Supplementary Appendix to Dynamic Asset Price Jumps: the Performance of High Frequency Tests and Measures, and the Robustness of Inference

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1 Supplementary Appendix to Dynamic Asset Price Jumps: the Performance of High Frequency Tests and Measures, and the Robustness of Inference Worapree Maneesoonthorn, Gael M. Martin, Catherine S. Forbes August 15, 2018 Abstract This document provides supplementary material for Dynamic Asset Price Jumps: the Performance of High Frequency Tests and Measures, and the Robustness of Inference. Section A provides an additional power surface for the LM test using the form of simulated critical value proposed by Dimitru and Urga (2012). Section B provides further empirical size and power results for all tests in the presence of microstructure noise; while Section C documents the prior specifications used in the Bayesian analysis of the dynamic jump models given in Sections 5 and 6 of the main text. This research has been supported by Australian Research Council Discovery Grants No. DP and DP We thank a co-editor and two anonymous referees for very helpful and constructive comments on an earlier draft of the paper. We are also grateful to John Maheu, Herman van Dijk, Maria Kalli and Jim Griffin, plus participants at the 11th Annual RCEA Bayesian Econometric Workshop (Melbourne, 2017), for very helpful comments on an earlier version of the paper. Melbourne Business School, The University of Melbourne, Australia. Corresponding author. Department of Econometrics and Business Statistics, Monash University, Australia. gael.martin@monash.edu. Department of Econometrics and Business Statistics, Monash University, Australia. 1

2 A. Power curve for the LM test using the simulated critical value In addition to the power surface for the LM test reproduced in Figure 1, Panel H of the paper, we produce here the corresponding surface based on the form of simulated critical value proposed in Dimitru and Urga (2012). From Figure A1 below it is evident that power of the LM test behaves in a broadly similar way to when the asymptotic critical value is used, apart from a greater degree of empirical size distortion as the magnitude of the volatility jump increases. Power curve for the LM test using simulated critical value Z p Z v Figure A1: Power curves over changing price and volatility jump sizes (Z p and Z v, respectively) for the LM test using the form of simulated critical value proposed by Dimitru and Urga (2012). B. Test performance in the presence of microstructure noise We investigate the power of each test in the presence of microstructure noise. We conduct the same experiment as in Section 4.1 of the main text, but with equation (22) modified to dp t = µ t dt + V t dw p t + dj p t + ε t, where ε t denotes the microstructure noise. Two specifications of the microstructure noise are considered: independent and identically distributed (i.i.d.) noise and autocorrelated noise. In the case of i.i.d. noise, we specify ε t to be normally distributed with mean zero and variance that is half the size of the unconditional variance of the price process. We follow Aït-Sahalia and Mancini (2008) and define the autocorrelated noise process to be ε t = U t + V t, with U t being i.i.d. and V t an autoregressive process of order one, with the first-order coefficient denoted by ρ. Both U t and V t are assumed to be normally distributed and independent of one 2

3 another. For the case of autocorrelated noise, we specify U t and V t such that the variance contributions from both components are equal and, as in the i.i.d. noise case, the noise process has a total variance that is half the size of the unconditional variance of the price process. Following Aït-Sahalia and Mancini (2008) we employ ρ = As in Figure 1 of the main text, we document here the empirical size and power of each of the tests. The empirical size and power in the presence of i.i.d. noise are documented in Figure B2 and Figure B3 gives the comparable results for the case of autocorrelated noise. As is clear, the behavior of the empirical size and power for all tests, as a function of price and volatility jump sizes, is broadly in line with that documented in the paper, despite there being a slight reduction overall in power. 1 We also experimented with higher magnitudes of ρ, namely ρ = 0.5 and ρ = 0.8, and found the results to be robust to this choice. 3

4 4 Figure B2: Power curves over changing price and volatility jump sizes (Zp and Zv, respectively) in the presence of i.i.d. microstructure noise.

5 5 Figure B3: Power curves over changing price and volatility jump sizes (Zp and Zv, respectively) in the presence of autocorrelated microstructure noise.

6 C. Prior specifications used in Sections 5 and 6 The prior specifications used in the numerical exercise in Section 5 using artificial data. and the empirical illustration in Section 6, are given in Table C1. Note that the parameter σv 2 is blocked with the leverage parameter, ρ, via the reparameterization: ξ = ρσ v and ω = σv 2 ξ 2 ; see Jacquier, Polson and Rossi (2004). This reparameterization is convenient as, given V 1:T, it allows ξ and ω to be treated respectively as the slope and error variance coefficients in a normal linear regression model. Direct sampling of ξ and ω is then conducted using standard posterior results, based on conjugate prior specifications in the form of conditional normal and inverse gamma distributions, respectively, given by p (ξ ω) N ( ξ 0 = 0.05, σ0 2 = ω/4.0) and p (ω) IG (a = 10, b = 0.03), where b denotes the scale parameter in the context of the inverse gamma distributions discussed here. The prior specifications for ξ and ω are chosen such that the implied prior distributions for ρ and σ v are relatively diffuse. Table C1: Prior specifications for each of the elements of the parameter vector φ Parameter Prior Spec µ U ( 10, 10) γ U ( 100, 0) ρ ρ, σ v joint κ U (0, 1) θ U (0, 10) σ v ρ, σ v joint µ p U ( 100, 100) γ p U ( 100, 100) σp 2 IG (a = 3, b = 2) π p β (a = 5, b = 5) δ p 0 β (a = 1, b = 9) α p U (0, 1) β p U (0, 1) µ v IG (a = 10, b = 8) δ0 v β (a = 1, b = 9) α v U (0, 1) β v U (0, 1) β vp U (0, 1) ψ 0 N (0, 0.1) ψ 1 N (1, 0.1) σbv 2 IG (a = 3, b = 2) β β (a = 7, b = 3) α β (a = 1, b = 100) σm 2 p IG (a = 3, b = 2) s n β (a = 1, b = 9) s p β (a = 7, b = 3) 6

7 References [1] Jacquier, E., Polson, N.G. and Rossi, P.E. (2004), Bayesian Analysis of Stochastic Volatility Model with Fat-tails and Correlated Errors, Journal of Econometrics, 122,