During the past decade, the popularity of nonlinear models in econometrics has been increasing quite rapidly. Nonlinear models are now widely used for modeling macroeconomic relationships, and they also are used frequently in financial econometrics. The most popular nonlinear models have been univariate. Threshold autoregressive, Markov switching autoregressive, and smooth-transition autoregressive models, just to name a few popular families of models, have been widely applied to modeling of macroeconomic series. Even nonlinear multivariate single-equation models have found application in areas where linear single-equation models traditionally have been used, such as modeling the demand for money, real exchange rates, consumption–income relationship, and house prices. Interest in nonlinearities in the Phillips curve also has grown recently.

A survey of nonlinear multivariate macro empirical models is attempted. Although theory may suggest that nonlinearity is to be expected, empirical studies have difficulty in discovering strong consistent effects. Regime switching techniques appear to be the most successful and evidence of nonlinearity is most found for interest rates. Most of the studies emphasize model fitting rather than model evaluation, which limits their usefulness.

We develop nonlinear leading-indicator models for GDP growth, with the interest-rate spread and growth in M2 as leading indicators. Since policy makers typically are interested in whether a recession is imminent, we evaluate these models according to their ability to predict the probability of a recession. Using data for the United States, we find that conditional on the spread, the marginal contribution of M2 growth in predicting recessions is negligible.

This paper investigates the potential for nonlinear Granger causality from money to output. Using a standard four-variable linear (subset) vector error-correction model (VECM), we first show that the null hypothesis of linearity can be rejected against the alternative of smooth-transition autoregressive nonlinearity. An interesting result from this stage of the analysis is that the yearly growth rate of money is identified as one of the variables that may govern the switching between regimes. Smooth-transition VECM's (STVECM's) are then used to examine whether there is nonlinear Granger causality in the money–output relationship in the sense that lagged values of money enter the model's output equation as regressors. We evaluate this type of nonlinear Granger causality with both in-sample and out-of-sample analyses. For the in-sample analysis, we compare alternative models using the Akaike information criteria, which can be interpreted as a predictive accuracy test. The results show that allowing for both nonlinearity and for money–output causality leads to considerable improvement in model's in-sample performance. By contrast, the out-of-sample forecasting results do not suggest that money is nonlinearly Granger causal for output. They also show that, according to several criteria, the linear VECM's dominate the STVECM's. However, these forecast improvements seldomly are statistically significant at conventional levels.

Previous studies investigating threshold behavior in real-exchange-rate and price difference data have used rather ad hoc statistical methods and have focused on univariate threshold models for relative prices. We utilize a general multivariate threshold cointegration model and develop a systematic testing and estimation strategy for this model, building on the work of others. Using Monte Carlo experiments, we systematically compare the use of univariate and multivariate techniques for testing threshold cointegration, estimating various threshold models, and testing specifications. We apply our methodology to a large set of U.S. disaggregated CPI data. We find evidence of threshold cointegration mainly for tradable goods. However, the type of threshold nonlinearity that we find generally does not support the transaction-cost view of commodity arbitrage.

We extend the conventional cointegrated VAR model to allow for general nonlinear deterministic trends. These nonlinear trends can be used to model gradual structural changes in the intercept term of the cointegrating relations. A general asymptotic theory of estimation and statistical inference is reviewed and a diagnostic test for the correct specification of an employed nonlinear trend is developed. The methods are applied to Finnish interest-rate data. A smooth level shift of the logistic form between the own-yield of broad money and the short-term money market rate is found appropriate for these data. The level shift is motivated by the deregulation of issuing certificates of deposit and its inclusion in the model solves the puzzle of the “missing cointegration vector” found in a previous study.

Clive W.J. Granger has summarized his personal viewpoint on testing for causality in numerous articles over the past 30 years and has outlined what he considers to be a useful operational version of his original definition of Granger causality, which he notes is partially alluded to in the Ph.D. dissertation of Norbert Wiener. This operational version of Granger causality is based on a comparison of the one-step-ahead predictive ability of competing models. However, Granger concludes his discussion by noting that it is common practice to test for Granger causality using in-sample F-tests. The practice of using in-sample type Granger causality tests continues to be prevalent. In this paper we develop simple (nonlinear) out-of-sample predictive ability tests of the Granger non-causality null hypothesis. In addition, Monte Carlo experiments are used to investigate the finite sample properites of the test. An empirical illustration shows that the choice of in-sample versus out-of-sample Granger causality tests can crucially affect the conclusions about the predictive content of money for output.

This paper analyzes the joint time-series properties of the level and volatility of expected excess stock returns. An unobservable dynamic factor is constructed as a nonlinear proxy for the market risk premia with its first moment and conditional volatility driven by a latent Markov variable. The model allows for the possibility that the risk–return relationship may not be constant across the Markov states or over time. We find an overall negative contemporaneous relationship between the conditional expectation and variance of the monthly value-weighted excess return. However, the sign of the correlation is not stable, but instead varies according to the stage of the business cycle. In particular, around the beginning of recessions, volatility rises substantially, reflecting great uncertainty associated with these periods, while expected return falls, anticipating a decline in earnings. Thus, around economic peaks there is a negative relationship between conditional expectation and variance. However, toward the end of a recession expected return is at its highest value as an anticipation of the economic recovery, and volatility is still very high in anticipation of the end of the contraction. That is, the risk–return relation is positive around business-cycle troughs. This time-varying behavior also holds for noncontemporaneous correlations of these two conditional moments.