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Given four time series $W_i$, $X_i$, $Y_i$ and $Z_i$ for $i=1,2,3,...,T$ observations we define:

$$\bar W:= \frac{1}{T}\sum_{i=1}^T w_i, \quad \bar X:= \frac{1}{T}\sum_{i=1}^T x_i, \quad \bar Y:= \frac{1}{T}\sum_{i=1}^T y_i, \quad \bar Z:= \frac{1}{T}\sum_{i=1}^T z_i$$

$$\begin{array}{rcl} m_n^W &:= & \frac{1}{T} \sum_{i=1}^T (w_i-\bar W)^n \\ m_n^X &:= & \frac{1}{T} \sum_{i=1}^T (x_i-\bar X)^n \end{array} \quad\quad \quad\begin{array}{rcl} m_n^Y &:= & \frac{1}{T} \sum_{i=1}^T (y_i-\bar Y)^n \\ m_n^Z & := &\frac{1}{T} \sum_{i=1}^T (z_i-\bar Z)^n \end{array}$$

From standard statistics we know that for the variable $X$ the following holds:

$$M_2^X=\frac{1}{T-1} \sum_{i=1}^T (x_i-\bar X)^2 = \frac{T}{T-1} m_2^X$$ is an unbiased estimator for $\mathbb{E}[(X-\mathbb{E}(X))^2]$,

$$M_3^X=\frac{T}{(T-1)(T-2)} \sum_{i=1}^T (x_i-\bar X)^3 = \frac{T^2}{(T-1)(T-2)} m_3^X$$ is an unbiased estimator for $\mathbb{E}[(X-\mathbb{E}(X))^3]$,

$$M_4^X = \frac{T^2}{(T-2)(T-3)}\bigg(\frac{T+1}{T-1}m_4^X - 3(m_2^X)^2 \bigg) $$ is an unbiased estimator for $\mathbb{E}[(X-\mathbb{E}(X))^4]$.

Question: Now what would be an unbiased estimator for the central comoments

$$\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y))(Z-\mathbb{E}(Z))]$$

and especially for

$$\mathbb{E}[(W-\mathbb{E}(W))(X-\mathbb{E}(X))(Y-\mathbb{E}(Y))(Z-\mathbb{E}(Z))] \,?$$

I can't imagine the answer is just

$$M_4^{WXYZ}=\frac{T}{(T-2)(T-3)} \sum_{i=1}^T (w_i-\bar W)(x_i-\bar X)(y_i-\bar Y)(z_i-\bar Z) \,?$$

EDIT:

The question has to do with the estimation of financial return moments. In finance you normally assume IID data and you calculate the sample Covariance matrix in the following way:$$\sigma_{ij} =\frac{1}{T-1} \sum_{i=1}^T (x_i-\bar X)(y_i-\bar Y)$$ for all $i,j$. As i know it is not necessary to assume a specific distribution.

Have a look at the Paper of Ledoit and Wolf.They do only assume IID Data: http://www.ledoit.net/ole2.pdf. (Notice that Ledoit and Wolf define the sample estimator differently.)

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  • $\begingroup$ What you say we "know" is not generally true. It seems you are implicitly assuming (1) all the terms in each time series have the same mean and variance and (2) the terms are all uncorrelated. One rarely uses "time series" to refer to such situations, which causes one to wonder whether you have actually formulated the problem you want to solve: is this question about time series or is it about moment estimators? $\endgroup$ – whuber Mar 21 '17 at 14:07
  • $\begingroup$ I've edited the question. Hope its clear now. $\endgroup$ – mwater Mar 21 '17 at 22:54

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