Bài 1: Các dạng hàm hồi quy

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FUNCTIONAL FORMS

Truong Dang Thuy

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Linear model



Consider a linear regression function



: change in Y when X increases by 1 unit.



Sometimes the relationship is not linear.



Common functional form:



Log-linear



Log-lin



Lin-log



Reciprocal



Polynomial

0

1

<i>Y</i>









<i>X</i>





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Functional forms

<b>Linear model </b>

<b>Log-linear </b>

<b>Lin-log </b>

<b>Log-lin </b>

0 1

<i>Y</i>









<i>X</i>





0 1

ln

<i>Y</i>









ln

<i>X</i>





0 1

ln

<i>Y</i>









<i>X</i>





0 1

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Functional forms

<b>Reciprocal (negative beta) </b>

<b>Reciprocal (positive beta) </b>

0 1 1

1

0

<i>Y</i>

<i>X</i>

















1



0 1 1

1

0

<i>Y</i>

<i>X</i>









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Example dataset

Viet Nam Provincial data on (file ‘

gdpprov.xlsx

’)



gdp

:

provincial GDP (mil. VND)



labfo

:

number of laborers of provinces (1000

persons)

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<b>Record of </b>

<b>commands </b>

<b>Record of results </b>

<b>Variables </b>

<b>(data) </b>

<b>Commands </b>

<b>Taskbar </b>

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Import data



<b>Copy from Excel </b>

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Data description

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Linear function

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LOG-LINEAR MODEL



The Cobb-Douglas Production Function:

can be transformed into a linear model by taking natural

logs of both sides:



The slope coefficients can be interpreted as elasticities.



<i>_{If (B}</i>

<i>₂</i>

<i>_{+ B}</i>

<i>₃</i>

_{) = 1, we have constant returns to scale.}



<i>_{If (B}</i>

<i>₂</i>

<i>_{+ B}</i>

<i>₃</i>

_{) > 1, we have increasing returns to scale.}



<i>_{If (B}</i>

<i>₂</i>

<i>_{+ B}</i>

<i>₃</i>

_{) < 1, we have decreasing returns to scale.}

3

2

1

<i>B</i>

<i>B</i>

<i>i</i>

<i>i</i>

<i>i</i>

<i>Q</i>



<i>B L K</i>

1

2

3

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Log-linear model

_cons 3.06333 .4515804 6.78 0.000 2.174233 3.952426
linvest .644785 .0405325 15.91 0.000 .5649824 .7245876
llabor .508612 .0643267 7.91 0.000 .381962 .635262

lgdp Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 224.910559 270 .833002069 Root MSE = .42886
Adj R-squared = 0.7792
Residual 49.2915017 268 .183923514 R-squared = 0.7808
Model 175.619057 2 87.8095284 Prob > F = 0.0000
F( 2, 268) = 477.42

Source SS df MS Number of obs = 271
. reg lgdp llabor linvest

(17 missing values generated)
. gen linvest = ln(rinvest)
. gen llabor = ln(labfo)

(10 missing values generated)
. gen lgdp = ln(rgdp)

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LOG-LIN OR GROWTH MODELS



The rate of growth of real GDP:

can be transformed into a linear model by taking natural logs

of both sides:



<i>Letting B</i>

₁

<i> = ln RGDP</i>

₀

<i> and B</i>

₂

<i> = ln (l+r), this can be </i>

rewritten as:

<i>ln RGDP</i>

_t

<i> = B</i>

₁

<i> +B</i>

₂

<i> t </i>



<i>B</i>

<i>₂</i>

<i> is considered a semi-elasticity or an instantaneous growth rate. </i>



<i>The compound growth rate (r) is equal to (e</i>

<i>B2</i>

<i> – 1). </i>

0

(1

)

<i>t</i>

<i>t</i>

<i>RGDP</i>



<i>RGDP</i>



<i>r</i>

0

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LOG-LIN MODEL

t 290 3 1.416658 1 5

Variable Obs Mean Std. Dev. Min Max

. sum t

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LOG-LIN MODEL

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LIN-LOG MODELS



Lin-log models follow this general form:



<i>Note that B</i>

<i>₂</i>

<i> is the absolute change in Y responding to a </i>

<i>percentage (or relative) change in X </i>



<i>If X increases by 100%, predicted Y increases by B</i>

<i>₂</i>

units

1

2

ln

<i>i</i>

<i>i</i>

<i>i</i>

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Exercise – lin-log model



Data: from VHLSS 2010



income

: individual annual income (1000 VND)



healthcost

: individual annual cost for health care

(1000 VND)



Use the data in ‘healthcost.dta’ to run the

regression

where

hcshare

is the share of health cost in income.





0

1

ln

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Health cost with Lin-log model

_cons .421608 .0322026 13.09 0.000 .35847 .484746
lincome -.0341629 .0029364 -11.63 0.000 -.0399202 -.0284056

hcshare Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 75.7996618 3474 .021819131 Root MSE = .14494
Adj R-squared = 0.0372
Residual 72.9563097 3473 .021006712 R-squared = 0.0375
Model 2.84335206 1 2.84335206 Prob > F = 0.0000
F( 1, 3473) = 135.35

Source SS df MS Number of obs = 3475
. reg hcshare lincome

. gen lincome = ln(income)

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RECIPROCAL MODELS



Lin-log models follow this general form:



Note that:



<i>As X increases indefinitely, the term approaches zero and Y approaches </i>

<i>the limiting or asymptotic value B</i>

<i>₁</i>

.



The slope is:



<i>Therefore, if B</i>

₂

<i> is positive, the slope is negative throughout, and if B</i>

₂

is negative,

the slope is positive throughout.

1

2

1

(

)

<i>i</i>

<i>i</i>

<i>i</i>

<i>Y</i>

<i>B</i>

<i>B</i>

<i>u</i>

<i>X</i>







2

1

(

)

<i>i</i>

<i>B</i>

<i>X</i>

2

₂

1

(

)

<i>dY</i>

<i>B</i>

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Exercise – Reciprocal model



Use the data in ‘

healthcost.dta

’ to run the

regression

0

1

1

<i>hcshare</i>

<i>income</i>







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Exercise – Reciprocal model

_cons .023971 .0032251 7.43 0.000 .0176478 .0302943

invincome 942.4843 81.65964 11.54 0.000 782.3786 1102.59

hcshare Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 75.7996618 3474 .021819131 Root MSE = .14498

Adj R-squared = 0.0367

Residual 72.9997153 3473 .02101921 R-squared = 0.0369

Model 2.79994649 1 2.79994649 Prob > F = 0.0000

F( 1, 3473) = 133.21

Source SS df MS Number of obs = 3475

. reg hcshare invincome

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POLYNOMIAL REGRESSION MODELS



The following regression predicting GDP is an example of a

quadratic function, or more generally, a second-degree

<i>polynomial in the variable time: </i>



The slope is nonlinear and equal to:



Exercise: run the above model with ‘gdpprov.dta’

2

1

2

3

<i>t</i>

<i>t</i>

<i>RGDP</i>



<i>A</i>



<i>A time</i>



<i>A time</i>



<i>u</i>

2

2

3

<i>dRGDP</i>

<i>A</i>

<i>A time</i>

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SUMMARY OF FUNCTIONAL FORMS

<b>MODEL </b>

<b>FORM </b>

<b>SLOPE </b>

<b>ELASTICITY </b>

<b>(</b>

<i>dY</i>

<i>dX</i>

<b>) </b>

.

<i>dY X</i>

<i>dX Y</i>

Linear

<i>Y =B</i>

<i>1</i>

<i> + B</i>

<i>2</i>

<i> X </i>

<i>B </i>

2 2

(

)

<i>Y</i>

<i>X</i>

<i>B</i>

Log-linear

<i>lnY =B</i>

<i>1</i>

<i> + ln X </i>

2

(

)

<i>Y</i>

<i>B</i>

<i>X</i>

<i>B </i>

2

Log-lin

<i>lnY =B</i>

<i>1</i>

<i> + B</i>

<i>2</i>

<i> X </i>

<i>B Y </i>

2

( )

<i>B</i>

2

(

<i>X</i>

)

Lin-log

<i>Y</i>



<i>B</i>

1



<i>B</i>

2

ln

<i>X</i>

2

1

(

)

<i>B</i>

<i>X</i>

)

1

(

2

<i>Y</i>

<i>B</i>

Reciprocal

1 2

1

(

)

<i>Y</i>

<i>B</i>

<i>B</i>

<i>X</i>





<i>B</i>

₂

(

1

₂

)

<i>X</i>



₂

(

1

)

<i>XY</i>

<i>B</i>



2

ln

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COMPARING ON BASIS OF R

2



We cannot directly compare two models that have

different dependent variables.



We can transform the models as follows and compare RSS:



<i>Step 1: Compute the geometric mean (GM) of the dependent </i>

<i>variable, call it Y</i>

*

_.



<i>Step 2: Divide Y</i>

<i>_i</i>

<i> by Y</i>

*

to obtain:



<i>Step 3: Estimate the equation with lnY</i>

<i>_i</i>

as the dependent variable

<i>using in lieu of Y</i>

<i>_i</i>

as the dependent variable (i.e., use ln as the

dependent variable).



<i>Step 4: Estimate the equation with Y</i>

<i>_i</i>

as the dependent variable

<i>using as the dependent variable instead of Y</i>

<i>_i</i>

.

<i>i</i>

<i>i</i>

<i>Y</i>

<i>Y</i>

<i>Y</i>

~

*



<i>i</i>

<i>Y</i>

~

<i>Y</i>

~

<i>_i</i>

<i>i</i>

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MEASURES OF GOODNESS OF FIT



<i>R</i>

2

: Measures the proportion of the variation in the regressand

explained by the regressors.



<i>Adjusted R</i>

2

: Denoted as , it takes degrees of freedom into account:



Akaike’s Information Criterion (AIC): Adds harsher penalty for adding

more variables to the model, defined as:



<i>The model with the lowest AIC is usually chosen. </i>



Schwarz’s Information Criterion (SIC): Alternative to the AIC criterion,

expressed as:



<i>_{The penalty factor here is harsher than that of AIC.}</i>

2

<i>R</i>



_

2 2

1

1 (1

)

<i>n</i>

<i>R</i>

<i>R</i>

<i>n k</i>



  



2

ln

<i>AIC</i>

<i>k</i>

ln(

<i>RSS</i>

)

<i>n</i>

<i>n</i>





ln

<i>SIC</i>

<i>k</i>

ln

<i>n</i>

ln(

<i>RSS</i>

)

<i>n</i>

<i>n</i>

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