Question

(1.2) Marks available: 5 Consider a regression model of the form y-Ao +AX, + U, where corr(X,,U) O. Now, define Zi = cXi, where c is a non-zero constant. Is Zi a valid instrument for Xi? What is the general implication of this exercise?

Answer 1

Two conditions of instrument variable;

1) instrument relevance , cov(Z, X) not equals to zero.

Z is related to X.

2) instrument exogeneity = cov( Z, u) = 0

Thus cov (Z,X) = cov(cX, X) = E(cX*X) - E(cX)*E(X)

= c*E(X^{​​​​​​2})-c[E(X)]²

= c*[E(X^{​​​​​​2})- E(X)²] = c* Var( X)

Hence covariance of Z & X not equals zero.

Also Cov(cX, u) = zero.

So yes Z is a valid instrument.

Thus scaling of a variable can be used as a instrument variable.