1. 1 7. Multiple Regression IV ECON 251 Research Methods

2. 2 Functional forms  The logarithmic and exponential functions are two of the most commonly used functions in model formulations.  With the logarithmic and exponential functions we can capture a variety of effects: marginal effects diminishing returns returns to scale growth rates odds

3. 3 Properties of logs  Y = A + B  ln Y = ln(A + B)  Y = AB  ln Y = ln(AB) = ln A + ln B  Y = A/B  ln Y = ln(A/B) = ln A - ln B  Y = A B  ln Y = ln(A B ) = B * ln A  A B  e B * ln A

4. 4 Marginal effects and elasticities  What is an elasticity? % change in Y with respect to a % change in X for a small change in X  What is a marginal effect? the change in Y per unit change in X example: price elasticity of demand  if price increases by 1% quantity demanded changes by η q,p.

5. 5 Marginal effects and elasticities NameFunctional FormMarginal EffectElasticity LinearY=β 0 + β 1 Xβ1β1 β 1 X/Y Linear-logY=β 0 + β 1 lnXβ 1 /Xβ 1 /Y QuadraticY=β 0 + β 1 X + β 2 X 2 β 1 + 2β 2 X(β 1 + 2β 2 X)X/Y Log-linearlnY=β 0 + β 1 Xβ1Yβ1Yβ1Xβ1X Double-loglnY=β 0 + β 1 lnXβ 1 Y/Xβ1β1 Logisticln[Y/(1-Y)]=β 0 + β 1 Xβ 1 Y(1-Y)β 1 (1-Y)X

6. 6 Linear-log model  Y = β 0 + β 1 ln X + ε used to capture the diminishing marginal returns (product) X Y β 0 + β 1 ln X

7. 7 Log-linear (semilog) model  ln Y = β 0 + β 1 X + ε used to capture constant growth rates P t = (1+g)P t-1  P t = P 0 (1+g) t  ln P t = ln P 0 + t ln(1+g)  ln P t = β 0 + β 1 t where β 0 = ln P 0 and β 1 = ln(1+g) X Y e β 0 + β 1 X

8. 8 Human resource application  theory suggests that the rate of return to an extra year of education is r. Then, for the first period w 1 = (1 + r ) w 0. For, the second period w 2 = (1 + r ) 2 w 0, and for s years w s = (1 + r ) s w 0. Then, taking logs of both sides yields: ln(w s ) = s ln(1 + r ) + ln(w 0 ) = β 0 + β 1 s  model: ln(WAGE) = β 0 + β 1 EDUC + ε  download file: HR.xls

9. 9 Human resource application  ln(WAGE) = β 0 + β 1 EDUC + ε ln(WAGE)=7.1565+0.0479*EDUC

10. 10  ln( WAGE )= 7.1565 + 0.0479* EDUC  the rate of return to an extra year of education is r  ln(1 + r )=0.0479  1 + r = e 0.0479  r = 0.0491 Marginal Effect: say when EDUC = 5: Y = e 7.1565 + 0.0479 * 5 = e 7.3962 = 1629.8571 Maginal Effect (X=5) : β 1 Y = 0.0479*1629.8571=78.14 How about when EDUC = 6: Maginal Effect (X=6) : β 1 Y = 0.0479*1709.8988=81.98 Human resource application – Interpretation

11. 11  So, when a person has 5 year of education, an extra year of education would increase salary by $78.14  when a person has 6 years of education, an extra year of education would increase salary by $81.98  Can you find percentage changes in salaries?  % changes: when EDUC = 5  78.14/1629.8571=0.04794 when EDUC = 6  81.98/1709.8988=0.04794  so, the interpretation of β 1 is the expected % change in Y when X increases by 1.  would a linear model be better (higher adj. R 2 )? cannot compare R 2 if dependent variable is not the same Human resource application – Interpretation

12. 12 Log-log (double log) model  popular in estimating demand functions coefficients are constant elasticities  Example: Demand for bus travel depends on ―price ―income ―price of a substitute good ―other factors

13. 13 Example – Demand for bus travel  Variables (Bus Travel) BUSTRAVL : demand for urban transportation by bus in thousands of passenger hours FARE : bus fare in dollars GASPRICE : Price of a gallon of gasoline in dollars INCOME : Average income per capita in dollars POP : Population in city in thousands DENSITY : density of population (persons/sq. mile) LANDAREA : land area of the city (sq. miles)  Find relevant elasticities of demand!

14. 14 Example – Demand for bus travel Modeling:  How does the full model perform?  Are there any insignificant variables?  Try removing out the most insignificant variable. What happens? Are there any other insignificant variables?  Which variables will you keep for your final model?  How does the final model perform?  Is there an improvement from the initial to the final model?

15. 15 Example – Demand for bus travel  Final model What do the coefficients mean?

16. 16 Interpretation:  ln( BUSTRAVL ) = 45.85 – 4.73 * ln( INCOME ) +1.82 * ln( POP ) – 0.97 * ln( LANDAREA )  What is income elasticity of demand for bus travel? η BT,I = -4.73  Is demand for bus travel elastic/inelastic/unit elastic with respect to POPULATION and LANDAREA? (disregard the sign of the coefficient) if η < 1: inelastic if η = 1: unit elastic if η > 1: elastic Example – Demand for bus travel

17. 17 Example – Demand for bus travel  Question: Is demand for bus travel elastic with respect to city’s population?  POPULATION H 0 : β 2 = 1 H 1 : β 2 > 1  t 0.05,36 = 1.68  t-stat ______________ than the critical value  __________ null hypothesis and conclude that demand for bus travel is ___________ w.r.t. population

18. 18 Log-log (double log) model  popular in estimating production functions Cobb-Douglass production function  used to capture: elasticities returns to scale ln Q = ln A + β 1 ln K + β 2 ln L + ε

19. 19 Cobb-Douglass production function  β 1 – elasticity of output wrt capital  β 2 – elasticity of output wrt labor Returns to scale:  when you double inputs, what happens to output? decreasing RTS: β 1 + β 2 < 1 constant RTS: β 1 + β 2 = 1 increasing RTS: β 1 + β 2 > 1

20. 20 Example – US manufacturing  load the dataset ( US manufactring.xls )  The data consist of labor hours as the estimate for labor input ( L ), total capital expenditures as the estimate for capital input ( K ), and value added as the estimate for output ( Q ).  transform the variables into logs  run a regression with ln Y being a dependent variable and ln K and ln L as independent variables  what can you say about the elasticities?  does the US manufacturing exhibit increasing, constant, or decreasing returns to scale?  write down the US manufacturing production function

21. 21 Example – US manufacturing

22. 22 Example – US manufacturing Interpretation:  Elasticities: both β 1 and β 2 < 1  inelastic  Returns to scale β 1 + β 2 = 0.5213 + 0.4683 = 0.9896  CRTS  Production function ln Y = 3.8876 + 0.5213 * ln K + 0.4683 * ln L take antilogs of both sides: Y = 48.794 K 0.5213 L 0.4683