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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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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 = AB ln Y = ln(AB) = B * ln A
AB e B * ln A
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Marginal effects and elasticities
What is an elasticity?• % change in Y with respect to a % change in X for a small
change in X
Y
X
X
Y
XXYY
X
Yxy
%
%,
X
Y
Effect Marg
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 ηq,p.
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Marginal effects and elasticities
Name Functional Form Marginal Effect Elasticity
Linear Y=β0+ β1X β1 β1X/Y
Linear-log Y=β0+ β1lnX β1/X β1/Y
Quadratic Y=β0+ β1X + β2X2 β1 + 2β2X (β1 + 2β2X)X/Y
Log-linear lnY=β0+ β1X β1Y β1X
Double-log lnY=β0+ β1lnX β1Y/X β1
Logistic ln[Y/(1-Y)]=β0+ β1X β1Y(1-Y) β1(1-Y)X
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Linear-log model Y = β0 + β1 ln X + ε
• used to capture the diminishing marginal returns (product)
X
Y
β0+ β1 ln X
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Log-linear (semilog) model
lnY = β0 + β1X + ε
• used to capture constant growth rates• Pt = (1+g)Pt-1 Pt = P0(1+g)t
ln Pt = ln P0 + t ln(1+g) ln Pt = β0 + β1t
where β0 = ln P0 and β1= ln(1+g)
X
Y
eβ0+ β1X
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Human resource application
theory suggests that the rate of return to an extra year of education is r. Then, for the first period w1 = (1 + r) w0. For, the second period w2 = (1 + r)2 w0, and for s years ws
= (1 + r)s w0. Then, taking logs of both sides yields:
ln(ws) = s ln(1 + r) + ln(w0) = β0 + β1 s
model:• ln(WAGE) = β0 + β1 EDUC + ε
download file: HR.xls
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Regression StatisticsMultiple R 0.36393R Square 0.1324451Adjusted R Square0.1139865Standard Error 0.2943775Observations 49
ANOVAdf SS MS F Sig F
Regression 1 0.621793 0.62179 7.175245 0.010153Residual 47 4.072933 0.08666Total 48 4.694726
Coefficients S.E. t Stat P-value Lower 95%Upper 95%Intercept 7.1565383 0.119077 60.1 4.18E-46 6.916986 7.396091EDUC 0.0479419 0.017898 2.67866 0.010153 0.011936 0.083947
Human resource application
ln(WAGE) = β0 + β1 EDUC + ε
ln(WAGE)=7.1565+0.0479*EDUC
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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 = e0.0479 r = 0.0491
Marginal Effect:• say when EDUC = 5:
Y = e7.1565 + 0.0479 * 5 = e7.3962 = 1629.8571• Maginal Effect (X=5) : β1Y = 0.0479*1629.8571=78.14
• How about when EDUC = 6:• Maginal Effect (X=6) : β1Y = 0.0479*1709.8988=81.98
Human resource application – Interpretation
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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 β1 is the expected % change in Y when X increases by 1.
would a linear model be better (higher adj. R2)?• cannot compare R2 if dependent variable is not the same
Human resource application – Interpretation
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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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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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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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Regression StatisticsMultiple R 0.7992298R Square 0.6387682Adjusted R Square 0.6086656Standard Error 0.7241214Observations 40
ANOVAdf SS MS F Sig F
Regression 3 33.37971 11.12657 21.21966777 4.3463E-08Residual 36 18.87667 0.524352Total 39 52.25638
Coefficients S.E. t Stat P-value Lower 95% Upper 95%Intercept 45.845675 9.61411 4.768582 3.04076E-05 26.3473557 65.34399INCOME -4.730082 1.021192 -4.631923 4.60006E-05 -6.8011554 -2.659009POP 1.8203709 0.235733 7.722176 3.79544E-09 1.34228245 2.298459LANDAREA -0.970997 0.206807 -4.695192 3.79873E-05 -1.3904208 -0.551574
Example – Demand for bus travel
Final model
What do the coefficients mean?
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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? ηBT,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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Example – Demand for bus travel
Question:• Is demand for bus travel elastic with respect to city’s
population? POPULATION
• H0: β2 = 1
• H1: β2 > 1
t0.05,36 = 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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bS
bt
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Log-log (double log) model
popular in estimating production functions• Cobb-Douglass production function
used to capture:• elasticities• returns to scale
21),( LAKLKQ
ln Q = ln A + β1 ln K + β2 ln L + ε
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Cobb-Douglass production function
β1 – elasticity of output wrt capital
β2 – elasticity of output wrt labor
Returns to scale: when you double inputs, what happens to output?
• decreasing RTS: β1 + β2 < 1
• constant RTS: β1 + β2 = 1
• increasing RTS: β1 + β2 > 1
21),( LAKLKQ
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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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Example – US manufacturing
Regression StatisticsMultiple R 0.9819243R Square 0.9641754Adj. R Sq. 0.9626827Std Error 0.2667521Observations 51
ANOVAdf SS MS F Sig. F
Regression 2 91.924607 45.962303 645.9310665 1.99686E-35Residual 48 3.4155201 0.0711567Total 50 95.340127
Coeffs S.E. t Stat P-valueIntercept 3.8875995 0.3962283 9.8115137 4.70478E-13lnK 0.5212791 0.0968871 5.3802743 2.18316E-06lnL 0.4683322 0.0989259 4.7341701 1.98088E-05