Recursive Systems. A recursive system has two crucial features: B is a triangular matrix and is a diagonal matrix. For this special case of

Recursive Systems. A recursive system has two crucial features: B is a triangular matrix and

Σ is a diagonal matrix. For this special case of the simultaneous equations model, OLS is still consistent, and under normality of the disturbances still maximum likelihood. Let us consider a specific example:

y1t + γ11x1t + γ12x2t = u1t

β21y1t + y2t + γ23x3t = u2t In this case, B =

1 0

β21 1 is triangular and



=

σ11 0 0 σ22 is assumed diagonal.

(a) Check the identifiability conditions of this recursive system.

(b) Solve for the reduced form and show that y1t is only a function of the xt’s and u1t, while y2t is a function of the xt’s and a linear combination of u1t and u2t.

(c) Show that OLS on the first structural equation yields consistent estimates. Hint: There are no right hand side y’s for the first equation. Show that despite the presence of y1 in the second equation, OLS of y2 on y1 and x3 yields consistent estimates. Note that y1 is a function of u1 only and u1 and u2 are not correlated.

(d) Under the normality assumption on the disturbances, the likelihood function conditional on the x’s is given by L(B, Γ, Σ) = (2π)−T/2|B|T |Σ|−T/2 exp(−1 2

T t=1 u



tΣ−1ut)

where in this two equation case u

t = (u1t, u2t). Since B is triangular, |B| = 1. Show that maximizing L with respect to B and Γ is equivalent to minimizing Q =

T t=1 u

tΣ−1ut.

Conclude that when Σ is diagonal, Σ−1 is diagonal and Q =

T t=1 u2 1t/σ11 +

T t=1 u2 2t/σ22.

Hence, maximizing the likelihood with respect to B and Γ is equivalent to running OLS on each equation separately.