Block Bootstrap Prediction Intervals for Vector Autoregression

Transcription

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

2 Block Bootstrap Prediction Intervals for Vector Autoregression Jing Li Miami University Abstract This paper attempts to answer the question of whether the principle of parsimony can apply to interval forecasting for a multivariate series. Toward that end this paper proposes the block bootstrap prediction intervals based on the parsimonious first order vector autoregression. The new intervals generalize the standard bootstrap prediction intervals by allowing for serially correlated prediction errors. The unexplained serial correlation is accounted for by the block bootstrap, which resamples two-dimensional arrays of residuals. A Monte Carlo experiment shows that the new intervals outperform the standard bootstrap intervals in most cases. Keywords: Forecast; Vector Autoregression; Block Bootstrap; Prediction Intervals; Principle of Parsimony Jing Li, Department of Economics, Miami University, Oxford, OH 45056, USA. Phone: , Fax: , lij14@miamioh.edu. 1

3 1. Introduction Chatfield (1993) emphasizes that the interval forecasting can provide the associated uncertainty, so is more informative than the point forecasting. This paper is concerned with constructing the individual prediction intervals for each component of a multivariate time series, based on the vector autoregression developed by Sims (1980) 1. We focus on the bootstrap prediction intervals since they can automatically account for the sampling variability of coefficient estimators and non-normal prediction errors. For forecasting purpose the VAR has pros and cons. The VAR fully utilizes the acrossvariable correlation, which is ignored by univariate models. However, the number of unknown coefficients in a dynamically adequate VAR can be large, making the model not parsimonious. This is at odds with the principle of parsimony: it is well known that a parsimonious univariate model may produce superior out-of-sample point forecasts, see Enders (2009) for example. Recently Li (2013) shows a parsimonious univariate model may yield superior interval forecasts as well. The main objective of this paper is to investigate whether the principle of parsimony applies to the interval forecasting for components of a multivariate series. Toward that end this paper proposes the block bootstrap intervals (BBI) using the first order VAR, the most parsimonious VAR. Because the error terms are likely to be serially correlated, we need to implement the block bootstrap of Künsch (1989), instead of the standard bootstrap of Efron (1979). More explicitly, the BBI is characterized by resampling two-dimensional arrays of residuals. The space dimension intends to preserve the correlation between variables; while the time dimension can capture the serial correlation. The proposed BBI adds to the literature by generalizing Thombs and Schucany (1990) in two directions: from univariate to multivariate, and from the assumption of independent 1 There are other types of multivariate forecasting such as the prediction ellipsoid studied in Kim (1999). This paper concentrates on the individual prediction intervals in light of their popularity among practitioners. 2

4 error to relaxing that assumption. As a comparison, this paper also considers applying the standard bootstrap to a dynamically complete p-th order VAR, where p is sufficiently large so that the error becomes serially uncorrelated. This is necessary because the standard bootstrap assumes independent errors. The resulting VAR with white noise errors can be complicated. Our conjecture is that the BBI will have better performance since it is based on the parsimonious model. Later we conduct a Monte Carlo experiment to check this conjecture. In addition, this paper considers improving the BBI by correcting the bias in the coefficient estimated by the ordinary least squares (OLS). Early works such as Shaman and Stine (1988) and Kilian (1998) illustrate the importance of correcting the autoregressive bias. This paper extends those studies from the perspective of the multivariate interval forecasting. The literature of bootstrap prediction intervals is growing. An incomplete list includes Thombs and Schucany (1990), Grigoletto (1998), Kim (1999), Kim (2001), Clements and Taylor (2001), Kim (2002), Kim (2004) and Li (2011). This paper distinguishes from the existing literature by applying the block bootstrap to the VAR forecasting. The remainder of the paper is organized as follows. Section 2 specifies the BBI. Section 3 conducts the Monte Carlo experiment. Section 4 discuss the possible extension of this study and concludes. 2. Bootstrap Prediction Intervals for VAR Let y t denotes an m 1 vector: y t = (y 1,t,..., y m,t ), where y i,t, (i = 1,..., m), (t = 1,...) is a univariate series. The goal is to find the prediction intervals for each component of the future vectors (y n+1,..., y n+h ) based on the observed values (y 1,..., y n ). To focus on the main issue we assume y t has zero mean and is weakly stationary: Ey t = 0, E(y i,t y j,t k ) = σ ijk, (i, j = 1,..., m), (k = 0, 1,...). In practice y t may represent the differenced, detrended or demeaned series, depending on the order of integration and the deterministic terms 2. 2 The literature on forecasting non-stationary series is vast, see Clements and Hendry (2001) for example. 3

5 The VAR is a multivariate model that takes into account the across-variable correlation σ ijk, which is ignored by univariate models. There are at least two ways to build the bootstrap prediction intervals using VAR. Their difference is which bootstrap method is used. Then it boils down to how to model the serial correlation: by a dynamically adequate model with white noise errors, or by a parsimonious model with serially correlated errors. Standard Bootstrap Intervals We start with the standard bootstrap intervals (BI for short), which are the multivariate generalization of the one proposed by Thombs and Schucany (1990). The BI is based on a VAR(p) model y t = α 1 y t 1 + α 2 y t α p y t p + e t, (1) where α i, (i = 1,..., p) is an m m coefficient matrix, and e t is an m 1 vector of error terms 3. Unlike the classical Box-Jenkins type prediction intervals, the BI does not assume e t follows normal distributions. However, because the standard bootstrap of Efron (1979) can only be applied in the independent setting, the assumption that e t in (1) is independent of e t j, ( j 1) is needed for the BI. The independent assumption is restrictive in the sense that model (1) must be dynamically adequate, or the lag value p should be sufficiently large (so that no serial correlation is left in e t ). The detailed algorithm for constructing the BI is omitted here, and can be found in Thombs and Schucany (1990). For our purpose, it suffices to stress that the algorithm involves resampling with replacement the individual m 1 residual vector ê t, after (1) is fitted by OLS. 3 Intercept term is dropped in (1) because we assume y t has zero mean. Our simulation indicates that there is no qualitative change if the intercept term is added. 4

6 Block Bootstrap Intervals Basically the BI utilizes the serial correlation through (possibly many) lagged values of y. Notice that there are m 2 p autoregressive coefficients in (1) to estimate. If p is large, the resulting VAR can be very complicated. In fact, this must be the case when the series follow a vector moving average process, for which any chosen VAR is a finite-order approximation for VAR( ). The principle of parsimony is forgone if that happens. Having an eye for the parsimony principle, this paper proposes the block bootstrap intervals (BBI for short) based on the simplest VAR(1) model y t = β 1 y t 1 + v t. (2) It is obvious that model (2) has no more coefficients to estimate than (1). Moreover, the chance of multicollinearity is minimized by (2). The BBI is motivated by the intuition that the simplicity of model (2) may lead to superior out-of-sample forecasts. In particular, the improvement in the forecasting accuracy is expected to be substantial when the sample size n is small and when serial correlation is strong. One issue with model (2) is that the error term v t in general is serially correlated. To see this, suppose the true data generating process is (1) with the independent error e t. Then by construction v t a 2 y t a p y t p + e t, so is correlated with v t 1 and so on. The serial correlation in the new error term is not surprising: the VAR(1) model is somewhat semiparametric, and just one lagged value in general cannot capture all the dynamic structure. Now we hope to take advantage of the simple model (2), and meanwhile use the serial correlation in the error term to refine the forecast. The bootstrap method can be helpful in this context 4. Nevertheless, Efron s standard bootstrap cannot be applied to the dependent 4 The serial correlation in the error term may suggest methods such as generalized least squares (GLS). We do not consider GLS since that method intends to model the correlation structure in the error term. In other words, the GLS is essentially based on a model as complicated as (1). 5

7 series v t. Instead we need to rely upon the multivariate version of the block bootstrap proposed by Künsch (1989). Let ˆβ 1 denote the coefficient estimated by OLS, and ˆv t y t ˆβ 1 y t 1 the residual vector 5. The key idea of the block bootstrap is resampling with replacement random blocks of adjacent residual vectors. A typical block B(ˆv) j from the residual ˆv t looks like B(ˆv) j = (ˆv j, ˆv j+1,..., ˆv j+b 1 ) = ˆv 1,j... ˆv 1,j+b (3) ˆv m,j... ˆv m,j+b 1 m b where b is the width (size) of block, and the index number j is a random draw from the discrete uniform distribution between 1 and n b + 1. Notice that B(ˆv) j is a two-dimensional array. The vertical horizon measured by m can preserve the across-variable contemporaneous correlation σ ij0. The horizontal horizon measured by b can capture the temporal serial correlation σ iik. When b = 1 the block bootstrap reduces into the standard bootstrap. The block bootstrap resamples blocks of residuals (a vector in this case), rather than individual residual, so that the serial correlation in v t can be accounted for. In the literature the block bootstrap has been mostly used for purposes other than forecasting, see Politis (2003). One exception is Li (2013), which applies the block bootstrap to the univariate interval forecasting by letting m = 1 in (3). This study generalizes that one by allowing for m > 1. In principle, the block size b should rise as the sample size rises. For given sample size, there is a tradeoff when choosing the optimal block size. Rising b can capture more serial correlation. But meanwhile, there is increasing overlapping between blocks, which produces less variability. More discussion about this tradeoff is given in Li (2013). We let b = 4 throughout this paper, as our preliminary simulation indicates that the results may not be 5 One may re-center and re-scale the residual using the factor of Stine (1987). We do not use the rescaling factor since it is unclear how to modify the original one for VAR. 6

8 sensitive to the block size 6. Constructing the BBI takes five steps: In step one, model (2) is fitted by OLS, and the residual ˆv t is saved. In step two, the backward representation of the VAR(1) model is fitted by OLS: y t = ϕ 1 y t+1 + u t, (4) where the regressor is the first lead, not lag. Denote the estimated coefficient and residual by ˆϕ 1 and û t. This backward regression is intended to ensure the conditionality of the bootstrap replicate on the last observed value y n. See Figure 1 of Thombs and Schucany (1990) for an illustration of the conditionality. In step three, use the backward residual û t to obtain the first random block B(û) j1 = (û j1, û j1+1,..., û j1+b 1 ), the second block B(û) j2 = (û j2, û j2+1,..., û j2+b 1 ), and so on, by redrawing the index number j1, j2,... with replacement from the discrete uniform distribution between 1 and n b + 1. Then we stack up these blocks until the length of the stacked series becomes n, the size of the observed sample. Let û t denote the t-th observation of the stacked series. Next one bootstrap replicate series (y 1,..., y n) is generated in a backward fashion as y n = y n, y t = ˆϕ 1 y t+1 + û t, (t = n 1,..., 1) (5) It is instructive to emphasize that we generate the last observation first, then move backward one period a time. By doing this all the bootstrap replicate series have the 6 Alternatively, the stationary bootstrap of Politis and Romano (1994) can be used, which assumes b follows a geometric distribution. The question still exists since the new one becomes how to select the parameter for the geometric distribution. 7

9 same last observation, which is y n. That is what conditionality on the last observed value means. Now we use the bootstrap replicates (y1,..., yn) to refit (2), and get the so called bootstrap coefficient ˆβ 1. Later this step will be repeated many times. Then the variability of ˆβ 1 can be used to mimic the inherent variability of estimating ˆβ 1. This is another advantage of the bootstrap intervals over the Box-Jenkins intervals which fail to account for the sampling variability of the estimated coefficient. In step four, in a similar fashion, obtain the random blocks B(ˆv) j1 = (ˆv j1,..., ˆv j1+b 1 ), B(ˆv) j2 = (ˆv j2,..., ˆv j2+b 1 ),... using ˆv t, the residual of the forward VAR(1). But this time we stack up these blocks until the length of the stacked series is h, the maximum forecast horizon. Let ˆv l denote the l-th observation of the stacked series. Then we compute recursively the block bootstrap l-th step forecast ŷ n+l as ŷ n = y n, ŷ n+l = ˆβ 1ŷ n+l 1 + ˆv l, (l = 1,..., h) (6) The above equation shows that the randomness of the forecast value comes from two sources. The randomness due to estimation is captured by ˆβ 1, while the randomness of the future shock is represented by ˆv l. In step five, we repeat step three and step four D times. In the end, there is a series of the block bootstrap l-th step forecasts for y i,t, (i = 1,..., m) : {ŷ i,n+l (s) } D s=1, (l = 1,..., h) (7) where s is the index for bootstrapping. The l-th step BBI at the γ nominal level for 8

10 the i-th component y i,t is given by l step BBI = [ ŷ i,n+l ( 1 γ 2 ), ŷ i,n+l ( )] 1 + γ 2 (8) where ŷ i,n+l ( 1 γ 2 ) and ŷ i,n+l ( 1+γ 2 ) are the ( 1 γ 2 ) ( 100-th and 1+γ ) th percentiles of the empirical distribution of {ŷ i,n+l (s)}d s=1. Throughout this paper we let γ = 0. We use D = 999 to avoid the discreteness problem raised by Booth and Hall (1994). Here the percentile method of Efron and Tibshirani (1993) is applied to construct the BBI. Hall (1988) discusses other percentile methods. De Gooijer and Kumar (1992) emphasize that the percentile method performs well when the conditional distribution of the predicted values is unimodal. Our preliminary simulation conducts the DIP test of Hartigan and Hartigan (1985) and finds that the distribution is indeed unimodal. Bias-Corrected Block Bootstrap Intervals There is a well known fact that the autoregressive coefficient estimated by OLS can be biased, see Shaman and Stine (1988) for example. This suggests that the proposed BBI may be improved by using the method of Efron and Tibshirani (1993) or Kilian (1998). In particular, equation (6) indicates that the residual in the forward VAR(1) ˆv t has direct effect on the forecast. Therefore correcting the bias in ˆβ 1 may be more important than the backward coefficient ˆϕ 1. This is confirmed by Li (2013) in the setting of univariate autoregressive forecasting. It is straightforward to correct the bias in ˆβ 1. We need to nest a new bootstrapping into the original one. Suppose there are C series of the bootstrap replicates, which can be used to refit the forward regression (2). After obtaining a series of forward bootstrap coefficients { ˆβ 1(s)} C s=1, the bias-corrected forward coefficient and the bias-corrected forward residual are 9

11 computed as ˆβ 1 c = 2 ˆβ C 1 C 1 ˆβ 1(s) (9) s=1 ˆv c t = y t ˆβ c 1y t 1. (10) Throughout this paper we let C = 100. Next, the bias-corrected block bootstrap l-th step forecast ŷ c n+l is generated as ŷ c n = y n, ŷ c n+l = ˆβ 1ŷ c n+l 1 + ˆv c l, (l = 1,..., h) (11) where ˆv c l is obtained by block bootstrapping ˆv c t. The l-th step bias-corrected block bootstrap intervals (BCBBI) at the γ nominal level for the i-th component y i,t are given by l step BCBBI = [ ŷ c i,n+l ( 1 γ 2 ), ŷ c i,n+l ( )] 1 + γ 2 (12) where ŷ c i,n+l ( 1 γ ) ( 2 and ŷ c 1+γ i,n+l 2 ) are the ( 1 γ 2 ) ( 100-th and 1+γ ) th percentiles of the empirical distribution of {ŷ c i,n+l (s)}d s=1. The new intervals can be improved further, for instance, by correcting the bias in ˆϕ 1. However, Li (2013) shows that the benefit of biascorrecting ˆϕ 1 may be marginal. 3. Monte Carlo Experiment This section compares the performance of the BI and BBI using a Monte Carlo experiment. The criterion of comparison is the average coverage rate (ACR) of the prediction intervals for each component of a multivariate series: k 1 k 1(y i,n+l PI), (i = 1,..., m, l = 1,... h) (13) 1 10

12 where k = is the number of iteration, 1(.) is the indicator function that equals one when the event in the parenthesis is true, and PI denotes the prediction intervals. y i,n+l is the i-th component of y n+l = (y 1,n+l,..., y m,n+l ). The maximum forecast horizon h is 5. No qualitative change is found for larger h in preliminary simulation. The true data generating process (DGP) is a second order VAR for a bivariate series y t = (y 1,t, y 2,t ) : y 1,t = a 1 y 1,t 1 + a 2 y 2,t 1 + a 3 y 1,t 2 + e 1,t (14) y 2,t = b 1 y 2,t 1 + b 2 y 2,t 2 + e 2,t, (15) where the coefficients a i and b i are all scalars. The error vector e t = (e 1,t, e 2,t ) is an independent series generated as e t = Au t, u t i.i.d(0, 1), A = 1 ρ 2 ρ 0 1. (16) It is easy to show the variance-covariance matrix of e t is Ω E(e t e t) = AA = 1 ρ ρ 1. (17) Notice that y 2,t 1 appears on the right hand side of (14), but y 1,t 1 is absent in (15). That means y 2,t first-order Granger causes y 1,t, but not vice versa. Using this DGP of triangular form is without losing generality because we can always transform a non-triangular form into a triangular form by multiplying an invertible matrix. One advantage of using the triangular form is the clear interpretation of the parameter: b 1 and b 2 control the stationarity of y 2,t ; a 1 and a 3 control the stationarity of y 1,t ; the across-variable correlation is measured by a 2 11

13 and ρ. For instance, y 2,t is stationary when b 1 = 1.4, b 2 = 0.48 since the corresponding characteristic roots are λ 1 = 0.6 and λ 2 =, both less than one in absolute value. The positive definiteness of Ω requires that ρ < 1. Three bootstrap prediction intervals are considered. The proposed block bootstrap intervals (BBI) (8) are obtained by applying the block bootstrap to the residual of the parsimonious VAR(1) model. One standard bootstrap intervals (denoted by AR1BI) are obtained by applying the standard bootstrap to the residual of the VAR(1) 7 ; the other (denoted by AR2BI) is obtained by applying the standard bootstrap to the residual of the VAR(2). The AR2BI is theoretically correct, but not parsimonious. The AR1BI is parsimonious, but fails to account for the serial correlation in the error term. The BBI is parsimonious and fully utilizes the serial correlation. In total we generate n + h observations 8 of y t. The first n observations are used for the in-sample fitting and constructing the prediction intervals. Then the average coverage rate (13) is computed for the last h pseudo out-of-sample observations. That is, we evaluate whether the last h observations are inside the prediction intervals. The best method is the one that yields the prediction intervals with the average coverage rate closest to the nominal level γ = 0. Error Distributions In theory the bootstrap intervals should be robust to non-normal distributions of the prediction errors. Following Thombs and Schucany (1990) we examine three distributions for u t : the bivariate standard normal distribution; the bivariate exponential distribution with mean of two, which is skewed; and the bivariate mixed normal distribution N( 1, 1)+0.1N(9, 1), 7 In practice one may check the adequacy of the model (no serial correlation in the error term) by applying the Breusch-Godfrey type test to the residual. For simplicity we skip those tests in this section and proceed as if the VAR(1) or VAR(2) is adequate. 8 The initial value y 1 is set as zero, the unconditional mean. The pseudo random number generator of Matlab R2011a is used. 12

14 which is bimodal skewed. All distributions are standardized to have zero mean and unity variance. Then the prediction error e t is generated as in (16), where A is the Cholesky decomposition of the variance-covariance matrix Ω. The parameters are set as b 1 = 1.4, b 2 = 0.48, a 1 = 1.2, a 2 = 0.6, a 3 = 0.35, ρ = 0.4, and the sample size is n = 50. By construction both y 1,t and y 2,t are stationary. Figure 1 plots the ACR against the forecast horizon, as the error distribution varies. The three panels in the first row show the ACR for the first component y 1,n+l ; the second row shows the ACR for the second component y 2,n+l. There are several findings from Figure 1. First of all, the BBI (denoted by circle) in most cases has the best performance with the ACR closest to 0. The superiority of the BBI becomes more evident as the forecast horizon rises. In particular, the BBI is shown to outperform the AR2BI (denoted by square). This fact may serve as the evidence supporting the principle of parsimony. The performance of the AR1BI (denoted by diamond) is interesting. It has the worst performance (lowest coverage rate) for long run forecast, while has the seemingly best performance (highest coverage rate) for the first-step forecast. This finding can be explained by two facts. First, the AR1BI uses the standard bootstrap, which fails to account for the serial correlation in the error term but can add more variability than the block bootstrap. Second, the serial correlation in the prediction error does not matter in short run as much as in long run. The performance of the AR1BI also highlights the tradeoff of preserving serial correlation vs adding variability. For the first-step forecast, adding variability outweighs keeping serial correlation, so the AR1BI has first-step coverage rate greater than the BBI. We find that the coverage rates of all three intervals remain largely unchanged as the error distribution varies. This verifies the robustness of the bootstrap intervals to the nonnormality. Finally, the ACRs of all three intervals decrease as the horizon rises, a finding consistent with Thombs and Schucany (1990). 13

15 Across Variable Correlation: ρ and a 2 Next we investigate the effect of varying across variable correlation determined by ρ and a 2. The same values of b 1, b 2, a 1, a 3 and n as Figure 1 are used. The error follows the bivariate normal distribution. But now we let ρ = 0.4, 0.4,, and a 2 = 0.4, 0.4,. The results are shown in Figure 2 (where ρ varies and a 2 = 0.4) and Figure 3 (where a 2 varies and ρ = 0.4), respectively. We find that the varying ρ and a 2 have minimal effect on the coverage rate for y 2,n+l. This is expected because of the triangular form of the DGP or the fact that y 1,t does not Granger cause y 2,t. Varying ρ and a 2 do affect the coverage rate for y 1,n+l. But the BBI still dominates in most cases. Persistence: b 1, b 2, a 1, a 3 The persistence of the series (or the speed at which the autocorrelation decays) is determined by the parameters b 1, b 2, a 1 and a 3. Their effect on the ACR is illustrated by Figures 4, 5 and 6. We let a 2 = 0.6, ρ = 0.4, n = 50 and the error follows the bivariate normal distribution. Table 1 summarizes the values of b 1, b 2, a 1, a 3, and the corresponding characteristic roots. Figure 4 is concerned with the real characteristic roots. For example, when b 1 = 1.2, b 2 = 0.35, the characteristic roots for y 2,t are and 0.5; when a 1 =, a 3 = 0.2, the characteristic roots for y 1,t are 0.5 and 0.4. So in this case (DGP1) both series are stationary. y 2,t becomes more persistent (and its autocorrelation decays more slowly) when DGP1 changes to DGP2; y 1,t becomes more persistent when DGP2 changes to DGP3. Figure 4 shows that the superiority of the BBI is insensitive to the change in the persistence. The BBI maintains its superiority in Figure 5, where some or all characteristic roots are complex conjugates. In DGP4-6 the series are still stationary because the modulus of the characteristic root is less than one, but now the autocorrelation function has the sinusoidal pattern. 14

16 Figure 6 relaxes the assumption of stationarity. In DGP7 and DGP8, y 2,t has one unit root, so becomes nonstationary. Moreover, y 1,t and y 2,t are cointegrated, in the terminology of Engle and Granger (1987). In DGP9, y 1,t is nonstationary but y 2,t is stationary. As shown by Figure 6, despite the nonstationarity and cointegration, the BBI still delivers the best performance in most cases. Sample Size: n Figure 7 demonstrates the effect of the sample size n on the coverage rate. The error follows the bivariate normal distribution, and the parameter values are the same as the Figure 1. As the sample size rises, we find that the coverage rates of all three intervals improve, by moving upward to the nominal level 0. Furthermore, the improvement of the BBI seems to be the most evident. When n = 150, the ACR of the BBI becomes almost flat at the level of 8. Bias Correction Figure 7 implies that the small sample size is one of the reasons why the prediction intervals tend to undercover the true future values. Another reason is the bias in the autoregressive coefficient estimated by OLS, as shown by Figure 8. In that figure the coverage rate of the BBI is compared to the bias-corrected block bootstrap intervals (BCBBI, denoted by square), using the same DGP as Figure 1. It is clear to see that correcting the autoregressive bias leads to an increase in the coverage rate. 4. Conclusion This paper proposes the block bootstrap prediction intervals (BBI) for each component of a multivariate time series based on the parsimonious first order vector autoregression. One characteristic of the BBI is resampling big blocks or two-dimensional arrays of residuals. 15

17 Those big blocks aim to preserve both the across variable correlation and serial correlation. By contrast, the standard bootstrap intervals require independent prediction errors, which amount to a possibly complicated model, and involve redrawing small blocks or onedimensional arrays of residuals. The Monte Carlo experiment indicates that the principle of parsimony can be extended to multivariate interval forecasting. The BBI is shown to outperform the standard bootstrap intervals in most cases. Remarkably, the BBI always dominates for the long run forecast. The performance of the BBI can be enhanced by bigger sample sizes and correcting the bias of the estimated autoregressive coefficient. There are some noteworthy future studies. One possibility is following Kim (1999) and developing the block bootstrap prediction region (ellipsoid) based on the parsimonious VAR. The prediction region can provide joint forecast for a multivariate series, but may incur higher computational cost. Another one is considering an improved BBI for the cointegration system that explicitly factors in the non-stationarity and the error-correcting mechanism. 16

18 References Booth, J. G. and Hall, P. (1994). Monte carlo approximation and the iterated bootstrap. Biometrika, 81, Chatfield, C. (1993). Calculating interval forecasts. Journal of Business & Economic Statistics, 11, Clements, M. P. and Hendry, D. F. (2001). Forecasting Non-Stationary Economic Time Series. The MIT Press. Clements, M. P. and Taylor, N. (2001). Boostrapping prediction intervals for autoregressive models. International Journal of Forecasting, 17, De Gooijer, J. G. and Kumar, K. (1992). Some recent developments in non-linear time series modeling, testing, and forecasting. International Journal of Forecasting, pages Efron, B. (1979). Bootstrap method: Another look at the jackknife. Annals of Statistics, 7, Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. London: Chapman and Hall. Enders, W. (2009). Applied Econometric Times Series. Wiley, 3 edition. Engle, R. F. and Granger, C. W. J. (1987). Cointegration and error correction: Representation, estimation and testing. Econometrica, 55, Grigoletto, M. (1998). Bootstrap prediction intervals for autoregressions: some alternatives. International Journal of Forecasting, 14, Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals. Annals of Statistics, 16,

19 Hartigan, J. A. and Hartigan, P. M. (1985). The DIP test of unimodality. Annals of Statistics, 13, Kilian, L. (1998). Small sample confidence intervals for impulse response functions. The Review of Economics and Statistics, 80, Kim, J. (1999). Asymptotic and bootstrap prediction regions for vector autoregression. International Journal of Forecasting, 15, Kim, J. (2001). bootstrap-after-bootstrap prediction intervals for autoregressive models. Journal of Business & Economic Statistics, 19, Kim, J. (2002). Bootstrap prediction intervals for autoregressive models of unknown or infinite lag order. Journal of Forecasting, 21, Kim, J. (2004). Bias-corrected bootstrap prediction regions for vector autoregression. Journal of Forecasting, 23, Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Annals of Statistics, 17, Li, J. (2011). Bootstrap prediction intervals for SETAR models. International Journal of Forecasting, 27, Li, J. (2013). Block bootstrap prediction intervals for autoregression. Working Paper. Politis, D. N. (2003). The impact of bootstrap method on time series analysis. Statistical Science, 18, Politis, D. N. and Romano, J. P. (1994). The stationary bootstrap. Journal of the American Statistical Association, 89,

20 Shaman, P. and Stine, R. A. (1988). The bias of autoregressive coefficient estimators. Journal of the American Statistical Association, 83, Sims, C. (1980). Macroeconomics and reality. Econometrica, 48, Stine, R. A. (1987). Estimating properties of autoregressive forecasts. Journal of the American Statistical Association, 82, Thombs, L. A. and Schucany, W. R. (1990). Bootstrap prediction intervals for autoregression. Journal of the American Statistical Association, 85,

21 Table 1: Parameters Values and Characteristic Roots in Figures 4, 5, 6 Parameters y 2,t y 1,t b 1 b 2 a 1 a 3 λ 1 λ 2 λ 1 λ 2 DGP DGP DGP DGP i i 0.6 DGP i i DGP i i +0.5i -0.5i DGP DGP DGP The data generating process is (14) and (15) with a 2 = 0.6, ρ = 0.4, n = 50, u t i.i.d.n(0, 1) 20

22 Normal Distribution BBI AR2BI AR1BI 5 5 Exponential Distribution Mixed Normal Distribution 5 5 Normal Distribution Exponential Distribution Mixed Normal Distribution Figure 1: Error Distributions Note: the data generating process is (14) and (15) with b 1 = 1.4, b 2 = 0.48, a 1 = 1.2, a 2 = 0.6, a 3 = 0.35, ρ = 0.4, n =

23 4 rho = rho = rho = 2 BBI 4 AR2BI AR1BI rho = rho = rho = Figure 2: Across Variable Correlation: ρ Note: the data generating process is (14) and (15) with b 1 = 1.4, b 2 = 0.48, a 1 = 1.2, a 2 = 0.6, a 3 = 0.35, n = 50, and u t i.i.d.n(0, 1). 22

24 6 a 2 = a 2 = a 2 = BBI 4 AR2BI AR1BI a 2 = a 2 = a 2 = Figure 3: Across Variable Correlation: a 2 Note: the data generating process is (14) and (15) with b 1 = 1.4, b 2 = 0.48, a 1 = 1.2, a 3 = 0.35, ρ = 0.4, n = 50, and u t i.i.d.n(0, 1). 23

25 6 4 2 DGP1 BBI AR2BI AR1BI 5 5 DGP2 5 5 DGP DGP1 DGP2 DGP Figure 4: Serial Correlation: Real Characteristic Roots Note: the data generating process is in Table 1. 24

26 4 DGP4 DGP5 DGP6 2 4 BBI 2 AR2BI AR1BI DGP4 DGP5 6 DGP Figure 5: Serial Correlation: Complex Characteristic Roots Note: the data generating process is in Table 1. 25

27 5 DGP7 BBI AR2BI AR1BI 5 DGP8 5 DGP DGP7 5 DGP8 DGP Figure 6: Serial Correlation: Nonstationarity and Cointegration Note: the data generating process is in Table 1. 26

28 5 n=50 BBI AR2BI AR1BI 8 6 n=100 8 n= n=50 8 n=100 n= Figure 7: Sample Size Note: the data generating process is (14) and (15) with b 1 = 1.4, b 2 = 0.48, a 1 = 1.2, a 2 = 0.6, a 3 = 0.35, ρ = 0.4 and u t i.i.d.n(0, 1). 27

29 4 2 Normal Distribution BBI BCBBI 4 2 Exponential Distribution Mixed Normal Distribution Normal Distribution 4 Exponential Distribution Mixed Normal Distribution Figure 8: Bias Correction Note: the data generating process is (14) and (15) with b 1 = 1.4, b 2 = 0.48, a 1 = 1.2, a 2 = 0.6, a 3 = 0.35, ρ = 0.4, n =