Jan 8, 2002 - is present among some occupational groups, as for others, the PIH cannot ..... as food, tobacco and alcohol, clothing, housing, furniture, health, ... In the case of the IV estimator we used a procedure proposed by Davidson and.
The Swiss : Are They Prudent? Christophe Kolodziejczyk¤ Deep, University of Lausanne, CH-1015 Lausanne, Switzerland January 8, 2002
Abstract Using a cross-section of Swiss households we test whether savings behavior is driven by the Permanent Income Hypothesis (PIH) or by a precautionary motive. Panel data would facilitate this task but they are not available in the Swiss context. We therefore propose estimators for key parameters such as household income risk. The PIH model is nested within the precautionary saving model. We …nd that a precautionary motive is present among households as a whole but is not quantitavely important. A closer look at the data suggests that a precautionary motive is present among some occupational groups, as for others, the PIH cannot be rejected. Keywords : Consumption, permanent income, precautionary saving, Switzerland JEL Codes : C21, D91 ¤
I wish to thank Ramses Abul Naga, Dolores Collado, Damjan Kozamernik and seminar participants at the University of Alicante for discussions and comments. I am responsible for any errors.
1
1
Introduction
The study of household savings is important for a quite large number of economic debates, namely when we are interested in income redistribution, …scal policy or more generally capital accumulation. Through savings households supplies a large fraction of capital available for investment. Before we understand the links between savings and these di¤erent issues, we have to know how savings is formed. If we have to give recommandations in terms of economic policy, it is imperative to know empirically what kind of behavior drives household savings. The Keynesian consumption function assumes saving is exogenous and people save a constant fraction of their current disposable income. This formulation is not very helpful because it does not allow any e¤ects due to the dynamic or the uncertainty aspects of the life cycle. The former problem was explored by Friedman with his permanent income hypothesis and by Modigliani with his life cycle model (see Deaton [9]), and since a lot of theoretical and empirical work has been done in this way. One of the major contribution was to reconcile the permanent income theory and the life cycle model in considering the latter model under the hypothesis of certainty equivalence and consequently to show that agents tend to smooth consumption through savings. In this case, movements of consumption should not be correlated with those of income. Though the model is able to explain some stylised facts about consumption, for instance that consumption is less volatile than income, it remains some predictions of the model which are not veri…ed empirically, namely the excess of volatility of consumption observed in the data with respect to the predictions of the model and the empirical evidence that consumption tracks income. Because of these two objections, for the last ten years researchers have tried to enrich the life cycle model in considering the e¤ects of uncertainty on consumption and to test empirically the precautionary saving motive. In a model of life cycle with certainty equivalence, consumption is determined by the permanent income, i.e. the discounted value of life-time resources of the household. The main prediction of this model is that the intertemporal pro…le of consumption should not depend on the income’s pro…le. This result is due to the preferences used in this model which allows to treat the model as if there were no uncertainty and rules out any precautionary motive. Moreover agents in this model are not prudent in the sense of Kimball [12]. When we relax this assumption, the latter result is no longer valid and a precautionary motive appears. The predictions of these two models are very di¤erent and have di¤erent implications in terms of public policy and redistribution of income. In the certainty equivalence case, savings is a function of income and the risk associated to income does not in‡uence the consumption decision. In the precautionary saving model, the variance of income is an important explanatory variable. When uncertainty is higher and agents are prudent, savings will rise in order to isolate from unexpectable states of nature. In this case a reduction of income uncertainty can substantially increase welfare. This is one justi…cation to the implementation of a social insurance and supporting income policies to less favorised families. Our aim in this paper is to determine which model explains best households savings behavior. Moreover we want to identify what types of households are more subject to the precautionary motive and quantify it. Using a cross-section of Swiss households we test whether savings behavior is driven by the Permanent Income Hypothesis (PIH) or by a precautionary motive. We …nd that households savings are better explained by the precautionary motive but this one is not quantitatively important. When we estimate the model for di¤erent types of occupational or activity groups, some of them are precautionary savers as for others we cannot reject the PIH. In section 2, we present both theories of consumption, the permanent income hypothesis and the precautionary saving model in the life-cycle framework. We derive two equations for the consumption function in order to test empirically these two hypotheses. We show that under some conditions on preferences and the stochastic process of income that the permanent income model is nested within the precau2
tionary saving model, where in the latter we add the variance of income as an explanatory variable of consumption. In section 3, we present the econometric methodology to estimate both models and test these two theories. We show how to construct an estimator of the variance of income in the context of cross-section data. Section 5 presents the data used in this study and the results we obtained. Section 6 gives some concluding comments. The derivation of the precautionary saving model is given in an appendix.
2 2.1
Theoretical framework The life cycle model
The neoclassical consumption theory is based on the life cycle hypothesis. It assumes that a household maximizes its intertemporal expected utility under its intertemporal budget constraint. "1 # X max1 U = Et ¯ iU (ct+i ) (1) fct+i gi=0
i=0
s:t: wt+i = wt+i¡1 (1 + rt+i) + yt+i ¡ ct+i c t+i ¸ 0 lim wt+i ¸ 0
i!1
where yt is disposable income at time t; wt is …nancial wealth and rt is the ex post gross return on the assets held by the household. ¯ is the discount factor. We assume that the utility function is intertemporaly separable. This means that the marginal rate of substitution of consumption between two periods is independent of the level of consumption in other periods. This assumption greatly simpli…es the analysis, but it does not allow to model the consumption of durable goods or habit formations. Moreover, it imposes a strong structure on the intertemporal allocation process of consumption. The …rst order conditions of this program yield the so called Euler equation £ ¤ U 0 (ct ) = Et U 0 (ct+1) (1 + rt+1 ) ¯ : (2) When the utility function is intertemporally separable, the household whishes to maintain at each period the discounted marginal utility of consumption constant.
2.2
The permanent income hypothesis
The basic idea of the permanent income hypothesis (PIH) is that consumption depends on the permanent income of the household, i.e. a long term average or expected income, rather than current income and that households desire to smooth consumption during their life. In the basic version of the permanent income, consumption is equal to the permanent income. When the change in current income is temporary, the permanent income and then the consumption will vary little. When we use the life cycle model with the certainty equivalent assumption, this theory and the PIH are similar, because the intertemporal pattern of income does not a¤ect the evolution of consumption. However the income pro…le is important in the determination of savings, which is equal to transitory income, as it is used to smooth consumption. The PIH might be considered as a particular case of the life cycle model where we suppose a quadratic utility function and the interest rate is constant and equal to the discount rate. These assumptions are clearly unrealistic but might be justi…ed for some reasons. Particularly we can assume as a …rst approximation that consumption is equal to the value of the human and non human wealth and say that agents have a preference for a constant ‡ow of consumption. 3
When we suppose a quadratic utility function, U (ct ) = ct ¡ a2 c 2t ; and rt = r = ± for all t; equation 2 becomes Et [ct+1] = ct : (3) The wealth accumulation equation can be written as an intertemporal budget constraint 1 X i=0
1 X c t+i yt+i = w + t i i: (1 + r) (1 + r) i=0
Taking expectations at time t on both sides of 4 and substituting 3, we obtain ( ) 1 X ct = (1 ¡ ®) wt + ®iEt [y t+i ] ´ ytp;
(4)
(5)
i=0
with ® = 1= (1 + r) : y pt is the discounted wealth of the household over the life cycle and can be interpreted as the permanent income. This model shows that consumption is entirely determined by the permanent income and does not depend on the intertemporal pro…le of income. Any variation in the uncertainty of income which maintain the permanent income unchanged does not a¤ect consumption. This result is a consequence of the quadratic utility assumption. In this case the expected marginal utility of future consumption is equal to marginal utility of expected consumption and we have the same results as in a model without uncertainty. If the households do not anticipate changings in their permanent income, they will not change their consumption plan. If they are able to borrow, they can maintain their consumption level by using …nancial markets. In this case, savings is a residual of income. Because of the separibility of the utility function, the marginal utility of consumption must follow a martingale and it must be also the case for consumption when preferences are quadratic. Hall [10] in the context of time series data tested the martingale property of consumption, but rejected his model since he found a correlation between lagged values of stock market prices and consumption variation.
2.3
The precautionary saving motive
We saw, in the case of the PIH, that the assumption of a quadratic utility function limits the applicability of this theory since it rules out any precautionnary saving motive. Precautionary saving is in‡uenced by the degree of prudence of households, in the sense of Kimball, and by the probability distribution of the di¤erent states of nature. Intuitively, when households face new information concerning income uncertainty, they will revise their consumption plan. When households are prudent, i.e. their marginal utility is decreasing and convex, an exogenous increase of uncertainty, in the sense of a mean-preserving spread increase of the permanent income, implies a higher evaluation of future consumption and then an increase in savings. An increase in uncertainty raises the expected future marginal utility of consumption which in turn implies an increase of today’s marginal utility in order to maintain the Euler equation and a decrease of consumption today. In the case of a quadratic utility function we have the following equality £ ¤ u0 (ct ) = Et u0 (ct+1) = u0 (Et [ct+1]) : (6) When u0 is convex (u000 > 0), by Jensen’s inequality we have the following condition £ ¤ Et u0 (ct+1) > u0 (Et [ct+1]) :
(7)
Condition 7 tells us that consumption will be lower in the precautionary than in the certainty equivalent case. Figure 1 shows the implication of a convex marginal utility function on the optimal 4
u' (C )
()
πu ' (C ) + (1 − π )u ' C
[
u ' π C + (1 − π )C
] π C + (1 − π )C
C
C
C
Figure 1: Certainty equivalent of consumption when marginal utility is a convex function. consumption choice under uncertainty compared to the certainty equivalence case. In order to simplify, we assume consumption is stochastic and can take two values, C with probability ¼ and C with probability 1¡ ¼: We see that the expected marginal utility is lower than the marginal utility of expected consumption when the marginal utility is a convex function of consumption. Figure 2 shows the e¤ect of a mean-preserving spread increase of the probability distribution of C when 0 consumers are prudent. Consumption can now take values C 0 < C and C < C with the same probability distribution. When the variance of consumption increases and the marginal utility is convex, the certain equivalent increases and raises the incentive to save. 2.3.1
A model of precautionary saving
In this section we take a life cycle model due to Caballero [2]. We derive a consumption function where preferences exhibit a constant coe¢cent of absolute risk aversion (CARA) and consumers face uncertainty on their labor income only. We give in appendix the technical details of the derivation of the consumption function. We assume the utility function is of the CARA form 1 U (c t) = ¡ e¡µct ; (8) µ where µ is the coe¢cient of absolute risk aversion. We assume risk is only attributable to labor income and has the following ARMA process yt+i = ¹ +
i X j=1
5
ªi¡j !t+j :
(9)
u' (C )
( )
πu ' (C ') + (1 − π )u ' C '
()
πu ' (C ) + (1 − π )u ' C
C'
C
π C + (1 − π )C C
C'
C
Figure 2: The e¤ect of an increase in income risk on consumption when marginal utility is convex. When the ! t+j ’s are gaussian i.i.d with zero expectation and variance equal to ¾2! conditionnaly to the information available at time t; the consumption function is equal to ct = ytp ¡
® µª2 2 ¾ ; 1 ¡® 2 !
(10)
P i with ª ´ (1 ¡ ®) 1 i=0 ® ªi : Consumption depends on the permanent income and the variance of the innovations in income. This last term is the precautionary motive. The weight of this motive depends on the coe¢cient of risk aversion. The more the agents are risk averse, the more they will take the uncertainty of income into account. In this model the marginal propensity to consume out of the permanent income is equal to one. 2.3.2
The concavity of the consumption function
As we mentionned it before, when we relax the assumption of quadratic utility function and allow for general stochastic processes on labor income and interest rate, it is not possible to …nd an analytic solution to the consumption function. In order to be able to derive properties of the consumption function under uncertainty, some authors used numerical methods (see Zeldes [18], Carrol [4] and Carroll [5]). These papers show that under more realistic assumption, namely with preferences which exhibit constant relative risk aversion (CRRA) the consumption function has a di¤erent behavior than under the certainty equivalent case. For example, Skinner [16] used a second order Taylor approximation to derive a functional form for the consumption function. In section 2.3.1, we were able to derive a linear consumption function but with quite strong assumptions on preferences and income uncertainty. However, Carroll et Kimball [6] show that when we introduce uncertainty in a general way in the intertemporal optimisation plan of the consumer and she has
6
preferences which exhibit a positive precautionary motive in Kimball’s sense1 [12] , the consumption function will be a concave function of wealth, i.e. c 00t (wt ) · 0. Carroll and Kimball formalize the idea of the marginal propensity to consume as a function of wealth. People with limited wealth should have a higher marginal propensity to consume wealth than rich ones. This result of the concavity of the consumption function is valid for a general class of utiliy function which have hyperbolic absolute risk aversion (HARA)2 . Moreover, when prudence is proportional and greater than risk aversion, and labor income and interest rate are uncertain and imperfectly correlated the consumption function is strictly concave. The quadratic , the CARA and the CRRA utility functions are particular cases of the HARA utility function. They also show that in the CARA case when only labor income is uncertain the consumption function is linear. We obtain the same result with CRRA preferences but with only uncertain interest rates. The consumption function is indexed by t meaning its dependance on information available at date t: Potentially this can include expectations on future income and its uncertainty, and changes in the demographic structure of the household. We recall the de…nition of wealth wt = (1 + rt) (wt¡1 ¡ c t¡1) + yt : We de…ne wt ´ Et¡1 [wt ] = (1 + Et¡1 [rt ]) (wt¡1 ¡ ct¡1 ) + Et¡1 [yt]. When we make a second order approximation of the consumption around wt , we obtain. 1 c t (wt ) ¼ c (wt ) + c0 (wt ) (wt ¡ wt ) + c00 (wt) (wt ¡ wt )2 2
(11)
The last term of 11 can be interpreted as a precautionary motive since c00 (wt) < 0 and (wt ¡ wt )2 is the squarred of the deviation from expected wealth and may be interpreted as a proxy of the variance of income.
3
Econometric methodology
In this section, we show how we test econometrically these two hypotheses concerning savings behavior in the context of cross-section data. We saw during the theoretic discussion that the permanent income model is included in the precautionary saving model, where we add a term which takes into account income uncertainty. The idea of the econometric test is to estimate the precautionary saving model and to test whether the paramater of the variance of income is statistically di¤erent from zero. Generally, empirical studies on precautionary saving use a second order Taylor expansion of the Euler equations in order to obtain an empirically estimable functional form (see Browning and Lusardi [1]). They allow CRRA preferences and uncertainty on labor income and interest rate. To estimate such a model at the micro level we need panel data for two reasons. First, because this model gives predictions on the change of consumption between two periods. Second, because if we want to proxy the variance of income for the individuals we are more able to achieve this when we have time series data. Unfortunately, in the Swiss context we only have a cross-section of household survey. Our theoretical approach gives in both case a functional form for consumption which is linear and allows one model to be nested within the other. This simpli…es the test of these two hypotheses. In order to estimate the precautionary 1
Kimball introduces the notion of prudence. A concave utility function u (c) will exhibit a prudent behavior if 000
´ (c) = ¡
u (c) > 0; u00 (c)
i.e. the marginal utility is convex. Precautionary saving and prudence are closely linked. 2 A HARA utility function is de…ned by the following property ¡u00 (x) =u0 (x) = 1= (A + (k + 1) x) :
7
saving model, we have to …nd an estimator of the variance of income. This task is di¢cult to accomplish in the context of cross-section data. Nevertheless, we propose a solution to this problem by taking an approach similar to Miles[14]. This method consists in a …rst step to assume that the determinants of the permanent income are known to the econometrician. In a second step we regress these determinants on current income and use the squarred of the residuals of this regression to approximate the variance of income. We show successively how to estimate these two models and how to approximate the variance of income. We also show formally how to test these two hypotheses.
3.1
The permanent income hypothesis
From our discussion of section 2.3.1, the consumption function is linear in the permanent income. According to equation 5, we can write c i = ½yip + vi :
(12)
ci represents consumption expenditures of household i; yip is the permanent income and vi is an error term uncorrelated with ypi and can be interpreted as transitory consumption. We only observe current income y i which is de…ned as the sum of the permanent income yip and transitory income yit: yti is a random variable which is independantly £ p t¤and identically distributed across time with zero 2 expectation and variance ¾yt ;i: We assume E yi yi = 0: We cannot observe the permanent income and we use current income as a proxy for this variable. When we substitue yi in equation 12, we obtain c i = ½yi + ´ i (13) with ´ i = ½yit ¡ vi : Because E [´ iy i] = ½¾2yt;i 6= 0; if we estimate 13 by OLS, the estimator of ½ will be biased. We have to use an instrumental variable estimator with a set Wi ; which are assumed to be correlated with the permanent income, as instruments for y i: In order to obtain a consistent estimator of ½, Wi must be uncorrelated with ´ i , which implies no correlation between transitory income and the set of instruments.
3.2
Precautionary saving motive
Our precautionary saving model predicts that, for two households with the same permanent income, the one who faces the higher uncertainty will save more. This prediction suggests that we add the variance of income as an explanatory variable in the consumption function 12, i.e. ci = ½yip + '¾2y;i + vi with ' < 0:
(14)
We have no direct measure of the variance of income. To solve this problem, we assume that the determinants of the permanent income are known, i.e. p
y i = Xi¯ + ²i : (15) £ ¤ We make the following assumptions E [²i] = 0 and E ²2i = ¾2²;i . In order to satisfy the assumption £ ¤ of orthogonality between transitory and permanent incomes, we assume E ²i yit = 0. Current income is equal to yi = Xi ¯ + ei (16) where ei = ²i + yti : We deduce easily that E [ei] = 0 and V [ei ] = ¾2y;i = ¾2²;i + ¾2yt ;i ´ ¾2ei : Consequently, we take the squarred of the residuals of the regression 16 to approximate ¾2y;i; that
8
is ¾ b2ei = ebi2. We can write this approximation as the sum of the true variance of income and an error term »i , ¾2e;i = ¾2y;i + »i: b (17) We suppose that the variance of income depends on a set Zi of instruments plus an error term ¹i :
Substituting 17 in 18, we obtain
¾2y;i = Zi° + ¹i :
(18)
b2e;i = Zi° + ¸i; ¾
(19)
with ¸i = ¹i +»i : We estimate the following equation with an instrumental variable estimator using Zi as instruments ci = ½yi + 'b ¾2ei + ui ; (20) where ui = ½yti +'¾2y;i ¡vi : Consistency of the instrumental variable estimator requires E [Ziui] = 0:
3.3
Empirical test of the precautionary motive
According to our econometric analysis, the permanent income hypothesis is nested within the precautionary saving model and corresponds to a value of the parameter ' equal to zero. The test of the PIH versus the precautionary saving model only requires a t-test on ' equal to zero under the null and less than zero under the alternative. We will also use tests of overidentifying restrictions, to judge the quality of the instruments used, and tests of speci…cation such as Hausman’s test to compare the IV estimators to OLS estimator. We incorporate into the consumption function demographic variables in order to control for variations during the life-cycle of the demographic structure of the households and their heterogeneity. The life cycle model predicts that households will equalize the discounted marginal utility across periods. The marginal utility does not only depend on the goods consumed but also on demographic composition of the household. For example, the presence of children could a¤ect the marginal utiliy of one additional unit of consumption compared to a situation without children, because there might be a substitution from adult goods towards children goods. These are variations of consumption due to the life cycle motive. For the permanent income hypothesis, we estimate the following equation ci = ½y i + ±0 Hi + ´ i: (21) For the precautionary saving model, we estimate c i = ½yi + 'b ¾2ei + ±0Hi + ui;
(22)
where H i is a matrix of demographic variables. We estimate these two equations by two-stage least squares because we have to instrument yi and ¾ b2ei which are two variables measured with error. The sets of instruments are respectively Xi for yi and Zi for b ¾2ei: Provided the identi…cation conditions are met, Xi and Zi might have common variables.
4 4.1
Empirical Analysis Data description
Our empirical analysis uses swiss data from the ”Enquête sur le Revenu et la Consommation 1998”. This survey gives information about consumption expenditure and incomes of Swiss households for the year 1998. Though the survey was conducted through the year 1998, the data are standardized for a period of one month. We also …nd detailed information about demographic characteristics of 9
the household namely on labor supply, education3 , occupation status, the household composition and housing. We selected households whose saving rates4 were between -1 and 1 and the age of household head was less than 60. People who are retired or near retirement are less subject to labor income risk. Because of plausibility of data, we also limited our sample to households whose monthly disposable income was between 1000 and 50000 swiss francs. We give on table 1 summary statistics of the variables used in this study. Consumption is de…ned as the sum of monthly expenditures on usual groups of goods, such as food, tobacco and alcohol, clothing, housing, furniture, health, transports, communication, leisure, education and other goods5 . Disposable income is de…ned as the sum of monthly labor, wealth and transfer incomes less taxes and social security deductions. Table 1 : Summary statistics of variables
Consumption Disposable income Household size Nchild Married age Swiss Sex Education Number of vehicles
Average
Standard deviation
5559:51 6414:36 2:63 0:50 0:57 39:41 0:84 0:73 19:73 1:18
2771:81 3579:90 1:36 0:85 0:49 9:99 0:36 0:44 3:25 0:72
Number of obvservations : 6954 Source : ERC 98
4.2
Results
In this section, we present the results of the estimation of our models. For the two models, we estimated the consumption function by ordinary least squares (OLS), two stage least squares (IV) and by the generalized methods of moments (GMM). This last estimator takes into account a potential heteroskedasticity of the disturbance term and is more e¢cient than the estimator of the instrumental variables (IV). A complete description of this estimator can be found in Wansbeek and Meijer [17]. In order to control both models for the life cycle motive we included in the consumption function the age (Age), age squarred (Age2), the number of children which are less than ten years old (Nchild), the household size (Hhsize) and a dummy variable if the household is a married couple (Married). We used the following variables as regressors to construct the variance of income for the precautionary saving model6 : age, age squared, the highest eduction level attained, a dummy variable for the sex (Sex), a dummy variable for being Swiss (Swiss) and the number of workers present in the household (Nworkers). To instrument the permanent income we used some common instruments such as the number of years of education (Educ), education squared and the number of vehicles possesed by the household. In the case of the precautionary saving model we used the 3 Education is measured by the numb er of years sp ent at school according to the last degree obtained by the household head. 4 t Saving rate is de…ned as the fraction of disp osable of income not consumed, i.e. st ´ yty¡c t 5 This also includes durable goods 6 Every variable concerns the household head.
10
following variables to instrument the variance of income : the ranking of the variance of income and a variable which takes a value of -1 if the variance is below its median and 1 if it is above. In order to jugdge the quality of our instruments we used tests of overidentifying restrictions. When the statistic of this test rejects the null hypothesis, either the model is misspeci…ed or the instruments are not valid. In the case of the IV estimator we used a procedure proposed by Davidson and MacKinnon [7] which uses arti…cial regressions. The procedure of the test consists in regressing the residuals of the regression by instrumental variables on the instruments used and then to compute the F-statistic as the uncentered R2 of this regression multiplied by the number of observations. This statistic follows a Â2 distribution with degrees of freedom equal to the number of instruments less the number of explanatory variables of the model7 . For the GMM estimator, we used a Jtest (for a description of the J-test see Hayashi [11]). The statistic of this test also follows a Â2 distribution with the same number of degrees of freedom as for the test with the IV estimator. In table 2, we present results of both models for the whole sample considered in this study. In table A1, in the appendix, we give the intermediate results of the regression of disposable income on the determinants of permanent income. When we instrument the disposable income, the marginal propensity from the permanent income is higher indicating that a measurement error bias is present. It is well known, in the case of the PIH, that the OLS estimator of this marginal propensity to consume is downward biased. The OLS estimator gives a value of 0.43, as the GMM estimator gives a value of 0.69. Moreover, we note that when we estimate the precautionary saving model, whatever the method of estimation, the marginal propensity to consume from the permanent income is highest for the precautionary saving model. The GMM estimator gives a value of 0.836 which seems quite plausible. When we omit the variance of income there will be a bias which depends on the correlation between permanent income and the variance of income. Given that the correlation between income and the variance is positive, the omitted variable bias should be negative since the coe¢cient on the variance of income is of the same sign. This is why usually we …nd a marginal propensity to consume which is higher when we estimate the precautionary saving model. We note also that the precautionary motive is higher when we use an instrumental variable estimator. The OLS estimator is inconsistent in presence of measurement error and we should have the case of an attrition bias. We performed a Hausman speci…cation test on both models to justify the use of an IV estimator. Under the null, the most e¢cient estimator is consistent. So, if the test rejects the null it indicates that the e¢cient estimator may be inconsistent. The statistic of this test is distributed as Â2 with degrees of freedom equal to the number of parameters. We test the IV estimators versus the OLS estimator. For the permanent income, the Â2 statistic is equal to 156.90 and rejects the null at 1 percent. For the precautionnary saving we have a value of 109.77 and we can reject the null also at 1 percent. In both cases, we can conclude that it is important to instrument disposable income and the variance of income. But in the case of the permanent income the J test rejects the null, as in the precautionary saving model we cannot reject the null suggesting that the permanent income model is misspeci…ed since the instruments used seem to be valid in the other model. The coe¢cient of the variance of income is statistically di¤erent from zero. We conclude that the precautionary motive is present among Swiss households as a whole but is not quantitatively important since we obtain an elasticity of the consumption with respect of the variance of income of -0.049 in the case of the GMM estimation . The permanent income seems to be an important explanatory variable of consumption, and the e¤ect of a permanent shock is higher when we take income risk into account. We also computed the share of precautionray saving (the ratio between the diminution of consumption due to precautionary saving and consumption which would prevail if there were no uncertainty). We found values of 1.01 %, 3.24 % and 3.27 % when we estimated respectively the model by OLS, IV and GMM. The value obtained with the model 7 The number of degress of freedom is 1 for the permanent income and 2 for the precautionary saving model. With Â20:05 (2) = 5:99; Â20:01 (2) = 9:21; Â20:05 (3) = 7:81 and Â20:01 (3) = 11:3:
11
estimated by GMM constitutes a signi…cant reduction in consumption. Table 2 : Permanent Income and Precautionary Saving for the whole sample Permanent Income
Disposable Income
OLS
IV
0:434 ¤¤¤
0:681 ¤¤¤
(61:20)
Age
129:03 ¤¤¤
Age2
¡1:26¤¤¤
(32:54) 98:23¤¤¤ (4:37) ¡0:94 ¤¤¤
¡305:30¤¤¤
¡43:63
(6:25)
(¡4:97)
Nchild
Hhsize Married Constant
(¡7:47) 418:80 ¤¤¤ (13:99) 381:82 ¤¤¤ (5:73) ¡1409:41¤¤¤
(¡3:40)
GMM 0:690 ¤¤¤
(27:23)
101:28 ¤¤¤ (3:94)
¡0:95¤¤¤ (¡3:04)
¡10:77
(¡0:89) 229:10¤¤¤ (6:41)
229:10 ¤¤¤
119:46
120:75 ¤¤¤
(¡3:62)
¡1785:36¤¤¤ (¡4:23)
¡1893:01¤¤¤
F = 1094:01
J = 1:30
J = 0:76
(1:59)
(¡0:22)
(5:46) (1:59)
(¡3:99)
Precautionary Saving
Disposable Income Variance of Income Age Age2 Nchild
OLS
IV
0:529 ¤¤¤
0:828 ¤¤¤
¡8:80 ¤¤¤
¡27:77¤¤¤
(¡4:46)
(¡2:41)
(58:06)
(¡16:17) 115:93 ¤¤¤ (5:72) ¡1:11¤¤¤
¡226:49¤¤¤
(26:58)
(¡9:55) 76:02¤¤¤ (3:37) ¡0:67 ¤¤¤
42:49
Hhsize
356:13 ¤¤¤
Married
319:76 ¤¤¤
Constant
¡1502:97¤¤¤
(0:83) 149:20¤¤¤ (3:91) 86:69¤¤¤ (1:16) ¡1846:96¤¤¤
F = 1010:19
J = 2:39
(¡5:60) (12:02) (4:88)
(¡3:94)
(¡4:40)
GMM 0:836 ¤¤¤
(22:13)
¡27:92 ¤¤¤ (¡8:02) 71:11¤¤¤ (2:78)
¡0:58
(¡1:85)
69:83 (1:37)
136:84 ¤¤¤ (3:43)
85:73¤¤¤ (1:06)
¡1813:75¤¤¤ J = 2:28
(¡4:06)
Number of obvservations : 6954 N o t e : * * * =s ig ni…c a nt a t 1 % , * * =s ig ni…c a nt a t 5 % , * =s ig ni… ca nt a t l 1 0 % t -s t a t is tic s in p a re nt hes e
We now investigate the precautionary motive among the sample used and try to identify the categories of persons more subject to the precautionary motive. For the rest of this section we estimate the models for di¤erent categories of households. In table 3, we look at the models for 12
some occupational groups of the household head. We estimated both models for the following categories : self-employed, farmers, workers and unemployed. We note that the precautionary motive is present among all the categories considered except for the unemployed, where we cannot reject the PIH. It is hard to believe that unemployed people are permanent income consumers since these people are more subject to income risk, but this result can be explained by the small size of the sample and the fact we used an instrumental variable estimator. But we can notice that the value estimated is much higher than for the other groups and though the permanent income hypothesis cannot be rejected. We notice that the precautionary motive is highest for the farmers with an elasticity of -0.123. For the self-employed this elasticity is -0.086 and -0.041 for the workers. The farmers and the self-employed may likely be more subject to income risk and may explain why their precautionary motive is higher than for the workers. Alternatively, the workers represent the majority of the sample, around 86 percent of the whole sample, and is the category which is the most in conformity to the assumptions of the model. We now only consider groups of persons which belong to the workers. Table 3 : Permanent Income and Precautionary saving according occupation Permanent Income Self-employed
Farmer
Worker
Unemployed
Disposable Income
0:735¤¤¤
:435 ¤¤¤ (2:84)
0:676 ¤¤¤ (26:05)
0:644¤¤¤
Â2 p-value
0:524 N = 449
0:58 N = 103
0:24 N = 5766
0:47 N = 88
(4:88)
(4:17)
Precautionary Saving Self-employed
Farmer
Worker
Unemployed
Disposable Income
0:907 ¤¤¤
0:488¤¤¤
0:803 ¤¤¤
0:763¤¤¤
Variance of Income
¡29:40¤¤¤
¡30:04¤¤¤
¡24:66¤¤¤
¡91:31
-0.086 5.29
-0.123 6.09
-0.041 2.52
-0.041 3.12
0:66 N = 449
0:28 N = 103
0:97 N = 5766
0:51 N = 88
Elasticity of variance % of precautionary saving
Â2 p-value
(4:94)
(3:44)
(¡2:58)
(¡2:82)
(19:81)
(¡6:03)
(3:75)
(¡0:96)
In table 4, we look at the precautionary motive among di¤erent cohorts of age. We observe the same result as before ; the inclusion of the variance of income tends to increase the marginal propensity to consume the permanent income and it increases through the life cycle. This result is also valid for the parameter of the variance of income which increases when households get older. Young people, aged between 18 and 30, have a precautionary motive with an elasticity of consumption with respect to the variance of income of -0.044. There seems to be no di¤erence in the magnitude of the precautionary motive, since for every category the elasiticity is around 0.045. 13
Again this result is not intuitive since the young are more subject to income risk and should have a less clear picture of their permanent income. Table 4 : Permanent Income and Precautionary saving for di¤erent cohorts of ages Permanent Income Age
18-30
45-60
0:699 ¤¤¤
0:700 ¤¤¤ (17:03)
0:02 N = 2840
0:21 N = 1656
0:629¤¤¤
Disposable Income
Â2 p-value
30-45
(7:09)
0:05 N = 1270
(18:01)
Precautionary Saving 18-30
30-45
45-60
Disposable Income
0:800¤¤¤
0:819 ¤¤¤
0:864 ¤¤¤
Variance of Income
¡29:07¤¤¤
¡27:90¤¤¤
¡26:00¤¤¤
¡0:044 2.37
¡0:045 2.69
¡0:046 3.03
0:503 N = 1270
0:14 N = 2840
0:36 N = 1656
(7:85)
(¡3:83)
Elasticity of variance % of precautionary saving
Â2 p-value
(15:05)
(¡4:26)
(11:67)
(¡4:09)
In table 5, we explore the precautionary motive for workers according to their activity status. We notice that the size of the sample has an in‡uence on the results. When we estimate the model with a small sample, the t-test will not reject the PIH. It is surprising to …nd that people in the service, skilled workers and unskilled workers are PIH savers. This result may be explained by the size of the samples used. In this case, our variance estimator may be a bad approximation. When samples are larger, we see that the precautionnary motive is present.
14
Table 5 : Permanent Income and Precautionary saving according activity Permanent Income Income
Â2 p.v.
N
Government o¢cial
0:655¤¤¤
0:57
455
Scienti…c
0:504¤¤¤
0:24
1032
Intermediary activity
0:644 ¤¤¤ (10:56) 0:623¤¤¤ (7:93) 0:732¤¤¤ (7:32) 0:679¤¤¤ (8:70) 0:860¤¤¤ (5:32) 0:779¤¤¤ (5:18)
0:50
1274
0:50
644
0:45
516
0:16
806
0:69
330
0:86
174
(9:03) (4:89)
Clerical Service Craftsman Skilled worker Unskilled worker
Precautionary Saving
Government o¢cial Scienti…c Intermediary activity Clerical
Variance of Income
"c;¾2y
% of p.s.
Â2 p.-v.
0:875¤¤¤
¡29:04¤¤¤
-0.072
4.20
0:08
0:674¤¤¤
¡24:30¤¤¤
-0.048
2.88
0:32
0:749 ¤¤¤ (11:70) 0:737¤¤¤ (6:55)
¡21:32¤¤¤
-0.036
2.87
0:50
-0.033
2.20
0:74
-0.007
0.42
0:70
-0.055
3.29
0:27
(6:06) (5:11)
¤¤¤
(¡2:73) (¡3:44)
(¡7:23) ¡27:84¤¤ (¡2:14)
Service
0:771
Craftsman
0:758¤¤¤
(¡0:35) ¡57:60¤¤¤ (¡3:86)
Skilled worker
0:842¤¤¤
¡24:66
-0.021
1.41
0:31
0:765¤¤¤
¡30:65
-0.032
1.95
0:30
Unskilled worker
5
Income
(4:83)
(8:62) (5:04) (4:81)
¡4:44
(¡0:89) (¡0:83)
Conclusions
We showed in this paper that for the Swiss households as a whole the precautionary savings motive is present but is not quantitatively important. We also found that the consumption behavior is better explained by the precautionary model than by the permanent income hypothesis, as the di¤erent statistical tests suggested it and that the marginal propensity to consume the permanent income is higher when we estimate the former model. The precautionary motive is signi…cant for self-employed and farmers, categories which are more likely subject to income risk. Workers have also a precautionary motive but it is less imporant than for other occupation status. A closer look at this category suggests another conclusion, since for some of the activity categories considered the precautionary is not present. This might be explained by the size of the samples used but 15
also by the fact that people who are the most risk averse might choose activity which are less risky. The precautionary motive is present among the young but does not seem to be higher than for older households, a counter-intuitive result since the model predicts that older people should accumulate more wealth at the beginning of their life precisely because of the precautionary motive and because income risk should be higher for young people. This result may be explained by the fact that intertemporal horizon of households are much shorter than it is assumed usually. Although the assumptions regarding the risk aversion of the households and the stochastic process of income, and also the fact that income risk is only attributable to labor income might be judged unrealistic, we showed that the technical convenience of our approach allowed us to derive functional forms for consumption and to perform simple statistical tests for our two hypotheses. We showed how in an econometric model which uses cross-section data how to build an estimator of the variance of income. However, we need a quite large sample in order for this estimator to be a good approximation of the true variance. Panel data would have facilitated this task but they were not available in the Swiss context. We were unable to identify clearly at the microeconomic level what types of households are most subject to the precautionary motive. The results obtained and the theoretical result of Carrol and Kimball [6] on the concavity of the consumption function suggests the use of an empirical model where the marginal propensity to consume the permanent income depends on the level of wealth, the degree of income risk and the information available at the time the household decides its consumption plan.
6 6.1
Appendix Derivation of the precautionary saving model
The aim of this section is to give the technical details of the derivation of the consumption function given in section 2.3.1. This section closely follows the derivation given by Caballero [2]. The household maximise its expected intertemporal utility function U under its intertemporal budget constraint. µ ¶ 1 X 1 ¡µct+i i max U = ¯ E ¡ e (A1) t µ fci g1 i=0 i=0
subject to
1 X
i
® ct+i = wt +
i=0
1 X
®i yt+i
i=0
with ® = 1=(1 + r): r is the interest rate. We assume r = ±. The …rst order condition is h i e¡µct = Et e¡µct+1 :
(A2)
It is di¢cult to solve this problem analitically. In order to …nd a solution for ct for all t, we have to guess it. We assume that the solution is linear in the consumption, i.e. ct+i = ¡t+i¡1 + Át+i¡1ct+i¡1 + vt+i
(A3)
Substituting equation A3 in the FOC gives. exp fµ [(Át ¡ 1) c t + ¡t ]g = Et [exp f¡µvt+1g]
(A4)
This implies that Át = 1 for all t; else consumption is only determined by the FOC without taking into account the budget constraint. We …nd an expression for ¡ t: ¡t =
1 1 ln Et [exp f¡µvt+1g] > Et [¡µvt+1] = 0 µ µ 16
(A5)
Equation A3 becomes (A6)
ct+i = ¡t+i¡1 + ct+i¡1 + vt+i Solving recursively we obtain ct+i = ct +
i X
(A7)
(¡t+j¡1 + vt+j )
j =1
Now we assume that labor income is driven by the following ARMA process (A8)
Et [yt+i] ¡ Et¡1 [yt+i ] = ªi! t or put di¤erently yt+i = ¹ +
i X
(A9)
ªi¡j !t+j :
j=1
When we substitute A7 in the budget constraint and take expectations at time t, we have wt = ct
1 X
i
® +
i=0
+
1 X
1 X
®
i=1
®i
i=1
i X j=1
i
i X j=1
v t+j ¡
We …nd a solution for ct ct = ypt ¡ (1 ¡ ®)
¡t+j¡1 ¡
1 X
®i
i=1
1 X
®i
i=0
i X
1 X
®i Et [yt+i ]
(A10)
i=0
ªi¡j ! t+j
j =1
i X
¡t+j¡1;
(A11)
j=1
¡ ¢ P i where ytp ´ (1 ¡ ®) wt + 1 i=0 ® Et [yt+i] : We can identify the distribution of vt: From the budget constraint ??, we have 1 i X X ®i (vt+j ¡ ªi¡j !t+j ) = 0: (A12) i=0
j=1
This condition is satis…ed for all t if
1 X i=0
®i (vh ¡ ªi¡1 !h) = 0 8h:
(A13)
Finally, we obtain with ª ´ (1 ¡ ®)
P1
i i=0 ® ªi:
vt = ª!t;
(A14)
When we substitute A14 in A5, we obtain
1 ln Et+i¡1 [exp f¡µª!t+i g] : (A16) µ If the wt+i ’s are i.i.d.gaussian with zero expectation and variance ¾2w , then ¡ is constant and equal 2 2 to µª 2 ¾ ! and the consumption function is equal to ¡t+i¡1 =
ct = ytp ¡
® µª2 2 ¾ : 1 ¡® 2 !
17
(A17)
6.2
Intermediate results Table A1 : Results of the regression of current income on the determinants of the permanent income for the whole sample OLS Age
109:50 ¤¤¤
Age2
¡0:89¤¤¤
(3:58)
(¡2:39)
Sex
456:01
Swiss
305:43
(5:07) (3:05)
Nworkers
1631:14 (30:81)
no education
¡1764:65
obligatory school
¡1698:29
professional education 1
¡1552:86
professional education 2
¡844:33
professional education 3
¡446:74
professional education 4
247:69
professional education 5
555:70
(¡2:50) (¡2:83) (¡2:43) (¡1:44) (¡0:72) (0:35) (0:90)
General education 1
¡103:75
General education 2
¡380:70
Business school
¡729:52
(¡0:15) (¡0:63) (¡1:01)
Institute of technology
861:72
University
1889:04
Constant
7776:97
(1:43) (3:17) (0:93)
Number of obvservations : 6954 t -s t a t is tic s in p a re nt hes e
18
Table A2 : Averages of monthly consumption, variance of income and disposable income for the di¤erent samples used for the econometric results Consumption
Variance of Income
Disposable Income
9:68
6402:93
Whole Sample
5546:48
Self employed
6434:25 (3860:22)
(80:43)
18:93
6876:05
Farmer
4742:83 (2064:87)
19:46
(80:16)
5811:24
Worker
5624:26
9:39
6567:99
Unemployed
3724:97
1:66
3525:01
18-30
(2762:31)
(56:22)
(2666:09)
(51:41)
(1403:24)
(3:11)
4415:34
(3570:51)
(4679:51) (4501:80) (3441:61) (1438:37)
(1997:15)v
(50:26)
6:75
5604:19
30-45
5699:53 (2518:02)
9:21
(51:33)
6541:65
45-60
6474:23
11:57
7395:01
Governement o¢cial
7206:98 (3330:45)
(72:23)
17:98
8405:16
Scienti…c
6446:52 (2770:40)
12:79
(71:25)
7869:86
Intermediary activity
5738:71 (2636:07)
9:79
(51:27)
6704:02
Clerical
5062:60 (2286:83)
6:73
(33:54)
5890:14
Service
4922:31 (2922:96)
8:76
(63:16)
5553:02
Craftsman
4879:46 (1780:19)
4:66
(15:53)
5605:03
Skilled worker
5306:43 (2154:70)
4:43
(13:31)
5856:78
Unskilled worker
4741:05
4:94
5629:40
(3009:24)
(50:95)
(2361:02)
(11:73)
(2874:28) (3317:73) (3843:67)
(4738:02) (3869:37) (3474:39) (2878:37) (3310:84) (2269:72) (2302:41) (2426:73)
N o te : s t a nd ar d de via tio ns in p ar e nt hes e s
References [1] Browning; Martin, Lusardi, Annamaria, ”Household Saving : Micro Theories and Micro Facts”, Journal of Economic Literature 34 (December 1996) :1797-1855. [2] Caballero, Ricardo, ”Consumption Puzzles and Precautionary Savings”, Journal of Monetary Economics 25 (1990), 113-136. [3] Carroll, Christopher D., ”How does Future Income a¤ect Current Consumption?”, Quarterly Journal of Economics, CIX (1994), 111-48 [4] Carroll, Christopher D., ”Bu¤er Stock Saving : Some Theory”, 1996, mimeo, The Johns Hopkins Unifersity. [5] Carroll, Christopher D., ”Bu¤er-Stock Saving and the Life Cycle/Permanent Income Hypothesis”, Quarterly Journal of Economics, CXII (1997), 1-55 19
[6] Carroll, Christopher D., Kimball, Miles S., ”On the Concavity of the Consumption Function”, Econometrica 64, 1996, 989-992. [7] Davidson, Russel, MacKinnon, James G., ”Estimation and inference in Econometrics”, Oxford, Oxford University Press, 1993. [8] Deaton, Angus, ”Saving and Liquidity Constraints”, Econometrica 59, 1991, 1221-48 [9] Deaton, Angus, ”Understanding Consumption”, Oxford : Oxford U. Press, 1992. [10] Hall, Robert E., ”Stochastic Implications of the Life Cycle-Permanent Income Hypothesis : Theory and Evidence”, Journal of Political Economy 86 (December 1978) : 971-87. [11] Hayashi, Fumio, ”Econometrics”, Princeton : Princeton University Press, 2000. [12] Kimball, Miles S., ”Precautionary Saving in the Small and in the Large”, Econometrica 58, 1990, 53-73. [13] Lusardi, Annamaria, ”On the Importance of the Precautionary Saving Motive”, American Economic Review. Vol. 88 (2). p 449-53. May 1998 [14] Miles, David, ”A Household Level Study of the Determinants of Incomes and Consumption”, The Economic Journal 107 (janvier 1997), 1-25. [15] Romer, David, ”Advanced Macroeconomics”, international edition, Mc Graw-Hill, 1996. [16] Skinner, Johnathan, ”Risky Income, Life Cycle, Consumption and Precautionnary Savings”, Journal of Monetary Economics 22 (1988), 237-255. [17] Wansbeek, T., Meier, E., ”Measurement Error and Latent Variables in Econometrics”, Advanced Textbooks in Economics, 37, Elsevier Science, 2000 [18] Zeldes, Stephen P., ”Optimal Consumption with Stochastic Income : Deviations from Certainty Equivalence”, Quarterly Journal of Economics, CIV (mai 1989), 275-98.
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