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

In jonathancornelissen/highfrequency: Tools for Highfrequency Data Analysis

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, …,N) in period t.

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

Then the JOjumpTest is given by:

\mbox{JOjumpTest}_{t,N}= \frac{N BV_{t}}{√{Ω_{SwV}} ≤ft(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 ∑_{i=1}^{N}(R_{t,i}-r_{t,i})

Ω_{SwV}= \frac{μ_6}{9} \frac{{N^3}{μ_{6/p}^{-p}}}{N-p-1} ∑_{i=0}^{N-p}∏_{k=1}^{p}|r_{t,i+k}|^{6/p}

μ_{p}= \mbox{E}[|\mbox{U}|^{p}] = 2^{p/2} \frac{Γ(1/2(p+1))}{Γ(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,
  ...
)

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.

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}σ_{u}dW_{u} + Z_t

where a is the drift term, σ denotes the spot volatility process, W is a standard Brownian motion and Z is a jump process defined by:

\mbox{Z}_{t}= ∑_{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.

Value

list

Author(s)

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

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

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

jonathancornelissen/highfrequency documentation built on Jan. 10, 2023, 7:29 p.m.