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How can be used Monte Carlo simulation in order to demonstrate that there…
How can be used Monte Carlo simulation in order to demonstrate that there is omitted variable bias?
First we define and configure a population process.
Generate independent variables
X1
Given by some sort of normally distributed random variable
X2
Equal to X1 times some sort of small number plus some sort of normal random error
there are generating a second independent variable which is correlated with the first one
But it's not perfectly correlated because we've got this random error component to it as well.
correlated with X1
We're going to use these two independent variables to generate a set of Y.
Y is going to be equal to alpha plus beta1 times X1 plus beta2 times X2, plus some error (a normal random error as well). So it's not perfectly determined by X1 and X2.
All of these things together define our population and our population processes.
We're going to use these population processes to generate samples.
We're going to generate a number of different samples, so we're going to generate sample one all the way through to a speciffic number of samples.
We're going to use least squares estimators on each of these samples in order to come up with the least squares estimate of beta1.
We're actually going to do is estimate a regression model on each of these samples
Y = alpha + beta1 * X1 + ε
Omitting the second important independent variable, we're missing X2 from this relationship.
Because we omit X2 from our relationship, and because X2 is correlated with X1, we're going to expect some bias.
The true value of beta1 might be something in the middle of the domain.
But because we've omitted this important factor, our sampling distribution is actually going to be skewed (upwards) of beta1.
It's not going to be an unbiased estimator. And it's not going to be unbiased because, essentially, within this error term epsilon here, we're implicitly including X2.
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Roberto Wagner