This test examines the presence of jumps in highfrequency price series. It is based on the theory of Ait-Sahalia and Jacod (2009) (AJ). It consists in comparing the multipower variation of equispaced returns computed at a fast time scale (*h*), *r_{t,i}* (*i=1, …,N*) and those computed at the slower time scale (*kh*), *y_{t,i}*(*i=1, … ,\mbox{N/k}*).

They found that the limit (for *N* *\to* *∞* ) of the realized power variation is invariant for different sampling scales and that their ratio is *1* in case of jumps and *\mbox{k}^{p/2}-1* if no jumps.
Therefore the AJ test detects the presence of jump using the ratio of realized power variation sampled from two scales. 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 return (with *i=1, …,N*) in period *t*.

And there is *N/k* equispaced returns in period *t*. Let *y_{t,i}* be a return (with *i=1, … ,\mbox{N/k}*) in period *t*.

Then the AJjumptest is given by:

*
\mbox{AJjumptest}_{t,N}= \frac{S_t(p,k,h)-k^{p/2-1}}{√{V_{t,N}}}
*

in which,

*
\mbox{S}_t(p,k,h)= \frac{PV_{t,M}(p,kh)}{PV_{t,M}(p,h)}
*

*
\mbox{PV}_{t,N}(p,kh)= ∑_{i=1}^{N/k}{|y_{t,i}|^p}
*

*
\mbox{PV}_{t,N}(p,h)= ∑_{i=1}^{N}{|r_{t,i}|^p}
*

*
\mbox{V}_{t,N}= \frac{N(p,k) A_{t,N(2p)}}{N A_{t,N(p)}}
*

*
\mbox{N}(p,k)= ≤ft(\frac{1}{μ_p^2}\right)(k^{p-2}(1+k))μ_{2p} + k^{p-2}(k-1) μ_p^2 - 2k^{p/2-1}μ_{k,p}
*

*
\mbox{A}_{t,n(2p)}= \frac{(1/N)^{(1-p/2)}}{μ_p} ∑_{i=1}^{N}{|r_{t,i}|^p} \ \ \mbox{for} \ \ |r_j|< α(1/N)^w
*

*
μ_{k,p}= E(|U|^p |U+√{k-1}V|^p)
*

*U, V*: independent standard normal random variables; *h=1/N*; *p, k, α, w*: parameters.

1 | ```
AJjumptest(pdata, p=4 , k=2, align.by= NULL, align.period = NULL, makeReturns= FALSE, ...)
``` |

`pdata` |
a zoo/xts object containing all prices in period t for one asset. |

`p` |
can be chosen among 2 or 3 or 4. The author suggests 4. 4 by default. |

`k` |
can be chosen among 2 or 3 or 4. The author suggests 2. 2 by default. |

`align.by` |
a string, align the tick data to "seconds"|"minutes"|"hours" |

`align.period` |
an integer, align the tick data to this many [seconds|minutes|hours]. |

`makeReturns` |
boolean, should be TRUE when rdata contains prices instead of returns. FALSE by default. |

`...` |
additional arguments. |

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 Ait-Sahalia and Jacod test is that: Using the convergence properties of power variation and its dependence on the time scale on which it is measured, Ait-Sahalia and Jacod (2009) define a new variable which converges to 1 in the presence of jumps in the underlying return series, or to another deterministic and known number in the absence of jumps. (Theodosiou& Zikes(2009))

list

Giang Nguyen, Jonathan Cornelissen and Kris Boudt

Ait-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. The Annals of Statistics, 37(1), 184- 222.

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.

1 2 | ```
data(sample_tdata)
AJjumptest(sample_tdata$PRICE, p= 2, k= 3, align.by= "seconds", align.period= 5, makeReturns= TRUE)
``` |

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