Econ 240AOutlineReview: Big Picture 1Review: Big Picture 2Review: Big Picture 3Review: Big Picture 4Review: Big Picture 5Review: Big Picture 6Review: Big Picture 7Review: Big Picture 8Review: Big Picture 9SummarySlide 1311 Econ 240AEcon 240APower 17Power 1722OutlineOutline•Review•Projects33Review: Big Picture 1Review: Big Picture 1•#1 Descriptive Statistics–Numericalcentral tendency: mean, median, modedispersion: std. dev., IQR, max-minskewnesskurtosis–Graphical•Bar plots•Histograms•Scatter plots: y vs. x•Plots of a series against time (traces) Question: Is (are) the variable (s) normal?44Review: Big Picture 2Review: Big Picture 2•# 2 Exploratory Data Analysis–Graphical•Stem and leaf diagrams•Box plots•3-D plots55Review: Big Picture 3Review: 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),(~:,//ˆnpBxwherenxnsuccessesp 66Review: Big Picture 4Review: 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 meannxxExzx//][/][95.0]/96.1/96.1[ nxnxp77Review: Big Picture 5Review: 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 distributed95.0]//[025.0025.0 nstxnstxp)//(][,0:,0:0nsxExtHHA88Review: Big Picture 6Review: 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 equation99Review: Big Picture 7Review: 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 test1010Review: Big Picture 8Review: 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 Dummies1111Review: Big Picture 9Review: 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)1212SummarySummary•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
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