Hi, in this video I am going to be talking about the Gauss-Markov
assumptions in econometrics,
and what their significance is.
So, the Gauss-Markov
assumptions are a set of criteria
which were first created by the
mathematicians
Carl-Friedrich Gauss, and Andrei Markov,
which if they are upheld,
then that says something about our ability to use Least-Squared
estimators on the sample data.
Well, it says that those Least-squares estimators are in fact BLUE.
But, what does it mean for an estimator to be BLUE?
Well, it means that there are no other linear unbiased estimators which
have a
lower sampling variance than that particular estimator.
So, I can illustrate this graphically, imagine I have a
sampling distribution of one estimator which looks like that.
And then I have another one
which has the same sort of centre as the first one,
except that it is slightly steeper towards the centre
of the distribution.
So, assuming that both of these are unbiased, so they are both centered around the true
population parameter,
we can see that the second estimator
has a lower sampling variance that the first.
Well, that means that,
more often than not
when I use my
Least-squared estimators -
when I apply my Least-squared estimators to the sample data,
they are going to more often than not
provide estimates of the
true population parameter 'beta p',
which are closer to 'beta p' than I would have got by using
the first type of estimator.
So, that's the significance in econometrics of the Gauss-Markov assumptions. But,
what are the Gauss-Markov assumptions?
There's no particular order to the Gauss-Markov assumptions, but I am going to label them
here so that means that I can refer to them in the future.
The first Gauss-Markov assumption has to do with the population process, so
assuming that there is some population process which connects wages with
the number of years of education,
although education doesn't exactly determine wages,
because there's some sort of error term here.
This is an example of a model which is linear in parameters, so that means that
its linear in alpha and beta.
So this is the first Gauss-Markov assumption which says
that
our population process has to be linear in parameters.
Note that if I had this type of model where I had wages equal to alpha times beta times the
number of years of education plus alpha...well just alpha
on its own -
this would be nonlinear in parameters because this implies some sort of
multiplicative effect between alpha and beta.
Or if I had beta-squared here, that would also be nonlinear in parameters.
Note that however that being
linear in parameters does not mean that I cannot have
a variable in our model which is nonlinear.
So, actually just having education squared in our model, rather than just education,
that is absolutely fine under the assumption of 'linearity in parameters'.
It just means that I am not allowed to have a model which has nonlinear
parameters within it.
So, that's the first Gauss-Markov condition - the second condition is
that we have a set of sample data -
x and y which are a random sample
from the population.
So what does that actually mean? Well, it means that within our population,
a random sample occurs
if each individual within our population
is equally likely to be picked, when I take the sample. That's what we mean by a random
sample. But, it also implicitly means that
not only are each person in the population equally likely to be picked, but
it means that all of our points come from the same
population. So they come from the same
population process which in this context might be wages being equal to
alpha plus beta times education.
plus some error.
The third condition is perhaps the most important of the Gauss-Markov conditions,
which is the zero conditional mean of errors.
So what does this
actually mean? Well mathematically it means that the expectation
of our error term
in our population
given our x term, which in this case is education
has got to be equal to zero.
Well, what does this mean practically? Well it means that if I know
someone's level of education
that does not help me to predict
whether they will be above or below the average population regression line.
So that's what it means for there to be a zero conditional mean of error.
And this is perhaps the most important of the Gauss-Markov assumptions, for reasons which we'll come
onto later.
So, that concludes our first video looking into the Gauss-Markov assumptions.
I'm going to, in the next video, explain the next three Gauss-Markov assumptions.