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Unformatted text preview: , can be chosen in a number of different ways VER. 10/23/2012. © P. KOLM 78 Remarks:
• Lasso estimates are not available in closed form and have to be solved for numerically
• Lasso is somewhat indifferent to highly correlated predictors, and will tend to pick one and ignore the rest. This is due to the 1norm penalty:
o The 1norm penalty can be viewed as the most selective shrinkage function that is also convex
o Selectivity tends to create sparse models since many of the parameters
ˆ
β will be set to 0
k o Convexity guarantees that there is one global minimum solution for a given set of data VER. 10/23/2012. © P. KOLM 79 Example (C9.9 in Wooldridge) In this exercise, you are to compare OLS and LAD estimates of the effects of
401(k) plan eligibility on net financial assets. The model is nettfa = β0 + β1inc + β2inc 2 + β3age + β4age 2 + β5male + β6e401k + u
(i) Use the data in 401KSUBS.RAW to estimate the equation by OLS and report the results in the usual form. Interpret the coefficient on e401k.
Solution: The equation estimated by OLS is
nettfa = 21.2 − 0.270inc + 0.0102inc 2 − 1.94age
(9.99) (0.075) (0.0006) (0.483) +0.0346age 2 + 3.37male + 9.71e401k
(0.0055) (1.49) (1.28) n = 9,275, R2 = 0.202 VER. 10/23/2012. © P. KOLM 80 The coefficient on e401k means that, holding other things in the equation
fixed, the average level of net financial assets is about $9,710 higher for a
family eligible for the 401(k) than for a family not eligible.
Ordinary Leastsquares Estimates
Dependent Variable =
net tfa
Rsquared
=
0.2022
Rbarsquared
=
0.2017
sigma^2
= 3266.0118
DurbinWatson =
1.9394
Nobs, Nvars
=
9275,
7
***************************************************************
Variable
Coefficient
tstatistic
tprobability
intercept
21.197792
2.121432
0.033912
inc
0.270224
3.621800
0.000294
incsq
0.010216
17.400183
0.000000
age
1.939771
4.012127
0.000061
agesq
0.034566
6.230229
0.000000
male
3.369048
2.267478
0.023384
e401k
9.713482
7.605730
0.000000 VER. 10/23/2012. © P. KOLM 81 (ii) Use the OLS residuals to test for heteroscedasticity using the Breusch Pagan test. Is u independent of the explanatory variables?
Solution:
2
ˆ
The OLS regression of ui 2 on inci, inci 2 , agei, agei 2 , malei, and e401ki gives Ru 2 =
ˆ 0.0374, which translates into F = 59.97. The associated pvalue, with 6 and 9,268
df, is essentially zero. Consequently, there is strong evidence of heteroscedasticity,
which means that u and the explanatory variables cannot be independent (even
though E(ux1,x2,…,xk) = 0 is possible). VER. 10/23/2012. © P. KOLM 82 Ordinary Leastsquares Estimates
Dependent Variable =
resid^2
Rsquared
=
0.0374
Rbarsquared
=
0.0368
sigma^2
= 1943590333.1872
DurbinWatson =
1.9858
Nobs, Nvars
=
9275,
7
***************************************************************
Variable
Coefficient
tstatistic
tprobability
intercept
14762.886530
1.915209
0.055497
inc
433.656753
7.534470
0.000000
incsq
5.798022
12.801419
0.000000
age
525.265463
1.40834...
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