Testing the Optimality of Consumption Decisions of the Representative Household: Evidence from Brazil

This paper investigates whether there is a fraction of consumers that do not behave as fully forward-looking optimal consumers in the Brazilian economy. The generalized method of moments technique was applied to nonlinear Euler equations of the consumption-based capital assets model contemplating utility functions with time separability and non-separability. The results show that when the household utility function was modeled as constant relative risk aversion, external habits and Kreps–Porteus, estimates of the fraction of rule-of-thumb households was, respectively, 89%, 78% and 22%. According to this, a portion of disposable income goes to households who consume their current incomes in violation of the permanent income hypothesis.


Introduction
The permanent income hypothesis (PIH), described by Friedman (1957), states that transitory changes in income have little e¤ect on consumer spending, while permanent income is responsible for most of the variation in consumption. In his seminal work, Hall (1978) founded a new approach to study aggregate consumption. By using Euler equations for the optimal choice of a representative consumer, he showed that consumption should follow a random walk and argued that this holds in empirical applications, for instance that postwar U.S. data are consistent with this implication. In contrast, Flavin (1981), using a rational expectations structure, argued that consumption is sensitive to current income and it is greater than that predicted by the permanent income hypothesis. This conclusion has been widely interpreted as evidence of the existence of liquidity constraint. Empirical evidence shows that liquidity constraint is one of the main reasons why it is di¢cult to observe consumption smoothing in the data. Based on this evidence, Campbell andMankiw (1989, 1990) suggested that aggregated data on consumption would be better characterized if there were two types of consumers. They nested the PIH in a more general model in which a proportion of consumers follow the rule of thumb, 1 consuming their current income (myopic spenders), while the remaining (1 ) individuals consume optimally (optimizing savers). Using log-linearization of the model and instrumental variable estimates, they established by empirical application that there was a strong violation of the permanent income hypothesis because a signi…cant fraction of the households have suboptimal behavior. Cushing (1992) and Weber (2002) used intertemporally non-separable utility functions to study the behavior of American consumers. Cushing used a quadratic utility function modeled with current consumption and once-lagged consumption. Weber (2002) generalized Cushing's analysis by modeling the rule of thumb in nonlinear Euler equations and using the generalized method of moments (GMM) estimation technique. In particular, he tested if the lifetime utility function is time non-separable and concluded that the e¤ect of the rule of thumb was small and not statistically signi…cant.
In this article, we follow the insight of Weber, who considered that consumption of the optimizing agent is aggregate consumption minus rule-of-thumb consumption. In addition, we use the consumption-based asset pricing model (CCAPM) of Breeden (1979) and Lucas (1978) as a base of modeling and testing. The CCAPM setup considers not only an interest 1 These consumers are restricted to consuming their current income, with no optimizing behavior rate as studied in Hall (1978), Flavin (1981), Campbell and Mankiw (1989) or Weber (2002), but several assets in the economy. For instance, Hansen andSingleton (1982, 1983) developed and tested the empirical implications of the PIH when asset returns are time-varying and stochastic. They used the S&P 500 index and Treasury Bill yield as a risk-free rates of return. Epstein and Zin (1989) used …ve individual stock return indexes which give value-weighted returns for broad groups of industrial stocks and Treasury Bill yields as a risk-free rates of return.
Regarding Brazilian data, some authors have tested the PIH by incorporating rule-ofthumb behavior, but no one has used this procedure of testing in the CCAPM setup 2 . Among the papers that have studied the rule-of-thumb proportion of consumption for the Brazilian economy are the articles of Cavalcanti (1993), Reis et al. (1998), Issler and Rocha (2000),  and Gomes and Paz (2004). Cavalcanti (1993), studying the intertemporal elasticity of substitution with data from 1980 to 1989, contemplated a budget constrained consumer in one of the models. He found that 32% of the population followed the rule of thumb. Reis et al. (1998), as Campbell andMankiw (1989), used a model in which a portion of the population was restricted to consume only current income in order to test the validity of the PIH. Their study ranged from 1947 to 1994. The econometric tests revealed that about 80% of the population was restricted to consume only their current income. Issler and Rocha (2000) conducted a study on the temporal series of consumption in Brazil from 1947 to 1994, aiming to examine theoretical issues of the PIH. The main results pointed to the acceptance of cointegration between consumption and income, and they also found that about 74% of the individuals are restricted in terms of liquidity.  used Beveridge and Nelson's decomposition to disclose a cyclical component in consumption when testing the PIH. When he adopted the habit formation speci…cation he found similar results to those of Reis et al. (1998). Gomes and Paz (2004)  This article makes some contributions to the literature on aggregate consumption. First, we use a new procedure to test the rule-of-thumb behavior for Brazilian consumers. We use the consumption-based asset pricing model (CCAPM), which allows more than one interest rate. Second, this paper generalizes the rule-of-thumb model to allow intertemporal nonseparability in the representative household's preferences considering external habits (Abel, 1990). In addition, the Kreps-Porteus (Epstein;Zin, 1989Zin, , 1991 expected lifetime utility function, which separates the coe¢cient of relative risk aversion from the intertemporal elasticity of substitution, is employed. As complementary analysis, we study the traditional utility functional forms of constant relative risk aversion (CRRA). These di¤erent types of utility functions permit estimating the structural parameters: intertemporal discount factor, intertemporal elasticity of substitution, relative risk aversion coe¢cient and the habit formation parameter. The purpose of this work is not to criticize the methods employed in previous articles, but instead to show a new procedure to test rule-of-thumb behavior for the Brazilian economy.
The empirical results in this paper provide evidences of rule-of-thumb behavior in the Brazilian case. In other words, there is a proportion of the individuals consuming their current income, and another group of individuals that consume optimally in each period.
Therefore, there was a strong violation of the permanent income hypothesis.
The remainder of this paper is organized as follows. In section 2, the model with rule-ofthumb behavior is brie ‡y discussed in the Euler equations for three di¤erent speci…cations.
The estimation and results are detailed in section 3. Finally, the conclusions are in section 4.

Testing rule of thumb in the CCAPM framework
The idea behind the consumption-based capital assets model (CCAPM), established by Lucas (1978) and Breeden (1979), is that agents accumulate assets to ensure their future consumption plan, so the asset return series are related with the consumption series. The maximization problem faced by the agents is: where U t is the utility function in period t, C 2;t is the aggregated household's consumption that consumes according to optimizing behavior, t is a vector of the N assets, P t is the assets' pricing vector for each period, and d t is the assets' dividends vector 3 . In each period, the agent receives an exogenous income Y t , which is a state variable in the consumer problem.
Solving this problem for U t = E t [ 1 X s=0 s u(C 2;t+s )] yields the Euler equations: ) ; for j = 1; 2:::; N and 8t where u t ( ) is the instantaneous utility function, is the intertemporal discount coe¢cient, the index j refer to each available asset, and @u t+1 =@C 2;t+1 @ut=@C 2;t is the stochastic discount factor at t + 1. Dividing both sides by P j;t and placing the rights side under (P j;t+1 + d j;t+1 ), it is possible to replace (P j;t+1 +d j;t+1 ) P j;t by R j;t+1 , the gross return of asset j at t + 1, so that: 1 = E t @u t+1 =@C 2;t+1 @u t =@C 2;t R j;t+1 , for j = 1; 2:::; N and 8t (3) Hall (1978), using a quadratic utility functional form and …xed return rate, reached the conclusion that the aggregate consumption series behaves as a random walk: where C 2;t is the variation in consumption and t was called innovation. Campbell and Mankiw (1989) divided consumers into two groups. The …rst group receives a share, , of the disposable income and consumes all their current income Y 1;t ; the second group receives a share (1 ) of the disposable income, follows the PIH and their income is Y 2;t . Hence, the total income of the economy is , Y t = Y 1;t + Y 2;t or: The consumers from the …rst group have C 1;t = Y 1;t = Y t , while the consumers from the second group follow equation (4). The total variation in consumption can be stated as C t = C 1;t + C 2;t , and replacing this yields Campbell and Mankiw's test equation: This equation says that the variation in consumption is a weighted average between the variation of the income of the …rst group and the unpredictable variation in the permanent income of the second group. They speci…ed their hypotheses as: When = 0, the permanent income hypothesis holds. Under the alternative hypothesis the change in consumption is a weighted average of changes in current income. Equation (6) should not be estimated by ordinary least squares (OLS) since the error component may be correlated with changes in income.
Weber (2002) modeled consumption in nonlinear Euler equations, by isolating the consumption of the second group, C 2;t 4 . So, let C t = C 1;t + C 2;t , then C 2;t = C t C 1;t , and C 1;t = Y t , then: The Euler equations of the CCAPM problem are only valid for optimizing consumers, replacing (8) in equation (3), and yields: , where W is the weighting matrix which acts to weight the various moment conditions to build the distance measure.
A test for the over-identifying restrictions (TJ-test) allows checking whether the model´s moment conditions match the data well or not. The TJ statistic employed is asymptotically chi-squared with r k degrees of freedom, where r is the number of orthogonality conditions and k the number of parameters in the structural model.

Utility functions
The utility's functional forms Constant Relative Risk Aversion Preferences (CRRA), external habits and Kreps-Porteus address time separability and non-separabibility.

The Constant Relative Risk Aversion Preferences
In the …rst model, the instantaneous utility funciton is parameterized as: where is the relative risk aversion coe¢cient and the reciprocal of the consumption's intertemporal elasticity of substitution = 1= .
The Euler equations are: # , for j = 1, 2:::; N and 8t (12) Replacing (8) in (12), yields: # , for j = 1, 2:::; N and 8t (13) Stationary regressors are obtained dividing through C t , therefore: # , for j = 1, 2:::; N and 8t (14) Let X t be a vector of chosen instruments, thus the orthogonality conditions are: The External Habits Preferences This parametric form of the individual preferece assume that individual keeps the history of her own consumption, viewed as consumer's habit, allowing for non-separability of the utility function over time. The instantaneous utility funciton for External Habits used is: Following Abel (1990), we specify the function t ( ) here as t = C D 2;t 1 C 1 D 2;t 1 . In order to have "external habit", we set D = 0 and > 0. Therefore t = C 2;t 1 and the utility funtion U t is: where C 2;t is the individual consumption at t; C 2;t 1 is the per capita aggregated consumption at t 1; is a parameter controlling the time separability in the function.

The Kreps-Porteus Preferences
The third utility preference treated here follow the Epstein and Zin (1989), being a generalization of the utility function proposed by Kreps and Porteus (1978). The aggregating function is parameterized as a constant elasticity of substitution (CES) function: where E t is the conditional expectation operator given the information avaliable to the agent in the planning period andŨ t+1 is the agent's future utility. The consumption's intertemporal elasticity of substitution is = 1 1 . The relative risk aversion coe¢cient is constant, = 1 where the parameter re ‡ects the agent's behavior towards risk.
In particular, when = 0, we are back to the expected utility function with logarithmic preference. When = , we have and additively separable utility function.
Replacing (8) in (22), yields: # , for j = 1, 2:::; N and 8t Stationary regressors are obtained dividing through C t , therefore: 3 5 , for j = 1, 2:::; N and 8t in the unconditional form representation we have: The data range from 1995.Q1 to 2011.Q2. This period starts with the implementation of the Plano Real, the plan the Brazilian government launched that …nally managed to end the persistently high in ‡ation (with bouts of hyperin ‡ation) that had held sway over the previous two decades. Another factor that contributed to this choice was that Reis et al. (1998) and  suggested that the high value of they found was due to the credit constraint the Brazilian population encountered (high and unpredictable in ‡ation with indexation not necessarily matched with salary indexation, making debt service as a proportion of household income extremely volatile). The lower in ‡ation rates through the period studied in this paper resulted in credit expansion, the availability of funding to …nance consumption was not at the same level as in the developed countries but was much higher than in the periods of the others studies. Therefore, a smaller part of the population following the rule of thumb was expected.
The series of the household consumption was calculated the same way as in Reis et al. (1998), where the gross …xed investment and current account balance series were subtracted from the GDP series to obtain a consumption of non-durable goods series. 5 .
The returns of the IBOVESPA index were used to represent the returns of risky assets, because it is the most important index of average returns of the Brazilian stock market.
Another interesting option would be IBrX, an index comprising more stocks that is widely used in the …nancial market for the static CAPM. However, the IBOVESPA series is longer and more suitable for the studied period. In order to represent the returns of the riskless asset in the Brazilian economy, the rate paid on government debt (SELIC rate) was used 6 .
The general price index (IGP-DI) calculated by Fundação Getulio Vargas (FGV) was used to de ‡ate income, consumption and returns of both assets. The consumption and income data were also subject to seasonal adjustments. Figure 1 shows the data in quarterly frequencies.

Empirical Results
In this section, we present GMM estimates of the rule-of-thumb models for the utility preferences shown in the previous section. In order to estimate the orthogonality conditions, generated by the Euler equations, we use several sets of instruments. The instruments correspond to lagged values of the growth in consumption and real interest rate 7 . We also use the tests of the overidentifying restrictions (TJ-test) to assess the joint validity of each model and the set of instruments. In this paper, several sets of instruments were tested and none were rejected at the 5% level.
The results are presented in Tables 1, 2 and 3, where only display those where the parameter estimate was between 0 and 1.
For the CRRA utility, Table 1, the median estimate of the parameter was^ = 0:8945 7 A large number of instruments or a high number of assets can cause problems to …nd the optimal weighting matrix or in ‡uence the quality of asymptotic approximation, therefore the data must meet the following condition: N M (N M +1) 2 < N T (Driscoll;Kraay, 1998), where N are the number of Euler equations and T are the number of observations. In this study, the number of instruments in the worst case the relation is 78 < 122. and all but one of them were signi…cant at the 5% level. That is, results for show that around 89% of the population follows the rule of thumb. All the estimates of the intertemporal discount coe¢cient, , were signi…cant at the 5% level, and their median waŝ = 0:9783. The median for the relative risk aversion coe¢cient was^ = 0:0974. Table 2 reports the …ndings of the estimation of Euler equations (20) which correspond for the external habits utility model. Results show that overall estimations of the parameter for the most part were signi…cant at the 5% level. The median of all valid estimates waŝ = 0:7817. It shows that around 78% of the Brazilian population follows the rule of thumb. All the results for the intertemporal discount coe¢cient, , were signi…cant at the 5% level and their median was^ = 0:9793. For the parameter this study found^ = 0:1518, but only three of the estimates were signi…cant at the 5% level. One positive feature of this estimation is that it does not violate the external habit basic assumption: if > 0 than > 0 or if < 0 than < 0. The relative risk aversion coe¢cient estimate was^ = 0:0548 and only four of them were signi…cant at the 5% level and only one at the 10% level. (1 where N=2, and R 1;t+1 =Ibovespa returns and R 2;t+1 = Returns on Selic Inst./Mtx P value (1) *,** and *** denote, respectively, signi…cance of parameter by the t-test at the 10%, 5% and 1% levels.
(2) The number in parentheses are the respective standard-deviation estimates, robust to heteroscedasticity and to serial correlation.
(3) The last line of the table shows the median of all estimates. (4) List of instruments: I1 uses R 2;t ; R 2;t 1 ; C t /C t 1 and C t 1 /C t 2 ; I3 uses R 2;t 1 ; R 2;t 2 ; C t 1 /C t 2 and C t 2 /C t 3 ; I4 uses R 2;t 1 ; R 2;t 2 ; R 1;t 1 ; R 1;t 2 ; C t 1 /C t 2 and C t 2 /C t 3 ; I5 uses R 1;t 1 ; R 1;t 2 ; C t 1 /C t 2 and C t 2 /C t 3 ; I6 uses R 2;t ; R 2;t 1 ; R 1;t ; R 1;t 1 ; C t /C t 1 and only reports the results that reached convergence in less than 1000 iterations and was not rejected by Hansen's (1982) test. (7) The …rst column shows the method by which the weighting matrix was reached, ASI denotes Andrews(1991), NWFSI denotes Newey and West(1987) with …xed windows, and NWVSI denotes Newey and West (1994), with variable windows.
(2) The number in parentheses are the respective standard-deviation estimates, robust to heteroscedasticity and to serial correlation.
(2) The number in parentheses are the respective standard-deviation estimates, robust to heteroscedasticity and to serial correlation.
(3) The last line of the table shows the median of all estimates. (4) List of instruments: I1 uses R 2;t ; R 2;t 1 ; C t /C t 1 and C t 1 /C t 2 ; I2 uses R 1;t ; R 1;t 1 ; C t /C t 1 and C t 1 /C t 2 ; I3 uses R 2;t 1 ; R 2;t 2 ; C t 1 /C t 2 and C t 2 /C t 3 ; I4 uses R 2;t 1 ; R 2;t 2 ; R 1;t 1 ; R 1;t 2 ; C t 1 /C t 2 and C t 2 /C t 3 ; I5 uses R 1;t 1 ; R 1;t 2 ; C t 1 /C t 2 and C t 2 /C t 3 ; I6 uses R 2;t ; R 2;t 1 ; R 1;t ; R 1;t 1 ; C t /C t 1 and C t 1 /C t 2 ; I7 uses R 2;t ; R 2;t 1 ; R 2;t 2 ; R 1;t ; R 1;t 1 and R 1;t 2 ; I8 uses R 2;t 2 ; R 2;t 3 ; R 1;t 2 and R 1;t 3 . (5) The p-value of Hansen's overidentifying restrictions test are shown in the last column. (6) The table only reports the results that reached convergence in less than 1000 iterations and were not rejected by Hansen's (1982) test. (7) The …rst column shows the method by which the weighting matrix was reached, ASI denotes Andrews(1991), NWFSI denotes Newey and West(1987) with …xed windows, and NWVSI denotes Newey and West (1994), with variable windows. (8) The parameters and in the Kreps-Porteus utility function model were estimated indirectly using the delta method (see Greene, 2008).
In Table 4 we compare the estimates for , and with studies that also used the CCAPM framework but did not contemplate the rule of thumb parameter in the model, such as Issler and Piqueira (2000), Bonomo and Domingues (2002), Catalão and Yoshino (2006) 11 . Table 4 shows the results of these studies for , and .
The results for the intertemporal discount coe¢cient, , were all signi…cant at the 5% level, for the three functional forms and in line with the previous studies.   One possible reason pointed out for a high was the lack of credit available to the Brazilian population during the period of study. After the end of the hyperin ‡ation in 1994, the Brazilian economy experienced a strong expansion of credit, so some agents who followed the rule of thumb due to credit constraint in the previous studies could have started to optimize their consumption decisions.  Gomes and Paz (2004) and Arreaza (2000) results for suggests rejection of the PIH for Latin American data. This study comes to the same conclusion for Brazilian data, but reached to slightly di¤erent values for the fraction of myopic consumption.

Conclusion
This paper investigated whether there is a fraction of consumers that do not behave as fully forward-looking optimal consumers in the Brazilian economy. We used di¤erent utility functional forms in the CCAPM framework. Beginning from Euler equations of the optimizing consumer utility problem, we estimated the structural parameters using the generalized method of moments (GMM) and tested the model's over-identifying restrictions using Hansen's (1982) TJ test.
Regarding the model's performance, we conclude that in the Brazilian case there is a proportion of the individuals consuming their current income, and another group of individuals that consume optimally in each period. These …ndings suggest that a signi…cant fraction of the Brazilian disposable income went to households who consumed their current income, following the rule of thumb. Therefore, there was a strong violation of the permanent income hypothesis.
The results found can be summarized as follows: 2. For the Kreps-Porteus utility function almost all estimates of were statistically insigni…cant, therefore the 22% median estimate is not robust enough to say there was a fraction of myopic consumers.
3. The results for the intertemporal discount coe¢cient, , were all signi…cant at the 5% level, for the three functional forms and in line with the previous studies.
4. Comparing the relative risk aversion results to the previous studies, we obtained lower values for the CRRA and the external habits utility models.
There are two possible explanations of the higher reached in the present study. One possible reason for a high was the lack of credit available to the Brazilian population during the period of study. After the end of the hyperin ‡ation in 1994, the Brazilian economy experienced strong expansion of the credit, so some agents who followed the rule of thumb due to credit constraint in the previous studies could have started to optimize their consumption decisions. On the other hand, the long period with no funds to …nance consumption caused a large pent-up demand during the period of this study. Another explanation is that great increase in income experienced by the lower social classes, especially after 2002, caused them to increase spending in a Keynesian way, assuming that those social classes spend their current income.
Interesting extensions of this paper could be to use factor model analysis to build portfolios in order to consider more than two assets or to explore other functional forms.