Measuring the Wealth Elasticity of Risky Asset Demand ... · 1 is the wealth elasticity of risky assets demand. Increasing ( 1 1)

Measuring the Wealth Elasticity of RiskyAssets Demand:

Evidence from the Wealth and AssetsSurvey

Christian Bontemps, Toulouse School of Economics,

Thierry Magnac, Toulouse School of Economics,

and

David Pacini, University of Bristol

Motivation

If household financial wealth increases by 1%,How much is going to change the household demand for

risky assets?

I Answering this question poses many challenges.1

I There is one which has received little attention:

Survey data on financial wealth and risky assets holdings areoften interval-censored.

I What is interval-censoring?

1Heterogeneity, participation and adjustment costs, indivisibilities,measurement error,...

Objective and Challenges

We aim to explore the effects of interval-censoring inestimates of the financial wealth elasticity of risky assetsdemand using data from the Wealth and Assets Survey

(WAS).

I Why should we care about interval-censoring in this context?

I How can we deal with the detrimental effects ofinterval-censoring?

Content

I Illustration using Wealth and Assets Survey.

I Imputation works under ”specific” assumptions.

I Interval-censoring is a type of measurement error (better thannon-response).

I Instrumental variable approach is not feasible.

I Set-identification approach seems to be the natural way toproceed.

(Some) Related Literature

I Risky Assets Demand:Uhler and Cragg (1971); Friend and Blume (1975); Siegel andHoban (1982); King and Leape (1998); Perraudin andSorensen (2000); Brunnermeir and Nagel (2008); Chiapporiand Paiella (2011); Calvet and Sodini (2013).

I Interval-Censoring:Stewart (1983); Manski and Tamer (2002); Bontemps,Magnac and Maurin (2011); Beresteanu, Molchanov andMolinari (2011).

Organization

I Research Question and the Econometric Model.

I Data and Problem.

I Identification Analysis.

I Summary, Conclusion and Extensions.

1. Econometric Model

Research Question

What is the wealth elasticity of risky assets demand in theUK?

I Why? Policy advice may depend on this elasticity.2

I No evidence for the UK.3

I What do we need?

I A model of household portfolio decisions.

I Data on wealth and risky assets holdings: Wealth and AssetsSurvey.

2Calvet and Sodini (2013)3Evidence from other countries is difficult to extrapolate to UK.

Approach(es)

I Recover the parameter β1 in:

ln(rit) = β1ln(wit) + uit

from data on risky asset holdings rit and financial assetholdings wit .

I Two possible justifications:

I Simply assume the log-log specification.

I Derive from primitive assumptions.

I We opt for the second approach. Why?

Econometric Model

I Log-log specification: ln(rit) = β1ln(wit) + uit

I Two-way disturbance term: uit = ηi + δt + vit .

I The idiosyncratic term vit is predetermined:

E (∆vitzit) = 0

where zit := ln(wit), ln(wit−1), ... as opposed to strictlyexogenous.4

I β1 is the wealth elasticity of risky assets demand. Increasing(β1 < 1), constant (β1 = 1), or decreasing (β1 > 1) relativerisk aversion.

4Disturbance vit can affect future log financial wealth ln(wit+1). In this case,OLS and FE estimators have undesirable theoretical properties.

2. Data and Problem

Wealth and Assets Survey

I We use data on risky assets holdings and financial assetsholdings from the Wealth and Assets Survey (WAS). Why?

I Risky assets holdings rit include stock- and mutual fund-likeassets.

I Financial assets holdings wit include risky assets holdings andcash- and saving account-like assets.

I Existing literature (for countries other than UK) tends tofavor the hypothesis of constant or decreasing relative riskaversion (β1 ≥ 1).5

5Brunnermeir and Nagel (08) constant for the US; Chiappori and Paiella(’11) constant for Italy; Calvet and Sodini (’13) decreasing for Sweden.

Imputed Case Analysis

Table 1. OLS, FE and IV Estimates

Imputed DataOLS-IM FE-IM FE-IM FE-IM IV-IM

(1) (2) (3) (4) (5)

β1 1.047 1.005 .978 1.048 .658(.008) (.030) (.036) (.036) (.097)

Demographics Yes Yes Yes Yes YesTime Dummies Yes Yes Yes Yes YesOnly Large Changes No No Yes No NoMills Ratio No No No Yes NoConstant Yes No No No NoObservations 15,228 4,139 825 2,564 4,139R2 .61 .34 .55 .35 .01F - First Stage - - - - 1,901

Interval-Censoring

I Results based on imputed data suggests an elasticity between.4 and .8 (meaning increasing relative risk aversion). Differentfrom evidence for other countries.

I To avoid non-response, households in the WAS may report aninterval rather than an exact amount for asset holdings.6

I Does interval-censoring affect the previous preliminaryconclusion?

6Similar strategy used in other surveys. e.g., HRS, PSID, SWIH and HFCS.

Interval-Censoring in WAS

Table 2. Number and Proportion of Interval-Censored Observations

Wave 1 Wave 2log -risky log-wealth log-risky log-wealth

Numeric 7,386 6,304 5,452 4,857Censored 3,281 (30%) 4,363 (41%) 2,316 (30%) 2,911 (37%)Total Obs. 10,667 10,667 7,768 7,768

Complete Case Analysis

Table 3. OLS, FE and IV Estimates

Uncensored DataOLS-CC FE-CC FE-CC FE-CC IV-CC

(6) (7) (8) (9) (10)

β1 1.017 .975 .943 .981 .377(.011) (.054) (.069) (.054) (.123)

Demographics Yes Yes Yes Yes YesTime Dummies Yes Yes Yes Yes YesOnly Large Changes No No Yes No NoMills Ratio No No No Yes NoConstant Yes No No No NoObservations 9,095 1,715 286 1,690 1,715R2 .61 .31 .51 .31 .01F - First Stage - - - - 350

Problem

I Complete data indicate an elasticity between .1 and .6(meaning increasing relative risk aversion).

I Differences between imputed and complete case analysissuggest paying more attention to interval-censoring.

I What is the explanation for these differences?Interval-censoring is not random.

Should We ...

I Ignore interval-censoring?We need more assumptions, which do not seem sensible in ourcontext.

I Impute values within the interval?Again we need more assumptions, which do not seem sensiblein our context.

I Use an instrument?Actually we cannot.

What We Could Do Is ...

I Interval measurements yield interval estimates.

Investigate the identified set delivered by the EconometricModel under interval-censoring.

I Two imprecise could make one precise.

Ask another measurement for risky and financial assets?7

7As in the Enquete des Patrimoine.

3. Identification Analysis

Set Identification

Does the Econometric Model impose enough restrictions to’recover’ the elasticity of interest β1 from interval-censored data?

I It depends on what we understand by ’recover’.

I If ’recover’ means point-identification, the answer is no.

I If ’recover’ means set-identification, the answer is yes.

The Identified Set

I Characterize all the values of β1 compatible with the intervalsobserved in the data (the identified set).

I We are not aware of techniques accomplishing this task.

I We have a characterization of the identified set based on theconcept of support function.

I We are in the process of implementing an estimator for theidentified set.

I Conceptual and computational challenges.

4. Summary, Conclusion and Extensions

Summary

I We aim to estimate the wealth elasticity of household riskyasset demand from the WAS.

I Imputed and complete case analysis deliver different results.

I We are in the process of exploring alternative solutions to dealwith interval-censoring.

Conclusion

I Economic theory supplies identifying assumptions.

I We cannot ignore interval-censoring.

I The wealth elasticity of risky asset demand may be one.

Extensions

I Implementing the estimator of the identified set.

I International comparisons.

I Length of intervals?

Additional Slides

Expected Utility Model

I Primitive Assumptions:I Individuals choose assets holdings by maximizing utility subject

to a budget restrictionI Utility Function: Risk-free asset and risky asset entering a

HARA function.I Risky Asset Price: evolves in time according to an stochastic

process.I Budget Restriction: Wealth evolves according to an stochastic

process.

I The demand function is linear in wealth.

I Decomposition of the disturbance term uit : uit = ηi + δt + vitI By-product: test for CRRA assumption. Why?

Can We Ignore Interval-Censoring?

I Complete case analysis is valid when interval-censoring israndom.

I Using only complete cases involves a loss in precision ofestimators.

I If interval censoring is not random, interval-censoring rendersestimators based on complete cases inconsistent and tests donot control for size.

Can We Solve the Problem by Imputing?

I Imputation is valid when interval-censoring is random.

I This assumption may be too restrictive: Imputed andcomplete case estimates should be similar

I If interval-censoring is not random, interval-censoring rendersestimators based on imputed data inconsistent, tests do notcontrol for size, and confidence intervals have confidence leveldifferent from the one advertised.

Can We solve the Problem by Using anInstrument?

I Interval-censoring is a measurement error problem.

I Common strategy to deal with measurement error is to findan instrument

I Instruments are infeasible when measurement error comesfrom interval censoring (if the ”instrument” is correlated withthe covariate then so is with the disturbance by construction)

y = x?β + u?

x = dx? + (1− d)xm

E (zu) = 0 and E (zx) 6= 0