There has been much interest in whether the presence of 401(k) pension plans, available to many

Name: Problem 9, Chapter 7: Introductory Econometrics
Uploaded: 2023-08-31T01:26:16-08:00
Duration: 3 min 8 s
Channel: Akash M

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Question

There has been much interest in whether the presence of 401$(\mathrm{k})$ pension plans, available to many U.S. workers, increases net savings. The data set 401 $\mathrm{KSUBS}$ contains information on net financial assets (netta), family income (inc), a binary variable for eligibility in a 401$(\mathrm{k})$ plan (e40lk), and several other variables. (i) What fraction of the families in the sample are eligible for participation in a 401$(\mathrm{k})$ plan? (ii) Estimate a linear probability model explaining 401$(\mathrm{k})$ eligibility in terms of income, age, and gender. Include income and age in quadratic form, and report the results in the usual form. (iii) Would you say that 401$(\mathrm{k})$ eligibility is independent of income and age? What about gender? Explain. (iv) Obtain the fitted values from the linear probability model estimated in part (ii). Are any fitted values negative or greater than one? (v) Using the fitted values $\widehat{e 401 k_{i}}$ from part (iv), $\widehat{e 401 k_{i}}$ $=1$ if $\widehat{e 401 k_{i}}$ $\geq .5$ and $\widehat{e 401 k_{i}}$ $=0$ if $\widehat{e 401 k_{i}}$ $<.5 .$ Out of $9,275$ families, how many are predicted to be eligible for a 401$(\mathrm{k})$ plan? (vi) For the $5,638$ families not eligible for a 401(k), what percentage of these are predicted not to have a $401(\mathrm{k}),$ using the predictor $\widehat{e 401 k_{i}}$ For the $3,637$ families eligible for a 401$(\mathrm{k})$ plan, what percentage are predicted to have one? (It is helpful if your econometrics package has a "tabulate" command.) (vii) The overall percent correctly predicted is about 64.9$\% .$ Do you think this is a complete description of how well the model does, given your answers in part (vi)? (viii) Add the variable pira as an explanatory variable to the linear probability model. Other things equal, if a family has someone with an individual retirement account, how much higher is the estimated probability that the family is eligible for a 401(k) plan? Is it statistically different from zero at the 10$\%$ level?

   There has been much interest in whether the presence of 401$(\mathrm{k})$ pension plans, available to many U.S. workers, increases net savings. The data set 401 $\mathrm{KSUBS}$ contains information on net financial assets (netta), family income (inc), a binary variable for eligibility in a 401$(\mathrm{k})$ plan (e40lk), and several other
variables.
(i) What fraction of the families in the sample are eligible for participation in a 401$(\mathrm{k})$ plan?
(ii) Estimate a linear probability model explaining 401$(\mathrm{k})$ eligibility in terms of income, age, and gender. Include income and age in quadratic form, and report the results in the usual form.
(iii) Would you say that 401$(\mathrm{k})$ eligibility is independent of income and age? What about gender? Explain.
(iv) Obtain the fitted values from the linear probability model estimated in part (ii). Are any fitted
values negative or greater than one?
(v) Using the fitted values $\widehat{e 401 k_{i}}$ from part (iv), $\widehat{e 401 k_{i}}$ $=1$ if $\widehat{e 401 k_{i}}$ $\geq .5$ and $\widehat{e 401 k_{i}}$ $=0$ if $\widehat{e 401 k_{i}}$ $<.5 .$ Out of $9,275$ families, how many are predicted to be eligible for a 401$(\mathrm{k})$ plan?
(vi) For the $5,638$ families not eligible for a 401(k), what percentage of these are predicted not to have a $401(\mathrm{k}),$ using the predictor  $\widehat{e 401 k_{i}}$ For the $3,637$ families eligible for a 401$(\mathrm{k})$ plan, what percentage are predicted to have one? (It is helpful if your econometrics package has a "tabulate" command.)
(vii) The overall percent correctly predicted is about 64.9$\% .$ Do you think this is a complete description of how well the model does, given your answers in part (vi)?
(viii) Add the variable pira as an explanatory variable to the linear probability model. Other things
equal, if a family has someone with an individual retirement account, how much higher is the
estimated probability that the family is eligible for a 401(k) plan? Is it statistically different
from zero at the 10$\%$ level?

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 7, Problem 9

Instant Answer

Solved by Expert Akash M

08/31/2023

Step 1

To do this, we count the number of cases where the variable e401k equals one. We find that there are 3,637 cases. The total number of observations in the sample is 9,275. Therefore, the fraction of families eligible for participation in a 401(k) plan is Show more…

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There has been much interest in whether the presence of 401$(\mathrm{k})$ pension plans, available to many U.S. workers, increases net savings. The data set 401 $\mathrm{KSUBS}$ contains information on net financial assets (netta), family income (inc), a binary variable for eligibility in a 401$(\mathrm{k})$ plan (e40lk), and several other variables. (i) What fraction of the families in the sample are eligible for participation in a 401$(\mathrm{k})$ plan? (ii) Estimate a linear probability model explaining 401$(\mathrm{k})$ eligibility in terms of income, age, and gender. Include income and age in quadratic form, and report the results in the usual form. (iii) Would you say that 401$(\mathrm{k})$ eligibility is independent of income and age? What about gender? Explain. (iv) Obtain the fitted values from the linear probability model estimated in part (ii). Are any fitted values negative or greater than one? (v) Using the fitted values $\widehat{e 401 k_{i}}$ from part (iv), $\widehat{e 401 k_{i}}$ $=1$ if $\widehat{e 401 k_{i}}$ $\geq .5$ and $\widehat{e 401 k_{i}}$ $=0$ if $\widehat{e 401 k_{i}}$ $<.5 .$ Out of $9,275$ families, how many are predicted to be eligible for a 401$(\mathrm{k})$ plan? (vi) For the $5,638$ families not eligible for a 401(k), what percentage of these are predicted not to have a $401(\mathrm{k}),$ using the predictor $\widehat{e 401 k_{i}}$ For the $3,637$ families eligible for a 401$(\mathrm{k})$ plan, what percentage are predicted to have one? (It is helpful if your econometrics package has a "tabulate" command.) (vii) The overall percent correctly predicted is about 64.9$\% .$ Do you think this is a complete description of how well the model does, given your answers in part (vi)? (viii) Add the variable pira as an explanatory variable to the linear probability model. Other things equal, if a family has someone with an individual retirement account, how much higher is the estimated probability that the family is eligible for a 401(k) plan? Is it statistically different from zero at the 10$\%$ level?

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Key Concepts

Classification Accuracy and Model Evaluation

Classification accuracy refers to the percentage of correct predictions the model makes when categorizing outcomes based on a specified threshold. Although overall accuracy provides an initial measure of model performance, it may not fully capture the model’s effectiveness, especially when the distribution of outcomes is imbalanced. Therefore, further evaluation methods, such as confusion matrices and other metrics, can be necessary to understand model strengths and weaknesses.

Indicator Variables (Dummy Variables)

Indicator variables, or dummy variables, are used in regression models to represent categorical information, such as gender or program eligibility. They take on values of 0 or 1 to denote the absence or presence of a particular attribute, thus enabling the model to capture differences in the dependent variable associated with different groups or categories.

Hypothesis Testing and Statistical Significance

Hypothesis testing is a statistical method used to decide whether to reject a null hypothesis in favor of an alternative hypothesis. In econometric models, this typically involves testing whether the coefficients of certain explanatory variables differ significantly from zero, using significance levels (such as 10%) and p-values to ascertain the strength of the evidence against the null hypothesis.

Fitted Values and Prediction Cutoffs

Fitted values are the predicted values obtained from a regression model. In the context of a linear probability model, these fitted values represent estimated probabilities of the outcome occurring, which can sometimes be below zero or above one. To transform these continuous probability predictions into binary outcomes, a threshold (often 0.5) is applied—a common practice in classification tasks to decide between the two outcomes.

Binary Dependent Variable

A binary dependent variable takes on only two possible outcomes, often coded as 0 and 1. This kind of variable is common in classification tasks, such as determining eligibility for a program or the occurrence of a particular event. The analysis of binary dependent variables requires methods that account for the dichotomous nature of the data, such as the linear probability model or logistic regression.

Linear Probability Model

The linear probability model (LPM) is a regression approach used when the dependent variable is binary. In this model, ordinary least squares (OLS) is applied to estimate the probability of a particular outcome by assuming a linear relationship between the independent variables and the probability of the event occurring. Although it provides a simple framework for interpretation, one of its major drawbacks is that predicted probabilities can fall outside the [0, 1] range.

Quadratic Terms in Regression

Quadratic terms are included in a regression model to capture non-linear relationships between the independent and dependent variables. When a linear model might miss curvature in the data, introducing a squared term for a variable, like income or age, allows for a more flexible relationship. This adjustment helps in detecting and modeling diminishing or increasing marginal effects that are not adequately described by a simple linear specification.

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