First step: macro parameter estimation

The first indirect inference estimation step thus only deals with the estimation of the macro parameters ξ^M. For that purpose, we must specify an auxiliary model that captures the properties of the LRR model-implied macro process y^(b)_M(ξ^M). Let us collect the first-step auxiliary model parameters in the vector θ^M ∈ Ξ^M ⊂ R^kM, where k_M is at least as large as the number of macro parameters, and presume that auxiliary parameter estimates ˆθ^M can be obtained by maximizing the criterion function Q^M_T ({w^(b)_M,t}^T_t=1,θ^M).

The challenge for the first-step auxiliary model is to account for the predictable growth componentxt, which induces small but very persistent serial correlations in

INDIRECT INFERENCE METHODOLOGY

the growth series. These deviations from i.i.d. growth let the asset pricing implica-tions of the LRR model unfold. A parsimonious way to capture the autocorrelation structure of a persistent process is the HAR specification proposed by Corsi (2009).

It is used in the realized volatility literature to capture the long-memory properties of squared and absolute returns by accounting for different sampling frequencies in an autoregressive model. To set up the first-step auxiliary model, we therefore use the following HAR specification for log consumption and dividend growth observed at the base frequency:² noise innovations. In an empirical application, the base frequency (b) could be quarterly (as in Hasseltoft, 2012) or annual (as in Constantinides and Ghosh, 2011).

f(h1) andf(h2) denote lower frequencies that result from a time aggregation of the base frequency data over h_i periods. With a quarterly base frequency, we would use h₁ = 4 and h₂ = 12 to obtain annual and triannual data. The time aggregation of consumption and dividend growth is based on the formulas given in Appendix 4.A.1.

Compared with a standard vector-autoregressive process, the HAR specification can account for the long-run impact of shocks to consumption and dividend growth in a parsimonious way, as the large required number of lagged growth rates gets replaced by few aggregates. The auxiliary parameters that result from the HAR specification are collected in the vector

θ^HAR = (c₁, c₂,vec(Φ₁)⁰, . . . ,vec(Φ_τ+2)⁰,vec(Σ_ζ)⁰)⁰, (4.2)

2 We are grateful to George Tauchen for suggesting the use of the HAR specification as an auxiliary model.

where Σ_ζ is the covariance matrix of ζ_t. The flexibility of the first-step auxiliary model is enhanced by extending the auxiliary parameter vector to include the means and standard deviations of the two growth processes and their time-aggregates,

g_t=

g_t^(b), g_d,t^(b), g_t^(f(h1)), g^(f_d,t^(h1)), g_t^(f(h2)), g_d,t^(f(h2))0

, (4.3)

which we collect in the vectorsµ_gandσ_g. The complete vector of first-step auxiliary parameters is then given by θ^M =

θ^HAR0,µ_g⁰,σ_g⁰⁰

. OLS regressions yield the estimates ofθ^HAR, and sample moments are used to estimateµg andσg. Assuming for the auxiliary model that w^(b)_M is a Gaussian process (a natural assumption as the innovations in Equations (2.1)-(2.4) are i.i.d. N(0,1)), the elements of ˆθ^M_T can be interpreted as pseudo-maximum likelihood estimates, and the criterion Q^M_T as a pseudo-likelihood function.

The number of auxiliary parameters exceeds the number of macro parameters, so that we use the following first-step indirect inference estimator of ξ^M:

ˆξ^M_T = argmin ξ^M ∈ Θ^M

∆^M(ξ^M)⁰ W^M_T ∆^M(ξ^M), (4.4)

where ∆^M(ξ^M) = ˆθ^M_T − θ˜^M_S (ξ^M). θ˜^M_S (ξ^M) denotes the estimate of θ^M that is obtained when the auxiliary parameters are estimated on simulated LRR model-implied data of sample sizeS, whereS is chosen as a fixed multipleH of T. W^M_T is a symmetric and positive definite weighting matrix, W^M_T →

p W^M, a non-stochastic positive definite matrix. The weighting matrix W^M_T can be used to enforce precise matches of elements of ˆθ^M_T and ˜θ^M_S .

Under the assumptions stated by Gourieroux et al. (1993), the first-step indirect inference estimator in Equation (4.4) is a consistent estimator ofξ^M₀ . In addition to

INDIRECT INFERENCE METHODOLOGY

stationarity and ergodicity of the data generating processes, we have to assume that the criterion function Q^M_T ({w^(b)_M,t}^T_t=1,θ^M) converges uniformly and almost surely to a non-stochastic limit function Q^M_∞(F₀,ξ^M₀ ,θ^M), where F₀ denotes the true distri-bution function of the fundamental innovations in Equations (2.1)-(2.4). Moreover, we have to assume that the limit function is continuous in θ^M and has θ^M₀ as the unique maximum. Defining

b(F,ξ^M) = argmax θ^M∈Ξ^M

Q^M_∞(F,ξ^M,θ^M), (4.5)

we have θ^M₀ =b(F₀,ξ^M). Consistency requires that the binding function

b(F₀,·) :ξ^M →b(F₀,ξ^M) (4.6)

is injective and that ^{∂b(F0,ξM0 )}

∂ξ^M0 is of full column rank.

While the rank condition is fulfilled, as can be assessed by simulation, the injec-tivity condition cannot be formally checked since the binding function is not available in closed form. Connections between auxiliary and structural parameters are obvi-ous, though. The autoregressive parameter matrices Φ should provide information about the persistence parameter ρ and the leverage ratio on expected consumption growth φ. The parameters c1, c2, and µg are linked to the unconditional expected values of log consumption and dividend growth, µ_c and µ_d, while the second mo-ments in Σ_ζ and σ_g should contribute to the identification of the variance-scaling parameters ϕ_e and ϕ_d and the parameters of the stochastic volatility process. To assess the feasibility of the estimation approach and to provide simulation-based evi-dence on the injectivity of the binding function, we conduct a Monte Carlo study and check whether the indirect inference strategy can reliably recover the true structural parameters ξ^M₀ when a large sample size is available (see Section 4.3.2).

Under the regularity conditions and assumptions stated by Smith (1993) and Gourieroux et al. (1993), the first-step indirect inference estimator ˆξ^M_T in Equa-tion (4.4) is asymptotically normal. As an alternative to using the large sample formulas, we rely on bootstrap-based inference, which we describe in Section 4.2.5.