MULTIVARIATE FRACTIONAL RESPONSE MODELS IN A PANEL SETTING WITH AN
APPLICATION TO PORTFOLIO ALLOCATION
By
Michael Anthony Carlton

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics – Doctor of Philosphy
2013

ABSTRACT
MULTIVARIATE FRACTIONAL RESPONSE MODELS IN A PANEL SETTING WITH AN
APPLICATION TO PORTFOLIO ALLOCATION
By
Michael Anthony Carlton
Several papers use subjective survival probabilities as a measure of mortality risk in studying
economic behavior. The first chapter “Wealth Holdings, Asset Allocation and Mortality: A Test
of the Information Content of Subjective Survival Probabilities” studies whether subjective
survival probability measures contain any additional information that can explain differential
wealth holdings and asset allocation among households. We find some evidence that survival
probabilities can explain differences in household wealth holding and allocation once we control
for other factors that affect decision-making. We also find that the estimated impact of subjective
survival is sensitive to the inclusion of reported survival probabilities of one.
Some fractional response variables, like the proportion of financial wealth allocated
across multiple assets, must satisfy an adding up restriction. In the second chapter “A Model for
Multivariate Fractional Responses with an Application to Asset Allocation”, we develop a twostep procedure where we estimate a model with multiple fractional response variables exploiting
the fact that these variables sum to one in each period and are correlated over time. The first step
entails estimation of the multivariate fractional responses using the multinomial quasi-likelihood
function which explicitly imposes the adding-up restriction and the second step uses the
Classical Minimum Distance estimator to account for serial correlation.
Many panel data estimators implicitly assume that we have a balanced panel at our
disposal. Unfortunately this is rarely the case and dropping observations is an unsatisfactory
solution to the problem. Estimation of fractional responses in a panel requires assumptions about

the distribution of the unobserved effect and its relationship with observables, which requires
special treatment in an unbalanced panel. In the third chapter, “Estimation of a Multivariate
Fractional Response Model with Unbalanced Panel Data”, we extend the approach in
Wooldridge (2010) to the case of multiple fractional responses and apply this to unbalanced
panel data on the allocation of financial wealth across several assets.

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TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... iv
LIST OF FIGURES ...................................................................................................................... vii
ABBREVIATIONS ..................................................................................................................... viii
CHAPTER 1
WEALTH HOLDINGS, ASSET ALLOCATION, AND MORTALITY: A TEST OF THE
INFORMATION CONTENT OF SUBJECTIVE SURVIVAL PROBABILITIES ......................1
1.1 Introduction ....................................................................................................................1
1.2 Theory and Literature Review .......................................................................................5
1.3 Model Specification, Descriptive Statistics and Estimation Strategy..........................10
1.4 Results and Discussion ................................................................................................17
1.5 Conclusion ...................................................................................................................27
CHAPTER 2
A MODEL FOR MULTIVARIATE FRACTIONAL RESPONSES WITH AN APPLICATION
TO ASSET ALLOCATION ..........................................................................................................30
2.1 Introduction ..................................................................................................................30
2.2 Fractional Response Variables .....................................................................................32
2.3 Single Equation Fractional Probit ................................................................................33
2.4 The Multinomial Quasi-Likelihood .............................................................................37
2.5 Classical Minimum Distance Estimation .....................................................................41
2.6 An Alternative Specification........................................................................................46
2.7 Results ..........................................................................................................................48
2.8 Conclusion ...................................................................................................................56
CHAPTER 3
ESTIMATION OF A MULTIVARIATE FRACTIONAL RESPONSE MODEL WITH
UNBALANCED PANEL DATA ..................................................................................................58
3.1 Introduction ..................................................................................................................58
3.2 Single Equation Fractional Probit in a Balanced Panel ...............................................60
3.3 Single Equation Fractional Probit in an Unbalanced Panel .........................................63
3.4 The Multinomial Quasi-Likelihood .............................................................................65
3.5 Results ..........................................................................................................................69
3.6 Conclusion ...................................................................................................................76
APPENDIX ....................................................................................................................................78
BIBLIOGRAPHY ........................................................................................................................118

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LIST OF TABLES

Table 1: Summary Statistics for Single Households......................................................................79
Table 2: Summary Statistics for Married Households ...................................................................80
Table 3: Average Wealth and Proportion of Financial Wealth Allocated by Pr(Live to 75),
Single Households .........................................................................................................................81
Table 4: Average Wealth and Proportion of Financial Wealth Allocated by Husband’s
Pr(Live to 75), Married Households ..............................................................................................82
Table 5: Average Wealth and Proportion of Financial Wealth Allocated by Wife’s
Pr(Live to 75), Married Households ..............................................................................................82
Table 6: Wealth Holding Estimation Results, Single Households ................................................83
Table 7: Average Partial Effects (Reported as Percentages) for Allocation of Financial
Wealth, Single Households, Estimated by Single Equation Fractional Probit ..............................84
Table 8: Wealth Holdings Estimation Results, Married Households ............................................86
Table 9: Average Partial Effects (Reported as Percentages) for Allocation of Financial
Wealth, Married Households, Estimated by Single Equation Fractional Probit............................88
Table 10: Sensitivity Analysis, Removing Permanent Income,
Single and Married Households, Wealth Holding and Asset Allocation.......................................91
Table 11: Sensitivity Analysis, Treating Pr(Live to 75) = 1 Differently,
Single Households, Wealth Holding and Asset Allocation ...........................................................92
Table 12: Sensitivity Analysis, Treating Pr(Live to 75) = 1 Differently,
Married Households, Wealth Holding and Asset Allocation.........................................................93
Table 13: Means and Medians of Relevant Variables ...................................................................94
Table 14: Proportion of Variation Between Households for Relevant Variables ..........................94
Table 15: Average Partial Effects from Pooled Linear Fixed Effects OLS Estimation
on Entire, Unbalanced Panel ..........................................................................................................95
Table 16: Average Partial Effects from Single Equation Pooled Bernoulli Quasi-Maximum
Likelihood Estimates for each Balanced Panel Subset, Averaged Across Balanced Panel
Subsets ...........................................................................................................................................96

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Table 17: Average Partial Effects from Pooled Multinomial Quasi-Maximum Likelihood
Estimates for each Balanced Panel Subset, Averaged Across Balanced Panel Subsets ................97
Table 18: Procedure 2: Classical Minimum Distance Estimation Applied to Multinomial
Quasi-Maximum Likelihood Estimates for each Year Within Balanced Panel Subsets ...............98
Table 19: Comparing Average Partial Effect Estimates from Pooled Single Equation
Bernoulli Quasi-Maximum Likelihood Estimates for each Balanced Panel Subset,
Averaged Across Balanced Panel Subsets to Procedure 2 Average Partial Effect Estimates .......99
Table 20: Comparing Average Partial Effects Estimates from Pooled Multinomial
Quasi-Maximum Likelihood Estimates for each Balanced Panel Subset, Averaged Across
Balanced Panel Subsets to Procedure 2 Average Partial Effect Estimates ....................................99
Table 21: Comparing Average Partial Effect Estimates from Pooled Linear Fixed Effects
OLS on Entire, Unbalanced Panel to Procedure 2 Average Partial Effect Estimates..................100
Table 22: Means and Medians of Relevant Variables .................................................................101
Table 23: Proportion of Variation Between Households for Relevant Variables ........................101
Table 24: Average Partial Effect Estimates from Pooled Linear Fixed Effects OLS
Estimation on Entire Unbalanced Panel ......................................................................................102
Table 25: Average Partial Effects from Single Equation Pooled Bernoulli Quasi-Maximum
Likelihood Estimates on Entire Unbalanced Panel,
Ignoring Unbalanced Nature of the Panel ....................................................................................103
Table 26: Average Partial Effects from Single Equation Pooled Bernoulli Quasi-Maximum
Likelihood Estimates for each Balanced Panel Subset,
Averaged Across Balanced Panel Subsets ...................................................................................104
Table 27: Average Partial Effects from Pooled Bernoulli Quasi-Maximum Likelihood
Estimates on Entire, Unbalanced Panel, Adjusting for Unbalanced Nature of the Panel ............105
Table 28: Average Partial Effects from Pooled Multinomial Quasi-Maximum Likelihood
Estimates for Entire, Unbalanced Panel, Ignoring Unbalanced Nature of the Panel ...................106
Table 29: Average Partial Effects from Pooled Multinomial Quasi-Maximum Likelihood
Estimates for each Balanced Panel Subset, Averaged Across Balanced Panel Subsets ..............107
Table 30: Average Partial Effects from Multinomial Quasi-Maximum Likelihood Estimates
on Entire, Unbalanced Panel, Adjusting for Unbalanced Nature of the Panel ............................108

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Table 31: Average Partial Effects from Multinomial Quasi-Maximum Likelihood Estimates
using the Multinomial Logit Functional Form on the Entire, Unbalanced Panel,
Adjusting for the Unbalanced Nature of the Panel ......................................................................109

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LIST OF FIGURES

Figure 1: Average Structural Function for the Proportion of Financial Wealth Allocated
to Stocks Against Husband Word Recall .....................................................................................110
Figure 2: Average Structural Function for the Proportion of Financial Wealth Allocated
to Bonds Against Husband Word Recall .....................................................................................111
Figure 3: Average Structural Function for the Proportion of Financial Wealth Allocated
to CDs Against Husband Word Recall ........................................................................................112
Figure 4: Average Structural Function for the Proportion of Financial Wealth Allocated
to Checking Against Husband Word Recall ................................................................................113
Figure 5: Average Structural Function for the Proportion of Financial Wealth Allocated
to Stocks Against Wife Word Recall ...........................................................................................114
Figure 6: Average Structural Function for the Proportion of Financial Wealth Allocated
to Bonds Against Wife Word Recall ...........................................................................................115
Figure 7: Average Structural Function for the Proportion of Financial Wealth Allocated
to CDs Against Wife Word Recall...............................................................................................116
Figure 8: Average Structural Function for the Proportion of Financial Wealth Allocated
to Checking Against Wife Word Recall ......................................................................................117

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ABBREVIATIONS

AHEAD............................................................. Asset and Health Dynamics among the Oldest Old
ANOVA ........................................................................................................... Analysis of Variance
APE ................................................................................................................ Average Partial Effect
ASF ...................................................................................................... Average Structural Function
CAMS .............................................................................. Consumption and Activities Mail Survey
CDF ............................................................................................. Cumulative Distribution Function
CMD ................................................................................................... Classical Minimum Distance
HRS ..................................................................................................... Health and Retirement Study
IRA ................................................................................................... Individual Retirement Account
OLS .............................................................................................................. Ordinary Least Squares
P75 ................................................................................ Subjective Probability of Living to Age 75
Pr(Live to 75) ............................................................... Subjective Probability of Living to Age 75
QMLE ................................................................................. Quasi-Maximum Likelihood Estimates

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CHAPTER 1
WEALTH HOLDINGS, ASSET ALLOCATION, AND MORTALITY: A TEST OF THE
INFORMATION CONTENT OF SUBJECTIVE SURVIVAL PROBABILITIES

1.1 Introduction
Economic theory predicts that individuals save to finance their consumption in
retirement. An important factor in deciding the level and allocation of wealth to different assets
is the length of time the individual expects to live after retirement. The risk that an individual
faces if they do not properly account for their survival expectations is that they will outlive their
assets. Social Security and pensions can help to alleviate this risk by providing a fixed stream of
income in retirement, but if individuals are attempting to smooth consumption there is a potential
for a significant decrease in utility later in life from consuming too much early in life. Therefore,
we should see that forward-looking individuals adjust their current behavior to their individual
life expectancy.
It is straightforward to determine whether mortality has an impact on wealth holding and
asset allocation. With repeated observations on individuals, we can simply see if wealth holding
and asset allocation is different for individuals with longer actual lifetimes. The drawback to this
approach is that it is looking at actual lifetimes, not expected lifetimes. In any period, an
individual does not know their actual date of death but only has some idea of their probability to
reach a target age. One can use life-table probabilities as a proxy for individual survival, but
since these are averages over the entire population, they only vary by age, race, and gender. The
ideal measure that we would like to use is individual’s expected survival probabilities at the
point in time that they are making their decision to save and allocate wealth.
Our goal in this paper is to determine whether we can explain any of the heterogeneity in

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wealth holdings and asset allocation using subjective survival probabilities elicited in the Health
and Retirement Study (HRS). The HRS collects detailed information on household asset
holdings as well as measures of subjective survival probabilities. Therefore, we have a source of
data that contains the information that we would need to determine the impact of subjective
survival probabilities on wealth holding and allocation behavior.
Several studies have analyzed the validity of these subjective survival probabilities in
terms of actual mortality risk. Hurd and McGarry (1995) find that these measures are internally
consistent, match closely to life table averages and covary with known correlates of actual
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mortality . In a follow-up study, Hurd and McGarry (2002) find that these probabilities are also
good predictors of actual mortality experiences of the HRS respondents. Smith, Taylor and Sloan
(2001) support this finding but also point out that there is a large portion of the sample that report
very small changes in their survival probabilities over time.
Elder (2012) looks closely at the subjective survival probabilities in the HRS and finds
systematic differences as compared to life tables probabilities. He finds that the subjective
survival probabilities do not account for yearly increases in mortality rates and that individuals
do not update their survival probabilities as expected. He also finds that the life-table survival
probabilities are considerably better predictors of actual survival than the subjective survival
probabilities. Perhaps most concerning is that Elder (2012) provides compelling evidence that
subjective survival probabilities in the HRS contain significant noise; in fact so much random
noise that it may overwhelm any individual information that reflects actual heterogeneity.
While the above studies provide evidence that subjective survival probabilities match
aggregate life-table probabilities but systematically differ from how mortality rates actually vary
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Hurd and McGarry (1995) do note that blacks report higher subjective survival probabilities
than their white counterparts. This is inconsistent with the life-table probabilities.
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over the lifetime it is still important to determine whether they are useful in explaining
heterogeneity in economic behavior. What is important to note is that the findings in Elder
(2012) suggest that we are likely to see very little impact from subjective survival probabilities
simply because there is little signal in these measures.
Our empirical work focuses on estimating reduced-form relationships of wealth holdings
and asset allocation with subjective survival probabilities elicited from individuals. We attempt
to control for the heterogeneity in factors that economic theory tells us should affect the wealth
holding and allocation decision including basic demographics (age, education, gender, etc.),
income (both current and permanent), as well as proxies for an individual’s time preference,
cognition, and risk aversion. Since the HRS collects financial data at the household level, we
model the household’s wealth holding and the proportion of financial wealth allocated to stocks,
bonds, CDs, and checking/savings/money market accounts. We estimate different models for
single and married households since multi-person households have to take into account the
survival expectations of both members. To estimate our allocation equations we use the QuasiMaximum Likelihood approach proposed in Papke and Wooldridge (1996, 2008) which is
appropriate for estimating fractional response variables.
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We find mixed evidence for the impact of subjective survival on wealth holdings . For
single households, we find that a one-percentage point difference in reported subjective survival
probability (a difference of 0.01) leads to about $266 more Net Worth and about $124 more
Financial Wealth holdings, both significant at the 5% level. For married households we find that
a one-percentage point difference in the husband’s subjective survival probability leads to

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We find mixed evidence in terms of statistical significance of our estimates. Regardless of
statistical significance, all of our estimated impacts for subjective survival probabilities are very
small in magnitude.
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increased Net Worth of about $241 and about $29 more Financial Wealth but both estimates are
statistically insignificant at standard levels. A one-percentage point difference for the wife’s
subjective survival probability is associated with about $425 more in Net Worth (statistically
significant at the 5% level) but, contrary to theory, leads households to hold about $22 less in
Financial Wealth, though this estimate is not statistically significant.
There appears to be no impact of survival probability on the proportion of financial
wealth allocated across assets for single households though we note that we see the correct sign
of the impact of survival probabilities on the proportion of wealth allocated to stocks and
checking. For married households, we find that a one-percentage point difference in the
husband’s subjective survival probability leads the household to allocate about 0.026% more to
stocks and a one-percentage point difference in the wife’s subjective survival probability leads
the household to allocate around 0.022% more financial wealth to stocks; these are significant at
the 5% and 10% levels. The average partial effects for the subjective survival probabilities in our
checking equation are negative for both husband and wife but are very small and statistically
insignificant at standard levels.
We find no evidence that our estimated impacts of subjective survival probabilities are
sensitive to the inclusion of a measure of permanent income. If we drop our measure of
permanent income, we find that the estimated impact of subjective survival increases slightly for
wealth holdings and asset allocation equations. We also show that our estimated impacts for
subjective survival on wealth holding and asset allocation look very different when we treat
reported survival probabilities of one differently. This is consistent with the findings from Elder
(2012) which suggest that there is significant measurement error in these reported survival
probabilities.

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The outline of this paper is as follows. Section 2 briefly discusses the underlying theory

and reviews studies that use subjective survival probabilities to explain economic behavior.
Section 3 briefly explains the HRS, provides descriptive statistics and patterns in the data, and
details the analysis of the informational content of survival probability measures in terms of
wealth holding and allocation. Section 4 presents the results of our analysis and Section 5
concludes.
1.2 Theory and Literature Review
Economic theory indicates that forward-looking individuals with higher survival
probabilities will hold more wealth and allocate a larger portion of their wealth to the risky asset
than individuals with lower survival probabilities. This is because with time-separable utility the
introduction of non-zero survival probabilities effectively multiplies a time varying factor to the
individual’s constant discount rate. Bernheim, Skinner, and Weinberg (2001) perform a simple
simulation and demonstrate that individuals with lower discount rates hold more wealth than
individuals with higher discount rates. Cocco, Gomes, and Maenhout (2005) and Sahm (2007)
show that the underlying policy function that defines the optimal portfolio choice of the
household is a function of total wealth: the sum of discounted future income and current
financial wealth holdings. With uncertain lifetimes, the individual will discount future income by
incorporating the probability that they will earn that future income stream. They show that the
proportion of financial wealth allocated to the risky asset is positively related to how “certain”
their future income stream is. DeNardi et al. (2009) also document that higher survival
probabilities (even if they are small) will lead even the oldest and sickest individuals to spend
down their retirement wealth very slowly. DeNardi et al. (2009) also show that the impact of
lower survival probabilities leads individuals to decrease their wealth holdings.

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The basic premise here is that individuals will smooth the marginal utility of

consumption over time by setting the marginal utility of current consumption to the marginal
utility of the discounted future consumption stream. As long as individuals discount the future by
including the probability of survival we will see that a person with higher survival probabilities
will hold more wealth and allocate a larger proportion of wealth to the risky asset.
Several studies attempt to link subjective survival probabilities to economic behavior and
these tend to focus on the areas of retirement, bequests, wealth holding, and consumption. Hurd
et al. (2002) analyze the decisions to retire and claim Social Security benefits early. Using
responses to subjective survival to age 85 and Social Security earnings data matched to the first
four waves of the HRS, they estimate their model on two different samples: those who are retired
prior to age 62 and those not retired by 62. They find that there is a small, statistically significant
increase in retirement and claiming of benefits but only for those individuals that report that their
probability of living until age 85 is zero.
Delevande et al. (2006) revisit the retirement and claiming issue with the hope of
obtaining estimates that are more accurate by instrumenting for measurement error in the
reported survival probabilities. They use the response to the question of survival to the age of 75
instead of 85 used by Hurd et al. (2002). As instruments for the subjective survival probability,
they use demographic information, mortality experience of parents, and an optimism index,
which is a predicted value generated from a factor analysis of the remaining subjective
probability questions. This optimism index reflects the correlation of the responses to all
probability expectation questions in the HRS and potentially represents the unobserved
heterogeneity in individual expectations. Notably this constructed optimism index has a large
and statistically significant impact on their instrumental variable for the subjective survival

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probability. Similar to Hurd et al. (2002) they estimate the impact of survival probabilities on
claiming behavior for the group of respondents that are retired by age 62 and the impact of
survival probability on early retirement and early claiming for the sample of respondents that are
still working at age 62. Using the raw subjective survival probability measures they find no
statistically significant impact of survival on either claiming or retirement. When they instrument
for subjective survival they find that there is a statistically significant impact on early claiming
but not on retirement. Their results suggest that a five-percentage point increase in the predicted
survival probability will lead to a 1.9 percentage point decrease in the number of people that will
claim Social Security benefits early. Delevande et al. (2006) claim that using instrumental
variables to rid the subjective survival probability of measurement error is the reason that they
are able to find a statically and economically significant effect of subjective survival on claiming
behavior.
Gan, Gong, Hurd, and McFadden (2004) study the impact of subjective survival
probability measures on the bequest behavior of elderly households. They derive estimable
equations from a life-cycle model so that they can estimate structural parameters that can
describe the individual’s preferences for bequests. To account for focal points and to calculate a
survival curve they construct a measure of yearly mortality rates based on responses to subjective
survival questions in the Asset and Health Dynamics among the Oldest Old (AHEAD) study. In
addition, they estimate their equations using the life-table survival curves to compare the
predictive power of the subjective survival curves. Since they are estimating a life-cycle model,
they are able to simulate consumption and wealth trajectories and compare model predictions to
actual decisions within the AHEAD panel. They find that their predicted consumption and
wealth trajectories using the survival curves derived from the subjective survival probabilities

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outperform the predicted values using life-table survival. They also note that their estimates
suggest that bequest motives of the older population represented in the AHEAD are very small
and that most bequests are involuntary or accidental.
Salm (2006) attempts to estimate the structural parameters from the Euler equation for
consumption derived from a simple life-cycle model. He uses data from single households that
completed the HRS interview in 2000 and 2002 and completed the Consumption and Activities
Mail Survey (CAMS) in 2001 and 2003. Using the subjective survival probabilities, he
constructs yearly survival rates following Gan et al. (2004). He interprets the inverse of his
estimated coefficient on the subjective survival probability as his estimated value for the risk
aversion parameter. To allow for precautionary savings he estimates the model including an
estimated variance of out-of-pocket medical expenses to proxy for consumption risk. Salm
(2006) finds that higher subjective survival probabilities lead to decreases in the growth rate of
consumption.
Perry (2005) estimates an empirical model derived from the Euler equation for
consumption. He constructs yearly survival rates from the subjective survival probability
responses in the HRS. Perry (2005) constructs his measure of consumption by looking at
differences between wealth holdings across periods. He finds that there is no statistically
significant relationship between the constructed subjective survival probabilities and his measure
of consumption. Perry (2005) points to substantial measurement error in his measure of
consumption as the culprit for the lack of statistical significance in his estimates.
Bloom et al. (2006) studies the impact of subjective survival probabilities on retirement
and wealth holdings of both single and married households. Bloom et al. (2006) take a sample of
individuals that were aged 50-70 in 1992 and estimate the relationship between their retirement

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decisions and wealth holding and subjective survival probability to age 75. To correct for
potential measurement error they instrument subjective survival probabilities using mortality risk
factors and parental mortality experience. They find no statistically significant impact of
subjective survival on either retirement or wealth holding for single households. Looking at
married households, they find no statistically significant relationship for the retirement decision
of either the husband or wife. For married households they find that once they instrument for the
subjective survival probability they estimate a statistically significant impact of survival
probabilities of both spouses. Their estimates suggest that a ten percentage point change in the
husband’s survival probability leads to a $27,600 increase in wealth (significant at the 10%
level) and a ten percentage point increase in the wife’s survival probability leads to a $32,600
increase in wealth. Bloom et al. (2006) interpret these findings as evidence that households save
more in the face of higher expected lifetimes because there is no incentive to postponing
retirement.
DeNardi, French, and Jones (2009) look at the impact of survival uncertainty, medical
expenses, and health uncertainty on the wealth holdings of elderly individuals in the AHEAD.
They estimate a structural model using the Method of Simulated Moments and matching the
medians that were in the data to the medians estimated by the structural model. They did not use
the subjective survival probabilities that are collected in the AHEAD survey; instead, they
3

estimate future survival relying on the actual mortality experience in the panel . They find that
the structural model fits the data very well. This lends much more credibility to the simulations
that they perform to isolate the impact of differential mortality on wealth holdings. Since they
have modeled their survival probabilities as a function of gender, health status, and permanent
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The model for survival probabilities includes the prior period’s health status, permanent
income, and gender.
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income, they perform simulations that show the impact of changes within the components of
survival. It appears that the impact of health, gender, and permanent income all have similar
impacts on the wealth holdings of elderly households. The simulations clearly point to the fact
that the slow spend down of wealth in old age is due to uncertain lifetime. As long as there is a
possibility of outliving one’s assets there will be a significant precautionary savings motive.
With the exception of Hurd et al. (2002), Bloom et al. (2006), and Delevande et al.
(2006), all of the cited works estimate either an Euler equation or a structural model using either
the HRS or the AHEAD. Most of the studies that use the subjective survival probabilities either
transform them into yearly survival curves or use instrumental variable techniques to correct for
measurement error. Our work is probably closest to Bloom et al. (2006), although we take
serious the findings of the other studies that we have reviewed. We estimate models for wealth
holdings of both single and married households and find a statistically significant but small
impact on wealth holdings for single and married households. While there are a few empirical
studies of asset allocation in the HRS they look at health status and not survival probabilities
(Rosen and Wu (2004) and Berkowitz and Qiu (2006)). Lillard and Willis (2001) estimate a
model of asset allocation including a measure of the number of focal point responses across all
subjective probability questions, which they interpreted as a measure of cognition. They found
that fewer focal point responses correlate with increased allocation to the risky asset.
1.3 Model Specification, Descriptive Statistics, and Estimation Strategy
The goal of our empirical exercise is to determine whether subjective survival
probabilities help to explain the differences in wealth holding and allocation behavior between
households in the Health and Retirement Study. We estimate reduced form equations to
determine the relationship between household wealth holdings, allocation of wealth and survival

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probabilities. In our model specification we include basic demographic characteristics: age,
education, working status, household size, gender, marital status, and current income. Theory
dictates that the household’s wealth holding and allocation decisions are a function of not only
their current resources but also what they expect their future resources will be. To this end, we
include a measure of permanent income in our model specifications. Differences in risk aversion
among households can affect the amount of wealth and the allocation of wealth across assets. To
proxy for risk aversion we include responses to questions regarding a household’s willingness to
accept different income gambles. Households can also differ in their discount rates. We include
responses to questions about the length of time that the individual considers for financial
planning as a proxy for discount rates. Previous research (Elder (2012), McArdle et al. (2009)
and Lillard and Willis (2001)) suggest that an individual’s cognition can also affect the outcome
of the wealth holding and asset allocation decisions; we include measures of word recall and
simple numerical calculations captured in the HRS as proxies for cognition.
We will estimate separate equations for single and married households. We present
equations for single households; married households will include the same set of regressors for
both the husband and wife. Let i represent the household and t represent the year of the survey.
To estimate the impact of survival probabilities on wealth holding we specify the following
equation:
Wit = β1 ⋅ Pr(Live to 75) + βXit + uit

(1.3.1)

where Wit is a measure of wealth, either Net Worth or Financial Wealth, Pr(Live to 75)
represents our measure of subjective survival probability from the HRS (see below for the
specific question used to solicit the survival probability), and Xit are control variables that we
detail below. We also estimate the relationship between survival probabilities and the allocation
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of Financial Wealth across four types of assets; stocks, bonds, CDs, and checking/savings/money
market accounts:

yitg = Φ(θ1 ⋅ Pr(Live to 75) + θZit ) + ε it

(1.3.2)

where yitg represents the proportion of financial wealth allocated to asset g in year t by
household i . Here Zit represents control variables that will include the same variables we use in
the wealth holding equations with the addition of a measure of total wealth available to the
household at time t . Due to the fractional nature of the asset share equation in (1.3.2) we have
followed Papke and Wooldridge (1996, 2008) and specified that the conditional mean function is
nonlinear; Φ represents the standard normal cumulative distribution function.
We include controls for standard demographic variables that can affect behavior such as
2

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age, age , age , household size, education, as well as controls for year effects. Theory tells us
that it is necessary to control for current income as well as the value of lifetime income. We
measure current income as the sum of all non-capital income of the household as reported in all
waves. As a measure of lifetime resources we follow Altonji and Doraszelski (2005) and
calculate a household’s permanent income. To do this we regress current household income on
2

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age, age , age , age , a set of year dummies, household size, and indicators for marital status,
4

gender and education level. For each household we calculate the average of the residuals from
this regression and then compute the permanent income of the household as the sum of the
average residual and the predicted value of income for the education level of the individual and

4

For married households, we do not include indicators for gender or marital status.
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5

assuming that the individual is at the average age in our sample. Since there is likely a nonlinear
impact of these income measures we also include squares of current and permanent income as
6

well as their cross product in our wealth holding equations .
Theory dictates that total wealth has an impact on the allocation of wealth across
available assets. Ideally, the measure of wealth that we would include in our model would
include pension wealth, Social Security wealth and the present value of all future income. For
our measure of wealth we use the log of Net Worth which is the sum of Financial Wealth (the
value of holdings in stocks, bonds, CDs and checking/savings/money market accounts) and NonFinancial Wealth (the value of holdings in IRA/Keogh accounts, housing, vehicles, other real
7

estate, and trusts) minus any debt associated with these asset holdings. Our hope is that the
combination of Net Worth and the estimated permanent income measure will act as a good proxy
for the total wealth measure that theory dictates will affect decision-making.
In addition to controlling for differences in household demographics, income, and wealth
there is also a need to control for risk aversion and discount rates of households. To proxy for
discount rates we will use responses to HRS question regarding the financial planning horizon of
8

the household . To control for risk aversion we will use questions meant to solicit aversion to
income risk and we will categorize individuals based on their responses to a series of unfolding

5

For single households we use calculate permanent income at age 58. For married households
we use 58 for the husband and 54 for the wife, both of these are the average ages in our sample.
6
In our asset allocation equations we actually use permanent log income, which we estimate in a
similar manner to permanent income with the exception that we regress the log of current income
on the demographic variables.
7
We use RAND imputations for missing values of income and asset holdings when the
respondent was unable to give exact values.
8
The choices available from the planning horizon question are: Next Few Months, Next Year,
Next Few Years, 5-10 Years and 10+ Years. We use Next Few Months as our base category.
13

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questions regarding different income scenarios following Barsky, et al. (1997). This essentially
categorizes an individual into one of four groups: High Risk Aversion, Medium-High Risk
Aversion, Medium-Low Risk Aversion and Low Risk Aversion. We use High Risk Aversion as
our base group.
We include two additional controls to proxy for the cognitive ability of the individual.
The HRS collects several measures of individual cognition, but not all measures are available in
all waves of the survey. In every wave of the HRS, respondents are given a list of nouns and then
asked to repeat this list immediately and at the end of the cognition section. Our first measure of
cognitive ability is the proportion of words recalled at the end of the cognition section. In
addition, each individual performs a series of five simple numerical calculations. First, the
individual subtracts 7 from 100. The next question in the series asks the individual to subtract 7
from the answer to the previous calculation. The individual performs this calculation a total of
five times. As a second measure of cognition, we use the number of correct calculations, ranging
from 0 to 5. By no means do we think that these two measures will completely capture the
cognitive ability of the individual. Our hope is that accounting for education level and some
time-varying measure of cognitive ability that we can accurately proxy for the cognitive ability
of an individual.

9

We use data from the Health and Retirement Study (HRS). The HRS began in 1992 and
was nationally representative of all non-institutionalized individuals aged 51-61 in that year. In
1998 the HRS combined with the Asset and Health Dynamics of the Oldest Old (AHEAD) and
added several new cohorts to be nationally representative of the population of non9

McArdle, Smith, and Willis (2009) study the ability of cognitive measures in the HRS to
explain wealth holdings and allocation. They find that word recall is positively correlated with
allocation to the risky asset (stocks) and the amount of wealth held. Their analysis does not
account for survival probabilities.
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institutionalized individuals aged 51 and older. In addition to initial respondents, spouses are
interviewed and followed in subsequent interviews. We use data for all cohorts from Waves 1-7
(1992-2004) of the HRS. In addition to basic demographic variables, the HRS collects detailed
information on wealth, allocation, income and its sources, health and measures of the probability
that future events occur. The main variable of interest in this analysis is the response to the
following question:
On a scale from 0 to 100, where 0 is no chance and 100 is absolutely certain, what are
the chances that you will live to age 75 or older?
We use responses to this question as a measure of subjective survival probabilities.
We construct our data set by household. For married households we combine spouses in
each wave so that our panel consists of household observations by year. To select individuals for
our sample we include observations that satisfy the following criteria:
(i)

respondent is younger than 65 at the time of the interview,

(ii)

respondent does not have missing values for subjective survival probability,

(iii)

responses are not from a proxy interview and

(iv)

the respondent is still living at the time of the interview.

These criteria are applied at the individual level so married households that do not have both
spouses are dropped. The decision to allocate wealth across assets is conditional on holding
positive financial wealth; therefore, for estimating allocation relationships we drop observations
where financial wealth is zero. Imposing the above criteria for married households leaves us with
6,608 unique households (18,603 observations) to estimate wealth holding equations (we call this
our Wealth Sample). If we drop those observations where households are holding zero financial
wealth we are left with 6,048 households (16,690 total observations) which we will call our

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Allocation Sample. Applying the criteria to single households we are left with 5,022 households
(14,275 total observations) for our Wealth Sample and dropping observations with zero financial
wealth we are left with 4,053 households (10,573 total observations) in our Allocation Sample.
We calculate means and medians for the Wealth and Allocation Samples for both single
and married households. Table 1 displays these descriptive statistics for single households. We
can see that single households are predominately white females. A majority of individuals have
not completed college (a little over 80%). Most single households hold most of their net worth in
non-financial wealth. Between the Wealth and Allocation sample, we can see that individuals
that hold positive financial wealth tend to report higher survival probabilities, have slightly more
education, income, and wealth. On average, it appears that single households tend to hold higher
amounts of their financial wealth in checking, savings, and money market accounts.
Table 2 displays descriptive statistics for the Wealth and Allocation Samples for married
households. Compared to single households, married household members tend to have slightly
higher education. Married households also report holding a large portion of their net worth in
non-financial wealth. Married households earn more income on average and they have higher
wealth holdings relative to single households. Comparing the Wealth Sample to the Allocation
Sample, it appears that households with positive financial wealth are better educated, earn
slightly more income, and hold more wealth. In comparison to single households it appears that
married households allocate about 10% more of their financial wealth to stocks and ten percent
less to checking, savings, and money market accounts.
Table 3 displays the raw relationship between average wealth holdings and average asset
allocation and survival probabilities for single households. Tables 4 and 5 show the same
relationship for married households; Table 4 uses the husband’s reported survival probability and

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Table 5 uses the wife’s reported survival probability. We can see that average wealth holdings
are higher for higher values of survival with the exception of those individuals that report a
survival probability of one. We can also see that the average proportion of financial wealth to
stocks is increasing with higher survival probabilities and the allocation to checking is
decreasing with higher survival probabilities. Since stocks and checking are the most risky and
least risky of the four financial assets this finding seems to fit well with what theory predicts.
The objective of this study is to determine whether this correlation still exists once we have
removed the impact of other factors that affect wealth holding and allocation behavior.
1.4 Results and Discussion
Using equation (1.3.1) we estimate models of wealth holding for single households;
results are presented in Table 6.

10

We see a positive impact of survival probabilities on both Net

Worth and Financial Wealth; a one-percentage point difference in survival probability is
associated with about $256 more Net Worth and $124 more Financial Wealth, both estimates are
significant at the 5% level. Education plays a key role in explaining differences in wealth
holdings among single households. Households that have a college degree have nearly $111,000
more in Net Worth (and nearly $51,000 more Financial Wealth) than individuals that have not
completed High School. There also appears to be a significant positive impact of a longer
financial planning horizon. Those households that plan for more than ten years into the future
have about $95,000 more in Net Worth and $39,000 more Financial Wealth than households that
only plan for the next few months.
We now turn to the results of estimating equation (1.3.2) using the Quasi-Maximum
10

Throughout the paper (including the tables) we refer to a one-percentage point change in the
subjective survival probability. This is equivalent to a 0.01 change in our measured variable. The
reported coefficients and average partial effects in the tables are already calculated for a 0.01
change in subjective survival probability.
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Likelihood Estimation proposed in Papke and Wooldridge (1996, 2008) for fractional response
variables. Table 7 presents the estimated average partial effects from estimation of this fractional
probit on the proportion of Financial Wealth allocated to the four different assets: stocks, bonds,
CDs, and checking/savings/money market accounts.
There appears to be no impact of survival probabilities on the allocation of wealth for
single households. While the signs on the average partial effects of survival probability on stocks
and checking are what theory predicts (positive and negative, respectively) they are estimated to
be nearly zero and are all statistically insignificant at any standard level. We see that households
with a college degree allocate 11% more of their financial wealth to stocks and 10% less to
checking accounts than households that have not completed High School, both significant at the
5% level. Financial Planning Horizon also plays a role in asset allocation, we estimate that
households with a horizon of ten years or more will allocate 5% more financial wealth to stocks,
1% more to bonds, and 6% less to checking accounts than those households that have a horizon
of only a few months.
Table 8 contains the results from estimation of the wealth holding equation (1.3.1) for
married households. We see a positive impact of a wife’s subjective survival on Net Worth
holdings; a one-percentage point difference in a wife’s survival probability is associated with
about $425 more in Net Worth, this is significant at the 5% level. A one-percentage point
difference in the husband’s subjective survival is associated with about $241 more Net Worth,
but this estimate is not statistically significant. Looking at the impact of subjective survival on
Financial Wealth, we see that the estimated impact for a husband’s survival is very small and
insignificant, and that the estimate of the impact of a wife’s survival probability is actually
negative, though both estimates have very large standard errors. The estimated impact of

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education on wealth holdings is not as clear-cut for married households as for single households.
We see that the husband’s education has no statistically significant relationship with Net Worth;
it appears that having less than high school education leads to more wealth holdings than if the
husband completed high school or some college. The wife’s education level does have the
expected impact on Net Worth. If the wife has a college degree, we estimate that household to
hold about $84,000 more in Net Worth than a similar household with a wife that did not
complete high school. Our estimates for education in the financial wealth regressions make a
little more sense. We find that a college degree for the husband leads to about $37,000 more in
financial wealth. If the wife also completed college, we estimate that the household’s financial
wealth will increase by $38,000. We also see that having the longest financial planning horizon
(10 + years) has a positive impact on wealth holdings. If both spouses have horizons of ten years
or more we see them holding about $110,000 more in Net Worth and about $52,000 more in
Financial Wealth.
Table 9 displays the results of the fractional probit estimation of equation (1.3.2) on the
allocation of Financial Wealth across the four asset types. We see that there is a statistically
significant impact of both husband and wife survival probabilities on the allocation to stocks and
CDs. A one-percentage point difference in the husband’s subjective survival leads households to
hold about 0.026% more of their financial wealth in stocks and about 0.013% less in CDs, these
estimates are significant at the 5% and 10% level respectively. A one-percentage point difference
in the wife’s survival probability leads the household to hold about 0.022% more financial
wealth in stocks and 0.019% less financial wealth in CDs, both these estimates are significant at
the 10% level. While the estimated average partial effects of husband and wife survival
probabilities are statistically insignificant for the allocation of wealth to checking we can see that

19

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we estimate negative, though small impacts of higher survival.
Similar to our findings for single households we see that higher education levels of both
spouses lead to increased allocation to stocks and decreased allocation to checking. Not only is
the estimated impact of a college degree statistically significant for both spouses for all assets but
the magnitude is also large compared to the estimated impact of subjective survival to age 75.
We see that the financial planning horizon of the husband only is positively associated with the
allocation of wealth to stocks and negatively associated with the allocation to checking. This is
not the case for the wife’s planning horizon. For stock allocation, we see that longer planning
horizons have a negative impact relative to the shortest horizon of only a few months. Looking at
the results for the allocation of wealth to checking we can see that only the longest planning
horizon (10+ years) has a negative impact, though none of the estimated average partial effects
for the wife’s planning horizon are statistically significant.
One main reason that our results for single and married households differ is that married
households need to consider the impact of the characteristics of both household members.
Women live longer than men and therefore would be the more likely of the two spouses to have
to face the risk of outliving assets. While we find an impact of the wife’s increased survival
probabilities on Net Worth, it is very small and does not appear to carry over to the financial
wealth of the households. Looking at the allocation decision, we see that the impact of the wife’s
survival probabilities is smaller than the estimated impact of the husband’s. In addition, it
appears that only the financial planning horizon of the husband has any impact on allocation. We
see that the longest planning horizon (relative to the shortest horizon) for the husband increases
the proportion allocated to stocks by about 3% and decreases the proportion allocated to
checking by 4%. It appears that the characteristics of the wife play a small role in the decision of

20

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how much to save and how to allocate wealth across assets. This could be consistent with the
idea that the household maximizes their joint utility, i.e. they only consider the lifetime of the
household when both spouses are alive. If this is the case, we would see that divorced or
widowed women would have less wealth at their disposal. From our analysis of single
households, we see that divorced women have less wealth but widowed individuals have greater
wealth. The asset allocation of divorced and widowed households in the analysis of singles
indicates that they allocate less to stocks and more to checking, but none of these estimates are
statistically significant. A more detailed analysis into the decision making of married households
and the way that they pool their characteristics to make wealth and allocation decisions is
definitely an avenue for future research.
We test the sensitivity of our results for subjective survival responses for wealth holding
and asset allocation for single and married households to dropping variables that include our
constructed measure of permanent income. We remove permanent income, its square, and the
cross product of permanent income and current income from the wealth holding equations and
we remove permanent log income from the asset equations.
For single households, we can see that removing permanent income from our wealth
holding equations leads to an increased impact of subjective survival probabilities. Without
controlling for permanent income, we see that a one-percentage point difference in subjective
survival leads single households to hold $339 more Net Worth and $156 more Financial Wealth.
Removing permanent log income from our asset allocation equations we see that the impact of
subjective survival does not change much. There is no statistically significant impact of
difference in subjective survival, though we do see that the estimated impacts of higher
subjective survival increased slightly for stocks and bonds and decreased slightly for CDs and

21

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checking.
When we remove permanent income variables from the wealth holding equations for
married households, we estimate that a one-percentage point difference in husband’s subjective
survival probability leads the household to hold $356 more in Net Worth and $66 more in
Financial Wealth. A one-percentage point difference in the wife’s subjective survival probability
increases Net Worth by $522 and Financial Wealth by $12. While the estimated impact for
Financial Wealth remains statistically insignificant and economically small for both spouses we
see that the estimated impact of a one-percentage point difference in subjective survival
probability for either spouse is generating about $100 more Net Worth and that the estimate for a
husband’s subjective survival probability is now significant at the 10% level. Removing
permanent log income from the asset allocation equations has a similar effect as in single
households. The estimated impact of subjective survival probabilities slightly increases for
stocks and bonds and slightly decreases for CDs and checking. We see that our estimated impact
for a one-percentage point difference in the husband’s subjective survival probability leads to
0.027% more of the households wealth allocated to stocks and only 0.015% less wealth allocated
to checking. We see that the one-percentage point difference in the wife’s subjective survival
probability is now driving a larger impact on the allocation of wealth to stocks. Including
permanent log income we see that the impact was around 0.022% more in stocks (significant at
the 10% level), but dropping our permanent log income measure we see that this estimate
increases slightly to 0.023% and remains statistically significant at the 10% level. In addition,
including permanent log income we estimated a negative (statistically insignificant) impact of
increased subjective survival probabilities of both spouses on the proportion of wealth allocated
to checking. When we drop permanent log income, the estimated impacts become slightly more

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negative.
Overall, excluding our measure of permanent income from our wealth holding and
allocation equations leads us to find nearly identical coefficients and average partial effects as
estimates from models that include permanent income. Removing permanent income, we see the
estimated impacts of subjective survival on wealth holdings and asset allocation move away
from zero. This suggests that the permanent income is likely weakly positively correlated to the
reported subjective survival probabilities. Controlling for a measure of expected lifetime
resources appears to have a negligible effect on the estimated impact of reported survival
probabilities. It is interesting though to note that the inclusion of a measure of permanent income
shrinks the estimated impacts for survival probabilities to zero.
Bunching at focal points has been a concern of the subjective survival probabilities in the
HRS since Hurd and McGarry (1995). The worry is that focal point responses may reflect either
difference in cognition (Lillard and Willis (2001)) or measurement error (Bloom et al. (2006)).
We provide evidence that the relationship between wealth holdings and asset allocation in the
raw data appears to support theoretical predictions; average wealth holding and allocation to the
risky asset increase as subjective survival probabilities increase, with the exception of
households that report survival probabilities of one. Bloom et al. (2006) points out that
households that report survival probabilities of one appear to have mortality rates similar to those
people responding with survival probabilities closer to 0.7 or 0.8.
To assess the impact of these focal points on our estimates we treat focal point responses
of one differently. First, we estimate our wealth holding and asset allocation equations dropping

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observations where the respondent reports a survival probability of one.

11

Second, recognizing

that a focal point response may reflect some underlying time-constant heterogeneity of the
household, we estimate our models dropping the all observations for the household if any
respondent has ever reported a survival probability of one.
For single households, excluding observations where the respondent reported a survival
probability of one leads us to estimate a larger impact of subjective survival probabilities on
wealth holdings. Despite increases in the standard errors from reducing the number of
observations we can see that we are estimating that a one-percentage point difference in
subjective survival probability translates into $394 more Net Worth and $182 more Financial
12

Wealth . If we drop households that ever reported a survival probability of one our estimate of
the impact of a one-percentage point difference in subjective survival on Net Worth is $466 and
$143 on Financial Wealth.
More interesting is the impact that dropping focal point responses has on the estimated
impact of subjective survival probabilities for asset allocation. When we drop observations
where the respondent gives a survival probability of one we estimate that a one-percentage point
difference in subjective survival increases the allocation of financial wealth to stocks by 0.032 %
and decreases allocation to checking by 0.035%; both estimates are statistically significant at the
5% level despite the increases in standard error due to fewer observations. We see an even larger
impact of removing households that ever reported a survival probability of one: we estimate that
a one-percentage point difference in survival probabilities leads to 0.05% more financial wealth
11

For married households, we drop the observation if either member of the household reports a
survival probability of one.
12
Though not statistically significant, this is $13,837 more Net Worth and $5,877 more
Financial Wealth than estimates including observations with reported survival probabilities of
one.
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allocated to stocks and 0.047% less allocated to checking; both estimates are significant at the
5% level.
Treating reported survival probabilities of one differently leads to significantly different
estimates of the impact of subjective survival on wealth holdings and asset allocation for married
households. Removing observations where either the husband or the wife reported a survival
probability of one leads us to estimate that a one-percentage point difference in the husband’s
survival probability results in only $57 more Net Worth and $5 less Financial Wealth (both
estimates are statistically insignificant). A one-percentage point difference in the subjective
survival probability of the wife leads to an estimated $698 (significant at the 5% level) difference
in Net Worth and $129 difference in Financial Wealth. Removing focal point responses causes
the estimate for the wife’s subjective survival to increase for wealth holding equations and the
estimate for the husband’s subjective survival to decrease for both wealth holding equations.
Removing households where either spouse ever reported a survival probability of one
leads us to estimate that a one-percentage point difference in the husband’s subjective survival
probability results in $171 less Net Worth and $14 more Financial Wealth. A one-percentage
point difference in the wife’s subjective survival probability leads to $497 more Net Worth and
$215 more Financial Wealth. Interestingly the estimated impacts of subjective survival
probabilities are statistically insignificant once we remove focal point households.
When we remove observations where either spouse reported a survival probability of one
we see that a one-percentage point difference in the husband’s survival probability results in
0.035% more financial wealth allocated to stocks (significant at the 5% level), 0.021% less
allocated to CDs (significant at the 5% level) and 0.017% less in checking (statistically
insignificant). A one-percentage point difference in the wife’s survival probability results in

25

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0.032% more financial wealth allocated to stocks (significant at the 5% level), and 0.026% less
allocated to CDs (significant at the 10% level). When we drop households that ever report a
survival probability of one we estimate that a one-percentage point difference in the husband’s
subjective survival leads us to find a statistically significant (5% level) increase of 0.012% of
financial wealth to bonds. Perhaps more interesting is the fact that we now estimate that a onepercentage point difference in subjective survival for the wife leads the household to hold 0.06%
more financial wealth in stocks and 0.036% less in CDs, while still small in magnitude they are
considerably larger than our estimates that include all households.
It is apparent that the estimated impact of subjective survival probabilities on wealth
holding and asset allocation are sensitive to the inclusion of reported survival probabilities of
one. Our sensitivity analysis suggests that households that ever report a survival probability of
one act very differently than the rest of our sample. Removing all observations for households
that ever report a survival probability of one leads us to find that single households tend to
increase their Net Worth more than their Financial Wealth in response to an increase in longevity
risk. We also find that single households increase their allocation of financial wealth to stocks
and decrease their allocation to checking. For married households we find that once we remove
the focal point households the wife’s survival probabilities becomes more important in the
wealth holding and asset allocation decisions. While the estimates of the impact on wealth
13

holding of wife’s survival probabilities are not statistically significant at standard levels , they
do coincide well with the allocation equation findings that the wife’s survival leads to a
significant positive impact on stock allocation. In fact, despite the lack of statistical significance
the estimated impact of the wife’s subjective survival probability is not much different than the
13

The estimated impact for Net Worth is significant at the 14% level and for Financial Wealth at
the 12% level.
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estimated impacts for the single household’s survival probability.
It appears that including households that ever report survival probabilities of one has a
significant impact on the estimated impacts of subjective survival on wealth holding and asset
allocation. Excluding these “focal point” households leads us to find estimated impacts that seem
reasonable from a theoretical perspective and are generally statistically significant, despite
increased standard errors due to smaller sample sizes. While there is still likely significant
measurement error present in the subjective probabilities, we have identified that at a minimum
the inclusion of households that report survival probabilities of one greatly affects estimates of
wealth holding and allocation equations.
1.5 Conclusion
Theory points to the need to include estimates of an individual’s subjective survival
probability in estimating models of economic decision-making. Several studies use the subjective
survival probabilities in the HRS to explain economic behavior with mixed results. We present
here a very simple analysis that attempts to explain differences in wealth holding and asset
allocation behavior using differences in subjective survival probabilities.
We find that there does appear to be a small, statistically significant impact of subjective
survival probabilities on wealth holding (Net Worth, particularly) and asset allocation (stocks, in
general). We find that our results for subjective survival probabilities are insensitive to the
inclusion of a measure of permanent income, though we do see that including a measure of
permanent income causes the estimated impact of subjective survival probabilities to move
towards zero. If we exclude our measure of permanent income, we find that our estimated impact
of subjective survival increases (but not by the total impact of permanent income) and the
estimated impact on asset allocation moves away from zero. We also find that inclusion of

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households that ever report a survival probability of one has a significant detrimental impact on
the estimated impact of subjective survival for asset allocation equations for single households
and for wife’s in married households. It appears that longevity risk plays a statistically
significant role for single households, which contradicts the findings of Bloom et al. (2006).
By no means does our study attempt to contradict the hypothesis that there is
measurement error in these subjective survival probabilities. In fact, it appears that bunching
does represent a form of measurement error but it may be distinctly different (or additive) to the
measurement error that is most likely present in all the subjective survival responses. Elder
(2012) finds that life table probabilities are better predictors of in-sample mortality and that
subjective survival probabilities probably reflect more measurement error than they reflect
heterogeneity in mortality expectations of HRS respondents. These findings raise questions about
both the validity of these subjective survival probabilities as proxies for mortality risk and how
we can interpret the estimated relationships between these measures and decision-making. Our
results suggest that controlling for other factors that can affect a household’s wealth holding and
allocation decisions (such as permanent income) almost completely removes the correlation with
subjective survival that we see in the raw data. In addition, we find that excluding households
that ever report a survival probability of one significantly affects our findings. We are not sure
what these focal point responses mean in terms of mortality or if they represent heterogeneity in
cognition but it does suggest that we are probably not able to interpret these probabilities as
reflecting actual mortality risk.
Future avenues for research would be to try different specifications for married
households, particularly around different types of objective functions that combine attributes of
the spouses differently. In addition, including the cohort survival probability module as a

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standard part of the HRS expectations survey would help to tease out measurement error and the
learning process that respondents go through when estimating their expectations. It might also be
beneficial to look into the impact of bequest intentions as they may play a large role for some
households compared to others.

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CHAPTER 2

A MODEL FOR MULTIVARIATE FRACTIONAL RESPONSES WITH AN APPLICATION
TO ASSET ALLOCATION
2.1 Introduction
Many interesting economic variables are fractional in nature. By definition, some
fractional response variables are related, such as the budget share of goods in a demand system
or the allocation of wealth across available assets. Since these fractional responses are shares of a
total, they must satisfy an adding up restriction. Due to the limited nature of the fractional
responses we are often interested in estimating relationships using nonlinear conditional mean
functions. Analogous to the case where we estimate linear conditional means, the adding up
restriction imposes constraints on the marginal effects of the covariates in our model. Since the
marginal effects of covariates in a nonlinear conditional mean are themselves nonlinear, this
causes some difficulty in estimation as it amounts to imposing nonlinear constraints on our
coefficients. We get around this issue by imposing the adding up restriction using the
multinomial quasi-likelihood.
Sivakumar and Bhat (2002) and Mullahy (2010) show how this approach works when
using cross-sectional data. In fact, this approach extends very naturally to the panel data case,
even if we specify a time-invariant unobserved effect for each cross-sectional unit. We can
appeal to the Correlated Random Effects approach to estimation (Chamberlain (1980) and
Mundlak (1978)) and specify that the conditional mean of the unobserved effect is a parametric
function of the time-averages of the independent variables. As pointed out in Wooldridge (2010),
this approach is only appropriate when we have a balanced panel. With an unbalanced panel, at a
minimum, we violate the assumption that the unobserved effect has a constant variance across

30

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cross-section units, since we have a different number of observations for each unit. We address
this concern by exploiting the independence of our cross-section units. To do this we estimate the
multinomial quasi-likelihood on balanced panel subsets of the full, unbalanced panel. We create
the balanced panels by combining households that are observed in the sample for the same
number of time periods. Since the households in different balanced panel subsets are independent
of each other we are able to take the weighted average of the average partial effects (and their
variances) estimated in each balanced panel subset to get the average partial effect for the entire
unbalanced panel.
This approach ignores the potential serial correlation present within the cross-sectional
unit. A natural next step is to correct for this serial correlation by using a weighting matrix that
accounts for serial correlation. To do this we use Classical Minimum Distance estimation.
Simply put, this method takes the estimated coefficients from the multinomial quasi-likelihood
estimation on each balanced panel subset of data and estimates a single coefficient for each
covariate that is a weighted average of all the balanced panel subset coefficients. The weighting
matrix that we construct contains the estimated variance-covariance matrices from the
multinomial quasi-likelihood estimated on each balanced panel subset and estimates of the serial
correlation within and across equations over time.
This paper takes the single equation fractional probit technique laid out in Papke and
Wooldridge (2008) and extends it to cover multiple fractional responses that must meet an
adding up restriction. In addition, we estimate an unbalanced panel using the Correlated Random
Effects framework by exploiting the independence of our cross-sectional units. Lastly, we also
show how we can apply Classical Minimum Distance to this procedure to potentially derive
more efficient estimates. Section 2 briefly covers some of the work done on fractional response

31

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variables. Section 3 presents the single equation specification of our multiple fractional
responses. Section 4 combines the single equations together using the multinomial quasilikelihood. Section 5 lays out the Classical Minimum Distance estimator. Section 6 describes an
alternative estimator that may be more parsimonious. Section 7 describes the data that we use in
our analysis and presents results of estimation and section 8 concludes.
2.2 Fractional Response Variables
Gourieroux, Monfort, and Trognon (1984) summarizes the theory underlying the quasimaximum likelihood estimator. They show that if a random variable comes from the linear
exponential family and we correctly specify the conditional mean, then estimation by quasimaximum likelihood generates consistent, though potentially inefficient, estimates for the
impacts of our covariates. This finding holds under the general conditions that we require for MEstimation. Under the standard assumptions, the quasi-maximum likelihood estimators are
asymptotically normal and converge at the rate of N −1/ 2 .
Papke and Wooldridge (1996) estimate a single fractional response variable in the crosssection using the logit functional form for the conditional mean function. Papke and Wooldridge
(2008) estimate a single equation fractional response variable in the panel data context using the
probit functional form for the conditional mean. They insert the time-averages of their
coefficients in their conditional mean function, thus parametrically “removing” the unobserved
effect. They also explore the possibility of improving efficiency by constructing a “working”
correlation matrix that allows for some correction of the serial correlation within cross-sectional
units. This technique essentially follows the logic that even if we incorrectly specify the
correlation structure that we can gain some efficiency by not ignoring its presence.

32

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≺⊙ ⊙ ց
Sivakumar and Bhat (2002), Mullahy (2010) and Koch (2010) study the multivariate

fractional response case in cross-sectional data. They follow a similar approach that we take in
this paper and exploit the robustness properties of the multinomial quasi-likelihood. They specify
their conditional means using the multinomial logit specification that imposes the adding up
restriction across equations. In this paper, we use the probit specification of the conditional mean
functions since it more easily accommodates the use of the Chamberlain-Mundlak device to
control for the presence of an unobserved effect.
2.3 Single Equation Fractional Probit
A natural starting point for our analysis is to ignore the relationship between our
fractional response variables and proceed as though we will estimate each equation separately.
We do this for two reasons; first our procedure essentially takes our single equation framework
and extends it to the multiple fractional response case. Second, the single equation framework
gives us a baseline set of estimates to compare to the results of the procedure that we present
here.
Assume that at each time period t , we observe household i allocate a proportion of their
financial wealth to each asset g . We assume that g = 1, 2,..., G exhausts all possible investment
options so that in each period the household invests all their financial wealth. This assumption
implies that

G

∑ g yitg = 1 for each i = 1, 2,..., N

and t = 1, 2,..., T . In addition to our response

vector, we observe a set of covariates xit and an unobserved effect cig . Our goal is to estimate a
model of the conditional mean of y given x to determine the impact of each covariate. By
construction, yitg is bounded between zero and one. Assuming that we use the same set of
covariates for each equation we can specify the conditional mean as:

33

ւ

≺⊙ ⊙ ց

E ( yitg | xit , cig ) = Φ(xit β g + cig ); i = 1,..., N ; t = 1,..., T ; and g = 1,..., G (2.3.1)
We point out that the above specification completely ignores the fact that these fractional
responses sum to one. It is simply acting as though we are treating each asset equation by itself.
Nevertheless, it is useful to study what we can derive from these single equation models since
our approach here essentially takes the specification in (2.3.1) for each equation and uses the
multinomial quasi-likelihood to impose the adding up constraint.
Our interest lies in estimating the average partial effects of the covariates, xit on the
proportion of financial wealth allocated to each asset. Following the results for a probit model
the direction of the partial effect is determined by the sign of β g . For a continuous variable xtk
(dropping the i subscript);
∂E ( ytg | xt , c g )
(2.3.2)
= β gk φ ( xt β g + c g )
∂xtk
To obtain the average partial effect we can average this over the distribution of cg and xt .
To consistently estimate the coefficients in this single equation framework we require
that the covariates are not correlated with the error term. We assume that xit is strictly
exogenous conditional on the unobserved effect,

E ( yitg | xi , cig ) = E ( yitg | xit , cig ).

(2.3.3)

Following Chamberlain (1980) and Mundlak (1978) we assume that
2
cig = ψ g + xi ξ g + aig ; where aig | xi ~ Normal (0, σ a ).

(2.3.4)

Using (2.3.4) we can rewrite (2.3.1) as:

E ( yitg | xi , aig ) = Φ(ψ g + xit β g + xi ξ g + aig ).
So the mean of yitg conditional on xi is:

34

(2.3.5)

ւ

≺⊙ ⊙ ց

ψ + x β + x ξ
g
it g
i g
E ( yitg | xi ) = E (Φ (ψ g + xit β g + xi ξ g + aig | xi ) = Φ 
 (1 + σ 2 )−1/ 2
ag



 (2.3.6)



or

(

)

E ( yitg | xi ) = E (Φ (ψ g + xit β g + xi ξ g + aig | xi )) = Φ ψ ga + xit β ga + xi ξ ga (2.3.7)
where we use the a subscript to denote that our estimated coefficients are scaled by
2
(1 + σ ag ) −1/ 2 . Woodridge (2002, Section 15.8.2) shows that the mixing properties of the

normal distribution lead to equation (2.3.7).
For identification of the scaled coefficients in (2.3.7) we require that there is no perfect
collinearity between the elements of xit and that there is enough variation in xit over time.
With the above parametric specification of the unobserved effect we can now write the Average
Structural Function (ASF) following Blundell and Powell (2003):

(

)

ASFg (xt ) = Ex  Φ ψ ga + xt β ga + xiξ ga 
i


(2.3.8)

A consistent estimator of ASFg ( xt ) is:

N
−1 Φ ψ + x β + x ξ
ˆ
ˆ
ˆ ga
ASF g (xt ) = N ∑
t ga i ga .
i =1

(

)

(2.3.9)

ˆ
ˆ
ˆ
where ψ ga , β ga , and ξ ga are consistent estimates of the scaled coefficients in equation (2.3.7).

For a continuous variable x j we can calculate the average partial effect by taking the derivative
of (2.3.9) and average over both i and t :

APE g ( x j ) =

ˆ
ˆ
ˆ
∂Φ(ψ ga + xit β ga + xi ξ ga )
∂x j

= NT −1

T N

ˆ
ˆ
∑ ∑ βˆga, jφ (ψˆ ga + xit β ga + xiξ ga )

t =1i =1

(2.3.10)
35

ւ

≺⊙ ⊙ ց

For a binary x j we can calculate the average partial effect by calculating the difference in the
Average Structural Function evaluated at zero and one.
The above presentation closely mirrors that of Papke and Wooldridge (2008) for the
single equation fractional probit model. We laid out the basic specification of the conditional
mean functions for each of our asset equations. We can use the individual conditional mean
functions and the previous results to estimate each equation assuming that there is no correlation
between equations. Estimation of the individual equation conditional means would entail
maximizing the Bernoulli quasi-likelihood function. While this works, it ignores the fact that
there is a definite relationship between the fractional responses, i.e. that they sum to one for each
household in each time period.
Analogous to the linear case, imposing the adding up restriction places a constraint on the
marginal effects of the covariates. The adding up restriction states that

G

∑ g =1 yitg = 1 . Taking

expectations of this conditional on xit and cig gives us the following restriction:

E ( yitg | xit , cig ) = 1.

(2.3.11)

Taking the derivative of (2.3.11) with respect to xk :

G ∂E ( y | x , c ) G
itg it ig
= ∑ β gk φ (xit β g + cig )
∑
∂xk
g =1
g =1
=

(2.3.12)

G

∑ β gk φ (ψ g + xit β g + xiξ g ) = 0.

g =1

Equation (2.3.12) shows that the adding up restriction implies that the sum of the marginal
effects of a specific covariate across equations is zero. Therefore, the effect of a change in a
covariate will lead to a reallocation of financial wealth across the available assets.
36

ւ

≺⊙ ⊙ ց
A very simple way to impose the adding up restriction in estimation is to exploit the

multinomial quasi-likelihood function.
2.4 The Multinomial Quasi-Likelihood
We use the multinomial quasi-likelihood function to allow for correlation across
equations and impose the restriction that our dependent variables must sum to one in each period
for each household. We assume our fractional responses ( yit1, yit 2 ,..., yitG ) are exhaustive and
mutually exclusive categories representing the population and that yitg ∈ [0,1] and

G

∑ g =1 yitg = 1 . Let xit

be a set of covariates that affect our quantity of interest, E ( yitg | xi ) .

Assume that

E ( yitg | xi ) = Φ(ψ o,t , g ,a + xit βo,t , g ,a + xi ξo,t , g ,a ) = Φ(xit , θo,t , g ,a )

(2.4.1)

For identification of the parameters the multinomial quasi-likelihood function requires that the
following constraint holds:
Φ (xit , θo,t ,G , a ) = E ( yitG | xi )
= 1 − Φ (xit , θo,t ,1, a ) − Φ (xit , θo,t ,2, a ) − ⋅⋅⋅ − Φ (xit , θo,t ,G −1, a ).

(2.4.2)

So that for any θt = {θt ,1,a , θt ,2,a ,..., θt ,G −1,a } ,

G

∑ Φ g (xit , θt ) = 1.

g =1

Assuming that the conditional mean functions are continuously differentiable in θt the adding up
restriction implies
∇θ ΦG (xit , θt ) = −∇θ Φ1(xit , θt ) − ∇θ Φ 2 (xit , θt ) − ⋅⋅⋅ − ∇θ ΦG −1(xit , θt ).
t
t
t
t

37

ւ

≺⊙ ⊙ ց
Assuming that we have a random draw i the multinomial quasi-likelihood equation for

each observation is:

ℓ it (θt ) = yit1 log[Φ1(xit , θt )] + yit 2 log[Φ 2 (xit , θt )] + ⋅⋅⋅ + yitG log[ΦG (xit , θt )]. (2.4.3)

14

The score function is

sit (θt ) =

yit1
yit 2
∇θ Φ1(xit , θt )′ +
∇θ Φ 2 (xit , θt )′ + ⋅⋅⋅
t
Φ1(xit , θt )
Φ 2 (xit , θt ) t

(2.4.4)

yitG
+
∇θ ΦG (xit , θt )′
ΦG (xit , θt ) t
Assuming the correct specification of the conditional mean
E (sit (θt ) | xi ) =
+

E ( yit1 | xi )
E ( yit 2 | xi )
∇θ Φ1(xit , θt )′ +
∇θ Φ 2 (xit , θt )′ + ⋅⋅⋅
Φ1(xit , θt ) t
Φ 2 (xit , θt ) t

E ( yitG | xi )
∇θ ΦG (xit , θt )′
ΦG (xit , θt ) t

= ∇θ Φ1(xit , θt )′ + ∇θ Φ 2 (xit , θt )′ + ⋅⋅⋅ + ∇θ ΦG (xit , θt )′
t
t
t
= 0.
Let Ht (xi , y i , θt ) be the Hessian, it can be shown that

− E[Ht (xit , yit , θo,t ) | xi ] =
+

∇θ Φ1(xit , θo,t )′∇θ Φ1(xit , θo,t ) ∇θ Φ 2 (xit , θo,t )′∇θ Φ 2 (xit , θo,t )
t
t
t
+ t
+ ⋅⋅⋅
Φ1(xit , θo,t )
Φ 2 (xit , θo,t )

∇θ ΦG (xit , θo,t )′∇θ ΦG (xit , θo,t )
t
t
ΦG (xit , θo,t )

and is consistently estimated by

14

We point out that using the Normal cumulative distribution function does not explicitly
restrict the predicted values of the “omitted” equation from falling outside of [0,1]. Using the
logarithm function in the quasi log-likelihood implicitly imposes this restriction. In our
application, we do not see this as a cause for concern as all predicted values of the “omitted”
equation fall within [0,1], though this may be a concern where there is a greater frequency of
zeros and ones in the response variable.
38

ւ

≺⊙ ⊙ ց

N
ˆ = N −1 ∇ Φ(x , θ )′W (x , θ )∇ Φ(x , θ )
At
∑ θt it ˆ t t it ˆ t θt it ˆ t
i =1
where

ˆ
 ∇θ Φ1(xit , θt ) 
t


ˆ
 ∇ Φ (x , θ ) 
θt 2 it t 
ˆ
∇θ Φ(xit , θt ) = 
t


⋮


 ∇ Φ (x , θ ) 
ˆ 
 θ G it t
 t

and

1

 Φ (x , θ )
ˆ
 1 it t


0
ˆ )=
Wt (xit , θt 

⋮



0







1

⋱
⋮
ˆ
.
Φ 2 (xit , θt )


⋱
⋱
0


1

…
0
ˆ 
ΦG (xit , θt ) 
0

⋯

0

Define

N
ˆ
ˆ
ˆ
Bt = N −1 ∑ sit (θt )sit (θt )′.
i =1
Then an estimator of the variance matrix that only relies on the correct specification of the
conditional means is given by

−1
N
 N

ˆ− ˆ ˆ−
ˆ
ˆ
ˆ
ˆ
ˆ
At 1Bt At 1 / N =  ∑ ∇θ Φ(xit , θt )′Wt (xit , θt )∇θ Φ(xit , θt )   ∑ sit (θt )sit (θt )′ 
t
t

 

 i =1
  i =1

N

ˆ
ˆ
ˆ
 ∑ ∇θ Φ(xit , θt )′Wt (xit , θt )∇θ Φ(xit , θt ) 
⋅
t
t


 i =1

39

−1
.

(2.4.5)

ւ

≺⊙ ⊙ ց
In the above presentation we laid out the basic approach to estimating the multinomial

quasi-likelihood for each time period in a balanced panel. With a few minor changes we can
rewrite the above equations to estimate a pooled version of the multinomial quasi-likelihood
where we do not allow the estimated parameter vector to vary by time period. We can write the
partial multinomial quasi-likelihood function as

ℓ i (θ) =

T

∑ yit1 log[Φ1(xit , θ)] + yit 2 log[Φ 2 (xit , θ)] + ⋅⋅⋅ + yitG log[ΦG (xit , θ)].

t =1

Then the score function is
yit1
yit 2


 Φ (x , θ ) ∇θ Φ1(xit , θ )′ + Φ (x , θ ) ∇θΦ 2 (xit , θ )′ + ⋅⋅⋅ 
T
T
2 it
 1 it

si (θ) = ∑ sit (θ) = ∑ 

yitG
t =1
t =1

⋅⋅⋅ +
∇θΦG (xit , θ )′


ΦG (xit , θ )


and the negative of the expected Hessian is

 ∇θ Φ1(xit , θo )′∇θ Φ1(xit , θo ) ∇θΦ 2 (xit , θo )′∇θ Φ 2 (xit , θo )

+
+ ⋅⋅⋅ 
T 
Φ1(xit , θo )
Φ 2 (xit , θo )


− E[ H ( xi , y i , θ o ) | xi ] = ∑ 
.
∇θ ΦG (xit , θo )′∇θ ΦG (xit , θo )
t =1

⋅⋅⋅ +


ΦG (xit , θo )



The consistent estimator of the Hessian is

N T
ˆ = NT −1
ˆ
ˆ
ˆ
A
∑ ∑ ∇θΦ(xit , θ)′W(xit , θ)∇θΦ(xit , θ)
i =1t =1
where

40

ւ

≺⊙ ⊙ ց
ˆ
 ∇θ Φ1(xit , θ) 


ˆ
 ∇ Φ ( x , θ) 
θ 2 it
ˆ

∇θΦ(xit , θ) = 


⋮


 ∇ Φ ( x , θ) 
ˆ
 θ G it 

and

1

 Φ (x , θ )
ˆ
 1 it


0
ˆ) = 
W(xit , θ 

⋮



0







1

⋱
⋮
ˆ)
.
Φ 2 (xit , θ


⋱
⋱
0


1

…
0
ˆ 
ΦG (xit , θ ) 
0

⋯

0

Define

N T
ˆ
ˆ
ˆ
B = NT −1 ∑ ∑ sit (θ)sit (θ)′.
i =1t =1
Then a consistent estimator of the variance matrix that only relies on the correct specification of
the conditional means is

−1 N T
N T



ˆ ˆˆ
ˆ
ˆ
ˆ
ˆ
ˆ
A −1BA −1 / N =  ∑ ∑ ∇θΦ(xit , θ )′W(xit , θ )∇θΦ(xit , θ )  ⋅  ∑ ∑ sit (θ )sit (θ )′  ⋅




 i =1t =1

 i =1t =1

−1
N T

ˆ
ˆ
ˆ
⋅  ∑ ∑ ∇θΦ(xit , θ)′W(xit , θ)∇θΦ(xit , θ)  .


 i =1t =1

2.5 Classical Minimum Distance Estimation
Classical Minimum Distance estimation (CMD) is similar to Generalized Method of
Moments and involves minimizing the Euclidean distance between a set of “reduced-form”

41

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≺⊙ ⊙ ց

parameters and their structural counterparts. The basic exposition is as follows and can be found
in Wooldridge (2002, Section 14.6).
Suppose that we have a P × 1 vector of structural parameters θo related to πo a S ×1
(where S > P ) vector of reduced-form parameters such that:
πo = h(θo )

(2.5.1)

where h is a known, continuously differentiable function that maps the structural parameters to
the reduced-form parameters. To perform CMD estimation we first obtain estimates of the

ˆ
ˆ
reduced-form parameters, say π , and then find an estimator θ that minimizes the weighted
ˆ
ˆ
Euclidean distance between π and h(θ) . Analogous to GMM, we can use any weighting matrix

so long as it is positive semi-definite, such as an S × S identity matrix, but the optimal weighting
matrix is the one that makes the CMD estimator the minimum chi-square estimator.
The CMD estimator solves the following problem:
ˆ −1 ˆ
ˆ
min {π − h(θ)}′Ξ {π − h(θ)}
θ∈Θ

(2.5.2)

ˆ
ˆ
where plim N →∞ Ξ = Ξo . Assuming that π is a consistent estimator of πo and that
a
ˆ
ˆ
N ( π − πo ) ~ Normal(0, Ξo ) then Ξ−1 is the inverse of any consistent estimate of the
asymptotic variance of

ˆ
ˆ
N ( π − πo ) . The solution to the optimization problem in (2.5.2) is θ

ˆ
ˆ
which minimizes the weighted Euclidean distance between π and h(θ) it can be shown that

a
ˆ − θ ) ∼ Normal[0, H′ Ξ−1H ] where H = H (θ ) and H (θ) ≡ ∇ h(θ) the S × P
N (θ o
o o o
o
o
θ

ˆ
Jacobian of h(θ) . The appropriate estimator of Avar(θ)
ˆ
ˆ
ˆ
Avar(θ) ≡ ( H ′Ξ−1H ) −1 / N = ( H ′[Avar( π )]−1 H ) −1

42

(2.5.3)

ւ

≺⊙ ⊙ ց
If the mapping function h is linear then we can state the general form of the CMD

estimator as follows (Wooldridge, 2002, Problem 14.7):
ˆ
ˆ
ˆ ˆ
θ = ( H′Ξ−1H ) −1 H′Ξ−1π.

(2.5.4)

This implies that the optimal CMD estimator of the structural parameter vector is a weighted
average of the reduced form parameter vector where the weighting matrix is the estimated
asymptotic variance matrix of the reduced-form parameters.
The mapping function that we use in the CMD estimation step is indeed linear and
therefore we can use (2.5.4) to construct the estimates of the structural parameter vector. To
obtain our reduced-form parameter vector we maximize the multinomial quasi-likelihood for
each time period within each balanced panel subset. Using these estimates we can then construct
the weighting matrix that we will use in the CMD estimation. The weighting matrix will be a
block matrix where the estimated variance matrices from each time period are on the diagonal
and the off-diagonal block matrices contain estimates of the covariance within and across
equations over time.
Given the results in section 3 and the specifications of the conditional mean functions in
section 2 we define the diagonal elements of the weighting matrix as

−1
N
 N

−1B A −1 / N =  ∇ Φ(x , θ )′W (x , θ )∇ Φ(x , θ )   s (θ )s (θ )′ 
ˆ
At ˆ t ˆ t
∑ θt it ˆ t t it ˆ t θt it ˆ t   ∑ it ˆ t it ˆ t 

 i =1
  i =1

N

ˆ
ˆ
ˆ
⋅  ∑ ∇θ Φ(xit , θt )′Wt (xit , θt )∇θ Φ(xit , θt ) 
t
t


 i =1


−1
.

We construct the off-diagonal elements of the weighting matrix in a similar manner

43

(2.5.5)

ւ

≺⊙ ⊙ ց

−1
N
 N

ˆ− ˆ ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
At 1Bts A −1 / N =  ∑ ∇θ Φ(xit , θt )′Wt (xit , θt )∇θ Φ(xit , θt )   ∑ sit (θt )sis (θ s )′ 
s
t
t

 

 i =1
  i =1

−1
N

ˆ
ˆ
ˆ
⋅  ∑ ∇θ Φ(xis , θ s )′Ws (xis , θ s )∇θ Φ(xis , θ s ) 
s
s


 i =1


(2.5.6)

for s ≠ t.
The weighting matrix that we use in CMD estimation is

 A −1B A −1 A −1B A −1
ˆ ˆ ˆ
ˆ ˆ ˆ
1 12 2
 1 11 1
 −1
ˆ ˆ ˆ
ˆ ˆ ˆ
 A B A −1 A −1B 22 A −1
2
2
ˆ =  2 21 1
Ω

⋮
⋮

 ˆ −1 ˆ ˆ −1 ˆ −1 ˆ
ˆ−
 A BT 1A
AT BT 2 A 2 1
 T
1

ˆ− ˆ ˆ−
A1 1B1T AT 1 


ˆ ˆ
ˆ−
⋯ A −1B 2T AT 1 
2
.

⋱
⋮

−1B A −1 
ˆ

⋯ AT ˆ TT ˆ T 
⋯

(2.5.7)

We calculate this weighting matrix for each balanced panel subset in our data. The mapping
function that we use is linear, it maps the T reduced-form coefficient vectors to the single
structural parameter vector for each balanced panel subset. Therefore, we can state that the CMD
ɶ
estimator β takes the following form

ɶ
ˆ
ˆ
β = ( H′Ω −1H ) −1 H ′Ω −1β

(2.5.8)

ˆ
ˆ
where Ω is defined above, β represents the T (G − 1) K × 1 ( K covariates, G − 1 equations, T

time periods in each balanced panel subset) vector of parameter estimates from the multinomial
quasi-likelihood maximization and H is a T (G − 1) K × (G − 1) K matrix that maps the reducedform parameter estimates to the structural parameter estimates:

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 It =1
 (G −1) K

 It = 2
 (G −1) K
H=

⋮

 t =T
 I (G −1) K

where I













t= j
(G −1) K is a (G − 1) K × (G − 1) K identity matrix for each time period j .

After deriving the CMD estimates of the structural parameters in (2.5.8), we can calculate
ɶ
the average partial effects of our covariates as in Section 2 using β . Therefore, our proposed

procedure is
Procedure 1:
1. For each time period obtain the multinomial quasi-likelihood estimates of the
coefficients, the gradient vector, and the robust variance matrix.
2. Create the weighting matrix as in (2.5.7), and the matrix H
3. Obtain the MD coefficient estimates using (2.5.8) and the asymptotic variance matrix
of these estimated coefficients as in (2.5.3).
The above presentation necessarily assumes that we have a balanced panel; this is due to
the use of the Chamberlain-Mundlak device employed to remove the unobserved effect. If we
have an unbalanced panel then, at a minimum, our assumption of a common variance across
households, conditional on the number of time periods, is incorrect. Households in the panel for
fewer time periods will necessarily have a larger estimated variance for their time-averaged
observables. One solution to this problem is to drop observations so that we have a balanced
panel. While this works technically, it has the potential to cause problems that are even more

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worrisome. Fortunately, there is a very simple way to allow us to use this procedure in the
context of an unbalanced panel.
Since we have independence within the cross-section, i.e., households are randomly
sampled and independent within each time period, we can simply subset our data based on the
number of times that we observe households and then average the estimated average partial
effects across these subsets. This entails creating balanced panels that contain households that are
observed for the same number of time periods. We estimate the multinomial quasi-likelihood on
each of these balanced panels by time period, followed by the minimum distance estimation to
account for serial correlation. We then calculate the average partial effects for each balanced
panel and average these across all the balanced panels.
We adapt Procedure 1 in the following way to allow for estimation on an unbalanced
panel:
Procedure 2:
1. Subset the panel into balanced panels which contain all households that have exactly j
observations, where j = 2,..., T .
2. For each time period in each balanced panel subset maximize the multinomial quasilikelihood obtaining the estimates of the coefficients, the gradient vector, and the robust
variance matrix.
3. Apply CMD estimation as detailed above to each subset of the data.
4. Calculate the average partial effects and their variances for each balanced panel subset
and then average these across all balanced panel subsets.
2.6 An Alternative Specification
The procedure presented above essentially takes single equation fractional probits and

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combines them into a system of equations. While this is the logical first step in estimating
multiple fractional response variables, it assumes that there are G sources of heterogeneity. This
is a very general assumption and acts as though we have a different distribution for the
unobserved effect for each equation. We know that heterogeneity is at the household level, and
we want to control for these time-invariant unobserved characteristics. Suppose that what we
cannot measure is household risk aversion and that risk aversion is time-invariant but correlated
with our covariates. If we do not control for risk aversion then our estimated coefficients will be
inconsistent. Although risk aversion is constant over time and across assets, we anticipate that
the expected value of the impact of risk aversion will vary depending on the asset under
consideration. A very risk-averse household will allocate a smaller proportion of their wealth to
the risky asset and a larger proportion to less risky assets. Therefore, the expected value of the
unobserved effect should vary by equation. Our model allows for this in a very general way. One
concern is that the model that we propose is too general and leads to estimation of too many
parameters.
A potential solution is to specify a slightly different model where instead of estimating a
different parameter for each time average and intercept by equation we estimate a single
parameter vector for the time averages and then estimate parameters to allow the effect of the
time-averaged covariates to vary by equation. We can adapt the above model as follows:

E ( yitg | xit , ci ) = Φ(xit β g + δ g ci ).
Here we introduce an equation specific parameter δ g that allows the impact of the unobserved
effect to vary across equations. In addition, we make the assumption that there is a single source
of heterogeneity.

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ci = Ψ + ξxi + ai
2
ai | xi ~ N (0, σ a )

Again, we assume strict exogeneity conditional on the unobserved effect:

E ( yitg | xi , ci ) = E ( yitg | xit , ci )
Plugging ci into E ( yitg | xit , ci ) gives us the mean conditional on observed and unobserved
factors:
E ( yitg | xit , ai ) = Φ (δ g Ψ + xit β g + δ g ξxi + δ g ai )
2 2
where δ g ai | xi ~ Normal (0, δ g σ a )
Therefore the conditional mean of yitg given xi is

δ Ψ + x β +δ ξx
g
it g
g i
E ( yitg | xi ) = Φ 
2 2

(1 + δ gσ a )1/ 2







This is nearly identical to equation (2.3.6) above, except that we have a different scale factor.
Regardless, we will still be able to derive the consistent estimates of the average partial effects.
We can estimate the above conditional means using the multinomial quasi-likelihood.
Since we cannot identify a parameter for each equation, we would need to normalize one say,

δG to one. We do not estimate this specification here, but we point out that in other applications
this can be considered since it is more parsimonious and may be more efficient.
2.7 Results
We use data from the Health and Retirement Study (HRS). The HRS began in 1992 and
was nationally representative of all non-institutionalized individuals aged 51-61 in that year. In
1998, the HRS combined with the Asset and Health Dynamics of the Oldest Old (AHEAD) and
added several new cohorts to be nationally representative of the population of non-

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institutionalized individuals aged 51 and older. Both initial respondents and their spouses are
interviewed and followed in subsequent waves. We use data for all cohorts from Waves 1-7
(1992-2004) of the HRS. In addition to basic demographic variables, the HRS collects detailed
information on wealth holdings, allocation of wealth, income and its sources, health, and
measures of the probability that future events occur. We use subjective survival probability, age,
word recall, household income, and non-financial wealth as regressors in our model.
Our measure of subjective survival probabilities comes from responses to the following
question:
On a scale from 0 to 100, where 0 is no chance and 100 is absolutely certain, what are
the chances that you will live to age 75 or older?
Our measure of income is calculated as the sum of all non-capital income received by the
respondent and spouse during the year. To calculate wealth variables we use responses to
questions about the value of holdings within various financial assets. The assets that are
measured are stocks and/or mutual funds, bonds (corporate and government), Certificates of
Deposit (CDs) and checking, saving and/or money market accounts. We also have information
about other assets owned by the household including housing, vehicles, other real estate,
IRA/Keogh accounts, and trusts. We distinguish between two measures of wealth holdings. Nonfinancial wealth consists of the value of housing, real estate, IRA/Keogh accounts, vehicles, and
trust holdings, less any associated debt. Financial Wealth is the value of all holdings in
stocks/mutual funds, bonds, Certificates of Deposit and savings, checking and money market
accounts. HRS respondents are given a list of nouns and then asked to repeat this list
immediately and then again at the end of the cognition section. We use the proportion of words
recalled at the end of the section as a proxy for cognition.

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We construct our data set by household. We combine spouses in each wave so that our

panel consists of household observations by year. To select individuals for our sample we drop
observations that satisfy the following criteria:
(i)

older than 65,

(ii)

missing values for subjective survival probability,

(iii)

is a proxy interview or

(iv)

is deceased.

We apply these criteria at the individual level so households that do not have both spouses are
dropped. The decision to allocate wealth across assets is conditional on holding positive financial
wealth; thus, we drop observations where financial wealth is zero. In addition, we drop
households that are only observed for one time period. Imposing the above criteria for married
households leaves us with 3,872 households (14,514 total observations).
In Table 13, we display means and medians of the relevant variables in our analysis. We
can see from Table 13 that the average allocation of wealth generally falls into two assets: stocks
and checking. There is some investment in CDs but very little in Bonds. There is some concern
regarding the time variation in the HRS in particular with subjective survival probability
measures.

15

Table 14 presents the proportion of the variation that is between households in our

dataset. We can see that the bulk of the variation in our variables is between households, but
there is some variation over time that we may be able to exploit.
As a first step, we estimate our allocation equations by linear fixed effects. In general
linear fixed effects coefficients should provide fairly good estimates of the average partial effects
and provide a reasonable baseline to compare other estimation techniques. The results of the
15

Smith et al. (2001) and Elder (2010) point to the lack of appropriate variation in the subjective
survival probabilities collected in the HRS.
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linear fixed effects estimation are presented in Table 15.
We now turn to different methods of estimating the average partial effects of our
covariates using a nonlinear conditional mean specification. As a first pass, we know that we can
estimate each equation using a single equation nonlinear method. We follow Papke and
Wooldridge (2008) and use the procedure that they laid out for the estimation of a pooled
fractional probit. Since we have an unbalanced panel we cannot simply estimate the single
equation fractional probit on the entire panel. Instead we subset the unbalanced panel as
described above, maximize the Bernoulli quasi-likelihood for each balanced panel subset,
calculate the average partial effects and then average these across the balanced panel subsets.
The results from this estimation approach are presented in Table 16. With the exception of the
estimated average partial effects on Husband and Wife Word Recall, the estimated average
partial effects are similar to those estimated by linear fixed effects but we can see that estimating
using balanced panel subsets significantly increased our estimated standard errors so that none of
our covariates are statistically significant. The difference in the estimated average partial effects
for both Husband and Wife Word Recall may suggest that there is some nonlinearity in the
impact of this variable on the allocation of wealth
Next, we maximize the multinomial quasi-likelihood on each balanced panel subset,
calculate the average partial effects for each balanced panel subset, and then average these over
the balanced panel subsets. Table 17 contains the estimated average partial effects and the
standard errors from this approach. Overall, we can see that the estimated average partial effects
for all equations are very similar to those estimated by the single equation fractional probit
technique. Looking at the standard errors, we can see that they are nearly identical to the
fractional probit standard errors. It appears that imposing the adding up restriction does not affect

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the estimates of either the average partial effects or the standard errors.
Table 18 presents the estimated average partial effects and standard errors from
Procedure 2 outlined above. Our main points of comparison here are Tables 15, 16 and 17, the
average partial effects estimated by linear fixed effects, maximizing the Bernoulli quasilikelihood and maximizing the multinomial quasi-likelihood on the balanced panel subsets.
Looking first at the estimated standard errors, we can see that they are smaller than those in both
Tables 16 and 17 but larger than those estimated by linear fixed effects. This suggests that
accounting for serial correlation within and between equations has led to more precise estimates
relative to the estimation that uses balanced panel subsets. In particular, the estimated standard
errors for the allocation of wealth to Bonds and CDs are almost half the size of the standard
errors estimated by maximizing the pooled multinomial quasi-likelihood.
Turning to the estimated average partial effects we can see that there are some differences
between the results in Table 18 compared to Tables 15, 16 and 17. Tables 19, 20 and 21 show the
p-values of testing the difference in estimated average partial effects across the different
estimation approaches. Tables 19 and 20 compare the estimated average partial effects of
Procedure 2 to the estimated partial effects from maximizing the Bernoulli quasi-likelihood for
each asset equation on balanced panel subsets and from maximizing the multinomial quasilikelihood on balanced panel subsets. Given the large standard errors from estimation on
balanced panel subsets with either the single equation or multiple equation approaches we only
see a statistically significant difference (at the 5% level) for the estimated average partial effect
of Log(Income) on the allocation of wealth to Bonds. Looking at Table 21, which compares
estimated average partial effects between Procedure 2 and linear fixed effects we can see that the
estimated average partial effect of Log(Income) on bonds is also statistically significantly

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different than the linear fixed effect estimate. It is unclear whether we should expect a negative
correlation between income and bond holdings, since it is not certain what increasing income
levels mean for the riskiness of that income. Theoretically a more risky income stream would
lead to an increased incentive to allocate wealth to less risky investments, but this will depend
upon the correlation between the riskiness of the return from each asset and income. Since the
estimated average partial effects from linear fixed effects, maximizing the Bernoulli quasilikelihood for each equation and maximizing the multinomial quasi-likelihood are relatively
close to zero with large standard errors it seems plausible that the effect is negative but not
controlling for serial correlation leads us to estimate a very small impact from income.
Simply looking at the magnitude of the estimated average partial effects it appears that
the largest differences occur within the stock and checking equations. The estimated average
partial effects that are the most different are for Husband and Wife Age, Husband and Wife
Pr(Live to 75), and Husband and Wife Word Recall. While the impacts of these covariates have
changed for both Husband and Wife there are distinct differences between the two members of
the household. The average partial effect of a ten percentage point increase in survival
probability on the allocation of wealth to stocks increased nearly tenfold for the Husband, but
decreased twofold for the Wife. We can also see that the estimated average partial effect on
stocks from Husband Word Recall is about 0.003 percentage points greater in Table 18
compared to the linear fixed effects estimate in Table 15, but this is not the case when comparing
Wife Word Recall. Nevertheless, though the magnitudes of the estimated average partial effects
are larger from Procedure 2 we cannot reject the null hypothesis that they are statistically
equivalent to the estimates from maximizing the single equation Bernoulli quasi-likelihoods or
the multinomial quasi-likelihood.

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Turning to Table 21, which compares the estimated average partial effects between linear

fixed effects and Procedure 2, the covariates with statistically significant different estimates (at
the 5% level) are Husband Age for stocks, Wife Pr(Live to 75) and Log(Income) for bonds and
Husband Age and Wife Age for checking. The differences in the estimated average partial
effects for Husband and Wife Age are not surprising given the fact that Procedure 2 is using the
between household variation from time averages within years of the balanced panel subsets to
estimate the impact of age. Perhaps more surprising is the fact that we now estimate a negative
average partial effect for Wife Pr(Live to 75) for bonds. Though this estimated average partial
effect from Procedure 2 is not statistically different from the linear fixed effect estimate it
suggests that the Wife may prefer less risky assets (and potentially more liquid in the case of
CDs and Checking) when facing an increased lifespan. While this is purely conjecture it may
suggest that there is an avenue for research into the different preferences of household members
against both the expected risk and return from assets and the liquidity of these assets.
There are several reasons that we would expect to see differences in the estimated
average partial effects from our Procedure 2. First, the two-step procedure maximizes the
multinomial quasi-likelihood for each time period within each balanced panel subset; this
technique exploits the variation in the differences from the time averages between households.
Second, Procedure 2 is weights the coefficients based on the serial correlation in the allocation of
wealth to each asset, the serial correlation in the covariates and the correlation of the covariates
across equations within each time period and also across time. Our procedure combines all of
these correlations into weights that are then applied to the coefficients from each time period
within the balanced panel subset. What is promising is that while we do see some statistically
different estimated average partial effects from Procedure 2 compared to maximizing single

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equation Bernoulli quasi-likelihoods, maximizing multiple equation multinomial quasilikelihoods and single equation linear fixed effects the actual impact from these covariates all
point to relatively small effects on the allocation of wealth across these assets.
Overall it seems that our estimated impacts from the procedure we laid out here are
reasonably close to estimates from models that we know will generate consistent but inefficient
results. In addition, despite slightly different estimated average partial effects our proposed
procedure still estimates small impacts of these covariates on the allocation decision which is
consistent with the other approaches that we used.
One of the advantages of the method that we proposed here is that it allows the effect of
the covariates to affect our response variables in a non-linear way. Even though we have
estimated our multinomial QMLE on time periods within balanced panel subsets we can still
construct the Average Structural Function (ASF) for the entire sample:

N T
−1
ˆ
ˆ
ˆ
ASF (xt ) = N ∑ ∑ 1[Ti = r ]Φ Ψ r + xt β gr + xi ξ gr for r = 2,..., T .
i =1 r = 2

(

)

We plot ASF (xt ) for Husband Word Recall for the allocation of wealth to Stocks, Bonds, CDs,
and Checking in Figures 1 – 4 and for Wife Word Recall in Figures 5 – 8. As a point of
reference, we also plot the corresponding ASF from the linear fixed effects estimation, which are
straight lines because the coefficient estimates are constant over the entire range of the covariate.
From figure 1 we can see that there is some nonlinearity in the impact of Husband Word Recall
on stocks. It seems that the impact of remembering additional words increases slightly as more
words are recalled. Figure 2 displays the average structural function for bonds. There appears to
be a steep decline in the predicted proportion of wealth allocated to bonds as the number of
words recalled increases, but this impact dampens as more and more words are recalled. For the
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allocation of wealth to CDs we see a relatively stable small impact of Husband Word Recall that
gets slightly larger as the number of words recalled increases. In figure 4 we can see that the
impact of Husband Word Recall on the proportion of wealth allocated to checking is initially
small, but increases as the number of words recalled increases.
Turning to Wife Word Recall we can see in Figure 5 that the average structural function
is fairly linear across the range of word recall. Figure 6 shows that there is some significant
nonlinearity in the impact of word recall on the allocation of wealth to bonds. It appears that
impact of recalling additional words increases once the wife has recalled greater than 50% of the
words given in the survey. Figure 7 shows that there is a slight dampening effect of recalling
additional words on the allocation of wealth to CDs, but that it sets in near the end of the range
of possible values. As we can see from Figure 8, there appears to be “diminishing returns” to the
impact of additional words recalled by the wife and it seems that this begins to set in slightly past
60% of the words recalled.
2.8 Conclusion
In this paper, we have presented a two-step estimation procedure for cases where we have
multiple fractional responses that must satisfy an adding up restriction. We leveraged the
properties of the multinomial quasi-likelihood and the normal cumulative distribution function to
account for the adding-up restriction and the restricted range of our multiple response variables.
In our application, we were able to exploit the random sampling of cross-sectional units to derive
consistent and more slightly more efficient estimates of the average partial effects of our
covariates on the allocation of financial wealth to various assets in an unbalanced panel while
still controlling for a time-invariant unobserved effect. Since the multinomial quasi-likelihood
function is a member of the linear exponential family we are able to achieve consistent results

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under the assumption that we have correctly specified the conditional mean. Our second step
allows us to increase the efficiency of our estimates from using nonlinear methods on balanced
panel subsets of our data while still estimating average partial effects that are (for the most part)
not statistically different from these other approaches.
We also laid out an alternative estimation approach that makes a further assumption on
the impact of our time averages. This refinement is a clear next step along this line of research.
There is also much to be gained in applying the two-step procedure to other problems where we
have multivariate fractional responses that must satisfy an adding-up restriction. In particular, it
would be useful to see how well this estimation technique performs in balanced panels and also
in panels where there is more within variation in the covariates. It would also be fruitful to see
the impact that different assumptions about the serial correlation have on the estimates from the
CMD step. One assumption that is obvious and that it used in the Generalized Estimating
Equation literature is to assume that the correlation within units over time is exchangeable.
Papke and Wooldridge (2008) find that there is almost no difference between their estimates
when assuming an exchangeable correlation structure compared to a completely unstructured
correlation.

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CHAPTER 3
ESTIMATION OF A MULTIVARIATE FRACTIONAL RESPONSE MODEL WITH
UNBALANCED PANEL DATA

3.1 Introduction
Estimation using panel datasets is more prevalent with the increase of available data on
repeated observations on micro-level units. With panel data we are careful to remove unobserved
effects by differencing the data or using fixed-effects. With a linear model this is straightforward
and allows for the unobserved effect to be correlated with observables and having a balanced
panel is not necessary. Once we begin to look at estimation of nonlinear models on panel data
this issue becomes a little more complicated.
Treating the unobserved effect as a parameter to estimate leads to inconsistent estimates,
referred to as the “incidental parameters problem”. In general it is assumed that the unobserved
effect is part of the conditional mean that we specify for our dependent variable. As this
unobserved effect is folded into a nonlinear function we are not able to remove it via differencing
or time-demeaning. We can avoid this problem either by ignoring it or using a random-effect
technique that assumes that the unobserved effect is not correlated with any observables. While
the random effects procedure provides a solution, the assumption that the unobserved effect is
uncorrelated with the independent variables can lead to inconsistent estimates if violated.
Another approach to controlling for the unobserved effect in panel data is through
Correlated Random Effects models. The approach makes parametric assumptions about the way
that the time-invariant effect is related to other independent variables in the conditional mean
model. The usual set up assumes that there is a linear relationship between the unobserved effect

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and the time-averages of the regressors. This Chamberlain-Mundlak device is relatively simple
to implement assuming that we have a balanced panel.
In the context of an unbalanced panel care needs to be taken when employing the
Chamberlain-Mundlak device to control for unobserved effects. The standard assumption is that
the unobserved effect ci is distributed Normal (ψ + ξ xi , σ c ) . This assumption is not accurate if
some cross-section units have fewer observations than others since the estimated variance of the
unobserved effect will be larger for units that are observed fewer times. The most obvious
approach to remedy this problem is to simply choose the largest balanced panel, essentially
dropping cross-section units that we do not observe for a set number of periods. This approach is
dangerous for two reasons. First, reducing the number of observations decreases the accuracy of
our estimates. Second, even if we assume that selection is uncorrelated with the idiosyncratic
errors there is a potential for selection to be correlated with the observables and unobserved
heterogeneity which will lead to inconsistent estimates of their effects on our dependent variable.
Wooldridge (2010) presents a modification of the Correlated Random Effects approach that
allows selection to be correlated with the covariates and the unobserved effect. This modification
allows us to apply the Correlated Random Effects approach to be applied to unbalanced panel
data.
Wooldridge (2010) presents an example using a standard probit (or fractional probit)
model and suggests that we can extend this modification to multivariate fractional responses.
Sivakumar and Bhat (2002) and Mullahy (2010) use the multinomial quasi-likelihood to estimate
models of multivariate fractional responses that have an adding up restriction and estimate their
models using cross-sectional data. Extending the models to balanced panel data is relatively
straightforward as we can simply estimate pooled models on repeated cross-section observations.

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Of course, we would want to make sure that we use clustered standard errors to make inference
robust. In a balanced panel we could use the Correlated Random Effects approach and control for
the unobserved heterogeneity by making parametric assumptions of the moments of its
distribution.
In this paper we combine the approach for multivariate fractional responses and the
modifications to the Correlated Random Effects approach to estimate models on multiple
fractional responses on cross-section units that are observed over time, but not for the same
number of time periods. We do this by combining the conditional mean specifications of each of
our fractional responses and estimate using the multinomial quasi-likelihood function which
explicitly imposes the adding up restriction of the multivariate fractional responses. We assume
that selection is uncorrelated with the idiosyncratic error but allow it to be correlated with the
unobserved effect and the covariates in our conditional mean.
The outline of the paper is as follows. Section 2 presents the single equation fractional
probit model discussed in Papke and Wooldridge (2008), section 3 extends the single equation
fractional probit to allow for selection following Wooldridge (2010), section 4 presents the
multinomial quasi-likelihood and details how we will estimate the multivariate fractional
response models on an unbalanced panel simultaneously, section 5 presents results from
estimation of this model comparing it to results from other estimation methods and section 6
concludes.
3.2. Single Equation Fractional Probit Models in a Balanced Panel
We begin by laying down the framework for estimating a model of portfolio allocation on
a balanced panel. We proceed in this manner since it makes obvious the modifications that we

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need to make for the unbalanced panel case and demonstrates the impact that these modifications
will have on the estimation of the average partial effects.
Assume that at each time period t , we observe household i allocate a proportion of their
financial wealth to each asset g . We assume that g = 1, 2,..., G exhausts all possible investment
options so that in each period the household invests all their financial wealth. This assumption
implies that

G

∑ g yitg = 1 for each i = 1, 2,..., N

and t = 1, 2,..., T . In addition to our response

vector, we observe a set of covariates xit and an unobserved effect cig . Our goal is to estimate a
model of the conditional mean of y given x to determine the impact of each covariate. By
construction, yitg is bounded between zero and one. Assuming that we use the same set of
covariates for each equation we can specify the conditional mean as:

E ( yitg | xit , cig ) = Φ(xit β g + cig ); i = 1,..., N ; t = 1,..., T ; and g = 1,..., G (3.2.1)
We point out that the above specification completely ignores the fact that these fractional
responses sum to one. It is simply acting as though we are treating each asset equation by itself.
Nevertheless, it is useful to study what we can derive from these single equation models since
our approach here essentially takes the specification in (3.2.1) for each equation and uses the
multinomial quasi-likelihood to impose the adding up constraint.
Our interest lies in estimating the average partial effects of the covariates, xit on the
proportion of financial wealth allocated to each asset. Following the results for a probit model
the direction of the partial effect is determined by the sign of β g . For a continuous variable xtk
(dropping the i subscript);
∂E ( ytg | xt , c g )
= β gk φ (xt β g + cg )
∂xtk
61

(3.2.2)

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≺⊙ ⊙ ց

To obtain the average partial effect we can average this over the distribution of cg and and xt .
To consistently estimate the coefficients in this single equation framework we require
that the covariates are not correlated with the error term. We assume that xit is strictly
exogenous conditional on the unobserved effect,

E ( yitg | xi , cig ) = E ( yitg | xit , cig ).

(3.2.3)

Following Chamberlain (1980) and Mundlak (1978) we assume that
2
cig = ψ g + xi ξ g + aig ; where aig | xi ~ Normal (0, σ a ).

(3.2.4)

Using (3.2.4) we can rewrite (3.2.1) as:

E ( yitg | xi , aig ) = Φ(ψ g + xit β g + xi ξ g + aig ).

(3.2.5)

So the mean of yitg conditional on xi is:

ψ + x β + x ξ
g
it g
i g
E ( yitg | xi ) = E (Φ (ψ g + xit β g + xi ξ g + aig | xi ) = Φ 
 (1 + σ 2 )−1/ 2
ag



 (3.2.6)



or

(

)

E ( yitg | xi ) = E (Φ (ψ g + xit β g + xi ξ g + aig | xi )) = Φ ψ ga + xit β ga + xi ξ ga (3.2.7)
where we use the a subscript to denote that our estimated coefficients are scaled by
2
(1 + σ ag ) −1/ 2 . Woodridge (2002, Section 15.8.2) shows that the mixing properties of the

normal distribution lead to equation (3.2.7).
For identification of the scaled coefficients in (3.2.7) we require that there is no perfect
collinearity between the elements of xit and that there is enough variation in xit over time.
With the above parametric specification of the unobserved effect we can now write the Average
Structural Function (ASF) following Blundell and Powell (2003):
62

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(

)

ASFg (xt ) = Ex  Φ ψ ga + xt β ga + xiξ ga 
i


(3.2.8)

A consistent estimator of ASFg ( xt ) is:

N
ˆ
ˆ
ˆ
ASF g (xt ) = N −1 ∑ Φ ψ ga + xt β ga + xi ξ ga .
i =1

(

)

(3.2.9)

ˆ
ˆ
ˆ
where ψ ga , β ga , and ξ ga are consistent estimates of the scaled coefficients in equation (3.2.7).

For a continuous variable, x j we can calculate the average partial effect by taking the derivative
of (3.2.9) and average over both i and t :

APE g ( x j ) =

ˆ
ˆ
ˆ
∂Φ(ψ ga + xit β ga + xi ξ ga )
∂x j

T N
ˆ
ˆ
ˆ
ˆ
= NT −1 ∑ ∑ β ga, jφ (ψ ga + xit β ga + xi ξ ga )
t =1i =1
(3.2.10)

For a binary x j we can calculate the average partial effect by calculating the difference in the
Average Structural Function evaluated at zero and one.
3.3 Single Equation Fractional Probit Models in an Unbalanced Panel
Once we are dealing with an unbalanced panel we need to make explicit the fact that each
cross section unit in our sample is not observed for the full span of time. Let Ti represent the
number of periods that we observe each cross sectional unit. We also introduce a selection
indicator to index whether an observation is used in estimation. Let sit be an indicator that takes
on a value of one if the observation is used in estimation and zero otherwise. To get consistent
estimates of the average partial effects we need to make explicit the assumption that selection is
not correlated with the idiosyncratic error term. Therefore, in addition to the usual strict
exogeneity assumption we assume that selection is conditionally ignorable. We write this
assumption as follows:
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E ( yitg | xi , cig , si ) = E ( yitg | xit , cig ) = Φ(xit β g + cig )

(3.3.1)

Now we need to specify a model for the distribution of the unobserved effect. In the typical case
this amounts to specifying that the expected value of the unobserved effect varies linearly with
the time averages of the observed covariates.
In an unbalanced panel, we need to specify that the mean and the variance of the
unobserved effect vary by Ti since intuitively the estimated moments will vary simply because
we are using a different number of observations to estimate the parameters of the distribution.
Following Wooldridge (2010), we define wi to represent functions of the covariates and
selection {( sit , sit xit : t = 1,..., T } . A straightforward extension of the standard ChamberlainMundlak device would be to allow E (cig | wi ) to vary by the number of time periods that we
16

observe our cross-section units :

E (cig | wi ) =

T

∑ ψ gr ⋅1[Ti = r ] + xiξ g

(3.3.2)

r =1

We allow the variance of cig to vary across the unbalanced panel through the following
specification:
 T −1

Var (cig | w i ) = exp  τ + ∑ 1[Ti = r ] ⋅ ω gr 


r =1



(3.3.3)

The specification in (3.3.3) allows the variance of the unobserved effect to vary with the number
of time periods that we observe a cross-section unit in our panel.

16

Wooldridge (2010) actually specifies that the conditional mean of the unobserved effect
includes the interaction between the indicator for number of time periods in sample and the time
averages of the covariates. We only include the indicator variables here due to non-convergence
of our likelihood functions in estimation.
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Given equations (3.3.2) and (3.3.3) and assuming cig conditional on wi is distributed

normal we can rewrite the conditional mean for a given asset as

E ( yitg | xit , wi , sit = 1) =


 x β + T ψ ⋅1[T = r ] + x ξ 
it g ∑ r = 2 gr
i
i g 
Φ

1/ 2 
 1 + exp τ + ∑T −11[T = r ] ⋅ ω 


gr 
r =1 i







{

(3.3.4)

}

We can reparameterize equation (3.3.4) so that the denominator is unity if we have a balanced
panel:
E ( yitg | xit , w i , sit = 1) =



 x β + T ψ ⋅1[T = r ] + x ξ 
it g ∑ r =1 gr
i
i g
Φ
1/ 2 

T

 exp  ∑

 r = 21[Ti = r ] ⋅ ω gr 





(3.3.5)

Given the specification in (3.3.5) we follow Blundell and Powell (2003) and construct the
Average Structural Function as:

 
  x β + T ψ ⋅1[T = r ] + x ξ
t g ∑ r =1 gr
i
i g
ASFg (xt ) = Ex Φ 
i 
1/ 2
  exp  ∑T 1[Ti = r ] ⋅ ω gr 
 r =2

 












(3.3.6)

A consistent estimate of this is


T ˆ
 ˆ
ˆ
−1 Φ  xt β g + ∑ r =1ψ gr ⋅1[Ti = r ] + xi ξ g
ASF g (xt ) = N ∑
1/ 2


i =1  exp  T 1[T = r ] ⋅ ω 
ˆ gr 
∑ r =2 i



N

65








(3.3.7)

ւ

≺⊙ ⊙ ց

ˆ
ˆ
ˆ
ˆ
where β g , ψ gr , ξ g and ω gr are consistent estimates of the coefficients in (3.3.5). For a

continuous variable xk we can calculate the average partial effect by taking the derivative of
(3.3.7):





ˆ
ˆ
N  xt β g + ∑T ψ gr ⋅1[Ti = r ] + xi ξ g  
ˆ
r =1
ˆ 

β gk  N −1 ∑ φ 
1/ 2  



i =1  exp  T 1[T = r ] ⋅ ω 
ˆ gr 

 ∑ r =2 i







(3.3.8)

where φ [ i] represents the standard normal probability distribution function. Once again we can
ˆ
see that β gk determines the direction of the partial effect.

3.4 The Multinomial Quasi-Likelihood: Multivariate Fractional Responses
We now describe the multinomial quasi-likelihood function and detail the how estimation
proceeds given the specifications for the conditional mean and variance. Since an observation
can only have a log-likelihood value if it is included in estimation the following exposition
necessarily assumes that the likelihood is only defined for observations where sit = 1 .
The multinomial quasi-likelihood function for a random draw i in a specific time period
t is given by:
ℓ it (θ) = yi1t log [ m(xit , w i , θ1) ] + yi 2t log [ m(xit , wi , θ2 )] + ⋯
⋯ + yiGt log [ m(xit , wi , θG ) ]

Where

66

(3.4.1)

ւ

≺⊙ ⊙ ց


 x β + T ψ ⋅1[T = r ] + x ξ
it g ∑ r =1 gr
i
i g
m(xit , wi , θ g ) = Φ 
1/ 2


 exp  ∑T 1[Ti = r ] ⋅ ω gr 

 r =2





 . 17




For identification of the parameters the multinomial quasi-likelihood requires that the following
constraint hold:
m(xit , w i , θG ) = 1 − m(xit , w i , θ1) − m(xit , w i , θ 2 ) − ⋅⋅⋅ − m(xit , w i , θG −1) (3.4.2)
Since we will be pooling observations the partial quasi-likelihood function is

ℓ i (θ) =

T

∑ yi1t log [ m(xit , wi , θ1)] + yi 2t log [ m(xit , wi , θ2 )] + ⋯ + yiGt log [ m(xit , wi , θG )]

t =1

(3.4.3)
The score function is then
∇ m(xit , w i , θ 2 )′


∇ m(xit , w i , θ1)′
+ yit 2 ⋅ θ
+ ⋯
 yit1 ⋅ θ
T
T 
m(xit , w i , θ1)
m(xit , w i , θ 2 )

si (θ) = ∑ sit (θ ) = ∑ 
 (3.4.4)
∇ m(xit , w i , θG )′


t =1
t =1
θ
⋯ + yitG ⋅

m(xit , w i , θG )


Define Hi (xi , y i , θ) as the Hessian then

17

We note that using the Normal cumulative distribution function does not explicitly restrict the
predicted value of the “omitted” equation to fall in the unit interval, though it is implicitly
imposed by the logarithm function. In our application, the predicted values of the omitted
equation remain in the unit interval. We also estimate an approximate multinomial logit form of
the system which does explicitly impose the adding-up restriction of the equations, the results are
nearly identical.
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 ∇ m(x , w , θ )′ ∇ m(x , w , θ )

θ
it i 1 θ
it i 1 + ⋯ 

m(xit , w i , θ1)




T  ∇ m(x , w , θ )′∇ m(x , w , θ )
it i 2 θ
it i 2 + ⋯ 
− E[Hi (xi , y i , θ | xi )] = ∑  θ

m(xit , w i , θ 2 )


t =1


⋯ + ∇θ m(xit , w i , θG )′∇θ m(xit , w i , θG ) 


m(xit , w i , θG )



(3.4.5)

this is consistently estimated by:

A = ( NT )−1

N T

ˆ
ˆ
ˆ
∑ ∑ µ(xit , wi , θ)′Wit (xit , wi , θ)µ(xit , wi , θ)

(3.4.6)

ˆ
 ∇θ m(xit , w i , θ1) 


ˆ
 ∇ m( x , w , θ ) 
θ
it i 2 
ˆ
µ(xit , w i , θ) = 


⋮


 ∇ m( x , w , θ ) 
ˆ
it i G 
 θ

(3.4.7)

i =1t =1
where

and

1

 m(x , w , θ )
ˆ
it i 1



0
ˆ) = 
Wit (xit , wi , θ 

⋮



0


ˆ
Define B =





1

⋯
0
ˆ )
 . (3.4.8)
m(xit , wi , θ2


⋮
⋱
⋮


1

0
⋯
ˆ 
m(xit , wi , θG ) 
0

N T

⋯

0

ˆ
ˆ
∑ ∑ sit (θ )sit (θ)′ , then we can construct the robust sandwich form of the

i =1t =1

asymptotic variance-covariance matrix as

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N T

ˆ ˆˆ
ˆ
ˆ
ˆ
A −1BA −1 / N =  ∑ ∑ µ(xit , wi , θ)′Wit (xit , wi , θ)µ(xit , wi , θ) i


 i =1t =1

N T

ˆ
ˆ
∑ ∑ sit (θ )sit (θ)′i

(3.4.9)

i =1t =1

−1
N T

ˆ
ˆ
ˆ
 ∑ ∑ µ (xit , wi , θ)′Wit (xit , wi , θ)µ(xit , wi , θ) 


 i =1t =1

3.5 Results
We use data from the Health and Retirement Study (HRS). The HRS began in 1992 and
was nationally representative of all non-institutionalized individuals aged 51-61 in that year. In
1998, the HRS combined with the Asset and Health Dynamics of the Oldest Old (AHEAD) and
added several new cohorts to be nationally representative of the population of noninstitutionalized individuals aged 51 and older. Both initial respondents and their spouses are
interviewed and followed in subsequent waves. We use data for all cohorts from Waves 1-7
(1992-2004) of the HRS. In addition to basic demographic variables, the HRS collects detailed
information on wealth holdings, allocation of wealth, income and its sources, health, and
measures of the probability that future events occur. We use subjective survival probability, age,
word recall, household income, and non-financial wealth as regressors in our model.
Our measure of subjective survival probabilities comes from responses to the following
question:
On a scale from 0 to 100, where 0 is no chance and 100 is absolutely certain, what are
the chances that you will live to age 75 or older?
Our measure of income is calculated as the sum of all non-capital income received by the
respondent and spouse during the year. To calculate wealth variables we use responses to

69

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questions about the value of holdings within various financial assets. The assets that are
measured are stocks and/or mutual funds, bonds (corporate and government), CDs, and checking,
saving and/or money market accounts. We also have information about other assets owned by
the household including housing, vehicles, other real estate, IRA/Keogh accounts, and trusts. We
distinguish between two measures of wealth holdings. Non-financial wealth consists of the value
of housing, real estate, IRA/Keogh accounts, vehicles, and trust holdings, less any associated
debt. Financial wealth is the value of all holdings in stocks/mutual funds, bonds, CDs, and
checking, savings, and money market accounts. HRS respondents are given a list of nouns and
then asked to repeat this list immediately and then again at the end of the cognition section. We
use the proportion of words recalled at the end of the section as a proxy for cognition.
We construct our data set by household. We combine spouses in each wave so that our
panel consists of household observations by year. To select individuals for our sample we drop
observations that satisfy the following criteria:
(i)

older than 65,

(ii)

missing values for subjective survival probability,

(iii)

is a proxy interview or

(iv)

is deceased.

We apply these criteria at the individual level so households that do not have both spouses are
dropped. The decision to allocate wealth across assets is conditional on holding positive financial
wealth; thus, we drop observations where financial wealth is zero. In addition, we drop
households that are only observed for one time period. Imposing the above criteria for married
households leaves us with 3,872 households (14,514 total observations).
In Table 22, we display means and medians of the relevant variables in our analysis. We

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can see from Table 22 that the average allocation of wealth generally falls into two assets: stocks
and checking. There is some investment in CDs but very little in Bonds. There is some concern
regarding the time variation in the HRS in particular with subjective survival probability
measures.

18

Table 23 presents the proportion of the variation that is between households in our

dataset. We can see that the bulk of the variation in our variables is between households, but
there is some variation over time that we may be able to exploit.
As a first step we estimate our allocation equations by linear fixed effects. In general
linear fixed effects coefficients should provide fairly good estimates of the average partial effects
therefore they should provide us with a reasonable baseline to compare other estimation
techniques. The results of the linear fixed effects estimation are presented in Table 24.
We now turn to different methods of estimating the average partial effects of our
covariates using nonlinear estimation. As a first pass we know that we can estimate each
equation using a single equation nonlinear method. We follow Papke and Wooldridge (2008) and
use the procedure that they laid out for the estimation of a pooled fractional probit. In this
approach we are ignoring the unbalanced nature of the panel and simply maximizing the
Bernoulli quasi-likelihood function for each asset equation over the full dataset. The results from
this approach are presented in Table 25. While we have controlled for an unobserved effect in
this estimation we have not accounted for the fact that we have an unbalanced panel, in fact we
have simply assumed that each household is observed the same number of times.
Table 26 presents the results of maximizing the Bernoulli quasi-likelihood for each
balanced panel subset of the full, unbalanced panel and then averaging the estimated average
partial effects (and their variances) across the balanced panel subsets. We can see that the
18

Smith et al. (2001) and Elder (2010) point to the lack of appropriate variation in the subjective
survival probabilities collected in the HRS.
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estimated average partial effects are nearly identical to the estimates provided by completely
ignoring the unbalanced nature of the data. The difference in standard errors between these two
approaches is significant though. The increase in the estimated standard errors arises from the
fact that there are considerably fewer observations within each balanced panel subset and this
naturally leads to larger variances. While this approach assumes that there is a difference in the
estimated moments of the unobserved effect depending on the number of times that a household
is viewed in the panel it does not provide a way to test whether the number of times that we
observe a household has a statistically significant impact on the estimates.
Our next step is to estimate each asset equation on the full, unbalanced panel correcting
for the unbalanced nature of the panel by allowing the mean and the variance of the unobserved
effect to vary with the number of time periods that we observe a household. We maximize the
Bernoulli quasi-likelihood and include indicator variables for the number of time periods that a
household is in the sample in the specification of the conditional mean and the conditional
variance of the unobserved effect. This is what Wooldridge (2010) proposes; except that we have
made the assumption that there is a constant slope on the time averages of the covariates in the
model. The results of this estimation approach are displayed in Table 27.
There are some differences in the estimated average partial effects in Table 27 for some
covariates when we compare them to those estimated by the other methods in Tables 24, 25 and
26. This appears to be most prevalent in the estimated average partial effects for the covariates in
the bond equation. For the most part, the larger average partial effects are not that worrisome
since the estimated standard errors are large enough so that they are statistically insignificant at
standard levels. The only estimated average partial effect which stands out as statistically
different is Wife Pr(Live to 75). While the overall impact on allocation is still small – a ten

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percentage point increase in Wife Pr(Live to 75) leads to a 0.005 percentage point increase in the
allocation of financial wealth to bonds - it is still five times larger than the estimated average
partial effect from the other single equation estimates. Perhaps most striking is that in the other
single equation estimation approaches we very nearly have average partial effects that sum to
zero across the four equations, but this does not appear to be the case in Table 27. In fact it
appears that none of the marginal effects of any covariate sum to zero across equations.
We also test the joint significance of the indicator variables for the total number of time
periods that a household is observed in both the mean and the variance equation. From our test of
joint significance we can see that the allocation of wealth to CDs is the only equation where we
see that we obtain a test statistic of 52.16 which is significant at the 5% level, but this is only for
the mean equation. This implies that there is little evidence of variation in the moments of the
unobserved effects at the single equation level, with the exception of the mean of ci for the
allocation of wealth to CDs.
Our next step is to estimate the allocation equations by maximizing the multinomial
quasi-likelihood function. We first estimate the multinomial quasi-likelihood function on the
entire unbalanced panel ignoring the fact that we observe households for a different number of
time periods. The results of this approach are shown in Table 28. We can see that our estimated
average partial effects are very similar to those that we estimated using linear fixed effects and
maximizing the Bernoulli quasi-likelihood for each equation. It appears that the biggest
difference in estimated average partial effects is for Log (Income) which is larger in the single
equation approaches than it is for the multinomial approach. This is not surprising as the
multinomial approach leads us to estimated coefficients that maximize a likelihood that is
restricted to values that satisfy the adding up constraint.

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As a next step we acknowledge the unbalanced nature of our panel by maximizing the

pooled multinomial quasi-likelihood for each balanced panel subset and then average the
estimated average partial effects and their variances across the balanced panel subsets. Table 29
displays the results of this estimation approach. Once again it is clear that our standard errors
have increased significantly since the sample sizes of the balanced panel subsets are smaller than
the full panel. Nevertheless, there appears to be little difference in the estimated average partial
effects compared to the estimates in Tables 24, 25 and 26. In fact, see that the estimated impact
of Log (Income) are closer to those estimated using the single equation approaches.
Table 30 presents the results of maximizing the multinomial quasi-likelihood on the full,
unbalanced panel and specifying that the moments of the unobserved effect to vary based on the
number of times that we observe households in the panel. The estimated average partial effects
are nearly identical to those estimated by linear fixed effects. Unlike the estimates from the
maximization of the Bernoulli quasi-likelihood in table 27 the estimated average partial effects
for each covariate sum to zero (with some rounding error) in table 30. This is due to the fact that
the adding up restriction is naturally imposed by the specification of the conditional mean
function and therefore restricts the maximization routine to focus on coefficients that satisfy the
restriction. Looking at the joint significance test for the indicator variables for the total number
of times that a household is observed we can see that there are some slight differences from the
similar test presented in Table 27. First, since we are “excluding” the checking equation in the
multinomial approach we do not have coefficients to test so these are missing. Second, it appears
that both the conditional mean and conditional variance of the unobserved effect for the
allocation of wealth to bonds differs across the number of time periods that we observe
households. Similar to the estimates in table 27 we see that the conditional mean of the

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unobserved effect in the allocation of wealth to CDs does differ depending on the total number of
time periods that a household is in sample. The difference in the bond allocation equation is most
likely due to the imposition of the adding up restriction. Once we force the impact of these
covariates to cancel out across the four assets it restricts the search for coefficients to those that
satisfy the adding up constraint. Since the allocation of wealth to bonds is so small (in fact for a
majority of households it is zero more often than not) and the quasi-likelihood function is flatter
than for other assets, a single equation approach is likely to find many coefficients that maximize
the flat likelihood but that do not maximize the multinomial quasi-likelihood.
As another point of comparison, we include Table 31 which shows the estimated average
partial effects from estimating equation (3.3.5) but replacing the Normal Cumulative Distribution
Function with the multinomial logit functional form. We should note that the below estimation
equation is an approximation, the actual derivation of the appropriate specification of the
conditional mean and variance function when using the multinomial logit functional form is
more complicated. We perform this additional estimation since using the Normal Cumulative
Distribution Function does not explicitly impose the restriction that the predicted value of the
omitted equation must be in the unit interval; instead we are relying on the logarithm to impose
this constraint. Specifically we are estimate the following equation:

E ( yitg | xit , w i , sit = 1) =


 x β + T ψ ⋅1[T = r ] + x ξ 
it g ∑ r =1 gr
i
i g
exp 
1/ 2 

T

 exp  ∑

 r = 21[Ti = r ] ⋅ ω gr 




.


 x β + T ψ ⋅1[T = r ] + x ξ 
G −1
it g ∑ r =1 gr
i
i g
1 + ∑ exp 
1/ 2 

T
g =1
1[Ti = r ] ⋅ ω gr 
 exp  ∑

 r =2





75

(3.5.1)

ւ

≺⊙ ⊙ ց

We can see from Table 31 that the estimated average partial effects are nearly identical to those
reported in Table 30, though the estimated standard errors for all equations are smaller than those
estimated using the multinomial fractional probit approach. An interesting difference between
Table 30 and 31 arises in the χ 2 test for ψ gr and ω gr for the Bond equation. In table 30 we
can see that there the covariates are in general not statistically significant, but the indicator
variables for the length of time in sample are significant in both the conditional mean and
variance equation. Using the multinomial logit functional form we see that the these indicator
variables are no longer statistically significant in either the conditional mean or variance
equation, but that now the estimated partial effects are more precisely estimated. The fact that the
estimated average partial effects are nearly identical across all estimation approaches suggests
that we can leverage the flexibility (and potential efficiency) of non-linear estimation techniques
on fractional response variables in an unbalanced panel.
Overall our estimated average partial effects from the procedure laid out in this paper are
nearly identical to those estimated by methods that we know will give us consistent estimates,
but it appears that estimating a nonlinear model that appropriately imposes the adding up
restriction and also accounts for the unbalanced nature of our panel has led to a happy
compromise between ignoring this information and over controlling for it.
3.6 Conclusion
One of the major criticisms of estimating nonlinear models on panel data with a
correlated random effects approach is that it cannot handle unbalanced panels. Wooldridge
(2010) offers a simple solution to this problem that is parsimonious and easily estimable by
standard software. We have extended this approach to multiple fractional response variables and
have exploited the fact that we have an adding up restriction that allows us to use the
76

ւ

≺⊙ ⊙ ց

multinomial quasi-likelihood function. Since the multinomial quasi-likelihood is in the linear
exponential family we can be confident that we have consistent estimates of the average partial
effects of our covariates.
We have also compared our results from our procedure to other estimation methods that
will give us consistent estimates of the average partial effects. We have found that the estimation
procedure we laid out here lies somewhere in between ignoring the unbalanced nature of the
panel and more drastic corrections for it such as estimation of standard correlated random effect
models on balanced panel subsets. Since we have explicitly modeled that the conditional mean
and variance of our unobserved effect varies with the number of time periods we observe a crosssectional unit we are able to use the full data set to estimate our average partial effects. This has
allowed us to maintain consistency while increasing the precision of our estimated average
partial effects.
Possible avenues for future research would be to attempt to more parsimoniously specify
that the unobserved effect will differ across equations. Here we have allowed for each covariate
in the conditional mean and variance of the unobserved effect to be different for each equation. It
might be fruitful to instead specify that the unobserved effect is the same across each equation
but instead there is an overall scale factor for each equation’s conditional mean and variance
function. With the appropriate normalization we could then estimate fewer coefficients leading
to possible more efficiently estimated average partial effects.

77

APPENDIX

78

Table 1. Summary Statistics for Single Households
Wealth Sample
Allocation Sample
Mean
Median
Mean
Median
Age
58
58
58
59
White
0.62
0.71
Black
0.27
0.21
Hispanic
0.09
0.06
Other
0.03
0.03
Less than High School
0.28
0.19
High School
0.34
0.35
Some College
0.20
0.23
College
0.08
0.10
Post Grad
0.10
0.13
Male
0.29
0.29
Divorced/Separated
0.56
0.56
Widowed
0.30
0.30
Never Married
0.14
0.14
Working
0.57
0.63
Retired
0.18
0.20
Unemployed
0.04
0.03
Disabled
0.18
0.12
Household Size
1.81
1
1.72
1
Word Recall
0.63
0.71
0.67
0.71
Series 7 Correct
2.34
2
2.66
3
High Risk Aversion
0.62
0.63
0.12
0.13
Medium-High Risk Aversion
Medium-Low Risk Aversion
0.10
0.10
Low Risk Aversion
0.14
0.13
Plan for Next Few Months
0.25
0.21
Plan for Next Year
0.11
0.11
Plan for Next Few Years
0.27
0.28
Plan for 5-10 years
0.26
0.29
Plan for 10+ years
0.10
0.10
Pr(Live to 75)
0.63
0.70
0.65
0.75
Net Worth
$131,430 $38,530 $168,903 $64,186
Financial Wealth
$35,029 $1,485 $47,295 $4,733
Current Income
$22,764 $15,300 $27,155 $19,407
Log(Current Income)
9.55
9.64
9.77
9.87
Permanent Income
$21,405 $15,303 $25,585 $18,983
Permanent Log Income
9.44
9.50
9.65
9.71
Stocks
0.17
Bonds
0.02
CDs
0.08
Checking
0.74
79

Table 1 (cont’d).
Observations
14,275
10,573
Households
5,022
4,053
Note: Reported Wealth and Income measures are in 1992 dollars.
Table 2. Summary Statistics for Married Households
Wealth Sample
Allocation Sample
Mean
Median
Mean
Median
Husband Wife Husband Wife Husband Wife Husband Wife
Age
58
54
58
55
58
55
58
55
White
0.80
0.81
0.84
0.85
Black
0.10
0.10
0.08
0.08
Other
0.02
0.02
0.02
0.02
Hispanic
0.07
0.07
0.05
0.05
Less than
0.21
0.19
0.17
0.15
High School
High School
0.32
0.39
0.33
0.40
Some College
0.21
0.23
0.22
0.24
College
0.12
0.10
0.13
0.10
Post Grad
0.14
0.09
0.15
0.10
High Risk
0.60
0.63
0.60
0.63
Aversion
Medium-High
0.13
0.15
0.14
0.15
Risk Aversion
Medium-Low
0.10
0.10
0.10
0.10
Risk Aversion
Low Risk
0.14
0.10
0.14
0.10
Aversion
Plan for Next
0.13
0.15
0.12
0.13
Few Months
Plan for Next
0.10
0.11
0.09
0.10
Year
Plan for Next
0.29
0.31
0.30
0.31
Few Years
Plan for 5-10
0.35
0.32
0.36
0.33
Years
Plan for 10+
0.12
0.11
0.12
0.11
Years
Word Recall
0.58
0.65
0.59
0.71
0.60
0.66
0.71
0.71
Pr(Live to 75)
0.64
0.67
0.70
0.75
0.65
0.68
0.70
0.75
Working
0.68
0.59
0.69
0.61
Retired
0.25
0.11
0.26
0.11
Unemployed
0.02
0.02
0.02
0.02
Disabled
0.09
0.06
0.07
0.05
Series 7
2.44
2.26
3
2
2.54
2.36
3
2
Correct
80

Table 2 (cont’d).
Current
Income
Log(Current
Income)
Household
Size
Permanent
Income
Permanent
Log Income
Net Worth
Financial
Wealth
Stocks
Bonds
CDs
Checking
Observations
Households

$56,086

$43,750

$59,317

$46,498

10.58

10.69

10.68

10.75

2.16

2

2.16

2

$56,914

$46,670

$59,988

$49,218

10.64

10.72

10.73

10.77

$304,024

$140,111

$330,922

$159,045

$65,593

$10,887

$73,111

$15,200

0.25
0.02
0.09
0.63
18,603
6,608

16,690
6,048

Note: Reported Wealth and Income measures are in 1992 dollars.

Table 3. Average Wealth and Proportion of Financial Wealth Allocated by Pr(Live to 75),
Single Households
Fraction
Financial
Stocks Bonds CDs Checking
Net Worth
of
Wealth
Sample
$41,804
$7,063
0.08
0.01
0.06
0.86
0.0594
0 ≤ P75 < 0.1
$70,412
$14,898
0.13
0.01
0.07
0.79
0.0282
0.1 ≤ P75 < 0.2
$107,678
$24,816
0.16
0.01
0.08
0.75
0.0355
0.2 ≤ P75 < 0.3
$103,123
$17,245
0.13
0.02
0.09
0.76
0.0210
0.3 ≤ P75 < 0.4
$129,572
$35,292
0.14
0.03
0.08
0.75
0.0217
0.4 ≤ P75 < 0.5
$116,788
$28,681
0.15
0.01
0.09
0.75
0.2487
0.5 ≤ P75 < 0.6
$176,562
$57,327
0.19
0.02
0.08
0.70
0.0824
0.6 ≤ P75 < 0.7
$177,824
$44,265
0.21
0.02
0.10
0.68
0.0760
0.7 ≤ P75 < 0.8
$185,234
$51,814
0.21
0.02
0.09
0.68
0.2026
0.8 ≤ P75 < 0.9
$347,801
$168,255
0.24
0.02
0.07
0.67
0.0162
0.9 ≤ P75 < 1
$111,057
$26,632
0.15
0.01
0.01
0.77
0.2085
P75 = 1
Note: Wealth figures are in 1992 dollars.

81

Table 4. Average Wealth and Proportion of Financial Wealth Allocated by Husband’s
Pr(Live to 75), Married Households
Fraction
Net
Financial
Stocks Bonds CDs Checking
of
Worth
Wealth
Sample
$150,509 $30,026
0.17
0.01
0.09
0.73
0 ≤ P75 < 0.1
0.0524
0.18
0.01
0.07
0.73
0.1 ≤ P75 < 0.2 $195,464 $31,120
0.0255
0.21
0.02
0.1
0.67
0.2 ≤ P75 < 0.3 $226,670 $48,087
0.0367
0.21
0.02
0.11
0.67
0.3 ≤ P75 < 0.4 $200,316 $41,090
0.0259
0.22
0.02
0.09
0.68
0.4 ≤ P75 < 0.5 $238,363 $45,413
0.0241
0.22
0.02
0.1
0.66
0.5 ≤ P75 < 0.6 $273,417 $57,287
0.2490
0.3
0.03
0.09
0.58
0.6 ≤ P75 < 0.7 $336,502 $80,045
0.1067
0.31
0.03
0.09
0.57
0.7 ≤ P75 < 0.8 $439,344 $116,920
0.0795
$364,742 $76,434
0.3
0.03
0.09
0.59
0.8 ≤ P75 < 0.9
0.2026
$404,967 $91,239
0.39
0.03
0.05
0.53
0.9 ≤ P75 < 1
0.0143
$306,326 $61,103
0.23
0.02
0.09
0.65
P75 = 1
0.1833
Note: Wealth figures are in 1992 dollars.

Table 5. Average Wealth and Proportion of Financial Wealth Allocated by Wife’s Pr(Live to
75), Married Households
Fraction
Net
Financial
Stocks Bonds
CDs
Checking
of
Worth
Wealth
Sample
$118,553 $18,608
0.14
0.01
0.08
0.77
0 ≤ P75 < 0.1
0.0395
0.18
0.01
0.1
0.71
0.1 ≤ P75 < 0.2 $180,629 $48,247
0.0174
0.16
0.01
0.1
0.74
0.2 ≤ P75 < 0.3 $177,559 $33,642
0.0280
0.22
0.01
0.11
0.66
0.3 ≤ P75 < 0.4 $183,728 $44,373
0.0176
0.19
0.02
0.12
0.68
0.4 ≤ P75 < 0.5 $159,264 $28,859
0.0200
$264,519 $58,833
0.22
0.02
0.1
0.66
0.5 ≤ P75 < 0.6
0.2395
0.26
0.02
0.1
0.62
0.6 ≤ P75 < 0.7 $307,643 $68,165
0.0954
0.31
0.03
0.08
0.58
0.7 ≤ P75 < 0.8 $434,637 $97,380
0.0841
0.3
0.03
0.09
0.59
0.8 ≤ P75 < 0.9 $370,027 $83,681
0.2436
$450,742 $115,980
0.35
0.03
0.07
0.55
0.9 ≤ P75 < 1
0.0205
$310,064 $57,644
0.25
0.02
0.09
0.64
P75 = 1
0.1944
Note: Wealth figures are in 1992 dollars.

82

Table 6. Wealth Holding Estimation Results, Single Households
Financial
Net Worth
Wealth
-51,213
-14,279
Age
(29,894)*
(13,684)
1,153
350
2
Age
(590)*
(276)
-8
-3
3
Age
(4)**
(2)
-45,779
-5,396
Black
(17,874)**
(6,116)
19,166
17,318
White
(18,174)
(6,211)**
-23,964
-625
Hispanic
(18,216)
(6,337)
17,680
8,859
High School
(8,268)**
(2,833)**
58,107
23,101
Some College
(12,235)**
(4,975)**
111,375
50,635
College
(23,695)**
(10,958)**
165,583
79,858
Post Grad
(28,868)**
(13,114)**
25,972
12,850
Male
(10,429)**
(4,692)**
-2,113
-3,394
Divorced/Separated
(10,033)
(4,784)
46,520
9,966
Widowed
(11,615)**
(5,101)*
248
7,630
Working
(13,335)
(8,387)
36,706
16,949
Retired
(10,668)**
(5,008)**
3,935
-5,174
Unemployed
(21,743)
(6,628)
-4,772
-2,668
Household Size
(1,845)**
(769)**
26,900
9,479
Word Recall
(12,737)**
(4,822)**
4,466
2,328
Series 7 Correct
(1,924)**
(856)**
6,102
-1,879
Medium-High Risk Aversion
(12,438)
(3,570)
2,723
2,850
Medium-Low Risk Aversion
(15,650)
(8,821)
83

Table 6 (cont’d).
29,291
9,374
(13,970)**
(5,480)*
6,775
860
Plan for Next Year
(10,052)
(4,315)
26,510
9,814
Plan for Next Few Years
(6,049)**
(2,362)**
50,030
22,221
Plan for 5-10 Years
(9,069)**
(4,045)**
94,892
38,520
Plan for 10+ Years
(15,926)**
(8,780)**
256
124
Pr(Live to 75)
(95)**
(44)**
-25,613
-23,819
Current Income
(9,518)**
(6,887)**
330
228
2
Current Income
(153)**
(97)**
31,176
11,922
Permanent Income
(7,194)**
(3,949)**
267
108
2
Permanent Income
(62)**
(35)**
Current Income*Permanent
-74
19
Income
(301)
(185)
Observations
14,275
Households
5,022
Note: Robust standard errors reported in parentheses. ** p<0.05 and
* p<0.1. A full set of year dummies is included in all models.
Reported coefficients for Pr(Live to 75) are for a one-percentage
point difference (0.01).
Low Risk Aversion

Table 7. Average Partial Effects (Reported as Percentages) for Allocation of
Financial Wealth , Single Households, Estimated by Single Equation
Fractional Probit
Stocks
Bonds
CDs
Checking
-0.117
0.002
0.117
-0.027
Age
(0.092)
(0.022)
(0.067)*
(0.105)
-2.054
-0.167
0.989
0.787
Black
(2.616)
(0.688)
(2.086)
(3.103)
3.830
0.360
1.555
-5.819
White
(2.387)
(0.594)
(1.781)
(2.819)**
0.676
0.168
-0.390
-0.421
Hispanic
(3.269)
(0.907)
(2.187)
(3.635)
4.729
0.830
0.806
-4.637
High School
(1.511)** (0.462)*
(0.994)
(1.563)**
84

Table 7 (cont’d).
Some College
College
Post Grad
Male
Divorced/Separated
Widowed
Working
Retired
Unemployed
Household Size
Word Recall
Series 7 Correct
Medium-High Risk Aversion
Medium-Low Risk Aversion
Low Risk Aversion
Plan for Next Year
Plan for Next Few Years
Plan for 5-10 Years
Plan for 10+ Years
Pr(Live to 75)
Log(Current Income)
Permanent Log Income

9.291
(1.711)**
11.068
(2.203)**
10.005
(2.164)**
0.257
(0.961)
-0.820
(1.257)
-0.349
(1.373)
0.642
(1.131)
1.915
(1.201)
-1.392
(2.081)
-0.783
(0.400)*
1.630
(1.800)
0.116
(0.284)
-1.369
(1.113)
1.450
(1.269)
2.607
(1.114)**
0.291
(1.464)
4.222
(1.171)**
5.163
(1.203)**
5.750
(1.499)**
0.0133
(0.0136)
0.539
(0.440)
1.199
(0.756)
85

-0.038
(0.399)
0.911
(0.639)
1.019
(0.595)*
0.005
(0.231)
0.227
(0.285)
0.405
(0.326)
-0.086
(0.296)
0.710
(0.377)*
1.511
(1.021)
-0.199
(0.127)
-0.255
(0.441)
0.065
(0.090)
-0.125
(0.275)
0.193
(0.330)
0.372
(0.341)
0.699
(0.539)
0.459
(0.363)
0.317
(0.353)
1.134
(0.544)**
-0.0046
(0.0035)
-0.010
(0.139)
0.240
(0.214)

-0.609
(1.077)
0.396
(1.330)
-0.705
(1.251)
-1.227
(0.656)*
-0.451
(0.926)
-0.970
(0.986)
0.742
(0.848)
0.768
(0.932)
2.082
(1.738)
-0.686
(0.295)**
-2.310
(1.273)*
-0.189
(0.214)
-0.012
(0.802)
-1.427
(0.806)*
-3.175
(0.739)**
1.885
(1.133)**
1.920
(0.860)
1.005
(0.829)
-0.973
(0.980)
-0.0037
(0.0093)
0.067
(0.283)
-1.145
(0.541)**

-6.789
(1.740)**
-10.372
(2.192)**
-9.451
(2.167)**
0.866
(1.089)
1.319
(1.420)
1.263
(1.537)
-0.601
(1.314)
-2.906
(1.378)**
-1.538
(2.533)
1.617
(0.451)**
1.139
(2.053)
-0.033
(0.328)
1.499
(1.265)
-0.069
(1.431)
-0.146
(1.277)
-2.390
(1.650)
-5.943
(1.303)**
-6.049
(1.307)**
-6.096
(1.619)**
-0.0037
(0.0154)
-0.620
(0.508)
-0.391
(0.873)

Table 7 (cont’d)
0.693
1.900
-6.884
(0.129)** (0.165)** (0.329)**
Observations
10,573
Households
4,053
Note: Average partial effects reported. Robust standard errors are in
parentheses. ** p<0.05 and * p<0.1. A full set of year dummies is included in
all models. Reported Average Partial Effects for Pr(Live to 75) are for a onepercentage point difference (0.01).
Log(Net Worth)

5.031
(0.319)**

Table 8. Wealth Holding Estimation Reults, Married Households
Net Worth
Financial Wealth
Husband
Wife
Husband
Wife
-10,009
-4,413
-81,003
-122,988
Age
(14,993)
(37,969)** (54,066)** (14,784)
1,953
2,480
262
118
2
Age
(738)**
(1,071)**
(293)
(323)
-14
-16
-2
-1
3
Age
(5)**
(7)**
(2)
(2)
-98,615
18,022
-32,990
-6,577
Black
(38,207)**
(33,643)
(18,756)*
(18,957)
6,275
32,419
322
-13,212
White
(36,598)
(32,772)
(17,389)
(19,586)
-51,752
51,415
-6,556
-16,710
Hispanic
(38,673)
(37,185)
(16,973)
(19,363)
-11,679
21,975
-3,068
5,762
High School
(14,385)
(13,271)*
(3,803)
(3,783)
-1,573
14,839
-27,431
55,406
Some College
(20,164)
(23,091)**
(5,098)
(5,294)**
54,901
84,406
37,505
38,063
College
(39,712)
(36,931)** (12,501)** (15,272)**
8,167
67,598
30,609
41,897
Post Grad
(36,482)
(38,935)* (10,834)** (13,844)**
-18,926
-2,707
4,303
-300
Medium-High Risk Aversion
(19,420)
(17,566)
(7,331)
(6,772)
-48,768
23,058
-6,619
3,323
Medium-Low Risk Aversion
(18,796)**
(21,725)
(5,551)
(7,363)
-7,599
46,826
-1,033
-240
Low Risk Aversion
(18,355)
(27,255)*
(6,077)
(6,792)
-35,539
-12,209
-6,624
-785
Plan for Next Year
(21,740)
(16,122)
(9,458)
(5,215)
-21,460
20,647
-7,968
9,455
Plan for Next Few Years
(20,062)
(15,212)
(7,436)
(5,234)*

86

Table 8 (cont’d)
-25,427
14,505
-8,661
7,219
(21,058)
(17,162)
(7,321)
(4,579)
49,614
59,879
24,764
27,512
Plan for 10+ Years
(27,072)* (23,574)** (10,742)** (8,563)**
-14,951
-15,527
-13,304
4,178
Word Recall
(37,053)
(50,025)
(12,258)
(9,782)
14,161
821
4,867
2,327
Series 7 Correct
(4,998)**
(5,921)
(1,646)**
(1,644)
-11,193
-4,319
Household Size
(5,887)*
(2,196)**
-43,525
-98,558
-13,889
-31,661
Working
(13,399)** (17,784)** (5,442)** (6,289)**
-64,020
-80,664
-19,344
-18,303
Unemployed
(20,379)** (22,084)** (6,320)** (8,064)**
15,281
19,732
11,103
3,034
Retired
(15,312)
(24,190)
(5,732)*
(9,127)
241
425
28
-22
Pr(Live to 75)
(203)
(193)**
(62)
(65)
11,292
-2,335
Current Income
(5,504)**
(1,303)*
21
132
2
Current Income
(138)
(20)**
46,914
13,071
Permanent Income
(6,340)**
(1,764)**
-281
-55
2
Permanent Income
(117)**
(19)**
Current Income*Permanent
83
-145
Income
(381)
(57)**
Observations
18,603
Households
6,608
Note: Robust standard errors reported in parentheses. ** p<0.05 and * p<0.1. A full set
of year dummies are included in all models. Reported coefficients for Pr(Live to 75) are
for a one-percentage point difference (0.01).
Plan for 5-10 Years

87

Table 9. Average Partial Effects (Reported as Percentages) for Allocation of Financial Wealth, Married
Households, Estimated by Single Equation Fractional Probit
Stocks
Bonds
CDs
Checking
Husband
Wife
Husband
Wife
Husband
Wife
Husband
Wife
0.041
0.029
0.128
0.077
-0.012
-0.273
-0.175
0.167
Age
(0.022) (0.055)**
(0.047)
(0.098) (0.085)**
(0.091)* (0.079)** (0.027)
Black

-2.420
(4.658)

-5.679
(4.267)

-0.299
(0.769)

1.241
(1.126)

-1.809
(2.718)

2.880
(3.423)

4.746
(4.736)

1.714
(4.734)

White

-0.713
(3.121)

-0.592
(2.884)

-0.065
(0.881)

0.688
(0.636)

1.372
(2.121)

-0.726
(1.936)

-0.762
(3.292)

0.572
(3.028)

Hispanic

-3.732
(3.390)

-6.245
(3.064)**

-0.751
(0.803)

0.179
(1.055)

0.337
(2.607)

-2.656
(1.765)

3.419
(3.761)

8.234
(3.366)**

High School

4.215
(1.307)**

5.521
(1.364)**

0.663
(0.456)

0.989
(0.488)**

0.994
(0.804)

2.955
(0.855)**

-4.253
-8.104
(1.358)** (1.365)**

Some College

8.674
(1.502)**

5.378
(1.539)**

0.957
(0.518)*

0.869
(0.559)

0.016
(0.860)

2.447
(1.017)**

-7.612
-7.538
(1.544)** (1.578)**

College

8.929
(1.718)**

1.854
1.939
-1.858
8.163
(1.955)** (0.667)** (0.785)** (0.899)**

2.275
(1.276)*

-7.693
-11.581
(1.749)** (1.983)**

Post Grad

6.747
(1.745)**

1.863
1.742
7.056
(2.012)** (0.687)** (0.761)**

-0.282
(0.998)

0.837
(1.185)

-6.994
-9.431
(1.845)** (2.096)**

MediumHigh Risk
Aversion
Medium-Low
Risk
Aversion
Low Risk
Aversion

1.471
(0.974)

-0.764
(0.911)

0.306
(0.276)

0.170
(0.268)

0.470
(0.611)

0.108
(0.577)

-2.204
(1.061)**

0.373
(0.994)

1.683
(1.144)

-0.637
(1.053)

-0.218
(0.315)

-0.084
(0.308)

-0.788
(0.676)

-1.254
(0.646)*

-0.568
(1.291)

1.769
(1.194)

2.788
(1.039)**

-0.053
(1.144)

0.161
(0.288)

0.018
(0.326)

-1.622
(0.604)**

-0.418
(0.686)

-1.599
(1.120)

0.230
(1.257)

88

Table 9
(cont’d).
Plan for Next
Year

0.270
(1.494)

-1.711
(1.340)

-0.326
(0.386)

-0.285
(0.383)

0.421
(0.931)

0.095
(0.870)

-0.053
(1.568)

1.885
(1.462)

Plan for Next
Few Years

2.076
(1.189)*

-1.620
(1.107)

-0.159
(0.345)

0.033
(0.359)

1.204
(0.749)

-0.004
(0.681)

-2.892
(1.256)**

1.661
(1.180)

Plan for 5-10
Years

1.996
(1.157)*

-0.153
(1.124)

-0.124
(0.337)

-0.084
(0.351)

0.292
(0.716)

0.090
(0.700)

-1.986
(1.229)

0.224
(1.202)

Plan for 10+
Years

3.167
(1.434)**

-0.772
(1.297)

-0.448
(0.342)

0.184
(0.415)

1.164
(0.913)

0.967
(0.883)

-4.121
(1.506)**

-0.460
(1.434)

Word Recall

4.500
(1.753)**

1.829
(1.703)

-0.404
(0.542)

0.393
(0.579)

-0.006
(1.178)

-0.319
(1.139)

-3.745
(1.942)*

-1.550
(1.899)

Series 7
Correct

0.624
(0.319)*

0.374
(0.278)

-0.020
(0.101)

-0.081
(0.091)

-0.035
(0.195)

0.242
(0.188)

-0.674
(0.334)**

-0.559
(0.302)*

Household
Size

-0.827
(0.426)*

-0.031
(0.114)

0.395
(0.279)

0.499
(0.462)

Working

0.247
(1.024)

0.218
(0.801)

-0.101
(0.289)

-0.194
(0.228)

0.195
(0.664)

0.636
(0.512)

0.104
(1.139)

-0.101
(0.871)

Unemployed

0.833
(2.332)

0.647
(2.253)

-0.742
(0.511)

0.229
(0.827)

-1.642
(1.352)

0.213
(1.466)

1.748
(2.478)

-0.980
(2.428)

Retired

2.846
(1.052)**

0.984
(1.090)

0.315
(0.311)

-0.044
(0.294)

1.188
(0.665)*

-0.141
(0.651)

-3.879
(1.145)**

-0.726
(1.178)

0.0031
(0.0036)

0.0036
(0.0040)

-0.0130
-0.0189
(0.0075)* (0.0081)**

-0.0146
(0.0131)

-0.0037
(0.0139)

Pr(Live to 75)

0.0262
0.0224
(0.0121)** (0.0131)*

Log(Current
Income)

1.055
(0.492)**

0.123
(0.156)

0.300
(0.300)

-1.531
(0.549)**

Permanent
Log Income

2.300
(0.813)**

0.050
(0.222)

-1.209
(0.458)**

-1.359
(0.885)

89

Table 9
(cont’d).
Log(Net
6.670
0.968
1.256
-8.115
Worth)
(0.389)**
(0.112)**
(0.159)**
(0.400)**
Observations
16,690
Households
6,048
Note: Average partial effects reported. Robust standard errors are in parentheses. ** p<0.05 and * p<0.1. A full
set of year dummies included in all models. Reported Average Partial Effect for Pr(Live to 75) is for a onepercentage point difference (0.01).

90

Table 10. Sensitivity Analysis, Removing Permanent Income, Single and
Married Households, Wealth Holding and Asset Allocation
Single Households
Wealth Holdings
Pr(Live to 75)

Net Worth
339
(100)**

Financial Wealth
156
(50)**

Asset Allocation
Pr(Live to 75)

Stocks
0.0142
(0.0136)

Bonds
-0.0044
(0.0035)

CDs
-0.0046
(0.0093)

Checking
-0.0040
(0.0154)

Married Households
Wealth Holdings
Husband Pr(Live to 75)
Wife Pr(Live to75)

Net Worth
356
(208)*
522
(200)**

Financial Wealth
66
(63)
12
(66)

Asset Allocation
Stocks
Bonds
CDs
Checking
0.0274
0.0032
-0.0135
-0.0152
Husband Pr(Live to 75)
(0.0121)** (0.0036) (0.0075)*
(0.0131)
0.0230
0.0036
-0.0193
-0.0040
Wife Pr(Live to75)
(0.0131)* (0.0040) (0.0081)** (0.0139)
Note: We remove all variables that include our measure of Permanent
Income. ** p<0.05, * p<0.10. Average Partial Effects (reported as
percentages are shown for Stocks, Bonds, CDs, and Checking. Reported
coefficients and Average Partial Effects for Pr(Live to 75) are for a onepercentage point difference (0.01).

91

Table 11. Sensitivity Analysis, Treating Pr(Live to75)=1 differently, Single Households,
Wealth Holding and Asset Allocation
Treatment 1: Removing
Treatment 2: Removing
Observations
Households
Net Worth
394
466
Pr(Live to 75)
(124)**
(141)**
Financial Wealth
182
143
Pr(Live to 75)
(54)**
(43)**
Observations
11,176
8,022
Households
4,461
3,241
Stocks
Pr(Live to 75)

0.0320
(0.0174)*

0.0501
(0.0209)**

-0.0022
(0.0046)

0.0016
(0.0059)

0.0043
(0.0117)

-0.0019
(0.0139)

Bonds
Pr(Live to 75)
CDs
Pr(Live to 75)
Checking
-0.0351
-0.0473
(0.0195)*
(0.0231)**
Observations
8,369
6,117
Households
3,566
2,588
Note: In Treatment 1 we drop observations where the household reports a survival
probability of one. In Treatment 2 we drop households that ever reported a survival
probability of one. ** p<0.05, * p<0.10. Average Partial Effects (reported as
percentages) are shown for Stocks, Bonds, CDs, and Checking. Reported coefficients
and Average Partial Effects for Pr(Live to 75) are for a one-percentage point difference
(0.01).
Pr(Live to 75)

92

Table 12. Sensitivity Analysis, Treating Pr(Live to 75)=1 differently, Married
Households, Wealth Holding and Asset Allocation
Treatment 1: Removing
Treatment 2: Removing
Observations
Households
Net Worth
56
-171
Husband Pr(Live to 75)
(253)
(314)
698
497
Wife Pr(Live to 75)
(253)**
(337)
Financial Wealth
-5
13
Husband Pr(Live to 75)
(81)
(98)
129
215
Wife Pr(Live to 75)
(97)
(141)
Observations
12,436
7,610
Households
5,272
3,252
Stocks
Husband Pr(Live to 75)
Wife Pr(Live to 75)

0.0359
(0.0161)**
0.0326
(0.0170)*

0.0288
(0.0203)
0.0602
(0.0208)**

0.0053
(0.0047)
0.0012
(0.0053)

0.0121
(0.0061)**
0.0055
(0.0067)

-0.0212
(0.0101)**
-0.0262
(0.0106)**

-0.0249
(0.0126)**
-0.0363
(0.0132)**

Bonds
Husband Pr(Live to 75)
Wife Pr(Live to 75)
CDs
Husband Pr(Live to 75)
Wife Pr(Live to 75)
Checking
-0.0002
-0.0123
(0.0002)
(0.0220)
-0.0039
-0.0253
Wife Pr(Live to 75)
(0.0180)
(0.0225)
Observations
11,183
6,753
Households
4,812
2,912
Note: In Treatment 1 we drop observations where any member of the household
reports a survival probability of one. In Treatment 2 we drop households where any
member ever reported a survival probability of one. ** p<0.05, * p<0.10. Average
Partial Effects (reported as percentages) are shown for Stocks, Bonds, CDs, and
Checking. Reported coefficients and Average Partial Effects for Pr(Live to 75) are for
a one-percentage point difference (0.01).
Husband Pr(Live to 75)

93

Table 13. Means and Medians of Relevant Variables
Mean
Median
Husband Wife Husband Wife
Pr(Live to 75)
0.65
0.69
0.70
0.75
Age
58
55
59
55
Word Recall
0.61
0.67
0.71
0.71
Income
$59,578
$47,000
Non-Financial Wealth
$268,579
$131,590
Allocation to Stocks
0.26
Allocation to Bonds
0.02
Allocation to CDs
0.09
Allocation to Checking
0.62
Observations
14,514
Households
3,872
Note: All dollar values are stated in 1992 dollars. In the right
panel all households with only one time observation are dropped

Table 14. Proportion of Variation Between
Households for Relevant Variables
Husband
Wife
Independent Variables
Pr(Live to 75)
0.66
0.65
Age
0.59
0.76
Word Recall
0.43
0.40
Income
0.65
Non-Financial Wealth
0.82
Dependent Variables
Allocation to Stocks
0.61
Allocation to Bonds
0.42
Allocation to CDs
0.52
Allocation to Checking
0.61
Observations
14,514
Households
3,872
Note: Results obtained by running an ANOVA
using the Household as the categorical variable.

94

Table 15. Average Partial Effects from Pooled Linear Fixed Effects OLS
Estimation on Entire Unbalanced Panel
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.008
0.005
Husband Age
(0.007)
(0.003)
(0.005)
(0.008)
0.001
-0.000
-0.001
0.000
Husband Pr(Live to 75)
(0.001)
(0.001)
(0.001)
(0.002)
0.006
-0.000
0.002
-0.008
Husband Word Recall
(0.002)**
(0.001)
(0.001)
(0.002)**
-0.004
-0.003
0.009
-0.002
Wife Age
(0.007)
(0.003)
(0.005)*
(0.008)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.001)** (0.001)**
(0.001)
(0.002)
0.002
0.001
-0.002
-0.001
Wife Word Recall
(0.001)
(0.001)
(0.001)*
(0.002)
0.011
0.001
0.004
-0.017
Log(Income)
(0.005)**
(0.002)
(0.003)
(0.005)**
Log(Non-Financial
-0.007
0.001
0.001
0.005
Wealth)
(0.004)*
(0.002)
(0.003)
(0.005)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and
serial correlation. ** p-value < 0.05, * p-value < 0.10. Reported average
partial effect for Husband and Wife Pr(Live to 75) and Husband and Wife
Word Recall represent a 10 percentage point change.

95

Table 16. Average Partial Effects from Single Equation Pooled
Bernoulli Quasi-Maximum Likelihood Estimates for Each Balanced
Panel Subset, Averaged Across Balanced Panel Subsets
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.007
0.004
Husband Age
(0.019) (0.006) (0.012)
(0.020)
0.001
-0.000
-0.001
0.001
Husband Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.004
0.000
0.002
-0.006
Husband Word Recall
(0.004) (0.002) (0.003)
(0.005)
-0.003
-0.002
0.009
-0.005
Wife Age
(0.019) (0.006) (0.012)
(0.020)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.001
0.001
-0.002
0.000
Wife Word Recall
(0.004) (0.001) (0.003)
(0.005)
0.012
0.002
0.003
-0.016
Log(Income)
(0.012) (0.004) (0.007)
(0.014)
Log(Non-Financial
-0.006
0.002
0.001
0.005
Wealth)
(0.012) (0.004) (0.007)
(0.013)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and
serial correlation. ** p-value < 0.05, * p-value < 0.10. Reported average
partial effect for Husband and Wife Pr(Live to 75) and Husband and
Wife Word Recall represent a 10 percentage point change.

96

Table 17. Average Partial Effects from Pooled Multinomial QuasiMaximum Likelihood Estimates for Each Balanced Panel Subset,
Averaged Across Balanced panel Subsets
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.006
0.003
Husband Age
(0.019) (0.006) (0.012)
(0.020)
0.000
-0.001
-0.001
0.001
Husband Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.004
-0.000
0.002
-0.006
Husband Word Recall
(0.004) (0.002) (0.003)
(0.004)
-0.003
-0.002
0.009
-0.004
Wife Age
(0.019) (0.006) (0.012)
(0.020)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.000
0.001
-0.002
0.001
Wife Word Recall
(0.004) (0.001) (0.003)
(0.004)
0.009
0.001
0.003
-0.013
Log(Income)
(0.012) (0.004) (0.007)
(0.012)
Log(Non-Financial
-0.006
0.001
0.000
0.005
Wealth)
(0.012) (0.004) (0.007)
(0.013)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity
and serial correlation. ** p-value < 0.05, * p-value < 0.10. Reported
average partial effect for Husband and Wife Pr(Live to 75) and
Husband and Wife Word Recall represent a 10 percentage point
change.

97

Table 18. Procedure 2: Classical Minimum Distance applied to Multinomial
Quasi-Maximum Likelihood Estimates for Each Year Within Balanced Panel
Subsets
Stocks
Bonds
CDs
Checking
0.027
0.004
-0.005
-0.026
Husband Age
(0.013)**
(0.003)
(0.007) (0.013)**
0.005
-0.000
0.000
-0.005
Husband Pr(Live to 75)
(0.002)*
(0.001)
(0.002)
(0.003)*
0.009
-0.000
-0.000
-0.008
Husband Word Recall
(0.003)**
(0.001)
(0.002) (0.003)**
-0.026
-0.003
0.003
0.026
Wife Age
(0.013)**
(0.003)
(0.007) (0.013)**
-0.007
-0.001
0.003
0.005
Wife Pr(Live to 75)
(0.003)**
(0.001)
(0.002)
(0.003)*
-0.001
0.000
-0.001
0.002
Wife Word Recall
(0.003)
(0.001)
(0.002)
(0.003)
0.023
-0.007
-0.003
-0.013
Log(Income)
(0.008)** (0.002)** (0.005)
(0.008)
Log(Non-Financial
0.003
0.000
-0.004
0.001
Wealth)
(0.008)
(0.002)
(0.005)
(0.009)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and
serial correlation. ** p-value < 0.05, * p-value < 0.10. Reported average
partial effects for Husband and Wife Pr(Live to 75) and Husband and Wife
Word Recall represent a 10 percentage point change.

98

Table 19. Comparing Average Partial Effect Estimates from Pooled
Single Equation Bernoulli Quasi-Maximum Likelihood Estimates for
Each Balanced Panel Subset, Averaged Across Balanced Panel Subsets
to Procedure 2 Average Partial Effect Estimates
Stocks
Bonds
CDs
Checking
Husband Age
0.137
0.972
0.913
0.153
Husband Pr(Live to 75)
0.267
0.939
0.602
0.182
Husband Word Recall
0.237
0.762
0.332
0.679
Wife Age
0.228
0.952
0.588
0.117
Wife Pr(Live to 75)
0.348
0.141
0.324
0.393
Wife Word Recall
0.698
0.743
0.890
0.759
Log(Income)
0.344
0.023** 0.370
0.805
Log(Non-Financial Wealth) 0.466
0.637
0.534
0.782
Note: p-values are robust to heteroskedasticity and serial correlation.
** p-value < 0.05, * p-value < 0.10. Standard errors used to compute
the test statistic are from Table 4: Weighted Average of the standard
errors for Average Partial Effects estimated by Single Equation
Bernoulli Quasi-Maximum Likelihood for each balanced panel subset.

Table 20. Comparing Average Partial Effect Estimates from Pooled
Multinomial Quasi-Maximum Likelihood Estimates for Each Balanced Panel
Subset, Averaged Across Balanced Panel Subsets to Procedure 2 Estimated
Average Partial Effects
Bonds
CDs
Checking
Stocks
Husband Age
0.133
0.955
0.935
0.145
Husband Pr(Live to 75)
0.229
0.922
0.519
0.129
Husband Word Recall
0.219
0.815
0.348
0.641
Wife Age
0.219
0.902
0.592
0.122
Wife Pr(Live to 75)
0.342
0.139
0.310
0.428
Wife Word Recall
0.735
0.793
0.918
0.759
Log(Income)
0.241
0.026** 0.410
0.989
Log(Non-Financial Wealth)
0.435
0.758
0.576
0.752
Note: p-values are robust to heteroskedasticity and serial correlation. ** pvalue < 0.05, * p-value < 0.10. Standard errors used to compute the test
statistic are from Table 5: Weighted Average of the standard errors for
Average Partial Effects estimated by Multinomial Quasi-Maximum
Likelihood for each balanced panel subset.

99

Table 21. Comparing Average Partial Effect Estimates from Pooled Linear
Fixed Effects OLS on Entire Unbalanced Panel to Procedure 2 Estimated
Average Partial Effect Estimates
Stocks
Bonds
CDs
Checking
Husband Age
0.027**
0.985
0.708
0.019**
Husband Pr(Live to 75)
0.116
0.705
0.377
0.053*
Husband Word Recall
0.311
0.776
0.209
0.934
Wife Age
0.079*
0.921
0.391
0.030**
Wife Pr(Live to 75)
0.150
0.005**
0.136
0.224
Wife Word Recall
0.253
0.729
0.632
0.372
Log(Income)
0.132
0.000**
0.110
0.610
Log(Non-Financial Wealth)
0.223
0.552
0.328
0.688
Note: p-values are robust to heteroskedasticity and serial correlation. ** pvalue < 0.05, * p-value < 0.10. Standard errors used to compute the test
statistic are from Table 6: Standard errors for Average Partial Effects
estimated by Procedure 2 (Classical Minimum Distance applied to
Multinomial Quasi-Maximum Likelihood estimates for each year within each
balanced panel subset).

100

Table 22. Means and Medians of Relevant Variables
Mean
Median
Husband Wife Husband Wife
Pr(Live to 75)
0.65
0.69
0.70
0.75
Age
58
55
59
55
Word Recall
0.61
0.67
0.71
0.71
Income
$59,578
$47,000
Non-Financial Wealth
$268,579
$131,590
Allocation to Stocks
0.26
Allocation to Bonds
0.02
Allocation to CDs
0.09
Allocation to Checking
0.62
Observations
14,514
Households
3,872
Note: All dollar values are stated in 1992 dollars. In the right
panel all households with only one time observation are dropped

Table 23. Proportion of Variation Between
Households for Relevant Variables
Wife
Husband
Independent
Variables
Pr(Live to 75)
0.66
0.65
Age
0.59
0.76
Word Recall
0.43
0.40
Income
0.65
Non-Financial Wealth
0.82
Dependent
Variables
Allocation to Stocks
0.61
Allocation to Bonds
0.42
Allocation to CDs
0.52
Allocation to Checking
0.61
Observations
14,514
Households
3,872
Note: Results obtained by running an ANOVA
using the Household as the categorical variable.

101

Table 24. Average Partial Effects from Pooled Linear Fixed Effects OLS
estimation on entire unbalanced panel
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.008
0.005
Husband Age
(0.007)
(0.003)
(0.005)
(0.008)
0.001
-0.000
-0.001
0.000
Husband Pr(Live to 75)
(0.001)
(0.001)
(0.001)
(0.002)
0.006
-0.000
0.002
-0.008
Husband Word Recall
(0.002)**
(0.001)
(0.001) (0.002)**
-0.004
-0.003
0.009
-0.002
Wife Age
(0.007)
(0.003)
(0.005)*
(0.008)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.001)** (0.001)**
(0.001)
(0.002)
0.002
0.001
-0.002
-0.001
Wife Word Recall
(0.001)
(0.001)
(0.001)*
(0.002)
0.011
0.001
0.004
-0.017
Log(Income)
(0.005)**
(0.002)
(0.003) (0.005)**
-0.007
0.001
0.001
0.005
Log(Non-Financial Wealth)
(0.004)*
(0.002)
(0.003)
(0.005)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and serial
correlation. ** p-value < 0.05, * p-value < 0.10. Reported average partial effect
for Husband and Wife Pr(Live to 75) and Husband and Wife Word Recall
represent a 10 percentage point change.

102

Table 25. Average Partial Effects from Single Equation Pooled Bernoulli QuasiMaximum Likelihood Estimates on Entire Unabalanced Panel, Ignoring
Unbalanced Nature of the Panel
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.008
0.005
Husband Age
(0.007)
(0.003)
(0.005)
(0.008)
0.001
-0.000
-0.001
0.000
Husband Pr(Live to 75)
(0.001)
(0.001)
(0.001)
(0.002)
0.006
-0.000
0.002
-0.008
Husband Word Recall
(0.002)**
(0.001)
(0.001)* (0.002)**
-0.004
-0.002
0.008
-0.002
Wife Age
(0.007)
(0.003)
(0.005)
(0.008)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.002)** (0.001)**
(0.001)
(0.002)
0.002
0.001
-0.002
-0.001
Wife Word Recall
(0.002)
(0.001)
(0.001)
(0.002)
0.011
0.001
0.004
-0.016
Log(Income)
(0.005)**
(0.002)
(0.003)
(0.005)**
-0.007
0.002
0.001
0.005
Log(Non-Financial Wealth)
(0.005)
(0.002)
(0.003)
(0.005)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and serial
correlation. ** p-value < 0.05, * p-value < 0.10. Reported average partial effect for
Husband and Wife Pr(Live to 75) and Husband and Wife Word Recall represent a
10 percentage point change.

103

Table 26. Average Partial Effects from Single Equation Pooled Bernoulli
Quasi-Maximum Likelihood Estimates for Each Balanced Panel Subset,
Averaged Across Balanced Panel Subsets
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.007
0.004
Husband Age
(0.019) (0.006) (0.012)
(0.020)
0.001
-0.000
-0.001
0.001
Husband Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.004
0.000
0.002
-0.006
Husband Word Recall
(0.004) (0.002) (0.003)
(0.005)
-0.003
-0.002
0.009
-0.005
Wife Age
(0.019) (0.006) (0.012)
(0.020)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.001
0.001
-0.002
0.000
Wife Word Recall
(0.004) (0.001) (0.003)
(0.005)
0.012
0.002
0.003
-0.016
Log(Income)
(0.012) (0.004) (0.007)
(0.014)
-0.006
0.002
0.001
0.005
Log(Non-Financial Wealth)
(0.012) (0.004) (0.007)
(0.013)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and
serial correlation. ** p-value < 0.05, * p-value < 0.10. Reported average
partial effect for Husband and Wife Pr(Live to 75) and Husband and Wife
Word Recall represent a 10 percentage point change.

104

Table 27. Average Partial Effects from Pooled Bernoulli Quasi-Maximum
Likelihood Estimates on Entire Unbalanced Panel Adjusting for Unbalanced
Nature of the Panel
Stocks
Bonds
CDs
Checking
0.000
0.015
-0.007
0.005
Husband Age
(0.007)
(0.011)
(0.005)
(0.008)
0.001
-0.000
-0.001
0.000
Husband Pr(Live to 75)
(0.002)
(0.002)
(0.001)
(0.002)
0.006
-0.001
0.001
-0.008
Husband Word Recall
(0.002)**
(0.002)
(0.001)
(0.002)**
-0.004
-0.010
0.006
-0.002
Wife Age
(0.007)
(0.011)
(0.005)
(0.008)
-0.004
0.005
0.0001
0.002
Wife Pr(Live to 75)
(0.002)** (0.002)** (0.001)
(0.002)
0.002
0.003
-0.001
-0.001
Wife Word Recall
(0.002)
(0.002)
(0.001)
(0.002)
0.011
0.008
0.004
-0.016
Log(Income)
(0.005)**
(0.007)
(0.003)
(0.005)**
-0.007
0.007
-0.001
0.005
Log(Non-Financial Wealth)
(0.005)
(0.008)
(0.003)
(0.005)

χ 2 Test Statistic for ψ gr
p-value

1.34
0.931

6.75
0.240

52.16
0.000**

3.80
0.579

1.45
4.74
5.26
2.83
χ 2 Test Statistic for ω gr
0.919
0.448
0.385
0.726
p-value
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and serial
correlation. ** p-value < 0.05, * p-value < 0.10. Reported average partial effect
for Husband and Wife Pr(Live to 75) and Husband and Wife Word Recall
represent a 10 percentage point change.

105

Table 28. Average Partial Effects from Pooled Multinomial Quasi-Maximum
Likelihood Estimates for Entire Unbalanced Panel, Ignoring Unbalanced Nature
of the Panel
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.008
0.005
Husband Age
(0.007)
(0.002)
(0.005)* (0.008)
0.001
-0.000
-0.001
0.001
Husband Pr(Live to 75)
(0.001)
(0.001)
(0.001)
(0.002)
0.006
-0.000
0.002
-0.007
Husband Word Recall
(0.002)**
(0.001)
(0.001)* (0.002)**
-0.004
-0.003
-0.002
0.009
Wife Age
(0.007)
(0.002)
(0.005)* (0.008)
-0.004
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.002)** (0.001)** (0.001)
(0.002)
0.002
0.001
-0.002
-0.001
Wife Word Recall
(0.002)
(0.001)
(0.001)* (0.002)
0.008
0.000
0.002
-0.009
Log(Income)
(0.005)*
(0.002)
(0.003) (0.005)**
-0.008
0.001
0.000
0.007
Log(Non-Financial Wealth)
(0.005)*
(0.002)
(0.003)
(0.005)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and serial
correlation. ** p-value < 0.05, * p-value < 0.10. Reported average partial effect
for Husband and Wife Pr(Live to 75) and Husband and Wife Word Recall
represent a 10 percentage point change.

106

Table 29. Average Partial Effects from Pooled Multinomial QuasiMaximum Likelihood Estimates for Each Balanced Panel Subset,
Averaged Across Balanced Panel Subsets
Stocks
Bonds
CDs
Checking
-0.001
0.004
-0.006
0.003
Husband Age
(0.019) (0.006) (0.012)
(0.020)
0.000
-0.001
-0.001
0.001
Husband Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.004
-0.000
0.002
-0.006
Husband Word Recall
(0.004) (0.002) (0.003)
(0.004)
-0.003
-0.002
0.009
-0.004
Wife Age
(0.019) (0.006) (0.012)
(0.020)
-0.003
0.001
0.000
0.002
Wife Pr(Live to 75)
(0.004) (0.001) (0.002)
(0.004)
0.000
0.001
-0.002
0.001
Wife Word Recall
(0.004) (0.001) (0.003)
(0.004)
0.009
0.001
0.003
-0.013
Log(Income)
(0.012) (0.004) (0.007)
(0.012)
-0.006
0.001
0.000
0.005
Log(Non-Financial Wealth)
(0.012) (0.004) (0.007)
(0.013)
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and
serial correlation. ** p-value < 0.05, * p-value < 0.10. Reported average
partial effect for Husband and Wife Pr(Live to 75) and Husband and Wife
Word Recall represent a 10 percentage point change.

107

Table 30. Average Partial Effects from Multinomial Quasi-Maximum
Likelihood Estimates on Entire Unbalanced Panel, Adjusting for the
Unbalanced Nature of the Panel
Stocks
Bonds
CDs
Checking
-0.001
0.003
-0.006
0.004
Husband Age
(0.007)
(0.003) (0.005)
(0.007)
0.001
-0.000
-0.001
0.001
Husband Pr(Live to 75)
(0.001)
(0.001) (0.001)
(0.002)
0.006
-0.000
0.001
-0.007
Husband Word Recall
(0.002)** (0.001) (0.001) (0.002)**
-0.002
-0.002
0.006
-0.002
Wife Age
(0.007)
(0.003) (0.005)
(0.008)
0.001
0.001
0.002
-0.004
Wife Pr(Live to 75)
(0.002)
(0.002)** (0.001)* (0.001)
0.001
0.001
-0.001
-0.001
Wife Word Recall
(0.002)
(0.001) (0.001)
(0.002)
0.008
0.001
0.003
-0.012
Log(Income)
(0.005)* (0.002) (0.003) (0.005)**
-0.007
0.001
-0.001
0.007
Log(Non-Financial Wealth)
(0.005)
(0.002) (0.003)
(0.005)
2 Test Statistic for ψ
2.13
17.40
69.65
χ
gr
0.830
0.004** 0.000**
p-value
0.80
11.78
6.35
χ 2 Test Statistic for ω gr
0.977
0.038** 0.274
p-value
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and serial
correlation. ** p-value < 0.05, * p-value < 0.10. Reported average partial
effect for Husband and Wife Pr(Live to 75) and Husband and Wife Word
Recall represent a 10 percentage point change.

108

Table 31. Average Partial Effects from Multinomial Quasi-Maximum Likelihood
Estimates Using the Multinomial Logit Functional Form on Entire Unbalanced
Panel, Adjusting for Unbalanced Nature of the Panel
Stocks
Bonds
CDs
Checking
-0.000
0.003
-0.008
0.005
Husband Age
(0.000)
(0.000)**
(0.006)
(0.007)
0.001
-0.000
-0.001
0.000
Husband Pr(Live to75)
(0.000)**
(0.000)**
(0.001)**
(0.001)
0.006
-0.000
0.001
-0.007
Husband Word Recall
(0.000)**
(0.000)**
(0.001)*
(0.001)**
-0.004
-0.002
0.008
-0.001
Wife Age
(0.000)**
(0.000)**
(0.008)
(0.008)
-0.004
0.001
0.001
0.002
Wife Pr(Live to 75)
(0.000)**
(0.000)**
(0.001)
(0.001)**
0.002
0.001
-0.001
-0.001
Wife Word Recall
(0.000)**
(0.000)**
(0.001)**
(0.001)*
0.011
0.002
0.003
-0.016
Log(Income)
(0.000)**
(0.000)**
(0.007)
(0.009)*
-0.008
0.002
-0.000
0.007
Log(Non-Financial Wealth)
(0.000)**
(0.000)**
(0.006)
(0.008)

χ 2 Test Statistic for ψ gr
p-value

2.42
0.789

6.27
0.281

13.91
0.016**

-

1.52
6.75
4.86
χ 2 Test Statistic for ω gr
0.911
0.240
0.434
p-value
Observations
14,514
Households
3,872
Note: Standard errors in parentheses are robust to heteroskedasticity and serial
correlation. ** p-value < 0.05, * p-value < 0.10. Reported average partial effects for
Husband and Wife Pr(Live to 75) and Husband and Wife Word Recall represent a
10 percentage point change.

109

Figure 1. Average Structural Function for the Proportion of Financial Wealth Allocated to
Stocks Against Husband Word Recall

Linear Fixed Effects

0.34

Classical Minimum Distance on Multinomial QMLE for
each year within balanced panel subsets

0.32
0.3
0.28
0.26
0.24
0.22
0

0.1

0.2

0.3

0.4

0.5

0.6

Husband Word Recall

110

0.7

0.8

0.9

1

Figure 2. Average Structural Function for the Proportion of Financial Wealth Allocated to
Bonds Against Husband Word Recall

0.036
0.034
0.032
Linear Fixed Effects
0.03
Classical Minimum Distance on Multinomial
QMLE for each year within balanced panel
subsets

0.028
0.026
0.024
0.022
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Husband Word Recall

111

0.8

0.9

1

Figure 3. Average Structural Function for the Proportion of Financial Wealth Allocated to
CDs Against Husband Word Recall

0.13

0.12
Linear Fixed Effects
0.11
Classical Minimum Distance on Multinomial QMLE
for each year within balanced panel subsets
0.1

0.09

0.08
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Husband Word Recall

112

0.8

0.9

1

Figure 4. Average Structural Function for the Proportion of Financial Wealth Allocated to
Checking Against Husband Word Recall

0.7

Linear Fixed Effects
Classical Minimum Distance on Multinomial
QMLE for each year within balanced panel
subsets

0.65

0.6

0.55

0.5
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Husband Word Recall

113

0.8

0.9

1

Figure 5. Average Structural Function for the Proportion of Financial Wealth Allocated to
Stocks Against Wife Word Recall

0.32

0.3
Linear Fixed Effects
0.28

Classical Minimum Distance on
Multinomial QMLE for each year within
balanced panel subsets

0.26

0.24
0

0.1

0.2

0.3

0.4
0.5
0.6
Wife Word Recall

114

0.7

0.8

0.9

1

Figure 6. Average Structural Function for the Proportion of Financial Wealth Allocated to
Bonds Against Wife Word Recall

0.036

0.032
Linear Fixed Effects
0.028
Classical Minimum Distance on Multinomial
QMLE for each year within balanced panel
subsets

0.024

0.02
0

0.1

0.2

0.3

0.4
0.5
0.6
Wife Word Recall

115

0.7

0.8

0.9

1

Figure 7. Average Structural Function for Proportion of Financial Wealth Allocated to
CDs Against Wife Word Recall

0.125
Linear Fixed Effects
0.115
Classical Minimum Distance
on Multinomial QMLE for
each year within balanced
panel subsets

0.105

0.095

0.085
0

0.1

0.2

0.3

0.4
0.5
0.6
Wife Word Recall

116

0.7

0.8

0.9

1

Figure 8. Average Structural Function for Proportion of Financial Wealth Allocated to
Checking Against Wife Word Recall

0.64
0.62
Linear Fixed Effects

0.6
0.58

Classical Minimum Distance on Multinomial
QMLE for each year within balanced panel
subsets

0.56
0.54
0.52
0

0.1

0.2

0.3

0.4
0.5
0.6
Wife Word Recall

117

0.7

0.8

0.9

1

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118

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