This paper reviews the contributions of Rex Bergstrom to the development of continuous time dynamic disequilibrium macroeconomic modeling since the early 1960s. The models provide an elegant integration of economic theory with analysis of steady state and stability properties. The subsequent contributions of his Ph.D. students, spawned by Bergstrom’s work over the years, is also reviewed. It was Bergstrom’s early pioneering vision 40 years ago of formulating and estimating continuous time models that underlies much of the research in that area of econometrics and finance today.

It is well known that in standard linear regression models with independent and identically distributed data and homoskedasticity, adding “irrelevant regressors” hurts (asymptotic) efficiency unless such irrelevant regressors are orthogonal to the remaining regressors. But we have found that under (conditional) heteroskedasticity “irrelevant regressors” can always be found such that one can achieve the asymptotic variance of the generalized least squares estimator by adding the “irrelevant regressors” to the model.

It is well known that conventional Wald-type inference in the context of quantile regression is complicated by the need to construct estimates of the conditional densities of the response variables at the quantile of interest. This note explores the possibility of circumventing the need to construct conditional density estimates in this context with scale statistics that are explicitly inconsistent for the underlying conditional densities. This method of studentization leads conventional test statistics to have limiting distributions that are nonstandard but have the convenient feature of depending explicitly on the user’s choice of smoothing parameter. These limiting distributions depend on the distribution of the conditioning variables but can be straightforwardly approximated by resampling.

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

Building on work of Hansen (1992, Econometric Theory 8, 489–501), we show weak convergence for power transformations of integrated processes, with possibly serially correlated and heterogeneously distributed increments, to stochastic power integrals. The theory is applicable when testing the unit root or cointegration hypothesis in nonlinear systems by regression-based test statistics.

Standardized slowly varying regressors are shown to be Lp-approximable. This fact allows us to provide alternative proofs of asymptotic expansions of nonstochastic quantities and central limit results due to P.C.B. Phillips, under a less stringent assumption on linear processes. The recourse to stochastic calculus related to Brownian motion can be completely dispensed with.

The main uniform convergence results of Hansen (2008, Econometric Theory 24, 726–748) are generalized in two directions: Data are allowed to (a) be heterogeneously dependent and (b) depend on a (possibly unbounded) parameter. These results are useful in semiparametric estimation problems involving time-inhomogeneous models and/or sampling of continuous-time processes. The usefulness of these results is demonstrated by two applications: kernel regression estimation of a time-varying AR(1) model and the kernel density estimation of a Markov chain that has not been initialized at its stationary distribution.

Lyapunov exponents may be used to provide information on attractors of nonlinear models and, if strange, their aperiodic dynamics, in much the same way as eigenvalues of a linear model. An advantage of calculating these from an estimated model, rather than calculating the largest ones from a time series, is that economic theory is used to help distinguish between deterministic and stochastic behavior. Estimates of the Bergstrom/Wymer model of the United Kingdom, and of some other models of a similar form, were unstable in a classical sense, to some extent being caused by policy parameters. The Lyapunov exponents show that this model, and variants of it, have strange attractors in the neighborhood of the estimated parameter values and hence are stable but with aperiodic oscillations.

The central limit theorem for nonparametric kernel estimates of a smooth trend, with linearly generated errors, indicates asymptotic independence and homoskedasticity across fixed points, irrespective of whether disturbances have short memory, long memory, or antipersistence. However, the asymptotic variance depends on the kernel function in a way that varies across these three circumstances, and in the latter two it involves a double integral that cannot necessarily be evaluated in closed form. For a particular class of kernels, we obtain analytic formulas. We discuss extensions to more general settings, including ones involving possible cross-sectional or spatial dependence.

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter to ensure that the corresponding counting process N(t) satisfies Var N(t) ~ Ct2d+1 (C > 0) as t → ∞, with the same memory parameter that was assumed for the durations. Thus, these conditions ensure that the memory parameter in durations propagates to the same memory parameter in the counts. We then show that any autoregressive conditional duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, whereas any long memory stochastic duration model with d > 0 and all finite moments yields long memory in counts, with the same d. Finally, we provide some results about the propagation of long memory to the empirically relevant case of realized variance estimates affected by market microstructure noise contamination.

This paper shows how fractional unit root tests originally derived under stationarity can be made robust to heteroskedasticity. This is done by using existing tests nested in a regression framework and then implementing these tests using White’s heteroskedasticity consistent standard errors (White, 1980). We show this approach is effective both asymptotically and in finite samples. We also provide some evidence on the asymptotic local power of different implementations of the tests, under both homoskedasticity and heteroskedasticity.

It is undoubtedly desirable that econometric models capture the dynamic behavior, like trends and cycles, observed in many economic processes. Building models with such capabilities has been an important objective in the continuous time econometrics literature, for instance, the cyclical growth models of Bergstrom (1966); the economy-wide macroeconometric models of, for example, Bergstrom and Wymer (1976); unobserved stochastic trends of Harvey and Stock (1988 and 1993) and Bergstrom (1997); and differential-difference equations of Chambers and McGarry (2002). This paper considers continuous time cyclical trends, which complement the trend-plus-cycle models in the unobserved components literature but could also be incorporated into Bergstrom type systems of differential equations, as were stochastic trends in Bergstrom (1997).

This paper offers a perspective on A.R. Bergstrom’s contribution to continuous-time modeling, focusing on his preferred method of estimating the parameters of a structural continuous-time model using an exact discrete-time analog. Some inherent difficulties in this approach are discussed, which help to explain why, in spite of his prescience, the methods around his time were not universally adopted as he had hoped. Even so, it is argued that Bergstrom’s contribution and legacy is secure and retains some relevance today for the analysis of macroeconomic and financial time series.

Perron (1989, Econometrica 57, 1361–1401) introduced unit root tests valid when a break at a known date in the trend function of a time series is present. In particular, they allow a break under both the null and alternative hypotheses and are invariant to the magnitude of the shift in level and/or slope. The subsequent literature devised procedures valid in the case of an unknown break date. However, in doing so most research, in particular the commonly used test of Zivot and Andrews (1992, Journal of Business & Economic Statistics 10, 251–270), assumed that if a break occurs it does so only under the alternative hypothesis of stationarity. This is undesirable for several reasons. Kim and Perron (2009, Journal of Econometrics 148, 1–13) developed a methodology that allows a break at an unknown time under both the null and alternative hypotheses. When a break is present, the limit distribution of the test is the same as in the case of a known break date, allowing increased power while maintaining the correct size. We extend their work in several directions: (1) we allow for an arbitrary number of changes in both the level and slope of the trend function; (2) we adopt the quasi–generalized least squares detrending method advocated by Elliott, Rothenberg, and Stock (1996, Econometrica 64, 813–836) that permits tests that have local asymptotic power functions close to the local asymptotic Gaussian power envelope; (3) we consider a variety of tests, in particular the class of M-tests introduced in Stock (1999, Cointegration, Causality, and Forecasting: A Festschrift for Clive W.J. Granger) and analyzed in Ng and Perron (2001, Econometrica 69, 1519–1554).

This paper studies the asymptotic behavior of a Gaussian linear instrumental variables model in which the number of instruments diverges with the sample size. Asymptotic efficiency bounds are obtained for rotation invariant inference procedures and are shown to be attainable by procedures based on the limited information maximum likelihood estimator. The bounds are obtained by characterizing the limiting experiment associated with the model induced by the rotation invariance restriction.