This test examines the presence of jumps in highfrequency price series. It is based on theory of Barndorff-Nielsen and Shephard (2006). The null hypothesis is that there are no jumps.
BNSjumpTest(
rData,
IVestimator = "BV",
IQestimator = "TP",
type = "linear",
logTransform = FALSE,
max = FALSE,
alignBy = NULL,
alignPeriod = NULL,
makeReturns = FALSE,
alpha = 0.975
)a list or xts (depending on whether input prices span more than one day)
with the following values:
\(z\)-test value.
critical value (with confidence level of 95%).
\(p\)-value of the test.
Assume there is \(N\) equispaced returns in period \(t\). Assume the Realized variance (RV), IVestimator and IQestimator are based on \(N\) equi-spaced returns.
Let \(r_{t,i}\) be a return (with \(i = 1, \ldots, N\)) in period \(t\).
Then the BNSjumpTest is given by
$$
\mbox{BNSjumpTest}= \frac{\code{RV} - \code{IVestimator}}{\sqrt{(\theta-2)\frac{1}{N} {\code{IQestimator}}}}.
$$
The options for IVestimator and IQestimator are listed above. \(\theta\) depends on the chosen IVestimator (Huang and Tauchen, 2005).
The theoretical framework underlying the jump test is that the logarithmic price process \(X_t\) belongs to the class of Brownian semimartingales, which can be written as: $$ \mbox{X}_{t}= \int_{0}^{t} a_u \ du + \int_{0}^{t}\sigma_{u} \ dW_{u} + Z_t $$ where \(a\) is the drift term, \(\sigma\) denotes the spot volatility process, \(W\) is a standard Brownian motion and \(Z\) is a jump process defined by: $$ \mbox{Z}_{t}= \sum_{j=1}^{N_t}k_j $$ where \(k_j\) are nonzero random variables. The counting process can be either finite or infinite for finite or infinite activity jumps.
Since the realized volatility converges to the sum of integrated variance and jump variation, while the robust IVestimator converges to the integrated variance,
it follows that the difference between RV and the IVestimator captures the jump part only, and this observation underlines the BNS test for jumps (Theodosiou and Zikes, 2009).
Barndorff-Nielsen, O. E., and Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics, 4, 1-30.
Corsi, F., Pirino, D., and Reno, R. (2010). Threshold bipower variation and the impact of jumps on volatility forecasting. Journal of Econometrics, 159, 276-288.
Huang, X., and Tauchen, G. (2005). The relative contribution of jumps to total price variance. Journal of Financial Econometrics, 3, 456-499.
Theodosiou, M., and Zikes, F. (2009). A comprehensive comparison of alternative tests for jumps in asset prices. Unpublished manuscript, Graduate School of Business, Imperial College London.
bns <- BNSjumpTest(sampleTData[, list(DT, PRICE)], IVestimator= "rMinRVar",
IQestimator = "rMedRQuar", type= "linear", makeReturns = TRUE)
bns
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