Instrumental Variables

In one of its simplest formulations the problem of estimating the parameters of a system of simultaneous equations with unknown random errors reduces to finding a way of estimating the parameters of a single linear equation of the form Y = Xβ₀+ ε where β_o is unknown, Y and X are vectors of data on relevant economic variables and ε is the vector of unknown random errors. The most common method of estimating β₀ is the method of least squares: <math display='block'> <mrow> <msub> <mover accent='true'> <mi>&#x03B2;</mi> <mo>&#x005E;</mo> </mover> <mrow> <mi>O</mi><mi>L</mi><mi>S</mi> </mrow> </msub> <mo>&#x2261;</mo> </mrow> </math>$${\hat \beta _{OLS}} \equiv $$ argmin ε(β)′ε(β), where ε(β) ≡ Y - Xβ. Under fairly general assumptions <math display='block'> <mrow> <msub> <mover accent='true'> <mi>&#x03B2;</mi> <mo>&#x005E;</mo> </mover> <mrow> <mi>O</mi><mi>L</mi><mi>S</mi> </mrow> </msub> </mrow> </math>$${\hat \beta _{OLS}}$$ is an unbiased estimator of β₀ provided E(ε_t|X) = 0 for all t, where ε_t is the tth-coordinate of ε.