Adjusted R 2 , Residuals, and Review

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Adjusted R 2 , Residuals, and Review. Adjusted R 2 Residual Analysis Stata Regression Output revisited The Overall Model Analyzing Residuals Review for Exam 2. Exercise Review. Use the caschool.dta dataseet

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Presentation Transcript
Adjusted R2, Residuals, and Review
• Residual Analysis
• Stata Regression Output revisited
• The Overall Model
• Analyzing Residuals
• Review for Exam 2
Exercise Review
• 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
Adjusted R2: An Alternative “Goodness of Fit” Measure
• 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 more complex the model (the bigger K
• is), the larger the adjustment
• The bigger R2 is, the smaller the
Residual Analysis: Trouble Shooting
• 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

Review: Assumptions Necessary for Estimating Linear Models

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

The Ideal: Sneeze Splatter

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

Heteroscedasticity

e

Predicted Y

Residuals for model

with outliers deleted

Possible Outliers

Checking for Outliers

Residuals for

model using

all data

e

Predicted Y

Problem: Under-specified model; measurement error

Stata Regression Model:

Regressing “testscr” onto “avginc”

Examination of Residuals

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

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

Residuals v. Predicted Values

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

Formal Test for Non-linearity:Omitted Variables

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

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

Formal Tests for Heteroscedasticity

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

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

Case-wise Influence Analysis

The Leverage versus Squared Residual Plot

What to Do?
• 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
• Influential and Deviant Cases
• Evaluate the cases
• Run with controls (multivariate model)
• Omit cases (last option)
Next Week
• Review regression diagnostics
• Introduction to Matrix Algebra
• Review for Exam