Monte Carlo results: macro parameters

As an initial feasibility check we tried to estimate all macro parameters ξ^M using the first-step auxiliary model described in Section 4.2.2 based on simulated samples with T=100k. These experiments revealed that the subset of the macro parameters ξ^M∗ = (µc, µd, ρ, ϕe, σ, φ, ϕd)⁰ could be reliably recovered by maximizing the first-step objective function (4.4) but not the SV parameters ν1 and σw, for which we obtain vastly different estimates ˆν₁ and ˆσ_w when using different initial values. This result raises the concern that the first-step auxiliary model may be unable to identify ν₁ and σ_w. Extending the auxiliary model in various directions does not alleviate the

6 This is a safety measure to prevent reporting overly optimistic results, but it makes the optimization more difficult. As a result, the optimization could not be successfully accomplished for some replications, in particular for small T. The optimization algorithm either exceeded the maximum number of iterations, or converged to implausible values (more than ten times larger than the true value in absolute terms). We consider these cases as failed estimation attempts and exclude them in the tables and plots that summarize the simulation study results. In the second estimation step, an estimation is also classified as failed if the LRR model is not solvable at the parameter values to which the optimization converges. In an empirical study, such problematic data could receive special treatment, by increasing the maximum number of iterations, or by using alternative optimization algorithms. Such an expensive handling is not tenable in a Monte Carlo study.

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problem. Including a heterogeneous autoregressive conditional heteroscedasticity model, as discussed in Appendix 4.A.4, does not allow to identify ν₁ and σ_w either.

Figure 4.1 suggests a possible explanation. It shows that in BY’s calibrated LRR economy the volatility of volatility is indeed very small, which suggests that the signal-to-noise ratio may be too low to estimate ν1 and σw.

These findings suggest an alternative estimation strategy, in which the condi-tional variance σ_t² is predicted by its unconditional expectation, E(σ_t²) = σ². Es-timating σ² within the first-step indirect inference estimation procedure entails re-placingσ_t²byσ² when generating LRR model-implied data. Whileν₁ andσ_w are not estimated in the first step, the true data-generating process still exhibits stochastic volatility: we do not change the model, but deliver an alternative estimate of σ_t². The Monte Carlo study investigates the consequences for the quality of the other parameter estimates.

In each replication we therefore estimate the reduced set of macro parametersξ^M∗ by minimizing the indirect inference objective function in Equation (4.4). The aux-iliary parameter vectorθ^M is constructed as described in Section 4.2.2, and with the following customization. In the HAR specification in Equation (4.1) we account for consumption and dividend growth on the annual and the triannual level by choosing h1 = 12 and h2 = 36. The first few monthly lags should be particularly informative for the estimation of the persistence parameterρ, so we setτ = 6. Initial estimations indicated that a precise match of the means and standard deviations of consumption and dividend growth can enhance the precision of the estimates of µ_c and µ_d and that of the variance-scaling parametersϕ_eand ϕ_d, which prove difficult to estimate.

This match is accomplished by using a diagonal weighting matrix W^M_T with values of 1 on the main diagonal, except for the entries that correspond to the first two elements of µg and σg, which receive a large weight (10⁴).

As a benchmark, we also perform a GMM estimation that relies on moment matches inspired by the studies of Hasseltoft (2012) and Constantinides and Ghosh (2011). For that purpose we exploit that the population moments of log consumption and dividend growth implied by the LRR model can be expressed as functions of the parameter vector ξ^M. The GMM strategy is based on exact identification using the seven moments given in Appendix 4.A.2.

Table 4.1 reports the medians and root mean squared errors (RMSEs) of the first-step indirect inference estimates (Panels A and B) and the GMM estimates (Panel C). Figure 4.2 illustrates the indirect inference results using kernel estimates.

In addition, Appendix 4.A.5 provides a comparison of the two estimation approaches regarding the precision of model-implied moment matches. The T=100k results show that the proposed indirect inference estimation strategy is feasible and works well. Biases and the RMSEs shrink, there are no estimation failures, and the bell-shaped kernel estimates center closely around the true parameter values. Using σ² instead of σ_t² when simulating LRR model-implied data does not affect the quality of the other parameter estimates. Panel B of Table 4.1 shows the results assuming that ν₁ and σ_w are known. These results do not differ qualitatively from those in Panel A, which reports the results when σ_t² is predicted by σ². This conclusion holds for all simulated sample sizes. The estimation precision is different across macro parameters. Not surprisingly, the estimates of the parameterϕ_e, which scales the variance of the latent expected growth componentx_t, andφ, the parameter that leverages the effect of x_t on expected dividend growth, are less precise. However, compared with the GMM results reported in Panel C, the indirect inference RMSEs are much smaller. A considerably smaller RMSE is also obtained for the persistence parameterρ. Figure 4.3 shows that the distribution of the indirect inference estimate

ˆ

ρ is much more closely centered around the true value than the GMM counterpart.

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Precise estimation becomes more difficult using smaller sample sizes, as indicated by the increase in the RMSE and the wider distribution of the estimates around the true parameters. Efficiency varies across parameters in a similar way as in the large sample. As could be expected from the 100k results, the critical parameters ϕe and φ prove most difficult to estimate precisely. However, we do assert that the optimization of the indirect inference objective function yields reliable results in that the algorithm converges to the same minimum, independent of the starting values.⁷ We conclude that the indirect inference strategy is reliable. Using the currently available sample sizes one should not expect a high estimation precision for some of the structural parameters, though. We believe that the simulation study draws a realistic picture of the estimation precision that can be expected in an empirical study.