Econometrics: Bivariate population model

Hello I'm finally back from my extended break. I thought that we should start studying Econometrics.

Let's begin by analyzing a simple bivariate regression. Assume that this equation describes the relationship between two variables X and Y.

$Y= \beta_0 + \beta_1 \cdot X + \epsilon$

We say that Y is the dependent variable, whereas X is the independent variable. In other words, we assume that Y (the output) depends on X (the input). Epsilon is the error term, it represents other factors that affect Y.

The error term must be uncorrelated with the variable X so that we do not need to include them in our regression, and thus the coefficient of X (beta) should not change, even though Epsilon also determines Y.

beta-0 is the constant term. It tells us what would be Y if X=0.

beta-1 is the effect on Y if X changes by one unit.

To see this, assume X=education is a continuous function, let's take the derivative of Y=wage with respect to X:

$\frac{\partial Wage}{\partial educ }=\beta_1$

Thus, if education goes up by one unit, we should expect, on average, wage to go up by about beta-1.

Note that this principal can be extended to non-continuous functions as well.

To run this regression, all I need is set of (random) data with wage and the education for each participant in the dataset.

reg wage educ

The following line above is what you should use in order to run this bivariate regression in STATA.

We need one more condition in order to make sure that our bivariate model is correct:

$E[\epsilon|X]=E[\epsilon]=0$

This tells us that given X, the other factors are randomly distributed in the population, and thus does not bias the coefficient of X (beta-1).

Reference: Wooldridge, Jeffrey. Introductory Econometrics, 3rd edition. Thomson South-Western (2006). p 24-28.

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