Gauss-Markov assumptions part 2

This video details the second half of the Gauss-Markov assumptions, which are necessary for OLS estimators to be BLUE.
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. Check out ben-lambert.co... for course materials, and information regarding updates on each of the courses. Quite excitingly (for me at least), I am about to publish a whole series of new videos on Bayesian statistics on youtube. See here for information: ben-lambert.co... Accompanying this series, there will be a book: www.amazon.co....