# Does homoscedasticity imply that the regressor variables and the errors are uncorrelated?

By OLS regression equation:

$$Y = a + bX + e$$

My thoughts are that homoscedasticity by definition imply that $$Var(Y|X) = Var(e|X)=$$ constant, then this would imply that $$Var(e|X) = Var(e)$$ which implies that $$e$$ and $$X$$ are uncorrelated.

My question basically arises from different authors separately calling out the two assumptions in OLS:

1. Homoscedastic
2. Independent variates are uncorrelated with the errors

Edit :

I was referring to the true model and not the fitted model. The assumptions listed in most texts (in relation to the errors are:

1. Mean of structural error = 0, and constant variance
2. Errors are uncorrelated.
3. Independent variate's are uncorrelated with the errors .
4. $$V[Y|X=x]$$ = constant (homoscedasticity)
5. Normality of errors (Required to establish likelihood derivation of OLS and confidence intervals)

Therefore by construction, homoscedasticity and constant variance assumption of errors are same and hence the statement constant variance and homoscedasticity are redundant ?

• Aside: there are more assumptions than those two in OLS. (E.g., linearity, no omitted variables, normality of the error distribution). Apr 26, 2020 at 16:50
• @Alexis no omitted variables is included in assumption 2; normality of the error distribution is only optional. Apr 27, 2020 at 8:49
• @markowitz No omitted variables is not strictly contained by homoscedasticity, and normality is not optional for valid inference in small sample sizes. Apr 27, 2020 at 15:49
• About no omitted variables: I referred on assumption 2 not 1 (rorschach300 numeration). About normality: now you are right but the question posed is general and not precisely referred on small samples. Apr 27, 2020 at 17:17
• About normality assumption detailed answer is given here stats.stackexchange.com/questions/148803/… Apr 27, 2020 at 22:09

First of all there is a crucial point that we have to clarify. In many case also econometrics books are ambiguous about that and probably you refers on. The main assumption requested is like $$E[\epsilon|X] = 0$$ named exogenous condition. The concept of exogeneity is related to causal inference, large part of econometrics problems are referred on it. Unfortunately causal inference in econometrics are badly treated, as pointed out in: Chen and Pearl (2013). Many relevant problems are revealed by this article but in my opinion some others are not adequately addressed. Those are mostly related to the concept of true model. This concept is very used in econometrics literature but, almost never enough words are spent about it in the books. Often almost nothing is say about it.

In particular the crucial point is: the true model is write down like a regression, in most case linear, but it is not a regression (linear or not). True model is something else. In my opinion the best way to think about it is as a structural causal model, in most case linear structural causal model.

These discussions are related

Regression and causality in econometrics

Difference Between Simultaneous Equation Model and Structural Equation Model

What is a 'true' model?

Now, you unambiguously talk about OLS regression equation like:

$$Y = \alpha + X’ \beta + \epsilon$$

Note that the term $$\epsilon$$ is a residual. Then in this situation your assumption 2 are useless because the uncorrelatedness requested holds, in any case, by construction not by assumption. Often this uncorrelatedness is erroneously conflated with a sort of weak form of exogeneity.

Statistically speaking, mean independence, also holds by construction if the exact conditional expectation function is linear. If all the variables involved (dependent and independent) have jointly Normal distribution, also stochastic independence holds by construction. However, disentangled from improper conflations, nothing of the above say something about exogeneity in its proper meaning. Therefore, in any case, even if homoscedasticity can be an assumption yet, your question is senseless.

Now, if we consider the same equation as before like a true model we have to note that the term $$\epsilon$$ is no more a residual but a structural error term; completely another thing. Here your question become well posed. So, from homoscedasticity we have that:

$$V[Y|X]= V [ \alpha + X’ \beta + \epsilon |X] = V [ \epsilon |X] = \sigma^2$$ and

$$V [ \epsilon |X] = E[\epsilon^2|X] – (E[\epsilon|X])^2 = \sigma^2$$

Actually this imply that $$E[\epsilon|X]=c$$, however we cannot sure that this constant term is $$c=0$$. Infact we have to note that, different than in regression, nothing is true by construction about $$\epsilon$$.

Finally if we assume homoscedasticity and $$E[\epsilon]=0$$ also, after all a natural features for any kind of error term, the reply to your question is yes. Uncorrelatedness between dependent variables and error term is implied; mean independence also. In other words, exogeneity is implied.

Actually this seems me an interesting result and I don't know why no author underscored it. However seem me that authors have greater problems to solve.

Moreover can be useful to say that in regression analysis heteroscedasticity can reveal misspecification problems. Therefore homoscedasticity is a good thing.

• Thank you. I should have made it more clear in the question that I was referring to the true model case and not the fitted model. The assumptions listed in most texts i 1. Mean of structural error = 0, and constant variance 2. Errors are uncorrelated. 3. Independent variate's are uncorrelated with the errors . Apr 27, 2020 at 23:22
• About your confusion: if you are student, do not have to say sorry. Several authors are not clear about the exogeneity point during them entire presentation. If you textbook speak about error term as structural and clearly distinguish it from residual, please let me know what it is. Apr 28, 2020 at 7:24
• About your EDIT: I suppose that you in assumption 4 speak about $V[Y|X=x]=\sigma^2$ and not $E[]$. About the revised question: In your new setting, under homoscedasticity, assumption 3 is redundant and not only the second part of 1. Exogeneity, is the core of explanation. At the end, if you are satisfied of my reply and comments, you can “up” the reply and accept it. Apr 28, 2020 at 7:31