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## Monday, September 16, 2013

### concrete illustration - variance of OLS estimators

Now I feel b is a sample estimate of the population beta (a parameter in our explanatory linear "model" of Y), but we need to know how close that b is to beta. If our b turns out to be 8.85, then beta could be 9, or 90. That's why we work out and reassure ourselves that (under certain assumptions) b has a normal distribution around beta, and the variance is .... that var(b|X).

I just made up a concrete but fake illustration, that I will share with my friends. See if I got the big picture right.

Say we have a single sample of 1000 data points about some fake index Y = SPX1 prices over 1000 days; X = 3M Libor 11am rates on the same days. We throw the 2000 numbers into any OLS and get a b1 = -8.85 (also some b0 value). Without checking heteroscedasticity and serial correlation, we may see var(b1) = 0.09, so we are 95% confident that the population beta1 is between 2 sigmas of -8.85, i.e. -8.25 and -9.45. Seems our -8.85 is usable -- when the rate climbs 1 basis point, SPX1 is likely to drop 8.85 points or thereabout.

However, after checking heteroscedasticity (but not serial corr), var(b1) balloons to 9.012, so now we are 95% confident that true population beta1 is between 2 sigmas of -8.85 i.e. -2.25 and -14.25, so our OLS estimate (-8.85) for the beta1 parameter is statistically less useful. When the rate climbs 1 basis point, SPX1 is likely to drop... 3, 5, 10, 13 points. We are much less sure about the population beta1.

After checking serial corr, var(b) worsens further to 25.103, so now we are 95% confident that true beta is between +1.15 and -18.85. When the rate climbs 1 point, SPX1 may drop a bit , a lot, or even rise, so our -8.85 estimate of beta is almost useless. One thing it It does help -- it does predict that SPX1 is UNlikely to rise 100 points due to the a 1 basis point rate change, but we "know" this without OLS.

Then we realize using this X to explain this Y isn't enough. SPX1 reacts to other factors more than libor rate. So we throw in 10 other explanatory variables and get their values over those 1000 days. Then we hit multicolleanearity, since those 11 variables are highly correlated. The (X' X)^-1 becomes very large.