ARCHProcess
✖
ARCHProcess
represents an autoregressive conditionally heteroscedastic process of order q, driven by a standard white noise.
Details
- ARCHProcess is a discrete-time and continuous-state random process.
- A process x[t] is an ARCH process if the conditional mean Expectation[x[t] {x[t-1], …}]=0 and the conditional variance given by Expectation [x[t]2{x[t-1, …}] satisfies the equation .
- The initial data init can be given as a list or a single path TemporalData object with time stamps understood as .
- A scalar ARCH process can have non-negative coefficients αi and a positive coefficient κ.
- ARCHProcess[q] represents an ARCH process of order q for use in EstimatedProcess and related functions.
- ARCHProcess can be used with such functions as RandomFunction, CovarianceFunction, and TimeSeriesForecast.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Simulate an ARCHProcess:
https://wolfram.com/xid/0n4qojg7usy-se76vc
https://wolfram.com/xid/0n4qojg7usy-8cdroq
https://wolfram.com/xid/0n4qojg7usy-0lg2yf
Unconditional mean and variance of a weakly stationary process:
https://wolfram.com/xid/0n4qojg7usy-b7z11p
https://wolfram.com/xid/0n4qojg7usy-p99hv2
https://wolfram.com/xid/0n4qojg7usy-t8vwwp
https://wolfram.com/xid/0n4qojg7usy-lqhuff
The observations are uncorrelated but dependent:
https://wolfram.com/xid/0n4qojg7usy-kd62j1
https://wolfram.com/xid/0n4qojg7usy-gth2m9
The squared values of the data are correlated:
https://wolfram.com/xid/0n4qojg7usy-4m7h01
https://wolfram.com/xid/0n4qojg7usy-gxyezj
Scope (13)Survey of the scope of standard use cases
Basic Examples (8)
Simulate an ensemble of paths:
https://wolfram.com/xid/0n4qojg7usy-bi9bqw
https://wolfram.com/xid/0n4qojg7usy-iv2byr
Simulate with arbitrary precision:
https://wolfram.com/xid/0n4qojg7usy-big92j
Simulate a weakly stationary process with given initial values:
https://wolfram.com/xid/0n4qojg7usy-qz5qde
https://wolfram.com/xid/0n4qojg7usy-ihrt8b
https://wolfram.com/xid/0n4qojg7usy-l3esgs
https://wolfram.com/xid/0n4qojg7usy-2vkjih
A non-weakly stationary process:
https://wolfram.com/xid/0n4qojg7usy-849us2
https://wolfram.com/xid/0n4qojg7usy-gq6qdl
https://wolfram.com/xid/0n4qojg7usy-t1m0n8
An integrated ARCHProcess:
https://wolfram.com/xid/0n4qojg7usy-uruguz
https://wolfram.com/xid/0n4qojg7usy-qb91v0
Explosive ARCHProcess:
https://wolfram.com/xid/0n4qojg7usy-vdahex
https://wolfram.com/xid/0n4qojg7usy-ptahjr
Such a process is not second-order stationary:
https://wolfram.com/xid/0n4qojg7usy-b12vkp
Conditions for an ARCHProcess to be covariance stationary:
https://wolfram.com/xid/0n4qojg7usy-mutsnx
Region of second-order stationarity for an ARCHProcess[2]:
https://wolfram.com/xid/0n4qojg7usy-vxckai
https://wolfram.com/xid/0n4qojg7usy-i15pmg
Estimate an ARCHProcess:
https://wolfram.com/xid/0n4qojg7usy-ppxlwo
https://wolfram.com/xid/0n4qojg7usy-fsgvlg
https://wolfram.com/xid/0n4qojg7usy-jd9i7o
Use maximum conditional likelihood estimator:
https://wolfram.com/xid/0n4qojg7usy-0m6doe
https://wolfram.com/xid/0n4qojg7usy-eirvpx
Find the forecast 20 steps ahead:
https://wolfram.com/xid/0n4qojg7usy-ywfs4n
https://wolfram.com/xid/0n4qojg7usy-de0jr
Find mean squared errors of the forecast:
https://wolfram.com/xid/0n4qojg7usy-0uy7hx
The forecasted states are equal to zero, hence the forecasted standard deviation bounds are as follows:
https://wolfram.com/xid/0n4qojg7usy-wux3zt
Plot the values with mean squared errors:
https://wolfram.com/xid/0n4qojg7usy-pksy44
Process Slice Properties (5)
Moments of a weakly stationary ARCH of order 1:
https://wolfram.com/xid/0n4qojg7usy-u1d2k5
https://wolfram.com/xid/0n4qojg7usy-6vu1p7
https://wolfram.com/xid/0n4qojg7usy-1z4n08
Moment of an ARCH process with given initial conditions:
https://wolfram.com/xid/0n4qojg7usy-4vo40k
https://wolfram.com/xid/0n4qojg7usy-c76l3j
https://wolfram.com/xid/0n4qojg7usy-nz2z6e
https://wolfram.com/xid/0n4qojg7usy-vw5ylw
https://wolfram.com/xid/0n4qojg7usy-diexmz
Region where kurtosis is defined:
https://wolfram.com/xid/0n4qojg7usy-y5r0ud
https://wolfram.com/xid/0n4qojg7usy-9ealgg
https://wolfram.com/xid/0n4qojg7usy-zmyzm4
Probability density function of the sample:
https://wolfram.com/xid/0n4qojg7usy-s3nc1d
Use Monte Carlo method to calculate NProbability for slice distribution:
https://wolfram.com/xid/0n4qojg7usy-c7kmwg
https://wolfram.com/xid/0n4qojg7usy-00997i
Calculate NExpectation:
https://wolfram.com/xid/0n4qojg7usy-n91pf6
Compare to the second Moment:
https://wolfram.com/xid/0n4qojg7usy-yzt5wc
Properties & Relations (3)Properties of the function, and connections to other functions
The values of an ARCHProcess are uncorrelated:
https://wolfram.com/xid/0n4qojg7usy-8tryi9
Corresponding ARProcess:
https://wolfram.com/xid/0n4qojg7usy-dfyenx
For a process with given initial values:
https://wolfram.com/xid/0n4qojg7usy-z1tydk
Squared values of an ARCHProcess follow an ARProcess:
https://wolfram.com/xid/0n4qojg7usy-rhmavw
https://wolfram.com/xid/0n4qojg7usy-inxryx
CorrelationFunction and PartialCorrelationFunction of squared values:
https://wolfram.com/xid/0n4qojg7usy-ntjl10
The corresponding autoregressive process:
https://wolfram.com/xid/0n4qojg7usy-7z6h31
CorrelationFunction and PartialCorrelationFunction of the AR process:
https://wolfram.com/xid/0n4qojg7usy-nukw68
Wolfram Research (2014), ARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ARCHProcess.html.
Text
Wolfram Research (2014), ARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ARCHProcess.html.
Wolfram Research (2014), ARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ARCHProcess.html.
CMS
Wolfram Language. 2014. "ARCHProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ARCHProcess.html.
Wolfram Language. 2014. "ARCHProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ARCHProcess.html.
APA
Wolfram Language. (2014). ARCHProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ARCHProcess.html
Wolfram Language. (2014). ARCHProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ARCHProcess.html
BibTeX
@misc{reference.wolfram_2024_archprocess, author="Wolfram Research", title="{ARCHProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ARCHProcess.html}", note=[Accessed: 11-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_archprocess, organization={Wolfram Research}, title={ARCHProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/ARCHProcess.html}, note=[Accessed: 11-January-2025
]}