Ultra-high-frequency data is defined to be a full record of transactions and their associated characteristics. The transaction arrival times and accompanying measures can be analyzed as marked point processes. The ACD point process developed by Engle and Russell (1998) is applied to IBM transactions arrival times to develop semiparametric hazard estimates and conditional intensities. Combining these intensities with a GARCH model of prices produces ultra-high-frequency measures of volatility. Both returns and variances are found to be negatively influenced by long durations as suggested by asymmetric information models of market micro-structure.

Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance. A regression model is then introduced with disturbances following an ARCH process. Maximum likelihood estimators are described and a simple scoring iteration formulated. Ordinary least squares maintains its optimality properties in this set-up, but maximum likelihood is more efficient. The relative efficiency is calculated and can be infinite. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The test is based simply on the autocorrelation of the squared OLS residuals. This model is used to estimate the means and variances of inflation in the U.K. The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies.

[A set of standard dynamic disaggregated price equations are estimated to examine the relationship between changes in input prices and output prices. The equations perform satisfactorily by conventional criteria; however, when disaggregated by frequency, it is found that the high and low frequency components appear to satisfy different models. The differences are generally significant suggesting that the model is misspecified and that another lag distribution should be used. In particular, the sum of the lag coefficients for labor inputs is substantially larger when estimated with the low frequency component than the high. Therefore, such a price equation estimated during a regime of continued wage inflation would exhibit a much larger long run output price elasticity with respect to wages, than would one estimated during a period of stable or randomly fluctuating wages.]

[Necessary and sufficient conditions are determined under which a truncated approximation to generalized least squares is more efficient than ordinary least squares. For the general case, the necessary conditions are unlikely to be fulfilled. For a first order Markov model in a second order world, the sufficient conditions are satisfied only when one of the second order roots is very small, and therefore the first order assumption is approximately true. When the class of possible exogenous variables is limited to those typical of economic time series, the sufficient conditions are satisfied for a wider range of cases. Relative efficiencies are computed for a variety of cases.]

A common finding in many of the recent empirical studies with the ARCH class of models applied to high frequency financial data concerns the apparent persistence of shocks for forecast of the future conditional variances. It is likely that several different variables share this same implied long-run component, however. In that situation, the variables are defined to be copersistent in variance. Conditions for copersistence to occur in the linear multivariate GARCH model are presented. These conditions parallel the conditions for linear cointegration in the mean. A simple empirical example with foreign exchange rate data illustrates the ideas. Copyright 1993 by The Econometric Society.

[Any misspecification of the disturbance error process in a linear regression may lead to an inefficient estimator. Although spectral methods proposed by Hannan will always be asymptotically efficient, they are frequently used because they are computationally demanding and very large samples are presumably required. This paper presents Monte Carlo evidence from a variety of typical econometric situations which indicates that the estimators perform quite well for moderate-sized samples (100) when the error process is highly dependent, and even for small samples when the error process is simple. The results are used to estimate a second order term in the asymptotic expansion for the variance.]

The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples. If each element of a vector of time series x first achieves stationarity after differencing, but a linear combination a'x is already stationary, the time series x are said to be co-integrated with co-integrating vector a. There may be several such co-integrating vectors so that a becomes a matrix. Interpreting a'x,= 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance. The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application. In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.

The expectati on of the excess holding yield on a long bond is postulated to depend upon its conditional variance. Engle's ARCH model is extended to allow the conditional variance to be a determinant of the mean and is called ARCH-M. Estimation and infer ence procedures are proposed, and the model is applied to three interest rate data sets. In most cases the ARCH process and the time varying risk premium are highly significant. A collection of LM diagnostic tests reveals the robustness of the model to various specification changes such as alternative volatility or ARCH measures, regime changes, and interest rate formulations. The model explains and interprets the recent econometric failures of the expectations hypothesis of the term structure. Copyright 1987 by The Econometric Society.

This paper proposes a new statistical model for the analysis of data which arrives at irregular intervals. The model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates. The conditional intensity is developed and compared with other self-exciting processes. The model is applied to the arrival times of financial transactions and therefore is a model of transaction volume, and also to the arrival of other events such as price changes. Models for the volatility of prices are estimated, and examined from a market microstructure point of view.