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Linear Regression with Multiple Regressors

5 important questions on Linear Regression with Multiple Regressors

What is the problem with omitted variables?

Omitted factors, such as student characteristics, can in fact make the ordinary least squares (OLS) estimator of an effect misleading, or even biased. If we have data on them we should include them in a multiple regression. Via that way we can measure the "clean" effect by holding the other factors constant.

Estimators are random variables because they depend on data from a random sample; and in large samples the sampling distributions of the OLS estimators are approximately normal.

Omitted variables bias?

Ignoring some important determinants we get misleading estimators; i.e. not the true effect of a unit change (as something else changes as well). THereby we come to the definition of omitted variables: if the regressor is correlated with a variable that has been omitted from the analysis and that determines in part the dependent variable.

1) Correlated
2) Determinant (direct effect)

Both should hold to make the error term conditionally correlated to the regressor. Meaning that the first OLS assumption is violated.

What about the multiple regression model?

Isolate a certain effect (partial effect), works same as single regression. Think of B0 as a variable as well, a constant regressor with value 1. Dummy's can be control variables. Homo/heteroskedastic is similar but extended to all variables.

Error term remains existent!

What assumptions for OLS?

- Conditional distribution of error given X has a mean of zero
- i.i.d
- large outliers are unlikely (nonzero finite kurtosis, fourth moment)
- no perfect multicollinearity (same variable/effect)

What about imperfect multicollinearity?

Highly correlated, no problem to OLS but to precision of estimates. Imagine that it is hard to keep other things cosntant when it comoves a lot. BAsically there is little imformation to separate a partial effect. BAsically the variance of such an estimator increases. THink of variance of Beta1 = 1/n * (1/(1- corr(x1,x2)) * (variance error /variance x).

The question on the page originate from the summary of the following study material:

Part 2: Quantitative Analysis

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