Outline. 2. Logarithmic Functional Form and Units of Measurement. Functional Form. I. Functional Form: log II. Units of Measurement

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1 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 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 = sales (112.8) (0.0089) n=209; R 2 = See Table in Page 5 Interpret = Find the elasticity of CEO salary with respect to sales. 3 4

2 Descriptive Statistics from Eviews Elasticity: (log log model) Date: 05/10/03 Time: 11:30 SALARY SALES Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability 0 0 Observations ) log log form: both Y and X are in logarithmic form. This is called a constant elasticity model. log( ) = log(sales) (0.288) (0.035) n=209 R 2 = = 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 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 = log(sales) (771.5) (92.36) n=209 R 2 = 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)= This is an approximate increase of 11.8% The exact increase is 12.5 %. Interpret: =

3 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( ) = educ n=526, R 2 =.0186 Interpret: = 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: WAGE EDUC Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability 0 0 Sum Sum Sq. Dev Observations Logarithmic Functional Forms Approximate change vs. exact change Logarithmic Functional Forms log(x) : natural log of x Eg. log( ) = edu 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( ) = educ 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

4 Interpretation of Log Models Summary If the model is y = x + u 1 is the change in y for a unit change in x. If the model is log(y) = log(x) + u 1 is the percentage change in y for a percentage change in x. If the model is log(y) = x + u 100* 1 is approximately the percentage change in y for a unit change in x If the model is y = log(x) + u 1 /100 is approximately the change in y for a percentage change in x II. Units of Measurement In summary, for data scaling on y, Let salardol be salary in dollars. salardol = 1000salary = roe = 963, ,501roe s.e. (213,240) (11,123) n = 209 R 2 = 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

5 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. = roe = roedec (213.2) (1112) n = 209 R 2 = 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 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

6 Example: CEO Salary and Sales log( ) = log(sales) Dependent Variable: LOG(SALARY) sales = millions of dollars salary: thousands of dollars (1) log( ) = log(sales) (s.e.) (0.288) (0.035) n=209, R 2 = salarydol = 1000salary (2) log( ) = log(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 LOG(SALES) R-squared Mean dependent var Adjusted R-squared S.D. dependent var (3) log( ) = log(salesdol) (s.e.) (0.764) (0.035) S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Are R 2 different in three models? Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) log ) = log(sales) log( ) = log(salesdol) Dependent Variable: LOG(SALARY*1000) Method: Least Squares Included observations: 209 Variable Coefficient Std. Error t-statistic Prob. C LOG(SALES) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: LOG(SALARY) Method: Least Squares Included observations: 209 Variable Coefficient Std. Error t-statistic Prob. C LOG(SALES* ) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

7 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 Recap Functional Form: log Units of Measurement 2. Unites of Measurement and Logarithmic Functional Form. Quantitative Methods of Economic Analysis Chairat Aemkulwat 27