Hi there, in this video I wanted to provide an introduction to the matrix
formulation of econometrics.
Well, what do we mean by the matrix formulation of econometrics? Well,
up until now if I had some multivariate model,
then I would have to describe it 'y i is equal to let's say alpha
plus beta one times x one i
plus beta two times x two i,
so all the way through to beta p times x p i,
and also I have some sort of population error, e i'.
And, the idea here is that the subscript 'i' means that
implicitly I am assuming that this model holds for each individual within the
population.
And notice that this is quite a clunky way of writing things, right? So, I've got
my x one i, x two i... all the way up to x p i.
It would be quite nice if there was a simpler way to write this down,
for the case of a multivariate model,
and it turns out that the matrix formulation of econometrics,
is the way in which we can do this.
And the way we actually do this is we write down explicitly the relationship
between
all of our independent variables and all of our dependent variables.
So, the idea is that I have n observations of my dependent variable, so I have
'y one' through 'y n'.
And the idea is that I can write this
as a matrix equation
in terms of my coefficient vector beta.
So my coefficient vector, is in this case, going to
have to beta one
through to beta p, as its components.
And, I'm finally going to have some sort of
error here. Where the error has the same number of dimensions as my dependent variable.
So, I've got some sort of error 'u',
rather using the same sort of notation.
Error eta one through to eta n.
So the only thing we have to here is what happens in
this matrix here.
Well, the idea is that I can write each of the rows
as being a list of my
variables for that particular observation. So,
starting out with the first row, I can write my first
expression, or my first term as,
'x one one'.
Then my second term in the first row is just going to be x
one two, and I'm just going to continue all the way through to x one p.
So just checking that this adds up to what we had in our original
formulation here,
you can see the if I was to take this row of
independent variables,
and multiply it through by my coefficient vector: beta one through beta p,
then the idea is that I would get something which is x one once times
beta one, plus x one two times beta two,
all the way through to
x one p times beta p
right? And notice that this is exactly the same form with I had up here,
except that I have replaced the subscript i here with the subscript one.
Ok, so that's my first row, what
was the second row? Well, I index my second row by x two one as the first
term,
and then the second term is x two two, and
I continue all the way through to x two p.
Right? And we can check this out by writing our second row
times our
row vector of coefficients, will actually give us the same form as we had above here.
Except
the subscript i will be two.
Do, the idea with writing out components of our matrices,
is that the first subscript of terms determines the row,
and the second subscript determines the column. So, the idea behind our second term here
in that x one two, is the first row - because I have a
number one,
but is the second column for our x variable.
Ok, so I can now fill out the rest of this matrix. If I was to continue down
then our final row will have an index of n,
and the first term will have a column index of one,
and that will then go all the way down to x n p.
So, that's the final row, and the final column, where p is the
maximum number of independent variables in my model,
or in my population process rather,
and the idea is that all terms inbetween would follow similar sorts of
relationships.
So, note that we have
actually written down in matrix form a relationship between the entirety of
our dependent variables,
and the entirety of our independent variables.
And we can actually simplify this by writing down a matrix equation. So we can write
down that y, our vector y,
and the way in which I denote vectors in this
course is that I use a straight line to indicate a vector,
and I use a curly line to represent a matrix.
So, I've got that my vector of dependent variables y, is equal to x
times my coefficient vector beta,
plus my vector of errors u.
And, notice that this is a completely general form
for a linear model,
and it is really going to come into its own when we are looking to explain more complicated
estimators than least squares.
And also for least squares it provides a nice symmetry for the way in which we can write
least squared estimators, but in particular
we're really going to run into trouble if we try to explain more complicated estimators than
least squares,
using this subscript form which we have here. So the idea with
this course is that we are going to be dealing completely in matrices,
and we're actually going to go through and prove all the other relationships which we proved
for the case of the
bivariate model. Since that's usually the case which we proved things for in the other course. We going to prove it for the
multivariate case, and we are going to use the matrix formulation.
I hope you'll join me.