In this exercise, you are to compare OLS and LAD estimates...

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 $$=\beta_{0}+\beta_{1} i n c+\beta_{2} i n c^{2}+\beta_{3} a g e+\beta_{4} a g e^{2}+\beta_{5} m a l e+\beta_{6} e 40 l k+u$$ (i) Use the data in 401 $\mathrm{KSUBS}$ to estimate the equation by OLS and report the results in the usual form. Interpret the coefficient on $e 401 k .$ (ii) Use the OLS residuals to test for heteroskedasticity using the Breusch-Pagan test. Is u independent of the explanatory variables? (iii) Estimate the equation by LAD and report the results in the same form as for OLS. Interpret the LAD estimate of $\beta_{6}$ . (iv) Reconcile your findings from parts (i) and (iii).

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
$$=\beta_{0}+\beta_{1} i n c+\beta_{2} i n c^{2}+\beta_{3} a g e+\beta_{4} a g e^{2}+\beta_{5} m a l e+\beta_{6} e 40 l k+u$$
(i) Use the data in 401 $\mathrm{KSUBS}$ to estimate the equation by OLS and report the results in the usual form. Interpret the coefficient on $e 401 k .$
(ii) Use the OLS residuals to test for heteroskedasticity using the Breusch-Pagan test. Is u
independent of the explanatory variables?
(iii) Estimate the equation by LAD and report the results in the same form as for OLS. Interpret the
LAD estimate of $\beta_{6}$ .
(iv) Reconcile your findings from parts (i) and (iii).

Key Concepts

Interpretation of Regression Coefficients

Interpreting regression coefficients involves understanding the expected change in the dependent variable for a one unit change in an explanatory variable while holding other variables constant. In the context of comparing different estimation techniques, such as OLS and LAD, the focus is on how sensitive the coefficient estimates are to the estimation method, especially in the presence of heteroskedasticity or outliers.

Comparison of Estimation Methods

Comparing OLS and LAD estimates is essential in applied econometrics as it allows researchers to assess the robustness of their results under different assumptions about the error distribution. This comparison can reveal the impact of outliers and heteroskedasticity on the estimates and help in choosing the most appropriate estimator for the data at hand.

Breusch-Pagan Test

The Breusch-Pagan test is a diagnostic tool used to detect heteroskedasticity in a regression model. It involves regressing the squared residuals from the original model on the explanatory variables and evaluating whether the explanatory variables collectively have a significant effect on the variance of the errors.

Least Absolute Deviations (LAD) Estimation

LAD estimation, also known as median regression, minimizes the sum of absolute deviations between observed and predicted values. This method is less sensitive to outliers compared to OLS because it does not square the errors, making it a robust alternative in scenarios where the error terms may not be normally distributed or when heteroskedasticity is present.

Heteroskedasticity

Heteroskedasticity refers to a condition in a regression model in which the variance of the error terms is not constant across observations. This violates one of the key assumptions of OLS, potentially leading to inefficient estimates and biased standard errors, which in turn affect hypothesis testing and confidence intervals.

Ordinary Least Squares (OLS) Estimation

OLS is a statistical method used to estimate the parameters in a linear regression model by minimizing the sum of the squared differences between the observed values and the values predicted by the model. This method provides the best linear unbiased estimates under the Gauss-Markov assumptions and is commonly used for its simplicity and efficiency when errors are homoskedastic and normally distributed.

Transcript

00:01 All right, hello everybody.

00:02 Today we're going to be comparing the effects of ols and lad estimates.

00:07 So let's go right into it.

00:09 First things first, you'll notice i've installed the woldridge package as well as the l1 pack and lm test packages.

00:16 L1 pack will allow us to perform our lad estimates, and the lm test will allow us to perform the bruch -pagan test for heteroscadisticity.

00:25 Obviously, once you install all these packages, you will then need to essentially import them into your working environment.

00:30 Using this function.

00:32 And over here, i've commented in the formula that we will be using.

00:36 And our dataset for today is the k -401 -k subs from, as always, the woolridge package.

00:42 So let's get straight into it.

00:43 First thing i'm going to do, just as a matter of convenience, is i'm going to set that equal to k -401 -k subs.

00:51 That's just so i don't have to type it out every time, because i am inherently lazy by nature by virtue of being a programmer.

00:58 Okay.

00:59 So first thing we want to do is estimate this equation by ols.

01:04 So we're going to say ols is our lm, that's linear model, net tfa to ink, plus i, ink, squared, plus age, plus i age squared, plus male, plus e 401k.

01:24 And i forgot to include the data equals data.

01:29 Awesome.

01:31 Now, if we take a summary of ols, we will get this.

01:34 So among other things, obviously, there's a lot of interesting information here.

01:39 What we want to do is interpret the coefficient on e401k.

01:43 This is 9 .713.

01:46 That means that if a family is eligible for 401k, that's what this means, right, eligibility for 401k, then their net financial assets would be 9 ,770 ,000.

02:02 $113 .482 higher than the family that's not eligible for a 401k, on average, of course.

02:10 So that's something that you should be able to do by looking at the dataset, looking at, you know, the research or the person that conducted it and finding out what they mean by these different things.

02:23 That's honestly just, you know, taking the time and doing the basic research.

02:27 But yeah, coefficient interpretation, not a big deal.

02:29 We'll also notice that this is statistically significant at the 5 -1 and 0 .1 % confidence levels.

02:35 Obviously, 10 as well.

02:38 So, yeah, that's interpreting the coefficient there.

02:43 Now, the next thing we want to do is check the heteroscadasticity using the broishe -pagan test.

02:51 So the easiest way to do this is using our lm test package...