The Bootstrap: Theory and Applications. Biing-Shen Kuo National Chengchi University

Transcription

1 The Bootstrap: Theory and Applications Biing-Shen Kuo National Chengchi University

2 Motivation: Poor Asymptotic Approximation Most of statistical inference relies on asymptotic theory.

3 Motivation: Poor Asymptotic Approximation Most of statistical inference relies on asymptotic theory. eg. t-ratio d N(0,1), Wald test, Lagrange Multiplier test, and likelihood ratio test d χ 2 (d.f.) as sample goes to infinity.

4 Motivation: Poor Asymptotic Approximation The idea is at its best hope the finite-sample distribution of the statistics COULD be well approximated by their asymptotic counterparts.

5 Motivation: Poor Asymptotic Approximation The idea is at its best hope the finite-sample distribution of the statistics COULD be well approximated by their asymptotic counterparts. But the reality in some cases might not turn out to be hoped.

6 Motivation: Poor Asymptotic Approximation Simulated illustration y i = β 0 + x 1i β 1 + x 2i β 2 + e i ( ) x 1i x 2i N(0, I 2 ) e i N(0, 9) β 0 = 0, β 1 = 1, β 2 = 0.5, n(sample size) = 300

7 Motivation: Poor Asymptotic Approximation The parameter of interest which is estimated by θ = β 1 β 2 θ = β 1 β 2, where β 1, β 2 are OLS estimates.

8 Motivation: Poor Asymptotic Approximation Want to test by using t-ratio H 0 : θ 0 = 2 (true value) t 300 = ( θ θ 0 ) S( θ)

9 Motivation: Poor Asymptotic Approximation where S( θ) = (Ĥ β V Ĥβ) 1/2, Ĥ β = (0, 1 β2, β 1 β 2 ), V : covariance matrix based on Taylor expansion or delta method.

10 Motivation: Poor Asymptotic Approximation

11 Motivation: Poor Asymptotic Approximation Exact distribution of t 300 can be calculated by simulation. Replications: 100,000.

12 Motivation: Poor Asymptotic Approximation Exact distribution of t 300 can be calculated by simulation. Replications: 100,000. We found dramatic divergence between the exact and asymptotic distribution.

13 Motivation: Poor Asymptotic Approximation Exact distribution of t 300 can be calculated by simulation. Replications: 100,000. We found dramatic divergence between the exact and asymptotic distribution. Exact: skewed and non-normal Asym.: symmetric and normal

14 Motivation: Poor Asymptotic Approximation Pr(t 300 >1.645)=0.00, Pr(t 300 <-1.645)=0.115, and Pr( t 300 >1.96)=0.084

15 Motivation: Poor Asymptotic Approximation Pr(t 300 >1.645)=0.00, Pr(t 300 <-1.645)=0.115, and Pr( t 300 >1.96)=0.084 indicative of a significant type I error distortion.

16 The Bootstrap as an Alternative Approximation to Dist. of Statistics It is a re-sampling scheme by treating the data as if they were the population.

17 The Bootstrap as an Alternative Approximation to Dist. of Statistics It is a re-sampling scheme by treating the data as if they were the population. It is to estimate the finite-sample distribution of a statistic, and thus moments or quantiles.

18 The Bootstrap as an Alternative Approximation to Dist. of Statistics It is a re-sampling scheme by treating the data as if they were the population. It is to estimate the finite-sample distribution of a statistic, and thus moments or quantiles. It is often more accurate than first-order approximations, the so-called asymptotic refinement.

19 Notations and Definition of the Problem The data: {x i } n i=1 F 0, an unknown CDF.

20 Notations and Definition of the Problem The data: {x i } n i=1 F 0, an unknown CDF. The statistic T n = T n (x 1, x 2,..., x n ) whose finite-sample CDF is denoted by G n (τ, F 0 ) = Pr(T n τ).

21 Notations and Definition of the Problem The data: {x i } n i=1 F 0, an unknown CDF. The statistic T n = T n (x 1, x 2,..., x n ) whose finite-sample CDF is denoted by G n (τ, F 0 ) = Pr(T n τ). G n (τ, F) is a different function of τ for different F.

22 Notations and Definition of the Problem T n is pivotal when G n (τ, F) does not depend on F.

23 Notations and Definition of the Problem T n is pivotal when G n (τ, F) does not depend on F. In most applications, pivotal statistics are not available.

24 Notations and Definition of the Problem Therefore, G n (., F 0 ) usually depends on F 0, and also CAN T be calculated, due to unknown F 0.

25 Notations and Definition of the Problem Therefore, G n (., F 0 ) usually depends on F 0, and also CAN T be calculated, due to unknown F 0. Both the bootstrap and the asymptotic theory are ways to estimate G n (., F 0 ).

26 Important Notion The limit distributions of many econometric statistics are N(0, 1) or χ 2, independent of F 0 or unknown population parameters.

27 Important Notion The limit distributions of many econometric statistics are N(0, 1) or χ 2, independent of F 0 or unknown population parameters. These statistics are called asymptotically pivotal.

28 Important Notion The limit distributions of many econometric statistics are N(0, 1) or χ 2, independent of F 0 or unknown population parameters. These statistics are called asymptotically pivotal. In notation, let G (., F 0 ) denote the asymptotic CDF of T n.

29 Important Notion The limit distributions of many econometric statistics are N(0, 1) or χ 2, independent of F 0 or unknown population parameters. These statistics are called asymptotically pivotal. In notation, let G (., F 0 ) denote the asymptotic CDF of T n. If T n is asymptotically pivotal, G (., F 0 ) G (.)

30 Important Notion The idea of asymptotic approximation is when n is large enough, G (.) is used to estimate G n (., F 0 ) without knowing F 0.

31 Important Notion The idea of asymptotic approximation is when n is large enough, G (.) is used to estimate G n (., F 0 ) without knowing F 0. As illustrated before, G (.) can be a very poor approximation.

32 The Idea of the Bootstrap The bootstrap replaces F 0 with an estimator F n. (Note: asymptotic approximation replaces G n with G.)

33 The Idea of the Bootstrap The bootstrap replaces F 0 with an estimator F n. (Note: asymptotic approximation replaces G n with G.) A nature choice of F n : Empirical distribution function of the data F n (x) = 1 n n I(x i x) i=1 with I being the indicator function.

34 The Idea of the Bootstrap Theoretical underpinning of the choice sup x F n (x) F 0 (x) p 0 (by Glivenko-Cantelli theorem)

35 The Idea of the Bootstrap Theoretical underpinning of the choice sup x F n (x) F 0 (x) p 0 (by Glivenko-Cantelli theorem) So the bootstrap estimator of G n (., F 0 ) is G n (., F n ).

36 Monte-Carlo Procedures to Produce G n (.,F n ) Step 1. Generate a bootstrap sample {x i }n i=1 from the EDF {x i } n i=1 with replacement.

37 Monte-Carlo Procedures to Produce G n (.,F n ) Step 1. Generate a bootstrap sample {x i }n i=1 from the EDF {x i } n i=1 with replacement. Step 2. Compute T n = T n (x 1,..., x n).

38 Monte-Carlo Procedures to Produce G n (.,F n ) Step 1. Generate a bootstrap sample {x i }n i=1 from the EDF {x i } n i=1 with replacement. Step 2. Compute T n = T n (x 1,..., x n). Step 3. Repeat Step 1, 2 B times, and then compute Pr(T n τ).

39 Consistency of the Bootstrap The bootstrap estimator G n (., F n ) is consistent if for each ǫ > 0, [ ] lim Pr sup G n (τ, F n ) G (τ, F 0 ) > ǫ = 0. n τ

40 Consistency of the Bootstrap The bootstrap estimator G n (., F n ) is consistent if for each ǫ > 0, [ ] lim Pr sup G n (τ, F n ) G (τ, F 0 ) > ǫ = 0. n τ This is the minimal criterion for G n (., F n ) to be an adequate estimator.

41 Consistency of the Bootstrap The bootstrap estimator G n (., F n ) is consistent if for each ǫ > 0, [ ] lim Pr sup G n (τ, F n ) G (τ, F 0 ) > ǫ = 0. n τ This is the minimal criterion for G n (., F n ) to be an adequate estimator. The bootstrap consistency is expected in most econometric applications.

42 Subsampling This is an alternative re-sampling method.

43 Subsampling This is an alternative re-sampling method. It often estimates G n (., F 0 ) consistently even when the bootstrap does not.

44 Subsampling This is an alternative re-sampling method. It often estimates G n (., F 0 ) consistently even when the bootstrap does not. It is not a perfect substitute for the bootstrap, because it tends to be less accurate than the bootstrap.

45 Subsampling: Intuition and Operation t n = t n (x 1,..., x n ): an estimator of the parameter θ. Let T n = ρ(n)(t n θ), where ρ(n) is the normalization factor. eg. θ is the population mean, t n = x, ρ(n) = n 1/2.

46 Subsampling: Intuition and Operation Let {x ij } m j=1 be a subset of m < n drawn from {x i } n i=1. N n,m = Cm n : total sub-sample numbers. t m,k : the estimator t m evaluated at the k th subsample.

47 Subsampling: Intuition and Operation The subsampling method estimates G n (τ, F 0 ) by G n,m (τ) = 1 N n,m I[ρ(m)(t m,k t n ) τ] N n,m k=1

48 Subsampling: Intuition and Operation The subsampling method estimates G n (τ, F 0 ) by G n,m (τ) = 1 N n,m I[ρ(m)(t m,k t n ) τ] N n,m k=1 Intuition: G m (., F 0 ) is the exact sampling distribution of ρ(m)(t m θ), and G m (τ, F 0 ) = Pr(ρ(m)(t m θ) τ) = E{I[ρ(m)(t m θ) τ]}.

49 Subsampling: Intuition and Operation G n,m (τ) would estimate G m (τ, F 0 ) well, because if n is large but m/n is small (so N n,m is large), t n varies less than t m, and ρ(t m t n ) and ρ(m)(t m θ) are close.

50 Subsampling: Intuition and Operation G n,m (τ) would estimate G m (τ, F 0 ) well, because if n is large but m/n is small (so N n,m is large), t n varies less than t m, and ρ(t m t n ) and ρ(m)(t m θ) are close. Expectedly, G n,m (τ) G m (τ, F 0 ) G (τ, F 0 ) as n, m.

51 Subsampling: Intuition and Operation G n,m (τ) would estimate G m (τ, F 0 ) well, because if n is large but m/n is small (so N n,m is large), t n varies less than t m, and ρ(t m t n ) and ρ(m)(t m θ) are close. Expectedly, G n,m (τ) G m (τ, F 0 ) G (τ, F 0 ) as n, m. Applications: Confidence interval for highly persistent AR coefficient.

52 Asymptotic Refinement Under some regularity conditions, G n (τ, F n ) G n (τ, F 0 ) = [G (τ, F n ) G (τ, F 0 )] + O( 1 n )

53 Asymptotic Refinement Under some regularity conditions, G n (τ, F n ) G n (τ, F 0 ) = [G (τ, F n ) G (τ, F 0 )] + O( 1 n ) If T n is not asymptotically pivotal, G (τ, F n ) G (τ, F 0 ) = O(n 1/2 ) because F n F 0 = O(n 1/2 ).

54 Asymptotic Refinement Under some regularity conditions, G n (τ, F n ) G n (τ, F 0 ) = [G (τ, F n ) G (τ, F 0 )] + O( 1 n ) If T n is not asymptotically pivotal, G (τ, F n ) G (τ, F 0 ) = O(n 1/2 ) because F n F 0 = O(n 1/2 ). The bootstrap has the same accuracy as asymptotic approximations.

55 Asymptotic Refinement If T n is asymptotically pivotal, G n (τ, F n ) G n (τ, F 0 ) = G (τ) G (τ)] = 0.

56 Asymptotic Refinement If T n is asymptotically pivotal, G n (τ, F n ) G n (τ, F 0 ) = G (τ) G (τ)] = 0. The bootstrap now makes an error of size O(n 1 ), smaller than that by asymptotic approximations O(n 1/2 ).

57 Asymptotic Refinement If T n is asymptotically pivotal, G n (τ, F n ) G n (τ, F 0 ) = G (τ) G (τ)] = 0. The bootstrap now makes an error of size O(n 1 ), smaller than that by asymptotic approximations O(n 1/2 ). Note: The argument is valid for stationary data, but not necessarily true for nonstationary data.

58 Asymptotic Refinement If T n is asymptotically pivotal, G n (τ, F n ) G n (τ, F 0 ) = G (τ) G (τ)] = 0. The bootstrap now makes an error of size O(n 1 ), smaller than that by asymptotic approximations O(n 1/2 ). Note: The argument is valid for stationary data, but not necessarily true for nonstationary data. The smoothness condition has to be satisfied for the statistics to be asymptotically pivotal.

59 Application: Bias Reduction Step 1: Use {x i } n i=1 to yield θ n, the estimated parameter.

60 Application: Bias Reduction Step 1: Use {x i } n i=1 to yield θ n, the estimated parameter. Step 2: Generate a bootstrap sample {x i }n i=1 from {x i } n i=1 with replacement. Use {x i }n i=1 to compute θ n.

61 Application: Bias Reduction Step 1: Use {x i } n i=1 to yield θ n, the estimated parameter. Step 2: Generate a bootstrap sample {x i }n i=1 from {x i } n i=1 with replacement. Use {x i }n i=1 to compute θ n. Step 3: Compute E θn by averaging B repetitions of θn in Step 2. Set Bn = E θn θ n.

62 Application: Bias Reduction Step 1: Use {x i } n i=1 to yield θ n, the estimated parameter. Step 2: Generate a bootstrap sample {x i }n i=1 from {x i } n i=1 with replacement. Use {x i }n i=1 to compute θ n. Step 3: Compute E θn by averaging B repetitions of θn in Step 2. Set Bn = E θn θ n. Step 4: Form the bias-corrected estimate by computing θ n B n.

63 Application: Computing Bootstrap Critical Values Step 1-3 are the same as before, and set Z n,α equal to the (1 α) quantile of the bootstrap distribution of T n.

64 Application: Constructing Confidence Interval Given n 1/2 (θ n θ 0 )/S n d N(0, 1), Where Sn is a consistent estimate of std.

65 Application: Constructing Confidence Interval Given n 1/2 (θ n θ 0 )/S n d N(0, 1), Where Sn is a consistent estimate of std. An asymptotic (1 α) 2-sided confidence interval for θ 0 is [θ n Z,α/2 S n n 1/2, θ n + Z,α/2 S n n 1/2 ].

66 Application: Constructing Confidence Interval Given n 1/2 (θ n θ 0 )/S n d N(0, 1), Where Sn is a consistent estimate of std. An asymptotic (1 α) 2-sided confidence interval for θ 0 is [θ n Z,α/2 S n n 1/2, θ n + Z,α/2 S n n 1/2 ]. The bootstrap version is [ θ n Z n,α/2 S nn 1/2, θ n + Z n,α/2 S nn 1/2].

67 Bootstrap Sampling with Dependent Data Asymptotic refinement can t be obtained by independent bootstrap sampling, with dependent data.

68 Bootstrap Sampling with Dependent Data Asymptotic refinement can t be obtained by independent bootstrap sampling, with dependent data. Bootstrap sampling has to be conducted in a way that suitably captures the dependence of the DGP.

69 Dependent Data: Block Bootstrap Step 1: Pick a block length b < n.

70 Dependent Data: Block Bootstrap Step 1: Pick a block length b < n. Step 2: Form n blocks of b residuals, each starting with a different one of the n residuals, and wrapping around to the beginning if necessary.

71 Dependent Data: Block Bootstrap Step 1: Pick a block length b < n. Step 2: Form n blocks of b residuals, each starting with a different one of the n residuals, and wrapping around to the beginning if necessary. Step 3: Generate the bootstrap errors by resampling the blocks. If n/b is an integer, there will have n/b blocks. Otherwise, the last block will have to be shorter than b.

72 Dependent Data: Block Bootstrap Numerous other block bootstrap procedures exist.(see Lahiri, 1999, Annals of Statisitcs)

73 Dependent Data: Block Bootstrap Numerous other block bootstrap procedures exist.(see Lahiri, 1999, Annals of Statisitcs) The performance of the block bootstrap is far from satisfactory, and is very sensitive to block length b.

74 Dependent Data: Sieve Bootstrap Step 1: Fit AR(p) to the residuals, where p is chosen based on the AIC or BIC criterion. i.e. x i = α 0 + α 1 x t α p x i p + ǫ i where {x i } n i=1 are the residuals, { α i} p i=0 are the AR estimates, and {ǫ i } are the rediduals from the autoregression.

75 Dependent Data: Sieve Bootstrap Step 2: Generate the bootstrap samples {ǫ i } form the EDF {ǫ i }.

76 Dependent Data: Sieve Bootstrap Step 2: Generate the bootstrap samples {ǫ i } form the EDF {ǫ i }. Step 3: Generate the bootstrap residuals {x i } by x i = α 0 + α 1 x t α px i p + ǫ i

77 Dependent Data: Sieve Bootstrap The idea behind the procedures is that the DGP can be represented as an infinite-order autoregression that can be approximated by AR(p).

78 Dependent Data: Sieve Bootstrap The idea behind the procedures is that the DGP can be represented as an infinite-order autoregression that can be approximated by AR(p). The AR order p has to grow at some suitable rate as n.

79 Dependent Data: Sieve Bootstrap The idea behind the procedures is that the DGP can be represented as an infinite-order autoregression that can be approximated by AR(p). The AR order p has to grow at some suitable rate as n. Sieve bootstrap performs better than block bootstrap in many cases.

80 Wild Bootstrap to Deal with Heteroscedasticity Consider the regression model y i = x i β + u i, u i = σ i ǫ i, E(ǫ 2 i ) = 1 where σ 2 i the error variance, depends of the regressors in an unknown fashion.

81 Wild Bootstrap to Deal with Heteroscedasticity The wild bootstrap DGP would be y i = x i β + f(ũ i )ν i where β would be restricted or unrestricted estimates, f(ũ i ) is a transformation of the i th residual associated with β and νi is a random variable with mean 0 and variance 1.

82 Wild Bootstrap to Deal with Heteroscedasticity A simple choice for f(.) is f(ũ i ) = ũ i (1 h i ) 1/2

83 Wild Bootstrap to Deal with Heteroscedasticity A simple choice for f(.) is f(ũ i ) = ũ i (1 h i ) 1/2 Many ways to specify νi. One popular choice is { νi 1 with probability 1/2 = 1 with probability 1/2

84 Wild Bootstrap to Deal with Heteroscedasticity In some respects, the error terms for the wild bootstrap DGP do not resemble those of the true DGP.

85 Wild Bootstrap to Deal with Heteroscedasticity In some respects, the error terms for the wild bootstrap DGP do not resemble those of the true DGP. Nevertheless, the wild bootstrap does mimic the essential features of the true DGP well enough for it to be useful in many cases.

86 Bootstrap Inference It is important to generate a valid bootstrap distribution under the null for correctly making inference from the data.

87 Bootstrap Inference It is important to generate a valid bootstrap distribution under the null for correctly making inference from the data. For this purpose, the null hypothesis must be imposed when generating the bootstrap samples.

88 Bootstrap Unit Root Testing Unit root (ADF) test as an example: The ADF regression x t = β 0 + β 1 x t1 + p δ j x t j + u t j=1

89 Bootstrap Unit Root Testing Unit root (ADF) test as an example: The ADF regression x t = β 0 + β 1 x t1 + p δ j x t j + u t j=1 The hypothesis of interest is H 0 : β 1 = 0 (unit root)

90 Bootstrap Unit Root Testing The test employed is the ordinary t statistic whose limit distribution is non-standard.

91 Bootstrap Unit Root Testing The test employed is the ordinary t statistic whose limit distribution is non-standard. The asymptotic approximation works very bad in typical small samples. The bootstrap approach is called for.

92 Bootstrap Unit Root Testing The bootstrap procedures are very much the same as previously described, except that the bootstrap sample {x t } is generated by x t = β 0 + p δ j x t j + u t j=1 where the restriction β 1 = 0 is imposed.

93 The Problem of the Stationarity Tests KPSS (1992) documented that their tests are subject to the size problem in the presence of highly persistent process.

94 The Problem of the Stationarity Tests KPSS (1992) documented that their tests are subject to the size problem in the presence of highly persistent process. Caner and Kilian (2001) reaffirmed the problem.

95 Construction of the KPSS Tests regress y t against an intercept, or an intercept and time trend, and obtain the residuals, denoted by û t.

96 Construction of the KPSS Tests regress y t against an intercept, or an intercept and time trend, and obtain the residuals, denoted by û t. compute the partial sum of the residuals, S t = t i=1 ûi, and estimate the long-run variance of ǫ t : ˆσ 2 = 1 T T û 2 t T t=1 L T w(i, L) û t û t i i=1 t=i+1

97 Construction of the KPSS Tests where w(i, L) = 1 i/(1 + L) is Bartlett kernel, and L is bandwidth. compute the statistic: KPSS = T 2ˆσ 2 T t=1 S 2 t

98 The Dangerous Zone: AR coefficient near 1 Simulated illustration to replicate findings of Caner and Kilian (2001).

99 The Dangerous Zone: AR coefficient near 1 Simulated illustration to replicate findings of Caner and Kilian (2001). DGP: y t = αy t 1 + e t, e t iid N(0, 1). Nominal level: 5%, replications: 5000.

100 Size Behavior of the KPSS Test T α KPSS τ (4) KPSS τ (8) KPSS τ (12)

101 The Dangerous Zone: AR coefficient near 1 Size distortion worsens as samples increase, and as AR coefficient approaches to 1.

102 The Dangerous Zone: AR coefficient near 1 Size distortion worsens as samples increase, and as AR coefficient approaches to 1. For fixed α that is near 1, size problem is reduced by an increase in bandwidth number.

103 The DGP DGP: represented by a component model: y t = r t + u t u t = αu t 1 + ǫ t r t = r t 1 + e t iid where c < 0, ǫ t (0, σǫ 2), e iid t (0, σe 2).

104 The DGP DGP: represented by a component model: y t = r t + u t u t = αu t 1 + ǫ t r t = r t 1 + e t iid where c < 0, ǫ t (0, σǫ 2), e iid t (0, σe 2). The null of concern H 0 : σ 2 e = 0

105 Problems with Bootstrapping Tests for Stationarity It is not clear how to impose the null constraint H 0 : σ 2 e = 0 when bootstrapping in the context.

106 Problems with Bootstrapping Tests for Stationarity It is not clear how to impose the null constraint H 0 : σ 2 e = 0 when bootstrapping in the context. This is more due to the impossibility of untangling the stationary component from the data in components models on which the tests are built.

107 Our Solutions: Using the Reduced Form The component model is 2 nd -order observationally equivalent in moments to an ARMA(1,1) for the differenced data: y t = α y t 1 + (1 θl)η t, where η t iid (0, σ 2 η), σ 2 η = σ 2 ǫ/θ, and θ = 1 2 {σ2 e σ 2 ǫ + 2 ( σ2 e σ 2 ǫ + 4 σ2 e σ 2 ǫ ) 1/2 }.

108 Our Solutions: Using the Reduced Form Based on the equivalence, H 0 : σ 2 e = 0 θ = 1.

109 Our Solutions: Using the Reduced Form Based on the equivalence, H 0 : σ 2 e = 0 θ = 1. The equivalence renders the bootstrapping feasible by imposing θ = 1 in the resampling scheme.

110 The Re-sampling Algorithm Step 1: Give a sample {y t } T t=1, fit y t with an ARMA(1,1) model y t = α y t 1 + η t θη t 1 to obtain α, θ and { η t } T t=1. Step 2: Draw a bootstrap sample from EDF { η t } T t=1, denoted by {η t }T t=1, with replacement.

111 The Re-sampling Algorithm Step 3: Generate a bootstrap sample {y t }T t=1 by the following bootstrap DGP: y t = y t 1 + α y t 1 + η t η t 1 Step 4: Compute KPSS test statistics, based on {y t } T t=1. Step 5: Repeat step 2 to step 4 NB times, to produce the bootstrap null distribution. Step 6: Compute the bootstrap C.V. at desired levels.

112 Simulation Evidence: Size T α KPSS µ (4) KPSS µ (8) KPSS µ (12) asym. boot. asym. boot. asym. boot DGP : y t = αy t 1 + e t, e t iid N(0, 1). 2. Nominal significance: 0.05; asym. c.v ; replications: NB=100, ND=1000.

113 Simulation Evidence: Size T α KPSS µ (4) KPSS µ (8) KPSS µ (12) asym. boot. asym. boot. asym. boot DGP : y t = αy t 1 + e t, e t iid N(0, 1). 2. Nominal significance: 0.05; asym. c.v ; replications: NB=100, ND=1000.

114 Simulation Evidence: Power T σζ 2 KPSS µ (4) KPSS µ (8) KPSS µ (12) asym. boot. asym. boot. asym. boot DGP : y t = αy t 1 + e t, e t iid N(0, 1). 2. Nominal significance: 0.05; asym. c.v ; replications: NB=100, ND=1000.

115 Simulation Evidence: Power T σζ 2 KPSS µ (4) KPSS µ (8) KPSS µ (12) asym. boot. asym. boot. asym. boot DGP : y t = αy t 1 + e t, e t iid N(0, 1). 2. Nominal significance: 0.05; asym. c.v ; replications: NB=100, ND=1000.

116 Empirical Application Evidence in support of PPP in previous studies was mixed, based on the asymptotic C.V.

117 Empirical Application Evidence in support of PPP in previous studies was mixed, based on the asymptotic C.V. Stronger evidence for PPP in OECD real exchange rates series by the bootstrap tests are observed, accounting for size distortions.

118 Bootstrap KPSS tests for real exchange rates country KPSS value ˆα bcv(5%) bcv(10%) Austria Belgium Canada Denmark Finland France Germany Greece Italy Japan Netherlands Norway Portugal Spain Sweden Switzerland United Kingdom All real exchange rates are constructed from IFS data on consumer price and end-of-period US$ exchange rates. Monthly data are for (312 obs.). 2. At the 5 (10)% significance level, the asymptotic critical value for KPSS test is (0.347). ( ) denotes a rejection at 5 (10)% level.

119 Bootstrap Inference Kuo and Lee (2002), Kuo and Tsong (2002), Kuo and Lee (2002), and Kuo, Lee ans Tsong (2004) all utilize this notion in constructing bootstrap tests for panel unit root, and for equal forecast accuracy.

120 Bootstrap Inference Kuo and Lee (2002), Kuo and Tsong (2002), Kuo and Lee (2002), and Kuo, Lee ans Tsong (2004) all utilize this notion in constructing bootstrap tests for panel unit root, and for equal forecast accuracy. Kuo and Tsong (2004) transform the component model on wihch the KPSS is based into a reduced form to have the null hypothesis imposed when generating bootstrap samples.

121 Other Issues Some important estimators are not covered, including kernel density estimator, non-parametric regression, GMM estimator, and non-smooth estimator.