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Adjusted R 2 , Residuals, and Review

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- Adjusted R2
- Residual Analysis
- Stata Regression Output revisited
- The Overall Model
- Analyzing Residuals

- Review for Exam 2

- Use the caschool.dta dataseet
- Run a model in Stata using Average Income (avginc) to predict Average Test Scores (testscr)
- Examine the univariate distributions of both variables and the residuals
- Walk through the entire interpretation
- Build a Stata do-file as you go

- Recall that R2 is calculated as:
- Hypothetically, as K approaches n, R2 approaches one (why?) – “degrees of freedom”
- Adjusted R2 compensates for that tendency

“explained sum of squares”

“total sum of squares”

- The bigger the sample size (n), the smaller
- the adjustment
- The more complex the model (the bigger K
- is), the larger the adjustment
- The bigger R2 is, the smaller the
- adjustment

- Conceptual use of residuals
- e, or what the model can’t explain

- Visual Diagnostics
- Ideal: a “Sneeze plot”
- Diagnostics using Residual Plots:
- Checking for heteroscedasticity
- Checking for non-linearity
- Checking for outliers

- Saving and Analyzing Residuals in Stata

ei

ei=0

X

1.Errors have identical distributions

Zero mean, same variance, across the range of X

2.Errors are independent of X and other ei

3.Errors are normally distributed

e

Predicted Y

Problems: It is possible to “over-interpret” residual plots; it is also possible to miss patterns when there are large numbers of observations

Problem: Standard errors are not constant; hypothesis tests invalid

e

Predicted Y

Problem: Biased estimated coefficients, inefficient model

e

Predicted Y

Residuals for model

with outliers deleted

Possible Outliers

Residuals for

model using

all data

e

Predicted Y

Problem: Under-specified model; measurement error

Regressing “testscr” onto “avginc”

Use the case ID number to find the relevant observation in the data set

gsort e (or you can use “-e”)

list observat testscr avginc yhat e in 1/5

.list observat testscr avginc yhat e in 1/5

+---------------------------------------------------+

observat testscr avginc yhat e

---------------------------------------------------

1. 393 683.4 13.567 650.8699 32.53016

2. 386 681.6 14.177 652.0157 29.5842

3. 419 672.2 9.952 644.0789 28.12111

4. 366 675.7 11.834 647.6143 28.08568

5. 371 676.95 12.934 649.6807 27.26921

+---------------------------------------------------+

Using an “ocular test,” non-linearity seems probable, but heteroscedasticity is not obvious here. But should we trust our eyeballs?

Tests whether adding 2nd, 3rd and 4th powers of X will improve the fit of the model:

Y=b0+b1X+b2X2+b3X3+b4X4+e

Tests to see whether the squared standardized residuals are linearly related to the predicted value of Y:

std(e2)=b0+b1(Predicted Y)

The Leverage versus Squared Residual Plot

- Nonlinearity
- Polynomial regression: try X and X2
- Variable transformation: logged variables
- Use non-OLS regression (curve fitting)

- Heteroscedasticity
- Re-specify model
- Omitted variables?
- Use non-OLS regression (WLS)
- Use robust standard errors

- Re-specify model
- Influential and Deviant Cases
- Evaluate the cases
- Run with controls (multivariate model)
- Omit cases (last option)

- Review regression diagnostics
- Introduction to Matrix Algebra
- Review for Exam