# autoregressive conditional heteroskedasticity

• In econometrics, the autoregressive conditional heteroskedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error
term or innovation as a function of the actual sizes of the previous time periods’ error terms;[1] often the variance is related to the squares of the previous innovations.

• [3] Model specification To model a time series using an ARCH process, let denote the error terms (return residuals, with respect to a mean process), i.e.

• These are split into a stochastic piece and a time-dependent standard deviation characterizing the typical size of the terms so that The random variable is a strong white
noise process.

• ARCH-type models are sometimes considered to be in the family of stochastic volatility models, although this is strictly incorrect since at time t the volatility is completely
pre-determined (deterministic) given previous values.

• may be a standard normal variable or come from a generalized error distribution.

• Now given some initial condition , the system above has a pathwise unique solution which is then called the continuous-time GARCH (COGARCH) model.

• [8] IGARCH Integrated Generalized Autoregressive Conditional heteroskedasticity (IGARCH) is a restricted version of the GARCH model, where the persistent parameters
sum up to one, and imports a unit root in the GARCH process.

• Since the drift term , the ZD-GARCH model is always non-stationary, and its statistical inference methods are quite different from those for the classical GARCH model.

• [2] In that case, the GARCH (p, q) model (where p is the order of the GARCH terms and q is the order of the ARCH terms ), following the notation of the original paper, is
given by Generally, when testing for heteroskedasticity in econometric models, the best test is the White test.

• In the example of a GARCH(1,1) model, the residual process is where is i.i.d.

• The idea is to start with the GARCH(1,1) model equations and then to replace the strong white noise process by the infinitesimal increments of a Lévy process , and the squared
noise process by the increments , where is the purely discontinuous part of the quadratic variation process of .

• As an alternative to GARCH modelling it has some attractive properties such as a greater weight upon more recent observations, but also drawbacks such as an arbitrary decay
factor that introduces subjectivity into the estimation.

• GARCH(p, q) model specification The lag length p of a GARCH(p, q) process is established in three steps: Estimate the best fitting AR(q) model Compute and plot the autocorrelations
of by The asymptotic, that is for large samples, standard deviation of is .

• In contrast to the temporal ARCH model, in which the distribution is known given the full information set for the prior periods, the distribution is not straightforward in
the spatial and spatiotemporal setting due to the interdependence between neighboring spatial locations.

• The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated coefficients must be significant.

• Gaussian process-driven GARCH In a different vein, the machine learning community has proposed the use of Gaussian process regression models to obtain a GARCH scheme.

• [2] ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility and volatility clustering, i.e.

• In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic T’R² follows distribution with q degrees of freedom, where is the number of equations
in the model which fits the residuals vs the lags (i.e.

• [15] This results in a nonparametric modelling scheme, which allows for: (i) advanced robustness to overfitting, since the model marginalises over its parameters to perform
inference, under a Bayesian inference rationale; and (ii) capturing highly-nonlinear dependencies without increasing model complexity.

Works Cited

[‘Engle, Robert F. (1982). “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation”. Econometrica. 50 (4): 987–1007. doi:10.2307/1912773. JSTOR 1912773.
2. ^ Jump up to:a b Bollerslev, Tim (1986). “Generalized
Autoregressive Conditional Heteroskedasticity”. Journal of Econometrics. 31 (3): 307–327. CiteSeerX 10.1.1.468.2892. doi:10.1016/0304-4076(86)90063-1. S2CID 8797625.
3. ^ Brooks, Chris (2014). Introductory Econometrics for Finance (3rd ed.). Cambridge:
Cambridge University Press. p. 461. ISBN 9781107661455.
4. ^ Lanne, Markku; Saikkonen, Pentti (July 2005). “Non-linear GARCH models for highly persistent volatility” (PDF). The Econometrics Journal. 8 (2): 251–276. doi:10.1111/j.1368-423X.2005.00163.x.
JSTOR 23113641. S2CID 15252964.
5. ^ Bollerslev, Tim; Russell, Jeffrey; Watson, Mark (May 2010). “Chapter 8: Glossary to ARCH (GARCH)” (PDF). Volatility and Time Series Econometrics: Essays in Honor of Robert Engle (1st ed.). Oxford: Oxford University
Press. pp. 137–163. ISBN 9780199549498. Retrieved 27 October 2017.
6. ^ Jump up to:a b Engle, Robert F.; Ng, Victor K. (1993). “Measuring and testing the impact of news on volatility” (PDF). Journal of Finance. 48 (5): 1749–1778. doi:10.1111/j.1540-6261.1993.tb05127.x.
SSRN 262096. It is not yet clear in the finance literature that the asymmetric properties of variances are due to changing leverage. The name “leverage effect” is used simply because it is popular among researchers when referring to such a phenomenon.
7. ^
Jump up to:a b Posedel, Petra (2006). “Analysis Of The Exchange Rate And Pricing Foreign Currency Options On The Croatian Market: The Ngarch Model As An Alternative To The Black Scholes Model” (PDF). Financial Theory and Practice. 30 (4): 347–368.
Special attention to the model is given by the parameter of asymmetry [theta (θ)] which describes the correlation between returns and variance.6 …
6 In the case of analyzing stock returns, the positive value of [theta] reflects the empirically
well known leverage effect indicating that a downward movement in the price of a stock causes more of an increase in variance more than a same value downward movement in the price of a stock, meaning that returns and variance are negatively correlated
8. ^
Higgins, M.L; Bera, A.K (1992). “A Class of Nonlinear Arch Models”. International Economic Review. 33 (1): 137–158. doi:10.2307/2526988. JSTOR 2526988.
9. ^ St. Pierre, Eilleen F. (1998). “Estimating EGARCH-M Models: Science or Art”. The Quarterly
Review of Economics and Finance. 38 (2): 167–180. doi:10.1016/S1062-9769(99)80110-0.
10. ^ Chatterjee, Swarn; Hubble, Amy (2016). “Day-Of-The-Whieek Effect In Us Biotechnology Stocks—Do Policy Changes And Economic Cycles Matter?”. Annals of Financial
Economics. 11 (2): 1–17. doi:10.1142/S2010495216500081.
11. ^ Hentschel, Ludger (1995). “All in the family Nesting symmetric and asymmetric GARCH models”. Journal of Financial Economics. 39 (1): 71–104. CiteSeerX 10.1.1.557.8941. doi:10.1016/0304-405X(94)00821-H.
12. ^
Klüppelberg, C.; Lindner, A.; Maller, R. (2004). “A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour”. Journal of Applied Probability. 41 (3): 601–622. doi:10.1239/jap/1091543413. hdl:10419/31047. S2CID
17943198.
13. ^ Li, D.; Zhang, X.; Zhu, K.; Ling, S. (2018). “The ZD-GARCH model: A new way to study heteroscedasticity” (PDF). Journal of Econometrics. 202 (1): 1–17. doi:10.1016/j.jeconom.2017.09.003.
14. ^ Otto, P.; Schmid, W.; Garthoff, R.
(2018). “Generalised spatial and spatiotemporal autoregressive conditional heteroscedasticity”. Spatial Statistics. 26 (1): 125–145. arXiv:1609.00711. doi:10.1016/j.spasta.2018.07.005. S2CID 88521485.
15. ^ Platanios, E.; Chatzis, S. (2014). “Gaussian
process-mixture conditional heteroscedasticity”. IEEE Transactions on Pattern Analysis and Machine Intelligence. 36 (5): 889–900. arXiv:1211.4410. doi:10.1109/TPAMI.2013.183. PMID 26353224. S2CID 10424638.
Photo credit: https://www.flickr.com/photos/verino77/2294126841/’]