Null Hypothesis Of Linearity Research Articles

Test for Linearity in Non-Parametric Regression Models

The problem of checking the linearity of a regression relationship is addressed. The test uses nonparametric estimation techniques. The null hypothesis is that the regression function is linear; it is tested against the non-specic alternatives hypotheses. This test is based on a Hermite transform characterization of conditional expectations. A statistical test is derived, the distribution of this statisticunder the null hypothesis of linearity is determined. A power study using simulation shows the new statistic to be more sensitive to non-linearity.

Sieve IV estimation of cross-sectional interaction models with nonparametric endogenous effect

In this study, we consider cross-sectional interaction models including spatial autoregressive models and peer effects models as special cases. Our model allows the endogenous effect – the effect of others’ outcomes on one’s own outcome – to be nonlinear and nonparametric. For the model estimation, we propose a sieve instrumental variable estimator and establish both its consistency and asymptotic normality. Furthermore, we propose a nonparametric specification test for the linearity of the endogenous effect. Under the null hypothesis of linearity, we show that the test statistic is asymptotically distributed as normal. As an empirical illustration, we focus on the data on regional economic performance investigated by Gennaioli et al. (2013). This empirical analysis highlights the usefulness of the proposed model and method.

Modeling And Forecasting Ghana’s Inflation Rate Under Threshold Models

In this study, we model and forecast Ghana’s inflation rate using nonlinear models between the period January 1981 and August 2016. Nonlinearity tests were conducted on the logarithm of the monthly rates of inflation using Keenan and Tsay tests, and based on the results, we rejected the null hypothesis of linearity of monthly rates of inflation. We used threshold models and compared their fitness and forecasting performance with standard linear Autoregressive (AR) models. We found out that the Self-Exiting Threshold Autoregressive (SETAR) and Logistic Smooth Threshold Autoregressive (LSTAR) models fit the data better. The simple linear Autoregressive (AR) models however, out-performed the nonlinear models in terms of forecasting. Various research projects have been carried out in this area of inflation modeling in Ghana, but these researchers modeled inflation in Ghana using nonlinear models that did not account for the conditional heteroscedasticity in the model. These models have been used and empirical evidence of their relative performance has been given for the success of developed economies such as US and Europe. However, limited studies have been done in the case of Ghana. This indicates a gap in the literature and poses a challenge as to which of these models is the optimal choice for modeling economic and financial data such as inflation rates for developing countries. It is recommended that policy makers, industry players and all those interested in modeling future rates of inflation in Ghana should consider using the threshold models instead of the traditional Box and Jenkins models since the threshold models are able to capture the heteroscedasticity in the model. Also by using the threshold models, policy makers and industry players would be able to properly capture the variability persistence in the monthly rates of inflation and hence estimates would be more accurate. Lastly, from the upward trend of the out-sample forecasts, it can be predicted that Ghana would experience double digit inflation in 2017. This would have several impacts on many aspects of the economy and could erode the economic gains made in the year 2016.

Linearity tests and stochastic trend under the STAR framework

This study investigates the linearity test of smooth transition autoregressive models when the true data generating process is a stochastic trend process. Results show that, under the null hypothesis of linearity, the asymptotic distribution of the W statistic proposed by Terasvirta (J Am Stat Assoc 89:208–218, 1994) follows the χ2 distribution, whereas the finite sample distribution does not. A maximized Monte Carlo simulation-based test is used to perform the linearity test, and the results show good performance.

NONLINEAR SMOOTH TRANSITION AUTOREGRESSIVE (STAR)–TYPE MODELLING AND FORECASTING ON MALAYSIA AIRLINES (MAS) STOCK RETURNS

This study aims to apply nonlinear Smooth Transition Autoregressive (STAR)-type model to the Malaysia Airlines (MAS) Stock Returns, which consists of 4450 number of observations. The data taken started from 29th August 1996 until 26th September 2014. Following the STAR strategies by Terasvirta, the diagnostic plots of linear Autoregressive (AR) model revealed that AR (3) model is adequate in modelling the MAS returns series. However, the squared residuals of Autocorrelation Function (ACF) of returns series illustrates a slight presence of correlations in the model, hence the effort to apply nonlinear model was continued. Before proceed to nonlinear STAR modelling, the identification of delay parameter in the second stage of Terasvirta need to be determined. The results of Lagrange Multiplier (LM) tests revealed that delay parameter, d=3 is the best to choose. In addition, the null hypothesis of linearity from LM test is rejected. Furthermore, from the sequence of nested hypothesis of delay parameter, d=3 indicated that LSTAR model is preferred than ESTAR model. Finally, the forecasts and comparison stages was made to compare which models are best performed in forecasting the future series of MAS returns. It proved that LSTAR model performed better in term of forecasting accuracy when compared to ESTAR and AR model.

Testing for Neglected Nonlinearity Using Artificial Neural Networks with Many Randomized Hidden Unit Activations

AbstractThis paper makes a simple but previously neglected point with regard to an empirical application of the test of White (1989) and Lee, White, and Granger (LWG, 1993), for neglected nonlinearity in conditional mean, using the feedforward single layer artificial neural network (ANN). Because the activation parameters in the hidden layer are not identified under the null hypothesis of linearity, LWG suggested to activate the ANN hidden units based on the randomly generated activation parameters. Their Monte Carlo experiments demonstrated the excellent performance (good size and power), even if LWG considered a fairly small number (10 or 20) of random hidden unit activations. However, in this paper, we note that the good size and power of Monte Carlo experiments are the average frequencies of rejecting the null hypothsis over multiple replications of the data generating process. The average over many simulations in Monte Carlo smooths out the randomness of the activations. In an empirical study, unlike in a Monte Carlo study, multiple realizations of the data are not possible or available. In this case, the ANN test is sensitive to the randomly generated activation parameters. One solution is the use of Bonferroni bounds as suggested by LWG (1993), which however still exhibits some excessive dependence on the random activations (as shown in this paper). Another solution is to integrate the test statistic over the nuisance parameter space, for which however, bootstrap or simulation should be used to obtain the null distribution of the integrated statistic. In this paper, we consider a much simpler solution that is shown to work very well. That is, we simply increase the number of randomized hidden unit activations to a (very) large number (e.g. 1,000). We show that using many randomly generated activation parameters can robustify the performance of the ANN test when it is applied to a real empirical data. This robustification is reliable and useful in practice and can be achieved at no cost as increasing the number of random activations is almost costless given today’s computer technology.

Fusion of hypothesis testing for nonlinearity detection in small time series

The performances of parametric or non-parametric Hypothesis Testing (HT) for nonlinearity detection are fairly weak for small time series (typically between 128 and 512 samples). A natural idea to improve the results is to merge several HT to make a more robust decision. In this paper, we inspect the topology to perform this fusion. However three steps are needed before optimizing this fusion process. The first one is a rigorous estimate of the robustness of the 12 selected HT in order to only keep the more robust ones. The second one is the validation of the surrogate data method to estimate the index pdf under H0 (i.e. the observed time series is “linear”). In fact, this pdf is necessary to define the threshold to accept/reject the null hypothesis of linearity. The last step is also an estimate of the mutual information between the indices involved in the fusion process, since the fusion of close indices cannot improve the decision. Numerical results show that the method of fusion changes when the time series length increases.

REAL EXCHANGE RATE DYNAMICS IN TRANSITION ECONOMIES: A NONLINEAR ANALYSIS

ABSTRACTThis article examines the behaviour of the real exchange rates for 18 transition economies using nonlinear models. We find strong evidence of nonlinearities in 16 of the 18 countries. Contrary to widely held belief that the behaviour of real exchange rates should exhibit symmetrical adjustment for deviations above and below its equilibrium level, we find strong evidence of asymmetrical adjustment in 12 of the 16 countries for which the null hypothesis of linearity is rejected. Applying nonlinear unit root tests reveals evidence of stationarity in all the cases except for Poland. Copyright © 2012 John Wiley & Sons, Ltd.

Threshold effects in credit risk and stress scenarios

This paper focuses on the analysis and modelling of credit risk, measured through the value of nonperforming loans. Taking into account the new regulatory framework introduced by Basel II, possible stress scenarios in the context of Pillar 2 are also identified. The analysis is conducted for three countries—Portugal, Spain and Italy. The null hypothesis of linearity is rejected for all three countries, for both the self-exciting threshold autoregressive (SETAR) and threshold autoregressive (TAR) alternatives, and this feature is taken into account when credit risk is modelled, making SETAR and TAR models a plausible alternative to linear models. Copyright © 2010 John Wiley & Sons, Ltd.

Testing for linearity in Markov switching models: a bootstrap approach

Testing for linearity in the context of Markov switching models is complicated because standard regularity conditions for likelihood based inference are violated. In particular, under the null hypothesis of linearity, some parameters are not identified and scores are identically zero. Thus, the asymptotic distribution of the relevant test statistic does not possess the standard χ 2-distribution. A bootstrap resampling scheme to approximate the distribution of the relevant test statistic under the null of linearity is proposed. The procedure is relatively easy to program and computation requirements are reasonable. The performance of the bootstrap-based test is investigated by means of Monte Carlo simulations. Results show that this test works well and outperforms the Hansen test and the Carrasco et al. test.

A simple nonparametric test for diagnosing nonlinearity in Tobit median regression model

In many applications, the response variable is observed only when it is above or below a given threshold otherwise the threshold itself is observed. Tobit median regression model is a useful semiparametric procedure for analyzing this type of censored data. We propose a simple nonparametric test for assessing the common linearity assumption in this model. Compared to those existing methods in the literature, the new test has the advantage of allowing the alternative to be any smooth function. In addition, it does not require any knowledge of the parametric distribution of the random error. The test is asymptotically normal under the null hypothesis of linearity. A small Monte Carlo study demonstrates its performance.

Price Indices and Nonlinear Mean-Reversion of Real Exchange Rates

[Author Affiliation]Jyh-Lin Wu, Institute of Economics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan; jlwu2@mail.nsysu.edu.tw and the Department of Economics, National Chung-Cheng University, Chia-Yi, 106, Taiwan; ecdjlw@ccu.edu.tw; corresponding authorPei-Fen Chen, Department of International Business and Trade, Shu-Te University, Kaohsiung, Taiwan; cpf@mail.stu.edu.tw[Acknowledgment]We thank two anonymous referees for interesting and constructive comments on an earlier version of the paper. Any remaining errors are ours.1. Introduction Are price indices crucial for the existence of a nonlinear mean reversion of real exchange rates? Do prices or exchange-rate adjustments dominate when deviations from purchasing power parity (PPP) occur? Is the half-life implied by a nonlinear model reasonable? The purpose of this article is to address the above three questions for the UK and New Zealand over the period of recent float.The PPP hypothesis has been one of the most intensive research issues in empirical international finance over the past two decades. The rationale behind it is a simple arbitrage hypothesis, which results in a linear adjustment of deviations from PPP and the stationarity of real exchange rates. Empirically, existing evidence based on unit-root tests provide mixed results for PPP (Abuaf and Jorion 1990; Mark 1990; O'Connell 1998a).Theoretically there are several reasons for the nonlinear adjustment of deviations from PPP, such as the existence of market frictions or transaction costs (Sercu, Uppal, and Van Hulle 1995). In addition, models of pricing to market and exchange rate pass-through give rise to impediments to goods' arbitrage (Krugman 1987; Froot and Klemperer 1989). The implication is that the speed of adjustment of deviations from PPP depends on the magnitude of the deviations.On the other hand, the adoption of a price index is crucial in examining PPP. Several authors have argued that the consumer and producer price indices (CPI and PPI, respectively) do not correspond to their theoretical counterparts and contain measurement errors and aggregation biases (Cheung and Lai 1993; Imbs et al. 2005). In addition, the commodity basket for CPI and PPI includes nontradable goods, which impart a nonstationary component to real CPI or PPI exchange rates (Engel 1999). Recently, Xu (2003) argues that the price index of traded goods (TPI) is the appropriate one for PPP since it reflects the behavior of arbitrage better than the CPI or PPI.The purpose of this article, therefore, is to examine the nonlinear dynamics of TPI-based real exchange rates. Several authors have applied a smooth transition autoregressive (STAR) model to capture the nonlinear dynamics of real exchange rates, as it allows for a smooth adjustment between regimes (Michael, Nobay, and Peel 1997; Taylor, Peel, and Sarno 2001). There are several reasons for us to apply a band-threshold autoregressive (TAR) instead of a STAR-type model in our empirical analysis. First, our empirical evidence fails to reject the unit-root hypothesis against the hypothesis of a nonlinear STAR stationary process based on the test provided by Kapetanios, Shin, and Snell (2003).1 Second, the plots of CPI-, PPI-, and TPI-based real exchange rates in Figure 1 show that the TPI-based real rate has the largest variation among these three real rates. Third, our empirical evidence rejects the null hypothesis of linearity against TAR-type nonlinearity and supports the hypothesis that real exchange rates are TAR-type stationary.Figure 1 Plots of Real Exchange Rates for the UK and New Zealand (Figure omitted. See article image.)Several empirical studies have applied a threshold-type process to examine the nonlinear mean-reversion of real exchange rates (Obstfeld and Taylor 1997; O'Connell 1998b; Taylor 2001; Sarno, Taylor, and Chowdhury 2004). …

The Asymmetric Effects of Oil Shocks on Output Growth: A Markov-Switching Analysis for the G-7 Countries

In this paper we specify and estimate different Markov-switching (MS) regime autoregressive models. The empirical performance of the univariate MS models used to describe the switches between different economic regimes for the G-7 countries is in general not satisfactory. We extend these models to verify if the inclusion of asymmetric oil shocks as an exogenous variable improves the ability of each specification to identify the different phases of the business cycle for each country under scrutiny. Following the wide literature on this topic, we have considered six different definitions of oil shocks: oil price changes, asymmetric transformations of oil price changes, oil price volatility, and oil supply conditions. We measure the persistence of each economic regime, as well as the ability of each MS model to detect the business cycle dates as described by widely acknowledged statistical institutions. Our empirical findings can be summarized as follows. First, the null hypothesis of linearity against the alternative of a MS specification is always rejected by the data. This suggests that regime-dependent models should be used if a researcher is interested in obtaining statistically adequate representations of the output growth process. Second, three-regime MS models typically outperform the corresponding two-regime specifications in describing the business cycle features for each country. Third, the introduction of different oil shock specifications is never rejected. Fourth, positive oil price changes, net oil price increases and oil price volatility are the oil shock definitions which contribute to a better description of the impact of oil on output growth. Finally, models with exogenous oil variables generally outperform the corresponding univariate specifications which exclude oil from the analysis.

A time-domain test for some types of nonlinearity

The bispectrum and third-order moment can be viewed as equivalent tools for testing for the presence of nonlinearity in stationary time series. This is because the bispectrum is the Fourier transform of the third-order moment. An advantage of the bispectrum is that its estimator comprises terms that are asymptotically independent at distinct bifrequencies under the null hypothesis of linearity. An advantage of the third-order moment is that its values in any subset of joint lags can be used in the test, whereas when using the bispectrum the entire (or truncated) third-order moment is required to construct the Fourier transform. We propose a test for nonlinearity based upon the estimated third-order moment. We use the phase scrambling bootstrap method to give a nonparametric estimate of the variance of our test statistic under the null hypothesis. Using a simulation study, we demonstrate that the test obtains its target significance level, with large power, when compared to an existing standard parametric test that uses the bispectrum. Further we show how the proposed test can be used to identify the source of nonlinearity due to interactions at specific frequencies. We also investigate implications for heuristic diagnosis of nonstationarity.

Testing for neglected nonlinearity in regression models based on the theory of random fields

Abstract Within a flexible regression model (J.D. Hamilton, Econometrica 69 (3) (2001) 537) we offer a battery of new Lagrange multiplier statistics that circumvent the problem of unidentified nuisance parameters under the null hypothesis of linearity and that are robust to the specification of the covariance function that defines the random field. These advantages are the result of (i) switching from the L 2 to the L 1 norm; and (ii) assuming that the random field is sufficiently smooth for its covariance function to be locally approximated by a high order Taylor expansion. A Monte Carlo simulation suggests that our statistics have superior power performance on detecting bilinear, neural network, and smooth transition autoregressive specifications. We also provide an application to the Industrial Production Index of sixteen OECD countries.

MULTIVARIATE STAR ANALYSIS OF MONEY–OUTPUT RELATIONSHIP

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.

Tests de linéarité, spécification et estimation de modèles à seuil : une analyse comparée des méthodes de Tsay et de Hansen

L'objectif de cet article est d'étudier, dans une perspective économétrique, les modèles à seuil et leur application aux séries temporelles. La particularité de ces modèles est leur capacité à engendrer une dynamique différente selon l'état de l''économie. La non-linéarité qu''ils induisent rend inopérantes les méthodes d'estimation usuelles. Notamment, les tests de linéarité contre l'alternative de modèles à seuil dépendent de paramètres de nuisance non identifiables sous l'hypothèse nulle de linéarité. L'article présente deux procédures de spécification des modèles à seuil et les compare à l'aide d'exercices de taille et de puissance. Une application au taux de croissance du Produit Intérieur Brut est enfin présentée pour huit pays de l'OCDE.

Testing time series linearity via goodness-of-fit methods

We consider the problem of testing time series linearity. Existing time domain and spectral domain tests are discussed. A new approach relying on spectral domain properties of a time series under the null hypothesis of linearity is suggested. Under linearity, the normalized bispectral density function Z is a constant. Under the null hypothesis of linearity, properly constructed estimators of 2| Z | 2 have a non-central chi-squared distribution with two degrees of freedom and constant non-centrality parameter 2| Z | 2 . If the null hypothesis is false, the non-centrality parameter is non-constant. This suggests goodness-of-fit tests might be effective in diagnosing non-linearity. Several approaches are introduced.

A consistent nonparametric test for linearity of AR( p) models

We propose a simple nonparametric test for a linear AR( p) model against a general nonparametric AR( p) alternative. The asymptotic distribution of the test statistic under the null hypothesis of linearity is shown to be the standard normal. In addition, we show that the test is consistent against nonparametric AR( p) alternatives. © 1997 Elsevier Science S.A.

On the Use of Nonparametric Regression for Checking Linear Relationships

SUMMARY The problem of checking the linearity of a regression relationship is addressed through the idea of smoothing of a residual plot. A pseudolikelihood ratio test statistic, which measures the distance between the nonparametric and the parametric models, is derived as a ratio of quadratic forms. The distribution of this statistic under the null hypothesis of linearity is calculated numerically by using Johnson curves. A power study shows the new statistic to be more sensitive to non-linearity than the Durbin-Watson statistic.