Risk Aversion and the Elasticity of Intertemporal Substitution among Australian Households*

3.1 Risky Asset Share Regressions

The risky asset share regressions rely on HILDA's wealth module, which is included in only four survey years: 2002, 2006, 2010 and 2014. The sample is based on an unbalanced panel of these four years consisting of all responding household heads between ages 23 and 59 inclusive.

In the base model, risky assets are defined narrowly as equities, to allow comparability with the two existing Australian studies, although we consider broader definitions of risky assets in extended models. The regressions only include households with positive risky assets (however defined), and those with information on household size and age of household head.15 The mean value of the risky asset share is 16 per cent, based on the narrow definition of risky assets. The instrumental variable (IV) regressions also require non-missing data on the instruments, that is, household disposable income and household consumption (which we again proxy using expenditure on groceries and meals out).

3.2 Consumption Euler Equations

Again, our Euler equation analysis uses grocery and meals out expenditure as a proxy for overall household consumption, an approach also adopted in most overseas studies.16 The panel encompasses the period 2001–16 inclusive, except for 2002 when information on expenditure was not collected in HILDA.

Our sample selection choices closely follow Alan et al. (2009) to aid comparability with that study. We restrict the sample to households consisting of couples who were in a stable relationship over the period in which they feature in the survey. To restrict the sample to working-age households, we exclude households with heads aged under 22 in the first year of the survey and households with heads aged over 60 in the last year of the survey (2016). Also, because the Euler equation only holds for an interior solution, we exclude households who are liquidity constrained, which we define as having zero financial assets in any of the years they feature in the panel. In other words, if they ever report having zero financial assets in one of the wealth module years, they are dropped from all years of the analysis. The latter restriction removes only 36 observations. The real interest rate series is calculated as the annual average nominal 3-month bank accepted bill rate less inflation expectations pertaining to that period.17

4.1 Results

As suspected, the estimated relationship between risky asset share and wealth is sensitive both to whether we use IVs to adjust for the impact of measurement error and to whether the model is estimated in levels or FDs (Table 1). The basic ordinary least squares (OLS) regression in levels (column 1) shows a strong and highly significant negative relationship ( ) between risky asset share and wealth. However, once we adjust for the impact of measurement error by instrumenting for wealth, the coefficient on wealth falls in magnitude and becomes insignificant (column 2). As we argued above, there are good reasons to think that measurement error would contribute to a negative cross-sectional relationship even when none truly exists when the mean risky asset share is <50 per cent.19 Column 3 re-estimates the basic model in FDs. The coefficient on wealth is again negative, but is very imprecisely estimated and not significantly different from zero at standard confidence levels.

Columns 4 and 5 experiment with expanded definitions of risky assets. Column 4 adds cash investments (bonds and the like) and trusts to the measure of risky assets, while the regression in column 5 additionally includes business equity and superannuation. In the latter regression, the denominator is also adjusted to include business equity and superannuation. The coefficient on wealth remains insignificant, although it continues to be negative.

Given the difference between the OLS and IV estimates, we explore whether the results are robust to alternative instruments for wealth. One obvious alternative instrument for change in wealth is the cumulative difference between disposable income and consumption in the intervening periods between observations on wealth. In other words, this utilises income and consumption data in years that did not contain a wealth module in HILDA. For example, for 2006 we subtract cumulative food consumption from cumulative disposable income for each household over the period 2002–6, which yields a `flow' measure of change in wealth. When we use this variable as the instrument in models 3–5, the key results are unaffected.

We also conduct a weak instruments test, and find that our chosen instruments are well correlated with financial wealth and change in financial wealth.20 A test of overidentifying restrictions fails to reject the null hypothesis that the chosen instruments are exogenous for models 3–5, but does reject the null for model 2.21 We also try including labour force participation as a component of , and use its lag as an instrument. This addition does not affect the main results in Table 1.

4.2 Comparison with Tsigos and Daly's Results

In finding no significant relationship between risky asset share and wealth, our results confirm the results from Tsigos and Daly 's sensitivity analysis but contradict their main result. However, their main result (that relative risk aversion falls with wealth) is based on a different measure of risk that uses portfolio weights calculated using external information on asset returns. Tsigos and Daly do not attempt to explain why the two approaches yield different results in their paper. It is also worth noting that, compared with Tsigos and Daly, our test of the relationship between risk aversion and wealth benefits from an extra year of wealth data, giving our test additional power.

While we are unable to reject the CRRA assumption, the estimated coefficient on wealth is negative in all six specifications, albeit insignificant in four cases. We therefore view our results as providing tentative support for the CRRA, particularly given that the insignificant estimates only arise in the IV regressions. Additional years of HILDA data going forward will help us to work out whether risk aversion really is constant with respect to wealth or not.

4.3 Distribution of Implied Relative Risk Aversion Parameter

Equation (2) can be used to calculate the distribution of across the population. However, as Chiappori and Paiella (2011) note, this distribution is only identified up to a scale factor, given by the ratio of the excess return to the variance of the risky portfolio, . Taking the case where this ratio is equal to , as in Chiappori and Paiella (2011), the value of is just the inverse of the risky asset ratio for investor h. Clearly, the derived value of will depend inversely on the definition of ‘risky asset', and will be higher or lower for narrower and broader definitions, respectively. Additionally, very small values of will result in very large values of . Following Chiappori and Paiella, we therefore truncate the distribution of risky asset shares to exclude values <6 per cent and compute the median rather than mean value. The median estimates of are 1.1, 2.2 and 2.4 across the three definitions of risky assets used in our regressions. We also find little evidence that varies by wealth. Table B1 in Appendix II contains further details.

4.4 Correlation with Subjective Data on Risk Aversion

An interesting question is whether the risk aversion measure derived from risky asset shares correlates with subjective measures of risk aversion contained in the HILDA Survey. In particular, the two separate survey questions ask respondents to indicate the degree of financial risk they are prepared to take.22 The risky asset share is significantly correlated with both subjective measures, with a correlation coefficient of for one subjective measure (which measures risk aversion) and . for the second measure (which measures risk appetite). Both are statistically significant.

5.1 Exact Nonlinear Equation Versus the Linearised Version

Previous studies have typically proceeded in one of two ways: estimating the exact, nonlinear Euler equation as it appears in Equation (5) using GMM; or taking a first- or higher-order linear approximation of the Euler equation and using OLS or linear IV estimation. Both approaches have drawbacks.

To begin with, both approaches require instruments for the model's endogenous variables: consumption growth and the interest rate. In the nonlinear case, economic theory implies that any variables in the consumer's information set at time are valid instruments, including lags of consumption growth and the interest rate. However, Carroll (2001a) and Alan et al. (2009)) show that estimates based on the exact nonlinear equation are biased when the consumption data contain measurement error, which is almost certainly the case in practice. These studies suggest that estimates of will be particularly biased in the presence of measurement error, and this bias does not improve with a longer panel length Attanasio and Low (2004);Alan et al. (2009)).

The linearised approach overcomes the problem with measurement error,23 but creates its own problems. To see this, first note that the first-order approximation is given by.

(7)

where includes higher-order terms remaining from the approximation process.

The problem is that the composite error term is now likely to be correlated with consumption growth and interest rates via the higher-order terms in . Carroll (2001a) shows that these high-order terms are inherently endogenous in a standard model of life-cycle consumption behaviour with wage uncertainty, and that estimates of will be severely biased. A further disadvantage of using Equation (7) is that can no longer be recovered as it is subsumed into the constant along with the mean values of the measurement error and approximation errors.

Various solutions have been proposed to overcome the difficulties with each of these approaches. In the nonlinear case, Ventura (1994), Chioda (2004) and Alan et al. (2009) show that explicitly taking account of measurement error in consumption can yield relatively accurate estimates of the preference parameters and even in panels of only moderate length (that is, around 15 years of data). Each of these studies essentially proceeds by assuming that true consumption ( ) is subject to independent and identically log-normally distributed measurement error with equal variance across households, such that . The modified Euler equation is

(8)

While this single moment equation does not allow and to be separately identified, Alan et al. (2009) show that an analogous moment condition for two-period-apart consumption can be used to identify (and therefore ).

Alternatively, Alan et al. (2018) show that relatively precise estimates of can be obtained using the linearised model if a long enough time series on consumption is available. However, their paper suggests that `long enough' equates to perhaps 40 years of data, a requirement that few if any panel surveys satisfy. As a practical alternative, they show that a synthetic panel formed from repeated cross-sectional surveys over a long enough period can deliver reliable estimates of using the linearised Euler equation.

5.2 Our Approach

With (at most) 15 years of available consumption data in the HILDA Survey, we opt for GMM on the exact nonlinear Euler equation, allowing for measurement error in our preferred specifications. For reference, we also obtain an estimate of from a linearised version of the model, noting the caveats above in relation to biased estimates in small panels such as ours.

In dealing with measurement error, we consider two approaches. As we use a single moment equation, neither approach allows us to separately identify the discount rate and the variance of the measurement error without further moment conditions or an `external' estimate of .24 In the first approach, we estimate a single parameter, which is some unknown combination of and . In the second approach, we use separate estimates of the variance of the measurement error , which enables us to recover an estimate of . This analysis is reported in Appendix IV.

The vector can in principle consist of any variables that may affect the marginal utility of consumption, including endogenous variables such as labour supply choices (Attanasio & Low, 2004). While we experiment with other variables, our final specification for only includes change in household size as in Alan et al. (2009). Changes in household size should have a large and direct effect on marginal utility, and this has been confirmed empirically (Attanasio & Low, 2004. We also include a 2008 year dummy in some specifications. This dummy would capture a possible structural break in the relationship between consumption growth and interest rates that may have occurred with the onset of the global financial crisis (GFC).

5.3 Empirical Model

Taking all of the above gives the following moment conditions for each period (where the household superscript has been suppressed) whose sample counterparts are used in GMM estimation:

(9)

where is the set of instruments at time , as described in the next section.

As noted above, we also estimate a standard linearised Euler equation:

(10)

where now includes as well as the means of the higher-order approximation errors, and now includes the household's expectational error, the measurement error and the time-varying components of the approximation error.

5.4 Choice of Instruments

We follow many previous studies in using lags of the interest rate (in addition to a constant) and consumption growth as instruments in our nonlinear GMM estimation. There are a number of reasons why these are likely to be good instruments. First, rational expectations suggests that lagged variables, which are known to the household at time , will be uncorrelated with their forecast error the following period. Second, consumption and interest rates tend to be adequately correlated with lags of themselves, avoiding the problem of `weak instruments'. Finally, simulation exercises have shown that they are valid in a standard life-cycle model environment (Alan et al., 2009). We also use change in household size as an instrument for itself, and do likewise with the 2008 year dummy where present. We also experiment with dropping lagged consumption growth from our instrument list (see column 4 in Table D1). This leaves the lagged interest rate and a constant to identify the two parameters of interest, and , which is essentially the just-identified model applied in (Alan et al., 2009). Because the linearised model is a log transformation of the true model, we substitute log versions of the instruments used in the nonlinear equation.25

All estimates of the nonlinear model are based on a standard two-step GMM with robust weight matrix. Estimation based on an alternative weight matrix, such as iterative GMM, had very little impact on the results. The linearised model is also estimated using (linear) GMM.

5.5 Allowing for Non-separable Consumption and Leisure

The Euler equations (5) and (10) were derived on the assumption of separability between consumption and leisure. However, the few international studies that have formally tested separability have generally found against this assumption.26 As a sensitivity test, we estimate a modified Euler equation derived from non-separable Cobb–Douglas preferences where is leisure. Here, the coefficient of relative risk aversion is a combination of both utility parameters and is given by . The resulting Euler equation is

(11)

5.6 Results

The GMM point estimates of range from 1.19 to 1.39 across the four separable specifications tested; see Table 2. This equates to an estimated EIS of between 0.74 and 0.84. In the context of Euler equation-based studies, the GMM estimates are relatively precise, with standard errors between around 0.17 and 0.24.27 The estimate for of 2.38 from the linearised Euler equation is quite a bit higher than the GMM estimates, but is very imprecise. This specific finding is very similar to that in (Alan et al., 2009) using PSID data. Their simulation exercise in the same study highlights that estimates of based on linearised models are prone to severe biases in cases where the time dimension is <3s 0 years or so (see also Attanasio and Low (2004)).

Table 2. Euler Equation Estimates

	(1)	(2)	(3)	(4)	(5)	(6)
	Log. Lin.	EGMM	EGMM	EGMM	EGMM	EGMMn
Coeff. of relative risk aversion ( )	2.381 (1.097)	1.368 (0.166)	1.289 (0.202)	1.392 (0.236)	1.186 (0.199)	2.251 (0.73)
Discount factor (β)				0.92 (0.062)	0.961 (0.023)
β/Meas'nt error in consumption†		0.759 (0.065)	0.804 (0.082)			0.737 (0.228)
Household size ( )	0.529 (0.246)	0.362 (0.07)	0.334 (0.077)	0.381 (0.094)	0.298 (0.069)	1.455 (1.112)
2008 dummy	−0.075 (0.011)		−0.070 (0.051)
Constant	0.007 (0.003)
Inflation measure for real interest rate instruments‡	Survey	Survey	Survey	Survey	Bond M.	Survey
Lagged interest rate	Yes	Yes	Yes	Yes	Yes	Yes
Lagged consumption growth	No	Yes	Yes	No	Yes	Yes

• EGMM = exact GMM; EGMMn = exact GMM with non-separable preferences.
• Standard errors in parentheses.
• ^† Measurement error variance set to 80 per cent of variance in consumption growth, consistent with our findings in Appendix IV.
• ^‡ The set of instruments also includes a constant in all equations, as well change in HH size and the GFC dummy where present for columns (1) to (5). Lagged changes in household leisure time, and age and age-squared of the household head are also included for the model in column (6).

As noted above, and are not separately identifiable without further assumptions (or additional moment conditions). In models 4 and 5, we assume that measurement error accounts for 80 per cent of the overall variance in consumption growth, a proportion guided by our separate estimates in Table D1. Given the variance in the Australian consumption data, this implies a measurement error variance of around 0.095 and yields estimates for between 0.92 and 0.96 across two specifications.

The coefficient on change in household size ranges from 0.3 to 0.4 for the exact GMM estimation based on separable preferences and is statistically significant across all four specifications. This suggests that family size has an important effect on consumption growth, a standard finding in the life-cycle consumption literature (see Attanasio (1999)).

In the non-separable model, we estimate a coefficient of relative risk aversion of 2.25 (column 6). While somewhat higher than the separable model estimates, the much larger standard error indicates no statistically significant difference from the separable estimates at standard confidence levels.

We run a series of tests to check the sensitivity of our results to sample selection. We find that the exclusion of consumption growth outliers has very little impact on the results, as does introducing a requirement that a respondent appears in at least five consecutive waves (as per Alan et al., 2009)). However, we find that including households whose head has begun or ended a marriage or de facto relationship during their time in the panel leads to implausibly low estimates of (well below 1) and instability in the GMM estimation procedure. In column 5, we experiment with an alternative real interest rate series, which is calculated using the inflation rate derived from market pricing of inflation-linked bonds. This leads to an only slightly lower estimate of .

Food consumption measurement was changed in HILDA from 2006 (see footnote 3). Therefore, as an additional robustness check we re-estimated Table 1 using data from 2006 onwards. A major problem with this is that the shorter time period makes estimation more difficult. Estimates for the coefficient of relative risk aversion ( ) in columns 2–5 are slightly larger but statistically identical: 1.47, 1.45, 1.39 and 1.29. The estimation algorithm failed to converge for column 6, demonstrating the difficulty of estimating with a shorter panel. The estimate for column 1 was somewhat larger, at 2.77, but would be unreliable for the reasons already discussed.

We removed households that were liquidity constrained on the basis of having no financial assets as described in Section 3 above. Another option would be to remove households who report difficulty in raising funds for an emergency. We use a question from HILDA about how difficult it would be to raise $2,000 ($3,000 after 2008) in an emergency. We drop households who indicate that they ‘couldn’t raise the funds’ or that it would require ‘something drastic’. This definition of `liquidity constrained' results in our dropping 4,831 households, many more than what we presented above in our preferred approach. However, our results are not materially affected. The estimates for columns 2–6 of Table 1 are 1.36, 1.27, 1.37, 1.16 and 2.03, none of which are statistically different than what is reported in our main estimation.

5.7 Comparison with International Estimates

Overall, our point estimates of for Australia based on the nonlinear Euler equation are within the range of recent estimates obtained in the USA.28 In the most comparable US study, Alan et al. (2009) shows GMM estimates of ranging from 1.15 to 1.53 using data from the PSID, with a preferred point estimate of 1.45.29 Our estimates of are also within the range of estimates found in Gourinchas and Parker (2002), who use the quite different approach of simulated method of moments. However, as noted earlier, these moderate estimates of contrast with the very high values needed to explain the large equity premium in the USA (Kocherlakota, 1996).

5.8 Caveats

We use the combination of food purchased at home, non-food grocery items and meals eaten out as a proxy for overall consumption. We then calculate the EIS with respect to food expenditure. This will only be an unbiased estimate of the overall EIS if within-period preferences are homothetic Browning and Crossley (2000).

As a necessity, food consumption has an income elasticity below , violating the homotheticity requirement. We have tried to mitigate this somewhat through the combination of food purchased for consumption at home (which definitely has an income elasticity below ) with non-food grocery items purchased at grocery stores and meals eaten out which are usually found to have an income elasticity greater than .

We ran a few simple regressions in our data and found an income elasticity of 0.72 for our combined measure. So, although it is less than unity, it is much closer to unity than standard income elasticity estimates for food consumption purchased to be eaten at home.

When we looked at the relationship between wealth and the share of risky assets, we found some heterogeneity in the distribution of across the population. This does pose some challenge for our GMM estimation of the Euler equation, where is assumed to be common across households. While pooled estimates are often conducted in economics across sub-populations with different parameter values, it is will known (going back at least DuMouchel and Duncan (1983)) that such pooled regressions do not produce population- or sample-weighted averages of parameters.

Alan et al. (2012) look at the performance of linearised Euler equation estimates when takes two different values across households. They find that the estimated value of is roughly the mean of the two values. Theirs is a very stylised situation.

All that we can conclude is that potential heterogeneity in is an additional source of uncertainty for our estimates.

This also raises some questions about the instruments used in our risky assets regression. Looking back at Equation (2), if the returns process is common across households, then heterogeneity in shares must be driven by heterogeneity in . That heterogeneity will end up in the error term in Equation (3), but this invalidates the instrument in the levels equation since the growth rate of consumption and the error term will both depend upon . So despite the result from the overidentification test for column 3 of Table 1, we prefer the first-difference IV estimators of columns 4 and 5 where this influence of will be differenced away.