[U] User's Guide

[ U ] 20.16 Obtaining robust variance estimates 299. use http://www.stata-press.com/data/r11/auto7(1978 Automobile Data). regress mpg weight foreignSource SS df MS Number of obs = 74F( 2, 71) = 69.75Model 1619.2877 2 809.643849 Prob > F = 0.0000Residual 824.171761 71 11.608053 R-squared = 0.6627Adj R-squared = 0.6532Total 2443.45946 73 33.4720474 Root MSE = 3.4071mpg Coef. Std. Err. t P>|t| [95% Conf. Interval]weight -.0065879 .0006371 -10.34 0.000 -.0078583 -.0053175foreign -1.650029 1.075994 -1.53 0.130 -3.7955 .4954422_cons 41.6797 2.165547 19.25 0.000 37.36172 45.99768With vce(robust), we get another:. regress mpg weight foreign, vce(robust)Linear regression Number of obs = 74F( 2, 71) = 73.81Prob > F = 0.0000R-squared = 0.6627Root MSE = 3.4071Robustmpg Coef. Std. Err. t P>|t| [95% Conf. Interval]weight -.0065879 .0005462 -12.06 0.000 -.007677 -.0054988foreign -1.650029 1.132566 -1.46 0.150 -3.908301 .6082424_cons 41.6797 1.797553 23.19 0.000 38.09548 45.26392Either way, the point estimates are the same. (See [R] regress for an example where specifyingvce(robust) produces strikingly different standard errors.)How do we interpret these results? Let’s consider the model-based interpretation. Suppose thaty i = x i β + ɛ i ,where (x i , ɛ i ) are independently and identically distributed (i.i.d.) with variance σ 2 . For the modelbasedinterpretation, we also must assume that x i and ɛ i are uncorrelated. With these assumptions anda few technical regularity conditions, our first regression gives us consistent parameter estimates andstandard errors that we can use for valid statistical inference about the coefficients. Now suppose thatwe weaken our assumptions so that (x i , ɛ i ) are independently and—but not necessarily—identicallydistributed. Our parameter estimates are still consistent, but the standard errors from the first regressioncan no longer be used to make valid inference. We need estimates of the standard errors that arerobust to the fact that the error term is not identically distributed. The standard errors in our secondregression are just what we need. We can use them to make valid statistical inference about ourcoefficients, even though our data are not identically distributed.Now consider a non–model-based interpretation. If our data come from a survey design that ensuresthat (x i , e i ) are i.i.d., then we can use the nonrobust standard errors for valid statistical inferenceabout the population parameters b. For this interpretation, we do not need to assume that x i and e iare uncorrelated. If they are uncorrelated, the population parameters b and the model parameters βare the same. However, if they are correlated, then the population parameters b that we are estimating