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- United-kingdom
- University of Bristol
- Economics
- Economics 20011
- Gregory Jolivet
- Ch12: Limitations Of Ols

Harry W.

• 7

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Limitations of OLS estimations

We challenge an empirical approach on two grounds:

- Internal validity: Is the inference valid for the population being studied? Are the assumptions used for our analysis correct for the population of interest.
- External validity: Can we generalise our conclusions to other populations and setting than those we used in our estimation.

Internal Validity

There are essentially two types of threats to internal validity:

First, the estimator of the causal effects of interest may be biased or not consistent.

- This is the case when the exogeneity assumption does not hold.

Secondly, the estimated standard errors may be inconsistent.

- In this case, we cannot conduct tests on the parameters of interest.
- This may happen if errors have heterogeneous variance (heteroskedasticity) or is they are correlated (sample not iid).

Sources of Bias/Inconsistency of OLS

The problems we are about to discuss stand in the way of a causal interpretation of OLS:

- We can no longer say that the OLS estimate is the causal effect of a regressor on Y.
- However, OLS still has a descriptive purpose, it captures the correlations between variables.
- The challenge is to go from these correlations to a causal analysis.

Omitted Variable Bias

Consider the linear model Y = b_{0} + b_{1}X_{1} + U and assume that there is another variable X_{2} which has a direct effect on Y and is also correlated with X_{1}:

X_{2} = ρX_{1} + V, where E(V|X_{1}) = 0.

This variable lies in the error term:

U = b_{2}X_{2} + W, where E(W|X_{2}) = 0.

We can then rewrite the linear model as:

Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + W = b_{0} + b_{1}X_{1} + b_{2}(ρX_{1} + V) + W

= b_{0} + X_{1}(b_{1} + ρb_{2}) + (b_{2}V + W)

OLS will then not produce an estimate of b_{1} but of b_{1} + ρb_{2}. There is a bias.

Measurement Error

Assume: Y = b_{0} + b_{1}X_{1} + u.

Classical measurement error is where the measurement error is random and uncorrelated with any of the model's variables.

If a regressor is measure with error, OLS may be biased and inconsistent:

- The data provide X
^{~}_{1}which is an imprecise measure of X_{1}. Then

Y = b_{0}+ b_{1}X^{~}_{1}+ V, where V = b_{1}(X_{1}- X^{~}_{1}) + U. - If the error term V is correlated with X
^{~}_{1}then OLS will be biased.

Selection Bias

Selection Bias occurs when some data is missing as a result of a selection process that depends on Y, even after controlling for the regressors.

Simultaneity

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