1 Econ 240A Power 17

2 Outline Review Projects

3 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?

4 Review: Big Picture 2 # 2 Exploratory Data Analysis –Graphical Stem and leaf diagrams Box plots 3-D plots

5 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

6 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 –Confidence intervals for the unknown population mean

7 Review: Big Picture 5 Inferential Statistics –If population variance is unknown, use sample standard deviation s, and Student’s t-distribution –Hypothesis tests –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

8 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*t 2 + e(t) Elasticity estimation: lny(t) = a + b*lnx(t) +e(t) Returns Generating Process: r i (t) = c +  r M (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

9 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

10 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

11 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 , –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)

12 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

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