Geometric Interpretation of Ordinary Least Squares: An Introduction

This video provides an introduction to the geometric interpretation of Ordinary Least Squares.
Check out oxbridge-tutor.co.uk/graduate-... for course materials, and information regarding updates on each of the courses. Check out ben-lambert.com/econometrics-... 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.com/bayesian/ Accompanying this series, there will be a book: www.amazon.co.uk/gp/product/1...

[@vatsalamolly]

Sometimes when you're studying advanced stuff, you get confused about the very basics. Thank you for this quick and easy explanation!!

[@andresiganciomunozecheverr6071]

Among all the intuitions we can possibly find, this is exactly the one I was looking for, thx a lot Ben!!!

[@291291ify]

you are a life saver, thank you so much!!

[@murfalious]

Thank you very much. Clarified this subject very well :)

[@NoName-kg5rv]

Amazing!!

[@bertobertoberto3]

that was excellent

[@Angel1a89]

Thank you!!!!

[@ranjit1568]

Hey thanks for the video. Why at the end do you say u hat equals XBeta hat?
I thought it is y hat = XBeta hat.. thanks

[@javierwagner4410]

Would it be valid to interpret it instead as the row space, given that the row space defines the independent variables and pre-determines in which space the observations can move. Row vectors seem to make more sense as all observation are restricted to the # of independent variables present.

[@Marteenez_]

Is it possible for y to be in the column space of X?

[@samkim6933]

I think u hat, in this case, doesn't indicate the residual, it only means the orthogonal projection of y onto col(X)... is that right?

[@kurtvonschleicher]

no, u hat is the distance between the orthogonal projection of y over the space (in this example represented as mu hat on a plane) spanned (obtained through linear combinations) by the columns or vectors (column space) and the actual y. there is a unique linear combination obtained through the Ols method that minimizes the value of u hat. the parameters of such "optimal" combination are the betas of the regression. u hat can be also interpreted as the portion of y that cannot be expressed through a linear combination of the columns (i.e. the part of y that does not lie on the column space). why is u hat perpendicular to the row space? due to the fact that u hat lies in the so called "left null space", which as stated in the fundamental theorem of linear algebra is orthogonal to the column space.

[@aleksanderpasato6916]

Hi, thanks for great explanation. I've been looking for something like this for ages ;) I guess I don't get one thing. Isn't it so that span of 3 independent vectors in 3D space cover entire space (as we can get any point by using linear combination) instead of just a plane?

[@kottelkannim4919]

2:59 The space described in the video is 4D, albeit drawn on a 2D graphic tablet. So 3 independent vectors in 4 dimensional space do span a "plane".

[@tomasnobrega8087]

@@kottelkannim4919 yeah! Thats a hyperplane (sounds much fancier than it is)

[@fk866]

@@kottelkannim4919 n

[@vinsavi]

what bounds all x into 1 plane? . if y1 is a step explained by [1 x11 x12] then it means there are 3 steps needed to explain y1 thats all, no one is saying that those all are in one plane. kindly explani

[@m.h.4652]

The end of the video is not correct. When you write u_hat = XBeta_hat, thats not correct. It should be u_hat = y_hat - X Beta_hat

[@zhizhongpu8937]

Agreed; there's also another error: the vector in the column space should be XB (in the graph now it's u) and the difference vector should be y-XB = u (instead of y-u as in the video 4:54)