Method of moments for a correlation

Question

Two mean-zero return series have a joint distribution with unknown correlation $\rho$ . From your data the sample average of the product $X Y$ is $0.5$ , and each series has sample variance $1.0$ (so each sample standard deviation is $1.0$ ).

Use the method of moments to estimate $\rho$ .

Accepted Answer

Which moment identifies ho. Correlation is defined through a mixed moment, the covariance, scaled by the two spreads: ho = \frac{\mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]}{\sqrt{\operatorname{Var}(X)\,\operatorname{Var}(Y)}}. With both means zero this simplifies to ho = \mathbb{E}[XY]/\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}. Match sample to theory. Replace each theoretical moment by its sample counterpart. The sample cross-moment is 0.5, and each sample standard deviation is 1.0: \hatho = \frac{0.5}{\sqrt{1.0 	imes 1.0}} = \frac{0.5}{1.0} = 0.5. Why this is method of mome…

Method of moments for a correlation

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