I am pleased to offer a comment on this very interesting article by an author who is always in the forefront of the research on empirical financial models. This article presents data analysis that estabishes the stylized facts about stock market volatility around market crashes. He concludes that volatility is high during periods of stock market decline and that it gradually returns to more normal levels. In the case of 1987, the peak was higher than usual and the decline was more rapid. The article uses 28,000 daily observations but does not really estimate a usable model; instead, it explores the data by estimating highly overparameterized models that reveal important features of the data. I suggest that this be considered an exploratory investigation and that in the face of more parsimonious models, rather interesting and somewhat different conclusions are revealed. The basic model estimated by Schwert is a 22-order autoregression of daily returns with a heteroskedastic error standard deviation which is itself assumed to be a 22-order autoregression in the absolute errors. Even with 28,000 observations, there is apparently a lot of noise in the coefficients. To allow for a risk premium, the mean is related to the variance, and in this case it is therefore related to 22 lagged absolute residuals. This part of the model uses 66 parameters. An alternative model is a first-order generalized autoregressive conditionally heteroskedastic model with variance influencing the mean [GARCH (l, l)-m], with a first-order moving average to correct for non-synchronous trading as used in Engle, Lilien, and Robins (1987), French, Schwert, and Stambaugh (1987), or Chou (1988), following the earlier work of Engle (1982). This requires only four coefficients! In the context of the parsimonious model, the parameter regulating the risk-return trade-off can be interpreted as the median agent’s taste for risk or his coefficient of relative-risk aversion. One naturally asks whether this parameter is constant over time, and we then recognize that the Schwert parameterization cannot answer the question.

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.]

There have been two major classes of urban area models: nonspatial models of income, employment, and structural change; and land use models usually oriented toward transportation planning. Recent efforts have become relatively complicated and have employed quite sophisticated techniques, with particular attention being paid to the housing market. Nevertheless, most of the work done so far appears somewhat deficient; convincing behavioral relations forming the basic structure are absent; and there have been inadequate efforts to test and validate the models. Further, relatively few efforts have specified the institutional framework necessary to introduce policy actions directly, although some recent efforts have been made in this direction. We propose to construct a model of the Boston metropolitan area that contains three major parts: a macroeconomic nonspatial model of output, employment, and income distribution; a model of long-term adjustments of population and capital stocks; and a model of spatial allocation. The equations of the model will be econometrically estimated and the main thrust of our efforts will be devoted to specification and testing of structural relationships reflecting actions of households, busi nesses, and governments interacting within both market and nonmarket institutions. The purpose of building the model is to permit systematic evaluation of a very wide range of policy alternatives considered at national, state, metropolitan, or local jurisdiction levels. If this is to be accomplished, there are three requisites. First, the model must endogenously generate those variables that enter evaluative (social welfare) functions. In this model we consider income, income distribution, availability of public services to particular population groups, and residential segregation of racial and income groups to be such variables. Second, the model must be designed so that policy alternatives can be modelled by varying the levels of particular exogenous variables. Finally, the model structure and parameter estimates must provide a model with a high degree of predictive power if the enterprise is to be of any value for policy evaluation. This paper contains a general guide to our thinking about how to construct and implement such a model. Many crucial questions of specification remain unresolved. To date we have collected most of the data that will be needed for preliminary versions of the model and some equations have been estimated. Undoubtedly many compromises will have to be made between our plans and what * Massachusetts Institute of Technology. This research was supported by a grant from the Ford Foundation.

[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.]