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This paper begins with a brief review of the recent experience using nonlinear models and ideas of chaos to model economic data and to provide forecasts that are better than linear models. The record of improvement is at best meager. The remainder of the paper examines some of the reasons for this lack of improvement. The concepts of "openness" and "isolation" are introduced, and a case is made that open and nonisolated systems cannot be forecasted; the extent to which economic systems are closed and isolated provides the true pragmatic limits to forecastability. The reasons why local "overfitting," especially with nonparametric models, leads to worse forecasts are discussed. Models and "representations" of data are distinguished and the reliance on minimum mean-square forecast error to choose between models and representations is evaluated.

The random-walk (white-noise) model and the harmonic model are two polar models in linear systems. A model in between is color chaos, which generates irregular oscillations with a narrow frequency (color) band. Time-frequency analysis is introduced for evolutionary time-series analysis. The deterministic component from noisy data can be recovered by a time-variant filter in Gabor space. The characteristic frequency is calculated from the Wigner decomposed distribution series. It is found that about 70 percent of fluctuations in Standard & Poor stock price indexes, such as the FSPCOM and FSDXP monthly series, detrended by the Hodrick-Prescott (HP) filter, can be explained by deterministic color chaos. The characteristic period of persistent cycles is around three to four years. Their correlation dimension is about 2.5. The existence of persistent chaotic cycles reveals a new perspective of market resilience and new sources of economic uncertainties. The nonlinear pattern in the stock market may not be wiped out by market competition under nonequilibrium situations with trend evolution and frequency shifts. The color-chaos model of stock-market movements may establish a potential link between business-cycle theory and asset-pricing theory.

We present a macroeconomic growth model in which investment in physical capital exhibits positive externalities which raise the stock of knowledge. Treating physical capital and knowledge as two separate variables, we show that the model can generate endogenous growth. It is demonstrated that there exist at most two balanced growth paths (BGPs) with endogenous growth. If the BGP is unique it is always saddle-point stable. If there are two BGPs, the first is always stable in the saddle point sense, whereas the second cannot be a saddle point. Instead, the second is either totally stable, giving rise to local indeterminacy, or completely unstable. Further, we can demonstrate that a Hopf bifurcation may occur at the second BGP, leading to persistent fluctuations. Besides local indeterminacy, the model may also give rise to global indeterminacy.

This paper develops a new kernel test for neglected nonlinearity in the conditional expectation function, and compares this test to the Ramsey RESET test (1969) and the Neural Net test of Lee, White, and Granger (1993). Like the Neural Test and the Ramsey Reset Test, this Kernel test is a Lagrange Multiplier test based on the R-Square statistic from a regression of the estimated residuals on the regressors, and an additional nonlinear function of the regressors. Unlike the other tests, our test has a two stage approach where in the first stage we estimate the structure of the misspecification and in the second stage we test for whether or not the estimate of the misspecification can better predict the residuals than their mean. This two stage approach can give the researcher guidance on the nature of the misspecification, and should improve the power of the test since the added function in the regression of the residuals is itself an estimate of the conditional expectation of the residuals given the independent variables. In addition, because it uses simple, well known estimation methods it can be implemented by researchers when using linear models.