1. Final Review
Econ 240A

2. Outline The Big Picture
Processes to remember ( and habits to form) for your quantitative career (FYQC)
Concepts to remember FYQC
Discrete Distributions
Continuous distributions
Central Limit Theorem
Regression

3. The Classical Statistical Trail
Rates &
Proportions
Inferential
Statistics
Application
Descriptive Statistics
Discrete Random
Variables
Binomial
Probability
Discrete Probability Distributions; Moments

4. Where Do We Go From Here? Regression Contingency Tables Properties
Assumptions
Violations
Diagnostics
Modeling
Count
ANOVA
Probability

5. Processes to Remember Exploratory Data Analysis
Distribution of the random variable
Histogram Lab 1
Stem and leaf diagram Lab 1
Box plot Lab 1
Time Series plot: plot of random variable y(t) Vs. time index t
X-y plots: Y Vs. x1, y Vs. x2 etc.
Diagnostic Plots
Actual, fitted and residual

6. Concepts to Remember
Random Variable: takes on values with some probability
Flipping a coin
Repeated Independent Bernoulli Trials
Flipping a coin twice
Random Sample
Likelihood of a random sample
Prob(e1^e2 …^en) = Prob(e1)*Prob(e2)…*Prob(en)

7. Discrete Distributions
Discrete Random Variables
Probability density function: Prob(x=x*)
Cumulative distribution function, CDF
Equi-Probable or Uniform
E.g x = 1, 2, 3 Prob(x=1) =1/3 = Prob(x=2) =Prob(x=3)

8. Discrete Distributions
Binomial: Prob(k) = [n!/k!*(n-k)!]* pk (1-p)n-k
E(k) = n*p, Var(k) = n*p*(1-p)
Simulated sample binomial random variable Lab 2
Rates and proportions
Poisson

9. Continuous Distributions
Continuous random variables
Density function, f(x)
Cumulative distribution function
Survivor function S(x*) = 1 – F(x*)
Hazard function h(t) =f(t)/S(t)
Cumulative hazard functin, H(t)

10. Continuous Distributions
Simple moments
E(x) = mean = expected value
E(x2)
Central Moments
E[x - E(x)] = 0
E[x – E(x)]2 =Var x
E[x – E(x)]3 , a measure of skewness
E[x – E(x)]4 , a measure of kurtosis

11. Continuous Distributions
Normal Distribution
Simulated sample random normal variable Lab 3
Approximation to the binomial, n*p>=5, n*(1-p)>=5
Standardized normal variate: z = (x-)/
Exponential Distribution
Weibull Distribution
Cumulative hazard function: H(t) = (1/) t
Logarithmic transform ln H(t) = ln (1/) +  lnt

14. Central Limit Theorem
Sample mean,

15. Population
Random variable x
Distribution f(m, s2)
f ?
Pop.
Sample
Sample Statistic
Sample Statistic:

16. The Sample Variance, s2 Is distributed chi square with n-1 degrees of
freedom (text, 12.2 “inference about a population variance)
(text, pp , Chi-Squared distribution)

17. Regression Models Statistical distributions and tests Assumptions
Student’s t F
Chi Square
Assumptions
Pathologies

18. Regression Models Time Series
Linear trend model: y(t) =a + b*t +e(t) Lab 4
Exponential trend model: y(t) =exp[a+b*t+e(t)]
Natural logarithmic transformation ln
Ln y(t) = a + b*t + e(t) Lab 4
Linear rates of change: yi = a + b*xi + ei
dy/dx = b
Returns generating process:
[ri(t) – rf0] =  + *[rM(t) – rf0] + ei(t) Lab 6

19. Regression Models Percentage rates of change, elasticities
Cross-section
Ln assetsi =a + b*ln revenuei + ei Lab 5
dln assets/dlnrevenue = b = [dassets/drevenue]/[assets/revenue] = marginal/average

20. Linear Trend Model
Linear trend model: y(t) =a + b*t +e(t) Lab 4

21. Lab 4

22. Lab Four F-test: F1,36 = [R2/1]/{[1-R2]/36} = 196
= Explained Mean Square/Unexplained mean square
t-test:
H0: b=0
HA: b≠0
t =[ – 0]/ = -14

23. Lab 4

24. Lab 4

25. Lab 4
2.5%
-14
-2.03

26. Lab Four 5%
4.12
196

27. Exponential Trend Model
Exponential trend model: y(t) =exp[a+b*t+e(t)]
Natural logarithmic transformation ln
Ln y(t) = a + b*t + e(t) Lab 4

28. Lab Four

29. Lab Four

30. Percentage Rates of Change, Elasticities
Cross-section
Ln assetsi =a + b*ln revenuei + ei Lab 5
dln assets/dlnrevenue = b = [dassets/drevenue]/[assets/revenue] = marginal/average

31. Lab Five Elasticity b = 0.778 H0: b=1 HA: b<1
t-crit(5%) = -1.71

32. Linear Rates of Change Linear rates of change: yi = a + b*xi + ei
dy/dx = b
Returns generating process:
[ri(t) – rf0] =  + *[rM(t) – rf0] + ei(t) Lab 6

33. Watch Excel on xy plots!
True x axis: UC Net

34. Lab Six
rGE = a + b*rSP500 + e

35. Lab Six

36. Lab Six

37. View/Residual tests/Histogram-Normality Test

38. Linear Multivariate Regression
House Price, # of bedrooms, house size, lot size
Pi = a + b*bedroomsi + c*house_sizei + d*lot_sizei + ei

39. Lab Six
price
bedrooms
House_size
Lot_size

40. Price = a*dummy2 +b*dummy34 +c*dummy5 +d*house_size01 +e

41. C captures three and four bedroom houses
Lab Six
C captures three and four bedroom houses

42. See Project I PowerPoint application to lottery with Bern variable
Regression Models
How to handle zeros?
Labs Six and Seven: Lottery data-file
Linear probability model: dependent variable: zero-one
Logit: dependent variable: zero-one
Probit: dependent variable: zero-one
Tobit: dependent variable: lottery
See Project I PowerPoint application to lottery with Bern variable

43. Regression Models Failure time models Exponential Weibull
Survivor: S(t) = exp[-*t], ln S(t) = -*t
Hazard rate, h(t) = 
Cumulative hazard function, H(t) = *t
Weibull
Hazard rate, h(t) = f(t)/S(t) = (/)(t/)-1
Cumulative hazard function: H(t) = (1/) t
Logarithmic transform ln H(t) = ln (1/) +  lnt

44. Applications: Discrete Distributions
Binomial
Equi-probable or uniform
Poisson
Rates & proportions, small samples, ex. Voting polls
If I asked a question every day, without replacement, what is the chance I will ask you a question today?
Approximate the binomial where p→0

45. Aplications: Discrete Distributions
Multinomial
More than two outcomes, ex each face of the die or 6 outcomes

46. Applications: Continuous Distributions
Normal
Equi-probable or uniform
Students t
Rates & proportions, np>5, n(1-p)>5; tests about population means given 2
Tests about population means, 2 not known; test regression parameter = 0

47. Applications: Continuous DistributionsF
Ch-Square, 2
Regression: ratio of explained mean square to unexplained mean square, i.e. R2/k÷(1-R2)/(n-k); test dropping 2 or more variables (Wald test)
Contingency Table analysis; Likelihood ratio tests (Wald test)

48. Applications: Continuous Distributions
Exponential
Weibull
Failure (survival) time with constant hazard rate
Failure time analysis, test whether hazard rate is constant or increasing or decreasing

49. Labs 7, 8, 9 Lab 7 Failure Time Analysis
Lab 8 Contingency Table Analysis
Lab 9 One-Way and Two-Way ANOVA