First differenced estimator

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

Description

Consider a model as:

Y_i = β₀ + β₁X_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 = β₀ + β₁X_it + β₂Z_i + ϵ_it for all observations i and time periods t

• Y_i,t-1 = β₀ + β₁X_i,t-1 + β₂Z_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 = (β₀ + β₁X_it + β₂Z_i + ϵ_it) - (β₀ + β₁X_i,t-1 + β₂Z_i + ϵ_i,t-1)

Y_it - Y_i,t-1 = β₁X_it - β₁X_i,t-1 + ϵ_it - ϵ_i,t-1

Y_it - Y_i,t-1 = β₁(X_it - X_i,t-1) + ϵ_it - ϵ_i,t-1

ΔY_i = β₁ΔX_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.

CategoryRicottone