Chapter 1 Simultaneous Equation Models and Mitchell Chapter 9
1.1 Labor Supply of Married, Working Women
Let’s look at the data for married working women
Our labor supply equation \[ hours_i= \beta_{10}+ \alpha_1 ln(wage_i) + \beta_{11} educ_i + \beta_{12} age_i + \beta_{13} kidslt6_i + \beta_{14} nwifeinc + u_1 \]
Our labor demand equation in term of wages as a function of productivity \[ ln(wage_i)= \beta_{20} + \alpha_2 hours_i + \beta_{21}*educ_i+ \beta_{22} exper_i + \beta_{23} exper^2_i + u_2 \]
We will use the ivregress 2sls command o estimate our labor supply we use educ, age, kidslt6, nwifeinc, exper, and exper-squared.
Instrumental variables (2SLS) regression Number of obs = 428
Wald chi2(5) = 12.60
Prob > chi2 = 0.0274
R-squared = .
Root MSE = 1344.7
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| Robust
hours | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lwage | 1639.556 593.3108 2.76 0.006 476.6879 2802.423
educ | -183.7513 67.78742 -2.71 0.007 -316.6122 -50.89039
age | -7.806092 10.48746 -0.74 0.457 -28.36114 12.74896
kidslt6 | -198.1543 208.4247 -0.95 0.342 -606.6592 210.3506
nwifeinc | -10.16959 5.287486 -1.92 0.054 -20.53287 .1936911
_cons | 2225.662 603.0964 3.69 0.000 1043.615 3407.709
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Instrumented: lwage
Instruments: educ age kidslt6 nwifeinc exper expersqTest exper and expersq as instruments
Linear regression Number of obs = 428
F(6, 421) = 14.76
Prob > F = 0.0000
R-squared = 0.1633
Root MSE = .66622
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| Robust
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
exper | .0418643 .0151135 2.77 0.006 .0121569 .0715718
expersq | -.0007625 .0004065 -1.88 0.061 -.0015614 .0000365
educ | .1011113 .0141358 7.15 0.000 .0733257 .128897
age | -.0025561 .0059149 -0.43 0.666 -.0141826 .0090704
kidslt6 | -.0532185 .1047899 -0.51 0.612 -.2591951 .1527581
nwifeinc | .00556 .0027435 2.03 0.043 .0001673 .0109527
_cons | -.4471607 .2889008 -1.55 0.122 -1.015028 .1207069
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( 1) exper = 0
( 2) expersq = 0
F( 2, 421) = 6.17
Prob > F = 0.0023Note: Our instrument is rather weak. What is a possible problem with this first-stage?
Interpretation: Our results show that the labor supply curve slopes upward. The estimated coefficient is \[ \Delta hours = 1640/100 * (\% \Delta wages) \]
Our results can be interpreted through linear-log elasticity.
\[ 100*(\Delta hours/hours) \approx (1,640/hours)(\% \Delta wages) \] Or \[ \% \Delta hours \approx (1,640/hours)(\% \Delta wages) \]
This implies that the labor supply elasticity with respect to wages is \[ \eta = 1,640/hours \]
The elasticity is not constant since it is a linear-log model instead of log-log model. Using average hours worked of 1,303 Our estimated elasticity is \[ (1,640/hours)(\%\Delta wages) = 1,640/1,303=1.26 \]
1.2586339Our estimated elasticity around mean hours is 1.26%, which means a 1% increase in wages around mean hours increases hours by 1.26%. This mean wage elasticity of supply is elastic (>1) around mean hours.
With a linear-log model, the estimated elasticity is not constant elasticity.
When hours are higher, full time 40 hours every week, a 1% increase in wages increase the supply of hours around 0.79%
.78846154Our wage elasticity of supply is inelastic since (<1)
At lower hours, our estimated wage elasticity of supply is more elastic. When hours are equal to 800, then our wage elasticity of supply is almost 2
2.05A 1% increase in wages increases hours by 2.05%
1.2 Inflation and Openness
Romer (1993) proposes that more open countries should have lower inflation rates. Romer (1993) tries to explain inflation rates in terms of a country’s average share of imports in gross domestic (or national) product since 1973. Average share of imports in GDP is his measure of opennes.
\[ inf_{i,t} = \beta_{10} + \alpha_1 open_{i,t} + \beta_{11} ln(pcinc_{i,t}) + u_1 \] \[ open_{i,t} = \beta_{20} + \alpha_2 inf_{i,t} + \beta_{21} ln(pcinc)_{i,t} + \beta_{22} ln(land)_{i,t} + u_2 \]
Where open is the average share of imports in terms of GDP, pcinc is the 1980 per capita income, and land is land area of a country in square miles
We can look at the reduced form equation to see the instruments impact on openness.
Linear regression Number of obs = 114
F(2, 111) = 22.22
Prob > F = 0.0000
R-squared = 0.4487
Root MSE = 17.796
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| Robust
open | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lland | -7.567103 1.141798 -6.63 0.000 -9.829652 -5.304554
lpcinc | .5464812 1.436115 0.38 0.704 -2.299276 3.392238
_cons | 117.0845 18.24808 6.42 0.000 80.92473 153.2443
------------------------------------------------------------------------------Let’s test the instrument and use an F-test.
( 1) lland = 0
F( 1, 111) = 43.92
Prob > F = 0.0000We want to see if a country’s openness impacts a country’s inflation rate using natural log of square miles as an instrument.
Instrumental variables (2SLS) regression Number of obs = 114
Wald chi2(2) = 5.19
Prob > chi2 = 0.0745
R-squared = 0.0309
Root MSE = 23.52
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| Robust
inf | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
open | -.3374871 .1504296 -2.24 0.025 -.6323238 -.0426504
lpcinc | .3758247 1.360282 0.28 0.782 -2.29028 3.041929
_cons | 26.89934 10.77526 2.50 0.013 5.780212 48.01846
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Instrumented: open
Instruments: lpcinc llandInterpretation: For every percentage point increase in average share of imports of GDP, inflation decreases by 0.337 percentage points