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1 7. Multiple Regression IV ECON 251 Research Methods
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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
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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
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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.
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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
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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
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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
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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
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9 Human resource application ln(WAGE) = β 0 + β 1 EDUC + ε ln(WAGE)=7.1565+0.0479*EDUC
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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
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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
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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
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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!
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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?
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15 Example – Demand for bus travel Final model What do the coefficients mean?
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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
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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
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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
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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
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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
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21 Example – US manufacturing
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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
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