ECON2P91_Part_4 - NONLINEAR REGRESSION FUNCTIONS Read(Stock...

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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) Linear-log models (3)Log-linear models and (4) Log-log models (3)Log-linear 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 higher-degree 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 Third-degree 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 F-tests 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 LINEAR-LOG MODELS LOG-LINEAR MODELS LOG-LOG MODELS Also called double-log models LINEAR-LOG VS CUBIC REGRESSIONS: EXAMPLE The estimated cubic regression function and The the estimated linear-log function are nearly identical in this example identical LOGARITHMS IN REGRESSIONS: SUMMARY ...
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