Question

Use the data in 401KSUBS for this exercise. The equation of interest is a linear probability model: $$p i r a=\beta_{0}+\beta_{1} p 401 k+\beta_{2} i n c+\beta_{3} i n c^{2}+\beta_{4} a g e+\beta_{5} a g e^{2}+u$$ The goal is to test whether there is a tradeoff between participating in a 401$(\mathrm{k})$ plan and havin an individual retirement account (IRA). Therefore, we want to estimate $\beta_{1}$ . (i) Estimate the equation by OLS and discuss the estimated effect of $p 401 k$ (ii) For the purposes of estimating the ceteris paribus tradeoff between participation in two different types of retirement savings plans, what might be a problem with ordinary least squares? (iii) The variable $e 401 k$ is a binary variable equal to one if a worker is eligible to participate in a 401$(\mathrm{k})$ plan. Explain what is required for $e 401 k$ to be a valid IV for $p 401 k .$ Do these assumptions seem reasonable? (iv) Estimate the reduced form for $p 401 k$ and verify that $e 401 k$ has significant partial correlation with $p 401 k .$ Since the reduced form is also a linear probability model, use a heteroskedasticity-robust standard error. (v) Now, estimate the structural equation by IV and compare the estimate of $\beta_{1}$ with the OLS estimate. Again, you should obtain heteroskedasticity-robust standard errors. (vi) Test the null hypothesis that $p 401 k$ is in fact exogenous, using a heteroskedasticity-robust test.

Name: Problem 8, Chapter 15: Introductory Econometrics
Uploaded: 2021-03-11T04:26:27-08:00
Duration: 11 min 40 s
Channel: David Gagnon

   Use the data in 401KSUBS for this exercise. The equation of interest is a linear probability model:
$$p i r a=\beta_{0}+\beta_{1} p 401 k+\beta_{2} i n c+\beta_{3} i n c^{2}+\beta_{4} a g e+\beta_{5} a g e^{2}+u$$
The goal is to test whether there is a tradeoff between participating in a 401$(\mathrm{k})$ plan and havin an individual retirement account (IRA). Therefore, we want to estimate $\beta_{1}$ .
(i) Estimate the equation by OLS and discuss the estimated effect of $p 401 k$
(ii) For the purposes of estimating the ceteris paribus tradeoff between participation in two different
types of retirement savings plans, what might be a problem with ordinary least squares?
(iii) The variable $e 401 k$ is a binary variable equal to one if a worker is eligible to participate
in a 401$(\mathrm{k})$ plan. Explain what is required for $e 401 k$ to be a valid IV for $p 401 k .$ Do these assumptions seem reasonable?
(iv) Estimate the reduced form for $p 401 k$ and verify that $e 401 k$ has significant partial correlation with $p 401 k .$ Since the reduced form is also a linear probability model, use a heteroskedasticity-robust standard error.
(v) Now, estimate the structural equation by IV and compare the estimate of $\beta_{1}$ with the OLS estimate. Again, you should obtain heteroskedasticity-robust standard errors.
(vi) Test the null hypothesis that $p 401 k$ is in fact exogenous, using a heteroskedasticity-robust test.

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 15, Problem 8

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Solved by Expert David Gagnon

03/11/2021

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The equation of interest is a linear probability model: $$p i r a=\beta_{0}+\beta_{1} p 401 k+\beta_{2} i n c+\beta_{3} i n c^{2}+\beta_{4} a g e+\beta_{5} a g e^{2}+u$$ We use the data in 401KSUBS for this exercise. The goal is to test whether there is a tradeoff Show more…

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Use the data in 401KSUBS for this exercise. The equation of interest is a linear probability model: $$p i r a=\beta_{0}+\beta_{1} p 401 k+\beta_{2} i n c+\beta_{3} i n c^{2}+\beta_{4} a g e+\beta_{5} a g e^{2}+u$$ The goal is to test whether there is a tradeoff between participating in a 401$(\mathrm{k})$ plan and havin an individual retirement account (IRA). Therefore, we want to estimate $\beta_{1}$ . (i) Estimate the equation by OLS and discuss the estimated effect of $p 401 k$ (ii) For the purposes of estimating the ceteris paribus tradeoff between participation in two different types of retirement savings plans, what might be a problem with ordinary least squares? (iii) The variable $e 401 k$ is a binary variable equal to one if a worker is eligible to participate in a 401$(\mathrm{k})$ plan. Explain what is required for $e 401 k$ to be a valid IV for $p 401 k .$ Do these assumptions seem reasonable? (iv) Estimate the reduced form for $p 401 k$ and verify that $e 401 k$ has significant partial correlation with $p 401 k .$ Since the reduced form is also a linear probability model, use a heteroskedasticity-robust standard error. (v) Now, estimate the structural equation by IV and compare the estimate of $\beta_{1}$ with the OLS estimate. Again, you should obtain heteroskedasticity-robust standard errors. (vi) Test the null hypothesis that $p 401 k$ is in fact exogenous, using a heteroskedasticity-robust test.

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Key Concepts

Linear Probability Model

A linear probability model is a type of regression model used when the dependent variable is binary. It estimates the probability of an event by fitting a linear relationship between the predictors and the probability, even though the probability is bounded between 0 and 1. This model is favored for its simplicity and interpretability, though it comes with issues such as heteroskedasticity and potential predictions outside the [0,1] range.

Ordinary Least Squares (OLS) Estimation

OLS is a common estimation technique that minimizes the sum of squared residuals to estimate model coefficients. In the context of a linear probability model, it provides an intuitive approach to measure the average relationship between the dependent and independent variables. However, when key regressors are endogenous or correlated with the error term, OLS estimates can be biased and inconsistent, which undermines the causal interpretation of the estimated coefficients.

Endogeneity

Endogeneity occurs when an explanatory variable is correlated with the error term in a regression model. This violation of the OLS assumptions can stem from omitted variable bias, measurement error, or simultaneous causality, resulting in biased and inconsistent estimates. Identifying and addressing endogeneity is essential for making valid causal inferences in econometric models.

Instrumental Variables (IV) Estimation

IV estimation is used to obtain consistent estimates when one or more regressors are endogenous. An instrumental variable is a variable that is correlated with the endogenous regressor but uncorrelated with the error term. This two-stage process involves first predicting the endogenous variable using the instrument(s) and then using these predicted values in the structural model, thereby isolating the variation that is exogenous.

Instrument Validity

For an instrument to be valid, it must satisfy two key conditions: relevance and exogeneity. Relevance means that the instrument is strongly correlated with the endogenous variable, ensuring that it can explain sufficient variation. Exogeneity means that the instrument is not correlated with the error term in the structural equation, which avoids introducing its own source of bias. Establishing these conditions is crucial for the credibility of the IV estimates.

Reduced Form Regression

The reduced form regression is a model that expresses the potentially endogenous explanatory variable as a function of the instrument(s) and any exogenous controls. This step is critical in assessing the relevance of the instrument by examining whether the instrument has a statistically significant effect on the endogenous variable. It provides evidence that the instrument is capable of inducing exogenous variation in the variable of interest.

Exogeneity Testing

Exogeneity testing, often carried out using tests like the Durbin-Wu-Hausman test, checks whether the potentially endogenous regressors are in fact uncorrelated with the error term. The null hypothesis usually states that the suspected endogenous variable is exogenous. Rejecting the null suggests that the variable is endogenous, thereby justifying the use of IV estimation. This test is crucial for validating the estimation strategy.

Heteroskedasticity Robust Standard Errors

Heteroskedasticity robust standard errors are adjustments made to the standard errors of coefficient estimates to account for heteroskedasticity—when the variance of the error term is not constant across observations. This correction is particularly important in models like the linear probability model, ensuring that hypothesis tests and confidence intervals remain reliable despite the presence of heteroskedasticity.

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Transcript

00:01 Hello, everybody, and welcome to an econometrics tutorial using our studio.

00:06 So in this video, we're going to be looking into some endogenous variables and how you can try to use instrumental variables to fix that problem.

00:17 So let's begin.

00:20 First of all, you need to download the dataset that we're going to be using, which is called oldridge.

00:27 So go ahead and open up that dataset.

00:31 Open up the data for k4 .1.

00:33 Subs.

00:35 That is the data we're going to be using and it is data about retirement funds, age, income, and that kind of thing.

00:49 If you open it up in help, you can see what all the denotations mean in the 401 subs data file.

00:56 So for this example, we're going to want to compare what effect a bunch of these variables have on pira, which is whether or not you have an individual retirement arrangement.

01:10 We'll be comparing it to income, age, and then whether or not you have a 401k, which is a retirement savings plan that some firms offer their employees.

01:23 We're first going to start with an ordinary laced squares model to come up with some inferences about the data.

01:35 So you're going to want to do a linear regression on this formula here.

01:44 So we're going to be comparing p4 -1k and what effect that has on it? what effect income has on it? income squared, age and age squared.

01:55 And then of course an error term.

01:57 So let's get to that, shall we? oh, actually, before we get to that.

02:02 Of course, it is also common practice.

02:04 As you see in here, quite a few of these are factors.

02:06 So you have, you know, marriage is a factor zero or one.

02:11 E401k is a factor.

02:12 Male or female is a factor.

02:16 That's not a factor.

02:18 P401k is a factor and also peers a factor.

02:20 So it is pretty good practice to change the data, the data for these factors into actual factors in r.

02:29 Because right now, as you can see down here, peer is not being recognized as a factor.

02:34 Neither is male, marriage.

02:38 E401k or p411.

02:40 So it is good practice to do that.

02:44 If you have factors, if you have binary variables, go ahead and switch them.

02:50 Again, you don't technically need to do this.

02:53 The conclusions you can draw from the model, the same more or less.

03:03 So it's just good practice to do this.

03:07 All right, so now is a good time to run the regression.

03:10 So let's go ahead and name it, model 1.

03:16 A very original.

03:28 All right.

03:29 So, let's run the model and see what happens.

03:32 Boom.

03:34 Creating the factors didn't seem to work.

03:36 So yeah, it is good to create factors, but it's not imperative.

03:39 So we will avoid it here.

03:42 All right.

03:43 So, here we have our model, and for this example, we're going to be mainly focusing on the effect of p4 .1k on pira.

03:51 So that's right over here.

03:54 We have the estimation.

03:55 It is what appears to be 5%.

03:59 P value is very small.

04:00 So this is really significant.

04:01 This looks really good.

04:03 Also, because both pira and p4 .1c are factors.

04:08 The number here, the b1, is actually represented by this equation here.

04:16 So you have pira hat, pira 1 hat, and pera 0 hat.

04:19 So this would be the probability that you, the probability that you have an ira given you also have a 401k account.

04:33 And then this is the probability that you have an ira given that you don't have a 401k account.

04:37 And that's what your b1 is...

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