Does beta react to market conditions? Estimates of ‘bull’ and ‘bear’ betas using a nonlinear market model with an endogenous threshold parameter

Abstract

The authors use a logistic smooth transition market (LSTM) model to investigate whether ‘bull’ and ‘bear’ market betas for Australian industry portfolios returns differ. The LSTM model allows the data to determine a threshold parameter that differentiates between ‘bull’ and ‘bear’ states, and it also allows for smooth transition between these two states. Their results indicate that ‘bull’ and ‘bear’ betas are significantly different for most industries, and that up-market risk is not always lower than down-market risk. LSTM models indicate that the transition between ‘bull’ and ‘bear’ states is abrupt, supporting a dual-beta market modelling framework.

Keywords:

• Bull and bear betas
• Dual-beta market (DBM)
• Models
• Linearity tests
• Logistic smooth transition market (LSTM) models
• Sequential conditional least squares (SCLS)

Acknowledgements

This research was supported by a Monash Graduate Scholarship and the Financial Derivatives Unit at Monash University, Australia. Heather Anderson acknowledges the financial assistance of Australian Research Council Discovery Grant # DP0449995. The authors are grateful to Robert Brooks and Timo Teräsvirta for helpful suggestions.

Notes

†In particular, Kim and Zumwalt (1979), Pettengill et al. (1995) and Jagannathan and Wang (1996) each show that, when beta is allowed to vary with market conditions, the importance of beta for explaining the cross-section of realized stock returns increases.

†We choose to analyse industry portfolios because the existence of industry-specific risk is recognized, and because one can be more confident of the response of a portfolio beta to changes in market conditions than in the case of a single security beta.

†Our own simulation experiments, based on ‘realistic’ parameter values that had been calibrated to reflect our data set, support the published results.

‡Note that since we have only a finite number of observations on , we will only be able to estimate interval in which c _i lies with respect to the set of (ordered by size). Superconsistency will, however, ensure that the probability of choosing the correct interval approaches 1 very quickly as the sample becomes larger, so that we can expect an accurate choice, given our sample of 253.

†Our joint tests are based on a single regression of the (vertically stacked T × k) vector of returns for k industries on where I _k is an identity matrix of dimension k, and Z contains the set of test regressors.

†Over all industries, the average down-market beta is approximately 1 (as might be expected), and the average up-market differential is about zero (as might also be expected).

‡We considered only those that implied at least 15 observations in each regime.

†We used the WESTPAC Coincident Index series, obtained from the Melbourne Institute of Applied Economic and Social Research (University of Melbourne), and the interest rate spread between Australian 10-year and 3-month government bonds. The spread was lagged by four months, following the market timing work undertaken by Resnick and Shoesmith (2002).