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Understanding Multiple Regression Coefficients and Nonlinear Models in Statistical Analysis

This lecture covers key concepts in statistical analysis, focusing on multiple regression coefficients, multicollinearity, and the F-test for heteroskedasticity. It emphasizes the limitations of ordinary least squares and the necessity for nonlinear models when certain assumptions are violated. Students will learn how to transform variables to achieve linearity in equations and explore logarithmic models and constant elasticity. Practical demonstrations illustrate linear and power functions, equipping students with essential tools for effective data analysis.

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Understanding Multiple Regression Coefficients and Nonlinear Models in Statistical Analysis

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Presentation Transcript

  1. Lecture 5 HSPM J716

  2. Assignment 4 • Answer checker • Go over results

  3. Multiple regression coefficient and multicollinearity • If Z=aX+b, the denominator becomes 0.

  4. F-test

  5. Heteroskedasticity • Heh’-teh-ro – ske-das’ – ti’-si-tee • Violates assumption 3 that errors all have the same variance. • Ordinary least squares is not your best choice. • Some observations deserve more weight than others. • Or – • You need a non-linear model.

  6. Nonlinear models – adapt linear least squares by transforming variables • Y = ex • Made linear by New Y = log of Old Y.

  7. e • Approx. 2.718281828 • Limit of the expression below as n gets larger and larger

  8. Logarithms and e • e0=1 • Ln(1)=0 • eaeb=ea+b • Ln(ab)=ln(a)+ln(b) • (eb)a=eba • Ln(ba)=aln(b)

  9. Transform variables to make equation linear • Y = Aebxu • Ln(Y) = ln(AebXu) • Ln(Y) = ln(A) + ln(ebX) + ln(u) • Ln(Y) = ln(A) + bln(eX) + ln(u) • Ln(Y) = ln(A) + bX + ln(u)

  10. Transform variables to make equation linear • Y = AXbu • Ln(Y) = ln(AXbu) • Ln(Y) = ln(A) + ln(Xb) + ln(u) • Ln(Y) = ln(A) + bln(X) + ln(u)

  11. Logarithmic models Y = Aebxu Y = Axbu Constant elasticity model The elasticity of Y with respect to X is a constant, b. When X changes by 1%, Y changes by b%. Linear Form ln(Y) = ln(A) + bln(X) + ln(u) U is random error such that ln(u) conforms to assumptions • Constant growth rate model • Continuous compounding • Y is growing (or shrinking) at a constant relative rate of b. • Linear Form ln(Y) = ln(A) + bX + ln(u) U is random error such that ln(u) conforms to assumptions

  12. The error term multiplies, rather than adds. Must assume that the errors’ mean is 1.

  13. Demos • Linear function demo • Power function demo

  14. Assignment 5

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