Department of Econometrics and Business Statistics

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1 ISSN X Australia Department of Econometrics and Business Statistics ttp:// Bayesian Bandwidt Estimation in Nonparametric Time-Varying Coefficient Models Tingting Ceng, Jiti Gao and Xibin Zang February 2015 Working Paper 03/15 revised 07/13

2 Bayesian Bandwidt Estimation in Nonparametric Time Varying Coefficient Models 1 Tingting Ceng, Jiti Gao, 2 Xibin Zang Department of Econometrics and Business Statistics, Monas University, Australia Abstract: Bandwidt plays an important role in determining te performance of nonparametric estimators, suc as te local constant estimator. In tis paper, we propose a Bayesian approac to bandwidt estimation for local constant estimators of time varying coefficients in time series models. We establis a large sample teory for te proposed bandwidt estimator and Bayesian estimators of te unknown parameters involved in te error density. A Monte Carlo simulation study sows tat i te proposed Bayesian estimators for bandwidt and parameters in te error density ave satisfactory finite sample performance; and ii our proposed Bayesian approac acieves better performance in estimating te bandwidts tan te normal reference rule and cross validation. Moreover, we apply our proposed Bayesian bandwidt estimation metod for te time varying coefficient models tat explain Okun s law and te relationsip between consumption growt and income growt in te US. For eac model, we also provide calibrated parametric forms of te time varying coefficients. Key words: Local constant estimator, bandwidt, Markov cain Monte Carlo. JEL Classification: C11, C14, C15 1 We tank participants at several seminars and conferences, during wic earlier versions of tis paper were presented, for teir constructive comments and suggestions, particularly to Zongwu Cai, David Frazier, Han Hong, Maxwell King, Robert Kon, Gael Martin, Adrian Pagan, Anastasios Panagiotelis, Peter Pillips, Peter Robinson and Farsid Vaid. Tis researc is supported under te Australian Researc Council s Discovery Projects Sceme under Grant Numbers: DP , DP and DP Correspondence: Jiti Gao, Department of Econometrics and Business Statistics, Monas University, Caulfield East, Victoria 3145, Australia. Telepone: Fax: jiti.gao@monas.edu. 1

3 1 Introduction Time varying coefficient time series models ave attracted muc attention of econometricians and statisticians during te last two decades, following te publication of Robinson Recent studies tat are most relevant to our work include Gao and Hawtorne 2006 for a semiparametric modeling of a temperature trend function, Cai 2007 for a nonparametric trending time series model, Li, Cen and Gao 2011, and Cen, Gao and Li 2012 for semiparametric trending panel data regression and its applications in modeling economic, financial and climatological data. An example tat illustrates te importance of suc models in finance is te capital asset pricing model CAPM. A traditional CAPM usually assumes a constant linear relationsip between an asset s return and a market portfolio s return, and suc a relationsip is reflected by te beta coefficient. However, some recent studies sow tat te beta coefficients migt vary over time see for example, Jagannatan and Wang, 1996; Gysels, 1998; Wang, In tis paper, our investigation is focused on a time varying coefficient model given by y t x t βτ t + u t, t 1,2,,n, 1 were x t x t1, x t2,, x tk is a stationary time series, βτ t β 1 τ t,β 2 τ t,,β k τ t are unknown functions of τ t t n, and u t follows a stationary AR1 process u t αu t 1 + v t, in wic α α < 1 is an unknown parameter, {v t } is a sequence of independent and identically distributed i.i.d. continuous random errors wit a parametrically known density function f v;η caracterized by a parameter vector η, a vector of unknown parameters, E η [v 1 ] 0 and 0 < E η [v1 2 ] <. It is also assumed tat x t and u t are mutually independent. Te unknown coefficient function βτ t can be estimated by local constant metod. It is generally accepted tat te performance of te local constant estimator is mainly determined by te coice of bandwidt. Robinson 1989 proposed coosing bandwidt troug te cross validation CV metod, but even in some simple settings, cross validation may perform poorly and exibit a large magnitude of sample variation see Fan, Heckman and Wand, 1995, among oters. Cai and Tiwari 2000, Cai 2002 and Cai 2007 proposed a nonparametric version of Akaike information criterion AIC to select bandwidt for local linear estimators. Tis metod aims to derive an optimal bandwidt tat minimizes a measure of penalized logaritmic mean 2

4 squared error, and tus, is similar to te CV. A rule of tumb metod for coosing bandwidt is te normal reference rule NRR defined as 1.06σn 1/5, 2 were σ is te population standard deviation and is replaced by its sample measure in practice. It is a roug metod for coosing bandwidts and is used in te absence of no oter practical metods. In te current literature, tere exist some investigations on te advantages of Bayesian sampling approaces to bandwidt estimation against some competing metods; see Zang, King and Hyndman Motivated by recent investigations on sampling algoritms for bandwidt estimation, we propose a Bayesian metod for bandwidt estimation in te time varying linear models given by 1, were te coefficients are estimated by local constant estimation metod. Te idea of using Bayesian approac to bandwidt selection is not new. However, to te best of our knowledge, tere is no study available to provide teoretical support for tis metod. Terefore, we aim to fill tis gap in te literature by establising a large sample teory for Bayesian bandwidt estimators. In addition, we contribute to te current literature by investigating te large sample beavior of Bayesian estimators for parameters in te error density as te relevant studies in te current literature focus on te asymptotic beavior of posterior distribution rater tan investigating posterior mean. For example, Walker 1969 sowed tat under suitable regularity conditions, as n, te posterior distribution converges to a normal distribution. Based on Walker 1969, Cen 1985 furter introduced tree sets of conditions for asymptotic posterior normality. Pillips and Ploberger 1996 developed an asymptotic teory of Bayesian inference for stationary and nonstationary time series and provided te limiting form of te Bayesian data density for a general case of likelioods and prior distributions. Kim 1998 considered posterior normality in te situation of nonstationary time series. Te contribution of tis paper is summarized as follows. We propose a Bayesian approac to bandwidt estimation for local constant estimators of time varying coefficients in time series models. We establis a large sample teory for a Bayesian bandwidt estimator, and tis fills a gap 3

5 in te current literature. We also provide a large sample teory for Bayesian estimators of a vector of unknown parameters tat caracterize te error density function. We examine te finite sample performance of te proposed Bayesian approac to parameter estimation troug simulation studies. We compare te performance of our Bayesian bandwidt estimator in estimating time varying coefficient functions wit cross validation and normal reference rule troug simulation studies and empirical applications. Te rest of tis paper is organized as follows. Section 2 briefly describes te local constant estimators of time varying coefficients in te time series regression model and presents te large sample teory for te proposed Bayesian estimators of bandwidt and parameters in te error density. In Section 3, we present Monte Carlo simulation studies to examine te finite sample performance of our proposed metod for bandwidt estimation and evaluate te accuracy of estimated coefficient function wit our proposed Bayesian bandwidt estimator. In Section 4, two empirical examples are presented to illustrate te application of our proposed Bayesian metod for bandwidt estimation. Section 5 concludes te paper. Te key assumptions and an outline of te proof of te main results are given in Appendix A. A nonparametric specification testing procedure is detailed in Appendix B. Meanwile, te full proofs of te main results are given in Appendix C of a supplementary document. 2 Large sample teory for Bayesian bandwidt estimation In tis section, we briefly describe te local constant estimator of te time varying coefficient function in te time series model. We ten establis large sample properties for te proposed Bayesian estimator of te bandwidt involved in a local constant kernel metod, and for te unknown parameters tat caracterize te error density. 2.1 Local constant kernel metod Te principle idea of local constant estimation metod is tat if βτ is continuous, it beaves like a reasonable constant over its small neigbourood. So we approximate βτ by a constant a in 4

6 te neigborood of τ and minimize a locally weigted sum of squares to obtain a local constant kernel estimator, βτ;, of te form: βτ; argmin a y t xt a2 K τ t τ, 3 were K u K u//, K is a kernel function and > 0 is te bandwidt satisfying 0 and n as n. Solving 3, we can obtain tat 1 βτ; x t xt K τ t τ x t y t K τ t τ. 4 Note tat it is also possible to use a local linear kernel metod. Since bot te establisment and te proof of te main results based on te local constant kernel metod are already very tecnical, we just use te local constant kernel metod in tis paper. From 4, it is clear tat we need to specify a kernel function K and coose a bandwidt before we may study and compute te estimate of βτ. It is generally accepted in te current literature tat te coice of K is less important tan tat of. In tis paper, we propose a Bayesian metod to estimate te bandwidt, for wic we will establis a large sample teory in te following section. 2.2 Bayesian bandwidt estimation Denote X n x 1, x 2,, x n and Y n y 1, y 2,, y n, were x 1, x 2,, x n are observed values of te regressor and y 1, y 2,, y n are observed values of te response in 1. Let θ α,η. For notational simplicity, we will use f v;θ to denote f v;η trougout te rest of tis paper. Let a n λ wit {a n } being a sequence of real numbers satisfying a n 0 as n and λ being a random variable. Since {v t } is assumed to be i.i.d., we introduce te following likeliood function: n n L n λ,θ f v t ;θ f û t αû t 1 ;θ, were û t y t x t βτ t ; in wic β ; is being defined in 4. Te logaritm of likeliood function is ten l n λ,θ logl n λ,θ. Let π λ λ be te density function of λ and π θ θ be te density function of θ. Ten, we estimate λ,θ by E [ λ,θ X n,y n ] λ,θ e l nλ,θ π λ λπ θ θdλdθ e l n λ,θ π λ λπ θ θdλdθ E [λ X n,y n ] E [θ X n,y n ]. 5

7 Since te Bayesian estimates E [λ X n,y n ],E [θ X n,y n ] do not ave closed form expressions, we terefore approximate tem by λ mn and θ mn, respectively, were λ mn 1 m m j 1 λ j n and θ mn 1 m m j 1 θ j n, in wic λ j n and θ j n denote te respective j t posterior draws and m denotes te number of Markov cain Monte Carlo MCMC iterations. Consequently, we can obtain te Bayesian bandwidt estimator ĥmn by ĥmn a n λmn. Before we establis an asymptotic distribution for λ mn, we introduce te following notation. Let γv t ;θ f 1 v t ;θ f v t ;θ were f 1 v t ;θ is te first derivative of f v t ;θ. Let γ n v t E[θ] θγv t ;θa n θπθdθ and φ n v t γv t ;θa n θπθdθ wit A n θ e ne θ [log f v 1 ;θ] e ne θ [log f v 1 ;θ] π θ θdθ and E θ [log f v 1 ;θ] log f v 1 ;θf v;θdv. Denote G n θ n f v t ;θ and g n θ logg n θ. Let f n θ X n,y n G n θπ θ θ/ G n θπ θ θdθ and E[θ X n,y n ] θ f n θ X n,y n dθ. Let θ n denote te maximum likeliood estimator of θ and n θ n g n 2 θ n, were g n 2 θ n is te second derivative of g n evaluated at te maximum likeliood estimator θ n. Denote C n { θ : 1/2 n θ n θ n θ < c } and D n { θ : 1/2 n θ n θ n θ c }, were c is a positive constant. Let β i τ denote te i t order derivative of βτ. Trougout tis paper, we use E θ [Z ] to denote te conditional expectation of Z given θ. Let Σ x E [ x 1 x1 ], wt xt Σ 1 x x t, d 1n E[v 1 φ n v 1 ], d 2n E[v 1 γ n v 1 ], b 1n a 1 n E[w 1] d 1n, b 0n 1 K 0E[λ 1 ]b 1n and δ 2 n E[w 2 1 ] E [ v1 φ n v 1 d 1n 2 ]. Let d 1 K 0 1 E[λ]E[λ 1 ]. We ten establis te main asymptotic distributions in Teorems 1 3 below. Teorem 1. Let Assumptions 1 4 listed in Appendix A1 old. If, in addition, n a2 n +d 2 1n m δ 2 n m and n, ten we ave 0 as n δ 1 n a n b0n λmn E[λ] d 1 b 1n D N 0,d 2 1. Remark 1: Teorem 1 establises an asymptotic normality for λ mn wit a rate of convergence n δ 1 n a n. Note tat Teorem 1 implies tat te rate of convergence for te bandwidt estimator is n δ 1 n. Note also tat b 1n disappears asymptotically wen an 1d 1n 0 as n. In tis case, λ mn is an unbiased estimator of E[λ] wit a rate of n δ 1 n a n. Wen d 1n C φ 0 for C φ <, λ mn is a biased estimator of E[λ] wit a rate of n δ 1 n. Teorem 2. i Let Assumptions 1 3 and 5 listed in Appendix A1, If, in addition, na 2 n n θ n σ 2 n P 0 and 6

8 n a 2 n +d 2 2n m σ 2 n 0 as m and n, ten we ave n σ 1 n a n θmn E[θ] b 2n D N 0,K 2 0E 2 [λ 1 ]I p, were b 2n a 1 n K 0E[λ 1 ] E[w 1 ] d 2n, σ 2 n E[w 2 1 ]E [ v1 γ n v 1 d 2n 2 ], I p denotes te p p identity matrix, and p denotes te number of parameters in te vector θ. ii Let Assumptions 1i, 2i, 3i and 5iiiiiiv old. If, in addition, γv;η f 1 v;η f v;η does not depend on η, ten as n θ n m P 0, we ave as m and n γv n α n α mn E[α] D N 0,Σ 0, 5 were Σ 0 is a positive definite matrix as defined in Assumption 5iv. Remark 2: i Teorem 2i sows tat asymptotic normality is acievable for Bayesian estimator θ mn wit a rate of convergence of n σ 1 n a n. To te best of our knowledge, tis is a new finding about Bayesian estimation, particularly for a class of posterior means, as te current relevant literature as only establised some asymptotic properties for te posterior distribution; see Walker 1969, Cen 1985 and Kim 1998 etc. Teorem 2i also sows tat te conventional n rate of convergence is acievable wen a n σ n C 0 0. ii As expected, Teorem 2ii sows tat α can be consistently estimated wit te conventional f n rate of convergence wen γv;η 1 v;η γv does not depend on η and n θ n f v;η n α > 0, were α is a positive definite matrix. An asymptotically normal distribution for te local constant estimator βτ;ĥmn wit bandwidt estimated by our proposed Bayesian metod is given in te following teorem. Teorem 3. Let Assumptions 1 4 listed in Appendix A1 old. As m and n, we ave nan E[λ] βτ; ĥ mn βτ D N 0,Σ β, were Σ β σ 2 u K 2 vdv Σ 1 x wit σ 2 u E[u2 1 ]. Remark 3: Teorem 3 sows tat wit our proposed Bayesian bandwidt, an asymptotic normality for te local constant estimator of te coefficient function, β ;, is acievable. We can find tat te local constant estimator as a rate of convergence at an order of n a n, wic is equivalent to te rate of n for te case were is being treated as a fixed bandwidt. 7

9 Before we give an outline of te proof of Teorems 1 3 in Appendix A and ten te full proofs in te supplementary material, we examine te finite sample performance of our Bayesian estimators by simulation studies in Section 3. 3 Simulation Te purposes of te Monte Carlo simulation study are as follows. First, wit 1000 replications under different data generating processes, we examine te finite sample performance of te proposed Bayesian approac to parameter estimation. Second, we compare te performance of our proposed Bayesian approac to bandwidt estimation wit cross validation metod and normal reference rule by computing mean squared error of te estimated time varying coefficient functions. 3.1 Parameter estimation We consider te following time varying coefficient time series model y t x t βτ t + u t, t 1,2,,n 6 were x t 1, x t2, βτ β 1 τ,β 2 τ, β 1 τ 0.2exp τ and β 2 τ 2τ+exp 16τ In order to generate samples, we generate x t2 troug an AR1 model given by x t2 0.5x t 12 + ɛ x wit ɛ x being generated from N 0, We generate te error term of 6 from a stationary AR1 process u t αu t 1 + v t wit α being generated from α ɛ 1, were ɛ 1 comes from a truncated normal distribution wit mean zero and standard deviation 0.5, and values of ɛ 1 are bounded witin 0.4,0.4, and v t is generated independently from a specified parametric distribution. We first consider te following two distributions for v t. [DGP1] Gaussian error v t N 0,σ 2 g : σ g is generated from σ g ɛ 2, were ɛ 2 is generated from te truncated N 0,0.5 2 and are restricted between 0.4 and 0.4. [DGP2] A mixture of two Gaussian distributions: v t wn µ 1,σ wn µ2,σ 2 2, were w is te weigt assigned to te first component density, µ 1 and µ 2 are te mean parameters for tese two component densities, and σ 2 1 and σ2 2 are te variance parameters for tese two component densities. 8

10 w is generated according to w ɛ 3, were ɛ 3 is generated from te truncated N 0,0.5 2 and is restricted witin 0.4,0.4. σ 1 is generated from σ ɛ 4, and σ 2 is generated from σ ɛ 5, were bot ɛ 4 and ɛ 5 are independently generated from te truncated N 0,0.5 2 and are restricted between 0.3 and 0.3. µ 1 is generated according to µ ɛ 6, were ɛ 6 is generated from te truncated N 0,0.5 2 and is bounded between 0.3 and 0.3. µ 2 is computed by solving te equation µ 2 w µ 1 /1.0 w. According to te existing teoretical results in nonparametric estimation, is proportional to n 1/4+p, were p is te dimension of variables, wit wic we are smooting. In tis simulation study, as te unknown coefficient is a function of time, so we set a n n 1/5. Our objective is to coose λ by our proposed Bayesian metod. Based on eac DGP, we can obtain likeliood function. For te coice of prior, we coose inverse gamma prior denoted as IG1,0.05 for 2, σ 2 g, σ2 1 and σ2 2, uniform prior U 0.95,0.95 for AR coefficient α, uniform prior U 0.05,0.95 for weigt parameter w in DGP2 and Gaussian prior N 0,9 for µ 1 in DGP2. Ten by Bayes teorem, we can obtain posterior, from wic we can use Metropolis algoritm to sample parameters. We coose sample size n to be 200, 600 and 1200, respectively. For eac data generating process DGP and eac sample size, we generate 1000 samples. For DGP1, we estimate parameters θ g θ g 1,θ g 2 α g,σ 2 g, were αg and σ 2 g are te AR coefficient and variance of v t, respectively. For DGP2, we estimate parameters θ m θ m1,θ m2 αm,σ 2, m were αm and σ 2 m denote AR coefficient and variance of v t, respectively. Note tat σ 2 m is computed by σ 2 m w µ2 1 + σ w µ2 2 + σ2 2 given tat wµ wµ 2 0. We use our proposed Bayesian metod to estimate θ g and θ m. Te estimates are denoted as θ g θg 1, θ g 2 α g, σ 2 g and θ m θm1, θ m2 αm, σ 2 m. In eac case, λ was also estimated. Since it is involved in, its performance is incorporated in te estimation of te coefficient functions in Section 3.2 below. To examine te finite sample performance of te Bayesian metod, we compute te biases and te mean squared errors MSEs for eac of te components of θ g and θ m as follows: for i g,m 9

11 and j 1,2 Bias i j θ i j r θ i j r and MSE r 1 θ i j r θ i j r 2, were θ i j r is te value of θ i j in te r t replication and θ i j r is te estimated value of θ i j in te r t replication. Te results of parameter estimates are presented in Table 1. From Table 1, we find tat wit te sample size increasing, te bias and te MSE of te AR coefficient and variance of v t in bot DGPs decrease. Tis indicates tat our proposed Bayesian approac to parameter estimation as very good finite sample performance. Table 1: Bias and MSE of parameter estimates based on 1000 replications 1000 r 1 DGP 1 DGP 2 n α g σ 2 g α m σ 2 m Bias MSE Estimation of time varying coefficient functions As our objective is to obtain better estimates for unknown coefficient functions, we examine te performance of our proposed Bayesian metod to bandwidt estimation by cecking te accuracy of β ;. We considered DGP1, DGP2 and te following DGP3: [DGP3] Centralised Ci squared distribution: v t χ For eac DGP, we assume te following structure for te error term. [Case 1] Assume u t αu t 1 + v t and v t N 0,σ 2. [Case 2] Assume u t αu t 1 + v t and v t wn µ 1,σ wn µ 2,σ 2 2. For eac case, we estimate λ and furter get bandwidt by a n λ. We ten compute β ; using te estimated bandwidt. We plotted te median of estimated coefficient functions β 1 ; 10

12 and β 2 ; based on 1000 replications wit sample size 600 in Figure 1. Te plots wit sample size 200 and 1200 are available in Appendix D of te supplementary document. From Figure 1, we find tat under eac DGP, estimated curves for β 1 τ and β 2 τ are very close to te corresponding true function curves. In addition, we measure te accuracy of te estimated bandwidts by computing mean squared error for β 1 ; and β 2 ; separately. MSE β ĥmn n 1000 r 1 βr j τ t ;ĥr mn β j τ t 2, were β r j τ t ;ĥr mn is te estimate of β j τ t for te r t replication and j 1,2 in wic ĥr mn a n λ r mn and ĥr mn is te estimate of bandwidt for te r t replication. We also compare our Bayesian metod wit cross validation CV metod. Te optimal bandwidt by CV can be obtained by solving te following equation. 1 cv argmin n yt xt β 1 τ t ;, were β 1 τ t ; denotes te leave one out local constant estimator of βτ t. After we obtained te optimal cv, we can compute βτ t ;cv and ten get te corresponding mean square error MSE β cv. Table 2: Mean squared errors of β 1 ; and β 2 ;. β 1 ; β2 ; n Bayes 1 Bayes 2 CV NRR Bayes 1 Bayes 2 CV NRR DGP DGP DGP

13 Figure 1: Estimated curves for β 1 τ and β 2 τ under DGP1 3 in Case 1 te first two rows and Case 2 te last two rows wit sample size n 600 DGP1 DGP2 DGP3 Intercept True Bayes Intercept True Bayes Intercept True Bayes Scaled time Scaled time Scaled time Slope True Bayes Slope True Bayes Slope True Bayes Scaled time Scaled time Scaled time Intercept True Bayes Intercept True Bayes Intercept True Bayes Scaled time Scaled time Scaled time Slope True Bayes Slope True Bayes Slope True Bayes Scaled time Scaled time Scaled time 12

14 Table 2 presents te estimation results for coefficient functions in different data generating processes in bot Case 1 and Case 2. Te columns of Bayes 1 and Bayes 2 in Table 2 denote te mean square error of local constant estimator of coefficient functions wit bandwidt estimated by Bayesian approac in Case 1 and Case 2, respectively. Table 2 reveals te following results. Under eac DGP, wit eac bandwidt selection metod, wen te sample size increases, te mean squared error decreases. Under eac DGP, te mean squared error wit bandwidt selected by our Bayesian metod is smaller tan tat wit bandwidt selected by te cross validation metod and normal reference rule. An interesting fact is tat te mean squared error wit te bandwidt selected by te cross validation is larger tan tat wit normal reference rule. Tis is because tat our simulated data are dependent and cross validation metod does not work well if data under consideration are dependent. For DGP1, te mean squared errors in Case 1 and Case 2 are similar. Tis is because tat normal distribution is a special case of mixture normal distribution. For DGP2 and DGP3, te mean squared error in Case 2 is smaller tan tat in Case 1. Tis is reasonable because for DGP2, te mixture of two normal distributions is te correct assumption. It is expected to ave a smaller mean squared error tan its competitor. However, due to te small distance between te two normal components, te difference is not very obvious. For DGP3, as te centralized χ 2 distribution is asymmetric, a single normal distribution cannot capture tis feature. In contrary, te mixture of two normal distributions can. 4 Application 4.1 Okun s law Since te seminal work of Okun 1962, tere ave been many studies tat provide evidence of te correlation between variations in unemployment rate and real output over a business cycle. 13

15 However, many studies in te current literature on Okun s law tend to focus on te lack of robustness of Okun s coefficient witout questioning te constant linear relationsip see for example, Lee, 2000; Silvapulle, Moosa and Silvapulle, In tis section, we employ te time varying coefficient model to demonstrate te time varying feature of Okun s coefficient. Te sample contains yearly paired observations of unemployment rate and GDP of te U.S. over te period from 1948 to Te data were collected from te OECD database. Let y t denote te unemployment rate and x t denote te logaritm of GDP. Define y t y t y t 1 and x t x t x t 1. Okun s law is usually investigated troug te model given by y t α 1 + α 2 x t + u t, were y t is te yearly cange of te unemployment rate, x t is te yearly cange of te log GDP and u t is assumed to follow a stationary AR1 process. Te time series plots of unemployment rate, GDP, cange of unemployment rate and GDP growt rate are presented in Figure 2. Te ordinary least squares OLS estimates of α 1 and α 2 are, respectively, and wit corresponding p values given in te parenteses. We fitted te time varying coefficient model: y t β 1 τ t + β 2 τ t x t + u t to te sample data, were te errors are assumed to follow an AR1 process u t αu t 1 + v t wit v t being assumed to follow a mixture of two normal distributions. Local constant estimates of β 1 τ and β 2 τ were computed wit te bandwidt cosen troug NRR, CV and Bayesian sampling. Te resulting estimates β 1 τ and β 2 τ are presented in Figure 3. From Figure 3, we found tat te local constant estimates of β 1 τ and β 2 τ wit te bandwidt estimated troug Bayesian sampling are clearly different wit tose wit bandwidt selected troug CV and NRR. Te bandwidt selected troug CV tend to over smoot te unknown functions wile te bandwidt by NRR tend to under smoot te unknown functions. Te MSE values of te constant coefficient model and time varying coefficient model are respectively, and Te introduction of time varying coefficient improves model fitting by 21.46% in comparison to te constant coefficient model. In order to examine weter te two coefficients are time varying, we employ te metod proposed by Cai, Fan and Yao 2000 to test te null ypotesis of constant coefficients against te 14

16 Figure 2: Time series plots of unemployment rate, GDP, cange of unemployment rate and GDP growt rate Unemployment rate GDP billion Year Year Cange of unemployment rate GDP growt rate Year alternative of time varying coefficients. Te proposed bootstrap test see Appendix B for details wit 1000 repetitions produced a p value tat is approximately 0. Terefore, we rejected te null ypotesis of constant coefficients in te linear regression model at te 1% significance level. We also derived te 95% point wise confidence intervals of te two time varying coefficients based on te asymptotic results obtained in Cai 2007 and we plotted te confidence intervals in Figure 4. 15

17 Figure 3: Local constant estimates of β 1 τ and β 2 τ wit te bandwidt derived troug NRR, CV and Bayesian sampling Intercept Bayes_mixture CV NRR OLS Year Slope Bayes_mixture CV NRR OLS According to te patterns of te estimated time varying coefficients sown in Figure 3, a piecewise polynomial function of time migt be appropriate to approximate eac coefficient function. Consider te time varying coefficient model given by Year y t β 1 τ t + β 2 τ t x t + u t, 7 were β j τ t, for j 1 and 2, are known form functions of τ t t n given by β 1 τ t τ t I 1 t τ t τ 2 t I 49 t 62, β 2 τ t τ t τ 2 t τ3 t τ4 t I 1 t τ t τ 2 t I 49 t 62. Te p values of all te above estimated coefficients are all zeros, and tis indicates tat all coefficients are significant at te 1% significance level. Te graps of te calibrated coefficient functions are presented in Figure 4, were we could find tat eac piecewise polynomial function coefficient is very close to te corresponding local constant estimate. Figure 4 also sows tat te 95% point wise confidence intervals covered te calibrated time varying coefficients. Tus, te calibrated coefficient functions are appropriate to capture te time varying dynamics of Okun s coefficient. 16

18 Figure 4: Local constant estimates of β 1 τ and β 2 τ and te calibrated parametric functions wit 95% point wise confidence intervals Intercept Bayes_mixture C.I. Calibrated OLS Slope Bayes_mixture C.I. Calibrated OLS Year Year In order to validate te calibrated parametric time trends for te time varying coefficients, we furter test te null ypotesis tat te two time varying coefficients are piecewise polynomial functions of time against te alternative of local constant estimates. Te bootstrap test procedure is based on 1000 repetitions. Te proposed bootstrap test produced a p value of Terefore, we could not reject te null ypotesis at te 1% significance level. Our finding indicates tat in different time period, Okun s coefficient is varying. 4.2 Consumption growt Te relationsip between consumption growt and income growt as been extensively investigated in empirical macroeconomics during te past several decades. Suc a relationsip is usually investigated troug te linear model given by y t α 1 + α 2 x t + ε t, 8 were y t is te consumption growt, wic is te first difference of logaritm of te consumption expenditure, and x t is te income growt, wic is te first difference of logaritm of te disposable income. Carrol and Summers 1991 estimated te coefficient of income growt rate α 2, wic is based on te data for 15 OECD countries during te period from 1960 to Campbell and Mankiw 1989 estimated α 2 using te U.S. quarterly data during te period from 1953 to

19 Tey found tat te estimate of α 2 is troug OLS, and tis estimate becomes larger wen te lagged consumption growt rates are used as instrumental variables. However, te relationsip between consumption growt and income growt may not necessarily be a constant over time. For example, if a consumer receives additional information on is/er present or future income, e/se would adjust is/er level of consumption discontinuously to be consistent wit is/er new inter temporal budget constraint. If te consumer knows tat te income growt rate will increase, e/se will adjust is/er level and rate of consumption. However, te model given by 8 indicates tat te response only depends on te level of income growt rate. In tis section, we treat te coefficients of 8 are time varying, and investigate te time varying relationsip between consumers consumption expenditure growt and disposable income growt during te period from 1960 to 2009 in te U.S. Te data are quarterly and are available at te website of te Bureau of Economic Analysis. Te time series plots of y t and x t are presented in Figure 5. If te two coefficients are assumed to be constants, te OLS estimates of α 1 and α 2 are respectively, and , wit teir p values being bot zero. Assuming te two coefficients are time varying, we fitted te model: y t β 1 τ t + β 2 τ t x t + u t to te above sample, were te errors are assumed to follow an AR1 process u t αu t 1 + v t wit v t being assumed to follow a mixture of two normal distributions. Local constant estimates of β 1 τ and β 2 τ were computed wit te bandwidt estimated troug te proposed Bayesian sampling. For comparison purpose, we also used NRR and CV to coose te bandwidts. Te graps of β 1 τ and β 2 τ wit teir bandwidts estimated/cosen troug Bayesian sampling, NRR and CV, are presented in Figure 6. Note tat te local constant estimators of te time varying coefficients are determined by te bandwidt. Te time varying coefficients wit teir bandwidt estimated troug Bayesian sampling are clearly different from tose wit teir bandwidt cosen troug NRR and CV. We find tat wit te bandwidt estimated troug Bayesian sampling, te long term trend of te slope coefficient is decreasing. Tis reflects a overall weakening relationsip between consumption growt and disposable income growt during te sample period, altoug for eac 18

20 Figure 5: Time series plots of consumption growt and income growt Consumption growt Q Q Q Q Q Q4 Quarter Income growt Q Q Q Q Q Q4 Quarter Figure 6: Local constant estimates of β 1 τ and β 2 τ wit bandwidt derived troug NRR, CV and Bayesian sampling. Intercept Bayes_mixture CV NRR OLS Slope Bayes_mixture CV NRR OLS 1960.Q Q Q Q Q Q4 Quarter 1960.Q Q Q Q Q Q4 Quarter 19

21 specific segment during te sample period, te slope coefficient exibited up and down due to different rates of economy growt. Te slope coefficient can be interpreted as te marginal propensity to consume MPC. A possible explanation is tat in aggregation, ouseolds increase teir spending as teir disposable income rises and ouseolds consume a smaller and smaller proportion of disposable income as disposable income increases. Te MPC is te ratio of a cange in consumption over a cange in income tat caused suc a consumption cange. From te data, we found tat disposable income demonstrated an increasing trend, terefore te MPC or te slope coefficient ad a long term decreasing trend. Te mean squared error MSE of residuals is for te time varying coefficient model, and is for te constant coefficient model. Te introduction of time varying coefficients improves model fitting by 12.37% in comparison to te constant coefficient model. We derived te asymptotic 95% point wise confidence intervals of te time varying coefficients, wic are plotted in Figure 7. Te construction of te confidence interval is same as te one for te time varying coefficients in Okun s law. Figure 7: Local constant estimates of β 1 τ and β 2 τ and te calibrated parametric functions wit 95% point wise confidence intervals Intercept Bayes_mixture C.I. Calibrated OLS 1960.Q Q Q Q Q Q4 Quarter Slope Bayes_mixture C.I. Calibrated OLS 1960.Q Q Q Q Q Q4 Quarter In order to examine weter te coefficients are time varying, we tested te null ypotesis of constant coefficients against te alternative of time varying coefficients. Te test procedure is te same as te one used in Section 4.1 and is described in Appendix B. Te proposed bootstrap test 20

22 wit 1000 repetitions produced a p value tat is approximately 0. Terefore, we rejected te null ypotesis of constant coefficients in te linear regression model at te 1% significance level. According to te patterns of te estimated time varying coefficients sown in Figure 6, a piecewise polynomial function of time migt be appropriate to approximate eac coefficient function. Consider te time varying coefficient model given by y t β 1 τ t + β 2 τ t x t + u t, 9 were β j τ t, for j 1 and 2, are known form functions of τ t. Fitting tis model to te sample, we derived te coefficient estimates of te time varying function. Te two estimated time varying functions are β 1 τ t τ t τ 2 t τ3 t I 1 t τ t τ 2 t τ3 t τ4 t I 39 t τ t τ 2 t τ3 t τ4 t I 109 t 199, β 2 τ t τ t τ 2 t τ3 t I 1 t τ t τ 2 t τ3 t I 39 t τ t τ 2 t τ3 t I 109 t 199, were I is an indicator function. Te p values of all estimated coefficients are zeros, indicating tat all coefficients are significant at te 1% significance level. Te graps of te above calibrated coefficient functions are presented in Figure 7, were we could find tat eac calibrated coefficient function is very close to its corresponding local constant estimate. Figure 7 also sows tat te 95% point wise confidence intervals covered te calibrated time varying coefficients, respectively. Terefore, te calibrated coefficients are appropriate to capture te time varying nature of te coefficients. In order to validate te calibrated piecewise functions for te time varying coefficients, we tested te null ypotesis tat te time varying coefficients are piecewise polynomial functions of time against te alternative of local constant estimates. Te test procedure is te same as te one used in Section 4.1. Te proposed bootstrap test wit 1000 repetitions produced a p value of Terefore, we could not reject te null ypotesis at te 1% significance level. 21

23 5 Conclusions and discussion In tis paper, we ave proposed a Bayesian approac to bandwidt estimation for local constant estimation of time varying coefficient time series models. We ave establised a completely new large sample teory for te proposed Bayesian bandwidt estimator as well as Bayesian estimators of te unknown parameters involved in te error density. From Monte Carlo simulation studies, we ave found tat our proposed Bayesian bandwidt estimator as good finite sample performance and it can acieve better performance tan NRR and CV in estimating te bandwidts for local constant estimators in te regression function. Applying te proposed Bayesian metod to te estimation of bandwidts for te time varying coefficient regression models tat reflect Okun s law and te relationsip between consumption growt and income growt, we ave found tat te proposed Bayesian metod works very well for te time varying coefficient time series model. Furtermore, based on te corresponding nonparametric estimates, we ave proposed parametric functions as approximations to time varying Okun s coefficient and time varying slope coefficient of consumption growt model. In tis study, we assume an AR1 error process to take into account serial correlation. It would be straigtforward to extend tis error structure to more general cases. In addition, we will take one step furter to accommodate eteroscedasticity in our future work. In terms of nonparametric estimation metod, it would be natural to extend te local constant estimation to local polynomial estimation, suc as te local linear estimation metod. Since a local polynomial kernel estimation metod involves muc more tecnicalities tan te local constant kernel metod, and te proofs of te main results are already very tecnical, we wis to leave suc extensions to future researc. One of te advantages of using a local polynomial kernel metod is tat certain derivatives of te β functions may be estimated naturally. As a result, it is ready to propose a plug in metod. Anoter important issue is te assumption on f v;η. In tis paper, we assume te density f v; η is parametrically unknown possibly indexed by a vector of unknown parameters, and we ten employ te proposed Bayesian metod to estimate te unknown parameters. In future researc, we will consider te case wen f v is nonparametrically unknown. In tis case, we may estimate 22

24 f v by a kernel density estimator of te form: f v;b 1 n 1 b L vt v b and we will see weter b may be estimated in a similar way to wat as been done for. In addition, it is possible to examine te out of sample performance of te time varying coefficient models troug some forecasting sceme. In empirical studies, it would ten be interesting to investigate forecasting performance based on our calibrated parametric forms., References Cai, Z. 2002, A two stage approac to additive time series models, Statistica Neerlandica 56, Cai, Z. 2007, Trending time varying coefficient time series models wit serially correlated errors, Journal of Econometrics 136, Cai, Z., Fan, J. and Yao, Q. 2000, Functional coefficient regression models for nonlinear time series, Journal of te American Statistical Association 95, Cai, Z. and Tiwari, R. C. 2000, Application of a local linear autoregressive model to BOD time series, Environmetrics 11, Campbell, J. Y. and Mankiw, N. G. 1989, Consumption, income, and interest rates: Reinterpreting te time series evidence, in O. J. Blancard and S. Fiscer, eds, NBER Macroeconomics Annual 1989, MIT Press, pp Carrol, C. D. and Summers, L. H. 1991, Consumption growt parallels income growt: Some new evidence, in B. D. Berneim and J. B. Soven, eds, National Saving and Economic Performan, University of Cicago Press, pp Cen, C.-F. 1985, On asymptotic normality of limiting density functions wit Bayesian implications, Journal of te Royal Statistical Society. Series B Metodological 47, Cen, J., Gao, J. and Li, D. 2012, Semiparametric trending panel data models wit cross sectional dependence, Journal of Econometrics 171, Fan, J., Heckman, N. E. and Wand, M. P. 1995, Local polynomial kernel regression for generalized linear models and quasi likeliood functions, Journal of te American Statistical Association 90, Gao, J. 2007, Nonlinear Time Series: Semi and Non Parametric Metods, Capman & Hall/CRC, London. Gao, J. and Hawtorne, K. 2006, Semiparametric estimation and testing of te trend of temperature series, Te Econometrics Journal 9, Gysels, E. 1998, On stable factor structures in te pricing of risk: Do time varying betas elp or urt?, Te Journal of Finance 53, Jagannatan, R. and Wang, Z. 1996, Te conditional CAPM and te cross section of expected returns, Te Journal of Finance 51,

25 Kim, J.-Y. 1998, Large sample properties of posterior densities, Bayesian information criterion and te likeliood principle in nonstationary time series models, Econometrica 66, Lee, J. 2000, Te robustness of Okun s law: Evidence from OECD countries, Journal of Macroeconomics 22, Li, D., Cen, J. and Gao, J. 2011, Non parametric time varying coefficient panel data models wit fixed effects, Te Econometrics Journal 14, Okun, A. 1962, Potential GNP: Its measurement and significance, Proceedings of te Business and Economic Statistics Section of te American Statistical Association pp Pillips, P. C. B. and Ploberger, W. 1996, An asymtotic teory of Bayesian inference for time series, Econometrica 64, Robinson, P. M. 1989, Nonparametric estimation of time varying parameters, in P. Hackl, ed., Statistical Analysis and Forecasting of Economis Structural Cange, Springer, Berlin, pp Silvapulle, P., Moosa, I. A. and Silvapulle, M. J. 2004, Asymmetry in Okun s law, Canadian Journal of Economics 37, Walker, A. 1969, On te asymptotic beaviour of posterior distributions, Journal of te Royal Statistical Society. Series B Metodological 31, Wang, K. Q. 2003, Asset pricing wit conditioning information: A new test, Te Journal of Finance 58, Zang, X., King, M. L. and Hyndman, R. J. 2006, A Bayesian approac to bandwidt selection for multivariate kernel density estimation, Computational Statistics and Data Analysis 50, Appendix A1: Assumptions In tis appendix, we describe te conditions tat are required to establis te asymptotic results, altoug some of tem migt not be te weakest possible. Assumption 1. i Let {x t } be strictly stationary and α mixing wit mixing coefficient satisfying ρs cs c 0 for 0 < c <, 2 < c 0 < 4 and s large enoug. Let Σ x E[x 1 x1 ] be positive definite and E[ x 1 x1 4 ] <. ii Consider te case were a n λ. Let λ be a continuous random variable wit π λ being its density suc tat λ 1 π λ λdλ <, λ 1 c0/2 π λ λdλ < and λ 2 c0/2 π λ λdλ < for 2 < c 0 < 4, were c 0 is same as tat in i. iii As n, a n 0 and na n. 24

26 Assumption 2. i K u is a symmetric probability kernel function wit K 0 > 0, K udu 1, u 2 K udu <, K 2 udu <, u 1 c 0 2 K udu < and u 2 c 0 2 K udu < for 2 < c 0 < 4, were c 0 is same as tat in Assumption 1i. ii β j, for j 1,2,,k, are twice differentiable and eac of te second derivative of β j is continuous in [0,1]. Assumption 3. i Let {v t } be an i.i.d. continuous random variables wit E θ [v 1 ] 0 and 0 < E θ [v 2 1 ] <. In addition, v t and x t are independent of eac oter. ii Let f v;θ be te conditional density of v 1 given θ. Suppose tat f v;θ is differentiable wit respect to v. Let γv;θ f 1 v;θ f v;θ E θ [γ 4 v 1 ;θ] <. wit E θ [γv 1 ;θ] 0, E θ [v 1 γv 1 ;θ] 0 and iii As n, E[v 1 γ n v 1 ] C γ for C γ < and n σ 1 n a n, were σ 2 n E[w 2 1 ] E[ v1 γ n v 1 E[v 1 γ n v 1 ] 2 ], in wic γ n v t E[θ] θγv t ;θa n θπθdθ wit A n θ e ne θ [log f v 1 ;θ] e ne θ [log f v 1 ;θ] πθdθ, E[w 2 1 ] E[ x 1 Σ 1 x x 1 2] and Eθ [log f v 1 ;θ] log f v;θf v;θdv. Assumption 4. As n, E[v 1 φ n v 1 ] C φ for C φ <, a n δ 2 n 0 and n δ 1 n a n, were φ n v t γv t ;θa n θπθdθ and δ 2 n E[w 2 1 ] E [ v1 φ n v 1 E[v 1 φ n v 1 ] 2 ]. Assumption 5. i g n θ is twice differentiable wit respect to θ in some neigborood of θ n. ii Te density π θ θ is twice differentiable and te second derivative is continuous. iii θ D n θ f n θ X n,y n dθ o P 1/2 n θ n and { θ : 1/2 n θ n θ n θ c } for some positive c. Dn eg n θ π θ θdθ e gn θ π θ θdθ o P 1/2 n θ n, were D n iv As n, 1/2 n θ n θn E[θ] D N 0,Σ 0, were Σ 0 is a positive definite matrix. Remark: Assumption 1i assumes x t is stationary and α mixing, wic can be satisfied by many linear and nonlinear time series. Assumption 1ii imposes some moment conditions on λ. Assumption 1iii is required for te consistency of local constant estimators. If we furter assume tat {v s } and {x t } are independent for all s, t wit E[x 1 ] 0, we may relax c 0 to just c 0 2 in Assumption 1i. Tis point will be made in te proofs of Teorems 1 and 2. Assumption 2i is a standard assumption for kernel functions. Te commonly used Gaussian and uniform kernel functions satisfy Assumption 2i. Assumption 2ii imposes smootness 25

27 constraints on te coefficient functions β j, wic is commonly used in literature; see Cai 2007 and Robinson Assumption 3iii ensures tat f v;θ is differentiable and te fourt moment of γv;θ exists. Te conditions of E θ [γv 1 ;θ] 0 and E θ [v 1 γv 1 ;θ] 0 are automatically satisfied wen f v;θ is te density function of a Normal random variable. As pointed out in te proofs of Teorems 1 and 2, we need not assume E θ [γv 1 ;θ] 0 wen E[x 1 ] 0. Assumptions 3 and 4 are automatically satisfied wen f v;θ is te density function of a Normal random variable of te form f v;θ f v;η 1 exp and η as a density function of te form πη exp η I [η > 0]. 2π η v 2 2η Assumption 5 is similar to wat as been used by Kim Assumption 5 iii is used to make sure tat we can do Taylor expansion for g n θ and π θ θ. Assumption 5iii is anoter commonly used one. For example, if te posterior density is a Normal density, ten θ D n θ f n θ X n dθ o P n 1/2 and tis assumption is automatically satisfied. We illustrate Assumption 5 iv using te following example. Suppose we ave X i θ + e i, e i N 0,1. Note tat θ is a random variable in Bayesian analysis. If we estimate θ 0 E[θ] by te sample mean, denoted as θ n, ten tis assumption is actually a quite standard result for a conventional maximum likeliood estimation metod. Appendix A2: Proofs of Teorems 1 and 2 In tis paper, our objective is to estimate λ,θ by te conditional mean, wic is given by E [ λ,θ ] λ,θ e lnλ,θ π λ λπ θ θdλdθ X n,y n e l n λ,θ E [λ X n,y n ]. π λ λπ θ θdλdθ E [θ X n,y n ] Equivalently, we can write E [λ X n,y n ] and E [θ X n,y n ] as follows: E [λ X n,y n ] E [θ X n,y n ] λe l n λ,θ π λ λπ θ θdλdθ e l n λ,θ π λ λπ θ θdλdθ, θe l n λ,θ π λ λπ θ θdλdθ e l n λ,θ π λ λπ θ θdλdθ. Since te Bayesian estimates E [λ X n,y n ],E [θ X n,y n ] do not ave closed form expressions, we terefore approximate tem by λ mn and θ mn, respectively. We will sow te large sample properties of λ mn and E [λ X n,y n ] in te proof of Teorem 1 in Appendix C of te supplementary material. 26

28 Te large sample properties of θ mn and E [θ X n,y n ] involved in Teorem 2 will also be given in Appendix C of te supplementary material. Appendix A3: Proof of Teorem 3 Let 0 E[] and λ 0 E[λ]. It is easy to find tat 0 a n λ 0. From Teorem 1, we ave tat ĥ mn 0 O P n 1/2 δ n. Terefore, by Assumption 2 ii, we can get tat βτ;ĥmn βτ; 0 O P n 1/2 δ n. It follows from Robinson 1989 tat n 0 βτ;0 βτ D N 0,Σ β, were Σ β σ 2 u K 2 wdw Σ 1 x, in wic σ2 u E[u2 1 ]. Terefore, under Assumptions 1 4, as n, we ave n0 βτ; ĥ mn βτ n 0 βτ; ĥ mn βτ; 0 + n 0 βτ;0 βτ O P a n δ 2 n + n 0 βτ;0 βτ o P 1 + n 0 βτ;0 βτ D N 0,Σ β, 10 wic completes te proof of Teorem 3. Appendix B: Nonparametric Specification Testing To examine weter te coefficients are time varying, we test te null ypotesis H 0 : β 1 τ α 1, β 2 τ α 2, against te alternative ypotesis given by H 1 : β 1 τ and β 2 τ ave a nonparametric specification. Te test statistic is defined as T S RSS 0 RSS 1 RSS 1, 11 were RSS 0 is te residual sum of square RSS under te null ypotesis, and RSS 1 is te RSS under te alternative ypotesis. Te null ypotesis is rejected for a large value of T S. Te p value is computed by employing te following bootstrap procedure. 27