Gauss-Markov assumptions part 2 - YouTube

Hi thanks for joining me. Today we are going to be talking about the

second half of the Gauss-Markov assumptions.

If you missed the first half you may want to have a look at the previous video

which looks through assumptions one to three.

So just to reiterate, the Gauss-Markov assumptions are the set of conditions which if

they are upheld

then that means that least-square estimators are BLUE.

So, that means that they are the best, linear, unbiased estimators

which are possible.

So the fourth Gauss-Markov assumption is something which we refer to as no

perfect

collinearity.

And this is

referring to our particular sample, but by deduction it also refers to

the population.

So, what does it actually mean? Well no perfect collinearity - in regressors

I should say -

that means that if i have some sort of model that y equals alpha plus 'beta

one' times 'x one'

plus 'beta two' times 'x two',

plus some sort of an error.

That there cannot be an exact relationship between 'x one' and 'x two',

so I cannot write down in an equation that 'x one' is equal to 'delta

nought',

plus 'delta one'

times 'x two'.

That means that if I know 'x two', I

exactly know 'x one'.

In a sense 'x one' and 'x two' are

exactly the same event.

So, an example of this might be, if I was trying to determine

which factors affect the house price of a given house

from its attributes,

then if I was to include a regression

which included the square meterage of that house,

and also the square footage.

Well, obviously if I know square meterage,

I actually know square footage - they are both essentially the same thing.

Square footage is essentially equal to nine, times

the square meterage of the house.

So, obviously within a regression, I am going to have a hard time unpicking

square footage from square meterage,

because they're exactly the same thing.

And, the assumption of no perfect collinearity among regressions means that

I cannot include both of these things in my regression.

Assumption five

is

called 'homoskedastic errors'.

So, homoskedastic errors means that if I was to draw a process -

so let's say that I have the number of years of education

and the wage rate,

and this

again is referring to population rather than to the sample.

If I have errors which,

looks something like - when I draw the population line - like that

whereby the distribution of errors away from the line remain relatively constant,

that are lying between the error lines which I draw here.

There's no increasing or decreasing of errors along the

education variable,

then that means that errors are

homoskedastic.

So, mathematically that just means that I can write the variance of our

error in

the population process,

is equal to some constant, 'sigma squared', or writing it a little bit more

completely. The variance of 'u i' given 'x i' is equal to 'sigma squared'.

In other words the variance - how far the points are away from the line -

does not vary systematically

with x.

The last Gauss-Markov assumption is

called 'no serial

correlation'.

What this means is mathematically that the covariance between

a given error 'u i' and

another error 'u j',

must be equal to zero,

unless i equals j. In which case we are considering the covariance of the error with itself, in

which case we have variance, which is to do with assumption five.

So,

this last assumption of 'no serial correlation' means that

the errors essentially have to be independent of one another.

So, knowing one of the errors,

doesn't help me predict another error. So in other words if I know this error here in

my diagram this doesn't help me predict the error here for a higher level of

education.

This concludes my video summarising the Gauss-Markov assumptions.

I'm going to go and examine each of these assumptions in detail

in the next few videos.

I'll see you then.