Arbitrage costs and nonlinear adjustment in the G7 stock markets

Abstract

This article aims to study stock price adjustments towards fundamentals due to the existence of arbitrage costs defined as the sum of transaction costs and a risky arbitrage premium associated with the uncertainty characterizing the fundamentals. Accordingly, it is shown that a two regime Smooth Transition Error Correction Model (STECM) is appropriate to reproduce the dynamics of stock price deviations from fundamentals in the G7 countries during the period 1969 to 2005. This model takes into account the interdependences or contagion effects between stock markets. Deviations appear to follow a quasi random walk in the central regime when prices are near fundamentals (i.e. when arbitrage costs are greater than expected arbitrage profits, the mean reversion mechanism is inactive), while they approach a white noise in the outer regimes (i.e. when arbitrage costs are lower than expected arbitrage profits, the mean reversion is active). Interestingly, as expected when arbitrage costs are heterogeneous, the estimated STECM shows that stock price adjustments are smooth and that the convergence speed depends on the size of the deviation. Finally, using two appropriate indicators proposed by Peel and Taylor (2000), both the magnitudes of under and overvaluation of stock price and the adjustment speed are calculated per date in the G7 countries. These indicators show that the dynamics of stock price adjustment are strongly dependent on both the date and the country under consideration.

Notes

¹ For exchange rate market, see Yoon (2010) who identified a nonlinear mean-reversion towards the power parity (i.e. the fundamentals) using an Exponential Smooth Transition Autoregressive Model (ESTAR) model.

² Direct or explicit costs are largely composed of taxes, regulation costs and other commissions. They generally depend on the nature of the type of broker, the nature of the order and the stock market.

³ In particular, spreads between transaction costs supported by individual investors and those supported by institutional investors contribute to heterogeneity.

⁴  In another way, Berdin and Hyde (2005) also use STAR models to capture nonlinearity in the cyclical character of stock price dynamics for eight countries (Belgium, Canada, France, Germany, Ireland, Japan, the UK and the USA). The authors show that the process describing the stock price adjustment towards fundamentals depends on the state of the economy (two regimes are considered: growth and recession). Using STAR models, Hasanov and Omay (2008) also show strong evidence in favour of nonlinear adjustment stock returns for the Athens and the Istanbul Stock markets.

⁵ The gross index takes into account the dividend investment while the price index excludes it. All indexes are closing prices.

⁶ The stock return is defined as the stock price logarithmic first difference plus the dividend's yield.

⁷ (α ₀, α _1,… , α _p) and (β ₀, β ₁, … , β _p) are, respectively, the Autoregressive (AR) coefficients in the first and second regime, d the lag parameter defining the transition variable (), γ the transition speed between the regimes and c the threshold parameter. Ω(·) is the transition function which is continuous and bounded between 0 and 1. Ω(·) is either logistic () or exponential. (). It implies, respectively, a LogisticSTAR (LSTAR) model or an ESTAR model.

⁸ For more explanations about nonlinearity characterizing dividend dynamics, see Jawadi (2009).

⁹ The smooth character of fundamental values is implied by the DDM, not by the STAR model used to determine the expected dividend. Indeed, according to the DDM, the fundamental value is the sum of discounted future dividends, this sum leading to formally removing the short-term movements in dividend and interest rate.

Note: Y and PFA are, respectively, the observed price and its estimated fundamental value in logarithm.

¹⁰ For more details about these conditions, see Equations 10 and 11.

¹¹ We nevertheless applied a Seemingly Unrelated Regression (SUR) system estimate: estimates were insignificantly different from those obtained with the OLS. This result confirms that the seven equations can be estimated independently.

¹² For more details about mixing tests and conditions, see Dufrénot and Mignon (2002).

¹³ Using similar tests, Hasanov (2009) tests the weak efficiency form for the Australian and New Zealand stock markets.

¹⁴ In line with Teräsvirta (1994) and recently Van Dijk et al. (2002), we applied several Lagrange Multiplier (LM) tests (LM₁, LM₂, LM₃, LM^e ₃ and LM₄) for all possible values of d: . The optimal value of the delay parameter is such that linearity is rejected the most strongly. Thus, should maximize the LM statistics and minimize the p-values of the linearity tests. In practice, all tests unanimously support nonlinearity, so we focus only on the results of the LM₃ test that is available to test linearity against both exponential and logistic STECMs.

¹⁵ These authors only apply the standard LM linearity tests.

¹⁶ We briefly describe the STECM methodology and LM tests. More details can be found in Van Dijk et al. (2002) and Jawadi (2006).

¹⁷ Overall, our results are in line with Shen et al. (2007) who used a nonlinear cointegration test to examine long-run symmetric equilibrium relationships between the Chinese Shanghai and Shenzhen stock markets. Their findings also show evidence of nonlinear mean reversion with time-varying adjustment speed.

¹⁸ Note that the average adjustment delay from prices to fundamentals is about 5 months for the seven countries. This average is given by the sum of the optimal values of d for the G7–MSCI indexes divided by 7.

¹⁹ For more details, see Luukkonen et al. (1988).

²⁰ For more details about these linearity tests, see Van Dijk et al. (2002) and Jawadi and Koubbaa (2007) among others.