JOjumpTest: Jiang and Oomen (2008) tests for the presence of jumps in the price series.

Description

This test examines the jump in highfrequency data. It is based on theory of Jiang and Oomen (JO). They found that the difference of simple return and logarithmic return can capture one half of integrated variance if there is no jump in the underlying sample path. The null hypothesis is no jumps.

Function returns three outcomes: 1.z-test value 2.critical value under confidence level of $95\%$ and 3.p-value.

Assume there is $N$ equispaced returns in period $t$.

Let $r_{t,i}$ be a logarithmic return (with $i=1, \ldots,N$) in period $t$.

Let $R_{t,i}$ be a simple return (with $i=1, \ldots,N$) in period $t$.

Then the JOjumpTest is given by: $$ \mbox{JOjumpTest}_{t,N}= \frac{N BV_{t}}{\sqrt{\Omega_{SwV}} \left(1-\frac{RV_{t}}{SwV_{t}} \right)} $$ in which, $BV$: bipower variance; $RV$: realized variance (defined by Andersen et al. (2012)); $$ \mbox{SwV}_{t}=2 \sum_{i=1}^{N}(R_{t,i}-r_{t,i}) $$ $$ \Omega_{SwV}= \frac{\mu_6}{9} \frac{{N^3}{\mu_{6/p}^{-p}}}{N-p-1} \sum_{i=0}^{N-p}\prod_{k=1}^{p}|r_{t,i+k}|^{6/p} $$ $$ \mu_{p}= \mbox{E}[|\mbox{U}|^{p}] = 2^{p/2} \frac{\Gamma(1/2(p+1))}{\Gamma(1/2)} % \mbox{E}[|\mbox{U}|^p]= $$ $U$: independent standard normal random variables

p: parameter (power).

Usage

JOjumpTest(
  pData,
  power = 4,
  alignBy = NULL,
  alignPeriod = NULL,
  alpha = 0.975,
  ...
)

Value

list

Arguments

pData: a zoo/xts object containing all prices in period t for one asset.
power: can be chosen among 4 or 6. 4 by default.
alignBy: character, indicating the time scale in which alignPeriod is expressed. Possible values are: "ticks", "secs", "seconds", "mins", "minutes", "hours"
alignPeriod: positive numeric, indicating the number of periods to aggregate over. E.g. to aggregate based on a 5 minute frequency, set alignPeriod to 5 and alignBy to "minutes".
alpha: numeric of length one with the significance level to use for the jump test(s). Defaults to 0.975.
...: Used internally, do not set.

Author

Giang Nguyen, Jonathan Cornelissen, Kris Boudt, and Emil Sjoerup

Details

The theoretical framework underlying jump test is that the logarithmic price process $X_t$ belongs to the class of Brownian semimartingales, which can be written as: $$ \mbox{X}_{t}= \int_{0}^{t} a_udu + \int_{0}^{t}\sigma_{u}dW_{u} + Z_t $$ 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.

The the Jiang and Ooment test is that in the absence of jumps, the accumulated difference between the simple returns and log returns captures half of the integrated variance. (Theodosiou and Zikes, 2009). If this difference is too great, the null hypothesis of no jumps is rejected.

References

Andersen, T. G., Dobrev, D., and Schaumburg, E. (2012). Jump-robust volatility estimation using nearest neighbor truncation. Journal of Econometrics, 169, 75- 93.

Jiang, J. G., and Oomen, R. C. A (2008). Testing for jumps when asset prices are observed with noise- a "swap variance" approach. Journal of Econometrics, 144, 352-370.

Theodosiou, M., Zikes, F. (2009). A comprehensive comparison of alternative tests for jumps in asset prices. Unpublished manuscript, Graduate School of Business, Imperial College London.

Examples

Run this code

joDT <- JOjumpTest(sampleTData[, list(DT, PRICE)])

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