Volume 3, Issue 4

There has been increasing interest in continuous-time macroeconometric models. This research investigates stability of the Bergstrom, Nowman, and Wymer continuous-time model of the U.K. when system parameters change. This particularly well-regarded continuous-time macroeconometric model is chosen to assure the empirical and potential policy relevance of the results. Stability analysis is important with this model for understanding the dynamic properties of the system and for determining which parameters are the most important to those dynamic properties. The main objective of this paper is to determine the boundaries of parameters at which instability occurs. Two types of boundaries are found: the transcritical bifurcation boundary and the Hopf bifurcation boundary, corresponding to two different ways that instability occurs when parameter values cross the bifurcation boundary.The existence of the Hopf bifurcation boundary is particularly useful, since Hopf bifurcation may provide explanations for some cyclical phenomena in macroeconomy. Numerical algorithms are designed to locate the stability boundaries, which are displayed in three-dimensional diagrams. A notable and perhaps surprising fact is that both types of bifurcations can coexist with this well-regarded U.K. model--in the same neighborhood of the parameter space.

We investigate asymmetries in the conditional mean dynamics of four sectors of the U.S. GDP data. Since the statistical evidence on nonlinearities in the conditional mean could be influenced by the presence of outliers, or by a failure to model conditional heteroskedasticity, we explicitly account for outliers by assuming that the innovations are drawn from the stable family, and model time-varying volatility by a GARCH(1,1) process. We also allow for the possibility of long memory in the series with fractional differencing. Our results indicate only weak evidence of significant nonlinearities in the conditional mean in some sectors of the GDP.

This paper reconsiders the conventional use of econometric models, especially identified vector autoregressive models, in guiding monetary policy. The main question I explore is whether these models are seriously flawed because they ignore asymmetries in the business cycles. Toward that end, models that allow for asymmetric business cyclesÑdefined by the case where recessions and expansions are not mirror images of each otherÑare estimated. The results suggest that policy makers should worry about asymmetries in business cycles because most econometric models cannot capture empirically important asymmetries. In particular, estimated multiregime models show that the effects of monetary policy are stronger during turning points and outright recessions than in expansions. I conclude that the symmetry/asymmetry question has as much, and maybe even more, practical significance than debates over identification assumptions that have influenced much of the empirical macroeconomic literature over the past 20 years.

Recent theoretical and empirical work has cast doubt on the hypotheses of a linear Phillips curve and a symmetric quadratic loss function underlying traditional thinking on monetary policy. This paper studies the one-period optimal monetary policy problem under an asymmetric loss function corresponding to the "opportunistic approach" to disinflation and a convex Phillips curve. The policy-inaction range and its properties are derived analytically. Numerical simulations are then used to assess the implications of asymmetric loss for the distributional properties of the equilibrium levels of inflation and unemployment. For parameter values relevant to the U.S., it is found that the asymmetric loss function yields an average inflation rate in excess of the target, and that bias is larger than the standard symmetric loss function. For moderate policy-maker preferences, the asymmetric loss function also yields a smaller gap between average unemployment and the natural rate, and higher (lower) variance of inflation (unemployment) compared to the symmetric benchmark. Calibrating the model to match the observed average unemployment rate requires a high degree of inflation aversion and small asymmetry.

By design a wavelet's strength rests in its ability to localize a process simultaneously in time-scalespace. The wavelet's ability to localize a time series in time-scale space directly leads to the computationalefficiency of the wavelet representation of a N £ N matrix operator by allowing the N largest elements of thewavelet represented operator to represent the matrix operator [Devore, et al. (1992a) and (1992b)]. Thisproperty allows many dense matrices to have sparse representation when transformed by wavelets.In this paper we generalize the long-memory parameter estimator of McCoy and Walden (1996) to estimatesimultaneously the short and long-memory parameters. Using the sparse wavelet representation of a matrixoperator, we are able to approximate an ARFIMA model's likelihood function with the series' wavelet coefficientsand their variances. Maximization of this approximate likelihood function over the short and long-memoryparameter space results in the approximate wavelet maximum likelihood estimates of the ARFIMA model.By simultaneously maximizing the likelihood function over both the short and long-memory parameters andusing only the wavelet coefficient's variances, the approximate wavelet MLE provides a fast alternative to thefrequency-domain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLEto be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas thefrequency-domain MLE dramatically deteriorates as the moving average parameter approaches the boundariesof invertibility.