The common intercept value in a linear regression equation is always taken as one. The basis for this is that including a constant in a given model is the same as including an explanatory variable whose value is always equal to one.
When using dummy variables in econometrics, dummy variable will always takes the value of 1 or 0. So let’s take an example there are two categories (m) of gender, either male or female, so it is vital that only m-1 categories are used in the model. Otherwise, you will fall into the dummy variable trap.
The dummy variable trap occurs where there is perfect collinearity between the intercept term in given model and the value achieved by the dummy variables. This is explained using a matrix as below:
A B C
| 1 1 0 |
| 1 0 1 |
In the matrix above, in the A column we have the common intercept term, in the b column we have male and in column C we have female. The common intercept is always taken as 1 as in column A above. In the case above, if you were to add the values of the dummy variables in both column B and C, you will get 1. The value of the dummy variables plus the value of the intercept can be perfectly predicted by each other, therefore they are perfectly collinear.
The reason you should use m-1 categories is because perfect collinearity is not possible in this case as per the matrix below:
| 1 1 |
| 1 0 |
In the case above, the intercept or constant is again equal to one in column A and now only one dummy variable represents the two categories (m) of gender, m-1. Now, as we see above, the value of the dummy component is no longer perfectly collinear with the intercept value. In other words, the value of both can no longer be perfectly predicted from the value of the other.
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