hello,If I don't make any mistake, empirical covariance matrix is an unbiaised estimator of the real covariance matrix. It is the same for (unbiaised) empirical std deviation for each random variable.My question is: Do we have the same property for the correlation matrix i.e. Is the associated correlation matrix fo sure an unbiaised estimator of the real correlation matrix??thx and have a good day

My take on the problem is that real correlation = future correlation, while empirical = estimated and, if you look at time series, historical.For instance, in bad times, when liquidity drains and markets experience big sell-offs, correlations tend to be higher than those in good times.

thx and I agree with you but it doesn't answer my question.It was:the correlation coefficient estimator calculated from the unbiaised covariance estimator divided by the product of unbiaised standard error estimator is an unbiased estimator of correlation coefficient or not??and if yes, how do we get E(Corr_coeff)=Corr_Coeff mathematicallysee u

The sample covariance is not an unbiased estimate of the true covariance, nor are the sample variance or standard deviation unbiased estimates of the population parameters.For a simple example, suppose we have three equally likely possible outcomes, X = 0, Y = 0; X = 0, Y = 1; and X = 1, Y = 0. That means:E(X) = E(Y) = 1/3E(XY) = 0Cov(X,Y) = E(XY) - E(X)E(Y) = -1/9If I take one observation, my sample covariance will always be zero. You may regard this as a special case, so suppose I take two observations. My sample covariance will be zero unless I get the draw (0,1), (1,0) or the same thing in the reverse order. Those sample covariances will be -1/4. That will happen 2 times in 9, so the expected value of my sample covariance is -1/18 rather than -1/9.

QuoteOriginally posted by: AaronThe sample covariance is not an unbiased estimate of the true covariance, nor are the sample variance or standard deviation unbiased estimates of the population parameters.For a simple example, suppose we have three equally likely possible outcomes, X = 0, Y = 0; X = 0, Y = 1; and X = 1, Y = 0. That means:E(X) = E(Y) = 1/3E(XY) = 0Cov(X,Y) = E(XY) - E(X)E(Y) = -1/9If I take one observation, my sample covariance will always be zero. You may regard this as a special case, so suppose I take two observations. My sample covariance will be zero unless I get the draw (0,1), (1,0) or the same thing in the reverse order. Those sample covariances will be -1/4. That will happen 2 times in 9, so the expected value of my sample covariance is -1/18 rather than -1/9.thanks for your responseok. but with a big enough sample, covariance estimator and unbiaised stdev estimator are unbiaised. (it is why they are named like that?!?!!) so E(estimator)= the true value no???and if yes the sample correlation matrix is a unbiaised estimator of the true correlation matrix?

Hi. It is the case that the sample covariance is an unbiaised estimator of the true covariance, you're right. Regarding the specific Aaron's example, which is a very good one I think, I do not get the same results. For the pair of samples for which the cov is non zero, say (0,1) and (1,0), the cov is -1/2 : for this case1/(2-1)*[(0-1/2)(1-1/2)+(1-1/2)(0-1/2)]=-1/2. This explains the factor 2 he found : The expected value of thi estimator is 0*7/9 + (-1/2)*2/9 = -1/9, as it shouldBy contrast, it is not the case for the correlation matrix. The expected value of the estimator is E[r]=rho-1/(2n)*rho(1-rho^2)with n the number of samples. It is thus asymptotically unbiaised, and the bias is maximum, for a given n, at rho = sqrt(1/3).However, I've observed very poor performances of this estimator on siome testing cases, for which the true value of rho is known, and I cannot explain it....

Although the sample covariance matrix is known to be an unbiased estimator of the true covariance matrix, I have sthg weird.In the case where X is a pxn multivariate normal vector (p being the dimension and n the number of samples), the sample covariance matrix S has Whishart distributionBut the expectation of such a Whishart variable is such that there is a bias, although it vanishes asymptotically.Does anyone have any idea of where is the bug ?

Sample correlation is biased, and Fisher already presented (here) a bias correction procedure. More recent improvements are also available.