Dynamic causality between the U.S. stock market, the Chinese stock market and the global gold market: implications for individual investors’ diversification strategies

ABSTRACT

This paper proposes a generalization of the prior VAR and EGARCH model to explore the linkage between returns and volatility transmissions in the U.S. stock market, the Chinese stock market, and the global gold market from 10 July 1996 to 20 July 2018. We found that past returns of the U.S. stock market can predict the current returns of the other two markets, and that significant reciprocal volatility transmission existed within and across all three markets. We further implemented average out-of-sample (OOS) forecasting to show that a risk-adjusted portfolio, such as mean-variance with sample estimator, does not outperform an equal-weighted portfolio. This provides insights for individual investors and helps to explain the ongoing disagreement in the portfolio literature concerning the effectiveness of risk-adjusted portfolios and equal-weighted portfolios when the number of assets is small.

KEYWORDS:

• Cdcc-VAR-MEGARCH model
• asymmetry
• composite likelihood
• equal-weighted portfolio

JEL CLASSIFICATION:

• C32
• G11
• C51
• G15

Acknowledgments

We gratefully acknowledge comments from Nicholas Flores, Martin Boileau, Michael J. Stutzer, Dick Startz, and the Winter 2018 ECON245B classmates at UC Santa Barbara.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ The US-China trade battle has escalated significantly ever since.

² Recently, Beckmann, Berger, and Czudaj (2018) summarized the literature with regard to gold and global financial markets and provided a systematic analysis of the role of gold to the functions of hedges and safe havens by extending the uncertainty of their analysis to economic policy, macroeconomics, and inflation. Their research concludes that gold is still able to serve as a hedge or safe haven in global markets.

³ See section 3 for the details.

⁴ De Nard, Ledoit, and Wolf (2018) use a GARCH-based average OOS forecasting method.

⁵ Negative returns drag down the market value of firms which increases the debt-to-equity ratio and is associated with higher volatility.

⁶ An alternative approach to model the dynamic conditional correlation.

⁷ We use sample estimator here to distinguish the sophisticated estimator we obtained from our proposed model. More details will be discussed in section 5.2.

⁸ A strong (weak) hedge is defined as an asset that is negatively correlated (uncorrelated) with another asset on average.

⁹ A strong (weak) safe haven is defined as an asset that is negatively correlated (uncorrelated) with the stock market in periods of extreme stock market declines.

¹⁰ Our results are consistent with the latest gold literature (Beckmann, Berger, and Czudaj 2018).

¹¹ This is consistent with the general wisdom in the portfolio selection literature (DeMiguel, Garlappi, and Uppal 2009; Plyakha, Uppal, and Vilkov 2014; Novy-Marx and Velikov 2016).

¹² For example, Tse (2000) developed a Lagrange multiplier (LM) test and rejected the CCC under four major stock markets.

¹³ Such as the BEKK (Baba, Engle, Kraft and Kroner) model of Baba et al. (1990).

¹⁴ One of the two components jointly determines the asymmetric effects in the model.

¹⁵ Dynamic conditional correlation-exponential GARCH model.

¹⁶ Or the volatility targeting approach (Engle 2009).

¹⁷ In a sense of consistency.

¹⁸ When the number of assets (N) are the same size as the numbers of observations (T), it is impossible to estimate accurately O($N^2$) parameters from O($N^2$) observations.

¹⁹ Based on the criteria of loss function and mean square error.

²⁰ N is less than 100.

²¹ For example, a 1% positive innovation affects the volatility of market i from market j by $({\theta }_{i,j}+{\gamma }_{i,j})$%, whereas a 1% negative innovation from market j to market i is measured by $({\theta }_{i,j}-{\gamma }_{i,j})\mathrm{\%}$.

²² Negative asymmetry if $\frac{\left|{\theta }_{i,j}-{\gamma }_{i,j}\right|}{({\theta }_{i,j}+{\gamma }_{i,j})}>1$, symmetry if $\frac{\left|{\theta }_{i,j}-{\gamma }_{i,j}\right|}{({\theta }_{i,j}+{\gamma }_{i,j})}=1$, and positive asymmetry if $\frac{\left|{\theta }_{i,j}-{\gamma }_{i,j}\right|}{({\theta }_{i,j}+{\gamma }_{i,j})}<1$. See section 5.1 and Table 3 for the details.

²³ The conditional standard deviations of return series.

²⁴ To establish consistency (Aielli 2013).

²⁵ As an estimator of the population covariance matrix, Q.

²⁶ We can estimate the parameters separately because we can write the log likelihood as the sum of the volatility part (Equation 7), which is essentially the sum of all EGARCH likelihood, and the correlation part (Equation 9). Maximizing them separately is equivalent to join-maximization (Engle 2002).

²⁷ See Ledoit and Wolf (2004), Theorem 2.1.

²⁸ We use the subscript n to emphasize the asymptotic properties, which will be assumed in the following.

²⁹ The normalized returns of $(R_{1,t},R_{2,t}$),$(R_{2,t},R_{3,t})$, and $(R_{1,t},R_{3,t}$).

³⁰ The benchmark stock index in China.

³¹ The global benchmark price for gold trading.

³² As trade sanctions between the US and China have escalated, the Chinese stock market declined almost 20% by September 2018.

³³ The past value of one leading variable is cross-correlated with the current values of another(lagging) variable.

³⁴ This establishes Granger causality.

³⁵ By implementing a bootstrap rolling window Granger causality approach.

³⁶ The second moment of the markets’ interdependence.

³⁷ Since it is negative asymmetry if $\frac{\left|{\theta }_{i,j}-{\gamma }_{i,j}\right|}{({\theta }_{i,j}+{\gamma }_{i,j})}>1$.

³⁸ Since it is positive asymmetry if $\frac{\left|{\theta }_{i,j}-{\gamma }_{i,j}\right|}{({\theta }_{i,j}+{\gamma }_{i,j})}<1$.

³⁹ The $Q^{2}test$.

⁴⁰ See Appendix for details.

⁴¹ This decreases the effectiveness of portfolio diversification, discussed in the next section.

⁴² Which track the SP500 and SHCOMP.

⁴³ Most recent papers focus on high-dimension portfolio selections for the sake of portfolio managers and when the number of assets (N) is greater than 30 up to 1000.

⁴⁴ Mean and covariances.

⁴⁵ For example, DeMiguel, Garlappi, and Uppal (2009) show that the estimation window needed for the sample-based mean-variance strategy and its extensions to outperform the 1/N benchmark is around 3000 months for a portfolio with 25 assets.

⁴⁶ N = 3.

⁴⁷ Which emphasizes the expected means of returns.

⁴⁸ While using Equation 1 as the mean equation, which is a natural extension of our proposed model.

⁴⁹ Most recent papers consider monthly rebalancing, but it generates large transaction costs for the individual investors, as suggested by Novy-Marx and Velikov (2016), who argue instead that infrequent re-balancing is the most efficient cost mitigation strategy.

⁵⁰ The ratio of the first two.

⁵¹ It is uninformative to compare negative IRs, though it is commonly appear during financial disturbance periods. Thus, we instead use modified information ratio (MIR) to rank the performance of these portfolios in the three sub-periods. Specifically, $MIR=\frac{ADR}{S{D}^{(ADR/\left|ADR\right|)}}$. For negative ADRs, the lower (less negative) the MIR, the better the performance of the portfolios and the higher the rank. For positive ADRs, it has the same value as IR.

⁵² S&P500, SHCOMP, and GLOBEX.

⁵³ See the Appendix for the setup.

⁵⁴ And IR.

⁵⁵ 0.033% versus 0.034%.

⁵⁶ The trading cost of ETFs consists of three parts, and is 50 basis points on average.

⁵⁷ According to the MIRs.

⁵⁸ We refer to professional portfolio management re-balancing services that are commonly offered by brokerage firms. Individual investors are charged an annual advisory fee in addition to the transaction costs of the ETFs.

⁵⁹ As in our proposed model.

⁶⁰ For example, gold futures contracts, gold ETFs, or gold-mining stocks.