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  1. #1
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    Conditional expectation




    Hi everyone,
    I've usually seen the conditional mean of X_n given X_1,X_2,.X_n-1 expressed as shown on p. 5 of:
    http://www.math.uiuc.edu/~r-ash/Stat/StatLec21-25.pdf
    or in Appendix B of Greene's Econometrics textbook.

    In a Journal of Finance article, I've recently come across this alternative way of expressing the conditional mean:

    E(X_n|X_1,X_2) = E(X_n) + {Cov[X_n - E(X_n), X_2|X_1]/Var(X_2|X_1)}*[X_2 - E(X_2|X_1)]

    All variables are normal.

    Could anyone help me understand how they got there?

    Many thanks in advance.

    Chiara

  2. #2
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    Re: Conditional expectation

    First of all you need the standard result of conditional distribution:

    http://en.wikipedia.org/wiki/Multiva..._distributions

    Intuition: It will be easy to see that

    E[X_n|X_2 = x_2] = E[X_n] + \frac {Cov[X_n, X_2]} {Var[X_2]} (x_2 - E[X_2])

    which is a standard result seen in regression.

    Since this results holds \forall x_2 \in \mathbb{R}, and thus

    E[X_n|X_2] = E[X_n] + \frac {Cov[X_n, X_2]} {Var[X_2]} (X_2 - E[X_2])

    and it will be tempting to put the conditional X_1 inside to reach the conclusion. To verify this we can do the following calculation.

    To shorten the notation, first we write

    \begin{bmatrix} X_1 \\ X_2 \\ X_n \end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_n \end{bmatrix}, \begin{bmatrix} \sigma_1^2 & \sigma_{12} & \sigma_{1n} \\ \sigma_{12} & \sigma_2^2 & \sigma_{2n} \\ \sigma_{1n} & \sigma_{2n} & \sigma_n^2 \end{bmatrix} \right)

    Then following the formula,

    E[X_n|X_1 = x_1, X_2 = x_2]

    = \mu_n + \begin{bmatrix} \sigma_{1n} & \sigma_{2n} \end{bmatrix}\begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{bmatrix}^{-1}\begin{bmatrix} x_1 - \mu_1 \\ x_2 - \mu_2 \end{bmatrix}

    = \mu_n + \begin{bmatrix} \sigma_{1n} & \sigma_{2n} \end{bmatrix}\frac {1} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2}\begin{bmatrix} \sigma_2^2 & -\sigma_{12} \\ -\sigma_{12} & \sigma_1^2 \end{bmatrix}\begin{bmatrix} x_1 - \mu_1 \\ x_2 - \mu_2 \end{bmatrix}

    = \mu_n + \frac {(\sigma_{1n}\sigma_2^2 - \sigma_{2n}\sigma_{12})(x_1 - \mu_1) + (\sigma_{2n}\sigma_1^2 - \sigma_{1n}\sigma_{12})(x_2 - \mu_2)} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2}


    On the other hand, the covariance matrix of X_2, X_n|X_1 is

    \begin{bmatrix} \sigma_2^2 & \sigma_{2n} \\ \sigma_{2n} & \sigma_n^2 \end{bmatrix} - \begin{bmatrix} \sigma_{12} \\ \sigma_{1n} \end{bmatrix} \begin{bmatrix} \sigma_1^2 \end{bmatrix}^{-1} \begin{bmatrix} \sigma_{12} & \sigma_{1n} \end{bmatrix}

    = \begin{bmatrix} \displaystyle \sigma_2^2 - \frac {\sigma_{12}^2} {\sigma_1^2} & \displaystyle \sigma_{2n} - \frac {\sigma_{12}\sigma_{1n}} {\sigma_1^2} \\ \displaystyle \sigma_{2n} - \frac {\sigma_{1n}\sigma_{12}} {\sigma_1^2} & \displaystyle \sigma_n^2 - \frac {\sigma_{1n}^2}{\sigma_1^2} \end{bmatrix}

    which means that

    Cov[X_2, X_n|X_1] = \sigma_{2n} - \frac {\sigma_{12}\sigma_{1n}} {\sigma_1^2}

    Now consider

    \frac {Cov[X_n - \mu_n, X_2|X_1]} {Var[X_2|X_1]} \times (X_2 - E[X_2|X_1])

    = \frac {\displaystyle \sigma_{2n} - \frac {\sigma_{12}\sigma_{1n}} {\sigma_1^2}}{\displaystyle \sigma_2^2 - \frac {\sigma_{12}^2} {\sigma_1^2}}  \times \left\{X_2 - \left[\mu_2 + \frac {\sigma_{12}} {\sigma_1^2}(X_1 - \mu_1) \right]\right\}

    = \frac {\sigma_{2n}\sigma_1^2 - \sigma_{12}\sigma_{1n}} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2} \times \left[(X_2 -  \mu_2) - \frac {\sigma_{12}} {\sigma_1^2}(X_1 - \mu_1) \right]

    = \frac {\displaystyle \left(\frac {\sigma_{12}^2\sigma_{1n}} {\sigma_1^2} - \sigma_{2n}\sigma_{12}\right)(X_1 - \mu_1) + (\sigma_{2n}\sigma_1^2 - \sigma_{12}\sigma_{1n})(X_2 - \mu_2)} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2}

    Lastly,

    E[X_n|X_1] = \mu_n + \frac {\sigma_{1n}} {\sigma_1^2} (X_1 - \mu_1)

    Combining together,

    E[X_n|X_1] + \frac {Cov[X_n - \mu_n, X_2|X_1]} {Var[X_2|X_1]} \times (X_2 - E[X_2|X_1])

    = \mu_n + \frac {(\sigma_{1n}\sigma_2^2 - \sigma_{2n}\sigma_{12})(X_1 - \mu_1) + (\sigma_{2n}\sigma_1^2 - \sigma_{12}\sigma_{1n})(X_2 - \mu_2)} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2}

    which is the same as the expression calculated above.

  3. The Following User Says Thank You to BGM For This Useful Post:

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  4. #3
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    Re: Conditional expectation

    BGM,
    thank you so much. This is incredibly helpful.
    Is this result also valid for the case in which X2 is a single variable, while X1 is a subset of two (or more) variables?

    Thanks again.

    Chiara

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    Re: Conditional expectation

    The answer is Yes. The above calculation is a little bit tedious, but it help you to verify the simplest case.

    The reason is that we can regard the conditional probability measure as another new measure; and the good thing is that the conditional joint distribution of

    X_2, X_n|\mathcal{F}(\mathbf{X})

    is still bivariate normal. So you can still apply the formula for the bivariate normal, but in a conditional fashion.

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    Re: Conditional expectation

    BGM,
    please see attached.

    Thank you.

    Chiara
    Attached Images

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    Re: Conditional expectation


    It is even more tedious when you include one more term inside, so I have not check the detail , but the overall form should be alright.

    The reason why I have not mentioned the measure stuff in the very first place, is to let you do some elementary calculation fundamentally and get the intuition behind. I know that one may not be so easy to understand the stuff.

    The key thing is that, you just regard the conditional joint pdf of X_2, X_n|X_1 = x_1

    as the joint pdf of a pair of new variable X_2^*, X_n^*

    i.e. define

    f_{X_2^*, X_n^*}(x_2, x_n) = f_{X_2, X_n|X_1 = x_1}(x_2, x_n|x_1)

    (the x_1 somehow just act as a parameter)

    The induced pair of random variables X_2^*, X_n^* will jointly follow the distribution induced by this joint pdf. And we know that this is a bivariate normal which share the same properties from other bivariate normal as well.

    That's why we can do the generalized stuff, as the multivariate normal is closed under the conditional operation

    http://lapmal.epfl.ch/files/content/...dout_lect3.pdf

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