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In this paper, the authors defined uncorrelatedness in the following way:

Let $\mathbf{X}=(X_1,...,X_n)$, and $\mathbf{Y}=(Y_1,...,Y_n)$, where $X_i\sim X$ and $Y_i\sim Y$.

$\mathbf{X},\mathbf{Y}$ are uncorrelated iff $(\mathbf{X}-\bar{X}\mathbf{1})'(\mathbf{Y}-\bar{Y}\mathbf{1})=0$, where $\bar{X}=1/n \sum X_i$ and $\mathbf{1}=(1,...,1)$.

My question is what is the link between this definition of uncorrelatedness and the one usually used (I think), that of (for vectors r.v.) covariance matrix - a matrix where each entry $(i,j)$ is $\text{Corr}(X_i,X_j)$ - being an identity matrix?

Any help would be appreciated.

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  • $\begingroup$ Looks like the thing they're computing is a multiple of a sample correlation. If those terms are zero, the off-diagonal elements of the sample correlation matrix will be 0. $\endgroup$ – Glen_b Aug 15 '14 at 21:21
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The authors in the specific paper do not treat the case of statistical correlation and non-correlation, which is only defined in terms of expected values. They are interested in vectors of realizations of two random variables, and the conditions they define are with respect to specific vectors, not the r.v.'s in general. And they explicitly say so in the Introduction of the paper.

Their interest lies in providing geometric intuition and conceptual separation about the three concepts of linear dependence, orthogonality and uncorrelatedness, in a linear algebra context, not a statistical one. So their definitions are suited to this view of these concepts, rather than the statistical ones.

Note how much more strict is this condition for uncorrelatedness of vectors of numbers, compared to the conditions for statistical uncorrelatedness: the later requires only that the condition $(\mathbf{X}-\bar{X}\mathbf{1})'(\mathbf{Y}-\bar{Y}\mathbf{1})=0$ holds "on average"

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  • $\begingroup$ Thanks for your interest in this question Alecos. One doubt: Could you elaborate a bit further on the last part of your answer? How does the statistical definition, $E[(\mathbf{X}-E(\mathbf{X}))(\mathbf{Y}-E(\mathbf{Y}))']= I_n$ relate to $E[(\mathbf{X}-E(\mathbf{X}))'(\mathbf{Y}-E(\mathbf{Y}))]=0$? And what is the link with orthogonality in the econometric sense? I hope you don't mind. (Your profile said you're doing a macro phd, so I have to take advantage. :) ) $\endgroup$ – An old man in the sea. Aug 15 '14 at 18:21
  • $\begingroup$ I have the impression that you are confusing $E[(\mathbf{X}-E(\mathbf{X}))(\mathbf{Y}-E(\mathbf{Y}))']$ with $E[(\mathbf{X}-E(\mathbf{X}))(\mathbf{X}-E(\mathbf{X}))']$. Moreover this (last) expression gives covariance of the elements of vector $\mathbf X$, not correlation, so it is not equal to the identity matrix -what we want for non-correlation is that it is diagonal. $\endgroup$ – Alecos Papadopoulos Aug 15 '14 at 18:36
  • $\begingroup$ Yes, you're right, my mistake... The papers def. is the usual def of sample covariance = 0 (without the constant 1/(n-1)). But still, how come in many econometric texts it seems as if they use the definition of orthogonality as interchangeable with uncorrelatedness? (maybe I'm just confusing things... I I have already asked something similar before, but I didn't understand some of the comments, nor did I get an answer. stats.stackexchange.com/questions/111391/… ) $\endgroup$ – An old man in the sea. Aug 15 '14 at 18:56
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    $\begingroup$ I posted an answer there. $\endgroup$ – Alecos Papadopoulos Aug 15 '14 at 19:23

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