Linearisation and aggregation

The Euler equation under uncertainty (25) is non–linear and presupposes an appropriate formulation of the expectations formation process. It has become common practice to log–linearise this equation (as first proposed by Hansen and Singleton, 1983, see equation (7) in Capéau and De Rock, 2014a). This practice was heavily criticised by Carroll (2001b) and Ludvigson and Paxson (2001). Car- roll (1997) already showed that omitting the second order term from the linearisation was one of the reasons why it was overlooked that consumption growth tracking earned income growth can be reconciled with the life–cycle framework under uncertainty. Gourinchas and Parker (2002) notably argue that the estimated effect of household size on consumption growth would be biased using the linearisation approach, and takes over part of precautionary motive.28 _{Attanasio and Low (2004) de-}

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The recent contributions of Jørgensen (2014, 2015) argue that the bias is not so much a consequence of the linearisation, but of inappropriately taking into account implicit or explicit credit constraints in the Euler equation approach. Indeed, when households would like to increase consumption when new children arrive, but cannot because of credit constraints, one risks to underestimate for example the household composition effect.

fend the linearisation approach and argue that with a sufficiently long time period of observation and sufficient variation in the real interest rate, bias will be limited. Alan, Atalay and Crossley (2012) confirm this but are more sceptical about the applicability of their result on more realistic data structures. While we have quite some time variation in the real interest rate (see Figure 10 below), unfortunately our length of observation period (15 years) is rather short as to what these authors consider (30 to 60 years). Our point of departure is however different. In line with Browning and Ejrnæs (2009), we investigate in how far a richer specification of the effect of household demographics as commonly is the case, can explain observed patterns of consumption growth without invoking a precautionary motive for saving. Contrary to Browning and Ejrnæs (2009), we cannot completely reject the contribution of a precautionary motive (see the discussion of Figure 11 below). In that sense our results need to be interpreted with some care. We conjecture however that from a low frequency perspective as ours, the potential bias in the estimates caused by the linearisation would be limited, as life–cycle motives tend to predominate precautionary motives in the longer term. For the family of models introduced in Section 4.2, the log–linearised version of the Euler equation reads as: ln (ct+1)−ln (ct) =σ lnβ+ ln 1 +rt 1 +πt + ln (γ(nt+1, et+1))−ln (γ(nt, et)) +εt+1. (26) In this equation,εt+1 embodies deviations of specific realisations of possible future values from their expected value (which on average are equal to zero), but also the error term due to the linearisation, which might be different from zero in expectation. It is the last one which causes potential biases in the estimates.

Equation (26) is supposed to hold for individual agents. By lack of true panel data, we will need to es- timate it, however, at the cohort level. Aggregation should then be done carefully. In macroeconomic applications it was common practice to test this type of equations on aggregate data. That is, on per capita figures for consumption drawn from national yearly or quarterly accounts for consumption. The counterpart for our application would be to estimate the mean consumption of people belonging to a generation g in observation years t =1995/96, 1996/97, . . . , 2010, making similar calculations for the household size and equivalence scale, and then estimate equation (26). However, this would introduce an aggregation bias.

An additional advantage of the log–linearised approach is that we can follow the aggregation procedure suggested by Blundell, Browning and Weber (1994), and Attanasio and Browning (1995b), which is more closely connected to the microeconomic foundation of these type of equations.29 Indeed, as equation (26) characterises individual behaviour, it is more appropriate to aggregate the logarithmic transformations of the relevant variables. That is, we estimated, for each of the observation years t = 1995/96, 1996/97, 1997/98, and 1999 to 2010, means over all people belonging to the same

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generation, of the log of consumption, the log of household size and the log of the equivalence scale.30 By consumption, the household size and equivalence scale of a person, we mean the consumption, the household size and equivalence scale of the household to which that person belongs. If these means are good approximations of the population counterparts (which is a cohort at a particular age a= t₋g), and individual behaviour satisfies (one of the) model(s) characterised by equation (26), then the aggregates of the logs should obey equation (26) too (while this need not be the case for the log of the aggregates).

We deviate again from the usual practice by calculating these numbers not for an arbitrarily chosen person in the household, but for all household members. This is in line with the argument we gave for interpreting the Euler equation as an equally valid characterisation of the behaviour of all household members (under the assumption they have identical preferences). We will denote the means of these logs as follows: ln (x)_g,t:= P i∈gwt,iln (xt,i) P i∈gwt,i , (27)

whereirefers to anindividual in the survey;i_∈gmeans that this individual belongs to generationg (i.e., is born in year g);wt,i is the sample weight of the household to which that individual belongs; and xt,i is the value in yeart of the variable x for the household to which individualibelongs, with x being consumption, household size and equivalence scale.