Final Review

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? Contingency Tables Regression Properties Assumptions Violations Diagnostics Modeling ANOVA Count 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. 266-270, Chi-Squared distribution)

17. Regression • Models • Statistical distributions and tests • 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 =[ -0.000915 – 0]/0.0000653 = -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 • 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 t25 = [0.778 – 1]/0.148 = - 1.5 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. Lab Six C captures three and four bedroom houses

42. 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 vehicle type data

43. Regression Models • Failure time models • Exponential • 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