How does Random Effects work?

in this video I want to talk about what actually random effects estimation is and how does it work and in order to do that we're going to use the same example which you've been using in the last few videos which is investigating the various factors which influence house price in a given City I at a time period t and we said that that might depend on the crime rate in that particular city at that point in time and the unemployment and as well as these factors there are also some unobserved factors which are contained within alpha I if they're specific to that particular city but don't vary across time and as well as having some idiosyncratic factors which are contained within this UI to a UI t term here rather so the assumption under random effects is we assume that the covariance of alpha Ryne with any of the independent variables X ideas of written them here is equal to 0 and we spoke about how in these examples the random effects estimator is a better estimator than first differences or as it turned out fixed effects estimation and also it was better than poured OLS and random effects estimation as we spoke around was a type of feasible generalized least squares estimator because essentially what we were doing is we were correcting for the presence of serial correlation okay so what exactly is random effects estimation well what I'm actually going to do is I'm going to write down what the transformed equation is for the case of the random effects estimator so the random effects estimator is when we take our dependent variable house price into the eye at time T and then we take off some amount lambda times the time mean of the house price in that particular city I and doing the same to the right hand side we just have beta naught times 1 minus lambda then we have plus beta 1 open brackets the crime eight into the oil time t minus lambda times the time mean of the crime rate in that particular city and then we have plus beta2 times the unemployment rate in that particular city at that point in time minus lambda times the unemployment rate in that particular city averaged over all time and if we call this particular error term here ETA I T we can just write that this error term here is going to be e to I t minus lambda times e to I and I put a bar over the eater here to indicate that this is actually a time mean of this particular app okay so bearing in mind that we don't know what this particular lambda is we can already make some sort of comments on this particular transform system specifically if lambda is equal to zero then we're essentially we're going to recover this original system we had so if lambda equals zero we just uncover the original equation which we had up there and we're just estimating called OLS on that particular equation so when lambda equals zero random effects estimation is actually equivalent to just ordinary least-squares when we pull all the observations together because I should actually mentioned that random effects is just estimating this transform system via pulled OLS okay and similarly we should know that if lambda equals 1 then this random effects transform system is actually exactly equivalent to the fixed effects estimator to see that if so imagine replacing lambda with just 1 and it shouldn't take you too much time to notice that this is exactly the same as what we had in the fixed effects estimator example and if you're not immediately clear of that then you should go back and have a look at the fixed effect video typically however lunder is between naught and 1 so in the circumstance where zero and lambda so lambda is greater than zero and less than 1 we have the random effect is not a privilege to either just called OLS or fixed effects estimation so it's those particular examples that which we normally talk about when we talk about using random effects estimation so what is this mysterious parameter lambda well lambda as it turns out is equal to 1 minus Sigma mu squared all divided through by Sigma mu squared plus T times Sigma alpha or squared or to the power half okay so what do all these things mean in our expression for lambda well Sigma mu is just the variance of our idiosyncratic error term here Sigma alpha squared is the variance of our unobserved term alpha right now that we have this expression for lambda what we can do is we can talk about the circumstances under which lambda collapse collapses to be equal to 0 or equal to 1 if it's equal to 0 we know that random effects is equivalent to OLS and if lambda is equal to 1 we know the random effects is equivalent to fixed effect so how could we get lambda being equal to 0 well to see this notice that if this term on the denominator which contains Sigma alpha squared was removed from the denominator then what would be left with is we'd be left with Sigma mu squared divided by Sigma mu squared which is just 1 and then taking it the square root of that we just get 1 so we get 1 minus 1 which is equal to 0 so if we have Sigma alpha squared being equal to 0 that implies that lambda is going to be equal to 0 and in that circumstance we know that random effects is equivalent to ordinary least squares or pulled OLS and that shouldn't surprise us because essentially what we're saying is that this effect alpha right up here is unimportant so what we can do is we can actually forget about alpha right and just estimate the above equation by OLS without having to worry about the issue of serially correlated errors how would we get lambda is equal to 1 well we get lambda P equal to 1 if it was the case that T times Sigma Alpha squared tended to infinity and to see that that would create laundry put a 1 essentially what would be doing is this denominator would be getting very very large and hence when we divide any sort of number by a very very large practically infinite denominator this whole second term is going to disappear and we're just going to get lambda being equal to 1 and we know that lambda be equal to 1 implies that random effects is equivalent to fixed effect and again this shouldn't really surprise it because essentially if Sigma Alpha squared gets really really big then what random effects tries to do is it tries to remove as much of this effect as possible because if alpha is getting really really big then we probably shouldn't be using a random effects model in the first place what we should be doing is using a fixed effects model so fix our so random effects in this circumstance what it tries to do is it tries to remove as much of this alpha right as possible by creating lambda being equal to 1 in practice as I've said before lambda typically lies between naught and 1 so essentially the random effects system which is this system here is what we call a quasi time Devine system its quasi time domain because we taken off some fraction of the time domain values of the original values it's not fully time to mean unless lambda equals 1

Channel: Ben Lambert

Views: 62,140

Rating: 4.9498749 out of 5

Keywords: Random Effects Model, Econometrics (Field Of Study), fixed effects

Id: EbdBHJYbOrg

Length: 8min 31sec (511 seconds)

Published: Fri Oct 04 2013