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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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Adjusted r 2 residuals and review
Adjusted R2, Residuals, and Review

  • Adjusted R2

  • Residual Analysis

  • Stata Regression Output revisited

    • The Overall Model

    • Analyzing Residuals

  • Review for Exam 2


Exercise review
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 r 2 an alternative goodness of fit measure
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”


Calculating adjusted r 2
Calculating Adjusted R2

  • 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


Residual analysis trouble shooting
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


Review assumptions necessary for estimating linear models

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


The ideal sneeze splatter

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

Problem: Standard errors are not constant; hypothesis tests invalid

Heteroscedasticity

e

Predicted Y



Checking for outliers

Residuals for model invalid

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
Stata Regression Model: invalid

Regressing “testscr” onto “avginc”



Residual plot
Residual Plot invalid


Examination of residuals

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
Residuals v. Predicted Values the data set

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
Formal Test for Non-linearity: the data setOmitted 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
Formal Tests for Heteroscedasticity the data set

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
Case-wise Influence Analysis the data set

The Leverage versus Squared Residual Plot


What to do
What to Do? the data set

  • 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
Next Week the data set

  • Review regression diagnostics

  • Introduction to Matrix Algebra

  • Review for Exam


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