Block Bootstrap Prediction Intervals for Autoregression

Transcription

1 Department of Economics Working Paper Block Bootstrap Prediction Intervals for Autoregression Jing Li Miami University 2013 Working Paper #

2 Block Bootstrap Prediction Intervals for Autoregression Jing Li Abstract Tis paper provides evidence tat te principle of parsimony can be extended to interval forecasts. We propose new prediction intervals based on parsimonious autoregressions. Te serial correlation in te error term is accounted for by te block bootstrap. Te proposed intervals generalize te i.i.d bootstrap intervals by allowing for serially correlated errors. Simulations sow te proposed intervals ave superior performance wen te serial correlation in te error term is strong and wen forecast orizon is sort. By applying te proposed intervals to U.S. inflation rate, we igligt a tradeoff between preserving correlation and adding variation. Keywords: Forecast; Block Bootstrap; Stationary Bootstrap; Prediction Intervals; Principle of Parsimony EconLit Subject Descriptors: C15; C22; C53 Jing Li, Department of Economics, Miami University, Oxford, OH 45056, USA. Pone: , Fax: , lij14@miamio.edu. 1

3 1. Introduction It is well known tat a parsimonious model may produce superior out-of-sample point forecasts tan a complicated model. Te main contribution of tis paper is to examine weter te principle of parsimony can be extended to interval forecasts. Te proposed block bootstrap prediction intervals (BBPI) are based on a parsimonious first order autoregression. By contrast, te i.i.d or standard bootstrap prediction intervals developed by Tombs and Scucany (1990) (called TS intervals tereafter) are based on te autoregression of order p, were p can be large. Te TS intervals assume te error term is independent. Tus te BBPI generalize te TS intervals by allowing for serially correlated errors. A largely overlooked fact is tat te principle of parsimony is forgone by te classical Box- Jenkins prediction intervals and te TS intervals. Bot require serially uncorrelated error terms, and terefore dynamically complete models 1. Usually te Breusc-Godfrey type test is conducted to ensure te adequacy of model. On te oter and, wen te goal is te point forecast, te models selected by criteria of AIC and BIC are typically parsimonious, but not necessarily adequate, see Enders (2009). Te proposed intervals do not require dynamically complete models. Unlike te Box- Jenkins intervals, non-normality can be automatically accounted for. Tis is because te block bootstrap is employed to obtain te empirical conditional distribution, wic can be non-normal, of te out-of-sample forecast. More explicitly, te block bootstrap redraws wit replacement random blocks of consecutive residuals of a parsimonious (sort) autoregression. Te blocking is intended to preserve te time dependence structure in te error term. By contrast, te TS intervals entail te i.i.d bootstrap of Efron (1979) tat resamples individual residual. Te i.i.d bootstrap works well in te independent setup. To satisfy te independence of te error term, a complicated (long) autoregression is needed for te TS intervals. 1 See te footnote page. 2

4 Tis paper also applies te bootstrap metod to direct forecasts. Tere are two ways to obtain te -step forecast. Te first is to run just one autoregression, and ten compute te -step forecast based on ( i)-step forecasts. We call tese iterated forecasts. Te second way is to run a set of direct autoregressions. Tey all use y t as te dependent variable, but te regressors are different. In te simplest case, te only regressor is y t 1 in te first direct regression, y t 2 in te second direct regression, and so on. We call tese direct forecasts. Tis paper considers bot forecasts. Tere are tree steps to construct te BBPI. In step one te AR(1) regression is estimated by ordinary least squares (OLS), and te residual is saved. In step two a backward AR(1) regression is fitted, and random blocks of residuals are used to generate te bootstrap replicate. In step tree te bootstrap replicate is used to run te AR(1) regression again and random blocks of residuals in step one are used to compute te bootstrap out-of-sample forecast. After repeating steps two and tree many times, te BBPI are determined by te percentiles of te empirical distribution of te bootstrap forecast. We discuss tecnical issues suc as correcting te bias of autoregressive coefficients, selecting te block size (lengt), coosing between overlapping and non-overlapping blocks and using te stationary bootstrap tat sets te block size as random. Monte Carlo experiment compares te average coverage rate of te BBPI to te TS intervals. Tere are two key findings. Te first is tat te BBPI dominate wen te error term sows strong serial correlation after y t 1 being controlled for. Te second is tat te BBPI always outperform te TS intervals for te one-step forecast. For longer forecast orizon, it is possible tat te TS intervals perform better. Our findings igligt a tradeoff between preserving correlation and adding variation. Te block bootstrap acieves te former but sacrifices te latter. Tis tradeoff is also illustrated wen we apply te BBPI to te U.S. montly inflation rate. Te literature on te bootstrap prediction intervals is growing fast. Important works 3

5 include Tombs and Scucany (1990), Masarotto (1990), Grigoletto (1998), Clements and Taylor (2001), Kim (2001), and Kim (2002). Te block bootstrap and stationary bootstrap are fully developed by Künsc (1989) and Politis and Romano (1994). See Stock and Watson (1999) and Stock and Watson (2007) for more discussion about forecasting te U.S. inflation rate. Te remainder of te paper is organized as follows. Section 2 specifies te BBPI. Section 3 conducts te Monte Carlo experiment. Section 4 provides an application, and Section 5 concludes. 2. Block Bootstrap Prediction Intervals Iterated Block Bootstrap Prediction Intervals Let {y t, t Z} be a strictly stationary and weakly dependent time series wit mean of zero. In practice y t may represent te demeaned, differenced, detrended or deseasonalized series. Te goal is to find te prediction intervals for future values (y n+1, y n+2,..., y n+ ), were is te maximum forecast orizon, after observing Ω = (y 1,..., y n ). Tis paper focuses on te bootstrap prediction intervals because (i) tey do not assume tat te distribution of y n+i conditional on Ω is normal, and (ii) te bootstrap intervals can automatically take into account te sampling variability of te estimated coefficients. Te TS intervals of Tombs and Scucany (1990) are based on a long p-t order autoregression: y t = ψ 1 y t 1 + ψ 2 y t ψ p y t p + e t. (1) Te TS intervals assume te error e t is independent of (or at least uncorrelated wit) e t j, j 0. Tis assumption requires tat te model (1) be dynamically adequate. In oter words, sufficient number of lagged values sould be included. It is not uncommon tat in practice te final model can be complicated, wic is inconsistent wit te principle of parsimony. Actually te model (1) is just a finite-order approximation if te true process is 4

6 ARMA process tat as AR( ) representation. In teory te error term e t can be serially correlated no matter ow large p is. Tis implies te independent assumption can be too restrictive. Tis paper relaxes te assumption of independent errors, and proposes te block bootstrap prediction intervals (BBPI) based on a sort autoregression. Consider te AR(1) model, te most parsimonious autoregression y t = ϕ 1 y t 1 + v t. (2) Te error v t is allowed to be serially correlated, so Model (2) can be inadequate. Neverteless, te serial correlation in v t sould be utilized to improve te forecast. Toward tat end te block bootstrap later will be applied to te residual ˆv t = y t ˆϕ 1 y t 1, (3) were ˆϕ 1 is te coefficient estimated by OLS. But first, any bootstrap prediction intervals sould account for te sampling variability of ˆϕ 1. Tis is accomplised by running repeatedly te regression (2) using te bootstrap replicate, a pseudo time series. Following Tombs and Scucany (1990) we generate te bootstrap replicate using te backward representation of te AR(1) model y t = θ 1 y t+1 + u t. (4) Note tat te regressor is lead not lag. Denote te OLS estimate by ˆθ 1, and te residual by û t : û t = y t ˆθ 1 y t+1, (5) 5

7 ten one series of te bootstrap replicate (y 1,..., y n) is computed in a backward fasion as (starting wit te last observation, ten moving backward) y n = y n, y t = ˆθ 1 y t+1 + û t, (t = n 1,... 1). (6) By using te backward representation we can ensure te conditionality of AR forecasts on te last observed value y n. Put differently, all te bootstrap replicate series ave te same last observation, yn = y n. See Figure 1 of Tombs and Scucany (1990) for an illustration of tis conditionality. In equation (6) te randomness of te bootstrap replicate comes from te pseudo error term û t, wic is obtained by te block bootstrap as follows: 1. Save te residual of te backward regression û t given in (5). 2. Let b denote te block size (lengt). Te first (random) block of residuals is B 1 = (û i1, û i1+1,..., û i1+b 1 ), (7) were te index number i1 is a random draw from te discrete uniform distribution between 1 and n b+1. For instance, let b = 3 and suppose a random draw yields i1 = 20, ten B 1 = (û 20, û 21, û 22 ). In tis example te first block contains tree consecutive residuals starting from te 20t observation. By redrawing te index number wit replacement we can obtain te second block B 2 = (û i2, û i2+1,..., û i2+b 1 ), te tird block B 3 = (û i3, û i3+1,..., û i3+b 1 ), and so on. We stack up tese blocks until te lengt of te stacked series becomes n. û t denotes te t-t observation of te stacked series. Resampling blocks of residuals is intended to preserve te serial correlation of te error term in te parsimonious model. Generally speaking, te block bootstrap can be applied 6

8 to any weakly dependent stationary series. Here it is applied to te residual of te sort autoregression. By contrast, te TS intervals resample te individual residual of a long autoregression. Tere are several issues. Te first is ow to determine te block size b. Tere is no definite answer due to a tradeoff between correlation and variation. We face te same tradeoff wen coosing between te block bootstrap and te i.i.d bootstrap. Longer block (bigger b) can capture more serial correlation. But longer block also reduces te variation in te bootstrap replicate because (i) te total number of blocks falls and (ii) te cance of overlapping blocks rises. Te general rule is tat b sould rise wen serial correlation gets stronger, or wen te sample size grows. For our purpose tis issue may not be as important as it appears, because simulations will sow te superiority of te BBPI over te TS intervals may be insensitive to b. Alternatively, one may resample blocks wit random sizes, were te block size follows a geometric distribution. Tat is te main idea of te stationary bootstrap suggested by Politis and Romano (1994). More explicitly, te distribution of b is specified as P (b = j) = p(1 p) j, (j = 0, 1, 2,...). (8) Wen p rises, te probability of generating small values of b also rises. Notice tat using stationary bootstrap does not solve te problem completely. Now te new problem is ow to select te probability parameter p. Te rule is we sould let p fall wen te serial correlation gets stronger. In te simulation we replace b = 0 wit b = 1 if tat appens. Te second issue is overlapping vs non-overlapping blocks. Te algoritm above indicates te blocks are possibly overlapping. For example, B 1 partially overlaps wit B 2 wen i1 < i2 < i1 + b. To generate non-overlapping blocks, we need to randomly redraw wit 7

9 replacement te index numbers i1, i2,... from te below set: {1, 1 + b, 1 + 2b,..., 1 + kb}, (9) were k n b 1. Note tere are gaps between te values in (9), wic ensure te blocks are not overlapping. Wen b rises, using non-overlapping blocks generates less randomness tan overlapping blocks, simply because k falls and tere are fewer values left in (9). We will revisit tis issue in te simulation. See Andrews (2004) for a discussion in te context of ypotesis testing. Finally, it is not required tat te modulo of dividing n by b be zero. Only a portion of te last block is included in te stacked series wit nonzero modulo. For example, it is ok tat te lengt of stacked series is 100, b = 3, and 100 is not a multiple of 3. A total of 34 blocks are drawn, but only te first observation of te 34-t block is included in te stacked series. After generating te bootstrap replicate series using (6), next we refit te model (2) using te bootstrap replicate (y2,..., yn). Denote te newly estimated coefficient (called bootstrap coefficient) by ˆϕ 1. Ten we can compute te iterated block bootstrap l-step forecast ŷn+l as ŷ n = y n, ŷ n+l = ˆϕ 1ŷ n+l 1 + ˆv l, (l = 1,..., ) (10) were te pseudo error ˆv l is obtained by block bootstrapping te residual (3). For example, let = 8, b = 4. Ten two blocks of residuals (3) are randomly drawn, and tey are B 1 = (ˆv i1, ˆv i1+1, ˆv i1+2, ˆv i1+3 ), B 2 = (ˆv i2, ˆv i2+1, ˆv i2+2, ˆv i2+3 ). Ten ˆv l in equation (10) is te l-t observation of te stacked series {ˆv l } l=1 = {ˆv i1, ˆv i1+1, ˆv i1+2, ˆv i1+3, ˆv i2, ˆv i2+1, ˆv i2+2, ˆv i2+3 }. (11) 8

10 Te ordering of B 1 and B 2 in te stacked series (11) does not matter. It is te ordering of te observations witin eac block tat matters. Tat witin-block ordering preserves te temporal structure. Notice tat te block bootstrap as been invoked twice: first it is applied to û t given in (5), ten applied to ˆv t given in (3). Te first application is to add randomness to te bootstrap replicate yt, wic is used to rerun te autoregression and simulate te sampling variability of te estimated coefficient. Te second application is to randomize te predicted value ŷn+l. Te block size wen resampling ˆv t is te same as û t since (4) and (2) sare te same structure of serial correlation, see Box et al. (2008). To get te BBPI, we need to generate C series of te bootstrap replicate using (6), fit te model (2) using te C bootstrap replicate series, and use (10) to obtain a series of te iterated block bootstrap l-step forecasts {ŷ n+l(i)} C i=1 (12) were i is te index. Te l-step iterated BBPI at te α nominal level are given by l step Iterated BBPI () = [ ŷ n+l ( 1 α 2 ), ŷ n+l ( )] 1 + α 2 (13) were ŷ n+l ( 1 α 2 ) and ŷ n+l ( 1+α 2 ) are te ( 1 α 2 ) ( 100-t and 1+α ) t percentiles of te empirical distribution of {ŷ n+l (i)}c i=1. Trougout tis paper we let α = To avoid te discreteness problem, one may let C = 999, see Boot and Hall (1994). In tis paper we use C = 1000 and find no qualitative difference. Basically we apply te percentile metod of Efron and Tibsirani (1993) to construct te BBPI. De Gooijer and Kumar (1992) empasize te percentile metod performs well wen te conditional distribution of te predicted values is unimodal. In preliminary simulation we conduct te DIP test of Hartigan and Hartigan (1985) and find tat te distribution is 9

11 indeed unimodal. Hall (1988) discusses oter percentile metods. Bias-Corrected Prediction Intervals It is well known tat te autoregressive coefficient estimated by OLS can be biased, see Saman and Stine (1988) for instance. Tat implies te BBPI (13) can be improved by correcting te bias. Following Kilian (1998) we generate D series of te bootstrap replicate using (6), ten refit te backward regression (4) using tese bootstrap replicate series and get a series of bootstrap backward coefficients {ˆθ 1(i)} D i=1. Next compute te bias-corrected coefficient, bias-corrected residual and bias-corrected bootstrap replicate as D ˆθ 1 c = 2ˆθ 1 D 1 ˆθ 1(i), (14) i=1 û c t = y t ˆθ c 1y t+1, (15) yn c = y n, yt c = ˆθ 1y c t+1 c + û c t, (t = n 1,... 1), (16) were û c t is obtained by block bootstrapping û c t. Finally, refit te model (2) using te biascorrected bootstrap replicate y c t and compute te bias-corrected bootstrap forecast as ŷ c n = y n, ŷ c n+l = ˆϕ c 1 ŷ c n+l 1 + ˆv l, (l = 1,..., ), (17) were ˆv l is obtained by block bootstrapping ˆv t (3). Te bias-corrected BBPI are determined by te percentiles of te distribution of ŷ c n+l. Actually we can go one step furter by bias correcting te residual ˆv of (3). To do so, we need to use te D series of te bootstrap replicate to refit te forward regression (2). After obtaining a series of bootstrap forward coefficients { ˆϕ 1(i)} D i=1, compute te bias-corrected 10

12 forward coefficient and bias-corrected forward residual as ˆϕ c 1 = 2 ˆϕ D 1 D 1 ˆϕ 1(i), (18) i=1 ˆv c t = y t ˆϕ c 1y t 1. (19) Ten te twice bias-corrected bootstrap forecast is computed as ŷ 2c n = y n, ŷ 2c n+l = ˆϕ c 1 ŷ 2c n+l 1 + ˆv c l, (l = 1,..., ), (20) were ˆv c l is obtained by block bootstrapping ˆv c t (19). Equation (20) makes it clear tat ˆv c l as direct effect on te bootstrap forecast, wile û c t only as an indirect effect troug ˆϕ c 1. So we conjecture tat bias correcting te forward residual ˆv t is more important tan te backward residual û t. Tis conjecture will be verified by te simulation. In practice one may also apply te stationary correction recommended by Kilian (1998) to ensure te series of bootstrap replicate y t is stationary. Direct Block Bootstrap Prediction Intervals We call te BBPI (13) iterated because te forecast is computed in an iterative fasion: in (10) te previous step forecast ŷn+l 1 is used to compute te next step ŷ n+l. Alternatively, we can use te bootstrap replicate (y 1,..., y n) to run a set of direct regressions using only one regressor. In total tere are direct regressions. More explicitly, te l-t direct regression uses yt as te dependent variable and yt l as te independent variable. Denote te estimated direct coefficient by ˆρ l. Te residual is computed as ˆη t,l = y t ˆρ l y t l. (21) 11

13 Ten te direct bootstrap forecast is computed as ŷ d n+l = ˆρ l y n + ˆη l (22) were ˆη l is a random draw wit replacement from te empirical distribution of ˆη t,l. Te l-step direct BBPI at te α nominal level are given by l step Direct BBPI (DBBPI) = [ ŷ d n+l ( 1 α 2 ), ŷ d n+l ( )] 1 + α 2 (23) were ŷ d n+l ( 1 α ) 2 and ŷ d n+l ( 1+α 2 empirical distribution of {ŷ d n+l (i)}c i=1. ) are te ( 1 α 2 ) ( 100-t and 1+α ) t percentiles of te Tere are oter ways to obtain te direct prediction intervals. For example, te bootstrap replicate (y 1,..., y n) can be generated based on te backward form of direct regression. It is also possible to bias correct te coefficient in te direct regression. Ing (2003) compares te mean-squared prediction errors of te iterated and direct point forecasts. In te next section we will compare te iterated and direct BBPIs. 3. Monte Carlo Experiment Error Distributions Tis section compares te performances of various bootstrap prediction intervals using te Monte Carlo experiment. First we investigate te distribution of error term. Following Tombs and Scucany (1990), te data generating process (DGP) is a stationary AR(2) process: y t = ϕ 1 y t 1 + ϕ 2 y t 2 + u t (24) were ϕ 1 = 0.75, ϕ 2 = 0.5, t = 1,..., 55. Te error u t follows an independently and identically distributed process. Tree error distributions are considered: te standard normal 12

14 distribution, te exponential distribution wit mean of 0.5, and mixed normal distribution 0.9N( 1, 1) + 0.1N(9, 1). Te exponential distribution is skewed; te mixed normal distribution is bimodal and skewed. All distributions are centered to ave zero mean. We compare tree bootstrap prediction intervals. Te iterated block bootstrap prediction intervals () are based on te sort AR(1) regression (2) and its backward form (4). Te TS intervals of Tombs and Scucany (1990) are based on te long AR(2) regression (24) and its backward form. Because we know te true DGP is AR(2), te error terms in te AR(1) regression are serially correlated, wile te errors are uncorrelated in te AR(2) regression. Finally te direct block bootstrap prediction intervals (DBBPI) are based on a series of first order direct autoregressions. Eac bootstrap prediction intervals are based on te empirical distribution of 1000 bootstrap forecasts. Tat is, we let C = 1000 in (12) for te, and so on. For te and DBBPI te block size b is set to 4. Te TS intervals use te i.i.d bootstrap, so block size is irrelevant. Te first 50 observations (n = 50) are used to run te regression. Ten we evaluate weter te last 5 observations are inside te corresponding prediction intervals. In oter words, we focus on out-of-sample forecasting. Te main criterion of comparing performance is te average coverage rate () given as: () = m 1 m i=1 1(y n+ Prediction Intervals) (25) were 1(.) denotes te indicator function. Te number of iteration is set as m = We find no qualitative difference if using m = Te forecast orizon ranges from 1 to 5. Te nominal coverage α is Te intervals wose is closest to 0.90 are deemed te best. Figure 1 plots te against wen te error distribution varies. Te s of te, TS intervals and DBBPI are denoted by circle, square and star, respectively. In 13

15 te leftmost grap te error follows te standard normal distribution. It is sown tat te of te is closest to te nominal coverage 0.90, followed by te TS intervals. Te DBBPI ave te worst performance. For instance, wen = 5, te ave of 3, te TS intervals ave of 4, and te DBBPI ave of 9. Te ranking remains largely uncanged wen te errors follow te exponential and mixed normal distributions, sown in te middle and rigtmost graps. Overall, Figure 1 indicates tat (i) te ave te best performance, and (ii) te DBBPI ave te worst performance. Finding (ii) is consistent wit previous works suc as Ing (2003) wic sows te iterated point forecast outperforms te direct point forecast. Finding (i) is new, and may be explained by te fact tat te are based on te parsimonious model. By comparing te tree graps, we see no big cange in wen te error distribution varies. Tis is expected because all intervals are bootstrap intervals tat do not assume normality. Autoregressive Coefficients Now we consider varying autoregressive coefficients in te DGP (24): ϕ 1 = 0.75, ϕ 2 = 0.5 (stationary AR(2)) (26) ϕ 1 = 1.0, ϕ 2 = 0.24 (stationary AR(2)) (27) ϕ 1 = 1.2, ϕ 2 = 0.2 (non-stationary AR(2)) (28) were t = 1,..., 55 and u t iidn(0, 1). Te leftmost grap in Figure 2 looks similar to tat in Figure 1 since te same DGP is used. In te middle grap we see no difference in te ranking. Te rigtmost grap is interesting. In tis case te sum of autoregressive coefficients is = 1. So te data are nonstationary (one caracteristic root is 1), violating te assumption of stationarity. Tis violation leads to distortion in te coverage rate, particularly wen is big. For example, at = 5, te of te is between 14

16 and wen data are stationary, but close to 0.76 wen data are nonstationary. In ligt of tis we recommend applying te prediction intervals to te differenced data if te data contain unit roots. We also see te direct intervals are te best wen data are nonstationary. Tis may be due to te fact tat te direct intervals are based on te direct regression. Sample Sizes Figure 3 is concerned wit te sample size. Te DGP is (24) wit ϕ 1 = 0.75, ϕ 2 = 0.5, u t iidn(0, 1). Tree sample sizes are used for in-sample fitting: n = 30, n = 60 and n = 100. From Figure 3 we see as te sample size rises, te lines in most cases move closer to te nominal level Terefore rising sample size improves te performance of all intervals. Block Sizes Now we focus on te, and investigate ow te block size affects performance. Te DGP is (24) wit t = 1,..., 55, u t iidn(0, 1). Te coefficients are given in (26), (27) and (28). Tere are two findings from Figure 4. First, using te block size of one (i.e., te i.i.d bootstrap, denoted by circle) leads to te worst performance. Te second finding is tat te difference between using te block sizes of tree (denoted by square) and five (denoted by star) is marginal in most cases. Tis suggests tat te performance of te BBPI may be insensitive to te block size. As long as blocks are used, te block size may be a secondary issue. Block Bootstrap Intervals vs Stationary Bootstrap Intervals Instead of using te block of a fixed size, we can employ te stationary bootstrap tat resamples blocks of random sizes. Using te same DGP as Figure 4, Figure 5 compares te (iterated) block bootstrap prediction intervals (BBPI) wit b = 4 (denoted by circle) to te 15

17 stationary bootstrap prediction interval (SBPI) of te block size tat follows te geometric distribution wit parameter p = 0.3 (denoted by square). Figure 5 sows tat in most cases te difference between te two intervals is not substantial. No one can dominate te oter. Overall, it is safe to say te stationary bootstrap, wic depends on te coice of p, is an alternative to te block bootstrap, wic depends on te coice of b. Overlapping vs Non-overlapping Blocks Using te same DGP as Figure 4, Figure 6 compares te (iterated) block bootstrap prediction intervals using overlapping (denoted by circle) and non-overlapping (denoted by square) blocks. Te block size is 4. We see tat using overlapping blocks yields better performance tan non-overlapping blocks. Te reason may be tat overlapping blocks can produce more variation in te bootstrap replicate tan non-overlapping blocks. Bias Correction Now we investigate weter correcting te bias of autoregressive coefficient can improve te performance of intervals. Figure 7 compares te iterated block bootstrap prediction intervals based on (10), (17) and (20). Basically (10) does not correct te bias; (17) only corrects te bias of ˆθ 1, te coefficient in te backward regression; (20) corrects bot ˆθ 1 and ˆϕ 1, te latter is te coefficient in te forward regression. In Figure 7 tese intervals are denoted by circle (no bias correction, or no BC), square (BC Once) and star (BC Twice), respectively. Te DGP is te same as Figure 4. From Figure 7 it is clear tat te BC Twice intervals ave te best performance. It appears tat correcting te coefficient in te backward regression alone results in almost no improvement. In ligt of tis we conclude tat bias-correcting te coefficient in te forward model (2) is far more important tan bias-correcting te coefficient in te backward model (4). Anoter finding is tat bias-correcting coefficients puses te coverage rate up toward 16

18 te nominal Tis indicates tat at least part of te downward bias in coverage rate we see in all graps is caused by te bias of autoregressive coefficients. Principle of Parsimony So far te DGP as been te AR(2) model (24). Next we cange te DGP to an ARMA(1,1) process y t = ϕy t 1 + u t + θu t 1 (29) were t = 1,..., 55 and u t iidn(0, 1). In teory for tis DGP tere exists AR( ) representation. Tus te AR(p) regression is finite-order approximation. We verify te principle of parsimony in tree ways. Figure 8 compares te iterated block bootstrap prediction intervals based on te AR(1) regression, to te TS intervals based on te AR(2) regression (TS2, denoted by diamond), te AR(3) regression (TS3, denoted by square) and te AR(4) regression (TS4, denoted by star). For te TS intervals we do not ceck weter te residual is serially correlated. Tat job is left to Figure 9. Figure 8 uses tree sets of ϕ and θ. Regardless of te coefficient, te block bootstrap intervals ave te best performance. Wen te autoregression becomes longer, te corresponding TS intervals sow worse performance. Tis is te first evidence tat te principle of parsimony may work for interval forecasts. Te second evidence is presented in Figure 9, were te TS intervals are based on te autoregression wose order is determined by te Breusc-Godfrey test. Te Breusc-Godfrey test is appropriate since te regressors are lagged dependent variables. We start from te AR(1) regression. If te residual passes te Breusc-Godfrey test, ten te AR(1) regression is cosen for constructing te TS intervals. Oterwise we cange to te AR(2) regression, apply te Breusc-Godfrey test again, and so on. In te end, te TS intervals are based on an adequate autoregression wit serially uncorrelated errors. 17

19 In te leftmost grap of Figure 9, ϕ = 0.4, θ = 0.2. We see te iterated block bootstrap intervals outperform te TS intervals wen equals 1 and 2. For greater teir ranking reverses. In te middle and rigtmost graps, more serial correlation is induced as θ rises from 0.2 to 0.6, and as ϕ rises from 0.4 to 0.9. In tose two graps te BBPI dominate te TS intervals. Te fact tat te ranking of te BBPI and TS intervals switces in te leftmost grap indicates a tradeoff between preserving serial correlation and adding variation. Remember tat te BBPI use te block bootstrap, so empasize preserving serial correlation. By contrast te TS intervals use te i.i.d bootstrap, wic can generate more variation in te bootstrap replicate tan te block bootstrap. Keeping tat in mind, ten te leftmost grap makes more sense. In tat grap θ is 0.2, close to zero. Tat means te ARMA(1,1) model is essentially an AR(1) model wit weakly correlated error. Terefore preserving correlation becomes less important tan adding variation, in particular for long-orizon forecasts. Figure 10 examines te principle of parsimony using te AR(2) model (24) as te DGP, but from te perspective of caracteristic roots of difference equations. Te relationsip between te autoregressive coefficients and caracteristic roots (λ 1, λ 2 ) is ϕ 1 = λ 1 + λ 2, ϕ 2 = λ 1 λ 2 (30) Note tat wen one of te caracteristic roots, say λ 2, is close to zero, ten ϕ 2 will be close to zero. In tat case λ 1 will dominate, and te second order difference equation beaves like a first order equation. In finite sample it is igly probable tat te Breusc-Godfrey will indicate te AR(1) regression is adequate, despite tat te true DGP is AR(2). Tat is te case in te leftmost grap of Figure 10, wen λ 2 = 0.1 and ϕ 2 = 0.05, bot being close to zero. We apply te Breusc-Godfrey test for model selection wen constructing te TS intervals. Most often te test picks te AR(1) regression. Te leftmost 18

20 grap sows te BBPI outperform te TS intervals only by a small margin. Actually tere is tendency tat as rises te ranking of two intervals may reverse. So we conclude tat in te presence of weak correlation, te TS intervals may perform better, in particular wen is large, since te i.i.d bootstrap generates more variation. Te BBPI may ave worse performance since te serial correlation sould oterwise be downplayed. It is instructive to consider te limit, wen serial correlation becomes 0 and data become independent. Ten te block size sould reduce to 1, and te block bootstrap sould degenerate to te i.i.d bootstrap, wic works best in te independent setting. Te middle and rigtmost graps in Figure 10 increase λ 1 and λ 2, respectively. Wit rising serial correlation, now te BBPI outperform te TS intervals by a large margin. Finally, from Figures 8, 9 and 10 we notice tat wen = 1, te BBPI always outperform te TS intervals, weter te serial correlation is weak or strong. Tis fact adds value to te BBPI for sort-orizon forecasts. Te next section uses real data to illustrate tis value. 4. Bootstrap Prediction Intervals for U.S. Montly Inflation Rate From te Federal Reserve Economic Data we download te montly 2 consumer price index (CPI) for all urban consumers. Te series is seasonally adjusted, and is from January 1947 to February Panel A of Figure 11 plots te CPI. Te series is trending and smoot. We ten compute te montly inflation rate (INF) as inf t = log(cpi t ) log(cpi t 1 ). Te skewness and kurtosis of INF are 0.55 and 6.99, respectively. Moreover, te skewness and kurtosis test rejects te normality at 0.01 level. Given te non-normality our goal is to obtain te bootstrap prediction intervals. We consider te BBPI, wic are based on te AR(1) regression, and te TS intervals, wic are based on te autoregression selected by te Breusc-Godfrey test. Using te wole sample (794 observations), te model tat passes te Breusc-Godfrey test is te AR(10) 2 See te footnote page. 19

21 regression 3 : inf t = 0.43 inf t inf t inf t inf t inf t inf t inf t inf t inf t inf t 10 + û t (31) were and denote significance at 0.01 and 0.05 level, respectively. Te p-value of te Breusc-Godfrey test applied to te residual û t is 0.22, so te null ypotesis of no AR(1) serial correlation is not rejected. Te AR(10) regression is obviously not parsimonious, and it is difficult to interpret te dynamics. Hence we try using te AR(2) regression as an approximation for te AR(10) regression: inf t = 0.49 inf t inf t 2 + û t (32) Te AR(2) model is not adequate; te p-value of te Breusc-Godfrey test is less tan However, te AR(2) model captures te essence of dynamics. It sows tat ˆϕ 2 = 0.13 is close to zero. Tat means (i) te remaining correlation is weak (but significant) after inf t 1 being controlled for; (ii) in a subsample wit small size te second lagged term inf t 2 may be ignored by te Breusc-Godfrey test. For example, if we use only te first 72 observations, te regression selected by te Breusc-Godfrey is te AR(1) regression inf t = 0.44 inf t 1 + û t (33) Te p-value of te Breusc-Godfrey test is After referring to te leftmost grap in Figures 9, we expect tat in small subsamples te BBPI will outperform te TS intervals wen is small. For long forecast orizon we expect vice versa. We construct te BBPI and TS intervals based on rolling windows. Te first window 3 See te footnote page. 20

22 W 1 contains te first n + observations (i.e., W 1 = {inf 1,..., inf n+ }), ten we move one period aead, and define te second window as te next n + observations (i.e., W 2 = {inf 2,..., inf n++1 }), and so on. We consider tree subsample sizes: n = 24, 48, 72, and we let te forecast orizon be = 1,..., 5. In eac window, te first n observations are used for in-sample fitting. Ten we evaluate weter te last observations are inside te prediction intervals. Figure 12 plots te average coverage rate (25) of te BBPI (wit b = 4) and TS intervals. It is sown tat te BBPI outperform te TS intervals only wen = 1. Tis finding is consistent wit our expectation. In tis application te error term in te AR(1) regression is weakly correlated. Preserving correlation becomes secondary, so te BBPI do not dominate te TS intervals. Figure 12 provides a macro picture. To get a micro view, we focus on n = 72 and te first-step forecast intervals. Panel B of Figure 11 igligts (using vertical grid lines) te observations were te inflation rate is uncovered by bot te BBPI and TS intervals. For example, te CPI decreased from to in October 2008, and ten fell to in November Te montly inflation rate was - percent and percent, respectively. Bot intervals fail to cover tese inflation rates. Panel C of Figure 11 igligts te observations tat are covered by te BBPI but not te TS intervals. Panel D does te opposite. In recent istory after year 2005, 6 observations are covered by te BBPI but not te TS intervals, wile 3 observations are covered by te TS intervals but not te BBPI. 5. Conclusion Tis paper proposes new prediction intervals by applying te block bootstrap to te first order autoregression. Te AR(1) model is parsimonious in wic te error term can be serially correlated. Ten te block bootstrap is utilized to resample blocks of consecutive 21

23 observations in order to maintain te time series structure of te error term. Te forecasts can be obtained in an iterated manner, or by running direct regressions. Te Monte Carlo experiment sows (1) tere is evidence tat te principle of parsimony can be extended to interval forecast; (2) tere is tradeoff between preserving correlation and adding variation; (3) te proposed intervals ave superior performance for one-step forecast; (4) bias-correcting te coefficient, particularly in te forward regression, can improve te intervals; (5) te performance of te proposed intervals may be insensitive to te block size; and (6) in most cases, te direct forecast performs worse tan te iterated forecast. We apply te proposed intervals to U.S. montly inflation rate. In small subsamples it is sown tat te inflation may follow te AR(1) process wit weakly correlated error. In tis application, te proposed intervals outperform te i.i.d bootstrap prediction intervals for one-step forecast. Wen te forecast orizon rises, adding variation outweigs preserving correlation. So te i.i.d bootstrap prediction intervals outperform te proposed intervals for long-orizon forecasts. 22

24 References Andrews D Te block-block bootstrap: Improved asymptotic refinements. Econometrica 72: Boot J G and Hall P Monte carlo approximation and te iterated bootstrap. Biometrika 81: Box G, Jenkins G M, and Reinsel G C Time Series Analysis Forecasting and Control. Wiley, Hoboken, New Jersey, 4 edition. Clements M P and Taylor N Boostrapping prediction intervals for autoregressive models. International Journal of Forecasting 17: De Gooijer J G and Kumar K Some recent developments in non-linear time series modeling, testing, and forecasting. International Journal of Forecasting (8): Efron B Bootstrap metod: Anoter look at te jackknife. Annals of Statistics 7:1 26. Efron B and Tibsirani R J An Introduction to te Bootstrap. Capman and Hall, London. Enders W Applied Econometric Times Series. Wiley, 3 edition. Grigoletto M Bootstrap prediction intervals for autoregressions: some alternatives. International Journal of Forecasting 14: Hall P Teoretical comparison of bootstrap confidence intervals. Annals of Statistics 16: Hartigan J A and Hartigan P M Te DIP test of unimodality. Annals of Statistics 13:

25 Ing C K Multistep prediction in autogressive processes. Econometric Teory 19: Kilian L Small sample confidence intervals for impulse response functions. Te Review of Economics and Statistics 80: Kim J bootstrap-after-bootstrap prediction intervals for autoregressive models. Journal of Business & Economic Statistics 19: Kim J Bootstrap prediction intervals for autoregressive models of unknown or infinite lag order. Journal of Forecasting 21: Künsc H R Te jackknife and te bootstrap for general stationary observations. Annals of Statistics 17: Masarotto G bootstrap prediction intervals for autoregressions. International Journal of Forecasting 6: Politis D N and Romano J P Te stationary bootstrap. Journal of te American Statistical Association 89: Saman P and Stine R A Te bias of autoregressive coefficient estimators. Journal of te American Statistical Association 83: Stock J H and Watson M W Forecasting inflation. Journal of Monetary Economics 44: Stock J H and Watson M W Wy as U.S. inflation become arder to forecast. Journal of Money, Credit and Banking 39:3 33. Tombs L A and Scucany W R Bootstrap prediction intervals for autoregression. Journal of te American Statistical Association 85:

26 0.89 Normal Distribution TS DBBPI 0.89 Exponential Distribution TS DBBPI Mixed Normal Distribution 0.9 TS 0.89 DBBPI Figure 1: Error Distributions 25

27 0.89 pi1 = 0.75; pi2 = 0.5 TS DBBPI pi1 = 1.0; pi2 = 0.24 TS DBBPI pi1 = 1.2; pi2 = 0.2 TS DBBPI Figure 2: Autoregressive Coefficients 26

28 n = 30 TS DBBPI n = 60 TS DBBPI n = 100 TS DBBPI Figure 3: Sample Sizes 27

29 pi1 = 0.75; pi2 = 0.5 blocksize=1 blocksize=3 blocksize= pi1 = 1.0; pi2 = 0.24 blocksize=1 blocksize=3 blocksize=5 pi1 = 1.2; pi2 = 0.2 blocksize=1 blocksize=3 blocksize= Figure 4: Block Sizes 28

30 0.9 pi1 = 0.75; pi2 = 0.5 BBPI SBPI pi1 = 1.0; pi2 = 0.24 BBPI SBPI pi1 = 1.2; pi2 = 0.2 BBPI SBPI Figure 5: Stationary Bootstrap vs Block Bootstrap 29

31 pi1 = 0.75; pi2 = 0.5 Overlapping BBPI Nonoverlapping BBPI 0.89 pi1 = 1.0; pi2 = 0.24 Overlapping BBPI Nonoverlapping BBPI pi1 = 1.2; pi2 = 0.2 Overlapping BBPI Nonoverlapping BBPI Figure 6: Overlapping vs Nonoverlapping Bootstrap 30

32 pi1 = 0.75; pi2 = 0.5 No BC BC Once BC Twice 5 pi1 = 1.0; pi2 = 0.24 No BC BC Once BC Twice 0.9 pi1 = 1.2; pi2 = 0.2 No BC BC Once BC Twice Figure 7: Bias Correction 31

33 0.89 pi = 0.4; teta = 0.2 TS2 TS3 TS4 pi = 0.4; teta = 0.6 TS2 TS3 TS4 pi = 0.9; teta = 0.6 TS2 TS3 TS Figure 8: Parsimony I 32

34 pi = 0.4; teta = 0.2 TS 5 pi = 0.4; teta = 0.6 TS pi = 0.9; teta = 0.6 TS Figure 9: Parsimony II 33

35 lambda1 = 0.5; lambda2 = 0.1 TS lambda1 = 0.5; lambda2 = TS lambda1 = 0.7; lambda2 = 0.4 TS Figure 10: Parsimony III 34

36 CPI Panel A: US Montly Consumer Price Index Panel B: Observations of INF Uncovered by BBPI and TS INF Panel C: Observations of INF Uncovered by TS Only INF Panel D: Observations of INF Uncovered by BBPI Only INF Figure 11: Time Series Plot of U.S. Montly Consumer Price Index and Inflation Rate 35

37 0.89 n = 24 TS n = 48 TS n = 72 TS Figure 12: Bootstrap Prediction Intervals for U.S. Montly Inflation Rate 36

38 Footnotes 1. Consider te AR(1) model y t+j = ϕy t+j 1 + e t+j wit te MA representation y t+j = e t+j + ϕe t+j 1 + ϕ 2 e t+j Only if e t is serially uncorrelated te variance of forecast error as a simplified form, wic is used by te Box-Jenkins intervals: E{[y t+j (Ey t+j Ω t )] 2 Ω t } = σe 2 j k=1 ϕ2(k 1) were Ω t = (y t, y t 1,...) is te information set at time t. Oterwise, covariance between e t+j and e t+j i is nonzero and sould be included. Te error term e t is serially uncorrelated wen te AR(1) model is dynamically complete, i.e., wen E(y t y t 1, y t 2,...) = E(y t y t 1 ). Wen E(y t y t 1, y t 2,...) E(y t y t 1 ), te error term in te AR(1) model is generally correlated. 2. We do not use te annual data because we need sufficient observations to compute te average coverage rate of prediction intervals based on rolling windows. 3. Te intercept term is not included because te demeaned inflation rate is used. 37