Noncausal autoregressions for economic time series

This paper is concerned with univariate noncausal autoregressive models and their potential usefulness in economic applications. In these models, future errors are predictable, indicating that they can be used to empirically approach rational expectations models with nonfundamental solutions. In the previous theoretical literature, nonfundamental solutions have typically been represented by noninvertible moving average models. However, noncausal autoregressive and noninvertible moving average models closely approximate each other, and therefore,the former provide a viable and practically convenient alternative. We show how the parameters of a noncausal autoregressive model can be estimated by the method of maximum likelihood and derive related test procedures. Because noncausal autoregressive models cannot be distinguished from conventional causal autoregressive models by second order properties or Gaussian likelihood, a model selection procedure is proposed. As an empirical application, we consider modeling the U.S. inflation which, according to our results, exhibits purely forward-looking dynamics

Parameter estimation in nonlinear AR–GARCH models

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.Nonlinear Autoregression, Generalized Autoregressive Conditional Heteroskedasticity, Nonlinear Time Series Models, Quasi-Maximum Likelihood Estimation, Strong Consistency, Asymptotic Normality

Stability of nonlinear AR-GARCH models

This paper studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and beta-mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance.-

Noncausal autoregressions for economic time series

This paper is concerned with univariate noncausal autoregressive models and their potential usefulness in economic applications. In these models, future errors are predictable, indicating that they can be used to empirically approach rational expectations models with nonfundamental solutions. In the previous theoretical literature, nonfundamental solutions have typically been represented by noninvertible moving average models. However, noncausal autoregressive and noninvertible moving average models closely approximate each other, and therefore,the former provide a viable and practically convenient alternative. We show how the parameters of a noncausal autoregressive model can be estimated by the method of maximum likelihood and derive related test procedures. Because noncausal autoregressive models cannot be distinguished from conventional causal autoregressive models by second order properties or Gaussian likelihood, a model selection procedure is proposed. As an empirical application, we consider modeling the U.S. inflation which, according to our results, exhibits purely forward-looking dynamics.Noncausal autoregression; expectations; inflation persistence

Parameter Estimation in Nonlinear AR-GARCH Models

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a functional coefficient autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.AR-GARCH, asymptotic normality, consistency, nonlinear time series, quasi maximum likelihood estimation

On the Estimation of Euler Equations in the Presence of a Potential Regime Shift

The concept of a peso problem is formalized in terms of a linear Euler equation and a nonlinear marginal model describing the dynamics of the exogenous driving process. It is shown that, using a threshold autoregressive model as a marginal model, it is possible to produce time-varying peso premia. A Monte Carlo method and a method based on the numerical solution of integral equations are considered as tools for computing conditional future expectations in the marginal model. A Monte Carlo study illustrates the poor performance of the generalized method of moment (GMM) estimator in small and even relatively large samples. The poor performance is particularly acute in the presence of a peso problem but is also serious in the simple linear case.peso problem; Euler equations; GMM; threshold autoregressive models

A Skewed GARCH-in-Mean Model: An Application to U.S. Stock Returns

In this paper we consider a GARCH-in-Mean (GARCH-M) model based on the so-called z distribution. This distribution is capable of modeling moderate skewness and kurtosis typically encountered in financial return series, and the need to allow for skewness can be readily tested. We apply the new GARCH-M model to study the relationship between risk and return in monthly postwar U.S. stock market data. Our results indicate the presence of conditional skewness in U.S. stock returns, and, in contrast to the previous literature, we show that a positive and significant relationship between return and risk can be uncovered, once an appropriate probability distribution is employed to allow for conditional skewnessConditional skewness, GARCH-in-Mean, Risk-return tradeoff

Modeling Conditional Skewness in Stock Returns

In this paper we propose a new GARCH-in-Mean (GARCH-M) model allowing for conditional skewness. The model is based on the so-called z distribution capable of modeling moderate skewness and kurtosis typically encountered in stock return series. The need to allow for skewness can also be readily tested. Our empirical results indicate the presence of conditional skewness in the postwar U.S. stock returns. Small positive news is also found to have a smaller impact on conditional variance than no news at all. Moreover, the symmetric GARCH-M model not allowing for conditional skewness is found to systematically overpredict conditional variance and average excess returns.Conditional skewness, GARCH-in-Mean, Risk-return tradeoff