67. Estimating the Correlation of Asset Returns: A Quantile Dependence Perspective

For instance, equity returns are more highly correlated during business cycle downturns and bear markets (Erb et al. 1994; Longin and Solnik 1995, 2001; Ang and Chen 2002); the same is true for exchange rates returns (Patton 2006; Bouyè and Salmon 2009). Likewise, the deviation in the stock-bond correlation during bear markets is documented by Guidolin and Timmermann (2005).

For example, Erb et al. (1994) examine the dependence of G-7 equity stock markets by computing semicorrelations and find that the correlation is generally larger when the equity returns and output growth of these countries are below than above their respective means, thus when the economies and financial markets in these countries are bearish. Longin and Solnik (1995) examine the dependence of the market returns of Switzerland plus G-7, less Italy, by using a version of the multivariate GARCH model to show that the correlation is larger in times of greater market uncertainties.

Introduction to copula models can be found in Nelsen (2006) and Trivedi and Zimmer (2007).

Take the US-Australia return pair, for example. The 81 correlation estimates consist of estimates of the correlation between the tenth US and tenth Australian return percentiles, the tenth US and 20th Australian return percentiles, and so on, up to the 90th US and 90th Australian return percentiles.

According to the seminal work of Chen, Roll, and Ross (1986), asset prices could also be linked to information about the macroeconomic aggregates as they could influence the discount rate or dividend stream, given that asset price is the sum of discounted future dividend stream (e.g., McQueen and Roley 1993; Flannery and Protopapadakis 2002; Shanken and Weinstein 2006).

For example, Balvers et al. (1990) show that the general equilibrium framework with a logarithmic utility function and full capital depreciation can deliver a linear econometric model of stock return on the log of output.

See Balvers et al. (1990) for a theoretical justification of the importance of output as a determinant of stock return. In the empirical study of Shanken and Weinstein (2006), industrial production is found to be an important determinant of stock return. In the empirical study of Shanken and Weinstein (2006), industrial production is found to be an important determinant of stock return among other factors, such as expected and unanticipated inflation, the spread in corporate bonds, and the spread in the treasury yields.

Monotonicity ensures that conditional on x _t and v _t , the quantile of y can be mapped from the quantile of u.

See p. 726 in Bouyè and Salmon (2009).

When x _t is replaced by y _t−1, the model becomes an autoregression in Φ^{− 1}(F _y (y _t )), leading to the nonlinear copula quantile autoregression model of Chen et al. (2009).

φ(τ _y , τ _x ) can be motivated from \( \tilde{\varphi}\left({u}_t,{v}_t\right) \) by anchoring the u , v-arguments in \( \tilde{\varphi}\left({u}_t,{v}_t\right) \) at F _u ⁻¹(τ _y ) and F _v ⁻¹(τ _x ).

Since Q _x (τ _x |z _t ) = b ^⊤ z _t + F _v ⁻¹(τ _x ), we may express the auxiliary regression of (67.1), i.e., x _t = b ^⊤ z _t + v _t , as x _t = b ^⊤ z _t + F _v ⁻¹(τ _x ) + v _t − F _v ⁻¹(τ _x ) = Q _x (τ _x |z _t ) + v _t − F _v ⁻¹(τ _x ) = Q _x (τ _x |z _t ) + v _t (τ _x ). Therefore, we may estimate υ _t (τ _x ) as the residual from a τ _x -quantile regression on (67.1).

For instance, the Steps 1 and 2 regressions can be implemented using the rq and nlrq commands of the quantreg package in R.

The monthly returns are constructed as 100 multiplied by the change in the log of the index.