The assumption of conditional symmetry is often invoked to validate adaptive estimation and consistent estimation of ARCH/GARCH models by quasi maximum likelihood. Imposing conditional symmetry can increase the efficiency of bootstraps if the symmetry assumption is valid. This paper proposes a procedure for testing conditional symmetry. The proposed test does not require the data to be stationary or i.i.d., and the dimension of the conditional variables could be infinite. The size and power of the test are satisfactory even for small samples. In addition, the proposed test is shown to have non-trivial power against root-T local alternatives. Applying the test to various time series, we reject conditional symmetry in inflation, exchange rate and stock returns. These data have previously been tested and rejected for unconditional symmetry. Among the non-financial time series considered, we find that investment, the consumption of durables, and manufacturing employment also reject conditional symmetry. Interestingly, these are series whose dynamics are believed to be affected by fixed costs of adjustments.
