I have a question regarding the omitted variable bias. Its properties of it:
- the omitted variable is a determinant of the dependent variable,$y$
- $X$ is correlated with the omitted variable.
I already know that it satisfies the first property. But I have a problem with the second. In the data, I found the correlation coefficient is between $-1$ and $+1$. Meaning if the coefficient is $-1$ it will be a negative correlation and if it's $+1$ it's a positive correlation if it's zero there will be no correlation at all. So basically all of the numbers I found are either negatively or positively correlated. For example, I found the number $-0.24$. this indicates that there is a negative correlation between two of the variable (student-teacher ratio, Average reading score).
When it's a negative correlation. does this mean that it satisfies the second property? since there is a correlation but it's negative.
Most of the numbers are like $0.02$ or $0.21$. It's close to zero but it is not zero. Does that mean that the second condition is satisfied?
It's very unlikely that the number will be exactly $0$, as you're talking about an estimate of the true correlation. Any nonzero value of the true correlation means the second condition is satisfied. So if the estimated correlation/covariance is significantly different from $0$, you can probably assume there is an omitted variable bias.
You should also use what you know about the data. If you would intuitively expect one variable to affect the other, you need to deal with omitted variable bias, even if the data you have is inconclusive on whether a correlation actually exists.
Whether $0.21$ is significant or not will depend on your sample size, but I think it probably will be.