Comprehensively testing linearity hypothesis using the smooth transition autoregressive model

Abstract This article examines the null limit distribution of the quasi-likelihood ratio (QLR) statistic for testing linearity condition against the smooth transition autoregressive (STAR) model. We explicitly show that the QLR test statistic weakly converges to a functional of a multivariate Gaussian process under the null of linearity, which is done by resolving the issue of identification problem arises in two different ways under the null. In contrast with the Lagrange multiplier test that is widely employed for testing the linearity condition, the proposed QLR statistic has an omnibus power, and thus, it complements the existing testing procedure. We show the empirical relevance of our test by testing the neglected nonlinearity of the US fiscal multipliers and growth rates of US unemployment. These empirical examples demonstrate that the QLR test is useful for detecting the nonlinear structure among economic variables.


Introduction
The smooth transition autoregressive (STAR) model has been widely used in many areas including economics. It has a property which it shares with other nonlinear models such as the threshold autoregressive model or the hidden Markov or Markov-switching autoregressive model: it nests a linear autoregressive model. Even more importantly, these models are not identified when the data-generating process is a linear model. This lack of identification was first studied by Davies (1977Davies ( , 1987; see also Watson and Engle (1985). If one estimates the STAR model before testing the linearity hypothesis, one may end up estimating an unidentified model. This typically causes numerical problems in estimation, and even when the estimation algorithm converges, the results are not reliable. It is therefore necessary to test linearity before estimating the STAR model. If the null of linearity is not rejected, the model builder can simply settle for a linear model and avoid the potential problems arising from fitting a more complicated nonlinear model.
The testing problem can be tackled head on by constructing an empirical null distribution by simulation or bootstrap, see, for example, Hansen (1996). Another popular approach consists of circumventing the identification problem by replacing the alternative by a Taylor series approximation around the null hypothesis and constructing a Lagrange multiplier (LM) test against this approximate alternative. For this solution, see Saikkonen and Luukkonen (1988) and Luukkonen, Saikkonen, and Ter€ asvirta (1988). Later, Granger and Ter€ asvirta (1993) and Ter€ asvirta (1994) made this test a part of their strategy for building STAR models.
However, the LM test statistic does not comprehensively test for the nonlinearity entailed by the STAR model. As will be detailed below, the STAR model violates the linearity condition in two different ways, and the LM statistic tests against only one of these two violations. The main goal of this study is to develop a testing procedure that complements the existing test by a test that has non-negligible power against arbitrary nonlinearity. Specifically, we resolve the foregoing identification issue by testing for nonlinearity in two different ways and combining the results into a single test.
An indication of the identification problem is that the model to be tested can be defined by more than one set of parameter restrictions on the alternative. When one such set is selected, some of the parameters of the alternative model remain unidentified under the null hypothesis. In this situation it is possible to choose a different set of restrictions such that the null model is defined using some or all of the parameters that were unidentified in the previous case. This implies that a different set of parameters, including the one or ones that defined the previous null model, are now unidentified when this null hypothesis holds. Following the previous literature, we call this the twofold identification problem. Although the models under the null are observationally equivalent, testing procedures without any consideration of the twofold identification problem may not have omnibus power against the null hypothesis of linearity.
We resolve the associated issues by considering the quasi-likelihood ratio (QLR) test statistic, which is known to have omnibus power against arbitrary nonlinearity. As Stinchcombe and White (1998) pointed out, a linearity test acquires omnibus power if it is based on an analytic function, which is the case of the STAR model. As already mentioned, in the LM statistic this analytic function is approximated by a polynomial with the result that the omnibus power is not achieved. Nevertheless, the LM test is easy to compute and has an asymptotic v 2 distribution under the null of linearity, which explains its popularity.
The QLR statistic in the context of linearity test is not novel. For instance, White (2011, 2014), Cho and Ishida (2012), White and Cho (2012), Baek, Cho, and Phillips (2015), and Cho and Phillips (2018) among others, study testing for neglected nonlinearity using analytic functions and note that the null of linearity can arise in two or three different ways, each of which carries its own identification problem. They propose a QLR test statistic to resolve the identification issues. We generalize the results in the previous literature to testing linearity against the STAR model and develop a testing procedure that is readily available for applications.
A similar testing procedure can be found in the literature on sub-vector inference. In particular, when the identification of some parameters depends on the identifiability of others, Cheng (2015) considers an inference method that remains valid uniformly on the parameter space by applying the methodology in Andrews and Cheng (2014). However, similarly to the LM test, only one side of the alternative hypothesis is concerned. The Wald test examined by Cheng (2015) does not comprehensively examine the null of linearity as for the QLR test examined by Andrews and Cheng (2014).
Once the QLR test has been constructed, it is appropriate to study its behavior by using a Monte Carlo simulation study. To this end, we consider the case where the exponential smooth transition autoregressive (ESTAR) model is given as an alternative model, and compare the performance of the QLR test with other tests available in the literature. The simulation results show that the QLR test has excellent size control and power and further that the QLR and score-based tests, especially the LM tests detailed in Section 2.2, can complement each others.
We then revisit two published empirical studies and examine popular nonlinearity assumptions imposed in applied macroeconomic literature. We first reexamine the macroeconomic data in Auerbach and Gorodnichenko (2012) who examined the government multiplier effect using the vector smooth transition autoregressive (VSTAR) model. They used nonstationary data in their analysis. Following Candelon and Lieb (2013), we transform Auerbach and Gorodnichenko's (2012) nonstationary VSTAR model into a stationary vector smooth transition error-correction (VSTEC) model. This makes it possible for us to apply the QLR (and the LM) test statistic. As it turns out, the QLR test statistic rejects linearity, which supports the use of the VSTEC model in studying nonlinear effects of fiscal policy in the US.
In addition, we extend the quarterly US unemployment rate series that has been previously studied by van Dijk, Ter€ asvirta, and Franses (2002). They tested linearity by the LM statistic, and in this study we illustrate the use of the QLR test statistic alongside the LM statistic and find nonlinear features in the series that could not have been found by the LM or the QLR statistic alone.
The plan of the article is as follows. Testing linearity in the STAR framework is discussed in Section 2, where the null limit distribution of the QLR test statistic is derived. Section 3 provides Monte Carlo simulation results and compare the performance of the QLR test with other tests. Section 4 contains applications of the QLR test statistic to the multiplier effect of US government spending and the US unemployment rate. Section 5 concludes.
The detailed proofs can be found in the Supplement. There our theory is applied to the ESTAR model and the logistic smooth transition autoregressive (LSTAR) model. Results on Monte Carlo simulations are reported. In particular, we demonstrate the use of Hansen's (1996) weighted bootstrap in the context of the QLR statistic.

Preliminaries
In this subsection, we clarify the difference between the STAR model and the artificial neural network (ANN) model, in which the QLR test has hitherto been studied (e.g., White, 2011, 2014;White and Cho, 2012;Baek, Cho, and Phillips, 2015;Cho and Phillips, 2018). This helps to explain how the current study contributes to the literature on the QLR test by tackling the twofold identification problem within the STAR family of models.
The standard single-hidden layer (univariate) ANN model is specified for stationary variables and has the following form: where z t :¼ ð1,z 0 t Þ 0 withz t :¼ ðy tÀ1 , :::, y tÀp Þ 0 , f ð0Þ ¼ constant, and p 0 , p, h j , c j , j ¼ 1, :::, q, are parameters. The ANN model (1) thus contains a linear combination of continuous and bounded functions (a hidden layer), typically logistic ones, although other bounded functions are possible. Nowadays, ANN models in applications often contain more than one hidden layer, but the single-hidden layer ANN model serves as a benchmark against which a STAR model may be compared. The twofold identification problem becomes obvious from (1). The model becomes linear by assuming either h j ¼ 0 or c j ¼ 0 (j ¼ 1, :::, q), so that if h j ¼ 0, c j disappears from the model; and if c j ¼ 0, p 0 and h j are not separably estimable from it. Therefore, Davies' (1977Davies' ( , 1987 identification problem arises in two different ways. This makes the Wald test inapplicable, and so the previous studies focusing on the QLR test apply the likelihood-ratio principle. In contrast, the following STAR model of order p is frequently specified as a prediction model of a time-series data y t (e.g., Granger and Ter€ asvirta, 1993;Ter€ asvirta, 1994): z t :¼ ð1,z 0 t Þ 0 is a ðp þ 1Þ Â 1 vector of regressors with a transition variablez 0 t a: Here,z t :¼ ðy tÀ1 , y tÀ2 , :::, y tÀp Þ 0 , and a ¼ ð0, :::, 1, 0, :::, 0Þ 0 denotes a selection vector chosen by the researcher.
The other parameter vectors p :¼ ðp 0 , p 1 , :::, p p Þ 0 and h :¼ ðh 0 , h 1 , :::, h p Þ 0 are the transition parameters, and c is used to describe the smooth transition from one extreme regime to the other. Symbols P, H, and C denote the parameter spaces of p, h, and c, respectively. The transition function f ðÁ, cÞ is a nonlinear, continuously differentiable, and uniformly bounded function. It is typically either exponential, f E ðz 0 t a, cÞ :¼ 1 À exp ðÀcðz 0 t aÞ 2 Þ, or logistic, f L ðz 0 t a, cÞ :¼ f1 þ exp ðÀcz 0 t aÞg À1 ; if the transition function is f E ðz 0 t a, cÞ (resp. f L ðz 0 t a, cÞ), the model is called the ESTAR (resp. LSTAR) model. In both cases, c > 0: The STAR model in (2) is a special case of the original STAR model in which the transition function f ðz 0 t a À c, cÞ with a constant c is substituted for f ðz 0 t a, cÞ in M 0 : We set c ¼ 0 in M 0 as in the regular exponential autoregressive model in Haggan and Ozaki (1981) and Auerbach and Gorodnichenko (2012) because the essential property in testing linearity is that f ðz 0 t a, ÁÞ is an analytic function. As we detail below, if c is estimated along with the other parameters, the inference becomes more complicated than ours, and this complexity limits its applicability.
The main difference between these two models is that the single hidden-layer ANN model contains a linear combination of several transitions that are themselves functions of linear combinations of elements of z t , whereas in the standard STAR model a linear combination of these elements is multiplied by a transition function usually with a single argument. 1 Due to these differences, the analysis of the QLR test statistic needs to be generalized in order to make the QLR test statistic applicable in the STAR framework. Specifically, White (2011, 2014) characterized the null limit distribution of the QLR test statistic in the ANN context as a functional of a univariate Gaussian process. This limit distribution cannot, however, be applied to the STAR case because as it turns out, a multivariate Gaussian process is required for the null limit distribution when testing for the STAR model.
The STAR model has a continuum number of regimes defined by transition functions obtaining values between zero to unity. This feature makes the model an appealing alternative in empirical studies because the behavior of economic agents can often be best described by multiple regimes and smooth transitions between them. The concept of smooth transition was introduced by Bacon and Watts (1971), in the econometrics literature by Goldfeld and Quandt (1972, pp. 263-264) and first applied in the time series context by Chan and Tong (1986). For more discussion on the STAR model the reader is referred to Granger and Ter€ asvirta (1993), Ter€ asvirta (1994), van Dijk, Ter€ asvirta, and Franses (2002), and Ter€ asvirta, Tjøstheim, and Granger (2010), among others.
The ESTAR and LSTAR models are specified by transforming the exponential function that is analytic, so it is generically comprehensively revealing for model misspecification as pointed out by Stinchcombe and White (1998). Therefore, the estimated parameters in the transition function become statistically significant such that the nonlinear component necessarily reduces the mean squared error of the model even when the assumed STAR model is misspecified. This implies that if the linear model is misspecified, the mean square error obtained from estimating the corresponding STAR model becomes smaller than that from the linear model. This in turn motivates testing linearity by comparing the estimated mean squared errors from the STAR and the linear model nested in the STAR. This process delivers an omnibus testing procedure for nonlinearity. 1 This argument is most often an element ofz t , although it can also be a weighted sum of several variables where the weights are assumed known. STAR models can also contain more than one additive transition, but this seems to be uncommon in applications.

Brief review of the LM test
Before discussing the QLR test, we briefly review the model framework for the LM test statistics to make a comparison with the QLR test. The following auxiliary model is first estimated for the LM test statistics: t Þ, and t t is the transition variable, viz.,z 0 t a: This model is obtained by applying a fourth-order Taylor expansion to the analytic function as an intermediate step to compute the LM test statistics conveniently which test c Ã ¼ 0: Luukkonen, Saikkonen, and Ter€ asvirta (1988) and Ter€ asvirta (1994) provided detailed rationales of carrying out testing linearity by examining the coefficients of nonlinear components of the approximate alternative.
To be specific, Ter€ asvirta (1994) and Escribano and Jord a (1999) specify the following four sets of hypotheses which are commonly considered in empirical studies: For later purpose, we denote the LM test statistics testing H 0, i as LM i, n , i ¼ 1, :::, 4: Here, LM 1, n and LM 2, n are general tests against STAR, whereas LM 3, n and LM 4, n are tests against the LSTAR and ESTAR models, respectively.

Data generating process and the QLR test statistic
We consider a univariate STAR model and study the null limit behavior of the QLR test statistic in this framework. In order to proceed, we make the following assumptions: Assumption 1. fðy t ,z 0 t Þ 0 2 R 1þp : t ¼ 1, 2, :::g ðp 2 NÞ is a strictly stationary and absolutely regular process defined on the complete probability space ðX, F , PÞ, with E½jy t j < 1 and mixing coefficient b s such that for some q > 1, P 1 s¼1 s 1=ðqÀ1Þ b s < 1: w Here, the mixing coefficient is defined as b s :¼ sup s2N E½sup A2F 1 sþs jPðAjF s À1 Þ À PðAÞj, where F s s is the r-field generated by ðy s , :::, y sþs Þ: Many time series models satisfy this condition, and the autoregressive process is one of them. It is general enough to cover the stationary time series we are interested in. We impose the following regular STAR model condition: Assumption 2. Let f ðz 0 Using Assumption 2, the linearity hypothesis and the alternative are specified as follows: These hypotheses are the same as the ones in White (2011, 2014), Baek, Cho, and Phillips (2015), and Cho and Phillips (2018). As in the previous literature, the focus is on developing an omnibus test statistic but now against STAR, and the QLR test statistic is a vehicle for reaching this goal. The QLR test statistic is formally defined as and f t ðcÞ :¼ f ðz 0 t a, cÞ À f ðz 0 t a, 0Þ: We let the nonlinear least squares (NLS) estimator ðp n ,ĥ n ,ĉ n Þ minimize the squared errors with respect to ðp, h, cÞ: Furthermore, ðp Ã , h Ã , c Ã Þ denotes the probability limit of the NLS estimator: ðp Ã , h Ã , c Ã Þ :¼ arg min p, h, c E½fy t À z 0 t p À f t ðcÞðz 0 t hÞg 2 is the pseudo-true parameter. Note that this limit is not unique under the null.
The main reason for proceeding with the QLR statistic is that linearity leads to a twofold identification problem, and this statistic is able to handle both parts of it. If E½y t jz t is linear with respect to z t with coefficient p Ã , we can generate a linear function from hðÁ, cÞ instead of f ðz 0 t a, cÞ for M as in the original STAR model, neither c Ã nor the additional c Ã is identified under h Ã ¼ 0, which leads to a more complicated identification problem. We fix our interest in the current derivative model M that excludes c Ã : This makes the type II identification problem more complicated as p Ã and h Ã are not separately identified. Centering thus transforms this complication into a relatively straightforward identification problem. Besides, mainly due to the invariance principle, the null limit distribution does not change by this centering. Note that p in M 0 is reparametrized to p À f ðz 0 t a, 0Þh in M, so that the QLR test obtained by this reparameterization becomes identical to that before the reparameterization. Without it, the null model investigation has to be separately conducted by discerning the parameters with the type II identification problem. So, we avoid the involved complication by the centering and obtain the null limit distribution of the QLR test efficiently. This centering is also indirectly applied in the literature when the null limit distribution of the LM test statistic is being derived. If z t contains a constant, this limit distribution is not affected by the centering, because the centered parameter is merged into the linear component in the Taylor expansion that forms the basis of the LM test statistic. Now, the null holds for the following two sub-hypotheses: H 01 : h Ã ¼ 0 and H 02 : c Ã ¼ 0: The limit distribution of the QLR test statistic can be derived under both H 01 and H 02 , leading to different null limit distributions even for the same statistic. We call these derivations type I and type II analyses, respectively. The null hypothesis of linearity against STAR is properly tested by tackling both H 01 and H 02 simultaneously, and we shall demonstrate that the QLR test is capable of doing this. For this purpose, we derive its null limit distribution from the separately obtained null weak limits in the spirit of likelihood-ratio principle. Specifically, we show how the two different weak limits are related to the null limit distribution of the QLR statistic.
Our view to testing linearity by accommodating type I and II analyses differs from the other tests in the literature. For example, the LM test statistic focuses on testing H 02 : The main argument for the LM test is that its asymptotic null distribution is chi-squared, which makes the test easily applicable. As another example, Cheng (2015) assumes the standard STAR model and analyses the standard Wald statistic for testing H 01 in the vein of the type I identification problem. None of them accommodates the twofold identification problem.

The null limit distribution of the QLR test
We now derive the null limit distribution of the QLR test and highlight the difference between the STAR-based approach and the ANN-based one. We first study the limit distributions of the QLR test under H 01 and H 02 separately, combine them and, finally, obtain the limit distribution under H 0 : For this, we let our quasi-likelihood (QL) function be 2.4.1. Type I analysis: Testing H 01 : h Ã ¼ 0 In this subsection, we discuss the limit distribution of the QLR test under H 01 : h Ã ¼ 0: The problem is that c Ã is not identified under this hypothesis. We obtain the NLS estimator by maximizing the QL function with respect to c in the final stage for the purpose of testing H 01 : L ð1Þ n : n denote the QLR statistic obtained by this optimization process. That is, where u :¼ ½u 1 , u 2 , :::, u n 0 , u t :¼ y t À E½y t jz t , Z :¼ ½z 1 , z 2 , :::, z n 0 , M :¼ I À ZðZ 0 ZÞ À1 Z 0 , and FðcÞ : ¼ diag½f 1 ðcÞ, f 2 ðcÞ, :::, f n ðcÞ: Therefore, we found that n : Here, we cannot let c ¼ 0 when deriving QLR ð1Þ n : If c ¼ 0, the alternative model reduces to the linear model, so that the QLR statistic cannot test the null model by letting c ¼ 0. We therefore examine its null limit distribution by supposing c 6 ¼ 0: We now derive the limit distribution of QLR n under H 01 : For this and to ensure a regular null limit distribution, we impose the following conditions: Assumption 5. There exists a sequence of stationary ergodic random variables m t such that for i ¼ 1, 2, :::, p, jz t, i j m t , ju t j m t , jy t j m t , and for some x ! 2ðq À 1Þ, E½m 6þ3x t < 1, where q is in Assumption 1, and z t, i is the i-th row element of z t . w Assumption 6. For each c 6 ¼ 0, V 1 ðcÞ and V 2 ðcÞ are positive definite, where for each c, V 1 ðcÞ :¼ E½u 2 tr t ðcÞr t ðcÞ 0 and V 2 ðcÞ :¼ E½r t ðcÞr t ðcÞ 0 withr t ðcÞ :¼ ðf t ðcÞz 0 t , z 0 t Þ 0 : w Assumption 3(i) implies that the model in Assumption 2 is not dynamically misspecified, and Assumption 3(ii) means that the errors are conditionally homoscedastic. Here, conditional homoscedasticity is not essential in achieving the main goal of this study, but this assumption will be assumed whenever it facilitates understanding the theoretical results intuitively. Assumption 4 plays an integral role in applying the tightness condition in Doukhan, Massart, and Rio (1995) to the QLR test statistic. Here it can be easily verified for the ESTAR and LSTAR models by noting t a, cÞÞjðz 0 t aÞj jðz 0 t aÞj, so that we can let m t in Assumption 4 be ðz 0 t aÞ 2 and jðz 0 t aÞj, respectively. The moment condition in Assumption 5 is stronger than those in White (2011, 2014), and it also implies that E½u 6 t and E½y 6 t are finite. The multiplicative component f t ðcÞz 0 t h in the STAR model makes the stronger moment condition necessary in the current study. Assumption 6 is imposed for the invertibility of the limit covariance matrix. This makes our test statistic non-degenerate. We have the following lemma: Lemma 1. Given Assumptions 1, 2, 3(i), 4, 5, 6, and H 01 , (i)r 2 n, 0 ! a:s: r 2 Ã :¼ E½u 2 t ; (ii) fn À1=2 Z 0 FðÁÞMu,r 2 n, 0 n À1 Z 0 FðÁÞMFðÁÞZg ) fZ 1 ðÁÞ, A 1 ðÁ, ÁÞg on CðÞ and CðÞ Â CðÞ, respectively, where CðÞ :¼ fc 2 C : jcj ! g, Z 1 ðÁÞ is a continuous multivariate Gaussian process with E½Z 1 ðcÞ ¼ 0, and for each c andc, E½Z 1 ðcÞZ 1 ðcÞ 0 ¼ B 1 ðc,cÞ such that B 1 ðc,cÞ :¼ There is a caveat to Lemma 1. It is clear from QLR ð1Þ n that its limit distribution is determined by the limit behavior under H 01 of both n À1=2 Z 0 FðÁÞMu and n À1 Z 0 FðÁÞM FðÁÞZ: Furthermore, lim c!0 Z 0 FðcÞMu ¼ a:s: Z 0 Fð0ÞMu ¼ 0 and lim c!0 Z 0 FðcÞMFðcÞZ ¼ a:s: Z 0 Fð0ÞMFð0ÞZ ¼ 0: This implies that it is not straightforward to obtain the limit distribution of QLR n around c ¼ 0: We therefore assume for the moment that 0 is not included in C by considering CðÞ instead of C and accommodate this effect by restricting the QLR test statistic to QLR ð1Þ n ðÞ :¼ max We relax this restriction when the limit distribution is examined under H 0 : Lemma 1 is central in deriving the null limit distribution of QLR ð1Þ n ðÞ and corresponds to Lemma 1 of Cho, Ishida, and White (2011). Despite being similar, the two lemmas are not identical; note that Z 1 ðÁÞ is mapped to R pþ1 , i.e., ðp þ 1Þ-variate Gaussian process, whereas their lemma obtains a univariate Gaussian process. This multivariate Gaussian process Z 1 ðÁÞ distinguishes the STAR model-based testing from the ANN-based approach. By this, the STAR model has a different null limit distribution, and the QLR test based upon the STAR model has power over alternatives in directions different from those of the ANN-based approach.
2.4.2. Type II analysis: Testing H 02 : c Ã ¼ 0 Here, the focus is on the limit distribution under H 02 : c Ã ¼ 0: This hypothesis is tested using the LM statistic. As we know, h Ã is not identified under H 02 : We therefore maximize the QL function with respect to h at the final stage: L ð2Þ n :¼ sup h sup c sup p À P n t¼1 fy t À z 0 t p À f t ðcÞðz 0 t hÞg 2 , and denote the QLR test defined by this maximization process by QLR ð2Þ n : Several remarks are in order. First, maximizing the QL with respect to p is relatively simple due to linearity. We let the concentrated QL (CQL) function be L ð2Þ n ðc, hÞ :¼ sup p L n ðp, h, cÞ ¼ À½y À FðcÞZh 0 M½y À FðcÞZh, where y :¼ ½y 1 , y 2 , :::, y n : Here, we have to assume h 6 ¼ 0: If h ¼ 0, the STAR model becomes linear, so the QLR test statistic cannot compare the null model with the alternative. Second, L ð2Þ n ðÁÞ is not linear with respect to c, so that the next stage CQL function with respect to c cannot be analytically derived. We approximate the CQL function with respect to c around c Ã ¼ 0 and capture its limit behavior under H 02 : The first-order derivative of L ð2Þ n ðc, hÞ with respect to c is ðd=dcÞL ð2Þ n ðc, hÞ ¼ 2½y À FðcÞZh 0 Mð@FðcÞ=@cÞZh, where ð@FðcÞ=@cÞ :¼ ð@=@cÞðf ðz 0 1 a, cÞ, :::, f ðz 0 n a, cÞÞ: For the LSTAR model, @f L ðz 0 t a, cÞ= @c ¼ f L ðz 0 t a, cÞð1 À f L ðz 0 t a, cÞÞz 0 t a and @Fð0Þ=@c ¼ ð1=4Þðz 0 1 a, :::,z 0 n aÞ 0 , whereas for ESTAR, it follows that @c ¼ ðz 0 t aÞ 2 ð1 À f E ðz 0 t a, cÞÞ, so @Fð0Þ=@c ¼ ððz 0 1 aÞ 2 , :::, ðz 0 n aÞ 2 Þ 0 , implying that we can approximate the CQL function by a second-order approximation. Nevertheless, as Luukkonen, Saikkonen, and Ter€ asvirta (1988), Ter€ asvirta (1994), and White (2011, 2014) pointed out, the first-order derivative of the CQL is often zero for many other models. For example, in M A :¼ fpy tÀ1 þ hf1 þ exp ðcy tÀ1 Þg À1 : ðp, h, cÞ 2 P Â H Â Cg, the first-order derivative of the CQL is zero when c Ã ¼ 0: Due to this, we need a higher-order approximation. Cho, Ishida, and White (2014) adopt a sixth-order Taylor expansion, whereas Luukkonen, Saikkonen, and Ter€ asvirta (1988), Ter€ asvirta (1994), and Cho, Ishida, and White (2011) use fourth-order Taylor expansions to obtain the null limit distributions of their tests. The order of expansion is determined by the functional form of f ðz 0 t a, ÁÞ: As we do not assume a specific form for our STAR model, we simply let j (j 2 N) be the smallest order such that the j-th order partial derivative with respect to c is different from zero at c ¼ 0, so that for all j < j, ð@ j =@c j ÞL ð2Þ n ð0, ÁÞ 0: For example, j ¼ 3 for M A : Then, the CQL function is expanded as L ð2Þ n ðc, hÞ ¼ L ð2Þ n ð0, hÞ þ 1 j! @ j @c j L ð2Þ n ð0, hÞc j þ ::: þ Note that for j ¼ 1, 2, :::, j À 1, ð@ j =@c j ÞL ð2Þ n ð0, hÞ ¼ 0 by the definition of j. If j ¼ 1, the firstorder derivative differs from zero, so that none of the derivatives is zero, meaning that j ¼ 0. The limit behaviors of the partial derivatives in (3) are given in the following lemma: Lemma 2. Given Assumption 2, the definition of j, and H 02 , for each h 6 ¼ 0, @ j @c j L ð2Þ n ð0, hÞ ¼ 2h 0 Z 0 H j ð0ÞMu for j j < 2j; and @ j @c j L ð2Þ n ð0, hÞ ¼ 2h for j ¼ 2j, where H j ðcÞ :¼ ð@ j =@c j ÞFðcÞ: w Using Lemma 2, we can specifically rewrite (3) as L ð2Þ n ðc, hÞ À L ð2Þ n ð0, hÞ ¼ To reduce notational clutter, we further let G j :¼ ½g 1, j , g 2, j , :::, g n, j 0 :¼ MH j ð0ÞZ, where g t, j :¼ h t, j ð0Þz t À Z 0 H j ð0ÞZ ðZ 0 ZÞ À1 Z 0 z t and 1 n :¼ n 1=2j c with h t, j ð0Þ being the t-th diagonal element of H j ð0Þ: Then, L ð2Þ n ðc, hÞ À L ð2Þ n ð0, hÞ ¼ We note that if j ¼ j, n Àj=2j G 0 j u ¼ O P ð1Þ by applying the central limit theorem. Furthermore, for j ¼ j þ 1, :::, 2j À 1, n Àj=ð2jÞ ð@ j =@c j ÞL ð2Þ n ðc, hÞ ¼ o P ð1Þ and h 0 G 2j u ¼ o P ðnÞ by the ergodic theorem, so that they become asymptotically negligible, implying that the smallest j-th component greater than j and surviving at the limit becomes the second-final term in the right side of (4). Note that n À1 G 0 j G j ¼ O P ð1Þ, if the ergodic theorem applies, and the terms with j > 2j belong to o P ðc 2j Þ by Taylor's theorem, so that they are asymptotically negligible under the null. Due to this fact, L ð2Þ n ðÁ, hÞ is approximated by the 2j-th degree polynomial function in (4), and we provide the following condition for the asymptotic analysis of the polynomial function: Assumption 7. For j ¼ j, j þ 1, :::, 2j and i ¼ 0, 1, :::, p, (i) E½ju t j 8 < 1, E½jh t, j ð0Þj 8 < 1, and E½jz t, i j 4 < 1; or (ii) E½ju t j 4 < 1, E½jh t, j ð0Þj 8 < 1, and E½jz t, i j 8 < 1: w Using Assumption 7, we can apply the CLT to n À1=2 G 0 j u for j ¼ j, j þ 1, :::, 2j: Note that G 0 j u ¼ P n t¼1 ðu t g t, j Þ, and E½ðu t g t, j Þ 2 < 1 by the moment conditions in Assumption 7 and Cauchy-Schwarz inequality, so that for j ¼ j þ 1, :::, 2j À 1, n Àj=2j G 0 j u ¼ o P ð1Þ: Although the QLR test statistic is approximated by the 2j-th degree polynomial function, the moment conditions in Assumption 7 are sufficient to apply the CLT to the first term in (4).
We establish the following lemma by collecting the asymptotically surviving terms: Lemma 3. Given Assumptions 1, 2, 7, and H 02 , QLR and1 j n ðhÞ maximizes the given objective function, so that1 j n ðhÞ ¼ W n ðhÞ, if j is odd; and1 j n ðhÞ ¼ max½0, W n ðhÞ, if j is even, where W n ðhÞ :¼ j!n 1=2 fh 0 G 0 j ug=fh 0 G 0 j G j hg: w Lemma 3 implies that the functional form of QLR ð2Þ n ðÁÞ depends on j: for each h 6 ¼ 0, QLR ð2Þ n ðhÞ ¼ 1 n ðÁÞ with respect to h. We now derive the regular limit distribution of QLR test statistic under H 02 : The following additional condition is sufficient for this: Assumption 8. V 3 ð0Þ and V 4 ð0Þ are positive definite, where for each c, V 3 ðcÞ :¼ E½u 2 t r t ðcÞ r t ðcÞ 0 and V 4 ðcÞ :¼ E½ r t ðcÞ r t ðcÞ 0 with r t ðcÞ :¼ ðh t, j ðcÞz 0 t , z 0 t Þ 0 : w We note that the nuisance parameter c does not play a significant role in Assumption 8 as it does in the type I analysis, because QLR n ðÁÞ has already concentrated the QL function with respect to c. Given these regularity conditions, the key limit results of the components that constitute QLR ð2Þ n ðÁÞ appear in the following lemma: Lemma 4. Given Assumptions 1, 2, 3(i), 4, 7, 8, and H 02 , (i) n À1=2 G 0 j u ) Z 2 , where E½Z 2 ¼ 0 and E½Z 2 Z 0 2 ¼ E½u 2 t g t, j g 0 t, j ; (ii) n À1 G 0 j G j ! a:s: A 2 , where A 2 :¼ E½g t, j g 0 t, j ; and (iii) if, additionally, Assumption 3(iii) holds, E½u 2 t g t, j g 0 t, j ¼ r 2 Ã E½g t, j g 0 t, j : w Using Lemma 4, Theorem 2 describes the limit distribution of QLR w Some remarks are in order. First, the covariance kernel of G 2 ðÁÞ is bilinear with respect to h andh: This implies that G 2 ðhÞ is a linear Gaussian process with respect to h. Therefore, if z $ Nð0, E½u 2 t g t, j g 0 t, j Þ, z 0 h as a function of h is distributionally equivalent to G 2 ðÁÞ: This fact relates the null limit distribution to the chi-squared distribution. Corollary 1 of Cho and White (2018) shows that max h2H G 2 2 ðhÞ ¼ d v 2 pþ1 if G 2 ðÁÞ is a linear Gaussian process and E½u 2 t g t, j g 0 is a chi-squared distribution with p þ 1 degrees of freedom. Second, the chi-squared null limit distributions of the LM test statistics in Luukkonen, Saikkonen, and Ter€ asvirta (1988), Granger and Ter€ asvirta (1993), and Ter€ asvirta (1994) follow from the fact that the LM test statistic is equivalent to the QLR test statistic under H 02 : Third, if h ¼ 0, G 2 ðhÞ is not well defined because the weak limit in Theorem 2 is obtained by assuming that h 6 ¼ 0: Nevertheless, the null limit distribution of the QLR test is well represented by Theorem 2 as it obtains the alternative model by letting h 6 ¼ 0: As Theorem 2 follows from Lemma 4 and continuous mapping, its proof is omitted.

Limit Distribution of the QLR test statistic under H 0
In this subsection, we derive the limit distribution of the QLR test under H 0 by examining the relationship between QLR ð1Þ n and QLR ð2Þ n : Specifically, using the arguments similar to those of White (2011, 2014), we show that QLR ð1Þ n ! QLR ð2Þ n , which means the limit distribution under H 0 equals that of QLR n ðhÞ is asymptotically equal tor À2 To show this, we first note that the two terms are idempotent and symmetric matrices, and make use of Exercise 8.58 in Abadir and Magnus (2005, p. 233). Then, Given that CðÞ was considered in Theorem 1 to remove c ¼ 0 from C, if we select as small as possible to have QLR n ¼ QLR n ðÞ þ o P ð1Þ so that we can let c ! 0 as posited in Lemma 5, it is now straightforward to show that the null limit distribution of the QLR test is characterized by the Gaussian process in Theorem 1. That is, under the conditions in Theorems 1 and 2, the null limit distribution of the QLR test statistic is obtained by combining Theorems 1 and 2. For this purpose, we first combine Assumptions 6 and 8 into a new assumption: Assumption 9. For each c 6 ¼ 0, V 5 ðcÞ and V 6 ðcÞ are positive definite, where V 5 ðcÞ :¼ E½u 2 t € r t ðcÞ€ r t ðcÞ 0 , V 6 ðcÞ :¼ E½€ r t ðcÞ€ r t ðcÞ 0 , and € r t ðcÞ : w Next, we provide the limit distribution of the QLR test under H 0 in the following theorem: Theorem 3. Given Assumptions 1, 2, 3(i), 4, 5, 7, 9, and H 0 , (i) QLR n ) sup c2C G 1 ðcÞ 0 G 1 ðcÞ, where G 1 ðÁÞ is a Gaussian process such that for each c andc, E½G 1 ðcÞ ¼ 0 with E½G 1 ðcÞG 1 ðcÞ 0 ¼ Theorem 3 immediately follows from Theorems 1 and 2 and from our earlier argument that n þ o P ð1Þ, which is why we do not prove it in the Supplement. Note that the consequence of Theorem 3 is the same as that of Theorem 1, although the null hypothesis is extended from H 01 to H 0 by enlarging the parameter space from CðÞ to C with sufficiently small . Also note that the Gaussian process G 1 ðÁÞ is obtained by supposing that c 6 ¼ 0: Otherwise, a meaningful QLR test statistic is not properly defined.
The null limit distribution in Theorem 3 is derived as in White (2011, 2014) and Baek, Cho, and Phillips (2015). Nevertheless, our proofs generalize theirs due to the complexity associated with the STAR model. Furthermore, from Section 2.2, it can be easily seen that the LM test statistics only test H 02 : In addition, it also extends the Wald test principle exploited by Cheng (2015) to test H 01 but not H 02 : The QLR statistic tests the linear model hypothesis by combining the null hypotheses neglected by the LM and Wald tests separately.

Simulations
In this section we report results of two simulation studies. To begin with, we apply our theory to the ESTAR and LSTAR models and check its validity by simulation. The idea of this set of simulations is simply to see how well the empirical null distribution of the QLR statistic matches the theoretical distribution under different assumptions on the parameter space C: Assuming that the errors of the model are standard normal, we derive the covariance kernel in Theorem 3 analytically using the ESTAR model, which enables us to represent the null limit distribution as an infinite sum of functions of c multiplied by Gaussian random coefficients. We can then compare the null limit distribution with the empirical null distribution of the QLR test obtained under various assumptions on the model. The detailed results, which are given in the Supplement, indicate that Theorem 3 is valid. In the Supplement, we also report another simulation result using the LSTAR model, obtaining the same conclusion as for the ESTAR model. In addition to this, we apply Hansen's (1996) weighted bootstrap to the QLR test and show that it can be usefully exploited when the covariance kernel of the Gaussian process in Theorem 3 is not available (see also . The purpose of the second simulation study is to evaluate the relative performance of the QLR test by comparing its empirical size and power with those of some other tests applied in the literature. First, we examine the LM tests presented in Section 2. Second, a referee prompted us to make comparisons with the score-based test proposed by Ling and Tong (2011), LT test for short. It has the advantage that it is straightforward to compute and applicable under rather mild regularity conditions. Furthermore, in contrast with the QLR test whose critical values are obtained by the bootstrap, the ones for the LT test are readily available.
We detail the second simulation. For this simulation, the time series {y t } T t¼1 is assumed to follow the ESTAR process: where u t $ IIDNð0, 1Þ, and the following ESTAR model is specified for {y t g : Notes: The empirical rejection rates of the linearity tests are reported. The rates of QLR E n and QLR E n, Ã are computed using 300 bootstrap replications and the critical values obtained from the null limit distribution in the Supplement, respectively. The parameter space of c is set to be ½0, 2: The parameter value a for the LT test is let to be 5%-quantile of data ðy 1 , :::, y nÀ1 Þ for each replication. The number of replications is 2,000.
M ESTAR :¼ fy t ¼ py tÀ1 À hy tÀ1 f1 À exp ðÀcy 2 tÀ1 Þg: p 2 P, h 2 H and c 2 Cg: The nonlinear analytic function f t ðcÞ ¼ 1 À exp ðÀc Ã y 2 tÀ1 Þ satisfies the aforementioned conditions and is widely employed in empirical applications. We simulate (5) using a number of designs. We set p Ã ¼ 0:5 and consider four different values of h Ã , viz. h Ã ¼ 0 (under H 0 ) and h Ã ¼ 0:2, 0:4, 0:6: The transition parameter c Ã either equals zero (under H 0 ) or is drawn from the uniform Uðr 0 À 1, r 0 Þ distribution for r 0 ¼ 1, 2, 3: The QLR test is computed over the parameter space C ¼ ½0, 2, which means that for r 0 ¼ 3, the model M ESTAR is misspecified. This in turn could have an adverse effect on the performance of the test.
As is done in the first set of simulations, the QLR test is computed in two different ways. First, given that the data generating process in (5) is equivalent to that in the first simulation study under H 0 , we can use the null distribution in the Supplement to compute the asymptotic critical values of the QLR test. The test statistic is denoted by QLR E n, Ã : Sometimes, the error distribution may not be known, however, and then, as already mentioned, we can use Hansen's (1996) weighted bootstrap for obtaining the critical value of the test, see also . The test statistic obtained via the bootstrap is denoted as QLR E n : The parameter a required for the LT statistic is set to equal the 0.05 quantile of the empirical cumulative distribution of the lagged dependent variable y tÀ1 : Furthermore, following Ling and Tong (2011), the vector b is a vector of ones. Two sample sizes are used, 200 and 500 observations, and the number of replications is 2,000.
The results are reported in Table 1 using the 5% level of significance. As already mentioned, if h Ã ¼ 0 or c Ã ¼ 0, linearity holds with probability 1, so that for both cases, the p-values represent the empirical size of the tests. We see that there is no difference between QLR E n, Ã and QLR E n : These two tests and the two LM tests, LM 1, n and LM 2, n , have the size under control in all cases considered. Meanwhile, the LT test is undersized, which agrees with the results in Ling and Tong (2011). When the alternative is true, the results show that the QLR tests are somewhat more powerful than the LM tests, which suggests that testing the two null hypotheses simultaneously pays off. Another observation is that for both tests, the power reaches its maximum when c Ã is drawn from U(0, 1) distribution and begins to decline when the distribution shifts to the right. This is due to the fact that the ESTAR model approaches a linear one when c Ã ! 1: As can be expected, the power of the tests increases with the sample size. Perhaps not unexpectedly, the two QLR tests still have power when the model is misspecified, i.e., when c Ã is drawn from Uð2, 3Þ: We also considered the parameter spaces C ¼ ½0, i, i ¼ 3, 4, 5, but did not find any difference in results (not reported here). Finally, the LT statistic, denoted LT n in Table  1, does not perform particularly well, which may partly be due to the fact that the transition function is nonmonotonic in y tÀ1 under the alternative.

Two empirical examples
In this section, we apply the QLR tests against the ESTAR and LSTAR models, denoted by QLR E n and QLR L n , to two empirical examples. As in the previous section, these are compared with the LM tests against the four alternative hypotheses outlined in Section 2.2.

Testing linearity of the fiscal multiplier effect
We first test the hypothesis of nonlinear government spending effect to other macro economic variables by revisiting the empirical work of Auerbach and Gorodnichenko (2012) who specified a VSTAR model for y t :¼ ðg t , s t , q t Þ 0 with g t , s t , and q t being log real government spending, log real government net tax receipts, and log real GDP deflated by the 2012 GDP deflator, respectively.
Because the time series used by Auerbach and Gorodnichenko (2012) are not stationary, we cannot apply our tests to their VSTAR model (they do not test linearity). Following Candelon and Lieb (2013), we bypass this difficulty by first converting this model into the following VSTEC form: where u t ¼ ðu 1t , u 2t , u 3t Þ 0 , w tÀ1 :¼ ½y Ã0 tÀ1 , Dy 0 tÀ1 0 , W 1Ã ðLÞ :¼ ½a RÃ ,P RÃ ðLÞ, and W 2Ã ðLÞ :¼ ½a DÃ ,P DÃ ðLÞ with a DÃ :¼ a RÃ À a EÃ andP DÃ ðLÞ :¼P RÃ ðLÞ ÀP EÃ ðLÞ: Here,P RÃ ðLÞ andP EÃ ðLÞ are the VSTEC coefficients associated with the recession and expansion periods, respectively; b Ã is the mÀdimensional cointegrating vector, which is invariance to the economic state; and y Ã tÀ1 ¼ b 0 Ã y tÀ1 (e.g., Rothman, van Dijk, and Franses, 2001;Hubrich and Ter€ asvirta, 2013). In addition, a Ã denotes the adjustment coefficient, which is a 3 Â m matrix, and it is assumed that a Ã ¼ ð1 À f t ðc Ã ÞÞa EÃ þ f t ðc Ã Þa RÃ , where a EÃ is not necessarily equal to a RÃ : 2 We now test the nonlinear effect of government spending in the following order. First, we marginalize the model (6) under the normality condition of u t as assumed by Auerbach and Gorodnichenko (2012). That is, for j ¼ 1, 2, 3, where h 0 jÃ :¼ E½u jt u Àjt E½u Àjt u 0 Àjt À1 , and, further, n j1Ã ðLÞ 0 :¼ w j1Ã ðLÞ 0 À h 0 jÃ w Àj1Ã ðLÞ 0 and n j2Ã ðLÞ 0 :¼ w j2Ã ðLÞ 0 À h 0 jÃ w Àj2Ã ðLÞ 0 : Here, u jt and u Àjt denote the j-th row element of u t and the 2 Â 1 vector obtained by removing u jt from u t , respectively. Furthermore, for each i ¼ 1 and 2, w jiÃ ðLÞ 0 and w ÀjiÃ ðLÞ 0 are the j-th row vector of W iÃ ðLÞ and 2 Â ðm þ 3Þ matrix obtained by removing the j-th row from W iÃ ðLÞ, respectively. Second, we estimate the model. For this, we first letb n denote the maximum likelihood estimator for b Ã estimated from (6), which is super-consistent (see Johansen, 1995), making it possible to estimate the other parameters by NLS by replacing y Ã tÀ1 withŷ tÀ1 :¼b 0 n y tÀ1 : Finally, we use the marginal model of (7) as our baseline model for testing for nonlinearity, where w tÀ1 is replaced withŵ tÀ1 :¼ ½ŷ 0 tÀ1 , Dy 0 tÀ1 0 : We apply the QLR and LM tests for each j ¼ 1, 2, 3. Note that rejecting the linearity hypothesis in at least one individual equation is sufficient for rejecting the linearity hypothesis of the whole VSTEC equation.
We now report our empirical findings using the data of Auerbach and Gorodnichenko (2012), which is comprised US quarterly macroeconomic variables. Their sample ranges from 1947Q1 to 2008Q4. In order to estimate the cointegration rank m, we apply Johansen's (1988Johansen's ( , 1991 trace The model above is similar to the one considered in Candelon and Lieb (2013), but our model is different from theirs as our model assumes a continuum of states and is not restricted to have a EÃ ¼ a RÃ : In addition, we test the linearity hypothesis under the assumption of conditional heteroscedasticity, without imposing a restriction on the error covariance matrix.
testing procedure with lag equal to 3 selected by AIC and BIC 3 and cannot reject the hypothesis that m ¼ 2 at the 5% significance level. Using this rank, we estimate the cointegration coefficient b Ã to obtainŷ tÀ1 : Next, we apply the diagnostic testing procedure to validate the assumption on the nonlinearity. As explained above, we replace w tÀ1 in the marginal model withŵ tÀ1 :¼ ½ŷ 0 tÀ1 , Dy 0 tÀ1 0 and test the linearity hypothesis by the QLR tests based upon ESTAR and LSTAR models. We report the p-values of the QLR and LM tests in Table 2, where the p-values for our tests are obtained by applying Hansen's (1996) weighted bootstrap with 20,000 replications. In general, both the QLR and LM tests strongly reject the linearity hypothesis. However, when the dependent variable is given by g t , the estimated p-value of LM 3, n is far above the significance level. A similar result is found when we apply LM 4, n to the model whose dependent variable is given by s t . Although the latter could be caused by specifying the exponential transition function under the alternative, this is in contrast with the testing results of QLR statistic that reports substantially small p-values irrespective of the dependent variables. This suggests, at least in this particular case, the QLR test would be more robust, and both tests could be used complementary to each other. In general, the linearity testing results imply that the linear error-correction model is not adequate and the VSTEC model can better capture the dynamic interrelationship among the variables.

Application to US unemployment rates
We now examine the performance of the tests when the QLR test is applied to the monthly US unemployment rate. van Dijk, Ter€ asvirta, and Franses (2002) tested linearity of the series running from June 1968 to December 1999. We perform the tests both with their time series and the same series extended to August 2015. 4 van Dijk, Ter€ asvirta, and Franses (2002) point out that the US unemployment rate is a persistent series with an asymmetric adjustment process and strong seasonality. They specify a STAR model with monthly dummy variables mainly because first differences of the seasonally unadjusted unemployment rate of males aged 20 and over is used for Dy t : They test linearity against STAR assuming that the transition variable is a lagged twelve-month difference of the unemployment rate. The alternative (STAR) model has the following form (the lag length has The lag order is also identical to that selected by Auerbach and Gorodnichenko (2012). To test serial correlation in the errors of the specified VSTEC model, we applied the multivariate Ljung-Box test and failed to reject the null of no serial correlation at 5% level of significance. Testing results are available from the authors upon request. 4 The data set used by van Dijk, Ter€ asvirta, and Franses (2002) is available at <http://swopec.hhs.se/hastef/abs/hastef0380. htm> that was originally retrieved from the Bureau of Labor Statistics.
been determined by AIC): h 17þk d t, k 2 4 3 5 f ðD 12 y tÀd , cÞ þ u t , where y t is the monthly US unemployment rate in question; Dy t is the first difference of y t ; f ðÁ, ÁÞ is a nonlinear transition function; D 12 y t is the twelve-month difference of y t ; d t, k is the dummy for month k; and u t $ IID ð0, r 2 Þ: The twelve-month difference D 12 y tÀd is not included as an explanatory variable in the null (linear) model. The theory in Section 2 can nonetheless be used without modification as a null model including D 12 y tÀd can be thought of having a zero coefficient for this variable. Following van Dijk, Ter€ asvirta, and Franses (2002), we test linearity by letting D 12 y tÀd , d ¼ 1, 2, :::, 6, be the transition variable. Our test results using the same series as van Dijk, Ter€ asvirta, and Franses (2002) are reported in the top panel of Table 3. Both the LM tests and QLR L n reject linearity when d ¼ 2, and, besides, LM 3, n that has power against LSTAR yields p ¼ 0.037 for d ¼ 2. The p-values of QLR L n , however, lie at or below 0.05 for all six lags, suggesting that at least in this particular case this QLR test is more powerful than the LM tests. The smallest p-value is even here attained for d ¼ 2. The results from QLR E n are quite different in that they reject the null only for d ¼ 1, 2, but not for other lags. This makes sense as this statistic is designed for ESTAR, and asymmetry in the unemployment rate is best described by an LSTAR model.
The bottom panel of Table 3 contains the results from the series extended to August 2015. 5 Now there seems to be plenty of evidence of asymmetry: all p-values of LM 1, n are rather small. LM 3, n also has small values for the first three lags, as has LM 2, n : The p-values from QLR L n are smallest of all, which is in line with the results in the top panel. Even QLR E n rejects the null of linearity at the 5% level for d ¼ 1, 2, 3, 4, 5: This outcome may be expected as the QLR statistics are omnibus tests and as such respond to any deviation from the null hypothesis as the sample size increases. Note, however, that even LM 4, n now yields two p-values (d ¼ 2, 3) that lie below 0.05, although the test does not have the omnibus property. The behavior of the unemployment rate during and after the financial crisis (a quick upswing and slow decrease) has probably contributed to these results.

Conclusion
The current study examines the null limit distribution of the QLR test statistic for neglected nonlinearity using the STAR model. The QLR test statistic contains a twofold identification problem under the null, and we explicitly examine how the twofold identification problem affects the null limit distribution of the QLR test statistic. We show that the QLR test statistic is shown to converge to a functional of a multivariate Gaussian process under the null of linearity by extending the testing scope of the LM test statistic in Luukkonen, Saikkonen, and Ter€ asvirta (1988), Granger and Ter€ asvirta (1993), and Ter€ asvirta (1994).
Finally, two empirical examples are revisited to demonstrate use of the QLR test statistic. We test for neglected nonlinearity in the multiplier effect of US government spending and the growth rates of US unemployment using the QLR test statistic by revisiting the empirical data examined by van Dijk, Ter€ asvirta, and Franses (2002) and Auerbach and Gorodnichenko (2012), 5 The recent observations of the monthly US unemployment rate are available at <http://beta.bls.gov/dataViewer/view/ timeseries/LNU04000025>. respectively. Through these examinations, the QLR test statistic turns out useful for detecting the nonlinear structure among the economic variables and complements the Lagrange multiplier test statistic in Ter€ asvirta (1994).