A unified approach to nonlinearity,structural change,and outliers

This paper demonstrates that the class of conditionally linear and Gaussian state-space models offers a general and convenient framework for simultaneously handling nonlinearity, structural change and outliers in time series. Many popular nonlinear time series models, including threshold, smooth transition and Markov-switching models, can be written in state-space form. It is then straightforward to add components that capture parameter instability and intervention effects. We advocate a Bayesian approach to estimation and inference, using an efficient implementation of Markov Chain Monte Carlo sampling schemes for such linear dynamic mixture models. The general modelling framework and the Bayesian methodology are illustrated by means of several examples. An application to quarterly industrial production growth rates for the G7 countries demonstrates the empirical usefulness of the approach.     

PPP may hold better than you think: Smooth breaks and non-linear mean reversion in real effective exchange rates

This paper revisits some key topics in the literature on purchasing power parity (PPP). The study applies a set of newly developed unit root tests, which account for both nonlinearity and multiple smooth temporary breaks in series, to the real effective exchange rates (REERs) of 23 developed countries. The results suggest that PPP generally holds for various currency-based real rates. There is evidence in favor of linear stationarity in REERs for highly integrated economies. The REERs of most other countries tend to have nonlinear adjustment toward large long-swing type mean changes around constant equilibrium values.     

A flexible approach to parametric inference in nonlinear and time varying time series models

Many structural break and regime-switching models have been used with macroeconomic and financial data. In this paper, we develop an extremely flexible modeling approach which can accommodate virtually any of these specifications. We build on earlier work showing the relationship between flexible functional forms and random variation in parameters. Our contribution is based around the use of priors on the time variation that is developed from considering a hypothetical reordering of the data and distance between neighboring (reordered) observations. The range of priors produced in this way can accommodate a wide variety of nonlinear time series models, including those with regime-switching and structural breaks. By allowing the amount of random variation in parameters to depend on the distance between (reordered) observations, the parameters can evolve in a wide variety of ways, allowing for everything from models exhibiting abrupt change (e.g. threshold autoregressive models or standard structural break models) to those which allow for a gradual evolution of parameters (e.g. smooth transition autoregressive models or time varying parameter models). Bayesian econometric methods for inference are developed for estimating the distance function and types of hypothetical reordering. Conditional on a hypothetical reordering and distance function, a simple reordering of the actual data allows us to estimate our models with standard state space methods by a simple adjustment to the measurement equation. We use artificial data to show the advantages of our approach, before providing two empirical illustrations involving the modeling of real GDP growth.     

Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note*

In this paper we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. We propose a simple variation of existing bias‐corrected estimators, which can exploit the additivity of the effects for numerical optimization. We exhibit the performance of the estimators in simulations.     

Money‐output Causality Revisited – A Bayesian Logistic Smooth Transition VECM Perspective*

This article proposes a Bayesian approach to examining money‐output causality within the context of a logistic smooth transition vector error correction model. Our empirical results provide substantial evidence that the postwar US money‐output relationship is nonlinear, with regime changes mainly governed by the output growth and price levels. Furthermore, we obtain strong support for nonlinear Granger causality from money to output, although there is also some evidence for models indicating that money is not Granger causal or long‐run causal to output.     

The limit distribution of the estimates in cointegrated regression models with multiple structural changes

We study estimation and inference in cointegrated regression models with multiple structural changes allowing both stationary and integrated regressors. Both pure and partial structural change models are analyzed. We derive the consistency, rate of convergence and the limit distribution of the estimated break fractions. Our technical conditions are considerably less restrictive than those in Bai et al. [Bai, J., Lumsdaine, R.L., Stock, J.H., 1998. Testing for and dating breaks in multivariate time series. Review of Economic Studies 65, 395–432] who considered the single break case in a multi-equations system, and permit a wide class of practically relevant models. Our analysis is, however, restricted to a single equation framework. We show that if the coefficients of the integrated regressors are allowed to change, the estimated break fractions are asymptotically dependent so that confidence intervals need to be constructed jointly. If, however, only the intercept and/or the coefficients of the stationary regressors are allowed to change, the estimates of the break dates are asymptotically independent as in the stationary case analyzed by Bai and Perron [Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78]. We also show that our results remain valid, under very weak conditions, when the potential endogeneity of the non-stationary regressors is accounted for via an increasing sequence of leads and lags of their first-differences as additional regressors. Simulation evidence is presented to assess the adequacy of the asymptotic approximations in finite samples.     

Chi-squared tests for evaluation and comparison of asset pricing models

This paper presents a general statistical framework for estimation, testing and comparison of asset pricing models using the unconstrained distance measure of Hansen and Jagannathan (1997). The limiting results cover both linear and nonlinear models that could be correctly specified or misspecified. We propose modified versions of the existing model selection tests and new pivotal specification and model comparison tests with improved finite-sample properties. In addition, we provide formal tests of multiple model comparison. The excellent size and power properties of the proposed tests are demonstrated using simulated data from linear and nonlinear asset pricing models.     

Panel Stationarity Test with Structural Breaks*

In this paper, we extend the heterogeneous panel data stationarity test of Hadri [Econometrics Journal, Vol. 3 (2000) pp. 148–161] to the cases where breaks are taken into account. Four models with different patterns of breaks under the null hypothesis are specified. Two of the models have been already proposed by Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175]. The moments of the statistics corresponding to the four models are derived in closed form via characteristic functions. We also provide the exact moments of a modified statistic that do not asymptotically depend on the location of the break point under the null hypothesis. The cases where the break point is unknown are also considered. For the model with breaks in the level and no time trend and for the model with breaks in the level and in the time trend, Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175] showed that the number of breaks and their positions may be allowed to differ across individuals for cases with known and unknown breaks. Their results can easily be extended to the proposed modified statistic. The asymptotic distributions of all the statistics proposed are derived under the null hypothesis and are shown to be normally distributed. We show by simulations that our suggested tests have in general good performance in finite samples except the modified test. In an empirical application to the consumer prices of 22 OECD countries during the period from 1953 to 2003, we found evidence of stationarity once a structural break and cross‐sectional dependence are accommodated.     

Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root

This paper derives the limiting distribution of the Lagrange Multiplier (LM) test for threshold nonlinearity in a TAR model with GARCH errors when one of the regimes contains a unit root. It is shown that the asymptotic distribution is nonstandard and depends on nuisance parameters that capture the degree of conditional heteroskedasticity and non-Gaussian nature of the process. We propose a bootstrap procedure for approximating the exact finite-sample distribution of the test for linearity and establish its asymptotic validity.     

Multiplicative-error models with sample selection

This paper presents a simple approach to deal with sample selection in models with multiplicative errors. Models for non-negative limited dependent variables such as counts fit this framework. The approach builds on a specification of the conditional mean of the outcome only and is, therefore, semiparametric in nature. GMM estimators are constructed for both cross-section data and for panel data. We derive distribution theory and present Monte Carlo evidence on the finite-sample performance of the estimators.     

WEIGHTED SMOOTH TRANSITION REGRESSIONS

A new procedure is proposed for modelling nonlinearity of a smooth transition form, by allowing the transition variable to be a weighted function of lagged observations. This function depends on two unknown parameters and requires specification of the maximum lag only. Nonlinearity testing for this specification uses a search over a plausible set of weight function parameters, combined with bootstrap inference. Finite‐sample results show that the recommended wild bootstrap heteroskedasticity‐robust testing procedure performs well, for both homoskedastic and heteroskedastic data‐generating processes. Forecast comparisons relative to linear models and other nonlinear specifications of the smooth transition form confirm that the new WSTR model delivers good performance. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation‐based Finite Sample Linearity Test against Smooth Transition Models*

In this paper, we use Monte Carlo (MC) testing techniques for testing linearity against smooth transition models. The MC approach allows us to introduce a new test that differs in two respects from the tests existing in the literature. First, the test is exact in the sense that the probability of rejecting the null when it is true is always less than or equal to the nominal size of the test. Secondly, the test is not based on an auxiliary regression obtained by replacing the model under the alternative by approximations based on a Taylor expansion. We also apply MC testing methods for size correcting the test proposed by Luukkonen, Saikkonen and Teräsvirta (Biometrika, Vol. 75, 1988, p. 491). The results show that the power loss implied by the auxiliary regression‐based test is non‐existent compared with a supremum‐based test but is more substantial when compared with the three other tests under consideration.     

Optimal forecasts in the presence of structural breaks

This paper considers the problem of forecasting under continuous and discrete structural breaks and proposes weighting observations to obtain optimal forecasts in the MSFE sense. We derive optimal weights for one step ahead forecasts. Under continuous breaks, our approach largely recovers exponential smoothing weights. Under discrete breaks, we provide analytical expressions for optimal weights in models with a single regressor, and asymptotically valid weights for models with more than one regressor. It is shown that in these cases the optimal weight is the same across observations within a given regime and differs only across regimes. In practice, where information on structural breaks is uncertain, a forecasting procedure based on robust optimal weights is proposed. The relative performance of our proposed approach is investigated using Monte Carlo experiments and an empirical application to forecasting real GDP using the yield curve across nine industrial economies.     

Bayesian analysis of a Tobit quantile regression model

This paper develops a Bayesian framework for Tobit quantile regression. Our approach is organized around a likelihood function that is based on the asymmetric Laplace distribution, a choice that turns out to be natural in this context. We discuss families of prior distributions on the quantile regression vector that lead to proper posterior distributions with finite moments. We show how the posterior distribution can be sampled and summarized by Markov chain Monte Carlo methods. A method for comparing alternative quantile regression models is also developed and illustrated. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed two other common classical estimators.     

Testing for ARCH in the presence of nonlinearity of unknown form in the conditional mean

Tests of ARCH are a routine diagnostic in empirical econometric and financial analysis. However, it is well known that misspecification of the conditional mean may lead to spurious rejection of the null hypothesis of no ARCH. Nonlinearity is a prime example of this phenomenon. There is little work on the extent of the effect of neglected nonlinearity on the properties of ARCH tests. We investigate this using new ARCH testing procedures that are robust to the presence of neglected nonlinearity. Monte Carlo evidence shows that the problem is serious and that the new methods alleviate this problem to a very large extent. We apply the new tests to exchange rate data and find substantial evidence of spurious rejection of the null hypothesis of no ARCH.     

Quasi-maximum likelihood estimation and testing for nonlinear models with endogenous explanatory variables

I propose a quasi-maximum likelihood framework for estimating nonlinear models with continuous or discrete endogenous explanatory variables. Joint and two-step estimation procedures are considered. The joint procedure is a quasi-limited information maximum likelihood procedure, as one or both of the log likelihoods may be misspecified. The two-step control function approach is computationally simple and leads to straightforward tests of endogeneity. In the case of discrete endogenous explanatory variables, I argue that the control function approach can be applied with generalized residuals to obtain average partial effects. I show how the results apply to nonlinear models for fractional and nonnegative responses.     

Mixed random forest,cointegration, and forecasting gasoline prices

One of the most successful forecasting machine learning (ML) procedures is random forest (RF). In this paper, we propose a new mixed RF approach for modeling departures from linearity that helps identify (i) explanatory variables with nonlinear impacts, (ii) threshold values, and (iii) the closest parametric approximation. The methodology is applied to weekly forecasts of gasoline prices, cointegrated with international oil prices and exchange rates. Recent specifications for nonlinear error correction (NEC) models include threshold autoregressive models (TAR) and double-threshold smooth transition autoregressive (STAR) models. We propose a new mixed RF model specification strategy and apply it to the determinants of weekly prices of the Spanish gasoline market from 2010 to 2019. In particular, the mixed RF is able to identify nonlinearities in both the error correction term and the rate of change of oil prices. It provides the best weekly gasoline price forecasting performance and supports the logistic error correction model (ECM) approximation.     

Are unemployment rates in OECD countries stationary? Evidence from univariate and panel unit root tests

This paper revisits the dynamics of unemployment rate for 29 OECD countries over the period of 1980–2013. Numerous empirical studies of the dynamics of unemployment rate are carried out within a linear framework. However, unemployment rate can show nonlinear behaviour as a result of business cycles or some idiosyncratic factors specific to labour market (Cancelo, 2007). Thus, as a testing strategy, we first perform Harvey, Leybourne, and Xiao (2008) linearity unit root test and then apply the newly ESTAR nonlinear unit root test suggested by Kruse (2011). This test has higher power than conventional unit root tests when time series exhibits nonlinear behaviour. Our empirical findings provide significant evidence in favour of unemployment rate stationarity for 25 countries. For robustness purpose, we have also used panel unit root tests without and with structural breaks. The empirical results show that unemployment hysteresis hypothesis is strongly rejected, when taking into account the cross-sectional and structural break assumptions. Thus, unemployment rate is expected to return back to their natural levels without executing any costly macroeconomic labour market policies by the OECD’s governments.     

Testing for Panel Cointegration with Multiple Structural Breaks*

This paper proposes a Lagrange multiplier (LM) test for the null hypothesis of cointegration that allows for the possibility of multiple structural breaks in both the level and trend of a cointegrated panel regression. The test is general enough to allow for endogenous regressors, serial correlation and an unknown number of breaks that may be located at different dates for different individuals. We derive the limiting distribution of the test and conduct a small Monte Carlo study to investigate its finite sample properties. In our empirical application to the solvency of the current account, we find evidence of cointegration between saving and investment once a level break is accommodated.