Robust Regression

Regression Methods We are going to look at three approaches

to robust regression: Regression with robust standard errors Regression with robust standard errors

including the cluster option Regression with random effect Regression with fixed effect

We will look at a model that predicts the api 2000 scores Our focus is whether the average class size in K through 3 (acs_k3) and average class size 4 through 6 (acs_46) affect the academic performance

use a new data set http://www.ats.ucla.edu/stat/stata/webbooks/reg/elemapi2

4.1.1 Regression with Robust Standard Errors The Stata regress command includes a robust option for estimating the standard errors using the Huber-White sandwich estimators. Such robust standard errors can deal with a collection of minor concerns about failure to meet assumptions,

Minor problems about normality Heteroscedasticity Some observations that exhibit large residuals, leverage or influence.

With the robust option, the point estimates of the coefficients are exactly the same as in ordinary OLS, but the standard errors take into account issues concerning heterogeneity and lack of normality.

As with the robust option, the estimate of the coefficients are the same as the OLS estimates, but the standard errors take into account that the observations within districts are non-independent. 

If you have a very small number of clusters compared to your overall sample size it is possible that the standard errors could be quite larger than the OLS results. For example, if there were only 3 districts, the standard errors would be computed on the aggregate scores for just 3 districts. 

Using the Cluster Option The elemapi2 dataset contains data on 400 schools that come from 37 school districts. It is very possible that the scores within each school district may not be independent, and this could lead to residuals that are not independent within districts. We can use the cluster option to indicate that the observations are clustered into districts (based on dnum) and that the observations may be correlated within districts, but would be independent between districts.

Control for random effect (school district)

Control for fixed effect (school district)

2.4 Examine Distribution Assumption Classical regression assumption requires

that the outcome (dependent) to be normally distributed.

In large sample, this assumption is not that important because of Central Limit Theory

In small sample, however, the distribution assumption could be relevant

We will investigate issues concerning normality.

Here we check the normality of enroll We start with making some graphs

Hisgram Kdesnity

We can use the normal option to superimpose a normal curve on this graph and the bin(20) option to use 20 bins.  The distribution looks skewed to the right.

An alternative to histograms is the kernel density plot, which approximates the probability density of the variable. Kernel density plots have the advantage of being smooth and of being independent of the choice of origin, unlike histograms. Stata implements kernel density plots with the kdensity command.

Having concluded that enroll is not normally distributed, how should we address this problem? We may try to transform enroll to make it more normally distributed. Potential transformations include taking the log, the square root or raising the variable to a power. Stata includes the ladder and gladder commands to help selecting the right transformation. Ladder reports numeric results and gladder produces a graphic display.

This indicates that the log transformation would help to make enroll more normally distributed. Let's use the generate command with the log function to create the variable lenroll which will be the log of enroll. Note that log in Stata will give you the natural log, not log base 10. To get log base 10, type log10(var)

2. 5 Summary Simple Regression Multiple Regression Hypothesis Testing Examine the normality assumption

Quiz I Make graphs of api99: histogram, kdensity plot What is the correlation between api99 and meals? Regress api99 on meals. Create and list the fitted (predicted) values. Graph meals and api99 with and without the regression line.

Quiz II Look at the correlations among the variables api99 meals ell avg_ed using the corr and pwcorr commands. Perform a regression predicting api99 from meals and ell. Interpret the output.