Testing weak exogeneity in multiplicative error models

Abstract

Empirical market microstructure literature widely employs the non-linear and non-Gaussian multiplicative error class of models (MEMs) in modelling the dynamics of trading duration and financial marks. It routinely maintains the weak exogeneity of duration vis-à-vis marks in estimations. However, microstructure theory states that trade duration, volume and transaction prices are simultaneously determined. We propose Lagrange-multiplier (LM) tests for weak exogeneity for the MEMs. Our LM tests are extensions of the weak exogeneity tests applicable to VAR or VECM models with Gaussian distributions. Empirical assessments show that (i) weak exogeneity is widely rejected by the data in the MEMs and (ii) the failure of weak exogeneity seriously biases parameter estimates. We hope our tests will be of interest in future empirical applications.

Keywords:

• Weak exogeneity
• Multiplicative error model
• LM test
• Market microstructure

JEL Classification:

• C32
• C12
• G10

Acknowledgements

We would like to thank Luc Bauwens for his valuable comments and Simone Manganelli for providing us with the data used in this paper. We would also like to thank the participants in the 10th BMRC—DEMS Conference on Macro and Financial Economics and Econometrics in May 2014. Kul Luintel would like to thank CESifo Group Munich for their excellent hospitality during his research visit in July 2016 when a major part of the revision of this paper is completed

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Durations are typically the time elapsing between trades of financial assets, whereas market marks, commonly of most interest, are the trading volume, bid-ask spread and the return volatility.

² Engle (2002) and Cipollini et al. (2007) propose for the dynamics of non-negative value processes. The MEMs incorporate the autoregressive conditional duration (ACD) model (Engle and Russell 1998) for financial duration and the GARCH model for return volatility as a special case.

³ Engle (2000) analyses the dynamics of duration and volatility recursively and finds that the longer durations lead to lower volatilities, which is also confirmed by Manganelli (2005) who analyses duration, volume and volatility jointly. In contrast, Grammig and Wellner (2002) formulate interdependent intraday duration and volatility models and report that lagged volatility significantly reduces transaction intensity. Dufour and Engle (2000), under the recursive VAR model, show that prices, bid-ask spreads and price volatility all increase when traders observe short duration. Bowe et al. (2009) analysing a trivariate VAR, find that duration is affected positively by volatility, which is opposite to the findings of Engle (2000).

⁴ To our knowledge, the only available distribution for these processes is the Multivariate Gamma Distribution; however, it only admits positive error correlation which is too restrictive, as shown by Cipollini et al. (2007).

⁵ Although the specification (2) is of first order, this test principle can easily accommodate higher order autoregressive terms.

⁶ Note this test principle is similar to Hausman’s two-stage testing for weak exogeneity.

⁷ We would like to thank an anonymous referee for encouraging us to illustrate this point.

⁸ To avoid the negative value of volume, we use a logarithmic version of the ACD model (see e.g. Bauwens and Giot 2000) for data generating process and estimation.

⁹ We generate 5200 data and use the first 5000 data for the estimation and the remaining 200 for out of sample forecasting.

¹⁰ See subsection 4.1 in Manganelli (2005) for a concise description of data prepared; we follow the same approach.

¹¹ The formula is from Manganelli (2005) equation (12).

¹² The details of the AACD/APGARCH model and estimated results are in appendix 4. The estimation results show that the (asymmetric) log-ACD models are appropriate for the duration process, while the (asymmetric) linear GARCH models are appropriate for the volatility process in general.

¹³ The asymmetric effect of transaction is driven by a sell or buy process.