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Specify a function for Expectation: Specifically AR(2) Time Series?

Posted 4 months ago
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I'm new to Mathematica so having some issues with functional specifications. Basic help will suffice even if it's not specific to my problem below.

I'm trying to take the expectation of a product of functions and definitely doing it incorrectly. For instance how would I recreate variance such as:

$$ \sigma^2= \mathrm { E } [ X ^ { 2 }] - \mathrm { E } [ X ] ^ { 2 }$$

I'm dealing with a WhiteNoiseProcess with constant variance. I got something relevant with:

In[1]= Expectation[ $ y[t] * y[t] $, y [Distributed] WhiteNoiseProcess [ $\sigma$ ]]

Out[1]= $\sigma^2$

Any help with how to properly input functions would be helpful. But if specifically how to take expectations of their products that'd be great.


My specific problem of interest involves Yule-Walker case: The objective function is $$ y _ { t } = a _ { 1 } y _ { t - 1 } + a _ { 2 } y _ { t - 2 } + \varepsilon _ { t }$$

The assumptions for this AR(2) time series function is the error is white noise with a mean of 0, and constant variance equal to $ \sigma^2 $. The series $ y_t $ is stationary with a constant mean $ \mu $ and variance equal to $ \sigma^2 $. Both of are time invariant.

$$E y _ { t } y _ { t } = a _ { 1 } E y _ { t - 1 } y _ { t } + a _ { 2 } E y _ { t - 2 } y _ { t } + E \varepsilon _ { t } y _ { t }$$

So by Yule-Walker steps I'm trying to multiply this difference equation by $ y_t $ then take its expectation. The only other relevant output I got more specific to my problem is the following:

In[12]:=  Expectation[a[1]* y[t-1] *y[t] + a[2] * y[t-2]*y[t] + \[Epsilon][t]*y[t] , {y \[Distributed] NormalDistribution[\[Mu],\[Sigma]], \[Epsilon] \[Distributed] WhiteNoiseProcess [\[Sigma]]}]
Out[12]= a[2] y[-2+t] y[t]+a[1] y[-1+t] y[t]

Any help is appreciated.

POSTED BY: A.I. S
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