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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Autoregressive conditional heteroskedasticity ： ウィキペディア英語版 Autoregressive conditional heteroskedasticity 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 error term or 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 innovations. Such models are often called ARCH models (Engle, 1982), 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 financial time series that exhibit time-varying volatility clustering, i.e. periods of swings followed by periods of relative calm. ARCH-type models are sometimes considered to be part of the family of stochastic volatility models but strictly this is incorrect since at time ''t'' the volatility is completely pre-determined (deterministic) given previous values. ==ARCH(''q'') model Specification== Suppose one wishes to model a time series using an ARCH process. Let $~\backslash epsilon\_t~$ denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These $~\backslash epsilon\_t~$ are split into a stochastic piece $z\_t$ and a time-dependent standard deviation $\backslash sigma\_t$ characterizing the typical size of the terms so that :$~\backslash epsilon\_t=\backslash sigma\_t\; z\_t\; ~$ The random variable $z\_t$ is a strong white noise process. The series $\backslash sigma\_t^2$ is modelled by :$\backslash sigma\_t^2=\backslash alpha\_0+\backslash alpha\_1\; \backslash epsilon\_^2+\backslash cdots+\backslash alpha\_q\; \backslash epsilon\_^2\; =\; \backslash alpha\_0\; +\; \backslash sum\_^q\; \backslash alpha\_\; \backslash epsilon\_^2$ where $~\backslash alpha\_0>0~$ and $\backslash alpha\_i\backslash ge\; 0,~i>0$. An ARCH(''q'') model can be estimated using ordinary least squares. A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows: # Estimate the best fitting autoregressive model AR(''q'') $y\_t\; =\; a\_0\; +\; a\_1\; y\_\; +\; \backslash cdots\; +\; a\_q\; y\_\; +\; \backslash epsilon\_t\; =\; a\_0\; +\; \backslash sum\_^q\; a\_i\; y\_\; +\; \backslash epsilon\_t$. # Obtain the squares of the error $\backslash hat\; \backslash epsilon^2$ and regress them on a constant and ''q'' lagged values: #: #: $\backslash hat\; \backslash epsilon\_t^2\; =\; \backslash hat\; \backslash alpha\_0\; +\; \backslash sum\_^\; \backslash hat\; \backslash alpha\_i\; \backslash hat\; \backslash epsilon\_^2$ #: #: where ''q'' is the length of ARCH lags. #The null hypothesis is that, in the absence of ARCH components, we have $\backslash alpha\_i\; =\; 0$ for all $i\; =\; 1,\; \backslash cdots,\; q$. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated $\backslash alpha\_i$ coefficients must be significant. In a sample of ''T'' residuals under the null hypothesis of no ARCH errors, the test statistic ''T'R²'' follows $\backslash chi^2$ distribution with ''q'' degrees of freedom, where $T\text{'}$ is the number of equations in the model which fits the residuals vs the lags (i.e. $T\text{'}=T-q$). If ''T'R²'' is greater than the Chi-square table value, we ''reject'' the null hypothesis and conclude there is an ARCH effect in the ARMA model. If ''T'R²'' is smaller than the Chi-square table value, we ''do not reject'' the null hypothesis. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Autoregressive conditional heteroskedasticity」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.