Habits in Consumption-Based Empirical Asset Pricing with Heterogeneous Agents

Habits in Consumption-Based Empirical Asset Pricing with

Heterogeneous Agents

Research Master Thesis – Supervised by L. Dam

Samuël David Nelemans – s2550344

12th August 2019

Abstract

I re-examine whether habit formation improves the performance of consumption-based asset pricing models when fitted to disaggregate consumption data. Using a model which nests both internal and external habit formation, and data from the Consumer Expenditure Survey (CEX), I find that neither type of habit formation yields an improvement in empirical performance for the CCAPM. This finding is robust to various methods for aggregation, return horizons and asset classes. This discrepancy with the literature is likely due to the higher volatility of aggregate consumption in the CEX compared to consumption expenditure reported in the National Income and Product Accounts (NIPA).

1

Introduction

Despite the long list of proposed solutions to the equity premium puzzle proposed in the literature since Mehra and Prescott (1985) formulated it, little research has been conducted to investigate the interaction between these adjustments. Many popular modifications of the CCAPM are neither mutually exclusive in terms of their assumptions nor modelling features or interpretation. As these models often only resolve the equity premium puzzle partially, combining the merits of several approaches may be a fruitful direction for research.

thus allowing the observed relative risk aversion coefficient to vary over time and yielding more realistic parameter estimates. Agent heterogeneity as a priced factor in asset markets stems from the incompleteness of financial markets, with agents being unable to fully insure against all personal income risk (e.g. labour income risk) and thus experiencing greater consumption risk than captured in aggregate consumption time series. As these two specifications interfere in no way, I analyse them jointly in an empirical setting.

In this paper, I fit the CCAPM with a flexible specification of habit formation to disaggregate consumption data from the U.S. Consumer Expenditure Survey (CEX), conducted by the U.S. Bureau of Labor Statistics, and a set of U.S. stock portfolios and treasury bonds. Notably, I find that aggregate consumption when constructed from the CEX already greatly reduces estimation and model performance compared to common results obtained from NIPA data. Furthermore, including information on the cross-sectional volatility of consumption only marginally improves model performance, and no support for any type of habit formation is found in the CEX dataset. This finding is robust to different aggregation methods, return horizons, restrictions on preference parameters and specifications of the habit function.

I consider a broad specification of habit formation, first described by Grishchenko (2010). The merit of this habit function lies in its flexibility, as it nests many classical models for internal or external habit formation as special cases. This allows an investigation of habit formation with heterogeneous agents in the broadest possible context. As both aggregate and individual consumption feature in the habit function, this model specifically lends itself very well for analysing disaggregate consumption data. Grishchenko also points this out in her paper:

Testing the mixture model with heterogeneous agents would require a different data set (such as the Consumption Expenditure Survey (CEX) data set) and could be an interesting research question on its own. The major difficulty with this exercise is the short-term nature of household consumption data in the CEX data set: every household’s consumption enters into the sample for only four quarters, so it is impossible to construct long-run individual habit stock like in my paper. Therefore, some sort of aggregation is needed to perform this empirical task.

of the GMM objective function, in an attempt to find parameter settings which are feasible, realistic and supported by the data. In section 5, I describe the asset pricing implications of several combinations of parameter settings for aggregate and disaggregate data.

2

Related Literature

2.1 Habit Formation

The difference between external and internal habit preferences lies in the effect that current consumption has on future habit. Internal habit, as studied by e.g. Dunn and Singleton (1986) and Constantinides (1990), presupposes that consumers adjust their present-day consumption levels to incorporate the effect on their future personal habit levels. External habit formation, also known as "Catching up with the Joneses" utility1 and studied by Gali (1994) and Campbell and Cochrane (1999), assumes that consumers treat their habits as an exogenous force which they cannot control.

The first formal comparison of internal and external habit formation was Abel (1990), who formulated a model which may accommodate these preferences jointly. In his utility function, habits are a multiplicative factor for consumption level rather than being subtracted from it, which complicates identification of his model parameters based on data. By computing theoretical asset returns, he finds that external habit formation can somewhat demystify the equity premium puzzle, whilst internal habit formation yields very high sensitivity of expected returns to risk aversion.

Nowadays, most asset pricing papers investigating habit formation specify utility as a power function of surplus consumption over the habit level. The most cited paper analysing such a utility function with an internal habit process is Constantinides (1990), whilst external habit is championed by Campbell and Cochrane (1999). Both papers find that including their type of habits into the model greatly reduces or even has the potential to solve the equity premium puzzle.

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More recently, Grishchenko (2010) fits a CCAPM with utility function which nests those of Constantinides (1990) and Campbell and Cochrane (1999). She models an agent’s internal habit formation process as a function of their past private consumption c_t, whilst external habits react to aggregate consumption Ct. This way, both habit types can be analysed jointly within a single

utility function, with a parameter governing their relative importance. She finds that long-term internal habit is the more important determinant of asset prices. Chen and Ludvigson (2009) obtain a similar conclusion from a nonparametric habit specification based on past consumption: they test their model under solely internal and solely external habits and find that the former yields smaller pricing errors.

The aforementioned papers both use aggregate consumption data to fit their asset pricing models. Grishchenko (2010) does so by first deriving the first-order conditions of households taking the effect of own consumption on future habit levels into account. She then imposes the equilibrium condition c_t= C_t and continues to calculate a discount factor time series based on aggregate data. She finds that the data clearly favours internal habit formation, and obtains relatively low values for risk aversion, albeit nor identified very strongly.

Although the CEX has not been employed before to empirically test the CCAPM with habit formation, a few papers have used microdata to analyse habit formation in consumption data. For instance, Dynan (2000) finds no evidence for internal habit formation based on annual data one food expenditures, whilst Ravina (2007) investigates and finds evidence for both internal and external habit formation. These papers are motivated from a macroeconomic rather than an asset pricing perspective, using the real after-tax interest rate or the personal interest/credit rate faced by individuals respectively. Thus, these results are of limited interest in an asset pricing context.

2.2 Heterogeneous Agent Asset Pricing

personal income until all risk has been pooled into aggregate risk, implying that ex-post relative consumption growth is equal for all agents. However, disaggregate consumption data clearly indicates cross-sectional heterogeneity in consumption growth, suggesting that the representative agent framework may not be suitable for consumption-based asset pricing and.

In the past two decades, numerous papers have appeared which empirically investigate the CCAPM based on disaggregate consumption data, with varying results. Initially, Lettau (2002) and Cogley (2002) find evidence against idiosyncratic risks being the missing piece of the equity premium puzzle, the latter using data from the Consumer Expenditure Survey. However, Brav, Constantinides, and Geczy (2002) and Jacobs and Wang (2004) do find that cross-sectional consumption heterogeneity may partially explain the equity premium. A key difference between these papers is that Brav et al. and Jacobs and Wang divide households into cohorts in order to mitigate some of the measurement error in the disaggregate consumption time series, which may be able to explain the discrepancy in empirical results.

Balduzzi and Yao (2007) further develop the idea that measurement error disturbs empirical tests of CCAPM based on household-level consumption data by formulating a new approach to aggregate households into a single discount factor. Whereas for instance Brav et al. (2002) calculate the unweighted average of households’ marginal rate of substitution, Balduzzi and Yao instead compute average marginal utility, and use this time series to compute a discount factor. They show that their approach better mitigates measurement error that persists over time for individual households, and that their aggregation approach improves the empirical performance of the CCAPM.

More recently, Grishchenko and Rossi (2012) analysed the effect of a clustering approach on empirical analysis of the heterogeneous-agent CCAPM based on CEX data. They find that using

k-means clustering to group households based on age and education of the reference person and

household income before taxes can resolve the equity premium puzzle to a great extent, whereas using individual-level data yields very unreliable results.

2.3 Aggregate Consumption Volatility

National Income and Product Accounts (NIPA) are subject to filtration before publishing, in order to mitigate measurement error. Whilst resulting in smooth and stable time series for aggregate consumption, this procedure also drastically lowers the volatility of consumption growth, as well as its correlation with stock returns. Kroencke (2017) has shown that reverse filtration of this time series

Savov (2011) and Da and Yun (2010) aimed to resolve this problem by investigating data on garbage production and electricity consumption respectively. These variables may be considered proxies for aggregate nondurable goods consumption, showing a high correlation with the NIPA data but higher volatility and better asset pricing performance. Similarly, Kolev (2013) uses weekly survey-based consumption data collected by Gallup. Although these data series are all more subject to the measurement error which filtration can mitigate, the estimated risk aversion coefficient produced by the standard CCAPM based on these alternative consumption proxies is much more realistic, reducing the equity premium puzzle. Notably, no research seems to have been conducted estimating the CCAPM from aggregate consumption time series constructed from the CEX.

3

Empirical Strategy

3.1 Model

The analysis of this paper is based on the asset pricing model of Grishchenko (2010). Consumers maximise lifetime utility given by:

V0 = E "∞ X t=0 ρt(cit− xit) 1−γ_{− 1} 1 − γ      I0 # .

In this equation, cit represents consumption of household i at time t. xit denotes a consumer’s

individual habit level given by:

xi,t+1= b J X j=0 (1 − a)j{ωci,t−j+ (1 − ω)Ct−j} + b ∞ X j=J +1 (1 − a)jCt−j,

where C_treflects aggregate consumption per capita in period t. The derivative of lifetime utility with respect to cit is then given by ∂V0_∂cit = Φi,tρt, where

Let R_t,t+k denote a vector of asset returns between time t and t + k, and let 1 denote a vector of ones. The moment conditions of the consumption-based asset pricing model for agent i can then be written as:

E " 1 − ρkΦi,t+k Φi,t Rt,t+k # = 0. (2)

I apply this moment condition2 to estimate the five-dimensional parameter vector (a, b, ω, γ, ρ) using the Generalised Method of Moments (GMM) of Hansen (1982).

As b is a scaling parameter for the habit function as a whole, setting this parameter equal to 0 yields the classical CCAPM model without habits. In this framework, the parameters γ and

ρ have their classical interpretation and can be estimated on aggregate or disaggregate data

alike. Parameters a and b jointly determine the relevance of the habit function to the marginal utility of consumption, where 1 − a denotes the persistence of the habit: larger values for a imply that further consumption lags become relatively less important. Thus, when given a certain magnitude of the habit level, high a and b would imply a quick-moving habit with large marginal effects of present-day consumption on next period utility that rapidly diminish over time, whilst low a and b denote a small but persistent effect of consumption on future utility. This also implies a trade-off between the habit persistence parameters a and the time preference parameter ρ.

Whereas ω denotes the relative importance of internal habit formation as opposed to external,

J denotes the persistence of internal habit formation exclusively. Setting J to a finite level is

interpreted by Grishchenko (2010) as consumers not remembering their full consumption history, and in her specification, she assumes that aggregate consumption history replaces personal consumption history after this point. Whilst a strange assumption, Grishchenko needs it for

ω to be identified in her model after imposing the equilibrium condition cit = Ct. However,

as I don’t need this assumption due to the use of disaggregate data, I can let J go to infinity. Both ω and J play a double role in my analysis: first, they determine to which extent personal consumption history is more important for habit formation than aggregate consumption history. Second, they describe the extent to which consumers take into account the effect of their present consumption patterns on their future utility of consumption through the habit process. An interesting direction for future research would be to disentangle these effects.

2_{I also considered the moment condition on excess returns, for which ρ is not identified, whilst identifying ρ on}

3.2 Consumption Data

Contrarily to Grishchenko (2010), I use the Consumer Expenditure Survey (CEX) produced by the U.S. Bureau of Labor Statistics for consumption data. This survey follows a representative set of U.S. households for four consecutive quarters, reporting in detail on their consumption habits. I use this dataset to create an unbalanced panel of per capita quarterly nondurable goods consumption growth measured monthly between February 1996 and March 2018. Following Grishchenko and Rossi (2012), I define nondurable goods consumption as the sum of spending on: food; alcohol; tobacco and smoking supplies; gasoline and motor oil; utilities, fuels and public services; apparel and services; public and other transportation; household operations; personal care; fees and admissions for entertainment; education; reading; life and other personal insurance; and health care3_{. These variables are split up into two parts in the CEX dataset}

(previous and current quarter), which I add to get full nondurable goods consumption of every household over a period of three months.

I divide nondurable household consumption number by the number of people in the household (“FAM_SIZE”). Using consumption per capita rather than per household corrects for changes in consumption purely due to household composition on consumption patterns, which are hardly a consumption risk as most of such changes (births, children moving out) are anticipated. Furthermore, as consumption measured in the CEX is in nominal terms, I adjust consumption levels by constructing the seasonally unadjusted consumer price index. I obtain this series by compounding the monthly CPI return series provided by the Center for Research in Security (CRSP).

I only consider households for which both food and aggregate nondurables consumption are positive, as other data can be considered unrealistic and thus unreliable. Furthermore, I only consider households where the reference person is between 18 and 75 years and only consider households which were followed for at least three consecutive quarters. Following Brav et al. (2002), Balduzzi and Yao (2007) and Grishchenko (2010), I filter extreme values for consumption

growth out of the sample. Specifically, I omit all observations for which household consumption growth was either below 1₅ or above 5. Furthermore, if consumption growth for a household i in

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some period t satisfies both ci,t

ci,t−1 ≤ 2 and

ci,t+1

ci,t ≤ 2, then this observation is also omitted.

Theoretically, equation 2 should hold for each individual consumer, implying that all model parameters could be estimated using the consumption trajectory of each individual consumer as a separate moment condition. However, this approach is subject to two important caveats. First of all, individual-level consumption data from the CEX is subject to severe measurement error (Attanasio and Weber, 1995; Brav et al., 2002; Jacobs and Wang, 2004; Balduzzi and Yao, 2007; Grishchenko and Rossi, 2012). Second, the CEX follows households for a maximum of 4 consecutive quarters, implying that either the habit level of consumers and the effect of present-day consumption on future marginal utility (which depends on future consumption) cannot be estimated based on multiple periods.

Aggregation of households into cohorts both yields long consumption series and mitigates some of the measurement error in the data. In fact, a brief survey of literature seems to indicate that papers which group households in the CEX tend to be successful in explaining the equity premium at least to some extent (Brav et al., 2002; Jacobs and Wang, 2004; Grishchenko and Rossi, 2012) whilst papers which do not cluster households tend to reject cross-sectional consumption heterogeneity as a driving factor of asset prices Cogley (2002); Lettau (2002). Following Grishchenko and Rossi (2012), I divide consumers into cohorts based on the education level and age of the reference person, and the total household income before taxes (“FINCBTAX”). For education, I distinguish three groups: a high-school diploma; some college experience; and at least a bachelor degree. These conform to the values 10-12, 13-14 and 15-17 of the variable “EDUC_REF” in the CEX. For age and income, I calculate the full-sample median or terciles for every month and use these as cut-off points. Grishchenko and Rossi (2012) recommends 9 cohorts, formed by the intersection of these three education and age cohorts. I also apply the 12-cohort partition, using the intersection of three education cohorts, two age cohorts and two income cohorts. Finally, I create a partition of 10 income groups, also sorted per month.4 For every cohort, I calculate quarterly cohort consumption growth as the weighted average consumption growth of all households in the cohort, where the weights are given by previous quarter consumption levels of each household5. This yields very similar results as adding

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Grishchenko and Rossi (2012) advises using K-means clustering rather than demographic cohorts to group consumers. Whilst this approach works very well in the classical model with CRRA utility, unfortunately, I cannot use such an approach as the definitions of these clusters change every period, implying that the observations for a single cluster are not chain-linked and that thus no internal habit level can be constructed.

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con-consumption levels per period per cohort, and calculating the growth of total cohort con-consumption as in Balduzzi and Yao (2007) and Grishchenko and Rossi (2012). However, as the size and composition of cohorts change every period, such a series becomes very unstable. By first calculating growth rates, then aggregating and converting back to consumption levels, the effect of households entering and leaving cohorts is mitigated. The resulting time series displays stable and realistic consumption growth.

Furthermore, as the utility function with habit formation is sensitive to the relative scaling of individual and aggregate consumption, I need to convert my consumption growth series back to consumption levels and normalise them. Note that as every household is interviewed quarterly, the CEX dataset naturally splits into three subsamples: those interviewed in January, April, July and October yield a separate time series of average quarterly consumption than those interviewed in February or March and the respective following quarters. The same applies to my time series, as I calculate quarterly consumption growth, implying that the consumption series of a given cohort starting in January is unrelated to the consumption series of the same cohort starting in February. Thus, when normalising, I divide constructed consumption levels of each quarterly time series starting in January by the average consumption of that series in January, April, July and October only, and similar for the other two subsamples.

For the sake of internal consistency in terms of what constitutes nondurable consumption, I construct aggregate quarterly nondurables consumption measured monthly from the CEX sample as well, in the same fashion as I retrieve cohort consumption levels. This time series is displayed in Figure 1. The trend is comparable to the one reported by the Federal Reserve of St. Louis (Federal Reserve Bank of St. Louis, 2019), although I observe stagnated consumption growth during the recession of 2008 rather than a decline. However, the volatility of aggregate consumption growth estimated from CEX data is approximately 35 times higher than in the per capita consumption growth series from NIPA constructed as in Grishchenko (2010). The correlation between aggregate consumption growth from the CEX and the NIPA series is 19.1%, whilst the CEX series correlates 14.8% with quarterly returns on the market portfolio, whereas the market and NIPA consumption correlate 33%.

Figure 1: Aggregate Consumption over Time ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 0.6 0.8 1.0 1.2 1.4 1.6 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015 2017 ●●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●●●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● January February March

In Figure 2, I plot the between cohort volatility over time for the different divisions into cohorts that I employ. Conversely, Table 1 reports the weighted standard deviation of consumption growth between households within each cohort. These numbers are similar to those presented by Grishchenko and Rossi (2012). The latter type of cross-sectional volatility is lost in aggregation as a potential source of measurement error, whilst the cross-cohort volatility could be a priced factor in the model, and which forms the basis of the distinction between internal and external habit formation in the model.

Figure 2: Cross-Cohort Standard Deviation of Consumption over Time

Table 1: Moments of Within-Cohort Weighted Standard Deviation over Time

Cohort

Aggregate Age/Educ./Inc. Age/Educ. Income Mean St. Dev. Mean St. Dev. Mean St. Dev. Mean St. Dev. 1 0.778 0.091 0.697 0.117 0.690 0.104 0.813 0.231 2 0.641 0.110 0.656 0.105 0.748 0.206 3 0.636 0.246 0.607 0.209 0.732 0.186 4 0.580 0.107 0.686 0.157 0.689 0.142 5 0.714 0.178 0.643 0.122 0.654 0.104 6 0.601 0.117 0.636 0.155 0.674 0.158 7 0.622 0.152 0.650 0.113 0.656 0.106 8 0.631 0.168 0.730 0.174 0.653 0.121 9 0.663 0.130 0.729 0.164 0.642 0.135 10 0.645 0.137 0.758 0.182 11 0.692 0.182 12 0.734 0.182 3.3 Return Data

I use the data library of Kenneth French (French, 2019) to retrieve time series of monthly returns for the 2X3 stock portfolio sorts on size and book-to-market. I furthermore retrieve return series for the 90-day U.S. treasury bill and for the 5- and 10-year U.S. treasury bonds from CRSP. I also retrieve monthly returns on the U.S. market portfolio from the Fama-French research data. To this series, I add the risk-free rate from the same dataset, because I am interested in gross returns rather than excess returns. I subtract the monthly return on the CPI as reported by CRSP from all return series to obtain real returns. Next, I compute quarterly returns measured monthly for all these assets, to match the structure of my consumption dataset.

3.4 Estimation

marginal rate of substitution and a vector of seven assets, being 6 Fama-French portfolios sorted on size and book-to-market ratio and the 90-day U.S. treasury bill.

Each of these assets yields a single moment condition, yielding 9 restrictions for 5 parameters. By using the identity matrix as a weighting matrix, all assets get the same relative importance in estimation. This yields a discounting factor tailored to price all assets equally well, rather than giving extra weight to moment conditions based on low volatility assets (Cochrane, 1996). The second method, first proposed by Balduzzi and Yao (2007), computes the average marginal utility of consumption over consumers or consumer cohorts and then computes marginal rates of substitution based on these average marginal utilities (BY aggregation). This procedure is known to mitigate some of the persistent measurement error in the CEX data. As the measurement error elimination proposed in this method should already have been mitigated by the formation of consumer cohorts, I include this aggregation method as a robustness test. I use the same 9 assets for moment conditions, once again computing one-step GMM estimates.

Third, I use the marginal rate of substitution of each cohort as an individual set of moment conditions. I only apply this method to cohorts, for only these yield coherent time series for group marginal utility of consumption. Once again, I use the same set of nine assets to generate a total of 81, 90 or 108 moment conditions depending on the cohorts used. Additionally, I fit the set of moment conditions for the cohorts based on a single asset, being the market portfolio. This approach might also help identify the relative importance of household clusters in financial markets based on the relative fit of these moment conditions.

In the empirical set-up of Campbell and Cochrane (1999) and Grishchenko (2010), aggregate consumption should never lie below the habit level, for this would yield infinite marginal utility of consumption. This restriction is generally non-binding, for especially the scaling parameter of the habit function (b in my model) will be low enough such that it will be below observed aggregate consumption levels. However, in my empirical framework, individual consumers or consumer cohorts may experience a drastic decrease in consumption which would yield numerically unstable results due to taking a fractional negative power of a negative number which may dominate the data process.

infeasible. If I am able to retrieve a complete time series for the marginal rate of substitution, I take the number of cohorts which were omitted for each calculation of a marginal rate of substitution. Denoting this quantity #N A_t, I next multiply the respective row of the GMM error matrix by λ  1 + #N At #Cohorts + #N A_t  ,

where #Cohorts is the number of cohorts used in aggregation and λ a hyperparameter governing the scaling of this penalty function. Whilst this procedure should theoretically not be necessary, it forces numerical optimisation algorithms away from parameter values that yield infeasible consumption choices. I only apply this correction method in GMM estimation of the model parameters, setting λ = 0 in the final calculations of the objective function once a local optimum has been reached.

According to Daniel and Marshall (1997), using long-horizon returns rather than short ones improves parameter identification for the CCAPM and the model of Constantinides (1990) with internal habit formation. Grishchenko (2010) also reports an increase in the convexity of the objective function for longer return horizons. For this reason, I estimate the model based on 1-quarter, 1-year and 3-year returns with the corresponding marginal rates of intertemporal substitution.

4

Parameter Estimates

4.1 Risk Aversion and Time Preference Without Habits

I first consider that case without habit formation, achieved by setting b = 0 and using only aggregate consumption. This yields the original setting for which Mehra and Prescott (1985) derived the equity premium puzzle, albeit using an aggregate consumption series constructed from the CEX dataset. Figure 3 shows the GMM objective function for this setting as a function of log(γ) and log_1−ρρ  estimated6 based on moment condition 2 based on a return horizon of 1 quarter, 1 year and 3 years or k = 1, 4, 12 respectively.

Figure 3: GMM Objective Function against ρ and γ: Different Return Horizons

Top: return horizon k = 1. Middle: return horizon k = 4. Bottom: return horizon k = 12

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In these plots, two locally optimal regions can be distinguished. First of all, there is a curved section that passes through the line γ = 1 (log(γ) = 0) which is nearly perpendicular to the scaled ρ-axis for high values of the time preference parameter, and which bends towards very low values of risk aversion around ρ = 0.9882 (log_1−ρρ = 4.331), which represents value minimising the GMM objective function when restricting γ = 1 or log utility. The right side of this curve represents values for which ρ approximately matches the inverse average rate of return on the test assets, whilst gamma is minimised to eradicate the "noise" which consumption volatility causes in this moment condition.

As for the left side of the lower flat area in the objective function, as well as the leftmost local optimum curve around γ = 20.09, 60.48 and 118.44 respectively, ρ barely seems identified. Especially for the leftmost local optimum, this may be an indication that the γ parameter is chosen as to let the volatility of the discount factor match that of stock returns. Furthermore, the objective function is very steep at this point, and the actual optimum range here lies somewhat deeper than visible on the graph. This indicates that when restricting estimation to this section of the objective function, γ is actually identified rather strongly, as opposed to the findings of e.g. Hansen (1982). Similarly, Grishchenko (2010) also finds very weak identification of γ in her model with habits. The larger values of γ based on long-horizon returns also stand in contrast to the findings of Bansal, Kiku, and Yaron (2009).

Nevertheless, the flat regions in these objective curves give rise to the notion that γ and ρ are not jointly identified by GMM based on aggregate consumption in the CEX dataset together with 6 Fama-French stock portfolios and the 90-day T-bill yield. This finding is robust to several return horizons, as well as to a respecification of the objective function using excess returns. A potential cause for this could be the increased volatility of the aggregate consumption series based on the CEX dataset relative to the NIPA consumption series. The choice for return horizon is a trade-off between more realistic estimates of risk aversion for short horizons and slightly higher degrees of identification for long horizons. As these quantitative differences do not fundamentally change the shape of the objective functions, in the remainder I only report results based on a 1-year return horizon.

choose ρ_agg as the minimising parameter setting for ρ of the GMM objective function in the log utility setting, that is, when γ = 1. Next, I define γ_agg as the minimiser of the objective function when restricting ρ = ρ_agg. Thus, I retain the parameter settings (γ, ρ) ∈ {(1, ρ_agg); (γ_agg, ρ_agg)} as optimal for the aggregate model without habits. The red lines in the graphs indicate the location of the first optimum.

Next, I consider the same figures for disaggregate consumption data series. In Figure 4 I report the GMM objective function based on 12 cohorts of consumers sorted by education, age and income, with a 1-year return horizon and b = 0. The top two objective functions are based on a single aggregate discount factor using the methods of Brav et al. (2002) and Balduzzi and Yao (2007) respectively. The bottom graphs show the objective function when each cohort yields its own set of moment conditions, based on the set of 6 stock portfolios and the 90-day T-bill yield or the market return only. Using the different sets of cohorts described in this paper yields nearly exactly the same plots.

Figure 4: GMM Objective Function against ρ and γ: Different Aggregation Methods

Topleft: BCG aggregation. Topright: BY aggregation.

Bottomleft: individual moment conditions. Bottomright: individual moment conditions, one asset

In the remainder, I only use BCG aggregation as it yields a lower estimate for risk aversion and because it better preserves the cross-sectional consumption heterogeneity in the disaggregate data series. This way, I can optimally study the asset pricing information in disaggregate consumption data. I retrieve two parameter sets (γ, ρ) ∈ {(1, ρ_bcg); (γ_bcg, ρbcg)} in similar fashion to the

aggregate case.

4.2 Habit Parameters

Having obtained baseline models without habits, I next plot the GMM objective function against the habit parameters a and b, parametrised as log_1−aa  and log_1−bb  to keep then between 0 and 1. I only plot the lower parts of the objective function, as it contains several missing values for higher values due to the penalty function for consumption levels above habit levels kicking in. I choose J = 4 for all models with internal habit formation. I graph the edge cases of fully external and fully internal habit formation, that is, for ω = 0 and ω = 1, in Figure 5.

From these graphs, it becomes clear that it is optimal to exclude habits from the model. The graphs are nearly monotonous in all models, and the objective function is minimised by setting b as low as possible (low habit scaling) and a as high as possible (low habit persistence). The same graphs are obtained when considering different return horizons, changing the set of cohorts or aggregation functions, and using γ_agg and γ_bcg instead of log utility, or any other intermediate value for γ. However, for higher values of γ, the plot becomes more erratic, and the slope of the objective function relative to a and b sharply increases.

Figure 5: GMM Objective Function against a and b

Top: ω = 0. Bottom: ω = 1. Left: aggregate data. Right: 12 cohorts. All graphs are based on log utility.

NIPA consumption series to the habit CCAPM. Thus, the increased volatility alone may be insufficient to explain the lack of evidence for habit formation in this model, but further research is required to properly establish this finding.

Figure 6: GMM Objective Function against ω 0.0 0.2 0.4 0.6 0.8 1.0 −4 −2 0 2 4 6 8 ω Log GMM Objectiv e Aggregate 12 Cohorts

4.3 Risk Aversion and Time Preference With Habits

Having imposed the weight of the habit function at a = 1₃ and b = .2, I once again try to estimate the risk aversion and time preference parameters using GMM based on aggregate and disaggregate data, for ω = 0 and ω = 1. I also consider a framework in which I find the optimal value of ω for each combination of γ and ρ and use this to calculate the GMM objective function. The respective plots are shown in Figure 7.

These graphs show that under habit formation, the optimal value for γ is very small and ρ is fit to the same level as in Figure 3. This confirms the idea that for a poorly specified marginal utility function, the objective function fits ρ to the inverse average rate of return and lowers γ to minimise the volatility of the "noise" caused by the marginal rate of intertemporal substitution. Furthermore, the model seems to yield much more stable estimates for external habit formation, with the graphs related to internal habit formation showing a large rugged area around the most realistic region for risk aversion. The graphs for optimal ω are most irregular, though this can be explained by the variation in the optimal ω for different values of γ and ρ.

Figure 7: GMM Objective Function against ρ and γ: Model with Habits

5

Asset Pricing Performance

From the previous section, I retrieve 9-model settings based on which an aggregate discount factor can be calculated. I consider only models based on aggregate data, and models with 12 cohorts aggregated through the BCG method. For all aggregate models I set ρ = ρ_agg, and for the models with consumer heterogeneity I use ρ = ρ_bcg. I test these in a setting with and without habits. In the setting without habits, I calculate a discount factor time series based on

γ = 1 and γ = γ_agg for the aggregate model or γ = γ_bcg for the disaggregate model respectively.

In all settings with habits, I choose a = 1₃ and b = 0.2 respectively, and γ = 1. I test the habit models with aggregate and disaggregate consumption using purely external and purely internal habit formation (ω = 0 and ω = 1 respectively). Additionally, for the disaggregate model I consider the mixed model where ω = ω_bcg. For every model, I compute discount factors corresponding to a return horizon of one year, and the persistence of internal habit is set to

J = 4. The differences between the models are summarised in Table 2.

Table 2: Models for Asset Pricing Tests Aggregate data Aggregate data 12 Cohorts

γ = 1 γ = γ_agg γ = 1

No habits No habits No habits 12 Cohorts Aggregate data Aggregate data

γ = γ_bcg γ = 1 γ = 1

No habits External Habit Internal Habit 12 Cohorts 12 Cohorts 12 Cohorts

γ = 1 γ = 1 γ = 1

External Habit Internal Habit ω = ωbcg

between models. As expected, the volatility of the discount factor increases for higher values of

γ and with the inclusion of habits, and when taking into account cross-sectional consumption

heterogeneity.

Figure 8: Discount Factors over Time

2000 2005 2010 2015 0.80 0.85 0.90 0.95 1.00 m 2000 2005 2010 2015 −1 0 1 2 3 4 m 2000 2005 2010 2015 0.85 0.90 0.95 1.00 m 2000 2005 2010 2015 0.5 1.0 1.5 2.0 2.5 m 2000 2005 2010 2015 0.8 1.0 1.2 1.4 1.6 1.8 m 2000 2005 2010 2015 0.5 1.0 1.5 2.0 2.5 3.0 m 2000 2005 2010 2015 0.8 1.0 1.2 1.4 1.6 1.8 m 2000 2005 2010 2015 0 2 4 6 m 2000 2005 2010 2015 0 1 2 3 4 m

and aggregate consumption data the returns somewhat approach the line. Once again, habit formation seems unable to yield an improvement in model performance under these parameter settings.

In Figure 9, I also report the GRS test statistic (Gibbons, Ross, and Shanken, 1989) for every model and the corresponding p-value. All models are rejected by the GRS test except for those with log utility and without habit formation, although this is likely caused by the low volatility of the discount factor for these models rather than indicating a better model fit.

Figure 9: Asset Pricing Performance of Discount Factors

●● ●●●●●●●● ● ●● ●●●●●● ● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 1.017 p−val = 0.448 ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 252.414 p−val = 0 ●● ●●●●●●●● ● ●● ●●●●●● ● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 0.852 p−val = 0.683 ● _● ● ●● ● _● ● ● ● ● _● ● ●●● ● ●●● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 37.612 p−val = 0 ●● ●●●●●●●● ● ●● ●●●●●● ● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 15.436 p−val = 0 ● ● ● ●● ● ●_●●● ● _● ● ●●● ● ●●● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 57.347 p−val = 0 ●● ●●●●●●●● ● ●● ●●●●●● ● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 13.777 p−val = 0 ● ● ● ●● ● ●_● ● ● ● ● ● ●●● ● ●●● ● ●● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

n _● ● Stocks Bonds GRS = 118.361 p−val = 0 ● ● ● ●● ● ●●●● ● _● ● ●●● ● ●●● ● ● ● ● ● ● ● ● 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

Realised Expected Return

Predicted Expected Retur

6

Conclusion

Based on CEX data, the CCAPM performs rather poorly, and including habits, whether internal or external, only decreases its performance. The incorporation of cross-sectional as a potential source of consumption risk only marginally affects asset pricing performance. This finding is robust to different aggregation methods, return horizons and calibrations for preference parameters.

This finding is rather out of line with the literature and may be a feature of the CEX dataset. As no paper before has used the CEX to retrieve aggregate consumption series for cross-sectional asset pricing, this is clearly a gap in the literature which requires more extensive investigation. Specifically, it would be interesting to see how a time-series for aggregate consumption similar to the NIPA series could be constructed from the CEX dataset, and what the asset pricing power of such a time series would be.

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