SLIDE 1
Lecture 3: Euler-Equation Estimation
Simon Gilchrist Boston Univerity and NBER
EC 745
Fall, 2013
SLIDE 2 Euler equation tests
This lecture considers some further implications and tests of the “standard” asset pricing model: for any gross return Rt+1, we have the equation: Et βu′(Ct+1) u′(Ct) Rt+1
Hence for any gross return Rt+1, we have Et βu′(Ct+1) u′(Ct)
t+1
These equations hold for any asset which you can buy or sell (at the margin): stocks, bonds, options, commodities, etc. Key intuition: assets are priced according to their covariance with consumption growth. Assets which pay off when consumption growth is high (in good times) are risky, hence they will need to have high expected returns for investors to want to hold them.
SLIDE 3
Euler equation methods
Earlier lecture studies the implication of the Euler equation, plus an assumption that consumption growth is iid and dividends = consumption. Euler equation methods requires very few assumptions - no need to specify the shocks, the technology (as you would in a GE model), or the endowment process. The logic of Euler equation methods applies to models others than the CRRA representative agent model.
Example: Suppose utility depends on ct and other observable xt, say U(ct, xt), then we have Mt+1 = βU1 (ct+1, xt+1) U1(ct, xt) , and Et (Mt+1Rt+1) = 1 can similarly be tested.
SLIDE 4 Euler equation tests
Specify u(C) = C1−γ
1−γ . For any variable Zt known at time t, we
have: ZtEt
Ct+1 Ct −γ Rt+1 − 1
Since Zt known at t: Et
Ct+1 Ct −γ Rt+1 − 1
Take unconditional expectations: E
Ct+1 Ct −γ Rt+1 − 1
This is a set of restrictions that the theory imposes on the joint distribution of consumption growth, returns and Zt.
SLIDE 5 Alternative interpretation
Express the Euler equation as: βu′(Ct+1) u′(Ct) Rt+1 − 1
where εt+1 = βu′(Ct+1) u′(Ct) Rt+1 − 1
βu′(Ct+1) u′(Ct) Rt+1 − 1
- By construction εt+1 is a forecast error – the difference between
realized and expected values. Therefore it is uncorrelated with any variable known at time t: E (Ztεt+1) = 0 No need for distributional assumption, except that returns and consumption growth must be stationary.
SLIDE 6 Instrument Relevance
Any stationary variable Zt realized at time t is a valid instrument. if Zt is uncorrelated with future returns and consumption, then it does not add any restriction since = E
Ct+1 Ct −γ Rt+1 − 1
E(Zt)E
Ct+1 Ct −γ Rt+1 − 1
Instrument relevance: Zt must forecast future returns or future consumption growth.
SLIDE 7 Empirical Implementation
This equation holds for any returns Rt+1, Zt, so we have NK equations, where N = # of returns and K = # of “instruments” Zt, and we have 2 parameters (β, γ). Intuitively, we simply pick β and γ to set the sample version of these equations equal to zero, i.e. 1 T
T
Zt
Ct+1 Ct −γ Rt+1 − 1
SLIDE 8 Exact identification:
Let Zt = 1 and apply to excess aggregate stock returns: choose γ such that 1 T
T
Ct+1 Ct −γ Re
t+1 − Rf t+1
Note: this may not be possible! i.e. with the data there may not be a γ s.t. this holds. Let Zt = 1 find β and γ to match both stock returns and the risk-free rate: 1 T
T
Ct+1 Ct −γ Re
t+1 − 1
1 T
T
Ct+1 Ct −γ Rf
t+1 − 1
Two eqns in two unknowns. Again, this may not be possible to find both β and γ in sample.
SLIDE 9 Over-identification
If more than equations, then of course (generically) cannot set all the equations equal to zero. In practice minimize the weighted errors: min
β,γ g′ T WT gT ,
where gT = 1
T
T
t=1 Zt
Ct
−γ Rt+1 − 1
weighting matrix. Can pick WT = I, or WT so that moments have the same scale,
- r choose W in a statistically optimal way. Formula for WT is
the inverse of the asymptotic variance of the moment vector: W =
T
T
Ct+1 Ct −γ Rt+1 − 1 −1
SLIDE 10 Over-identifying tests
Overidentification: if NK > # of parameters the system is
Test that 1
T
T
t=1 Zt
Ct
−γ Rt+1 − 1
SLIDE 11 GMM general formulas
Express a model as E[f(xt, b)] = 0 The general GMM estimate ˆ b is given by: aT gT (ˆ b) = 0, where gT (b) ≡ 1 T
T
f(xt, b), and aT is a matrix that defines which linear combination of gT (b) will be set to zero.
SLIDE 12 Asymptotics
The asymptotic distribution of the GMM estimate is: √ T(ˆ b − b) → N[0, (ad)−1aSa′(ad)−1′], where d ≡ E ∂f ∂b′ (xt, b)
∂b′ , a ≡ aT , S ≡
∞
E[f(xt, b)f(xt−j, b)′]. Hansen (1982) gives the sampling distribution of the moments gT (b): √ TgT (ˆ b) → N[0, (I − d(ad)−1a)S(I − d(ad)−1a)′].
SLIDE 13 Efficieny
The efficient estimate is obtained by setting: a = d′S−1 In this case, √ T(ˆ b − b) → N[0, (d′S−1d)−1] The quadratic objective function provides a test of
- ver-identifying restrictions:
TJT = TgT (ˆ b)′S−1gT (ˆ b) = χ2(NK − x)
SLIDE 14
Testing parameter restrictions
Suppose we estimate a model that imposes R restrictions on b. We can test these restrictions using: T (JT,R − JT ) = χ2(R). where JT,R is the value of objective estimated under the restriction. Comment: when imposing the restrictions we should use the unrestricted weighting matrix. Otherwise the test statistic has a non-central χ2 distribution.
The test statistic is constructed as a quadratic function of the difference between the two moment vectors: (gT,R − gT )′WT (gT,R − gT ) If we used alternative weighting matrices the difference in the J statistics is not guaranteed to be positive.
SLIDE 15 Ordinary Least Squares
β solves: Minβ ET [(yt − β′xt)2] In this case: f(xt, β) = xt(yt − β′xt) = xtεt, gT (β) = ET [xt(yt − β′xt)], aT = I, d = −ET (xtx′
t),
ˆ β = ET [xtx′−1
t
ET (xtyt) var(ˆ β) = 1 T ET [xtx′−1
t
[
∞
E(εtxtx′
t−jε′ t−j)]ET [xtx′−1 t
SLIDE 16 An implication of Euler equations
With Euler equations we have gT (b) = 1 T
T
Ztεt+1(b) We then choose b to minimize the quadratic gT (b)WT gT This is equivalent to choosing b so that d(gT (b))′ db WT gT = 0 In this case, the optimal weighting matrix is WT =V−1
gT where
VgT = E
SLIDE 17 Variance-covariance matrix
Note that in Euler equation case: E (f(xt, b)f(xt−s, b)) = E
t−s+1Z′ t−s
- If we confine attention to one period returns then εt is serially
uncorrelated and this exprssion is zero for s = 0. In this case, the variance-covariance matrix can be constructed as E
= 1 T
T
Ztεt+1ε′
t+1Z′ t
If we use multi-period returns then we induce serial correlation and we need a Newey-West style estimator of the variance of the moment matrix.
