

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Partial adjustment mechanisms (2)

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

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Stavrinos results