Introduction to the matrix formulation of econometrics

in this video I want to provide an introduction to the matrix formulation of econometrics and I'm going to explain how in some ways is beneficial over the way in which we've normally written down Ecco metric models last bar so the idea here is that let's say we have a model whereby we've got some y:i and y:i is given by let's say beta naught plus beta 1 times X 1 I plus B 2 2 times X 2 I all the way through let's say we have P independent variables so we have plus P 2 P times X P I and then finally we have our current term here epsilon 9 okay so one thing I mention about this is that this is quite a clunky way of writing things down I mean if you've got P in dependables I have to write down P expressions right I have to write down P of these particular terms here and I have to include on each of these terms a subscript I where the subscript I here is implicitly going from 1 all the way through to the last point in our sample so this is quite a clunky way of writing things down and ideally we would like a slightly more compact and ammeter way of writing things down and it turns out that the matrix formulation of econometrics is a neat way of doing it so what we doing the matrix formulation of econometrics is that we stack each of our observations on top of one another so our top observation is y1 and then we have y2 and we sort of continue all the way down until we get to the end observation of the dependent variable which I write yn and we can write this as being equal to a nother matrix or this is a vector straightening we're going to write this in terms of a matrix which I'm going to define in a minute times a parameter vector so this parameter vector here is going to have components which are just the parameter values so it's first value is going to be beta naught its second value is going to be e 2 1 and it's going to go beta 2 all the way through to b2p and then finally to ensure that we actually have the same dimensions on both sides and to ensure that we've included our error term I'm going to include here now a vector of the error terms which funnily enough is going to start with epsilon one then it's going to have epsilon two and it's going to continue all the way through to epsilon M so this error term vector here is M by one this dependent variable vector is M by 1 and this parameter vector by contrast is P by 1 so in order to make the dimensions of both sides match up this thing this matrix here which I haven't defined yet better have dimensions n by P because then what we have is we have a cancelling of the inner dimensions of these matrices and we're going to be left with an M by 1 vector left overall so what are the inner components of this particular matrix well it actually turns out that it's quite simple to define so if I write down the first row the first row is going to have a first component 1 which I'm going to explain in a minute a second component is just going to be X 1 1 its second is third component rather is going to be X 2 1 and we're going to continue all the way up until X P 1 ok so why have I done that well to see why I've done that essentially what we need to do is we need to take this row and multiply it by this parameter vector here because that's what you do in matrix multiplication right if we do that we're going to get a 1 times beta naught plus if I can sort of write it in a different color here perhaps you can see it a bit better we're going to have X 1 1 times B 2 1 and then we're going to have X - 1 times B 2 etc all the way through to X P 1 times B 2 P so if I was to write out this first row explicitly what we would have is we would have 1 times beta naught which is just B 2 naught plus X 1 1 times beta 1 which is just B 2 1 times X 1 1 and then the second or the third part would then be plus B 2 2 times X 2 1 and we continue all the way to adding B to P times X P 1 and then if we were to add on now the error term for this first row will then just get an absolute 1 so writing out the first row in full we have the dependent variable for observation 1 y1 is equal to this linear combination of independent variables plus the error term epsilon 1 so we've actually just recovered exactly what we have in the original equation and but we've just done it for the first row so what's the second row of our matrix well it's not hard to figure out it's just going to be very similar except now we're going to be talking about having we're still going to have the first component being 1 because of the fact that we have this constant in our model the second part is then going to be x12 where the 2 here stands for the fat that we're talking about the second individual the third part is then going to be X 2 2 and we're going to continue all the way up to X P 2 and if you were to multiply out this row now times the parameter vector we would recover the exact same equation which we had off the top except now we would have the Y 2 is equal to beta naught plus beta 1 times X 1/2 plus b2 2 times X 2 2 plus B 2 P or all the way up to be 2 P times X P 2 and then finally we have our error term epsilon 2 so we've recovered exactly what we had before so in general what we would do is we would need to fill in this matrix for all n individuals so the last row will be for the nth individual and we will have the first component still being 1 because we've got this constant and then we would have X 1 n X 2 N and then all the way through up to X P n being the last term in the bottom right of the matrix and all the terms in between would be filled out appropriately and this is a really nice way of writing down and a metric model because essentially I can write this all down as a matrix equation I can write this down as having a dependent variable Y or a vector which is equal to a matrix of independent variables which I'm gonna call X but then tilde underneath it times the parameter vector PETA Plus this vector of the arrow terms which I'm calling here excellent with a line underneath there and I hope you agree that this down here is a lot nicer and a neater way of writing models rather than including explicitly all of these subscript terms here implicitly what we're assuming is that this matrix X contains all of our observations so that's also kind of a benefit because the fact that we're able to for each different individual in our sample we are essentially writing down that particular equation again and again and again and because of that it's kind of a complete way of writing down systems and actually this particular formation of economics or formulation of econometrics rather comes into its own when you're describing more complicated econometrics models and it's going to be are sort of forming a basis for all of our elementary descriptions of situations for the graduate course in econometrics

Channel: Ben Lambert

Views: 192,754

Rating: 4.9561043 out of 5

Keywords: Econometrics (Field Of Study), matrix econometrics, Linear Algebra (Field Of Study)

Id: GMVh02WGhoc

Length: 7min 38sec (458 seconds)

Published: Mon Oct 28 2013