Sargan Test and Hausman Test

Sargan Test	Hausman Test
H0: Valid Instrument	H0: X1 exogeneity
2SLS	y ~ X
e2SLS = y - Xb2SLS	e = y-Xb
e2SLS ~ Z	X1 ~ Z
m instruments in Z	v = X1 - Zb2SLS1
k explanatory variables in X	e ~ X + v
nR2 – χ2 (m-k)	nR2 – χ2 (k1)

Note:

X = (X1, X2)
Z = (Z*, X2)

X1: potentially endogenous variables
X2: exogenous variables
Z:   Instrumental variables

2SLS: Two Stage Least Square Method

Stage 1:  X1 = Z*ϒ1 + X2ϒ2 + η
Stage 2:  y = fitted_X1β + X2β + ε