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Assume that we have T measurements over the last T years of n various variables such as $X_1=$ GDP-growth, $X_2$=growth in private consumtion, ... . My question is whether it makes sense to estimate the Covariance-Matrix $Cov(X_1, ...., X_n)$ by calculating the Covariance-Matrix of the sample. Because is there even a "true" covariance Matrix to be estimated? I mean I dont really see why the values of BIP-growth can be modelled as being generated by a random process, so I would argue that the supposed covariance-matrix that ought to be estimated by the sample does not even exist. Is my argument correct or do I miss something?

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    $\begingroup$ For context, what is BIP-growth? In general, while the real world quantities represented by $X$ may not be generated by a random process, we consider their recorded values to be, with random error being introduced by the measuring instruments and/or techniques. $\endgroup$ – deasmhumnha Jun 11 '18 at 8:57
  • $\begingroup$ i meant GDP growth. BIP is the german word for it. $\endgroup$ – Sebastian Jun 11 '18 at 9:03
  • $\begingroup$ And yes I see that there is measurement error but in order to estimate a Covariance between two variables I guess I need n draws from the SAME population. This does not seem to be the case for the above example right? Every year I draw from a different box so to say. $\endgroup$ – Sebastian Jun 11 '18 at 9:06
  • $\begingroup$ I'd say the covariance matrix always exists in that it can be calculated, but your interpretation may change depending on the data. In this case, the covariance measures the linear association between variables in time, rather than linear association within a population. This is obviously still useful, as it shows which variables are more /less tightly coupled and the directionality of these relationships (though the correlation matrix may be easier to interpret). Ultimately, it might help to think of each year and its associated value(s) as one sample from a population of years. $\endgroup$ – deasmhumnha Jun 11 '18 at 9:32

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