This paper reviews recent developments in nonlinear time series analysis, focusing on stochastic nonlinear dynamical systems of the form $ x_{t+1}=f(x_{t},x_{t-1},\cdots,x_{t-p})+\sigma(x_{t})\varepsilon_{t} $, where $ \{x_{t}\} $ is a zero mean, covariance stationary process. The paper discusses representation theory, nonparametric approaches, ergodic properties, and parametric models such as piecewise linear models and autoregressive models for volatility. It also covers hypothesis testing and forecasting in nonlinear time series. The Volterra expansion is used to approximate nonlinear functions, and the paper discusses the use of recurrent neural networks and wavelets for nonparametric estimation. The paper also explores the ergodic properties of nonlinear time series, including Lyapunov exponents, which measure the sensitivity of dynamic trajectories to initial conditions. The sum of Lyapunov exponents provides a measure of the Kolmogorov–Sinai entropy, which indicates how quickly trajectories separate. The dimension of the dynamical system is also discussed, with Takens' theorem providing a method for estimating the dimension from a scalar history. The paper discusses piecewise linear models, including Markov switching models and threshold autoregressive (TAR) models, which are used to capture nonlinearities in economic data. It also covers models of volatility, such as the GARCH model and stochastic volatility (SV) models, which are used to describe volatility clustering and heavy-tailed returns in financial market data. The paper also addresses testing for linearity and Gaussianity in nonlinear time series, using tests such as the BDS test and bicorrelation and bispectrum tests. It discusses forecasting in nonlinear time series, noting that results are mixed, with some studies finding that nonlinear models can produce superior forecasts. The paper concludes by discussing non-nested hypotheses in economics, where competing models are tested against each other. It provides an overview of non-nested hypothesis tests and their application to classical regression models. The paper emphasizes the importance of testing for non-nested hypotheses in economics to determine which model better explains the data.This paper reviews recent developments in nonlinear time series analysis, focusing on stochastic nonlinear dynamical systems of the form $ x_{t+1}=f(x_{t},x_{t-1},\cdots,x_{t-p})+\sigma(x_{t})\varepsilon_{t} $, where $ \{x_{t}\} $ is a zero mean, covariance stationary process. The paper discusses representation theory, nonparametric approaches, ergodic properties, and parametric models such as piecewise linear models and autoregressive models for volatility. It also covers hypothesis testing and forecasting in nonlinear time series. The Volterra expansion is used to approximate nonlinear functions, and the paper discusses the use of recurrent neural networks and wavelets for nonparametric estimation. The paper also explores the ergodic properties of nonlinear time series, including Lyapunov exponents, which measure the sensitivity of dynamic trajectories to initial conditions. The sum of Lyapunov exponents provides a measure of the Kolmogorov–Sinai entropy, which indicates how quickly trajectories separate. The dimension of the dynamical system is also discussed, with Takens' theorem providing a method for estimating the dimension from a scalar history. The paper discusses piecewise linear models, including Markov switching models and threshold autoregressive (TAR) models, which are used to capture nonlinearities in economic data. It also covers models of volatility, such as the GARCH model and stochastic volatility (SV) models, which are used to describe volatility clustering and heavy-tailed returns in financial market data. The paper also addresses testing for linearity and Gaussianity in nonlinear time series, using tests such as the BDS test and bicorrelation and bispectrum tests. It discusses forecasting in nonlinear time series, noting that results are mixed, with some studies finding that nonlinear models can produce superior forecasts. The paper concludes by discussing non-nested hypotheses in economics, where competing models are tested against each other. It provides an overview of non-nested hypothesis tests and their application to classical regression models. The paper emphasizes the importance of testing for non-nested hypotheses in economics to determine which model better explains the data.