Statistical Methods for Economists Lecture 5 Correlation Analysis •David Bartl •Statistical Methods for Economists •INM/BASTE Outline of the lecture •Revision: Scalar product, Expected value, Covariance •Pearson’s Correlation Coefficient •Regression Coefficient •Multiple Correlation Coefficient •Coefficient of Partial Correlation •Hypothesis Testing •Non-parametric and robust methods: Spearman’s Rank Correlation Coefficient Simple Linear Regression: Summary Simple Linear Correlation: Motivation Multiple Linear Regression: Summary Multiple Linear Correlation: Motivation Revision: Scalar Product •Scalar Product & the Length of a vector •Geometrical interpretation •Useful conclusions • Revision: Scalar Product & the Length of a vector Revision: Scalar Product & the Length of a vector Revision: Scalar Product: Geometrical interpretation Revision: Scalar Product: Why cos φ ? axis 1 Revision: Scalar Product Revision: Scalar Product Revision: Expected value, Covariance •Expected value •Covariance •Variance •Standard deviation •Geometrical interpretation •Uncorrelated random variables • Revision: Expected value Revision: Expected value Revision: Covariance & Variance Variance: Geometrical interpretation Variance: Geometrical interpretation Variance: Geometrical interpretation scalar product Standard deviation: Geometrical interpretation Standard deviation: Geometrical interpretation axis 1 axis 2 Standard deviation: Geometrical interpretation axis 1 axis 2 Covariance: Geometrical interpretation Covariance: Geometrical interpretation We then have: Covariance: Geometrical interpretation axis 1 axis 3 axis 2 Covariance: Geometrical interpretation axis 1 axis 3 axis 2 ¡¡¡ Notice !!! axis 1 axis 2 The geometrical interpretations: ¡¡¡ Notice !!! Pearson’s Correlation Coefficient •Covariance •Independent random variables •Uncorrelated random variables •Pearson’s Correlation Coefficient • Revision: Covariance Pearson’s Correlation Coefficient is Pearson’s Correlation Coefficient Pearson’s Correlation Coefficient Pearson’s Correlation Coefficient Pearson’s Correlation Coefficient Regression Coefficient •Regression Coefficient •Regression Lines •Coefficients of Regression • Regression Coefficient Regression Coefficient Regression Coefficient Regression Coefficient Regression Coefficient The Regression Lines Coefficients of Regression Coefficients of Regression ` Coefficients of Regression Multiple Correlation Coefficient & Coefficient of Partial Correlation •Multiple Correlation Coefficient •Coefficient of Partial Correlation • Multiple Correlation Coefficient Multiple Correlation Coefficient ` Multiple Correlation Coefficient Coefficient of Partial Correlation Coefficient of Partial Correlation Coefficient of Partial Correlation Coefficient of Partial Correlation Hypothesis Testing •Motivation •Pearson’s sample correlation coefficient •Sample multiple correlation coefficient •Sample coefficient of partial correlation • Hypothesis Testing: Motivation Hypothesis Testing: Motivation Pearson’s sample correlation coefficient Pearson’s sample correlation coefficient Pearson’s sample correlation coefficient Pearson’s sample correlation coefficient: Theorem Pearson’s sample correlation coefficient: Hyp. Test Sample Multiple Correlation Coefficient and the random variable Sample Multiple Correlation Coefficient Sample Multiple Correlation Coefficient Sample Multiple Correlation Coefficient Sample Multiple Correlation Coefficient: Theorem Sample Multiple Correlation Coefficient: Hyp. Test Sample Coefficient of Partial Correlation Sample Coefficient of Partial Correlation Sample Coefficient of Partial Correlation Sample Coefficient of Partial Correlation Sample Coefficient of Partial Correlation Sample Coefficient of Partial Correlation Sample Coefficient of Partial Correlation: Theorem… Sample Coefficient of Partial Correlation: …Theorem Sample Coefficient of Partial Correlation: Hyp. Test Non-parametric and robust methods: Spearman’s Rank Correlation Coefficient •Introduction •Definition •Simplification •Theorems & Hypothesis Testing •Remarks • Spearman’s rank correlation coefficient: Introduction Spearman’s rank correlation coefficient: Introduction Spearman’s rank correlation coefficient: Introduction Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient: Simplification Secondly, Spearman’s rank correlation coefficient: Simplification Finally, Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient: Simplification Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient Spearman’s rank correlation coefficient: Hyp. testing Spearman’s rank correlation coefficient: Hyp. testing Spearman’s rank correlation coefficient: Hyp. testing Spearman’s rank correlation coefficient: Theorems Spearman’s rank correlation coefficient: Hyp. test Spearman’s rank correlation coefficient: Hyp. test Spearman’s rank correlation coefficient: Hyp. test Spearman’s rank correlation coefficient: Remarks