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
Extensions to the multiple regression model
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)
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 (4)
Log-linear regression models (5)
Log-linear regression models (6)
The parameters and can be estimated directly from a regression of the variable lnQ on lnL and lnK
Log-linear regression models (7)
Log-linear regression models (8)
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).
Dummy variables (4) Testing for differences in intercept.
Yi = (b1+ b3) + b2 Xi + ut
For men:Di= 1.
For women: Di = 0.
Yi = b1 + b2 Xi + ui
years of experience
Interactive dummies: Testing for differences in intercept and slope
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
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”
Partial adjustment mechanisms (1)
Partial adjustment mechanisms (2)
Illustration: cigarette consumption in Greece (see Stavrinos, Applied Economics, 1987 19, pp323-329)