econ-130 lecture 09 - kids in prison program · lecture 9 functional forms dummy or indicator...

Economics 130

Lecture 9

Functional Forms

Dummy or Indicator Variables

Projects (Teams)

Readings for This Week

Text:

• CH 6

• CH 2, 2.8 – 2.9

• CH 4, 4.4 – 4.6

• CH 5, 5.6 – 5.8, CH 7

Multiple Regression

• We continue with addressing our second issue

+ add in how we evaluate these relationships:

– Where do we get data to do this analysis?

– How do we create the model relating the data?

– How do we relate data to on another?

– How do we evaluate these relationships?

Introducing Functional Forms

• How to formulate and estimate nonlinear

relationships.

Functional Forms

• Introduce concept – world is not linear

1. Power: xp raises the variable to the power p;

– Quadratic (x2) and cubic (x3) transformations.

2. The reciprocal: 1/x.

3. The natural logarithm: ln(x). Variations using logs…

Functional Forms

Functional Forms

Y = 3x(x = 1...15)

Y = x2

(x = 1...15)

Y = lnx(x = 1...15)

0

10

20

30

40

50

0 5 10 15 20

0

50

100

150

200

250

0 5 10 15 20

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

Functional Forms

• Model with power functions:

– y = β1 + β2 x + β3x2 + β4x

3 + e

– Note: STILL LINEAR, just in non-linear functions of x.

• Model with reciprocal:

– y = β1 + β2 (1/x) + e

– Note: STILL LINEAR, just in non-linear function of x.

Functional Forms

• Example with reciprocal :

Functional Forms

• Gretl output:

• Model 1: OLS, using observations 1-40

• Dependent variable: y

• coefficient std. error t-ratio p-value

• ---------------------------------------------------------

• const 358.036 24.3579 14.70 2.91e-017 ***

• invx -1119.08 285.656 -3.918 0.0004 ***

• R-squared 0.287688

Functional Forms

• What is the impact of income on food expenditures when x=19.605?

A. 0.102

B. 10.21

C. -1119.078

D. 2.9116

E. 1119.078

F. None of the above

Functional Form

• Why?

• The “slope” = dy/dx =

- β2/x2

• b2 = -1119.08

• Therefore the marginal effect on Y =

• -(-1119.08/(19.605)2

• = 2.9116

A. 0.102

B. 10.21

C. -1119.078

D. 2.9116

E. 1119.078

F. None of the above

Functional Forms

1. The log-log model : The parameter β2 is the elasticity

of y with respect to x. I.e., a 1% increase in x leads to

a β2 % increase in y. (E.g., β2=1.)

2. The log-linear model : A one-unit increase in x leads

to (approximately) a 100×β2 percent change in y.

3. The linear-log model: A 1% increase in x leads to a

β2/100 (β2%) unit change in y.

Functional Forms

• Linear Log Model

• FOOD_EXP = -97.18 + 132.165 ln(INCOME)

1 2_ ln( )FOOD EXP INCOME e

Functional Forms

• A 1% increase in X

implies how many unit

changes in Y?

• A 1% increase in x leads

to a β2/100 unit change

in y.

• Therefore:

1% increase leads to

132.16/100 unit change

in y = $1.32

Functional Forms

• Log Log Model:

– LN (FOOD_EXP) = 3.96 + .555 ln(INCOME)

1 2_ ln( )FOOD EXP INCOME e Ln(FOOD_EXP)

Functional Forms

• A 1% increase in x

implies what change in

y?

• A 1% increase in x leads

to a β2 % change in y.

• Therefore:

1% increase leads to

.55% change in y

Functional Forms

• Why care about

functional form?

• Data may not look like

“linear.”

• Consider, for example,

our housing model.

Functional Forms• Gretl output for the housing model:

• Model 1: OLS, using observations 1-14• Dependent variable: price

• Coefficient Std. Error t-ratio p-value• const 121.179 80.1778 1.5114 0.15888

• sqft 0.148314 0.021208 6.9933 0.00002 ***

• bedrms -23.9106 24.6419 -0.9703 0.35274

• R-squared 0.834673 Adjusted R-squared 0.804613

Functional Forms

• Gretl output for Linear Log Housing Model:

• Model 2: OLS, using observations 1-14• Dependent variable: price

• Coefficient Std. Error t-ratio p-value

• const -1749.97 259.141 -6.7530 0.00003 ***

• l_sqft 299.97 39.9758 7.5039 0.00001 ***

• l_bedrms -145.094 84.7188 -1.7127 0.11478


Functional Forms

• Log linear models are

widely used in human

capital literature.

• Suppose we believe

there is a constant return

on education, r. Then

first period return on

education =

w1 = (1+r)w0

• Therefore in year s:

Wages = w0 (1+r)s

Taking the logarithms:

ln WAGE( ) = ln WAGE0( ) + ln 1+ r( )EDUC

= 1+

2EDUC

Functional Forms

• Now let’s compare two models, one linear and

the other log linear.

Functional Forms

• First, linear model:


• Dependent variable: wage


• ---------------------------------------------------------

• const -4.91218 0.966788 -5.081 4.48e-07 ***

• educ 1.13852 0.0715497 15.91 5.59e-051 ***

• Mean dependent var 10.21302 S.D. dependent var 6.246641

• Sum squared resid 31092.99 S.E. of regression 5.581693


Functional Forms


• Dependent variable: l_wage


• ---------------------------------------------------------

• const 0.788374 0.0848975 9.286 9.71e-020 ***

• educ 0.103761 0.00628307 16.51 2.40e-054 ***

• Mean dependent var 2.166837 S.D. dependent var 0.552806

• Sum squared resid 239.7676 S.E. of regression 0.490151


Functional Forms

• For old times: can you reject the null

hypothesis that 2 = 0 at a 1% significance

level?

coefficient std. error t-ratio p-value

educ 0.1038 0.00628307 16.51 2.40e-054 ***

Functional Forms

• So what’s the interpretation of these results?

• Remember: With log-linear model, a one-unit

increase in x leads to (approximately) a 100×β2

percent change in y.

Functional Forms

• So a 1 unit increase in

education results in:

A. 0.1038% increase in

wages

B. 10.38% increase in

wages

C. 100.38% increase in

wages

D. None of the above

Functional Forms

• So a 1 unit increase in

education results in:


wages


wages


wages


Answer:


wages


wages


wages


Functional Forms

• Also note that the R2 for the log linear model is

better than the linear model.

• In the two models, R2 is:

• Linear: 0.202366

• Log-linear: 0.214621

• And this is a slight underestimate of the log-linear R2

Functional Forms

• A final word about measuring elasticity.

• Log-log models can be used to calculate

elasticities.

• Let’s consider the Bus Travel Model

Functional Forms

• Here are the regression results of turning the Bus Travel Model into a Log-log model.

• Model 1: OLS, using observations 1-40• Dependent variable: l_BUSTRAVL

• Coefficient Std. Error t-ratio p-value• const 45.8457 9.61411 4.7686 0.00003 ***• l_INCOME -4.73008 1.02119 -4.6319 0.00005 ***• l_POP 1.82037 0.235733 7.7222 <0.00001 ***• l_LANDAREA -0.970997 0.206807 -4.6952 0.00004 ***

• Mean dependent var 7.023257 S.D. dependent var 1.157544• Sum squared resid 18.87667 S.E. of regression 0.724121• R-squared 0.638768 Adjusted R-squared 0.608666• F(3, 36) 21.21967 P-value(F) 4.35e-08• Log-likelihood -41.73848 Akaike criterion 91.47697• Schwarz criterion 98.23248 Hannan-Quinn 93.91955

Functional Forms

• Ln(BUSTRAVEL) = 45.85 – 4.73 ln(INCOME)

+ 1.82 ln(POP) - .971 ln(LANDAREA)

• Income elasticity of demand = -4.73

• This appears to be highly elastic (<-1).

• Question: is it significantly different than -1?

Functional Forms

• Let’s us a 2-sided t test determining if this

income elasticity number is significantly

different than 1.

• Test statistic: t = – c/se

• = 4.73 – 1/1.021 = 3.65.

• Critical value at a = .05 = 2.03

• Conclusion: Income is “significantly elastic”

Multiple Regression

• Dummy Variables (Indicator Variables)

Regressions with Dummy Variables

• Simple Introduction:

• Dummy variable is either 0 or 1.

• Use to turn qualitative (Yes/No) data into 1/0.

Multiple Regression

• Simple Regression with a Dummy Variable

Y = 1 + 2D + e

• OLS estimation, confidence intervals, testing, etc. carried out in standard way

• Interpretation a little different.

Dummy Variables

• Simple Regression with a Dummy

Variable

• Fitted value for ith observation (point on

regression line):

• Since Di = 0 or 1 either

• or37

Ŷi = b1 + b2Di

Ŷi = b1

Ŷi = b1 + b2Di

Dummy Variables• Example: Explaining house prices (continued)

• Regress Y = house price on D = dummy for air

conditioning (=1 if house has air conditioning, = 0

otherwise).

• Result:

• Average price of house with air conditioning is $85,881

• Average price of house without air conditioning is

$59,88538

b1 = 59,885

b2 = 25,996

b1 + b2 = 85,881

Dummy Variables• Multiple Regression with Dummy Variables

• Example: Explaining house prices (continued)

• Y = 1 + 2D1 . . . kDk + e

• Regress Y = house price on D1 = driveway dummy and

D2 = rec room dummy

•

• Four types of houses:

• Houses with a driveway and a rec room (D1=1, D2=1)

• Houses with a driveway but no rec room (D1=1, D2=0)

• Houses with a rec room but no driveway (D1=0, D2=1)

• Houses with no driveway and no rec room (D1=0, D2=0)

Dummy Variables

Example: Explaining house prices (continued)

• If D1=1 and D2=1, then

Ŷi = b1 + b2 + b3 = 47,099 + 21,160 + 16,024 = 84,283

• “The average price of houses with a driveway and rec room is

$84,283”.

Coeff. St.

Error

t Stat P-

value

Lower

95%

Upper

95%

Inter. 47099.1 2837.6 16.60 2.E-50 41525 52673

D1 21159.9 3062.4 6.91 1.E-11 15144 27176

D2 16023.7 2788.6 5.75 1.E-08 10546 21502

Dummy Variables• If D1 = 1 and D2=0, then

• “The average price of houses with a driveway but no rec room is $68,259”.


•

• “The average price of houses with a rec room but no driveway is $63,123”.

•


•

• “The average price of houses with no driveway and no rec room is $47,099”.

Ŷi = b1 + b2 = 47,099 + 21,160 = 68,259

Ŷi = b1 + b3 = 47,099 + 16,024 = 63,123

Ŷi = b1 = 47,099

Dummy Variables

• Multiple Regression with Dummy and non-Dummy

Explanatory Variables

• Regress Y = house price on D = air conditioning

dummy and X = lot size.

• OLS estimates: b1= 32,693

b2 = 20,175

b3 = 5.64

Dummy Variables

• For houses with an air conditioner D = 1 and

• For houses without an air conditioner D=0 and

• Two different regression lines depending on whether the house has an air conditioner or not.

• Two lines have different intercepts but same slope (i.e. same marginal effect)

Ŷi = 52,868 + 5.64X

Ŷi = 32,693 + 5.64X

Dummy Variables

• Verbal ways of expressing OLS results:

• “An extra square foot of lot size will tend to add $5.64 onto the price of a house” (Note: no ceteris paribusqualifications to statement since marginal effect is same for houses with and without air conditioners)

• “Houses with air conditioners tend to be worth $20,175 more than houses with no air conditioners, ceteris paribus” (Note: Here we do have ceteris paribusqualification)

• “If we consider houses with similar lot sizes, those with air conditioners tend to be worth an extra $20,175”

Dummy Variables

• Another House Price Regression

• Regress Y = house price on D1 = dummy variable for driveway, D2 = dummy variable for rec room, X1 = lot size and X2 = number of bedrooms

• OLS estimates:

Y = 1 + 2D1 + 3D2 + 4X1 + 5X2 + e

b1= -2,736b2 = 12,598b3 = 10,969b4 = 5.197b5 = 10,564

Dummy Variables

1. If D1=1 and D2=1, then

This is the regression line for houses with a driveway and rec room.


• This is the regression line for houses with a driveway but no rec room.


This is the regression line for houses

with a rec room but no driveway.


This is the regression line for houses

with no driveway and no rec room.

Dummy Variables


This is the regression line for houses with a driveway and rec room.


• This is the regression line for houses with a driveway but no rec room.

Dummy Variables


This is the regression line for houses with a rec

room but no driveway


This is the regression line for houses with no

driveway and no rec room.

Dummy Variables

• “Houses with driveways tend to be worth

$12,598 more than similar houses with no

driveway.”

• “If we consider houses with the same number

of bedrooms, then adding an extra square foot

of lot size will tend to increase the price of a

house by $5.197.”

• “An extra bedroom will tend to add $10,562 to

the value of a house, ceteris paribus”

Dummy Variables

• Interacting Dummy and Non-Dummy Variables

• Where Z=DX.

• Z is either 0 (for observations with D=0) or X (for observations with D=1)

• If D=1 then

• If D=0, then

Two different regression lines corresponding to D=0 and D=1 exist and have different intercepts and slopes. • The marginal effect of X on Y is different for D=0 and

D=1

Y = 1 + 2D + X + 4Z + e

Ŷi = (b1 + b2) + (b3 + b4)X

Ŷi = b1 + b3X

Dummy Variables

• Regress Y = house price on D = air conditioner dummy, X = lot size and Z = DX

• OLS estimates:

• The marginal effect of lot size on housing is 7.27 for houses with air conditioners and only $5.02 for houses without.

• Increasing lot size will tend to add more to the value of a house if it has an air conditioner than if it does not.

51

b1= 35,684

b2 = 7,613

b3 = 5.02

b4 = 2.25

Dummy Variables

• Issue here: Using dummy variables on the right side of regression, i.e., as INDEPENDENT variables

• Dummy variables are (0,1) variables

• The illustrative example:

• House prices depend upon house characteristics: size (square feet), location (may be dummy), # of bedrooms, bathrooms, age, whether has a pool (dummy), whether has a tile roof (dummy), etc.

• Note: These sorts of models are called “hedonic price” models.

52

Dummy Variables

Here again is our basic simple regression model:

1 2PRICE SQFT e

1 if characteristic is present

0 if characteristic is not presentD

1 if property is in the desirable neighborhood

0 if property is not in the desirable neighborhoodD

Dummy Variables

1 2PRICE D SQFT e

1 2

1 2

( ) when 1( )

when 0

SQFT DE PRICE

SQFT D

Dummy Variables

Dummy Variables

1 2 ( )PRICE SQFT SQFT D e

1 2

1 2

1 2

( ) when 1( )

when 0

SQFT DE PRICE SQFT SQFT D

SQFT D

2

2

when 1 ( )

when 0

DE PRICE

DSQFT

1 2

1 2

1 2

( ) when 1( )

when 0

SQFT DE PRICE SQFT SQFT D

SQFT D

2

2

when 1 ( )

when 0

DE PRICE

DSQFT

Dummy variables can also be used to determine whether the

marginal impact of one variable depends on the presence (or

absence) of the dummy characteristic. Does price per square

foot depend upon whether the house is in a good

neighborhood?

Dummy Variables

Dummy Variables

• Here’s a model with both intercept and slope

dummy effects. We will estimate it now.

Dummy Variables

• 1000 house sales from two similar neighborhoods,

one bordering a university, the other 3 miles away.

• Variables:

• price house price, in $1000

• sqft square feet of living area, in 100's

• age house age, in years

• utown =1 if close to university

• pool =1 if house has pool

• fplace =1 if house has fireplace

Dummy Variables

• The Model:

• Sample Data:

Dummy Variables

• Model 2: OLS, using observations 1-1000• Dependent variable: p

• coefficient std. error t-ratio p-value • ---------------------------------------------------------• const 24.5000 6.19172 3.957 8.13e-05 ***• sqft 7.61218 0.245176 31.05 1.87e-148 ***• utown 27.4530 8.42258 3.259 0.0012 ***• usqft 0.0129940 0.00332048 3.913 9.72e-05 ***• pool 4.37716 1.19669 3.658 0.0003 ***• fplace 1.64918 0.971957 1.697 0.0901 *• age -0.190086 0.0512046 -3.712 0.0002 ***

• Mean dependent var 247.6557 S.D. dependent var 42.19273• Sum squared resid 230184.4 S.E. of regression 15.22521• R-squared 0.870570 Adjusted R-squared 0.869788• F(6, 993) 1113.183 P-value(F) 0.000000

Dummy VariablesPremium for

lots near the

university

is….

Premium for

having a pool

is…

Premium for

having a

fireplace is …

Δ Price due to

Δ in house

size is . . .

A. $24,500

B. $76

C. $4,377

D. $1.2994

E. $1,649

F. $27,453

A. $24,500

B. $76

C. $4,377

D. $1.2994

E. $1,649

F. $27,453

A. $24,500

B. $76

C. $4,377

D. $1.2994

E. $1,649

F. $27,453

A. $24,500

B. $76

C. $4,377

D. $1.2994

E. $1,649

F. $27,45

Dummy Variables

Based on these regression results, we estimate:

the location premium, for lots near the university, to be

$27,453

the price per square foot to be $89.12 for houses near the

university, and $76.12 for houses in other areas.

that houses depreciate $190.10 per year

that a pool increases the value of a home by $4377.20

that a fireplace increases the value of a home by

$1649.20

Dummy VariablesVariable Coefficient X Units Y Units Marginal Effect

UTOWN (1 = close

to university)

27.453 1,0 1000s 1 x 2.7453 x 1000 =

$27453

SQFT (size in 100s) 7.612 100s of

SQFT

1000s 7.612 ÷ 100 = .07612 x

1000 = $76.12

USQFT ( x SQFT) .01299 1,0 1000s .01299 x 1000 = $12.99

+ $76.12 = $89.11

AGE (years) -.190 Years 1000s 1 x -.19 x 1000 = - $190

FPLACE (1 =

fireplace)

1.649 1,0 1000s 1 x 1.649 x 1000 =

$1,649

POOL (1 = pool) 4.377 1,0 1000s 1 x 4.377 x 1000 =

$4,377

Dummy Variables

• Other Interactions:

– Here we are interacting a dummy with a

continuous variable

– Also could interact with other dummies:

• Does gender discrimination in wages depend upon

race? Interact gender with race in wage equation.

– Can interact continuous variables

• Does price per square foot depend upon house age?

Interact AGE with SQFT.

Dummy Variables

• There are many examples of categorical data– Regions, seasons, time (years)

– Credit ratings

– School ratings

• In some cases, there is a natural ordering (higher is better, as with ratings)

• In others, there is no natural ordering (regions)

• Even in cases with natural ordering, however, there is no clear meaning to units– Going from rank 1 to 2 is not the same “quantity” as going

from 2 to 3

• For these cases, define multiple dummies to account for different regions, seasons, years, ratings

Dummy Variables

• Equivalence of two regressions

–First begin a base model that includes

interactions between dummy variables.

–A wage model, where wage may depend

upon

• Education

• Gender

• Race

Dummy Variables

1 2

1 1 2

1 2 2

1 1 2 2

( )

EDUC WHITE MALE

EDUC BLACK MALEE WAGE

EDUC WHITE FEMALE

EDUC BLACK FEMALE

Dummy Variables

• Base Model Estimate:

Dummy Variables

• Calculate the marginal effects on hourly wage

due to race and gender. Effects are additive!!

Coefficient for

Black Males = δ1

= -1.83

Therefore, the

marginal effect of

being a black male

on hourly wage is

-$1.83

Coefficient for

Females = δ2 = -

2.55

Therefore, the

marginal effect of

being a femal on

hourly wage is

-$2.55

Coefficient for Black

Females = γ = .59

Add δ1 + δ2 + γ = -

3.79

Therefore, the

marginal effect of

being a black female

on hourly wage is

-$3.79

Dummy Variables

• Do an F Test to test the hypothesis that Gender and Race Don’t Matter:

– 3 Restrictions (J), 5 Regressors (K), N=1000

R Model = WAGE = 1 + 2 EDUC + e

– Result: F = 3.80

– Critical Value = 2.60

– Reject the null.

Dummy Variables/Gretl Practice

• We are now going to return to a model we

looked at earlier for women’s labor force

participation

• This model used 1990 data from 50 states.

Gretl Practice

• Here is a model of the determinants of women’s labor force participation for all 50 states.

• WLFP = Participation rate (%) of women > 16 in the labor force

• YF = Annual median earnings by females (000s of $)

• YM = Annual median earnings by males (000s of $)

• EDUC = Female HS grads > 24 (%)

• UE = Unemployment rate (%)

• MR = Marriage rate (%) women over 16

• DR = Divorce rate (%)

• URB = % of state’s population that is urban

• WH = % of state’s female > 16 population who are white

Gretl Practice

• This is another “kitchen sink” model:

• WLFP = 1 + 2YF + 3YM + 4EDUC +

5UE + 6MR + 7DR + 8URB + 9WH e

Gretl Practice

• Model 1: OLS, using observations 1-50• Dependent variable: wlfp

• Coefficient Std. Error t-ratio p-value• const 44.5096 8.97496 4.9593 0.00001 ***• yf 0.987983 0.407583 2.4240 0.01985 **• ym -0.174345 0.306207 -0.5694 0.57221• educ 0.285129 0.0931647 3.0605 0.00389 ***• ue -1.61058 0.313617 -5.1355 <0.00001 ***• mr -0.0782145 0.173139 -0.4517 0.65383• dr 0.437371 0.258336 1.6930 0.09804 *• urb -0.0926339 0.0333355 -2.7788 0.00820 ***• wh -0.0874916 0.0398446 -2.1958 0.03382 **


Gretl Practice


• Coefficient Std. Error t-ratio p-value• const 41.346 5.55984 7.4365 <0.00001 ***• yf 1.06712 0.364515 2.9275 0.00550 ***• educ 0.258172 0.0708648 3.6432 0.00073 ***• ue -1.59099 0.307647 -5.1715 <0.00001 ***• dr 0.391632 0.235404 1.6637 0.10363• urb -0.0876356 0.0311463 -2.8137 0.00742 ***• wh -0.0850871 0.0391115 -2.1755 0.03527 **• ym -0.198418 0.298664 -0.6644 0.51010


Gretl Practice


• Coefficient Std. Error t-ratio p-value• const 41.8336 5.47528 7.6405 <0.00001 ***• yf 0.849264 0.158152 5.3699 <0.00001 ***• educ 0.249152 0.0690987 3.6057 0.00080 ***• ue -1.67758 0.276859 -6.0593 <0.00001 ***• dr 0.434104 0.22508 1.9287 0.06039 *• urb -0.0942172 0.0293363 -3.2116 0.00250 ***• wh -0.0960861 0.0352037 -2.7294 0.00916 ***



• We are going to look at data from both 1980

and 1990 because it’s possible there was a

structural change in WLFP over that decade.

• We are going to create a Dummy Variable for

1990, called D90. For 1990, D = 1.

• Now we are going to incorporate interaction

terms, by multiplying D90 by the other

variables.


• Here is the model we are starting with:

• WLFP = 1 + 2YF + 3YM + 4EDUC +

5UE + 6MR + 7DR + 8URB + 9WH

q1(D90*YF) + q2(D90*YM) + q3(D90*EDUC)

+ q4(D90*UE) + q5(D90*MR) + q6(D90*DR)

+ q7(D90*URB) + q8(D90*WH) + e



• Dependent variable: WLFP

• Omitted due to exact collinearity: D90YM


• ------------------------------------------------------------

• const 49.6235 10.5465 4.705 1.00e-05 ***

• YF 0.00470565 0.000948974 4.959 3.71e-06 ***

• YM -0.000133492 0.000273021 -0.4889 0.6262

• EDUC 0.286358 0.0586220 4.885 4.97e-06 ***

• UE -1.09155 0.269898 -4.044 0.0001 ***

• MR -0.210187 0.153263 -1.371 0.1739

• DR 0.208349 0.172456 1.208 0.2304

• URB -0.0665164 0.0299182 -2.223 0.0289 **

• WH -0.126810 0.0343940 -3.687 0.0004 ***

• D90 -4.85398 13.6340 -0.3560 0.7227

• D90YF -0.00376523 0.000880843 -4.275 5.08e-05 ***

• D90EDUC -0.00164444 0.111949 -0.01469 0.9883

• D90UE -0.537329 0.396399 -1.356 0.1789

• D90MR 0.127952 0.230950 0.5540 0.5811

• D90DR 0.239853 0.304639 0.7873 0.4333

• D90URB -0.0276884 0.0437808 -0.6324 0.5288

• D90WH 0.0369987 0.0517175 0.7154 0.4764



• Suspecting multicollinearity, we eliminate variables with insignificant coefficients one at a time.



• Dependent variable: WLFP


• ----------------------------------------------------------

• const 47.6366 6.57840 7.241 1.52e-010 ***

• YF 0.00477939 0.000733949 6.512 4.28e-09 ***

• EDUC 0.275070 0.0455059 6.045 3.43e-08 ***

• UE -1.06141 0.245591 -4.322 4.02e-05 ***

• MR -0.207293 0.104894 -1.976 0.0512 *

• DR 0.281618 0.133697 2.106 0.0380 **

• URB -0.0784652 0.0206237 -3.805 0.0003 ***

• WH -0.111495 0.0242421 -4.599 1.40e-05 ***

• D90YF -0.00405375 0.000682124 -5.943 5.36e-08 ***

• D90UE -0.569355 0.327225 -1.740 0.0853 *

• D90MR 0.126361 0.0509756 2.479 0.0151 **


• F(10, 89) 53.87052 P-value(F) 1.98e-33


• Here is the final model:

• WLFP = 47.63 + .00478 YF - .00405 (D90 *

YF) + .275 EDUC – 1.06 UE - .569 (D90 *

UE) - .207 MR + .126 (D90 * MR) + .282 DR

- .078 URB - .111 WH

• Adjusted R2 = .842


For 1980, set D90 = 0

• WLFP = 47.63

+ .00478YF + .275

EDUC – 1.06 UE - .207

MR +.282 DR - .078 URB

- .111 WH

For 1990, set D90 = to 1 and

combine terms

• WLFP = 47.63

+ .00073 YF + .275

EDUC – 1.63UE - .081

MR + .282 DR - .078

URB - .111 WH

Dummy Variables

Observations:

1. Marginal effect of marriage is smaller in 1990,

suggesting more women stayed in the labor after

marriage.

2. More dramatic unemployment coefficient

suggests the discouraged worker hypothesis is

stronger in 1990.

3. YF???

The Project

• Teams

• TOPICS

econ-130 lecture 09 - kids in prison program · lecture 9 functional forms dummy or indicator...

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