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Let's say $y=c+ax+by+error$ (where the error term fulfills all the assumptions) descibes reality. If we have $z=vx$ than $y=h+max+s$ will also describe reality. When reading about OVB I have seen statemets like "if the true modell is $y=c+ax+by+error$ and x and z are correlated than regressing only on x will overestimate(or underestimate) the true coefficient of 𝑥.” But this seems to be trying to establish causality? If we are only talking about descriptive models than there could be many different correct coefficients for 𝑥 depending on the variables included in the model couldn't there?

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This is not about causality. Causality is a statement that $x$ causes $y$ but it is not the value of the coefficient ($a$ in your model). Even if you care just about coefficient estimates you will get wrong answer. This is because the formula for estimate of the coefficient $a$, that is $\hat{a}$, in simple OLS can be expressed as:

$$\hat{a} = a + \beta \frac{\text{COV}(x,z) }{VAR(x)} $$

That is the estimate of $a$ you get from OLS is the true estimate $a$ + the true value of $\beta$ which would be the coefficient of $z$ if you would actually include it in a model by estimating $y=c+ax+\beta z + error$ multiplied by the covariance of $x$ and $z$ and divided by variance of $x$. Now variance of $x$ must be non-zero just to be able to estimate OLS in the first place and variance is always positive so the over or underestimation will be given by the sign of the covariance and the $\beta$.

Now whether you care about causality or no this bias is still important consider the following example. You want to estimate the effect of education of parents on number of children they have. You dont care about causality you just want to calculate whats the coefficient inside your sample i.e. you only care about whats the average number of children for a couple with average years of education, and in addition suppose that education is also correlated with income. Also to make calculations easy lets assume that VAR$(x)=0.5$ and COV$(x,z)=0.5$ and that average years of education of parents are 10 years. Now lets suppose the true model looks like this (I already replaced $c,a,\beta$ with actual numbers):

$$y_i= 10 -0.3x_i-0.2z_i + u_i$$

In this true model a family with average education of 10 would have on average 3 less kids compared to family with no education. Note I am not saying anything about causation this is just purely caring about descriptively describing the household with average education.

However, note what happens if we now omit $z$ from the regression model and calculate $a$ using the formula above:

$$y_i= 5 - 0.42x_i+ u_i$$

now the estimate suggest that family with average education has 4 less kids which is plain wrong - this estimate underestimates the number of kids for average family by $25\%$. In this case the sign stayed the same but thats just due to the numbers I chose in real life bias could easily change the sign of the coefficient altogether.

Hence, even if you dont care about causality and you just want to use regression for some description of dataset you care about omitted variable bias.

PS: Also in my answer I left out $by$ from your example: $ y=c+ax+by+error$ since there must be some typo you cant regress $y$ on itself, yet your formula includes $y$ on both sides. Or to be specific you could do that but mathematically coefficient is always set to $1$ when you regress dep variable on itself so there is never a reason to do that.

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  • $\begingroup$ But in the first model doesn’t the coefficient tell you the effect additional education holding income constant? So the first model would state that a family with an average education would have on average 3 children less than a family with no education and equal income. The second model would say that a family with average education has on average 4.2 children less than a family with no education. I don’t see how these statements contradict each other. $\endgroup$ – robot112 Dec 28 '19 at 12:22
  • $\begingroup$ @robot112 well since you are using regression analysis instead just simple means comparison I took it as implied that you want to hold other things constant. Your question was about regression where you don’t care about causality not where you don’t care about conditioning the results. This being said what you say is correct although there are additional problems if you don’t care only about averages but about knowing the actual slope. In that case the smaller regression will be still wrong. Note causality is concept not directly related to the bias of the coefficient. $\endgroup$ – 1muflon1 Dec 28 '19 at 12:48
  • $\begingroup$ What I mean with that last sentence is that just because there is no bias it does not imply relationship is causal and vice versa. For example, you can take rainfall as your dep. var (y) and regress it on crop yields (x) in a given year and I can imagine you can get an unbiased and precise estimate of this relationship which will even give you good and precise forecast of rainfall, but of course we know that that yields dont cause rainfall. If there would be some other variable correlated with yield and explaining rainfall like geography then the coefficient estimate would be biased $\endgroup$ – 1muflon1 Dec 28 '19 at 13:00
  • $\begingroup$ meaning it would give you bad predictions and would be for all practical purposes useless as you could just answer the question of whats the average rainfall given average yield just by comparing means, but again it would not be causal. (Also sorry for rather long comments I just wanted to clarify this) $\endgroup$ – 1muflon1 Dec 28 '19 at 13:01
  • $\begingroup$ What I meant by trying to infer causality is that you talk about the “true” parameter b. To me it seems there is only one true parameter in a causal sense if we are only being descriptive there could be many true parameters depending on the variable included. So my question remains where is the bias in the second equation? It seems to me to describe reality, the only difference is income is not held constant. $\endgroup$ – robot112 Dec 28 '19 at 13:21

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