Intercept is in the error term (dropping?)

The model is $$y_{it} = \delta_0d2_t + \delta_1 crm_{it} + (\alpha_i+u_{it})$$

Here the intercept is placed in the error term. Therefore if $\alpha_i$ is correlated with an independent variable would cause bias problems.

• What are the consequences of dropping $\alpha_i$?
• Why is it important to keep it?

• Why do we delete it under fixed effects or first differencing $\alpha_i$ would be eliminated.

When buidling a sum you can easily change the order (Cummutative property of addition) and the brackets (Associative property of addition).

$y_{it} = \delta_0d2_t + \delta_1 crm_{it} + (\alpha_i+u_{it})$

is the same as

$y_{it} = \delta_0d2_t + \delta_1 crm_{it} + \alpha_i+u_{it}$

and the same as

$y_{it} = \alpha_i + \delta_0d2_t + \delta_1 crm_{it} +u_{it}$.

Nonetheless I would like to outline why theequation is written in the way it is written:

These are random effects which depend on the individual and not on the time component. The first part of the equation equals the expected value und certain conditions.

$y_{it} = \delta_0d2_t + \delta_1 crm_{it}$

However the part in the bracket represents the noise. These are random effects which depend on the individual and not on the time component. The variance which does not depend on the regression $\delta_0$ and $\delta_1$ is captured by the term in the brackets. Thus neither time nor crm have an impact on the dependent variable.

$\alpha_i+u_{it}$

In other words if $\delta_0$ and $\delta_1$ are both zero than the following equation is always true.

$y_{it} = \alpha_i+u_{it}$

Consequences of dropping $\alpha_i$:

You delete any effect which is typical for the individual. For example is your sample consists of Sweden, Italy, Spain, Portugal and Malta and you delete the individual specific component. Than you do not have the "Sweden effect" (or the effect for the other countries any more. So you will not anymore capture the effect that Sweden is a northern european, protestant country with long winters while the other countries are different.

If we apply a FD estimation we are not interest in the effect of Sweden, but in the effect of the difference between the years. Therefore we do not have a close look at the Sweden effect at a certain point of the procedure.