much later
but that's just sort of in and introduction to
why we cannot assume causality of the time
jazz important
so this brings us to you
our I'm visual representation
in we so did this last time but we're going to do it a little bit more
formally
so we have our independent variable X
and are dependent variable y
in we have all this data
put some there on Sat here and then
we run our regression and we get this line that looks something like this:
where this would be are paid 0
the slope would be our paid one
its
and now we can teak some
X here will call it X I and
at this point say there's a the point here
at this point we can
go up to hear and this distance
is are
why I so it's what we actually observe
am it's our our data
our data says that this point as at why I
and XIII I'm but what we
estimate was all the way up here in so this distance from here to here
is actually our I am
R Y II hacked
so it's what we estimated and then the difference between the two
is are you I hot so this is a more formal
I guess a visual representation of what we're actually doing
so these are estimations because we can't actually
no the population but all we can do now is give
a formula for what residuals are now that we have this
so we know that
you II hacked
is this which is the difference between these two things
and so we have this is equal to you
why I minus why
I hacked and then
if we plug in are a a regression for why I had
what we obtain is why I minus
Peter 0 how minus
be dole one at X
I and so this
here is what our
est mean residual is at this point
and now
that we have this formula what we've been leading up to this entire time
is Sourav he in formal derivation
love our ordinary least-squares
estimates need so how can
what we're what i'm saying is how can we estimate this beta 0
in estimate this beta 1 how can we get a good
estimate for both love them am
so what we do is we minimize
the some
love all uv-vis squared in this year 'em just dunno the you i squared
summed up is called are summer squares residuals which is often
denoted as in Rs s or a.m. and s are it really depends on your professor book as well
so make sure you just get your terminology correct and so
when we minimize the summer squared residuals
word essentially trying to get the best line here in a in a regression
and when we do that
a.m. we're not going to go into the folder vation
avi a.m. the coefficients but in
or less we above team he beat her
one hat equal to you the covariance love axe and why all over the variance over tax
so in or less you'll generally be able to find this won't be given any data
cycle be able to find
the X in the covariate the very inspection the covariance that's why
and that is what will give you you pay to one in a simple linear regression
and it's as simple as that in then DB to zero
at once we get that we will be able to obtain
be 20 hat which is equal to the mean value for
minus the
I'm the tall one had
so what we obtained here we can play in right here
and then modified by the mean
Apax so that's how
we find our coefficients in our
estimated regression and I hope you understand
a little bit more on what they are and what they mean
and them I hope this visual representation helped a little bit too
understand but the residuals are and what they mean
and I'm I hope you sort of understand
the the interpretation of the model and what it is
what it takes to be able to assume causality so
I'm if if you remember causality was
we had to assume that the there is nothing in
the in this residual that
a.m. that was correlated with the axe
in then I also hope you understand
the difference between population sample regression so sample is what we can
estimate these
these hats and then the population is what we never observe it's the true
model that we're trying to find the functional form that we
we truly want to find in yeah
I hope you understand a little bit more about the simple linear regression
and our next video will be up soon
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