= First differenced estimator =

A '''first differenced''' ('''FD''') '''estimator''' is used to address omitted variable bias.

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== Description ==

Consider a model as:

''Y,,i,, = β,,0,, + β,,1,,X,,i,, + ϵ,,i,,''

If the data generating process is in fact based on both the observed ''X,,i,,'' and some unobserved ''Z,,i,,'', the regression will feature omitted variable bias.

One method for addressing this bias, if the confounding variable does not vary over time, is to collect multiple periods of data. There are now two models:

 * ''Y,,it,, = β,,0,, + β,,1,,X,,it,, + β,,2,,Z,,i,, + ϵ,,it,,'' for all observations ''i'' and time periods ''t''
 * ''Y,,i,t-1,, = β,,0,, + β,,1,,X,,i,t-1,, + β,,2,,Z,,i,, + ϵ,,i,t-1,,'' for all observations ''i'' and for time periods ''t=[2,T]''

Subtracting the latter from the former leads to a new model:

''Y,,it,, - Y,,i,t-1,, = (β,,0,, + β,,1,,X,,it,, + β,,2,,Z,,i,, + ϵ,,it,,) - (β,,0,, + β,,1,,X,,i,t-1,, + β,,2,,Z,,i,, + ϵ,,i,t-1,,)''

''Y,,it,, - Y,,i,t-1,, = β,,1,,X,,it,, - β,,1,,X,,i,t-1,, + ϵ,,it,, - ϵ,,i,t-1,,''

''Y,,it,, - Y,,i,t-1,, = β,,1,,(X,,it,, - X,,i,t-1,,) + ϵ,,it,, - ϵ,,i,t-1,,''

''ΔY,,i,, = β,,1,,ΔX,,i,, + ϵ,,it,, - ϵ,,i,t-1,,''

...for all observations ''i'' and for time periods ''t=[2,T]''.

In fact, all time-invariant variables are removed from the model by taking first differences.



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