The data generated from the agent-based computational consumption CAPM model is then used to fit the Euler consumption equation, which is derived from the assumption
of homogeneous agents under rational expectations.13 Below we shall follow Hansen and Singleton (1983) to re-derive this equation to fit our specific context.14
Consider the representative consumer with a CRRA (constant relative risk aversion) utility function:15
u(ct) = c1t−ρ/1− ρ, ρ > 0. (45) The representative consumer in this economy is assumed to choose a consumption plan so as to maximize the expected value of his time-additive utility function,
E[
∑
∞r=0βtu(ct+r) | Ωt], 0 < β < 1. (46) The mathematical expectation E(· |) is conditioned on information available to agents at time t,Ωt. Current and past values of real consumption and asset returns are assumed to be included inΩt.
Agents substitute present for future consumption by trading the ownership rights of M assets. As above, the vectorqt denotes the holdings of the M assets at the date t,pt
denotes the vector of prices of the M assets, andwtdenote the vector of M values of the dividends at date t. Then agents’ consumption and investment plan (ct, qt) maximize (46) subject to the sequence of budget constraints,
ct+1+ pt+1·q t+1≤ (pt+ wt) ·q t. (47) The first-order necessary conditions, that involve the equilibrium price of the M assets, are
u (ct) = β · E[u (ct+1) | Ωt] · Rm,t; m= 1, ..., M, (48) where Rm,t= pm,tp+wm,tm,tis the return on the mth asset expressed in units of the consumption good.
The definition of asset returns here is different from the usual derivation. This is be-cause we have a different time line for agents. Due to computational hardness for the fix point, our temporal equilibrium is not Walrasian. Agents submit their orders based on their estimated price pim,t, which in general is different from the realized temporal equilibrium price pm,t. In other words, the equilibrium prices only happen after their submission. By this time line, when they are making the decision for the period t+ 1 the effective return is actually the Rm,tdefined above.16
13There is also an assumption about the joint distribution of consumption and returns. We shall be back to this issue later.
14The Hall (1988)’s derivation is similar, and, therefore, is skipped.
15The CRRA utility function is what we need here. In fact, our agent-based simulation is further restricted to the case whereρ =1, i.e., the log utility function. See Table 1.
16In fact, an alternative measure which can capture the capital gain is
Rm,t= pm,t+ wm,t
pim,t = pm,t+ wm,t
pm,t−1 .
The second equality is based on the random-walk assumption, Equation (17). This discussion of different measure of returns points out the relevance of trading mechanisms. Is it traded with continuous double auction, or traded with Walrasian auctioneer, or traded with a rationing scheme? Obviously, empirical literature may not be interested in this distinction because the consumption Euler equation is only applied to the low-frequency
Substituting (45) to (49) and rearranging gives E[β(ct+1
ct )−ρ· Rm,t| Ωt] = 1; m = 1, ..., M, (49) Assuming the joint distribution of consumption and returns is lognormal, from (49), a restricted linear time-series representation of the logarithms of consumption and asset returns can be derived. Let
xt ≡ ct/ct−1, Um,t≡ x−ρt Rm,t−1. Then (49) can be rewritten as
E(Um,t | Ωt−1) = 1/β, m = 1, ...M. (50) Next, let
Δct ≡ logxt, rm,t≡ logRm,t, Yt ≡ (Δct, r1,t−1, ..., rM,t−1),
um,t≡ logUm,t= −ρΔct+ rm,t−1 (m = 1, ..., M),
and Ωyt−1 denote information set {Yt−s : s ≥ 1}. Further, assume that Yt is stationary Gaussian process. This distributional assumption implies that the distribution of um,t
conditional on Ωyt−1 is normal with a constant variance σm2 and a mean μm,t−1 that is a linear function of past observation on Yt.
Hence,
E(Um,t| Ωyt−1) = E(exp[um,t]) = exp[μm,t−1+ (σm2/2)], m = 1, ...M. (51) SinceΩyt−1 ⊆ Ωt−1, we can take expectations of both side of (50) conditional onΩyt−1to obtain
E(Um,t| Ωyt−1) = 1/β. (52)
Equating the right-hand sides of equation (51) and (52) and solving forμm,t−1yields μm,t−1= −logβ − (σm2/2). Define
Vm,t ≡ um,t− μm,t−1= −ρΔct+ rm,t−1+ logβ + (σm2/2), (53) Then,
E(Vm,t | Ωyt−1) = 0 and
E(rm,t−1| Ωyt−1) = ρE(Δct | Ωyt−1) − logβ − (σm2/2), (54) Because that
rm,t−1= E(rm,t−1| Ωyt−1) + εm,t−1 (55)
data, never to the daily data, not to mention the high-frequency data. It, therefore, raise the question: which time frame is actually the most appropriate one to examine Euler consumption regression. The issue may not be that important as far as the aggregate data is concerned. Nonetheless, when we are moving to individual data, as the current empirical research indeed does, this issue is no longer irrelevant. The advantage of agent-based modeling is that it allows to explore the possible existence of heterogeneity in the time frame.
Δct = E(Δct | Ωyt−1) + υt (56) Substituting (55) and (56) into (54) and rearrange, we obtain
rm,t−1= ρΔct− ρυt+ εm,t−1− logβ − (σm2/2), (57)
Define
ηt = −ρυt+ εm,t−1,
and μ = −logβ − (σm2/2),
the equation (57) becomes
rm,t−1= μ + ρΔct+ ηt, (58)
whereμ is a constant, and ηtis the random term.17Similarly, one could follow Hall (1988) to derive the inverse form of (58).
Δct= τ + ψrm,t−1+ ξt. (59)
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