Quantile Regression with Structural Changes

Transcription

1 Quantile Regression with Structural Changes Mei-Yuan Chen Department of Finance National Chung Hsing University Tel: mei August 15, 2017 M.-Y. Chen Chiang-Mai University August 15, / 44

2 Tests for Structural Change: Conditional Mean and Quantiles 1 Test for a break at a known time: Chow test, F-test 2 Tests for a break at an unknown time 3 Tests for multiple breaks at unknown times (a given τ and across τs) 1 how many breaks? 2 when they occur? M.-Y. Chen Chiang-Mai University August 15, / 44

3 Tests for Breaks in Conditional Mean: strucchange 1 Residual-based Tests 1 recursive residuals: Recursive and moving sums 2 OLS residuals: Recursive and moving sums 2 Estimate-based Tests: 1 Recursive-window OLS Estimates 2 Moving-window OLS Estimates M.-Y. Chen Chiang-Mai University August 15, / 44

4 Determine the Number and Estimate when Breaks Occur 1 Bai (1996), Testing for parameter constancy in linear regressions: An empirical distribution functional approach, Econometrica 64, Bai (2000), Vector autoregressive models with structural changes in regression coefficients and in variance-covariance matrices, Annals of Economics and Finance, 1, Bai and Perron (1998), Estimating and testing linear models with multiple structural changes, Econometrica, 66, Bai and Perron (2003), Critical values for multiple structural change tests, Econometrics Journal, 6, M.-Y. Chen Chiang-Mai University August 15, / 44

5 Tests for Structural Change: Conditional Quantiles 1 Test for a break on a quantile at an unknown time 2 Test for a break across quantiles at an unknown time 3 Test for multiple breaks on a quantile at unknown time 4 Test for multiple breaks across quantiles at unknown time M.-Y. Chen Chiang-Mai University August 15, / 44

6 References 1 Oka, Tatsushi and Zongjun Qu (2011), Estimating Structural Changes in Regression Quantiles, Journal of Econometrics, 162, Qu, Z. (2008), Testing for Structural Change in Regression Quantiles, Journal of Econometrics, 146, Qu, Z. and P. Perron (2007), Estimating and Testing Structural Changes in Multivariate Regressions, Econometrica, 75, M.-Y. Chen Chiang-Mai University August 15, / 44

7 Tests for Structural Breaks in Regression Quantiles Suppose the conditional quantile function of y t is linear and denote it by Q yt (τ x t ) = x tβ t (τ) (1) which may vary across t. Then, the response of y t to x t is different from that of y s to x s (hence structural change occurs) if and only if β t (τ) β s (τ) (2) for τ [0,1]. Tests can be constructed, as shown later, to evaluate constancy of the relationship (2), at a certain quantile, or across multiple quantiles. M.-Y. Chen Chiang-Mai University August 15, / 44

8 For the heteroskedastic linear model given by y t = x tβ t +(x tγ t )u t (3) where u t is a sequence of i.i.d. errors. A key feature of the model is that the shocks are allowed to change the location as well as the shape of the distribution. The model (3) has a linear conditional quantile function, i.e., Q yt (τ x t ) = x tβ t (τ) x t(β t +γ t F 1 t (τ)) where F( ) is the distribution of u t. Note that if there is no structural change in the conditional quantiles, i.e., β t = β 0, γ t = γ 0 and F t ( ) = F 0 ( ), then the conditional quantile function reduces to Q yt (τ x t ) = x tβ 0 (τ). M.-Y. Chen Chiang-Mai University August 15, / 44

9 Test for Changes in a given quantile The null and alternative hypotheses of testing for a change in a pre-specified quantile are given by H 0 : β t = β 0 for all t, for a givenτ (0,1) H 1 : β t = { β1 for t = 1,2,...,T 1 β 2 for t = T 1 +1,...,T for a given τ (0,1) Under the null, the parameter β 0 is estimated by solving the following minimization problem min β R p T t=1 with ρ τ (u) is the check function given by ρ τ (y t x tβ) (4) ρ τ (u) = u(1(u < 0) τ). M.-Y. Chen Chiang-Mai University August 15, / 44

10 Since the objection in (4) is not everywhere differentiable, hence subgradient is defined in place of gradient. Denote the partial sum of the subgradient as [Tλ] S n (λ,τ,β) = T 1/2 x t ψ τ (y t x tβ) t=1 where β are some parameter estimates and { 1 τ whenu 0 ψ τ (u) = (1(u 0) τ) = τ whenu > 0. M.-Y. Chen Chiang-Mai University August 15, / 44

11 By definition, P(u 0) = τ and P(u > 0) = 1 τ, therefore ψ τ (y t x tβ 0 (τ)) is a pivot statistic, i.e., a sequence of independent binary random variable with mean and variance E[ψ τ (y t x tβ 0 (τ))] = (1 τ)τ +( τ)(1 τ) = 0 var[ψ τ (y t x tβ 0 (τ))] = (1 τ) 2 τ +( τ) 2 (1 τ) = τ(1 τ). M.-Y. Chen Chiang-Mai University August 15, / 44

12 This implies, under some regularity conditions, S n (λ,τ,β 0 (τ)) d N(0,λτ(1 τ)j 0 ), where J 0 = plim T T 1 T t=1 x tx t. Define then T H λ,t (β 0 (τ)) = (T 1 x t x t) 1/2 S n (λ,τ,β 0 (τ)), t=1 H λ,t (β 0 (τ)) d N(0,λτ(1 τ)i p ). M.-Y. Chen Chiang-Mai University August 15, / 44

13 Now, replacing the unknown β 0 (τ) with quantile regression estimates using the full sample by imposing the null hypothesis, ˆβ T (τ), we have T H λ,t (ˆβ T (τ)) = (T 1 x t x t) 1/2 S n (λ,τ, ˆβ T (τ)). t=1 M.-Y. Chen Chiang-Mai University August 15, / 44

14 Re-centering H λ,t (ˆβ T (τ)) by the quantity H 1,T (ˆβ T (τ)) leads to the following statistic SQ τ = sup [τ(1 τ)] 1/2 [H λ,t (ˆβ T (τ)) λh 1,T (ˆβ T (τ))] (5) λ [0,1] where is the maximum norm. M.-Y. Chen Chiang-Mai University August 15, / 44

15 The Wald type statistic can also be constructed as follows. Let ˆβ 1 (λ) denote the solution from (4) using observations up to [Tλ] for some 0 < λ < 1 and ˆβ 2 (λ) denote the solution of (4) using the remaining sample. Denote their difference as ˆβ(λ,τ) = ˆβ 2 (λ,τ) ˆβ 1 (λ,τ). Then the Wald test for no structural change at date [Tλ] is given by T ˆβ(λ,τ) ˆV(λ,τ) 1 ˆβ(λ,τ) where ˆV(λ,τ) is a consistent of the variance of T ˆβ(λ,τ). M.-Y. Chen Chiang-Mai University August 15, / 44

16 Under the null hypothesis so that satisfies { 1 plim T ˆV(λ,τ) = [τ(1 τ)][f 0 (F0 1 (τ))] 2 λ + 1 } Ω 1 λ 0 where f 0 and F 0 are the density and cumulative distribution function of u t, Ω 0 = H0 1 J 0H0 1 with H 0 = plim T T 1 T t=1 x tx t/(γ 0x t ) and J 0 = plim T T 1 T t=1 x tx t. For the situation of unknown break point in priori, the statistic is modified as where SW τ = sup λ Λ ǫ T ˆβ(λ,τ) ˆV(λ,τ) 1 ˆβ(λ,τ), Λ ǫ = [ǫ,1 ǫ],for some0 < ǫ < 1/2. M.-Y. Chen Chiang-Mai University August 15, / 44

17 Qu (2008) derives the following asymptotic results for statistics SQ τ and SW τ : 1 SQ τ sup λ [0,1] B p (λ), where B p is a vector of p independent Brownian bridge process on [0,1] p. 2 SW τ sup λ Λǫ B p (λ) 2 /[λ(1 λ)]. M.-Y. Chen Chiang-Mai University August 15, / 44

18 In both cases, the limiting distribution depends on the model only through the number of parameters subject to change. For the SQ τ test, this feature may appear surprising at first sight since no normalization is used when constructing the test (in particular it does not require estimating the sparsity [f 0 (F 1 0 (τ))] 1. The explanation lies in the fact that the subgradient, when evaluated at the true parameter values β 0 (τ), does not depend on the distribution of the errors it is a weighted average of Bernoulli random variables. As a result, the test statistic SQ τ is self-normalized. This property is particularly desirable when the sample size is small and it is hard to estimate the sparsity to decent precision. The critical values provided in Table 1 by Qu (2008) via simulations are shown in the following table. M.-Y. Chen Chiang-Mai University August 15, / 44

19 Asymptotic critical values of the SQ τ M.-Y. Chen Chiang-Mai University August 15, / 44

20 Asymptotic critical values of the SW Test p = 1 p = 2 π 0 λ 10 % 5 % 1 % 10 % 5 % 1 % p = 3 p = 4 π 0 λ 10 % 5 % 1 % 10 % 5 % 1 % M.-Y. Chen Chiang-Mai University August 15, / 44

21 Asymptotic critical values of the SW Test p = 5 p = 6 π 0 λ 10 % 5 % 1 % 10 % 5 % 1 % p = 7 p = 8 π 0 λ 10 % 5 % 1 % 10 % 5 % 1 % M.-Y. Chen Chiang-Mai University August 15, / 44

22 Asymptotic critical values of the SW Test p = 9 p = 10 π 0 λ 10 % 5 % 1 % 10 % 5 % 1 % p = 11 p = 12 π 0 λ 10 % 5 % 1 % 10 % 5 % 1 % M.-Y. Chen Chiang-Mai University August 15, / 44

23 R code Part 1 in qrbreak.r M.-Y. Chen Chiang-Mai University August 15, / 44

24 Test for Change across Quantiles The tests SQ τ and SW τ can be extended for such a purpose, leading to the following tests: DQ = sup sup [H λ,t (ˆβ T (τ)) λh 1,T (ˆβ T (τ))] τ T ω and λ [0,1] DW = sup τ T ω sup λ Λ ǫ T ˆβ(λ,τ) ˆV(λ,τ) 1 ˆβ(λ,τ), where T is a closed set consisting of quantiles of interest and usually is set as T = [ω,1 ω] for some 0 < ω < 1/2. M.-Y. Chen Chiang-Mai University August 15, / 44

25 Test for Change across Quantiles Qu (2008) derives the following asymptotic results for these two tests: 1 DQ sup τ Tω sup λ [0,1] B(λ,τ). 2 DW sup τ Tω sup λ Λǫ B(λ,τ) 2 /[λ(1 λ)τ(1 τ)]. The useful critical values for DQ and DW are provided by Qu (2008) as follows. M.-Y. Chen Chiang-Mai University August 15, / 44

26 Asymptotic critical values of the DQ τ M.-Y. Chen Chiang-Mai University August 15, / 44

27 Asymptotic critical values of the DW τ M.-Y. Chen Chiang-Mai University August 15, / 44

28 Multiple Structural Changes Suppose there are m breaks occur under the alternative hypothesis. Denote the following two statistics as follows for testing structural changes in a given quantile and across quantiles: and SW τ (m) = sup Tˆβ(λ,τ) R (RŜ(λ,τ)R ) 1 Rˆβ(λ,τ), λ Λ ǫ(m) DW τ (m) = sup sup Tˆβ(λ,τ) R (RŜ(λ,τ)R ) 1 Rˆβ(λ,τ), τ T ω λ Λ ǫ(m) where λ = (λ 1,...,λ m ) denotes a partition of the sample; M.-Y. Chen Chiang-Mai University August 15, / 44

29 Λ ǫ (m) specifies the set of admissible partitions, i.e, for some small ǫ > 0, Λ ǫ (m) = {((λ 1,...,λ m ); λ j λ j 1 ǫ,λ 1 ǫ,λ m 1 ǫ}; where ˆβ(λ,τ) is the consistent quantile regression estimator vector for β 0 (τ) using subsamples, i.e., ˆβ(λ,τ) = (ˆβ 1 (λ,τ),..., ˆβ m+1 (λ,τ) ); R is the convectional matrix such that ˆβ(λ,τ) R = (ˆβ 2 (λ,τ) ˆβ 1 (λ,τ),..., ˆβ m+1 (λ,τ) ˆβ m (λ,τ) ) and Ŝ(λ,τ) is a consistent estimate of the variance Tˆβ(λ,τ) under the null hypothesis, i.e., plim T Ŝ(λ,τ) = τ(1 τ)diag(1/(λ 1 λ 0 ),..., with Ω 0 = H 1 0 J 0H 1 0, λ 0 = 0 and λ m+1 = 1. 1/(λ m+1 λ m )) Ω 0 (6) M.-Y. Chen Chiang-Mai University August 15, / 44

30 Qu (2008) provides the following asymptotic distributions for these two statistics under the null as follows: and SW τ (m) sup λ Λ ǫ(m) DW(m) sup τ T ω sup λ Λ ǫ(m) m λ j B p (λ j+1 ) λ j+1 B p (λ j ) 2 j=1 λ j λ j+1 (λ j+1 λ j ) m λ j B p (λ j+1,τ) λ j+1 B p (λ j,τ) 2. τ(1 τ)λ j λ j+1 (λ j+1 λ j ) j=1 M.-Y. Chen Chiang-Mai University August 15, / 44

31 Now suppose the alternative hypothesis specifies some unknown number of breaks between 1 and some upper bound m. The following statistics are suggested by Qu (2008): SSW τ (M) = max a msw τ (m) 1 m M SDW(M) = max b mdw(m), 1 m M where SSW τ (M) looks at structural changes in a given quantiles and SDW(M) does so across quantiles; and a m and b m are sets of weights such that the marginal p-values are equal across values of m. M.-Y. Chen Chiang-Mai University August 15, / 44

32 More precisely, let c(p,α,m) be the asymptotic critical value of SW τ (m) for a significance level α with p parameters allowed to change. The weights a m are then defined as a 1 = 1 and a m = c(p,α,1)/c(p,α,m) for m > 1. The weights b m are defined similarly except the critical values of SW τ (m) are replaced with those for DW(m). The null limiting distributions for these two statistics are provided by Qu (2008) as follows. and SSW τ (m) max SDW(M) max 1 m M 1 m M a m sup τ T ω sup λ Λ ǫ(m) sup λ Λ ǫ(m) m j=1 m j=1 λ j B p (λ j+1 ) λ j+1 B p (λ j ) 2 λ j λ j+1 (λ j+1 λ j ) λ j B p (λ j+1,τ) λ j+1 B p (λ j,τ) 2 τ(1 τ)λ j λ j+1 (λ j+1 λ j ) M.-Y. Chen Chiang-Mai University August 15, / 44

33 The critical values for SW τ (m) are already tabulated in Bai and Perron (2003). For the other statistics, Qu (2008) provides the response surface estimations for critical values calculation using the nonlinear regression CV i (α) = (z 1iδ 1 )exp(z 2iδ 2 )+e i, where CV i (α) is the simulated critical value, and α is the nominal size. The choice of regressors to include is dictated by overall significance subject to the requirement that the R 2 is no smaller than M.-Y. Chen Chiang-Mai University August 15, / 44

34 The regressors considered in the estimation of response surfaces are 1 for the SW τ (m) test: z 1 = {1,p,m,1/p,mp,mǫ}, z 2 = {1/m,1/(mǫ),ǫ,p}; 2 for the SSW τ (m) test: z 1 = {1,p}, z 2 = {ǫ,p}, M = 3; 3 for the DW(m) test: z 1 = {1,p,m,1/p,mp,mǫ}, z 2 = {1/m,1/(mǫ),1/(mω),ǫ,ω,p}; 4 for the SSW τ (m) test: z 1 = {1,p}, z 2 = {ǫ,ω,p}, M = 3; In the simulations, 1 p 10, m 3, 0.1 ǫ 0.2 and 0.1 ω 0.2 are considered and the estimated results are shown in Table 4 in Qu (2008). M.-Y. Chen Chiang-Mai University August 15, / 44

35 Determination the Number of Breaks Suppose a model with l breaks has been estimated and the break times have been estimated as ˆT 1,..., ˆT l. By checking each of the (l +1) segments for the presence of an additional break, the SQ τ and DQ tests applied to the jth segment are denoted as SQ τ,j = sup [τ(1 τ)] 1/2 [H λ,ˆt j 1,ˆT j (ˆβ j (τ)) λh 1,ˆT j 1,ˆT j (ˆβ j (τ))] λ [0,1] DQ j where = sup sup [τ(1 τ)] 1/2 [H λ,ˆt j 1,ˆT j (ˆβ j (τ)) λh 1,ˆT j 1,ˆT j (ˆβ j (τ))] τ T ω λ [0,1] H λ,ˆt j 1,ˆT j (ˆβ j (τ)) = T j T j 1 +1 x t x t 1/2 T j T j 1 +1 x t ψ τ (y t x tˆβj (τ)) M.-Y. Chen Chiang-Mai University August 15, / 44

36 The strategy proceeds by testing each of the (l +1) segments for the presence of an additional break. Then, SQ τ (l +1 l) and DQ(l +1 l) are defined as the maximum of SQ τ,j and DQ j over the l +1 segments, i.e., SQ τ (l +1 l) = max SQ τ,j 1 j l+1 DQ(l +1 l) = max DQ j. 1 j l+1 The null of l breaks is rejected and in favor of a model with (l +1) breaks if the resulting value is sufficiently large. M.-Y. Chen Chiang-Mai University August 15, / 44

37 Given suitable assumptions, Oka and Qu (2011) gives the asymptotic distributions for SQ τ (l +1 l) and DQ(l +1 l), i.e., P(SQ τ (l +1 l) c) G p (c) l+1 P(DQ(l +1 l) c) Ḡ p (c) l+1, where G p (c) is the distribution function of sup s [0,1] W 0 (s), Ḡ p (c) is the distribution function of sup τ Tω sup s [0,1] W 0 (λ,s), and W 0 is the p-dimensional Brownian bridge. Some useful critical values are studied and provided by exploring the response surface of the critical values from simulations in Oka and Qu (2011) M.-Y. Chen Chiang-Mai University August 15, / 44

38 Implementing the Tests For the tests SW τ and DW, a consistent estimate of the variance covariance matrix of T ˆβ(λ,τ) is needed. This in particular requires estimating the following quantity, ϕ 0 (τ) = [f 0 (F0 1 (τ))] 1 H0 1 ( T = plim T T 1 t=1 [f 0 (F 1 0 (τ))] γ 0 x x t x t t ) 1 for which two methods are often used. One is based on difference quotient, as discussed by Siddiqui (1960) and Hendricks and Koenker (1992). M.-Y. Chen Chiang-Mai University August 15, / 44

39 The other is based on kernel density estimation, proposed by Powell (1989). Qu (2008) suggest to use the first approach since it incorporates the linearity of the conditional quantile function and performs more stably in simulation studies. Specifically, since that the quantity f 0 [F 1 0 (τ)]/γ 0x t is the conditional density function of y t evaluated at the τ th conditional quantile, it can be consistently estimated by the difference quotient T = 2h T x tˆβ(τ +h T ) x tˆβ(τ h T ). M.-Y. Chen Chiang-Mai University August 15, / 44

40 圖 : Illustration for the estimator of conditional error density at τ M.-Y. Chen Chiang-Mai University August 15, / 44

41 圖 : Illustration for the estimator of conditional error density at τ M.-Y. Chen Chiang-Mai University August 15, / 44

42 To determine the bandwidth H T, the following rule suggested by Hall and Sheather (1988), based on Edgeworth expansions of studentized quantiles (and using Gaussian plugin), determines h T = T 1/3 z 2/3 α [1.5φ2 (Φ 1 (τ))/(2φ 1 (τ) 2 +1)] 1/3 where φ and Φ stand for the density and cumulative distribution function of the standard normal and z α satisfies Φ(z α ) = 1 α/2. Alternatively, the rule of Bofinger (1975), which is based on minimizing the mean squared error of the density estimator and the Gaussian plug-in, the following bandwidth that is widely used in practice h T = T 1/5 [4.5φ 4 (Φ 1 (τ))/(2φ 1 (τ) 2 +1)] 1/5. (7) Once T is obtained, ϕ 0 (τ) is estimated by (T 1 T t=1 Tx t x t ) 1 and J 0 by T 1 T t=1 x tx t. M.-Y. Chen Chiang-Mai University August 15, / 44

43 R code Part 2 in qrbreak.r M.-Y. Chen Chiang-Mai University August 15, / 44

44 R code: Empirical Study TWIRetVol.R M.-Y. Chen Chiang-Mai University August 15, / 44

45 code: TWIRetVol quant.r > ## > ## Authors: Oka and Qu > ## Date : 12/27/2010 > ## Updated: 01/28/2013 > ## > > ## Note: > ## This code applies the procedure developed in > ## "Estimating Structural Changes in Regression Quantiles" to anal > ## growth example considered in the paper. > ## Please be advised that this code uses an R-package, quantreg, > ## developed by Roger Koenker. If you have not installed this pack > ## please type the following command before executing our procedur > ## install.packages(³quantreg³) > > ## clear all existing objects > rm(list=ls(all=true)) > > setwd("d://rjobs//quzhongjun//qrbreak") M.-Y. Chen Chiang-Mai University August 15, / 44

46 > ## > ## Set up the model > ## > ## library and source files > library(quantreg) > source(³quant-chg.r³) > > ## read data > DATA = read.csv("d:/data/twi_stock/twiretvolm.csv", header=true) > > nn = length(data) > ##plot(data, type = ³l³) > > ## two lags are included as regressors > > y <- DATA$ret #dependent variable > > x <- cbind(data$ret1, DATA$ret2,DATA$dvol1, DATA$dvol2) > ## quantiles > vec.tau = seq(0.20, 0.80, by = 0.150) M.-Y. Chen Chiang-Mai University August 15, / 44

47 > ##vec.tau = seq(0.20, 0.80, by = 0.075) ## a finer grid to check > > ## DQ test > q.l = min(vec.tau) > q.r = max(vec.tau) > > ## the maximum number of breaks allowed > m.max = 3 > > ## the signifance level for sequenatial testing > ## 1, 2 or 3 for 10%, 5% or 1%, respectively > v.a = 2 > > ## the significance level for the confidence intervals of > ## estimated break dates. > ## 1 or 2 for 90% and 95%, respectively. > v.b = 2 > > ## the sample size of cross-section > N.size = 1 M.-Y. Chen Chiang-Mai University August 15, / 44

48 > > ## the trimming proportion for estimating the break dates (used to > ## the boundaries of the sample) > trim.e = 0.15 > > ## Global varible for the DQ test. Set d.sym=true if the trimming > ## for vec.tau is symmetric: i.e, q.l = (1 - q.r); Otherwise, set > d.sym = TRUE > > ## If d.sym = FALSE, then please set d.sim = TRUE for the first-ti > ## The simulation takes a while depdeing on the computer you use. > d.sim = FALSE > > > ## the time index > #vec.time = seq(1947, 2009, 1) %x% matrix(1, 4, 1) > #vec.time = vec.time * matrix(c(1, 2, 3, 4), length(vec.time > #vec.time = vec.time[2:(nn+1)] > #if( min(vec.time)!= ){ > # stop("wrong Begining") M.-Y. Chen Chiang-Mai University August 15, / 44

49 > #} > #if( max(vec.time)!= ){ > # stop("wrong End") > #} > #vec.time = vec.time[(n.lag+1):nn] > # > vec.time <- DATA$obs > > > > # > ## You should not have to modify the following > # > ## global variable > eps =.Machine$double.eps ^ (2/3) > options(warn=-1) > ## ====== Main Analysis ====== > rq.break(y, x, vec.tau, N.size, trim.e, q.l, q.r, vec.time, m.max, ================================================================== M.-Y. Chen Chiang-Mai University August 15, / 44

50 =====Analysis based on a single conditional quantile function ===== ================================================================== Quantile: At the significance level: 5 % SQ test: Critical values: The number of breaks detected: Break Dates and Confidence Intervals: 95 % - - [,1] [,2] [,3] [1,] [,1] [,2] [,3] [1,] "1990M09" "1988M07" "1998M04" - - Estimation Results - - (a)coefficients estimates for each regime - - < Regime: 1 > Value Std. Error t value Pr(> t ) (Intercept) temp$x temp$x temp$x M.-Y. Chen Chiang-Mai University August 15, / 44

51 temp$x < Regime: 2 > Value Std. Error t value Pr(> t ) (Intercept) e e-15 rem.x e e-01 rem.x e e-01 rem.x e e-01 rem.x e e-02 (b) Break sizes - - Value Std. Error t value Pr(> t ) big.x big.x big.x big.x big.x ================================================================== Quantile: 0.35 M.-Y. Chen Chiang-Mai University August 15, / 44

52 At the significance level: 5 % SQ test: Critical values: The number of breaks detected: 0 ================================================================== Quantile: At the significance level: 5 % SQ test: Critical values: The number of breaks detected: 0 ================================================================== Quantile: At the significance level: 5 % SQ test: Critical values: The number of breaks detected: 0 ================================================================== Quantile: At the significance level: 5 % SQ test: M.-Y. Chen Chiang-Mai University August 15, / 44

53 Critical values: The number of breaks detected: Break Dates and Confidence Intervals: 95 % - - [,1] [,2] [,3] [1,] [,1] [,2] [,3] [1,] "1991M04" "1991M03" "1991M12" - - Estimation Results - - (a)coefficients estimates for each regime - - < Regime: 1 > Value Std. Error t value Pr(> t ) (Intercept) e-06 temp$x e-02 temp$x e-01 temp$x e-02 temp$x e-01 < Regime: 2 > Value Std. Error t value Pr(> t ) (Intercept) M.-Y. Chen Chiang-Mai University August 15, / 44

54 rem.x rem.x rem.x rem.x (b) Break sizes - - Value Std. Error t value Pr(> t ) big.x e-06 big.x e-01 big.x e-01 big.x e-01 big.x e-01 ================================================================== ===== Analysis based on multiple conditional quantiles ===== ================================================================== At the significance level: 5 % DQ test: Critical values: The number of breaks detected: 0 M.-Y. Chen Chiang-Mai University August 15, / 44

55 ================================================================== Estimation Time (min) user system elapsed ================================================================== M.-Y. Chen Chiang-Mai University August 15, / 44