Measuring currency exposure with quantile regression

Abstract

In this paper, we explore an alternative explanation of the exposure puzzle, the missing variable bias in previous studies. We propose to correct the bias with the quantile regression technique invented by Koenker and Bassett (Econometrica 46:33–51, 1978). Empirically, as soon as we take into account the missing variable bias as well as time variation in currency exposure, we find that 26 out of 30 or 87 % of the US industry portfolios exhibit significant currency exposure to the Major Currencies Index, and 23 out of 30 or 77 % show significant exposure to the Other Important Trading Partners Index. Our results have important theoretical and practical implications. In terms of theoretical significance, our results strengthen the findings in Francis et al. (J Financ Econ 90:169–196, 2008), and suggest that methodological weakness, not hedging, may explain the insignificance of currency risk in previous studies. In terms of practical significance, our results suggest a simple yet efficient approach for managers to estimate currency exposure of their firms.

Appendix

The quantile regression coefficients can pick up the effects of missing variables. To illustrate the idea, consider a simple case in which only leverage affects currency exposure and takes value of one when leverage is high and zero otherwise. Then we can express the true model in Eq. (2) as.

$$ r_{it} = \alpha_{i} + \beta_{i,M} M_{t} + \beta_{i,FX} FX_{t} + u_{it} ,\,{\text{when leverage is low}} $$

(6a)

$$ r_{it} = \alpha_{i} + \beta_{i,M} M_{t} + \beta_{i,FX} FX_{t} + \beta_{i,leverage} FX_{t} + u_{it} ,\,{\text{when leverage is high}}\, $$

(6b)

When leverage is high, the conditional mean of r _it on FX _t given a specific value of M in Eq. (6b) will be higher or lower than the conditional mean of r _it on FX _t in Eq. (6a) when leverage is low depending on the signs of FX and β _i,leverage. When the least-squares regression is applied to the misspecified model in Eq. (1), however, the regression attempts to estimate the conditional mean of the misspecified model, which will obviously yield biased estimate for either of the two true conditional mean relationships depicted in either Eqs. (6a) or Eq. (6b). However, if Eq. (1) is estimated with the quantile regression instead, the effects of leverage will be captured by the quantile regression coefficients $\beta_{i,FX}^{\tau }$ near the tails.

Figure 1 is a simple simulation of a scenario depicted above where M _t in the true model specified by Eqs. (6a) and (6b) is generated as a normal random variable with a mean of 0 and a standard deviation of 5, FX _t is a uniform random variable between 0 and 5, u _it is a standardized normal, the leverage dummy variable is generated as a binomial random variable with a 0.5 probability of being in either state, α = β _i,FX  = 1, β _i,M  = 0, β _i,leverage  = 3 and the sample size is 1,000. In the figure, the solid dark line is the conditional mean of r _it on FX _t when the leverage is low, while the dash-dot dark line depicts the conditional mean of r _it on FX _t when the leverage is high. The dark dotted line is the least-squares regression applied to the misspecified model in Eq. (1). It is obvious in the figure that the least-squares regression line results in a biased estimate of both conditional mean relationships. The grey solid lines are the quantile regressions for the misspecified model in Eq. (1) for τ ∈ [0.1, 0.9] in increments of 0.1. The nine quantile regression lines manage to capture the heteroskedastic effect caused by the missing interaction effect between the leverage dummy and FX _t when the misspecified model in Eq. (1) is used. The different slopes, $\beta_{i,FX}^{\tau }$, of the quantile regression lines also reveal the effect of the omitted variable.

Fig. 1

A Simulation of the least-squares regression bias caused by missing variables The two conditional means of the true model specified in Eqs. (6a) and (6b) are represented by the solid dark line for low leverage and the dash-dot dark line for high leverage. The dark dotted line is the least-squares regression based on the misspecified model in Eq. (1). The least-squares regression applied to the model with missing variable yields a biased estimate of both conditional mean relationships. The grey solid lines are the quantile regressions for the misspecified model in Eq. (1) for τ ∈ [0.1, 0.9] in increments of 0.1. The nine quantile regression lines for the misspecified model in Eq. (1) capture the heteroskedastic effect caused by the omitted interaction effect between the leverage dummy and FX _t . The different slopes, $\beta_{i,FX}^{\tau }$, of the quantile regression lines also reveal the effect of the omitted variable