LinearRegression4

Raw to estimate the equation by ols and report the

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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 1-norm penalty: o The 1-norm 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 Least-squares Estimates Dependent Variable = net tfa R-squared = 0.2022 Rbar-squared = 0.2017 sigma^2 = 3266.0118 Durbin-Watson = 1.9394 Nobs, Nvars = 9275, 7 *************************************************************** Variable Coefficient t-statistic t-probability 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 p-value, 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(u|x1,x2,…,xk) = 0 is possible). VER. 10/23/2012. © P. KOLM 82 Ordinary Least-squares Estimates Dependent Variable = resid^2 R-squared = 0.0374 Rbar-squared = 0.0368 sigma^2 = 1943590333.1872 Durbin-Watson = 1.9858 Nobs, Nvars = 9275, 7 *************************************************************** Variable Coefficient t-statistic t-probability 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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