OLS Examples Page 1 Problem OLS Regression The Kelley Blue Book provides information on wholesale and retail prices of cars. Following are age and price data for 10 randomly selected Corvettes between 1 and 6 years old. Here, age is in years, and price is in hundreds of dollars. age 6 6 6 2 2 5 4 5 1 4 price 205 195 210 340 299 230 270 243 340 240 Page 2 1
OLS Regression Coding Sheet for Corvette Data Variable Age of Corvettes Price of Corvettes Possible Values Years Hundred of Dollars, Nominal Source Kelley Blue Book, various issued IBID. Mnemonic age price Page 3 Page 4 2
Note Mean and Std. Dev. of PRICE S XX S YY Note from Stat 101: ( Y Y ) = SYY s = n 1 n 1 = 25681.6 9 2 = 53.41827 Page 5 Page 6 3
OLS Specification LS price C age Form for OLS command: LS dependent variable C independent variable(s) or Equation eq1.ls dependent variable C independent variable(s) Page 7 Section 1 Section 2..Standard table layout Section 3 Page 8 4
Equation for Ŷ : Estimated Price = 371.6019-27.90291* AGE Page 9 Estimated Price = 371.6019-27.90291* AGE 360 PRICE of Corvettes vs. AGE of Corvettes 320 PRICE 280 240 Residual 200 160 0 1 2 3 4 5 6 7 AGE Page 10 5
SSE = Sum of Squared Residuals = 1623.709 s = Standard Error of Regression = 14.24653 Note: SSE 1623.709 = = = 202.96363 n 2 10 2 s = 202.96363 = 14.24653 s 2 Page 11 Standard Errors of Estimates Note: s AGE = s S XX 14.24653 = = 2.562889 30.9 Page 12 6
t c and p-values Both Highly Significant Note: t AGE 27.90291 = = 10.88729 2.562889 Page 13 Page 14 7
Sum is zero SSE Page 15 No Explanatory Variable Page 16 8
SST = Note: 2 ( Y - Y) = S = 25681. 60 YY R 2 F c and p-value R 2 Note: SSE SSE = 1 = 1 SST S 1623.709 = 1 25681.60 = 0.936775 YY Page 17 4.1 η Price = -27.90291 257.2 = -0.444798 Very Inelastic Page 18 9
OLS Regression Variable Price Elasticity Summary Table Estimate Mean Elasticity -27.90291 4.1-0.444798 Classification Inelastic Interpretation: Price of a corvette is inelastic with respect to the age of the corvette so that a 1% increase in age decreases the price by only 0.4%. Other factors are at play regarding the lower prices, but age is certainly a major factor as evidenced by the R 2 of 0.94. Page 19 Other Examples Problem 1: CAPM Page 20 10
Problem 1: CAPM Problem Estimate β, the systematic risk, for CAPM CAPM is E(r Ω i An empirical version is t ) = r f + β i [ E(r Ω ) r ] m t ~ r α β ~ i = + i r m + ε 2 ε ~ N(0, σ ) ~ ri = ri - rf ~ r = r - r m α = 0 m f f i Page 21 beta is key Problem 1: CAPM Each security has a beta Each portfolio has a beta Portfolio beta is weighted average of individual betas Page 22 11
Problem 1: CAPM beta is proportionality factor E(ri Ω t ) rf βi = E(r Ω ) r m Excess returns for security over riskless return is proportional to excess returns in market over riskless returns Excess is extra return to compensate for risk for not holding the market portfolio Premium on security is proportional to premium on the market portfolio t f Page 23 Problem 1: CAPM If E(ri Ω t ) rf βi = > 1 Premium i > Premiumm E(r Ω ) r m t f then asset i is viewed as riskier than the market Page 24 12
Problem 1: CAPM beta Interpretation Example 1 Moves With Market Conglomerates (AT&T) >1 More Volatile Companies Sensitive To Macro Events (Autos) 0<beta<1 Less Volatile Mildly Sensitive To Macro Events (Electric Utilities) <0 Move Counter To Goldmining Stocks Market Page 25 Coding Sheet for CAPM Data Variable Excess returns for an overall stock market index of the total market in the UK. Monthly, 1/80-12/99. Excess returns on an index of 104 stocks in the cyclical consumer goods sector in the UK. Monthly, 1/80-12/99. Excess returns on an index of 104 stocks in the noncyclical consumer goods sector in the UK. Monthly, 1/80-12/99. Excess returns on an index of 104 stocks in the information tech sector in the UK. Monthly, 1/80-12/99. Possible Values Percentages* Source DataStream (2000) Mnemonic rendmark Percentages* IBID. rendcyco Percentages* IBID. rendncco Percentages* IBID. rendit Excess returns on an index of 104 stocks in the telcom sector in the UK. Monthly, 1/80-12/99. Percentages* IBID. rendtel *Note: See calculations note. Page 26 13
Problem 1: CAPM Excess returns are calculated as follows Let p i be the closing price of the index at the last trading day in month i and let r i be the onemonth interest rate at the start of month i. Then the return v i of the index is v ( ) / i = pi pi 1 pi 1 and the excess return is v i r i. The reported numbers are 100(v i r i ). Page 27 Page 28 14
60 Excess Returns Distributions Stock Market Sectors (UK) 1980-1999 40 Excess Returns (%) 20 0-20 -40 RENDTEL RENDNCCO RENDMARK RENDIT RENDCYCO Note: Monthly data Page 29 20 Excess Returns Overall Stock Market Index (UK) 1980-1999 10 Excess Returns (%) 0-10 -20-30 80 82 84 86 88 90 92 94 96 98 Note: Monthly data Page 30 15
70 60 50 40 30 20 10 Excess Returns Overall Stock Market Index (UK) 1980-1999 0-30 -20-10 0 10 Series: RENDMARK Sample 1980M01 1999M12 Observations 240 Mean 0.808884 Median 1.204026 Maximum 13.46098 Minimum -27.86969 Std. Dev. 4.755913 Skewness -1.157801 Kurtosis 8.374932 Jarque-Bera 342.5191 Probability 0.000000 Page 31 30 Excess Returns Cyclical Consumers Goods Sector (UK) 1980-1999 20 Excess Returns (%) 10 0-10 -20-30 -40 80 82 84 86 88 90 92 94 96 98 Note: Monthly data Page 32 16
40 30 20 10 0 Excess Returns Cyclical Consumer Goods Sector (UK) 1980-1999 -30-20 -10 0 10 20 Series: RENDCYCO Sample 1980M01 1999M12 Observations 240 Mean 0.499826 Median 0.309320 Maximum 22.20496 Minimum -35.56618 Std. Dev. 7.849594 Skewness -0.230900 Kurtosis 4.531597 Jarque-Bera 25.59050 Probability 0.000003 Page 33 20 Excess Returns Noncyclical Consumer Goods Sector (UK) 1980-1999 10 Excess Returns (%) 0-10 -20-30 -40 80 82 84 86 88 90 92 94 96 98 Note: Monthly data Page 34 17
60 Excess Returns Information Technology Sector (UK) 1980-1999 40 Excess Returns (%) 20 0-20 -40 80 82 84 86 88 90 92 94 96 98 Note: Monthly data Page 35 40 Excess Returns Telecommunications Sector (UK) 1980-1999 30 Excess Returns (%) 20 10 0-10 -20-30 80 82 84 86 88 90 92 94 96 98 Note: Monthly data Page 36 18
Page 37 Problem 1: CAPM Interpretation The intercept is highly insignificant suggesting that the line goes through the origin The slope is highly significant and positive at the 0.0 level indicating that the market rate of return (excess returns) have a positive effect on the returns for the cyclical consumer goods sector in the UK The R 2 is 0.50 suggesting that 50% of the variation in returns in the cyclical consumer goods sector is accounted for by the market excess returns The model is significantly different from the naïve model as indicated by the p-value for F Page 38 19
Problem 1: CAPM Interpretation Since this is a CAPM, the slope has the interpretation of systematic risk Since it is greater than 1.0, this indicates that the cyclical consumer goods sector is riskier that the market as a whole When the market rises, the excess returns in this sector will rise more than the market Page 39 Problem 1: CAPM Model Portfolio Model Intercept RendMark R 2 F C -0.477 (0.2188) 1.17* (0.0000) 0.5035 241.3363 (0.0000) Notes: p-value in parenthese; *=significant Page 40 20
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Page 43 Other Examples Problem 2: Bank Wages Page 44 22
Problem 2: Bank Wages Problem Find the relationship between the salary of employees at a major US bank and their years of education completed Page 45 Coding Sheet for Bank Salary Data Variable Possible Values Source Mnemonic Current yearly salary of bank employees in US Nominal dollars SPSS, version 10 (2000) salary Number of years of education completed Years IBID. educ Natural log of salary logs Calculated as natural log of salary logsalary Page 46 23
140000 Current Yearly Salary vs. Education Completed 120000 Current Yearly Salary ($) 100000 80000 60000 40000 20000 0 6 8 10 12 14 16 18 20 22 Education (Years Completed) Page 47 Page 48 24
140000 Distribution of Current Yearly Salary Bank Employees (US) 120000 Current Yearly Salary ($) 100000 80000 60000 40000 20000 0 Page 49 70 60 50 40 30 20 10 0 Distribution of Current Yearly Salary Bank Employees (US) 25000 50000 75000 100000 125000 Series: SALARY Sample 1 474 IF GENDER=1 Observations 258 Mean 41441.78 Median 32850.00 Maximum 135000.0 Minimum 19650.00 Std. Dev. 19499.21 Skewness 1.629931 Kurtosis 5.702920 Jarque-Bera 192.7741 Probability 0.000000 Page 50 25
12.0 Distribution of Log of Current Yearly Salary Bank Employees (US) Natural Log of Current Year Salary 11.6 11.2 10.8 10.4 10.0 9.6 Page 51 36 32 28 24 20 16 12 8 4 0 Distribution of Log of Current Yearly Salary Bank Employees (US) 10.0 10.5 11.0 11.5 Series: LOGSALARY Sample 1 474 IF GENDER=1 Observations 258 Mean 10.54461 Median 10.39967 Maximum 11.81303 Minimum 9.885833 Std. Dev. 0.398579 Skewness 0.840303 Kurtosis 2.805992 Jarque-Bera 30.76731 Probability 0.000000 Page 52 26
Problem 2: Bank Wages Our model is β1educi εi Salary i = Ae e or ln ( SALARYi ) = β0 + β1educi + εi Interpret the slope parameter as the percentage increase in salary (S) due to one additional year of education d ln( S ) ds S = dx dx Page 53 Salary rises 9% for each additional year of education Page 54 27
Problem 2: Bank Wages Interpretation The intercept and slope parameters are highly significant at the 0.0 level This indicates that education has a significant, positive effect on salary The higher the level of education, the higher the salary For each extra 1 year of education, salary rises 9% Almost 49% of the variation in (log) salary is accounted for by education The model is significantly different from the naïve model as indicated by the p-value for F Page 55 Problem 2: Bank Wages Model Portfolio Model Intercept Educ R 2 F C 9.22* (0.0000) 0.092* (0.0000) 0.4680 225.2383 (0.0000) Notes: p-value in parenthese; *=significant Page 56 28
Other Examples Problem 3: Population and Economic Growth Page 57 Problem 3:Population and Economic Growth Population growth is an important factor for long-term economic growth Malthus assumed (correctly) that population will grow geometrically (e.g., 1, 3, 9, 27, 81, ) while the food supply will increase arithmetically (e.g., 10, 10, 30, 40 ) Population would double every 24 years As a result of the faster growth of population, population growth would overtake food supply growth but he didn t specify when Page 58 29
Problem 3:Population and Economic Growth Population Food Food Output and Population Time Page 59 Problem 3:Population and Economic Growth Digression on compound growth Compounding t ( 1 g) t FV = PV + Time to double What is the implied population growth rate, g, for Malthus? How long to double at 1.9%? 0 Page 60 30
Problem 3:Population and Economic Growth Curent population projections The Year 2000 population growth rate, g, was about 1.4% World population in 2000 was about 6.1Billion people Absolute growth: 85.4 Million in one year Page 61 Problem 3:Population and Economic Growth Page 62 31
Problem 3:Population and Economic Growth Page 63 Problem 3:Population and Economic Growth The future There is some hope ahead The UN forecasts a drop in fertility in developing countries Developed countries will grow by less than 1% annually Predicted world s population in 2300 Old: 12 Billion New: 9 Billion Page 64 32
Problem 3:Population and Economic Growth Malthus had checks on population growth Contraceptives Abortion Self-restraint Wars Malthus wrote right at the time of the Industrial and Green Revolutions, so he could not see their implications Page 65 Problem 3:Population and Economic Growth Data from the World Development Indicators, World Bank (2004) Calculated average annual growth for GDP and population 1 40 Value in 2003 i gi = 1, i = 1,...,107 countries Value in 1963 i Page 66 33
Coding Sheet for Growth Data Variable Real GDP in country (1995 $US), 1963 and 2003 Possible Values Real dollars Source World Development Indicators (World Bank, 2004) Mnemonic gdp63, gdp03 Average annual growth in GDP Decimal values Calculated g Population in country, 1963 and 2003 Count World Development Indicators (World Bank, 2004) pop63, pop03 Average annual growth in population Decimal values Calculated p Page 67 Page 68 34
.10 Real GDP and Population Average Annual Growth 1963-2003 Real GDP Average Annual Growth.08.06.04.02.00 -.02.00.01.02.03.04 Population Average Annual Growth by Country (n = 107) Page 69 Highly insignificant Terrible R 2 Page 70 35
.10 Real GDP and Population Average Annual Growth 1963-2003 Real GDP Average Annual Growth.08.06.04.02.00 -.02.00.01.02.03.04 Population Average Annual Growth by Country (n = 107) Page 71 36