The file PENSION contains information on...

The file PENSION contains information on participant-directed pension plans for U.S. workers. Some of the observations are for couples within the same family, so this data set constitutes a small cluster sample (with cluster sizes of two). (i) Ignoring the clustering by family, use OLS to estimate the model $p c t s t c k=\beta_{0}+\beta_{1}$ choice $+\beta_{2}$ prfshr $+\beta_{3}$ female $+\beta_{4} a g e$ $+\beta_{5} e d u c+\beta_{6} f i n c 25+\beta_{7} f i n c 35+\beta_{8} f i n c 50+\beta_{9} f i n c 75$ $+\beta_{10} f i n c 100+\beta_{11} f i n c l 0 l+\beta_{12}$ wealth $89+\beta_{13} s t c k i n 89$$+\beta_{14} i r a i n 89+u$ where the variables are defined in the data set. The variable of most interest is choice, which is a dummy variable equal to one if the worker has a choice in how to allocate pension funds among different investments. What is the estimated effect of choice? Is it statistically significant? (ii) Are the income, wealth, stock holding, and IRA holding control variables important? Explain. (iii) Determine how many different families there are in the data set. (iv) Now, obtain the standard errors for OLS that are robust to cluster correlation within a family. Do they differ much from the usual OLS standard errors? Are you surprised? (v) Estimate the equation by differencing across only the spouses within a family. Why do the explanatory variables asked about in part (ii) drop out in the first-differenced estimation? (vi) Are any of the remaining explanatory variables in part (v) significant? Are you surprised?

The file PENSION contains information on participant-directed pension plans for U.S. workers. Some
of the observations are for couples within the same family, so this data set constitutes a small cluster
sample (with cluster sizes of two).
(i) Ignoring the clustering by family, use OLS to estimate the model
$p c t s t c k=\beta_{0}+\beta_{1}$ choice $+\beta_{2}$ prfshr $+\beta_{3}$ female $+\beta_{4} a g e$
$+\beta_{5} e d u c+\beta_{6} f i n c 25+\beta_{7} f i n c 35+\beta_{8} f i n c 50+\beta_{9} f i n c 75$
$+\beta_{10} f i n c 100+\beta_{11} f i n c l 0 l+\beta_{12}$ wealth $89+\beta_{13} s t c k i n 89$$+\beta_{14} i r a i n 89+u$
where the variables are defined in the data set. The variable of most interest is choice, which is a
dummy variable equal to one if the worker has a choice in how to allocate pension funds among
different investments. What is the estimated effect of choice? Is it statistically significant?
(ii) Are the income, wealth, stock holding, and IRA holding control variables important? Explain.
(iii) Determine how many different families there are in the data set.
(iv) Now, obtain the standard errors for OLS that are robust to cluster correlation within a family.
Do they differ much from the usual OLS standard errors? Are you surprised?
(v) Estimate the equation by differencing across only the spouses within a family. Why do the
explanatory variables asked about in part (ii) drop out in the first-differenced estimation?
(vi) Are any of the remaining explanatory variables in part (v) significant? Are you surprised?

Key Concepts

Clustered Standard Errors

Clustered standard errors adjust the typical OLS standard errors to account for intra-cluster correlation. When data points within a cluster (like members of the same family) are not independent, failing to adjust standard errors can lead to misleading conclusions about statistical significance.

First-Differencing Estimation

First-differencing is a transformation technique used to remove fixed, unobserved effects that are constant within clusters or over time. By subtracting one observation from another within the same group, it eliminates potential omitted variable bias from time-invariant or group-invariant factors, facilitating a clearer estimation of the impact of variables that change within the cluster.

Cluster Sampling

Cluster sampling is a technique where the population is divided into clusters, such as families or groups, and the analysis accounts for this grouping. Since observations within the same cluster may be correlated, this approach requires special treatment in analysis to ensure that inference, particularly standard errors, is valid.

Ordinary Least Squares (OLS) Estimation

OLS is a fundamental method for estimating the parameters of a linear regression model by minimizing the sum of squared differences between observed and predicted values. It provides a way to quantify relationships between variables and underpins many statistical analyses, assuming normally distributed errors and homoskedasticity under classical assumptions.

Control Variables

Control variables are additional predictors included in a regression model to account for potential confounding factors. Their inclusion helps to isolate the impact of the main variable(s) of interest by statistically holding constant other influences that might otherwise bias the estimation of the relationship.

Statistical Significance Testing

Statistical significance testing involves assessing whether an observed effect in a data sample is likely to reflect a true effect in the population rather than being a product of random variation. This is typically done by evaluating p-values or confidence intervals to determine if the estimated relationships differ from zero in a meaningful way.

Dummy Variables

Dummy variables are binary indicators used to represent categorical data within a regression model. By coding qualitative distinctions as 0s and 1s, they allow researchers to capture the effect of categorical factors, such as the presence or absence of a choice, on a quantitative outcome.

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