Outline 2. Logarithmic Functional Form and Units of Measurement I. Functional Form: log II. Units of Measurement Read Wooldridge (2013), Chapter 2.4, 6.1 and 6.2 2 Functional Form I. Functional Form: log OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y. Note that SLR.1 (linear in the parameters) is not violated Can take the natural log of x, y or both Can use quadratic forms of x Can use interactions of x variables 1) level level form: Linear variables in simple regression models Eg. salary in thousands of dollars sales in millions of dollars = 1174 + 0.0155sales (112.8) (0.0089) n=209; R 2 =0.014369 See Table in Page 5 Interpret = 0.0155 Find the elasticity of CEO salary with respect to sales. 3 4
Descriptive Statistics from Eviews Elasticity: (log log model) Date: 05/10/03 Time: 11:30 SALARY SALES Mean 1281.12 6923.793 Median 1039 3705.2 Maximum 14822 97649.9 Minimum 223 175.2 Std. Dev. 1372.345 10633.27 Skewness 6.854923 4.999125 Kurtosis 60.54128 35.29968 Jarque-Bera 30470.1 9955.666 Probability 0 0 Observations 209 209 2) log log form: both Y and X are in logarithmic form. This is called a constant elasticity model. log( ) = 4.821 + 0.257log(sales) (0.288) (0.035) n=209 R 2 =0.210817 = 0.257 This is the elasticity of salary with respect to sales Find the partial effect of sales on salary in thousands of dollars. See Table in Page 5 5 6 Level log Model Digression: Percentage point vs. Percent 3) level log form: independent variable in logarithmic form salary in thousands of dollars log(sales) sales in millions of dollars = 898.9 + 262.9log(sales) (771.5) (92.36) n=209 R 2 =0.037672 Interpretation: percentage point change vs. percentage change Unemployment rate: 8% to 9% rate = 1. This is a one percentage point change. log(rate): log(9) log(8)= 0.118. This is an approximate increase of 11.8% The exact increase is 12.5 %. Interpret: = 262.9 7 8
Semi elasticity (log level) 4) log level form: dependent variable in logarithmic form log(wage) wages in dollars per hour educ years of education log( ) = 0.583773 + 0.082744educ n=526, R 2 =.0186 Interpret: = 0.083 Find the semi elasticity of wages with respect to education. Find the elasticity of wages with respect to education. See Table in Page 5 Descriptive Statistics from Eviews Date: 06/02/09 Time: 09:10 Sample: 1 526 WAGE EDUC Mean 5.896103 12.56274 Median 4.65 12 Maximum 24.98 18 Minimum 0.53 0 Std. Dev. 3.693086 2.769022 Skewness 2.007325-0.61957 Kurtosis 7.970083 4.884245 Jarque-Bera 894.6195 111.4653 Probability 0 0 Sum 3101.35 6608 Sum Sq. Dev. 7160.414 4025.43 Observations 526 526 9 10 Logarithmic Functional Forms Approximate change vs. exact change Logarithmic Functional Forms log(x) : natural log of x Eg. log( ) = 0.584 + 0.083edu When education increases by one year, wages increase approximately by 8.3% (or 100(0.083)) This estimate is approximated or inexact. Approximate percentage change log( ) = + educ log( ) = 0.584 + 0.083educ When education increases by one year, the hourly wage increases by approximately 8.3% (100(0.083)) Let y=wage; x=educ. This is because as the change in log(y) becomes larger and larger, the approximation % y 100 log (y) becomes more and more inexact. Exact percentage change in the predicted y is % y = 100[exp( ) 1] = 8.654% One more year of education increases the predicted wages exactly by 8.65% 11 12
Interpretation of Log Models Summary If the model is y = 0 + 1 x + u 1 is the change in y for a unit change in x. If the model is log(y) = 0 + 1 log(x) + u 1 is the percentage change in y for a percentage change in x. If the model is log(y) = 0 + 1 x + u 100* 1 is approximately the percentage change in y for a unit change in x If the model is y = 0 + 1 log(x) + u 1 /100 is approximately the change in y for a percentage change in x 13 14 II. Units of Measurement In summary, for data scaling on y, Let salardol be salary in dollars. salardol = 1000salary = 963.191 + 18.501roe = 963,191 + 18,501roe s.e. (213,240) (11,123) n = 209 R 2 = 0.0132 roe (in percentage points) The old values (residuals, coefficients, s.e.) are multiplied by c 1 =1000 to get the corresponding new values. Note that statistics involving ratios (R 2, t statistic) are unaffected. Interpret: = 18,501 The predicted CEO salary increases by $18,500 when roe increases by 1 percentage point. 15 16
Data Scaling on Independent Variable In summary, for data scaling on x, Independent variable roe roedec roedec = (1/100)roe in percent in fraction The OLS coefficient and its standard error (as well as residuals) are divided by c 2 =1/100 to get the corresponding new values. = 963.191 + 18.501 roe = 963.191 + 1850.1 roedec (213.2) (1112) n = 209 R 2 = 0.0132 Note that the intercept and other statistics involving ratios (R 2, t statistics) are unaffected. Interpretation: If roedec increases by 0.01, the salary is predicted to increase by 18.5 thousand dollars. 17 18 Redefining Variables Rescaling and Log Form Changing the scale of the y variable will lead to a corresponding change in the scale of the coefficients and standard errors, so no change in the significance or interpretation Changing the scale of one x variable will lead to a change in the scale of that coefficient and standard error, so no change in the significance or interpretation Let y i * = c 1 y i let x i * = c 2 x i In summary, if the dependent variable or independent variables are in logarithmic form, eg., log(y i *), log(x i *), changing the units of measurement does not affect the slope coefficient. Only OLS intercept is affected. 19 20
Example: CEO Salary and Sales log( ) = 4.821 + 0.257log(sales) Dependent Variable: LOG(SALARY) sales = millions of dollars salary: thousands of dollars (1) log( ) = 4.821 + 0.257log(sales) (s.e.) (0.288) (0.035) n=209, R 2 =0.210817 salarydol = 1000salary (2) log( ) = 11.73 + 0.257log(sales) (s.e.) (0.288) (0.035) salesdol = 1,000,000sales Method: Least Squares Date: 06/05/08 Time: 05:38 Included observations: 209 Variable Coefficient Std. Error t-statistic Prob. C 4.821996 0.288340 16.72332 0.0000 LOG(SALES) 0.256672 0.034517 7.436168 0.0000 R-squared 0.210817 Mean dependent var 6.950386 Adjusted R-squared 0.207005 S.D. dependent var 0.566374 (3) log( ) = 1.276 +0.257log(salesdol) (s.e.) (0.764) (0.035) S.E. of regression 0.504358 Akaike info criterion 1.478462 Sum squared resid 52.65600 Schwarz criterion 1.510446 Are R 2 different in three models? Log likelihood -152.4992 F-statistic 55.29659 Durbin-Watson stat 1.859599 Prob(F-statistic) 0.000000 21 22 log ) = 11.73 + 0.257log(sales) log( ) = 1.276 +0.257log(salesdol) Dependent Variable: LOG(SALARY*1000) Method: Least Squares Included observations: 209 Variable Coefficient Std. Error t-statistic Prob. C 11.72975 0.288340 40.68033 0.0000 LOG(SALES) 0.256672 0.034517 7.436168 0.0000 R-squared 0.210817 Mean dependent var 13.85814 Adjusted R-squared 0.207005 S.D. dependent var 0.566374 S.E. of regression 0.504358 Akaike info criterion 1.478462 Sum squared resid 52.65600 Schwarz criterion 1.510446 Log likelihood -152.4992 F-statistic 55.29659 Durbin-Watson stat 1.859599 Prob(F-statistic) 0.000000 Dependent Variable: LOG(SALARY) Method: Least Squares Included observations: 209 Variable Coefficient Std. Error t-statistic Prob. C 1.275946 0.763884 1.670341 0.0964 LOG(SALES*1000000) 0.256672 0.034517 7.436168 0.0000 R-squared 0.210817 Mean dependent var 6.950386 Adjusted R-squared 0.207005 S.D. dependent var 0.566374 S.E. of regression 0.504358 Akaike info criterion 1.478462 Sum squared resid 52.65600 Schwarz criterion 1.510446 Log likelihood -152.4992 F-statistic 55.29659 Durbin-Watson stat 1.859599 Prob(F-statistic) 0.000000 23 24
Log is cool... Rules of Thumb for taking logs Reasons why taking logs are preferable: 1. Gauss Markov assumptions (SLR.1 SLR.5) For example, heteroskedasticty. 2. Estimates less sensitive to outlying (extreme) values 3. Meaningful economic interpretation 1. a variable with positive dollar amount Eg. wages, salaries, firm sales, and market capitalization value 2. a variable with large integer values Eg. population, total number of employees 3. Maybe, a proportion or percent Eg. unemployment rate, participation rate Variables in their original form Variables measured in years Eg. education, experience, tenure, age 25 26 Recap Functional Form: log Units of Measurement 2. Unites of Measurement and Logarithmic Functional Form. Quantitative Methods of Economic Analysis. 2949605. Chairat Aemkulwat 27