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Language: en
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In this video I will expand the previous videos
principle to covariance matrixes. So a correlation
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matrix is the special case of covariance matrix
that has been scaled so that the variance of its
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variable is 1. So core correlation matrix is kind
of like a standardized version of a covariance
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matrix. Some features of linear models are
better understood in covariance metric,
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so understanding the same set of rules in
covariance form is useful. Let's take a look at
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the covariance between X 1 and Y. We calculate
the covariance X 1 y exactly the same way as
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we calculated correlation. So we take there are
the unstandardized regression coefficients here,
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so previous will be working with standardized
regression coefficients, these are now on
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standardized because we are working on the raw
metric instead of the correlation matrix. So we
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have X 1 to Y 1 path. We get the beta 1 goes
here. Then another way of X 1 to Y is to our
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travel 1 correlation a covariance X 1 to X 2 so
that's covariance and then ricca-san path. So
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we get that and then our X 1 2 X 3 1 covariance
and then 2 y so that's all there are after we sum
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those together. That gives us the covariance
between X 1 and Y and that's exactly the same
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math than we had in a correlation example but
instead of working with correlations we work
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with covariances. Things get more interesting
when we look at what is the variance of Y. So
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the variance of Y is given by that equation
here. So the idea is that on we go from Y,
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and then we go to each source of variance of Y
and then we come back. So we go from Y to X 1,
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we take the variance of X 1 and then we come back.
So its variance of X 1 times beta 1 squared in the
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correlation matrix we just take beta 1 and beta
1 squared because the variance in correlation
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matrix is one so we just ignore that. When
we go from Y to R to X 1 X 2 and X and beta
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2 then we get that here and we go it both ways.
So why that are this is a useful rule is because
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it allows us to see that the variance of Y is a
sum of all these different sources of variation,
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so we get variation due to X covariance due to X 1
X 2 we get variation due to the error term. So the
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variation of Y is the sum of all these variances
and covariances of the explanatory variables,
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plus the variance of U the error term that is
our uncorrelated all the explanatory variables.
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This covariance form of the model employed
a correlation matrix rule is useful when
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you start working more complicated models,
such as confront a factor analysis model.