Orthogonal Series Estimation In Nonlinear Cointegrating Models . - Monash

(c) Let xt xt 1 vt with x0 OP (1). Let et P j 0 j φj .(d) For any given u R, define h(u) P j 0φj t j with φ0 1,P j 0φj 6 0 andϕ0 (u).ϕ(u)Suppose that there is a nonnegative function k(λ)R Qsuch that maxj 0 h(λφj ) k(λ) and k(λ) Γ(λ) dλ , where Γ(λ) i 0 ϕ(λ φi )is the characteristic function of et .Condition (a) shows the requirement for the underlying process { j , j Z} that determinesthe properties of the regressor and the error term. The moment conditions are commonly usedin the literature. The integrability of λϕ(λ) in (1) is about to derive some properties forthe density functions related to xt , and the condition for h(u) is satisfied in many cases, suchas symmetric stable variables with α [1, 2], in which h(u) C1 uα 1 and k(u) C2 uα 1for some finite C1 and C2 . Meanwhile, Assumption 1(d) is also satisfied with the case whereφ(u) 2/(eu e u ) and then h(u) e u eueu e uand k(u) 1 (Lukacs, 1970, p.88).The regressor xt is integrated by the linear process vt , while the linear processes vt and ethave the same i.i.d. sequence { j , j Z} as building blocks. The endogeneity of the structuralcointegration model is incurred accordingly. While the same type of endogeneity is used inWang and Phillips (2014) for the kernel estimation method, the estimation method as well asthe establishment and the proof of the main results in this paper are quite different from thekernel method case.By the Beveridge-Nelson decomposition (Phillips and Solo, 1992, p. 972), vt ψ t ṽt 1 ṽtP P where ṽt k j 1 ψk . Note that ṽt is a stationary process sincej 0 ψ̃j t j with ψ̃j P 2j 0 ψ̃j due to (b) of Assumption 1. A similar condition is used in Phillips and Solop P(1992). It follows that xt ψ tj 1 j ṽ0 ṽt and hence dt : Ex2t ψ t(1 o(1)).Define, for 0 u 1,WT (u) 1x[T u] .dT(2.7)It is known that WT (u) D B(u), a standard Brownian motion. Straightforwardly, E[xt et ] PPtψ j 1 φt j E[ṽ0 et ] E[ṽt et ] where E[ṽ0 et ] t 1 and E[ṽt et ] j 0 ψ̃j φj is a constant. 1Generally, E[xt et ] 6 0 but E[d 1t xt et ] 0 as t . This implies dt xt and et are asymptoti-cally uncorrelated for large t. More importantly, in Lemma A.5 below we claim that d 1t xt andes are asymptotically independent for all large t and s. Our asymptotic theory is built uponthe asymptotic independence.Meanwhile, our asymptotic theory relies on the local time process LB (t, s) of B(u) definedby1LB (t, s) limε 0 εZtI{ B(u) s ε}du,(2.8)0where I(A) denotes the conventional indicator function. Roughly speaking, the local timecan be interpreted as a spatial occupation density in s for Brownian motion B(u). The localtime is a key tool in studying the intersection of nonlinearity and nonstationarity, e.g., Park5

and Phillips (1999, 2001), Wang and Phillips (2009a). Phillips (2001) provides some exampleswhere the tool of local time can be used to analyse economic time series which is called “spatialanalysis of time series”.Assumption 2. Let f (x) L2 (R) be differentiable. Moreover, there exists a positive integerr such that xi f (r i) (x) L2 (R) for all i 0, · · · , r.Assumption 2 requires that f (x) is sufficiently smooth with the thin tail such that theorthogonal expansion converges with a fast rate. See Lemma A.3 in Appendix A. The sameassumption in a different form is used by Lemma 3 of Schwartz (1967). The classes of f includesGaussian functions, Laplace functions and functions with compact support.Assumption 3. Let the truncation parameter p of the Hermite series expansion satisfy p [c · T α ] where c 0 is a constant and12(r 1) α 15 .Assumption 3 restricts the truncation parameter p to guarantee the convergence of theregression matrix Z τ Z and the smoothness order r to ensure the truncation residue γp (·) doesnot affect the limit distribution studied below. The condition for r and α also implies r 72 ,which can be satisfied by r 4 in Assumption 2.3Asymptotic theory3.1Consistency of series estimatorIn this subsection, we discuss the asymptotic consistency of the series estimator.Lemma 3.1. Under Assumptions 1–3, we have as T , kθb θk oP (1), and supx fb(x) f (x) oP (1).Lemma 3.1 shows that the estimated coefficients converge to the true coefficients and theseries estimator fb(x) for f (x) has a uniform convergence.When data are stationary time series, polynomials or splines are usually used as basisfunctions, e.g., in Andrews (1991), Newey(1997), and Gao (2007). In their cases, the uniformconsistency is usually based on more restrictive assumptions than those for the point–wiseconsistency. By contrast, in nonparametric and nonstationary context it is very difficult, if notimpossible, to obtain a uniform convergence on the entire real line using kernel method. Gaoet al. (2009), Chan and Wang (2014) and Wang and Chan (2014) study the uniform convergencethat, however, happen in a compact domain of the real line. In our study, due to the uniformboundedness of Hermite series, the uniform consistency requires the same conditions as those forthe point–wise consistency. This is one of advantages that series estimation has in comparisonwith kernel estimation.6

3.2Asymptotic distributionIn this subsection, we shall establish asymptotical distribution for the series estimator. Thereare two kinds of approximation of fb(x) to f (x): one is pointwise, fb(x) f (x) Z τ (x)(θb θ) p2γp (x) for any x R; another one is in the L -sense, kfb(x) f (x)k2L2 (R) kθb θk2 kγp (x)k2L2 (R) , 1/2R 2where by definition kg(x)kL2 (R) g (x)dxthe norm of g(x) L2 (R). The followingtheorem gives the asymptotic distribution of fb(x) in both the pointwise and L2 -norm sense.Theorem 3.1. Under Assumptions 1–3, we have as T s LB (1, 0) T bf (x) f (x) D N (0, 1),σe2 kZp (x)k2 dTand moreovers1 T b 1/2kf (x) f (x)kL2 (R) D LB (1, 0),· p dTσe2(3.1)(3.2)where LB (1, 0) is a local–time random variable with its cumulative distribution function beinggiven by 2Φ(x) 1, x 0,FL (x) P (LB (1, 0) x) 0,x 0,(3.3)in which Φ(x) is the cdf of N (0, 1).Since kZp (x)k2 O(p) uniformly in x and dT O(T 1/2 ), the rates of convergence of theseries estimator in both pointwise and L2 -norm sense are (T 1/4 p 1/2 ) 1 . Meanwhile, the rate ofconvergence of the kernel estimator is (T 1/4 h1/2 ) 1 (see, for example, Wang and Phillips 2014),where h is the bandwidth parameter. Thus, they are equivalent when we replace h by p 1 .Note also that there are three nuisance parameters involved in the large sample theory of (3.1), namely, ψ in dT ψ T (1 o(1)), σe2 and the local time LB (1, 0), which should bereplaced by their consistent estimates. However, noting the structure of LB (1, 0)/dT in (3.1) andPthe limit dTT Tt 1 φ(xt ) P LB (1, 0) in a rich probability space where φ(x) 12π exp( x2 /2),Pwe may estimate the ratio of LB (1, 0)/ ψ by 1T Tt 1 φ(xt ). Moreover, we estimate σe2 byσbe2T1X 2: eb ,T t 1 twhere ebt : yt fb(xt ).(3.4)It is also possible to estimate ψ individually if we stipulate a parametric structure forthe linear process vt in Assumption 1. See Dong and Gao (2014) for the details. Thus, inpractice the limit in (3.2) can also be used for inference by noting that LB (1, 0) follows thesame distribution as N where N is a standard normal variable. Nonetheless, we focus onlyon (3.1) since the limit is normal and it does not need an estimate of ψ.7

Corollary 3.1. Under Assumptions 1–3, we have as T , σbe2 P σe2 , 1TPTt 1φ(xt ) PLB (1, 0)/ ψ ; consequently,1σbe kZp (x)kvu T uXtbφ(xt ) f (x) f (x) D N (0, 1).(3.5)t 1The proofs of Lemma 3.1, Theorem 3.1 and Corollary 3.1, which are given in Appendix B,employ an asymptotic approximation of the regression matrix Z τ Z by a diagonal matrix listedin Theorem 3.2 in the next subsection.3.3Asymptotic property of Z τ ZAs mentioned in the introductory section and seen in the above discussion, the least squaresestimator of θ involves an inverse matrix of Z τ Z, which causes both theoretical and computational difficulties. In the literature, such difficulties are avoided through using a transformedversion of θb of the form θe Z τ Z · θb (see, for example, Dong and Gao 2014). As a consequence, itb although a rate of convergence of θe is available.is difficult to obtain a rate of convergence for θ,Therefore, we tackle this difficulty by studying the convergence ofdTTZ τ Z directly.Theorem 3.2. Let p [c · T α ] for c 0 and 0 α 51 . Suppose that Assumption 1 holds.Then, in an expanded probability space, we have as T dT τZ Z LB (1, 0)Ip P 0,T(3.6)where Ip is an identity matrix of dimension p p.It follows from the definition of Z thatdT τZ Z LB (1, 0) IpT2 p 1Xi 0TdT X 2F (xt ) LB (1, 0)T t 1 i!2 p 1Xi6 j 0TdT XFi (xt )Fj (xt )T t 1!2.Since p , existing results (Wang and Phillips 2009a, 2011, for example) regarding all termsin the bracket are not applicable. Thus, the proof of Theorem 3.2 is not trivial because the keysteps used in deriving the rates of convergence for the terms in the bracket use new ideas andvarious properties about the orthogonal series.As frequently encountered in the nonparametric nonstationary series estimation context,Theorem 3.2 is of independent interest. The implication is that the regression matrix Z τ Zfor the parameterized model after normalization is asymptotically a diagonal matrix withLB (1, 0) at its diagonal, and hence the eigenvalues satisfy λmin ( dTT Z τ Z) LB (1, 0) oP (1)and λmax ( dTT Z τ Z) LB (1, 0) oP (1). Our experience suggests that such convergence itselfmay be applicable to significantly simplify the construction of existing estimation and specification procedures, such as those discussed in Dong and Gao (2013, 2014).The proof of Theorem 3.2 is given in Appendix C of the supplementary material. In Section4 below, we examine the finite–sample performance of the series estimation.8

4Simulation studyIn this section, we conduct Monte Carlo experiments to assess the finite sample performanceof the proposed nonparametric series estimator. The data generation procedure is as follows.Let { t , et } be an independent and identically distributed sequence, { t , et } N (0, Σ) with Σ 0.12 ρ1 ρ1 . The regressor xt is integrated by an AR(1) process vt , i.e.xt xt 1 vtand vt 0.2 vt 1 t ,where x0 OP (1). The following models are used to investigate the performance:1 et , t 1, ., T ;1 x4tModel 2 : yt (1 sin(xt )) exp( x2t /2) et ,Model 1 : yt (4.1)t 1, ., T.(4.2)We shall consider two cases for ρ: ρ 0, implying the case of exogeneity, and ρ 0.9, implyingthe existence of endogeneity.4.1Bias and standard deviationLet T 400, 800, 1200 and 1800 be the sample sizes. The number of replications is 2000.Using a generalised cross–validation method proposed in Gao et al. (2002), the truncationparameter is chosen as p [2 · T 1/8 ] such that it varies along with the sample size and satisfiesthe theoretical requirement in Assumption 3.The sample bias, standard deviation (Std) and root mean square error (RMSE) are definedbyNT1 1 XXBias f (xn,t ) fb(xn,t ) ,N T n 1 t 1!1/2Std NT 21 1 XX bbf (xn,t ) f (xn,t )N T n 1 t 1!1/2RMSE TN 21 1 XX bf (xn,t ) f (xn,t )N T n 1 t 1,,respectively, where (xn,1 , · · · , xn,T ) denotes the simulated data in n th replication, and by which fb(·) is the series estimator of the regression function, and fb(·) Zp (·)τ θb with θb being theaverage of θbn over Monte Carlo replications, and N is the number of replications. The resultsof the simulation are summarised in Table 1.It should be pointed out that the sample size of simulation for nonstationary integrableregression models usually has to be much larger than that for stationary regression models.The reason is the slower rate of convergence in the former case.It can be seen from Table 1 that both the bias and the standard deviation decrease withthe increase of the sample size. These verify the approximation of the proposed estimator9

Table 1: Simulation Results for Bias, Std and RMSEρ 0BiasStdRMSEρ 0.9TModel 1Model 2Model 1Model 06618000.02520.00610.02470.0054to the true regression function. Comparing the results of the two models, however, Model 2outperforms Model 1 in all sample sizes. According to our experience this may be mainly dueto the difference in the tails of two regression functions, namely, the tail of 1/(1 x4 ) is muchheavier than that of (1 sin(x)) exp( x2 /2). It is known that the heavier tail results in a slowerconvergence of the orthogonal series expansion. Consequently, Model 2 has better results thanModel 1.Additionally, the results for the case of ρ 0 and the case ρ 0.9 have no evidenceto show how they are different. Based on these results, the endogeneity does not affect thenonparametric estimate in the proposed models, and more importantly, this coincides with ourtheoretical findings in the preceding section.4.2Normal approximation and confidence interval curvesCorollary 3.1 gives the normality of our estimator fb(x) with all nuisance parameters estimatedby the observation {(xt , yt ), t 1, · · · , T }. Accordingly, we are able to construct the confidenceinterval at a significance level and any point. This section devotes to the visualization of thenormality.To do so, using ksdensity function in MatLab we first estimate the density of a set offb(x ) f (x ) with normalization according to Corollary 3.1 for a particular point x 0 for10

Model 2 with T 200, 400 and 800 for ρ 0 and N 1000, where the truncation parameteris taken using the same formula as before, viz., p [2 T 1/8 ].Figure 1: Normal density approximation and confidence interval curves1.60.4T 200T 400T 800N(0,1)0.35true functionfitted functionupper boundlower bound1.41.20.310.250.80.20.60.150.40.10.20.050 40 3 2 101234 0.2(a) Normal density approximation 2 10123(b) Confidence interval curvesTechnically, we only use the replications that both the numbers of observations less thanand larger than zero are greater than 0.2 T . The reason is that, due to the divergence ofthe integrated data, it is possible that the generated data (xn1 , · · · , xnT ), where n correspondsto the n–th replication of the total number of replications, in one replication may be locatedmostly in one side of zero, which definitely gives a poor estimation of the density, particularlyfor the kernel method of ksdensity function in Matlab. Similar discussion is available in Section5 of Karlsen et al. (2007).Figure 1a shows three estimated density curves corresponding to the different sample sizeT . It can be seen that the densities gradually approach to the standard normal density withthe increase of the sample size. We may conclude that the theoretical result of the normalityin Corollary 3.1 is verified in this experiment.Second, for significance level 95%, we draw for Model 2 the lower bound and upper boundPconfidence curves based on the result of (3.5), namely, fb(x) 1.96 σbe kZp (x)k( T φ(xt )) 1/2 ,t 1where φ(·) is the density function of a standard normal variable. Here, T 800 and p isthe same as before. Figure 1b displays the true regression function, the estimated functionaveraging over replications and the confidence interval curves. As can be seen, the estimatedcurve fb(x) is located exactly between the lower bound and the upper bound, implying thereliability of inference based on our estimator.5DiscussionIt is worthy to discuss potential extensions of our method to deal with models where regressionfunctions are not in L2 (R). The following is a brief discussion on this issue. Consider yt 2f (xt ) et where xt and et still satisfy Assumption 1 but f (x) L2 (R, e x ). It follows that2f (x) : f (x)ϕ(x) L2 (R) where ϕ(x) e x /2 . This motivates multiplying the both sides of11

the model by ϕ(xt ), givingỹt f (xt ) ẽt ,t 1, · · · , T,(5.1)where ỹt yt ϕ(xt ), and ẽt et ϕ(xt ). Now, model (5.1) is completely the same as model (2.4).Expand f (x) into orthogonal series in terms of {Fi (x)}:f (x) Xθ̃i Fi (x) Zpτ (x)θ̃Z γ̃(x),with θ̃i f (x)Fi (x)dx(5.2)i 0Pwhere for any p 1, θ̃ (θ̃0 , · · · , θ̃p 1 )τ , Zp (x) is the same as before and γ̃p (x) i p θ̃i Fi (x).Suppose further that f (x) and truncation parameter p satisfy Assumptions 2 and 3. We areable to have an estimator of f (x) following exactly the same procedure as in Section 2.2,fb̃(x) Zpτ (x)b̃θ,where b̃θ (Z τ Z) 1 Z τ Ỹ ,(5.3)in which b̃θ is an estimate of θ̃, Z is the same as before and Ỹ (ỹ1 , · · · , ỹT )τ . Denote for lateruse that ẽ (ẽ1 , · · · , ẽT )τ and γ̃ (γ̃p (x1 ), · · · , γ̃p (xT ))τ .To derive the asymptotic distribution of fb̃(x), notice that, for any x R, fb̃(x) f (x) θ θ̃) γ̃(x) andZ τ (x)(b̃pkfb̃(x) f (x)k2L2 (r) Zθ θ̃k2 [fb̃(x) f (x)]2 dx kb̃Zγ̃p2 (x)dxZ 2θ θ̃)γ̃p (x)dxZpτ (x)(b̃ kb̃θ θ̃k2 kγ̃(x)k2L2 (R) .Hence, following a similar fashion we may be able to establish the asymptotic distribution offb̃(x) in both the point–wise and the L2 sense.Meanwhile, it is possible to extend the approach in Sections 2 and 3 to a partially linearsingle–index model of the form: yt xτt β0 f (xτt θ0 ) et , where xt is a vector of integrated timeseries, (β0 , θ0 ) is a vector of unknown parameters and f (·) is an unknown integrable function.In empirical applications, a vector of macro-economic time variables, such as the income andreal interest rate variable, may be chosen as xt and yt can be the expenditure variable whenare interested in establishing the relationship between yt and xt . In order to establish similarresults to Theorem 3.1 and Corollary 3.1, some new techniques may be needed. We thereforewish to leave such extensions to future research.6ConclusionsIn this paper, we have established the uniform consistency and asymptotic distribution inboth the point–wise and L2 sense for the Hermite series estimator of the proposed integrablecointegration model accommodating endogeneity. The endogeneity is of a general form. Possibleextensions from integrable models to non-integrable models have been discussed. The finitesample experiments show that the proposed series estimator performs well for models satisfyingour assumptions.12

Nonetheless, there are some problems that may be studied in our future research. The choiceof the truncation parameter should be discussed in more detail and a data driven choice of thetruncation parameter should be investigated. The theory may be extended to an additivemultivariate model with both stationary and nonstationary regressors or a partially linearcointegration model.7AcknowledgementsThe original version of this paper was presented at a lunch time seminar in the Department ofEconometrics and Business Statistics, Monash University, Australia. The first author acknowledges some useful comments by the audience. The authors also acknowledge comments byFarshid Vahid and the financial support by the Australian Research Council Discovery GrantsProgram under Grant Numbers: DP1096374 and DP1314229.ALemmasFive useful lemmas are given in this section. All their proofs can be found in Appendix C of thispaper. Throughout the rest of this paper, we use 0 C to denote a generic constant which mayhave different values at different places. Meanwhile, we use · L2 to simplify · L2 (R) in the proofs.We shall consider several versions of decomposition for xt . Without loss of generality, in whatfollows let x0 0 almost surely. It follows thatxt tX 1v t XXψ i i 1 i tX tXψ i i : i tXbt,i i .i max(1,i)Let j t be fixed. Thus we havext bt,j j xt/j ,with xt/j : tXbt,i i ,(A.1)i ,6 jwhere xt/j is the variable deducting the term containing j in xt . Obviously, xt/j and j are mutuallyindependent.Additionally, letting 1 s j t, xt also has the following decomposition:xt x s xts x s bt,j j xts/j ,(A.2)PPwhere x s xs x̄s with x̄s ti s 1 sa ψi a a containing all the information available up toPtPts and xts i s 1 bt,i i , while obviously xts/j i s 1,6 j bt,i i . Evidently, xts captures all theinformation contained in xt on the time periods (s, t], wh

Phillips (1999) discuss asymptotics for nonlinear transformation of unit root process and Park and Phillips (2001) for nonlinear regression with a unit root process. Furthermore, asymptotic properties for nonparametric estimation for nonlinear cointegration models have been derived by Wang and Phillips (2009a,b).