Download Power 17

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
pˆ  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  t0.025s / n    x  t0.025s / 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 + b*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