Hausman test for Random Effects vs Fixed Effects

in this video I once talked briefly about the Houseman test which is a test of essentially whether we should be using random effects estimation or fixed effects estimation so in order to talk about this test we're going to talk about a particular example so in this example we have a dependent variable Y which depends on an individual's characteristics where the individual is given by I and for a particular point in time and we say that this I of this Y rather is equal to beta naught plus beta 1 times some explanatory variable which varies across individual and across time and we also suppose that there is some sort of hidden unobserved factor alpha Y which we also should be including in our model even if we don't explicitly include it in our estimation strategy this alpha right is always going to be there okay so remember that random effects essentially assumes that the covariance of alpha Y with the independent variable X I T is equal to 0 and if this covariance is equal to 0 it is the case that both random effects and fixed effects of consistent estimators and in the circumstance where this particular condition is satisfied then not only do we know that random effects and fixed effects are both consistent it is the case that random effects is more efficient than fixed effects so the standard error of random effects should be less than the standard error of that which we would obtain by fixed effects ok so now that we know these two particular things we are almost in position to set up our particular test statistic the final thing we need to note that is that if this above assumption isn't true then it is not the case that random effects and fixed effects are consistent if this above assumption isn't true then fix effect is solely consistent and by writing solely here I mean that random effects is no longer consistent okay so on the basis of these three statements we're able to start talking about the Haussmann statistic and the Haussmann statistic for one particular explanatory factor is essentially constructed by the numerator is equal to the fixed effects estimated value of the parameter beta or in this example beta 1 and I put a star here to indicate that this is the actual value which is outputted from that estimation rather than the estimator which is a function and we take off the value which random effects outputs from that particular estimation strategy and then we square the difference between these two estimation strategies and/or these two particular estimated parameters by the two different estimated strategies rather and we divided through by the variance of the fixed effects estimate minus the various variants rather of the random effects estimate and it turns out under the null hypothesis being true and I'm defined what in the non poppers it is yet but bear with me then this is chi squared with one degree of freedom so what is the null hypothesis here well the null hypothesis which we are testing against here is that the covariance of RI with X I T is equal to zero in other words we're testing on multiple verses here is that we should be able to use random effects and the alternative is just that H naught isn't true okay so what's the intuition behind this particular test statistic well the way which it works is that in if the null hypothesis is true then we know that these two particular statements are true so we know that both random effects and fixed effects are consistent as a starting point and if they're both consistent then the difference between these two estimates which is kind of what this numerator is should be very very small secondly if this normal opposite is true then we know that the variance of fixed effects is greater than the variance of random effects so where is this numerator will be quite small this denominator will be relatively large because there should be quite a big difference between the variance of fixed effects with that of random effects which will mean that the value of double w-which will get out from this statistic will be quite small and to see how this is important when we're saying under H naught this test statistic is kind of spread with one degree of freedom we need to actually draw what a chi-squared statistic looks like with one degree of freedom or consecrate distribution rather with one degree of freedom so the y-axis here is the probability and the x-axis here is the value of W and for a chi-squared distribution with one degree of freedom it looks something like this so the overwhelming majority of values of W should be very very close to zero so if it actually turns out the W is close to zero so perhaps we get a value of W which is something like this then it is essentially quite likely that we would have got this value of W if the null hypothesis was true because this is the distribution under the null hypothesis being true whereas if the null hypothesis is false then essentially we're saying that the covariance of alpha right with X I T does not equal zero and if that is the case then it is this third statement which is going to come into play and the way in which it works is that if the third statement is true then the numerator now is going to be relatively large because there is going to be some difference between the point estimates of fixed effects versus random effects because fixed effects is the only one which is consistent so random effects is most likely going to be quite different to that and because of this large numerator the value of W the housement statistic which we get could be quite a long way away from zero so perhaps it's somewhere like this and it is actually quite like unlikely rather that we would get a value of W which is this far away from zero if the multiple position is true so if we get a large value of W we are going to reject the null hypothesis and it looks likely that fixed effects estimation is the way to go because essentially we have some covariance between alpha I and XIT which doesn't equal zero so just to summarize if we conduct this test and we do not reject the null hypothesis then we conclude the random effects and fixed effects are both consistent but we should use random effects because of the fact that it's variance is lower than that of fixed effects whereas if we reject the null hypothesis then we are going to conclude that fixed effects is the estimation strategy to go with because of the fact that only fits the best estimation is consistent in the circumstance essentially what this particular statistic W is doing is it is comparing the consistency of be two estimators which is the numerator with the relative gains of efficiency which can be obtained by using random effects a fixed effect

Channel: Ben Lambert

Views: 78,600

Rating: 4.9217219 out of 5

Keywords: Econometrics (Field Of Study), hausman test, panel data, random effects, fixed effects

Id: 54o4-bN9By4

Length: 8min 35sec (515 seconds)

Published: Fri Oct 04 2013