1. Describe 3 sources of bias/inconsistency in OLS estimators. Be sure to descri

Question

1. Describe 3 sources of bias/inconsistency in OLS estimators. Be sure to describe how the bias arises (i.e. how endogeneity arises) 2. For each source of bias, characterize the direction of bias with a table of possible cases or a simple description 3. For each of the 3 sources of bias, name at least one method that may help reduce or eliminate bias Consider the following equation : Yit = 0 + 1xit + Awit t (1) 4. List/describe the assumptions required for the following methods to give con- sistent estimators for , assuming wi Is unobserved: OLS, FE, and FD 5. What is the difference between FD and FE methods? 6. Describe how you could estimate the FE model without differencing? 7. Suppose t is exogenous and w; is endogenous. What does that mean? 8. Suppose you have an extraneous variable z). What assumptions must be met for z, to satisfy the conditions of a valid instrumental variable for the endogenous 9. Assuming z is a valid instrument, describe the 2SLS procedure which may be 10. Suppose ui is unobserved. Under what conditions will omitting u, cause bias 11. Suppose omitting w, does cause bias. Under what conditions would the extra- used to obtain consistent estimators for 1 and 2 in the OLS estimator of ? neous variable (z) satisfy the assumptions of a valid proxy variable? Given that it satisfies these assptions, how does one use the proxy to obtain a consistent estimator for 12. Suppose omitting wi doesn't cause bias, but Tit is measured with error which follows the classical errors-in-variables (CEV) assumptions. What are those assumptions and what is the consequence?

Explanation / Answer

1. OLS Estimators may be biased/inconsistent in the following cases:-

a) Underfitting or ommission of relevant variable

Let us consider the following model, Yt = B1 + B2X2i + B3X3i +ui

Instead of estimating the model as the above, we incorrectly specify as:

Yt = A1 + A2X2t + vt

Here variable X3i has been ommited which was relevant to the model. This can often occur when the appropriate theory is not applied to the model. In this case, A1 and A2 are biased, that is, their average values do not coincide with their true values and they are also inconsistent. The nature of this bias can be downward or upward.

b) Assumption of Cov(Xi, ui) = 0 does not hold

The OLS estimators are biased and inconsistent. This problem is termed as endogenity. This occurs when E(Xi, ui) is not equal to 0. Examples of this would occur in cases of ommitted variables, measurement errors and simultaneity in simultaneous equation models.

This is considered as follows:

1) Ommitted variables - This phenomenon is linked to cross-sectional data in particular. For example, if we collect data for education, experience for a sample of 800 men in the US to estimate wages, we will find that an important variable of ability has been excluded. This can be resolved by a proxy variable of IQ.

2) Measurement errors - The problem arises when measurement error occurs in the independant variable. For example, Yt = B1 + B2Xi + ui is such that Xi is wrongly measured.

3) Simultaneous equation models - This arises when one or more of the independant variables, Xj's is jointly determined with the determined variable Y, usually through an equilibrium mechanism. This may arise in the context of quantity and price by demand and supply.

c) Attenuation Bias

This occurs if we perform regression of Y on X, then the measurement error leads to a biased OLS estimator towards zero. This is more of a limiting case of the endogenity problem with measurement errors.

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