Bootstrapping Two-Stage Quasi-Maximum Likelihood Estimators of Time Series Models

Abstract

This article provides results on the validity of bootstrap inference methods for two-stage quasi-maximum likelihood estimation involving time series data, such as those used for multivariate volatility models or copula-based models. Existing approaches require the researcher to compute and combine many first- and second-order derivatives, which can be difficult to do and is susceptible to error. Bootstrap methods are simpler to apply, allowing the substitution of capital (CPU cycles) for labor (keeping track of derivatives). We show the consistency of the bootstrap distribution and consistency of bootstrap variance estimators, thereby justifying the use of bootstrap percentile intervals and bootstrap standard errors.

KEYWORDS:

• Copulas
• Inference
• Multivariate GARCH

Supplementary Materials

The supplemental appendix contains all proofs.

Notes

1 For a textbook discussion of inference in multi-stage estimation see White (1994) and Newey and MacFadden (1994).

2 Other artcles that have proposed fast resampling methods include Davidson and MacKinnon (1999), Andrews (2002), Gonçalves and White (2004), Hong and Scaillet (2004), and La Vecchia, Moor and Scaillet (2020), among others. However, all these articles consider one-step M or GMM estimators.

3 These results can easily be generalized to multistage QMLE. In our Monte Carlo simulations, we estimate a copula model involving a three-stage QMLE.

4 Under general heterogeneity and time series dependence, α₀ and ${\beta }_0$ could depend on n. We omit the index n to simplify the notation.

5 The validity of the bootstrap for LM statistics follows from similar arguments. To conserve space and because Wald tests are more popular than LM tests, we only report results for Wald statistics.