Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. in ation. Econometrica 50, 987-1007.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....volatility n describes the change of (conditional) variance. The autoregressive conditionally heteroscedastic (ARCH) models are one of the speci cations of (1.3) In this case the conditional variance n is a linear function of the squared past observations. ARCH(p) models introduced by Engle [37] are de ned by n = 0 j X n j ; 0 0 ; 1 ; p 1 0; p 0 ; n 2 ; 1.4) where p is the order of the ARCH process. There are two natural extensions of this model. Bollerslev [12] proposed the so called generalized ARCH (GARCH) processes. The conditional variance ....

Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. in ation. Econometrica 50, 987-1007.

....markets exhibit positive autocorrelation which is typically found in the empirical autocorrelation function of squared returns. Periods of higher and lower volatility alternate. This phenomenon is well known and generated a vast body of econometric literature after the seminal contributions by Engle (1982), Bollerslev (1986) and Taylor (1986) introducing the (generalized) autoregressive conditionally heteroskedastic ( G)ARCH) process and the stochastic volatility model, respectively. The large variety of existing (univariate) parametric models already indicates that particular specifications fail ....

Engle, R.F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation". Econometrica, 50, 987-1008.

....models that incorporate clusters of volatilities are more appropriate than ARMA speci cations. We consider here as a motivating example an application of nonlinear time series analysis to foreign exchange high frequency data. For these data the autoregressive heteroscedastic models (ARCH) by Engle (1982) have been studied extensively. An ARCH model for time series fY t g with ARCH error term of order q is de ned through X t = t t , where t are independent mean zero and variance one random variables and 2 t = 1 X 2 t 1 2 X 2 t 2 : q X 2 t q , with 0; i ....

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. in ation. Econometrica, 50, 987-1008.

....models. Furthermore, as originally pointed out by[37] and [12] the data exhibit non Gaussian kurtosis and dynamic asymmetries such as the leverage effect , expressed by a negative cross correlation between current returns (x t ) and future squared returns (x 2 t u with u 0) The ARCH model of [35] represents the most famous nonlinear time series model apt to account for such features, except for dynamic asymmetries. Denoting by F t the oe field of events generated by fx s : s tg and setting oe 2 t =var(x 2 t jF t;1 ) the ARCH(p) postulates that oe 2 t = ff 1 x 2 t;1 ....

Engle, R. (1982): "Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation," Econometrica,50, 987--1007.

....long memory is most important for volatilities of stock returns. This has important consequences for the valuation of options. A standard class of models introduced for modeling volatilities of stock returns consists of the Autoregressive Conditional Heteroscedasticity (ARCH) models (see Engle (1982)) These models assume that the conditional variance depends on the currently known information in a nontrivial way. But they do not allow for modeling long range dependence, because shocks to the conditional variance decay exponentially and thus have almost no influence for long time optimal ....

Engle, R. F. (1982): "Autoregressive conditional heteroscedasticity with estimates of the variance of U. K. inflation." Econometrica 50, 987 - 1008.

....mean are the RESET test (of Ramsey, 1974) and the neural net (NN) test (of Lee, White and Granger, 1993) results for both of which are included in Table 3. To investigate non linearities in the conditional variance we also report the autoregressive conditional heteroscedasticity (ARCH) test (Engle, 1982). All tests for non linearities are computed using the residuals of an AR(p) model for the outlier adjusted differences, or the level of the series for the stationary variables. The AR order is specified by testing down from 12 lags in PcGive and selecting the model that minimises the Schwartz ....

Engle, R. F. (1982). `Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation', Econometrica, Vol. 49, pp. 1057-1072.

....the volatility n describes the change of (conditional) variance. The autoregressive conditionally heteroscedastic (ARCH) models are one of the speci cations of (1.1) In this case the conditional variance 2 n is a linear function of the squared past observations. ARCH(p) models introduced by Engle (1982) are Received AMS 1991 subject classi cations. Primary 60H25; secondary 60G10, 60J05. Key words and phrases. ARCH model, autoregressive process, geometric ergodicity, heavy tail, heteroscedastic model, Markov process, recurrent Harris chain, regular variation, Tauberian theorem. 1 2 M. ....

Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. in ation. Econometrica 50, 987-1007.

....models having hyperbolic convergence rates, the FIGARCH, and a newly proposed generalization, the HYGARCH model. The latter models are applied to Asian exchange rates over the 1997 crisis period, and are shown to account well for the characteristics of the data. 1Introduction Many variants of Engles (1982) ARCH model of conditional volatility have been proposed, including GARCH (Bollerslev 1986) IGARCH (Engle and Bollerslev 1986) and FIGARCH (Baillie, Bollerslev and Mikkelsen, 1996, Ding and Granger, 1996) All of these models, and many other cases that might be devised, fall into the class in ....

Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom ination. Econometrica 50, 9871007.

....t = 1 # ## 2 t 1 ) 1 2 # t , where # = 0.5, 0.9 is taken. 7 # 0 # N(0, 1) and # # t # t # # N ## 0 0 ## 1 0 0 1 ## are assumed, which are also used in Simulations B D. The uniform distribution is taken for the prior density of #, i.e. P # (#) 1 for 0 # 1. See Engle (1982) and Bollerslev, Engle and Nelson (1994) for the ARCH model. Simulation B (Stochastic Volatility Model) Take an example of the following state space model: y t = exp (0.5# t ) # t and # t = ## t 1 # t , where # = 0.5, 0.9 is taken. The uniform prior P # (#) 1 for 0 # 1 is assumed. See ....

....= 1 for 7 Note from the transition equation that the unconditional variance of # t is assumed to be one. In this paper, # = 0.5, 0.9 is examined. # t is distributed with large tails in the case of # = 0.9, because the forth moment of the ARCH(1) process does not exist when # 2 1 # 3. See Engle (1982) and Bollerslev, Engle and Nelson (1994) 9 t = 61, 62, 80 and d t = 0 otherwise. The di#use prior is assumed for #. 8 This model corresponds to the case where the sudden shifts occur at time periods 21, 41, 61 and 81. Simulation D (Shifted Mean Model) The data generating process is given ....

Engle, R.F., 1982, " Autoregressive Conditional Heteroscedasticity with Estimates of Variance of U.K. Inflation, ", Econometrica, Vol.50, pp.987 -- 1008.

....random variable which depends on state or regime. 2.1. 7 Stochastic Variance Models In this section, we introduce two stochastic variance models (see Taylor (1994) for the stochastic variance models) One is called the autoregressive conditional heteroskedasticity (ARCH) model proposed by Engle (1982) and another is the stochastic volatility model (see Ghysels, Harvey, and Renault (1996) 9 Let # be a k 1 vector of unknown parameters to be estimated. y t and x t are assumed to be observable variables. The first order ARCH model is given by the following two equations: Measurement ....

Engle, R.F., 1982, " Autoregressive Conditional Heteroscedasticity with Estimates of Variance of U.K.

....examine the following state space model: y t = # t # t and # t = 1 # ## 2 t 1 ) 1 2 # t , where # = 0.5, 0.9 is taken. Note that in Simulation 2 the unconditional variance of # t is assumed to be one. This is called the ARCH (Autoregressive Conditional Heteroscedasticity) model (see Engle (1982), Harvey (1989) and Harvey and Streibel (1998) Simulation 4: Markov Switching Model (Table 4) Finally, we consider the following k variate state space model: y t = x t # t # t and # i,t = # # i,t 1 i,1 (1 # i,2 ) 1 # i,t 1 # i,t for i = 1, 2, k, which is very close to the Markov ....

Engle, R.F., 1982, " Autoregressive Conditional Heteroscedasticity with Estimates of Variance of U.K. Inflation, ", Econometrica, Vol.50, pp.987 -- 1008.

....univariate system: y t = # t # t and # t = ## t 1 # t . Simulation II (ARCH Model) The model is given by: y t = # t # t and # t = # 0 ## 2 t 1 ) 1 2 # t for # 0 0, 0 # # 1 and # 0 = 1 # are taken, where y t consists of the ARCH(1) process # t and the error term # t . See Engle (1982) and Bollerslev et al. 1994) for the ARCH model. Simulation III (Stochastic Volatility Model) Take the state space model: y t = exp(0.5# t )# t and # t = ## t 1 # t for 0 # # 1. See Ghysels et al. 1996) for the stochastic volatility model. Simulation IV (Nonstationary Growth Model) The ....

Engle, R.F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of Variance of U.K. Inflation,, Econometrica, 50, 987--1008.

....and asymmetric VaR models. These models are applied to daily stock indexes data in Section 3 where we assess their performances and characterize the long and short VaR. 2 2 VaR models In this section we present parametric VaR models of the ARCH class. ARCH class models were first introduced by Engle (1982) with the ARCH model. Since then, numerous extensions have been put forward, see Engle (1995) Bera and Higgins (1993) or Palm (1996) but they all share the same goal, i.e. modelling the conditional variance as a function of past (squared) returns and associated characteristics. Because quantiles ....

Engle, R. (1982): "Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation," Econometrica, 50, 987--1007.

....the past. In its basic form, the sequence fX n g can be described as X n = Y n p h n , where fY n g is an iid sequence of standard normal variables and h n is the conditional variance of Y n , h n = n ffX 2 n Gamma1 , with n 0 and 0 ff 1. This model is called an ARCH(1) model, see Engle (1982). The parameter ff remains constant over time. The intercept in the conditional variance equation depends on n but we assume that it is constant up to time m, 1 = Delta Delta Delta = m j 0 : We now aim to detect if a change occurs in when new data are observed, that is, we want to test ....

Engle, R. F. (1982), `Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation', Econometrica 50, 987--1007.

....Another approach due to A t Sahalia (1996) estimates the implied density of discrete changes in the spot rate implied by various continuous time models, and compares these with the empirical distribution of the discrete changes in the spot rate. 2. 2 GARCH Models The ARCH model was introduced by Engle (1982) and later extended by Bollerslev (1986) who developed the generalized ARCH, or GARCH model. In a GARCH(1, 1) 5 model, the conditional mean and conditional variance of a time series process are modeled simultaneously r t = a br t 1 e t where the conditional volatility of e t is given by ....

Engle, Robert F., 1982, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation, Econometrica 50, 987--1008.

....discounting approach. In recent years, research in nance and econometrics has devoted signi cant efforts to come up with more sophisticated stochastic variance models; two major types of volatility models ARCH GARCH models and stochastic volatility (SV) models have been developed. ARCH([27]) GARCH( 8] types of models appeared in the early 1980 s; they model the conditional variance as a moving average of the squared past observation errors and an autoregression of the past conditional variances. Numerous extensions with univariate structure were rapidly developed, among which ....

R F Engle. Autoregressive conditional heteroscedasticity with estimates of the variance of UK in ation. Econometrica, 50:987-1008, 1982.

....presence of skew in asset returns will also play an important role in developing hedging strategies. Modelling and forecasting volatility, skew and kurtosis Density forecasting in finance can be viewed as beginning with the literature that aims to model and forecast volatility. The ARCH model of Engle (1982) models the conditional variance as a linear function of squares of past observations, and thus delivers forecasts with time varying conditional variances. A Generalized ARCH(1,1) process includes the lagged conditional variance and can be written (with zero mean) as 12 y t e t h 1 2 t h ....

Engle RF. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50: 987-1008.

....case: The 1. McKinsey Professor, Santa Fe Institute, 1399 Hyde Park Rd. Santa Fe 87501, jdf santafe.edu. 2. Economics Department, Yale University, New Haven CT. 2 largest price movements often occur with little or no news (Cutler et al. 1989) price volatility is strongly temporally correlated (Engle 1982), short term price fluctuations are nonnormal 3 , and prices may not accurately reflect rational valuations (Campbell and Shiller 1988) This suggests that markets have nontrivial internal dynamics. Traders may be thought of as signal processing elements, that process external information and ....

Engle, R., 1982, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation, Econometrica 50, 987 - 1008.

....Applied Bayesian Data Analysis Using State Space Models 5 3 Applications 3.1 Stochastic Volatility Models in Econometrics The stochastic volatility (SV) model introduced by Tauchen and Pitts (1983) is used to describe nancial time series. It o ers an alternative to the ARCHtype model of Engle (1982) for the well documented time varying volatility exhibited in many nancial time series. The SV model provides a more realistic and exible modeling of nancial time series than the ARCH type model, since it essentially involves two noise processes, one for the observations, and one for the latent ....

ENGLE, R.F. (1982): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom In ation. Econometrica, 50, 987{ 1007.

....cautioned that the alternative model of interest rate process adopted here is only one special case, and a full scale study of misspeci cation issue is clearly outside the scope of this paper. Even in the following limited example, the performance of EMM is not unrelated with the established ARCH (Engle 1982) and GARCH (Bollerslev 1986) ltering of the stochastic volatility process. In fact, the power of EMM will be optimal if the rst stage SNP auxiliary model with an ARCH (earlier version) or GARCH (recent version) leading term adequately captures the interest rate dynamics. 21 plot the 5 ....

Engle, Robert F. (1982), \Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. In ation," Econometrica, vol. 50, 987-1008.

....volatility. Guillaume et al. 1994) have surveyed some stylized facts associated with intraday patterns in FX markets. Some early evidence on the cross sectional patterns in intraday FX data was provided by M uller et al. 1990) Early time series applications, which built on the ARCH model of Engle (1982) to model the dynamics of intraday FX volatility, included: Engle, Ito, and Lin (1990) who used four points during the 24 hour trading day to study the transmission (spillover) of information across markets; and Baillie and Bollerslev (1991) who utilized hourly observations in a seasonal GARCH ....

Engle, Robert F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom in ation. Econometrica 50(4), 987-1007.

....mean. The correlogram of the squared returns, however, indicates substantial dependence in the volatility of returns. 3. 2 A volatility model A widely used class of models for the conditional volatility is the autoregressive conditionally heteroskedastic class of models introduced by Engle (1982), and extended by Bollerslev (1986) Engle et al. (1987) Nelson (1991) Glosten et al. (1993) amongst many others. See Bollerslev et al. 1992) or Bollerslev et al. 1994) for summaries of this family of models. A popular member of the ARCH class of models is the GARCH(p,q) model: 18) ....

Engle, Robert F., 1982, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50(4), 987-1007.

....of ARCH models is necessary in order to understand how a measure of risk and an estimate for exchange rate variance can be obtained simultaneously. 3.1. Introduction to ARCH Models A body of literature has developed recently which applies particularly well to estimation of equation (3. 1) Engle (1982) first proposed a class of time series models which took account of systematic changes in the disturbance term. 13 More efficient estimation could result if past magnitudes of the disturbance term provided information about the variance of the current disturbance. Engle obtained a prediction of ....

....refinements to ARCH became necessary due to problems with estimation. In many instances the signs of the lagged squared residual parameters were negative clearly a result which violated the requirement of positive variance estimates. Ad hoc procedures were developed to overcome this problem. Engle (1982 and 1983) regressed squared residuals on a linearly declining weighted average of past squared error terms, reducing the parameters to be estimated to two. Other authors such as Diebold and Nerlove (October 1985) and Domowitz and Hakkio (1985) squared the ARCH parameters, imposing the ....

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Engle, Robert F. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica; July 1982; 50(4):987-1007.

....confidence level, c Length of a testing interval, T Admissible VAR exceedings, E Admissible VAR frequencies, E=T Significance level, x Significance level, x 5 1 5 1 95 500 [17,33] 14,36] 3.40 , 6.60 ] 2.80 , 7.20 ] 1500 [61,89] 56,94] 4.07 , 5.93 ] 3.73 , 6. 27 ] 99 500 [2,8] [0,10] [0.40 , 1.60 ] 0.00 , 2.00 ] 1500 [9,21] 6,23] 0.60 , 1.40 ] 0.40 , 1.53 ] We examined forecasting properties of stable and VAR models for data series described in Table 1. In testing procedures we considered the following parameters: 38 In nominal levels, an exceeding implies a case when ....

Engle, R. F., 1982, "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation", Econometrica 50 (July 1982), 987-1007.

....These distributions can be multimodal. It is also expected that the marginal distribution of the time series could be multimodal. Thirdly,asaby product of the changing shape of the conditional distribution, it can be shown that the MAR models are able to capture conditional heteroscedasticity #Engle, 1982# which is common in some #nancial time series. 2. The Mixture Autoregressive Model The K component mixture autoregressive #MAR# model is de#ned by F #y t jF t,1 #= K X k=1 # k # # y t , # k0 , # k1 y t,1 , # kp k y t,p k # k : We denote this model by MAR#K; p 1 ;p 2 ; p K #. ....

Engle, R. F. #1982#. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom In#ation. Econometrica, 50, 987-1007.

.... ned by (6) includes many of familiar nonlinear both parametric and nonparametric models commonly encountered in many applied elds such as econometrics and nance, for example, the threshold models and their various modi cations in Tong (1990) the ARCH and GARCH models as proposed and studied by Engle (1982) and Bollerslev (1986) the exponential autoregressive models introduced and extended by Haggan and Ozaki (1981) and Ozaki (1982) and the MARS models considered in Lewis and Stevens (1991) In particular, 4) covers the nonlinear additive autoregressive model with exogenous variables Y t = ....

Engle, R.F. (1982). Autoregressive conditional heteroscedasticy with estimates of the variance of U.K. in ation. Econometrica, 50, 987-1008.

....are presented in section 6. In the final section we conclude and provide a summary. 2 Model Specifications Generalised Autoregressive Conditional Heteroscedasticity (GARCH) models have thusfar been the most frequently applied class of time varying volatility model. Since its introduction by Engle (1982) and subsequent generalisation by Bollerslev (1986) this model has been extended in numerous ways which usually involved alternative formulations for the volatility process 4 . Although the Stochastic Volatility (SV) model has been recognised as a viable alternative to the GARCH model, the ....

Engle, R.F. (1982), Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50, 987-1006.

....to prove the results. The likelihood processes for the models are studied in sections 4 and 5, and theorems are proved in section 6. 4 2 ARCH models and diffusions 2. 1 ARCH models Probably the most important innovation in discrete time modeling of financial time series is the introduction by Engle (1982) of the ARCH model. The model makes the conditional variance of a series of prediction errors equal to some function of lagged errors, time, parameters, and predetermined exogenous and lagged endogenous variables. Specifically, the observed time series x k , k = 1; Delta Delta Delta ; n, are ....

....on k, oe 2 k , and x k Gamma1 ; x k Gamma2 ; Delta Delta Delta, ff is a vector of parameters, A k is a vector of exogenous and lagged endogenous variables. Any model of the form (1) 2) is referred to as an ARCH model. The existing ARCH models differ in their specification for oe 2 k . Engle (1982) chose the following function form for oe 2 k , oe 2 k = ff 0 p X j=1 ff j y 2 k Gammaj = ff 0 p X j=1 ff j oe 2 k Gammaj 2 k Gammaj ; 3) where ff s are nonnegative constants. The model specified by (3) is often called ARCH(p) The appeal of this model lies in the way ....

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987-1007.

....studies by Schwert (1989a, 1 1989b) find that stock market volatility has not increased in recent years. However, since Mandelbrot s (1963) paper, numerous studies indicate that the volatility of stock returns varies over time. The attempt to explain this variation has led to fairly wide use of Engle s (1982) autoregressive conditional heteroskedastic (ARCH) model and its derivatives. Recently, variation in volatility has been associated with regime shifts. For example, Glosten, Jagannathan and Runkle (1993, p. 1789) speculate on this possibility by suggesting the data may be explained by . ....

Engle, Robert F., 1982, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50, No. 4, 987-1007.

....especially to relax the assumption of constant volatility. The volatility smile phenomenon and term structure suggest that it is more reasonable to consider volatility as a stochastic process. Two classes stand out in the research of volatility calibration: i) ARCH GARCH time series models [Engle (1982), Bollerslev (1986) in which volatility is de ned as a function of past returns, make the inference and forecasting easy to implement, but fail to explain certain market behaviors such as the volatility smile; ii) Stochastic volatility (SV) models [see Taylor (1994) Shephard (1996) and ....

Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom in ation. Econometrica 50, 987-1007.

.... the same feature holds for many high frequency financial time series (such as daily stock market returns and interest rates) To describe this volatility clustering in empirical finance one often uses the so called Autoregressive Conditionally Heteroskedastic [ARCH] model put forward by Engle (1982). Over the last fifteen years, this model has been the subject of intensive research, see the surveys of Bollerslev, Chou and Kroner (1992) Bera and Higgins (1993) and Bollerslev, Engle and Nelson (1994) among others. A popular extension of the basic ARCH model is the Generalized ARCH [GARCH] ....

Engle, R.F., 1982, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, 987--1007.

....two Markov regimes. Although t is assumed to be serially uncorrelated, the conditional squared residual ( s ) can be serially correlated since the Markov variable s t is serially correlated. This is in the same line as the Autoregressive Conditional Heteroskedasticity models (ARCH) proposed by Engle (1982) and generalized by Engle and Bollerslev (1986) and the switching ARCH model (SWARCH) proposed by Hamilton and Susmel (1994) The first order assumption of the Markov chain implies that all the relevant information for predicting future states is included in the current state, i.e. pr[s t I ....

Engle, R., 1982, "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation." Econometrica, 50, 987-1007.

....properties of the model. Diagnostic tests are presented for all models. So, we report the Akaike information and the Schwartz information criteria for model comparison as well as conventional tests for skewness and kurtosis. Also, we report the Lagrange multiplier ARCH test of second order of Engle (1982) and the Lomnicki JarqueBera normality tests (Lomnicki, 1961; Jarque and Bera, 1980) In addition, we present the group of tests designed by Eitrheim and Tersvirta (1996) and Tersvirta (1998) specifically for STAR and STR models. These tests are all computed using F statistics for the significance ....

Engle, Robert F. (1982). "Autoregressive conditional heteroscedasticity, with estimates of the variance of the United Kingdom inflation", Econometrica, 50, 987-1007.

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Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. in ation. Econometrica 50, 987-1007.

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R. Engle, "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of the United Kington Inflations," Econometrica , 50, 987-1008 (1982).

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Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987--1007.

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Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of UK in ation. Econometrica 50, 987-1008.

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Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica, 50:987--1008.

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Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987--1007.

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Engle, R. (1982) Autoregressive conditional heteroscedastic models with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987--1007.

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Engle, R., #982, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation, Econometrica 50, 987-##08.

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Engle, Robert, (1982), "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50/4: 987-1006.

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Engle, R. F. (1982), Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom In ation, Econometrica 50, 9871007.

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Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimators of the variance of United Kingdom inflation. Econometrica, 50, 987-1008.

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Engle, R. (1982) : "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation", Econometrica, 40, 987-1008.

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Engle, R.F. (1982). Autoregressive Conditional Heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 9871006.

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Engle, R. (1982) : "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation", Econometrica, 50, 987 - 1008.

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Engle, R. (1982): "The Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation,"Econometrica, 45, 987--1007.

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Engle, R. (1982), Autoregressive conditional heteroscedasticity with estimates of the variance of United Kindom inflation, Econometrica, 50, 987-1008.

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R.F. Engle, 'Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation', Econometrica, 50, 987--1008, (1982).

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