Question

Econometrics Question

Consider the following relation between y and x, where u is an error term: y = β₀ + β₁x + u.

(a) Briefly comment on the properties of the OLS estimator for β₁ obtained from a random sample of x and y, when x and u are uncorrelated.

Suppose x and u are correlated. Let z be an instrument for x.

(b) Compared to part (a), what are the properties of the OLS estimator for β₁?

(c) Enumerate the properties that z must satisfy for it be a good instrument.

(d) Based on your answer in part (c), briefly explain why a proxy variable should not be used as an instrument when endogeneity is caused by an omitted variable.

Answer 1

Our model is given as:

1)

When x and u are uncorrelated, it means that the CLRM property of exogeneity is satisfied. As a result, the model is non endogenous. In this scenario, we will get unbiased and consistent OLS estimate of which is represented as .

This OLS estimate is BLUE (best linear unbiased estimator)

2)

When x and u are correlated, the CLRM property of non endogeneity or exogeneity is violated. As a result, the model is potentially endogeneous. Under this, if we run OLS on the model, the OLS estimator will be biased and inconsistent. This means this estimator will no longer have the BLUE property.

3)

In order for z to be a good instrument for x, the following properties should hold:

• cov (z,u) =0 , i.e. z and u are uncorrelated
• Cov (z,x) 0 , i.e. there exists sufficient correlation between the endogenous x and the instrument z
• cov (z,y) 0 , i.e. there exists sufficient correlation between the dependent variable y and instrument z
• var (z) exists and is a finite and positive number

4)

The idea of proxy variable is to replace the endogenous variable and use another variable which is closely related to it. However, often it might be the case (endogeneity arising due to omitted variables), the proxy variable itself might suffer from the problem of endogneity because of lack of information or insufficient ways of measuring it. ( For instance, you can't measure ability in a person just like that). Hence cov (proxy, u) might be non zero.

So, should such a case arise, proxy variables should not be used.