Fixed Effects, First Differences and Pooled OLS - intuition

in this video I want to provide some intuition as to why fixed effect and first differences estimators actually provide a much better estimate of parameter values than pulled OLS in general so the example which I'm going to be considering here is I'm interested in the determinants of crime in SETI at a time period of T and I think that crime in a city our time period T depends somewhat on some city specific characteristics which don't change through time and it depends on let's say the unemployment rate in City line at time period t plus some other factors which also vary both pre city and across time and the idea here is that we expect that beta is greater than zero because if the unemployment rate is higher that tends to be associated with a high prime rate so how does pulled OLS go about thinking about the situation well what pulled OLS does is it says let's just lump all our observations together and then let's just estimate the model as if it's just one big broad cross section so the idea is we might have some observations which look something like this so we have perhaps for three cities at two different points in time we have observations so we have these six observations which have indicated here and what pooled OLS does is it says well let's just treat all these observations as if it's one big cross section and that's let's then fit the line of best fit here and the line which pulled our less would fit would look something like this but notice that here what pulled OLS would output is it would output a value of beta which was less than zero and importantly this is completely nonsensical that says that if the unemployment rate increases then there tends to be a decrease in the crime rate and we know in practice that that's hardly ever the case so what's going on here and why does pulled OLS actually give such a stupid estimate well if we look at what fixed effects and first differences do is essentially what both of these estimators do is they actually think about observations from different cities as being different so in first differences essentially what happens is you regress the first different of the prime rate in CGI time period t on the first different of the unemployment rate in city i time period t plus the first difference of the error uit and notice that we have removed the unobserved heterogeneity term the alpha right okay but implicit in this scheme is the fact that we should treat observations from different cities differently because essentially we're still saying let's consider within a city what has been the difference of the crime rate between this period and the last period so essentially what first differences does as well as fixed effects is it says well let's look at the observations across time within a specific cities so these top two observations here might actually correspond to london and the bottom observation might be london in the year 2000 and the top one might be london in the year 2010 and these two observations here might correspond to a different city they might correspond to bristol and they correspond to bristol in the year 2000 and bristol in the year 2010 and then finally these last two observations might correspond to let's say and brighton and they might correspond to brighton in the year 2000 and year 2010 again and notice that because we've removed this alpha roi term here as well as we do in fixed effects essentially what fixed effects and first differences say is let's disregard the fact that there are differences in sort of average limit levels of crime rate between these three cities and let's assume that they are all due to city specific characteristics which don't change free time and that seems like a much more sensible idea than proceeding as called OLS those as actually neglecting these to these specific characteristics which don't change through time so what first difference is does is it essentially says well let's consider each of these pairs of observations and let's try and fit a line of best fits of these pairs of observations so for the first pair of observations we might fit a line which looks something like that for the Bristol case we might fit a line which looks like that and the Brighton case we'd also fit a line which is also up and sloping so notice that in each of these three circumstances we would find or we would conclude rather that beta is greater than zero and hence beta on average across each of the three cases would be greater than zero and hence we would actually understand or we would conclude rather the unemployment or increases in unemployment tend to be associated with increases in the crime rate so that's the first difference case what about the fixed effects case well the fixed effects case is a little bit more difficult difficult because the fact that the fixed effects regression is a little bit more complicated but not much more the idea is that you regress the time demeaned crime rate so that's the crime rate in city height time period t on the time averaged level of crime rate on the time averaged or time demeaned rather level of unemployment rate so that's the unemployment rate in city I a time period t minus the unemployment rate averaged across time plus some sort of time demean error you till the I T and essentially because we're only dealing with two periods here essentially what fixed effects is doing is it saying let's mark on each of these or in each of these pairs let's mark on there the average level of crime rate and the average level of unemployment rate and in each of these cases is actually going to correspond to the middle of these these lines which I've drawn which draw of which actually connect the two points and then what fixed effects does is it said essentially says draw a line which goes through this point which is in the center here and also passes is near to the other two points as possible and actually you can see in this case that a fixed effect is actually going to be exactly equivalent to first difference it because essentially we're only dealing with two time periods this point is going to lie exactly in the middle of this line bisecting the two point and hence when I draw and try and draw a line which actually gets us close to these three points as possible it's actually going to correspond exactly to that of the first difference is catch and hence just like the first difference is case we are going to conclude that beta is greater than zero in other words unemployment rate increases tempting prease the crime rate so notice that in both fixed effects and first differences we have done away with the unobserved heterogeneity term and by doing away with this unobserved heterogeneity term that has allowed us to actually get asked estimates of parameters which are much better than that which we would have obtained by using pooled OLS alone

Channel: Ben Lambert

Views: 101,846

Rating: 4.9464722 out of 5

Keywords: panel data, fixed effects, Econometrics (Field Of Study), first differences estimator, pooled ols

Id: 1SchyQ77VFg

Length: 7min 2sec (422 seconds)

Published: Sat Jan 18 2014