Timeline for Prove OLS consistency
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2022 at 18:59 | comment | added | mlofton | Note that, later on, in chapter 2, Hayashi covers the case where an MDS is assumed. So, it's no longer the straightforward case that just assumes that $\epsilon$ is normally distributed. I still have to read that chapter to understand what that means but it's always good to have other people's input also. TEX: Another good book ( for MDS ) that I unfortunately have in storage and not at my finget tips is White's "Asymptotic Theory For Econometricians". That will say something about the MDS cases. It's a terse but clear and good book. | |
| Jul 30, 2022 at 18:52 | comment | added | mlofton | Christoph: TEX is interested in the case when $E(\epsilon_{i}|X)$ doesn't equal zero. As you showed, except for very special cases, it won't be the case that $\beta$ is consistent if $E(\epsilon | X) = 0$ and there is no intercept. OTOH, Hayashi shows ( page 7) that, in the case, where there is an intercept, it will soak up the non-zero mean of the error term ( basically transform it to zero ) so , in that case, one does obtains consistency. But I agree with you that, theoretically speaking, a non-zero mean of the error term, ( and no intercept ) generally implies a lack of consistency. | |
| Jul 30, 2022 at 18:18 | history | edited | Christoph Hanck | CC BY-SA 4.0 | added 455 characters in body |
| Jul 30, 2022 at 18:18 | comment | added | Christoph Hanck | I did surely not mean to claim that inconsistency then is impossible (see also my point on $E(X_i)=0$), just that orthogonality then no longer implies that $E(U_i)=0$, and when the latter is not the case, whether or not we have consistency depends on other features of the DGP - which is also what you illustrate, afaics (I do not read MATLAB too well unfortunately - maybe you can write down the DGP?). I am sorry if that was not your question. I further added my answer to clarify in this regard. | |
| Jul 30, 2022 at 16:30 | comment | added | Star | I have added to my question a Matlab simulation where $E(X_iU_i)=0$, $E(U_i)\neq 0$, $cov(X_i, U_i)\neq 0$. Yet, we have consistency. | |
| Jul 30, 2022 at 16:25 | comment | added | Christoph Hanck | But you are right that edit was necessary | |
| Jul 30, 2022 at 16:25 | history | edited | Christoph Hanck | CC BY-SA 4.0 | deleted 24 characters in body |
| Jul 30, 2022 at 16:13 | comment | added | Christoph Hanck | Maybe I misunderstand your question. My point indeed precisely is that it is not clear how to get $E(X_iU_i)=0$ when $E(U_i)\neq0$. So clearly, orthogonality is sufficient for consistency, but my example shows that such orthogonality does not generally obtain for $E(U_i)\neq0$, so that your claim that we do not need $E(U_i)=0$ when there is no constant requires care. | |
| Jul 30, 2022 at 15:56 | comment | added | Star | I am not sure the example in your answer makes sense as you are simulating a DGP where $E(U_i X_i)$ is different from zero. You are just showing that if $E(U_i X_i)\neq 0$ we have inconsistency. I don't understand what you want to prove. | |
| Jul 30, 2022 at 15:32 | comment | added | Christoph Hanck | It is just that when $X_i$ does contain a constant (say, $X_i=(1, D_i)$), then (we are then in a multidimensional case!) $E(X_iU_i)=0$ also implies $E(1U_i)=E(U_i)=0$. Also, $E(X_iU_i)=0$ then implies $E(D_iU_i)=0$ and then also $Cov(D_i,U_i)=E(D_iU_i)-E(D_i)E(U_i)=0-0E(D_i)=0$. | |
| Jul 30, 2022 at 15:32 | comment | added | Christoph Hanck | My above answer indeed only adresses a), arguing that a nonzero-mean error term and a regression without constant generally do not produce consistency, as it will violate $E(X_iU_i)=0$. As to b), your derivation shows that consistency obtains only if $E(X_iU_i)=0$. There does not seem to be anything special to show in the multidimensional case. | |
| Jul 30, 2022 at 15:11 | comment | added | Star | Thanks. Can you conclude your answer by addressing my (a) and (b) questions? Thanks | |
| Jul 30, 2022 at 15:09 | history | answered | Christoph Hanck | CC BY-SA 4.0 |