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| {{Redirect|ARCH|other uses|Arch (disambiguation)}}
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| {{Review|date=May 2009}} <!-- the notation is not consistent //-->
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| In [[econometrics]], '''AutoRegressive Conditional Heteroskedasticity''' (ARCH) models are used to characterize and model observed [[time series]]. They are used whenever there is reason to believe that, at any point in a series, the error terms will have a characteristic size, or [[variance]]. In particular ARCH models assume the variance of the current [[Errors and residuals in statistics|error term]] or [[Innovation (signal processing)|innovation]] to be a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous [[innovation (signal processing)|innovation]]s.
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| Such models are often called '''ARCH''' models ([[Robert_Engle |Engle]], 1982),<ref>[http://www.jstor.org/stable/10.2307/1912773 Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation] Robert F. Engle, Econometrica , Vol. 50, No. 4 (Jul., 1982), pp. 987-1007. Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/1912773</ref> although a variety of other acronyms are applied to particular structures of model which have a similar basis. ARCH models are employed commonly in modeling [[mathematical finance|financial]] [[time series]] that exhibit time-varying [[volatility (finance)|volatility]] clustering, i.e. periods of swings followed by periods of relative calm.
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| ==ARCH(q) model Specification==
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| Suppose one wishes to model a time series using an ARCH process. Let <math> ~\epsilon_t~ </math> denote the error terms (return residuals, with respect to a mean process) i.e. the series terms. These <math> ~\epsilon_t~ </math> are split into a stochastic piece <math>z_t</math> and a time-dependent standard deviation <math>\sigma_t</math> characterizing the typical size of the terms so that
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| :<math> ~\epsilon_t=\sigma_t z_t ~</math>
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| The random variable <math>z_t</math> is a strong [[White noise]] process. The series <math> \sigma_t^2 </math> is modelled by
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| :<math> \sigma_t^2=\alpha_0+\alpha_1 \epsilon_{t-1}^2+\cdots+\alpha_q \epsilon_{t-q}^2 = \alpha_0 + \sum_{i=1}^q \alpha_{i} \epsilon_{t-i}^2 </math>
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| where <math> ~\alpha_0>0~ </math> and <math> \alpha_i\ge 0,~i>0</math>.
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| An ARCH(q) model can be estimated using [[Least squares|ordinary least squares]]. A methodology to test for the lag length of ARCH errors using the [[Lagrange multiplier test]] was proposed by [[Robert F. Engle|Engle]] (1982). This procedure is as follows:
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| # Estimate the best fitting [[autoregressive model]] AR(q) <math> y_t = a_0 + a_1 y_{t-1} + \cdots + a_q y_{t-q} + \epsilon_t = a_0 + \sum_{i=1}^q a_i y_{t-i} + \epsilon_t </math>.
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| # Obtain the squares of the error <math> \hat \epsilon^2 </math> and regress them on a constant and ''q'' lagged values:
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| #:
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| #: <math> \hat \epsilon_t^2 = \hat \alpha_0 + \sum_{i=1}^{q} \hat \alpha_i \hat \epsilon_{t-i}^2</math>
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| #:
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| #: where ''q'' is the length of ARCH lags.
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| #The [[null hypothesis]] is that, in the absence of ARCH components, we have <math> \alpha_i = 0 </math> for all <math> i = 1, \cdots, q </math>. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated <math> \alpha_i </math> coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic ''TR²'' follows <math> \chi^2 </math> distribution with ''q'' degrees of freedom. If ''TR²'' is greater than the Chi-square table value, we ''reject'' the null hypothesis and conclude there is an ARCH effect in the [[Autoregressive moving average model|ARMA model]]. If ''TR²'' is smaller than the Chi-square table value, we ''do not reject'' the null hypothesis.
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| ==GARCH==
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| If an [[autoregressive moving average model]] (ARMA model) is assumed for the error variance, the model is a '''generalized autoregressive conditional heteroskedasticity''' ('''GARCH''', Bollerslev (1986)) model.
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| In that case, the GARCH (p, q) model (where p is the order of the GARCH terms <math> ~\sigma^2 </math> and q is the order of the ARCH terms <math> ~\epsilon^2 </math> ) is given by
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| <math> \sigma_t^2=\alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \cdots + \alpha_q \epsilon_{t-q}^2 + \beta_1 \sigma_{t-1}^2 + \cdots + \beta_p\sigma_{t-p}^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2 + \sum_{i=1}^p \beta_i \sigma_{t-i}^2 </math>
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| Generally, when testing for [[heteroskedasticity]] in econometric models, the best test is the [[White test]]. However, when dealing with [[time series]] data, this means to test for ARCH errors (as described above) and GARCH errors (below).
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| EWMA is an alternative model in a separate class of exponential smoothing models. It can be an alternative to GARCH modelling as it has some attractive properties such as a greater weight upon more recent observations but also some drawbacks such as an arbitrary decay factor that introduce subjectivity into the estimation.
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| ===GARCH(p, q) model specification===
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| The lag length ''p'' of a GARCH(p, q) process is established in three steps:
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| # Estimate the best fitting AR(q) model
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| #:
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| #: <math> y_t = a_0 + a_1 y_{t-1} + \cdots + a_q y_{t-q} + \epsilon_t = a_0 + \sum_{i=1}^q a_i y_{t-i} + \epsilon_t </math>.
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| # Compute and plot the autocorrelations of <math> \epsilon^2 </math> by
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| #:
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| #:<math> \rho = {{\sum^T_{t=i+1} (\hat \epsilon^2_t - \hat \sigma^2_t) (\hat \epsilon^2_{t-1} - \hat \sigma^2_{t-1})} \over {\sum^T_{t=1} (\hat \epsilon^2_t - \hat \sigma^2_t)^2}} </math>
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| # The asymptotic, that is for large samples, standard deviation of <math> \rho (i) </math> is <math> 1/\sqrt{T} </math>. Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the [[Ljung-Box test]] until the value of these are less than, say, 10% significant. The Ljung-Box [[Q-statistic]] follows <math> \chi^2 </math> distribution with ''n'' degrees of freedom if the squared residuals <math> \epsilon^2_t </math> are uncorrelated. It is recommended to consider up to T/4 values of ''n''. The null hypothesis states that there are no ARCH or GARCH errors. Rejecting the null thus means that such errors exist in the [[conditional variance]].
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| ==NGARCH==
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| Nonlinear GARCH (NGARCH) also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH) was introduced by Engle and Ng in 1993. <br />
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| <math> ~\sigma_{t}^2= ~\omega + ~\alpha (~\epsilon_{t-1} - ~\theta ~\sigma_{t-1})^2 + ~\beta ~\sigma_{t-1}^2</math>
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| <math>~\alpha , ~\beta \geq 0 ; ~\omega > 0</math>.<br />
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| For stock returns, parameter <math>~ \theta</math> is usually estimated to be positive; in this case, it reflects the leverage effect, signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.<ref>{{cite journal|last=Engle|first=R.F.|coauthors=Ng, V.K.|title=Measuring and testing the impact of news on volatility|journal=Journal of Finance|year=1991|volume=48|issue=5|pages=1749–1778|url=http://papers.ssrn.com/sol3/papers.cfm?abstract_id=262096}}</ref><ref>{{cite journal|last=Posedel|first=Petra|year=2006|title=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|journal=Financial Theory and Practice|volume=30|issue=4|pages=347–368|url=http://www.ijf.hr/eng/FTP/2006/4/posedel.pdf}}</ref>
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| This model shouldn't be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.{{Clarify|date=November 2009}}
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| ==IGARCH==
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| Integrated Generalized Autoregressive Conditional Heteroskedasticity IGARCH is a restricted version of the GARCH model, where the persistent parameters sum up to one, and therefore there is a [[unit root]] in the GARCH process. The condition for this is
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| <math>
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| \sum^p_{i=1} ~\beta_{i} +\sum_{i=1}^q~\alpha_{i} = 1
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| </math>.
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| ==EGARCH==
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| The '''exponential general autoregressive conditional heteroskedastic''' (EGARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
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| <math>\log\sigma_{t}^2=\omega+\sum_{k=1}^{q}\beta_{k}g(Z_{t-k})+\sum_{k=1}^{p}\alpha_{k}\log\sigma_{t-k}^{2}</math>
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| where <math>g(Z_{t})=\theta Z_{t}+\lambda(|Z_{t}|-E(|Z_{t}|))</math>, <math>\sigma_{t}^{2}</math> is the [[conditional variance]], <math>\omega</math>, <math>\beta</math>, <math>\alpha</math>, <math>\theta</math> and <math>\lambda</math> are coefficients, and <math>Z_{t}</math> may be a [[standard normal variable]] or come from a [[generalized error distribution]]. The formulation for <math>g(Z_{t})</math> allows the sign and the magnitude of <math>Z_{t}</math> to have separate effects on the volatility. This is particularly useful in an asset pricing context.<ref>{{cite journal |last=St. Pierre |first=Eilleen F. |year=1998 |title=Estimating EGARCH-M Models: Science or Art |journal=The Quarterly Review of Economics and Finance |volume=38 |issue=2 |pages=167-180 |doi=10.1016/S1062-9769(99)80110-0 }}</ref>
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| Since <math>\log\sigma_{t}^{2}</math> may be negative there are no (fewer) restrictions on the parameters.
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| ==GARCH-M==
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| The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification:
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| <math>
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| y_t = ~\beta x_t + ~\lambda ~\sigma_t + ~\epsilon_t
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| </math>
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| The residual <math> ~\epsilon_t </math> is defined as
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| <math>
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| ~\epsilon_t = ~\sigma_t ~\times z_t
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| </math>
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| ==QGARCH==
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| The Quadratic GARCH (QGARCH) model by Sentana (1995) is used to model symmetric effects of positive and negative shocks.
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| In the example of a GARCH(1,1) model, the residual process <math> ~\sigma_t </math> is
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| <math>
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| ~\epsilon_t = ~\sigma_t z_t
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| </math>
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| where <math> z_t </math> is i.i.d. and
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| <math>
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| ~\sigma_t^2 = K + ~\alpha ~\epsilon_{t-1}^2 + ~\beta ~\sigma_{t-1}^2 + ~\phi ~\epsilon_{t-1}
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| </math>
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| ==GJR-GARCH==
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| Similar to QGARCH, The Glosten-Chris Hughton-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process. The suggestion is to model <math> ~\epsilon_t = ~\sigma_t z_t </math> where <math> z_t </math> is i.i.d., and
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| <math>
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| ~\sigma_t^2 = K + ~\delta ~\sigma_{t-1}^2 + ~\alpha ~\epsilon_{t-1}^2 + ~\phi ~\epsilon_{t-1}^2 I_{t-1}
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| </math>
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| where <math> I_{t-1} = 0 </math> if <math> ~\epsilon_{t-1} \ge 0 </math>, and <math> I_{t-1} = 1 </math> if <math> ~\epsilon_{t-1} < 0 </math>.
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| ==TGARCH model==
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| The Threshold GARCH (TGARCH) model by Nikolai (2013) is similar to GJR GARCH, and the specification is one on conditional standard deviation instead of [[conditional variance]]:
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| <math>
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| ~\sigma_t = K + ~\delta ~\sigma_{t-1} + ~\alpha_1^{+} ~\epsilon_{t-1}^{+} + ~\alpha_1^{-} ~\epsilon_{t-1}^{-}
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| </math>
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| where <math> ~\epsilon_{t-1}^{+} = ~\epsilon_{t-1} </math> if <math> ~\epsilon_{t-1} > 0 </math>, and <math> ~\epsilon_{t-1}^{+} = 0 </math> if <math> ~\epsilon_{t-1} \le 0 </math>. Likewise, <math> ~\epsilon_{t-1}^{-} = ~\epsilon_{t-1} </math> if <math> ~\epsilon_{t-1} \le 0 </math>, and <math> ~\epsilon_{t-1}^{-} = 0 </math> if <math> ~\epsilon_{t-1} > 0 </math>.
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| ==fGARCH==
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| Hentschel's '''fGARCH''' model,<ref>Hentschel, Ludger (1995). [http://ideas.repec.org/a/eee/jfinec/v39y1995i1p71-104.html All in the family Nesting symmetric and asymmetric GARCH models], Journal of Financial Economics, Volume 39, Issue 1, Pages 71-104</ref> also known as '''Family GARCH''', is an omnibus model that nests a variety of other popular symmetric and asymmetric GARCH models including APARCH, GJR, AVGARCH, NGARCH, etc.
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * [[Tim Bollerslev|Bollerslev, Tim]] (1986). "Generalized Autoregressive Conditional Heteroskedasticity", ''Journal of Econometrics'', 31:307-327
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| *Bollerslev, Tim (2008). [ftp://ftp.econ.au.dk/creates/rp/08/rp08_49.pdf Glossary to ARCH (GARCH)], working paper
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| *Enders, W. (1995). ''Applied Econometrics Time Series'', John-Wiley & Sons, 139-149, ISBN 0-471-11163-5
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| *[[Robert F. Engle|Engle, Robert F.]] (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation", ''Econometrica'' 50:987-1008. ''(the paper which sparked the general interest in ARCH models)''
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| * Engle, Robert F. (2001). "GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics", ''Journal of Economic Perspectives'' 15(4):157-168. ''(a short, readable introduction)'' [http://pages.stern.nyu.edu/~rengle/Garch101.doc Preprint]
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| *Engle, R.F. (1995) ARCH: selected readings. Oxford University Press. ISBN 0-19-877432-X
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| *Gujarati, D. N. (2003) ''Basic Econometrics'', 856-862
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| *Hacker, R. S. and Hatemi-J, A. (2005). [http://ideas.repec.org/a/taf/apeclt/v12y2005i7p411-417.html A Test for Multivariate ARCH Effects], ''Applied Economics Letters'', 12(7), 411–417.
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| *Nelson, D. B. (1991). "Conditional heteroskedasticity in asset returns: A new approach", ''[[Econometrica]]'' 59: 347-370.
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| {{Volatility}}
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| {{Stochastic processes}}
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| {{Statistics|analysis}}
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| {{DEFAULTSORT:Autoregressive Conditional Heteroskedasticity}}
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| [[Category:Time series analysis]]
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| [[Category:Nonlinear time series analysis]]
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| [[Category:Econometrics]]
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