Since the introduction of the autoregressive conditional heteroskedastic (ARCH) model in Engle (1982), numerous applications of this modeling strategy have already appeared. A common finding in many of these studies with high frequency financial or monetary data concerns the presence of an approximate unit root in the autoregressive polynomial in the univariate time series representation for the conditional second order moments of the process, as in the so-called integrated generalized ARCH (IGARCH) class of models proposed in Engle and Bollerslev (1986). In the IGARCH models shocks to the conditional variance are persistent, in the sense that they remain important for forecasts of all horizons. This idea is readily extended to a multivariate framework. Even though many time series may exhibit persistence in variance, it is likely that several different variables share the same common long-run component. In that situation, the variables are naturally defined to be co-persistent in variance, and the co-persistent linear combination is interpretable as a long-run relationship. Conditions for co-persistence to occur in the multivariate linear GARCH model are presented. These conditions parallel the conditions for linear co-integration in the mean, as developed by Engle and Granger (1987). The presence of co-persistence has important implications for asset pricing relationships and in optimal portfolio allocation decisions. An empirical example relating to the time series properties of nominal U.S. dollar exchange rates for the deutschemark and the British pound provides a simple illustration of the ideas.

This paper investigates the properties of the conditional covariance estimates generated by a misspecified ARCH model. For example, suppose that we observe a diffusion process at discrete time intervals of length h. For each h, we use a GARCH(1,1) model to estimate the instantaneous conditional covariance matrix of the diffusion. Under mild regularity conditions, the difference between these conditional covariance estimates and the true conditional covariance converges to zero in probability as h↓0. Many other ARCH models (for example, Exponential ARCH) have similar consistency properties. This may well account for the success of ARCH models in short-term forecasting using high-frequency data, since even misspecified ARCH models can produce ‘good’ estimates of volatility.

Suppose an observed time series is generated by a stochastic volatility model-i.e., there is an unobservable state variable controlling the volatility of the innovations in the series. As shown by Nelson (1992), and Nelson and Foster (1994), a misspecified ARCH model will often be able to consistently (as a continuous time limit is approached) estimate the unobserved volatility process, using information in the lagged residuals. This paper shows how to more efficiently estimate such a volatility process using information in both lagged and led residuals. In particular, this paper expands the optimal filtering results of Nelson and Foster (1994) and Nelson (1994) to smoothing and to filtering with a random initial condition.

It is widely known that conditional covariances of asset returns change over time. Researchers adopt many strategies to accommodate conditional heteroskedasticity. Among the most popular are: (a) chopping the data into short blocks of time and assuming homoskedasticity within the blocks, (b) performing one-sided rolling regressions, in which only data from, say, the preceding five year period is used to estimate the conditional covariance of returns at a given date, and (c) two-sided rolling regressions which use, say, five years of leads and five years of lags. GARCH amounts to a one-sided rolling regression with exponentially declining weights. We derive asymptotically optimal window lengths for standard rolling regressions and optimal weights for weighted rolling regressions. An empirical model of the S&P 500 stock index provides an example.

We consider the errors-in-variables model when more than one independent variable is measured with error, and show that the vector of corresponding OLS parameter estimates is asymptotically biased towards zero. The appropriate vector norm is not, in general, the euclidean norm.

A companion paper (Nelson, 1992) showed that in data observed at high frequencies, an ARCH model may perform well in estimating the conditional variance of a process, even when the ARCH model is severely misspecified. While such models may perform reasonably well at filtering (i.e., at estimating unobserved instantaneous conditional variances), they may perform disastrously at medium- and long-term forecasting of the process and its volatility. In this paper, we develop conditions under which a misspecified ARCH model successfully performs both tasks, filtering and forecasting. The key requirement (in addition to the conditions for consistent filtering) is that the ARCH model correctly specifies the functional form of the first two conditional moments of all state variables. We apply these results to a diffusion model employed in the options pricing literature, the stochastic volatility model of Hull and White (1987), Scott (1987), and Wiggins (1987).

We investigate the conditional covariances of stock returns using bivariate exponential ARCH (EGARCH) models. These models allow market volatility, portfolio-specific volatility, and beta to respond asymmetrically to positive and negative market and portfolio returns, i.e., ''leverage'' effects. Using monthly data, we find strong evidence of conditional heteroskedasticity in both market and non-market components of returns, and weaker evidence of time-varying conditional betas. Surprisingly while leverage effects appear strong in the market component of volatility, they are absent in conditional betas and weak and/or inconsistent in nonmarket sources of risk.

This paper investigates the convergence of stochastic difference equations (e.g., ARCH) to stochastic differential equations as the length of the discrete time intervals between observations goes to zero. These results are applied to the GARCH(1,1) model of Bollerslev (1986) and to the AR(1) Exponential ARCH model of Nelson (1989). In their continuous time limits, the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal, respectively. In addition, a class of diffusion approximations based on the Exponential ARCH model is developed.

This chapter evaluates the most important theoretical developments in ARCH type modeling of time-varying conditional variances. The coverage include the specification of univariate parametric ARCH models, general inference procedures, conditions for stationarity and ergodicity, continuous time methods, aggregation and forecasting of ARCH models, multivariate conditional covariance formulations, and the use of model selection criteria in an ARCH context. Additionally, the chapter contains a discussion of the empirical regularities pertaining to the temporal variation in financial market volatility. Motivated in part by recent results on optimal filtering, a new conditional variance model for better characterizing stock return volatility is also presented.

To keep the conditional variances generated by the GARCH (p, q) model nonnegative, Bollerslev imposed nonnegativity constraints on the parameters of the process. We show that these constraints can be substantially weakened and so should not be imposed in estimation. We also provide empirical examples illustrating the importance of relaxing these constraints.

We estimate and compare a variety of continuous-time models of the short-term riskless rate using the generalised method of moments. We find that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate. A number of well-known models perform poorly in the comparisons because of their implicit restrictions on term structure volatility. We show that these results have important implications for the use of different term structure models in valuing interest rate contingent claims and in hedging interest rate risk.

GARCH models have been applied in modelling the relation between conditional variance and asset risk premia. These models, however, have at least three major drawbacks in asset pricing applications: (i) Researchers beginning with Black (1976) have found a negative correlation between current returns and future returns volatility. GARCH models rule this out by assumption. (ii) GARCH models impose parameter restrictions that are often violated by estimated coefficients and that may unduly restrict the dynamics of the conditional variance process. (iii) Interpreting whether shocks to conditional variance “persist” or not is difficult in GARCH models, because the usual norms measuring persistence often do not agree. A new form of ARCH is proposed that meets these objections. The method is used to estimate a model of the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987.

This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift converges almost surely to zero, while IGARCH(1,1) with a positive drift is strictly stationary and ergodic. We examine the persistence of shocks to conditional variance in the GARCH(l.l) model, and show that whether these shocks "persist" or not depends crucially on the definition of persistence. We also develop necessary and sufficient conditions for the finiteness of absolute moments of any (including fractional) order.

Journal Article Simple Binomial Processes as Diffusion Approximations in Financial Models Get access Daniel B. Nelson, Daniel B. Nelson The University of Chicago Search for other works by this author on: Oxford Academic Google Scholar Krishna Ramaswamy Krishna Ramaswamy The Wharton School, University of Pennsylvania Search for other works by this author on: Oxford Academic Google Scholar The Review of Financial Studies, Volume 3, Issue 3, July 1990, Pages 393–430, https://doi.org/10.1093/rfs/3.3.393 Published: 30 April 2015