Unformatted text preview: NONLINEAR REGRESSION FUNCTIONS Read (Stock and Watson, Chapter 8) Linear regression can be used to capture nonlinear Linear relationships relationships In such cases, the interpretation of the slope In coefficients changes slightly coefficients We shall consider the following cases: (1) We Polynomial regressions (2) Linearlog models (3)Loglinear models and (4) Loglog models (3)Loglinear POLYNOMIAL REGRESSIONS
(1) Polynomial in a single independent (1) variable variable (2) In a polynomial regression, the In population regression is approximated by a quadratic, cubic, or higherdegree polynomial polynomial POLYNOMIAL REGRESSION: EQUATION POLYNOMIAL REGRESSIONS (Contd.)
These are just linear regression modelsexcept that the regressors are powers of x Estimation, hypothesis testing, etc. in Estimation, polynomial regressions proceeds as in multiple regression model using OLS multiple The coefficients are difficult to interpret, The but the regression function itself is interpretable interpretable QUADRATIC REGRESSION
Quadratic regression is also called a seconddegree polynomial regression The equation of a quadratic regression is QUADRATIC REGRESSION QUADRATIC REGRESSION: EXAMPLE QUADRATIC REGRESSION: EXAMPLE (Contd.)
The relationship between Test scores and The district income district There is a positive correlation between test There scores and district income (correlation=0.71), but the linear OLS regression line does not adequately describe the relationship between these variables (see scatter diagram below). (see SCATTERPLOT QUADRATIC REGRESSION: EXAMPLE (Contd.) QUADRATIC REGRESSION: EXAMPLE (Contd.) QUADRATIC REGRESSION: EXAMPLE (Contd.) CUBIC REGRESSION Thirddegree polynomial regression CUBIC REGRESSION
Cubic regression is also called a thirddegree polynomial r egression The equation of a cubic regression is CUBIC REGRESSION: EXAMPLE CUBIC REGRESSION: EXAMPLE (Contd.) CUBIC REGRESSION: EXAMPLE (Contd.) POLYNOMIAL REGRESSION: SUMMARY
Estimation: by OLS after defining new Estimation: regressors regressors Coefficients have complicated interpretations Hypothesis tests concerning degree r can be Hypothesis tested by t and Ftests on the appropriate (blocks of) variable(s) (blocks Choice of degree r (based on t/F tests, Choice judgement, model selection criteria, etc.) judgement, LOGARITHMIC TRANSFORMATIONS
Y and/or X is transformed by taking its and/or logarithm logarithm This gives the “percentages” interpretation This (see below) that makes sense in many applications applications LINEARLOG MODELS LOGLINEAR MODELS LOGLOG MODELS Also called doublelog models LINEARLOG VS CUBIC REGRESSIONS: EXAMPLE The estimated cubic regression function and The the estimated linearlog function are nearly identical in this example identical LOGARITHMS IN REGRESSIONS: SUMMARY ...
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 Winter '09
 Ogwang
 Econometrics, Linear Regression, Regression Analysis, Contd, quadratic regression

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