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Econ 240AEcon 240A

Power 17Power 17

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OutlineOutline

• Review

• Projects

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Review: 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?

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Review: Big Picture 2Review: Big Picture 2

• # 2 Exploratory Data Analysis– Graphical

• Stem and leaf diagrams• Box plots• 3-D plots

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Review: 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

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Review: 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 mean

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95.0]/96.1/96.1[ nxnxp

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Review: 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 distributed

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)//(][,0:,0:0 nsxExtHH A

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Review: 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 equation

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Review: 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 test

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Review: 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 Dummies

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Review: 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)

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SummarySummary

• 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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