Causal Inference
================

Random notes on causal inference.

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Panel Data
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Difference in Difference
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We can impute the potential outcome of the treatment group by adding the difference between :math:`T_{post} = 0` and :math:`T_{post} = 1` in control group to :math:`T_{post}=0` in the treatment group.

.. math::

   E[Y_{(0)} | D = 1, T_{post} = 1] = & E[Y | D = 1, T_{post} = 0] + \\
                                      & (E[Y | D = 0, T_{post} = 1] - E[Y | D = 0, T_{post} = 0])

The imputation is quite intuitive as we can consider :math:`E[Y | D = 1, T_{post} = 0]` as a baseline, and see the difference in control group as a trend that's universal to both of the treatment group and the control group.

The :math:`ATT` is defined as:

.. math::

   ATT &= E[Y_{it,(1)} - Y_{it,(0)} | D = 1, T_{post} = 1] \\
       &= E[Y_{it,(1)} | D = 1, T_{post} = 1] - E[Y_{it,(0)} | D = 1, T_{post} = 1] \\
       &= E[Y | D = 1, T_{post} = 1] - E[Y_{it,(0)} | D = 1, T_{post} = 1]

The first term is correct since :math:`Y_{(1)} = Y` in treatment group after treatment, while the second term is the quantity we are trying to impute. Substituting the imputed :math:`E[Y_{(0)} | D = 1, T_{post} = 1]` into `ATT` we can get a nice representation of difference-in-difference:

.. math::

   ATT &= (E[Y | D = 1, T_{post} = 1] - E[Y | D = 1, T_{post} = 0]) - \\
       &= (E[Y | D = 0, T_{post} = 1] - E[Y | D = 0, T_{post} = 0])