Econ 240A Power 17 1 Outline Review Projects 2 Review Big Picture 1 1 Descriptive Statistics Numerical central tendency mean median mode dispersion std dev IQR max min skewness kurtosis Graphical Bar plots Histograms Scatter plots y vs x Plots of a series against time traces Question Is are the variable s normal 3 Review Big Picture 2 2 Exploratory Data Analysis Graphical Stem and leaf diagrams Box plots 3 D plots 4 Review Big Picture 3 3 Inferential statistics Random variables Probability Distributions Discrete Equi probable uniform binomial Poisson Probability density Cumulative Distribution Function Continuous normal uniform exponential Density CDF Standardized Normal z N 0 1 Density and CDF are tabulated Bivariate normal Joint density marginal distributions conditional distributions Pearson correlation coefficient iso probability contours Applications sample proportions from polls p successes n x n where x B p n 5 Review Big Picture 4 Inferential Statistics Cont The distribution of the sample mean is different than the distribution of the random variable Central limit theorem z x Ex x x n Confidence intervals for the unknown population mean p x 1 96 n x 1 96 n 0 95 6 Review Big Picture 5 Inferential Statistics If population variance is unknown use sample standard deviation s and Student s t distribution p x t0 025 s n x t0 025 s n 0 95 Hypothesis tests H 0 0 H A 0 t x Ex s n Decision theory minimize the expected costs of errors Type I error Type II error Non parametric statistics techniques of inference if variable is not normally distributed 7 Review Big Picture 6 Regression Bivariate and Multivariate Time series Linear trend y t a b t e t Exponential trend ln y t a b t e t Quadratic trend y t a b t c t2 e t Elasticity estimation lny t a b lnx t e t Returns Generating Process ri t c rM t e t Problem autocorrelation Diagnostic Durbin Watson statistic Diagnostic inertial pattern in plot trace of residual Fix up Cochran Orcutt Fix up First difference equation 8 Review Big Picture 7 Regression Bivariate and Multivariate Cross section Linear y i a b x i e i i 1 n b dy dx Elasticity or log log lny i a b lnx i e i b dy dx y x Linear probability model y 1 for yes y 0 for no y a b x e Probit or Logit probability model Problem heteroskedasticity Diagnostic pattern of residual or residual squared with y and or x Diagnostic White heteroskedasticity test Fix up transform equation for example divide by x Table of ANOVA Source of variation explained unexplained total Sum of squares degrees of freedom mean square F test 9 Review Big Picture 8 Questions quantitative dependent qualitative explanatory variables Null No difference in means between two or more populations groups One Factor Graph Table of ANOVA Regression Using Dummies Null No difference in means between two or more populations groups Two Factors Graph Table of ANOVA Comparing Regressions Using Dummies 10 Review Big Picture 9 Cross classification nominal categories e g male or female ordinal categories e g better or worse or quantitative intervals e g 13 19 20 29 Two Factors mxn m 1 x n 1 degrees of freedom Null independence between factors expected number in cell i j p i p j n Pearson Chi square statistic sum over all i j of observed i j expected i j 2 expected i j 11 Summary Is there any relationship between 2 or more variables quantitative y and x graphs and regression Qualitative binary y and quantitative x probability model linear or non linear Quantitative y and qualitative x graphs and Tables of ANOVA and regressions with indicator variables Qualitative y and x Contingency Tables 12 13
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