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# Adjusted R 2 , Residuals, and Review - PowerPoint PPT Presentation

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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Adjusted R 2 , Residuals, and Review

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### 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

Problem: Standard errors are not constant; hypothesis tests invalid

### Heteroscedasticity

e

Predicted Y

Problem: Biased estimated coefficients, inefficient model

### Non-Linearity

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”

### Residual Plot

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

### 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