DBJ Discussion Paper Series, No.0905

Bagging and Forecasting in Nonlinear Dynamic Models

Mari Sakudo(Research Institute of Capital Formation,
Development Bank of Japan,
andDepartment of Economics, Sophia University)

December

2009

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings, conclusions or
recommendations expressed in these papers are those of the authors and do not reflect theviews of the Institute.

Bagging and Forecasting in Nonlinear Dynamic Models∗Mari Sakudo†
Development Bank of Japan,
and Sophia University
First Draft, September 2003
This Draft, December 2009

Abstract
This paper proposes new variants of point forecast estimators in Markov switchingmodels (Hamilton, 1989) utilizing bagging (Breiman, 1996), and applies them to study
real GNP in the U.S. The empirical and Monte Carlo simulation results on out-ofsample forecasting show that the bagged forecast estimators outperform the benchmark
forecast estimator by Hamilton (1989) in the sense of the prediction mean squared error.
The Monte Carlo experiments present that interactions between a Markov process
for primitive states and an innovation affect the relative performance of the bagged
forecast estimators, and that effectiveness of the bagging does not die out as sample
size increases.

Keywords: Bagging; Bootstrap; Forecast; Regime Switching; Time Series
JEL Classifications: C13; C15; C53; E37∗

I am truly indebted to Frank Diebold, Masayuki Hirukawa, Atsushi Inoue, Yuichi Kitamura, Kevin Songand seminar participants at Development Bank of Japan and University of Tokyo for helpful comments andsuggestions. The earlier version of this paper is entitled “The Bootstrap and Forecasts in Markov Switching
Models.”†Address: Development Bank of Japan, 9-3, Otemachi 1-chome, Chiyoda-ku, Tokyo, Japan 100-0004.
Tel.: +81-3-3244-1241; fax: +81-3-3270-7084. E-mail addresses: masakud@dbj.jp/sakudo@sophia.ac.jp.

11Introduction

The bagging (bootstrap aggregating) method (Breiman, 1996) has recently been applied inanalysis of forecasting time series data (Inoue and Kilian, 2003, 2008; Lee and Yang, 2006;
Stock and Watson, 2005). There, the presumed models are linear, or binary and quantile. In the existing literature on bagging and time series forecasting, genuine dynamics—
autoregressive component—of the process of interest does not play a key role. Markov
switching models (Hamilton, 1989) are widely used in analyses of time series economic andfinancial data (for example, Ang and Bekaert (2002); Garcia and Perron (1996); Guidolin
and Timmermann (2006, 2007); Perez-Quiros and Timmermann (2000)), where the nonlinear dynamics is an important element. However, to the best of my knowledge, no attempt
has been made to implement the bagging method in Markov switching models. This paper
proposes new variants of point forecast estimators in Markov switching models utilizing thebagging method, and applies them to forecast nonlinear dynamics of real GNP in the UnitedStates.
In many economic and financial data, nonlinearity (Teräsvirta, 2006) or structural breaks(Stock and Watson, 1996) exist, suggesting that economic agents’ forecasting must also be
nonlinear (Clements and Hendry, 2006). Markov switching models (Hamilton, 1989)—a
special case is the threshold autoregression (TAR) model (Tong, 1983) of observable shifts inregime—adapt nonlinearity by making inference on stochastic changes from the discrete-stateprocess. The nonlinearity from shifts in regimes is determined by the interaction between
the data and the Markov chain. As the sample size is often small in forecasting application,
Markov switching models provide us flexible nonlinearity in a parsimonious way.
Another key ingredient of forecasting estimators in this paper is bagging. Breiman (1996)
introduces bagging in the field of machine learning in random sample settings. The bagging
method uses the bootstrap (Efron, 1979; Horowitz, 2000), but is a method to improve theaccuracy of the estimators or predictors themselves—rather than to approximate distributions and access the accuracy of parameter estimators or predictions. In the context of crosssections, the existing literature on bagging studies estimators for nonlinear models of smoothfunctions (Friedman and Hall, 2000), indicator predictors (Bühlmann and Yu, 2001), andpredictive risk estimators for discontinuous loss functions (Kitamura, 2001).
Recently, bagging has been analyzed in time series context. Lee and Yang (2006) study
bagging binary and quantile predictors, where the improvement of accuracy in predictions
comes from non-differentiabilities of objective functions. Inoue and Kilian (2003, 2008) andStock and Watson (2005) apply bagging to construct forecasts among many possible predic2

tors in linear dynamic models. In their cases, bagging improves the accuracy of forecasts,
due to variance reduction in the part of predictors other than deterministic and autoregressive components. In contrast, this paper proposes bagged forecast estimators in nonlinear
dynamic models of stochastic regime switching. Improvement in accuracy of forecasts comesfrom nonlinear dynamics of the process of interest.1 In other words, it adopts the idea ofbagging in nonlinear models in cross section context by Friedman and Hall (2000) to dynamic
forecasting situations.
To implement bagging, we resample data by the bootstrap, construct a forecast—or an
estimate, or an objective function—for each bootstrap sample, and take an average overthe bootstrap replications. In the existing literature on bagging in time series forecasting,in order to capture the dependence structure of the data, Lee and Yang (2006) and Inoueand Kilian (2003, 2008) conduct the blocks bootstrap (Gonçalves and White, 2004; Hall andHorowitz, 1996). Stock and Watson (2005) implement the parametric bootstrap assuming
a normal distribution to the error term. The model is reduced to a linear model withindependent innovations.
In contrast, I implement bootstrapping in nonlinear dynamic models in which the nonlinearity comes from the changes in non-independent stochastic components. The nonlinear
dynamic model in which I implement the bootstrap can be reduced to an autoregressive moving average (ARMA) model with serially dependent innovations. Direct draws of the seriallydependent innovations in the ARMA model are not feasible. Conducting the nonparametricbootstrap without using information on the assumed model structure is not desirable. Therefore, for nonlinear dynamic models of stochastic regime shifts, to implement the bootstrap
using the model information, values of the stochastic states also need to be drawn from theestimated probability distribution. I propose new variants of the parametric bootstrap andthe residual bootstrap as I explain in detail later.
Empirical and Monte Carlo simulation results on out-of-sample point forecasting showthat the bagged forecast estimators outperform the benchmark forecast estimator by Hamilton (1989) in terms of the prediction mean squared error. Monte Carlo results present
that interactions between a Markov process for primitive states and an innovation have an
influence on the relative performance of the bagged forecast estimators. Moreover, the experimental results present that the accuracy improvement by utilizing the bagging does not
1

One recent paper by Li (2009) proposes bootstrapping in the TAR models. Both his paper and my paper

study forecasting in nonlinear dynamic models. However, Li (2009) uses the bootstrap to approximate the
distribution of predictors and improve confidence intervals of prediction, whereas I use bootstrap aggregating
to improve the forecasts themselves.

3die out even in large sample. A possible reason is that the nonlinearity, which comes from
regime shifts, is stochastic in Markov switching models. The uncertainty about nonlinearity exists regardless of sample size; hence, even in large sample. The bagging reduces thevariance of forecast that stems from stochastic nonlinearity.
This paper is organized as follows. The next section describes the model. Section 3
explains the bootstrap sampling procedure, the forecast estimators using the bagging, andthe forecast evaluation method. Section 4 explains Monte Carlo simulation procedure, andpresents Monte Carlo evidence on the numerical performance of the bagged forecast estimators in Markov switching models. In section 5, I estimate the business cycle model onreal GNP in the United States—the first difference obeys a nonlinear stationary process,construct forecasts, and present the results. Section 6 concludes.

2

Model

A process of y˜t obeys a special case of an ARIMA model, in order to study forecasts as in
Hamilton (1989) for comparison.
∆ỹt = cst + φ1 ∆ỹt−1 + φ2 ∆ỹt−2 + · · · + φr ∆ỹt−r + t ,(1)

where φj , j = 1, ..., r are parameters, ∆ỹt = y˜t − ỹt−1 , an innovation of t is i.i.d.(0, σ 2 ), andst represents the unobserved state at date t. The constant term cst depends on the state at
date t, st . For simplicity, non-dependency of φj on the state conditions for j = 1, ..., r, isassumed.
Furthermore, the process of y˜t is the sum of a Markov trend process of ARIMA(1,1,0),
nt , and an ARIMA(r,1,0) process without drift, z̃t .
y˜t = nt + z˜t ,(2)

where z̃t − z̃t−1 = φ1 (z̃t−1 − z̃t−2 ) + · · · + φr (z̃t−r − z̃t−r−1 ) + t , nt − nt−1 = α1 s∗t + α0 , and t is
independent of nt+j for all j. Here, s∗t ’s are primitive states that follow a first-order Markov
process. α0 and α1 are parameters. The primitive state, s∗t , is such thatP rob(s∗t = 1|s∗t−1 = 1) = p(3)

andP rob(s∗t = 0|s∗t−1 = 0) = q.

To simplify notations, let yt ≡ ∆ỹt and zt ≡ ∆z̃t . Detailed explanation on the model is in
the appendix.
4

3

Bagged Forecasts

3.1

The Bootstrap

First of all, ML estimates are obtained as in Hamilton (1989). For the bootstrap data, random draws for the independent innovations are obtained by either (1) the residual bootstrap,or (2) the parametric bootstrap with a normally distributed error.
Let Yτ = (yτ0 , yτ −1 , . . . , y−m )0 be a vector containing all observations obtained throughdate τ . The state st is a first-order J(= 2r+1 ) state Markov chain. Let P (st = j|Yτ ; θ)denote a probability of the state st conditional on data obtained through date τ given the
population parameters θ. Collect these conditional probabilities in a (J × 1) vector of ξt|τ .
Let X̂ denote an estimate of X.
3.1.1

Residual Bootstrap

I calculate residuals from estimates of the parameters in the first estimation and from observed data.
ˆt = yt − Ê(cst |Yτ ) − φ̂1 yt−1 − · · · − φ̂r yt−r ,t = r + 1, . . . , T(4)

In the above formula, I use the parameter estimates of φ, α’s, and the estimated state probabilities. The negative of the second term on the right hand side is Ê(cst |Yτ ) = (ĉ1 , . . . , ĉJ )0 ξˆt|τ ,where τ = T or τ = t. In other words, ξˆt|τ is either the vector of estimated smoothed probabilities, ξˆt|T , or the vector of estimated inferred probabilities, ξˆt|t . ĉj , for j = 1, . . . , J, areestimated constant terms for each state, st , in Equation (1).
Then, I repeat the following procedure B times, for b = 1, . . . , B. The subscript b denotes
the bootstrap replication, from 1 to B.
1. Draw bootstrapped residuals {ebt }Tt=r+1 with replacement from the original residuals{ˆt }Tt=r+1 .
2. For t = r + 1, . . . , T, construct blocks of ηt ’s such that








yt−r


 . 
ηt ≡  ..  .
yt(5)Assume that the distribution puts equal probability mass to each block. Draw an
initial block of (y1b , . . . , yrb )0 from the density distribution of the blocks.
53. For t = 1 + r, . . . , T , draw state at date t, sbt , from the estimated probabilities, ξˆt|τ :either from the estimated smoothed probabilities, ξˆt|T , or from the estimated inferredprobabilities, ξˆt|t .
4. Starting with the initial bootstrap observation block, (y1b , . . . , yrb )0 , construct the bootstrap sample of ytb ’s recursively bybb+ ˆbt , t = r + 1, ..., T,+ · · · + φ̂r yt−r
ytb = Ê b (cst |Yτ ) + φ̂1 yt−1(6)where Ê b (cst |Yτ ) = (ĉ1 , . . . , ĉJ )0 sbt . τ is T when the estimated smoothed state probabilities are used to draw states, and is t when the estimated inferred probabilities areused. Here, the bootstrap sample starts at date t = r + 1 rather than t = 1. As I
utilize the estimated smoothed probabilities that are obtained from date r + 1 to date
T , I set the time period from 1 to r as an initial time block.
3.1.2

Parametric Bootstrap with a Normally Distributed Error

For each bootstrap replication, b = 1, . . . , B, I draw the bootstrap errors, {ebt }Tt=r+1 , fromthe normal distribution with mean zero and the estimated variance from the first estimation,σ̂ 2 . The rest of the procedure is the same as in the case of the residual bootstrap above.

3.2

Forecasting Using Bagging

One-step-ahead optimal point forecast based on observable variables2 is a sum of productsof forecasts conditional on states and inferred state probability at date T + 1, ξT +1|T .
E(yT +1 |YT ; θ) =PJ

j=1 P (sT +1 = j|YT ; θ)E(yT +1 |sT +1 = j, YT ; θ)(7)

After resampling data by the bootstrap, I estimate parameters and state probabilities
for each bootstrap replication, b = 1, . . . , B. For each bootstrap sample of YT , b = 1, . . . , B,
the forecast estimator isÊ b (yT +1 |YT , θ) =PJ

j=1 P̂

b(sT +1 = j|YT ; θ)Ê b (yT +1 |sT +1 = j, YT ; θ).

(8)

In Equation (8), I calculate the estimates of inferred probabilities of states at date T + 1 bybξˆTb +1|T = Λ̂ ξˆTb |T ,2

(9)

We can consider longer than one-step-ahead point forecasts and see how the forecast horizon affects

relative performance of the bagged forecast estimator. In this paper, I analyze one-step-ahead forecasting asa start.

6

where Λ denotes the (J ×J) transition matrix of state st . I obtain the estimates of conditional
forecasts for j = 1, . . . , J, by
Ê b (yT +1 |sT +1 = j, YT ; θ) = ĉbj + φ̂b1 yTb + · · · + φ̂br yTb −r .

(10)

Taking an average of the maximum likelihood forecast estimators over the bootstrap
samples, the one-step-ahead bagged forecast estimator is defined as follows.3
Ê(yT +1 |YT ; θ) ≈ B1

PB

bb=1 Ê (yT +1 |YT ; θ)(11)

The expression on the right hand side in Equation (11) is a Monte Carlo estimate of thebagged forecast estimator, which approaches the true bagged forecast as B goes to infinity.
In estimation, I set the number of the bootstrap replications at 100.

3.3

Forecast Evaluation

The updating scheme I adopt is rolling horizon—rolling forecasts are constructed from only
the most recent fixed interval of the past data. The length of the data window, which is the
sample size for estimating and forecasting, T , remains the same as date is updated.
Let ŷt+h denote Ê(yt+h |Yt ; θ), an h-step-ahead forecasting estimator, where t is the time
the forecast is made. An optimal forecast here implicitly uses squared difference between aforecast and outcome as a loss function. The risk used is the expected loss—the expectation
is taken over the outcome of Yt+h given Yt and holding parameters θ fixed, as its value is
unknown at the time the forecast is made:E[(yt+h − ŷt+h )2 |Yt ; θ].

(12)

The classical approach to forecasting uses the same risk on evaluating the forecast. In
contrast, I use the population prediction mean squared error (PMSE) conditional on theoutcome variable at the forecasting date:E[(yt+h − ŷt+h )2 |yt+h , Yt ; θ].

(13)

Explanation on the PMSE and its sample analogue used is in the appendix.
3

For simplicity, I abuse notation and use the same notation E for the expectation with respect to thebootstrap probability measure.

7

4Monte Carlo Experiments

4.1

Monte Carlo Procedure

In Monte Carlo simulation, I set the number of lags in autoregression at one, r = 1, or atfour, r = 4 as in Hamilton (1989). For simplicity, I explain Monte Carlo simulation procedurein the case of one lag. Thus, the process of yt , that is, ∆ỹt , is rewritten as
yt = cst + φ1 yt−1 + t ,
(14)

where
cst = α1 s∗t + α0 − φ1 (α1 s∗t−1 + α0 ).

The primitive state at date 0 of s∗0 has a Bernoulli distribution such that the probability ofstate 1 is ps∗0 . I assume that the initial primitive state probability is ergodic. I also assume
that an initial observation y0 is normally distributed, y0 ∼ N (m0 , σ02 ), where the mean is
m0 , and σ02 is the variance. The mean of an initial observation is defined as the expectationof a constant term associated with the initial state, m0 ≡ ps∗0 · µ0 + (1 − ps∗0 ) · µ1 .
Given assumed parameter values of {φ1 , p, q, α0 , α1 , σ, ps∗0 , σ0 }, I generate the process ofyt , t = 1, . . . , T, as follows. First, I draw an initial primitive state, s∗0 , from the Bernoulli
distribution of (ps∗0 , 1 − ps∗0 )0 . An initial value of the process, y0 , is drawn from the normal
distribution of N (m0 , σ02 ).
Second, I recursively draw primitive states, s∗t , for t = 1, . . . , T . Primitive state, s∗t , isdrawn from Bernoulli distribution of (p, 1 − p)0 if primitive state at the previous date is instate 1 (i.e. s∗t−1 = 1), and is drawn from Bernoulli distribution of (q, 1 − q)0 if primitive
state at the previous date is in state 0 (i.e. s∗t−1 = 0). Then, state is defined as st ≡ (I{s∗t =1}, I{s∗t = 0}, I{s∗t−1 = 1}, I{s∗t−1 = 0})0 , t = 1, . . . , T, where I{A} is an indicator functionthat takes 1 if A is true and 0 otherwise.
Third, I draw errors of t ’s from the i.i.d. normal distribution of N (0, σ 2 ) and iterate on
yt = cst + φ1 yt−1 + t , for t = 1, . . . , T.

4.2

Results

Two benchmark parameter settings are as below. I perturb a presumed parameter value ora presumed setting from the benchmark setting and see how forecast performance changes
across different presumed settings. The first setting corresponds to the paper by Hamilton(1989). The second is a simplest setting.

8• Parameter setting as in Hamilton (1989):
r = 4, α0 + α1 = 1.1647, α0 = −0.3577, φ1 = 0.014, φ2 = −0.58, φ3 = −0.247, φ4 =−0.213, σ = 0.796, σ0 = 0.796, p = 0.9049, q = 0.755. Note that ps∗0 = 0.72037636 by
the assumption that the initial primitive state probability is ergodic.
• Base parameter setting:r = 1, α0 + α1 = 1.2, α0 = −0.4, φ1 = 0.1, σ = 0.7, σ0 = 0.7, p = 0.85, q = 0.7. Notethat ps∗0 = 0.66666667 by the assumption that the initial primitive state probability is
ergodic.
Monte Carlo simulation results on out-of-sample one-step-ahead point forecasts show thatthe bagged forecast estimators dominate the benchmark forecast estimator in terms of the
PMSE since the bagging reduces the variance. The tables of all the results are in the
appendix.
First, magnitude of the improvement by the bagging depends on uncertainty in the data
generating process. Table 7 compares the benchmark forecasts and the bagged forecasts that
use the parametric bootstrap with smoothed probability across different standard deviationsof the independent innovation term. The advantage to using the bagging becomes small when
the standard deviation is small. Note that ‘Percent difference in PMSE’ takes a negatively
large value when the bagging improves forecasts significantly.
Second, Tables 3 and 9 show the results across different Markov processes for the primitive state. If both states are less persistent (for example, if Markov transition probabilityof staying in the same state as at a previous date is 0.2) magnitude of the PMSE improvement becomes small.4 Note that persistence of states tends to be high in real data. For
instance, each Markov transition probability is larger than 0.7 in a study of real GNP in the
U.S. by Hamilton (1989). It is around 0.9 in an analysis of stock returns by Guidolin andTimmermann (2007).
Uncertainty from the Markov process and the independent innovations determines nonlinearity of the observed process. Hence, relative performance of the bagged forecast estimators
4

Overall, as the Markov transition probability of state 1 conditional on state 1 at the previous date,p, is larger, PMSE improvement by the bagging increases. Note that an absolute value of the constantterm is larger in state 1 than in state 0 in these examples. High persistence of the state that generates alarge constant increases PMSE improvement by the bagging. However, given small values of p (for example,
p = 0.2) the improvement is larger as another Markov transition probability of state 0 conditional on state
0 at previous date is larger. That is, if state 1 is less persistent, the PMSE improvement by the bagging islarger as persistence of state 0 increases.

9

across different Markov transition probabilities is interrelated to magnitude of uncertainty
from the independent innovations. If the standard deviation of the independent innovation
term is smaller than that in the base parameter setting (for example, σ = 0.5) the relative performance of the bagged forecast estimators depends more on the Markov transition
probability of states as in Table 11.5
Third, percentage improvement in the PMSE when using the bagging is similar acrossdifferent parameter values of coefficient
the bylagged
dependent
variable as in Table 8. Note
Percent Differenceof
in PMSEsample sizeof estimation:
100x[PMSE(Bagging)-PMSE(Benchmark)]/PMSE(Benchmark)that non-dependency of coefficient
onofthestate
is assumed
in the model.
Smoothed probability
states for
randomconditions
draws and for originalresiduals
M=100, r=1. Parameter values are those in the base parameter set.
-75
0

100

200

300

400

500

600

700

800

900

1000

-80

Percent

-85

-90

-95

-100
Sample size of estimation
Parametric Bootstrap

Residual Bootstrap

Figure 1: Percent difference in PMSE by sample size
Fourth, Figure 1 shows relative performance of the bagged forecast estimators by sample
size of estimation.6 Monte Carlo simulation results based on Markov switching models show
the bagging is also effective in large sample. A possible reason is that the nonlinearity comesfrom regime shifts and is stochastic in Markov switching models. The uncertainty about
5

If the standard deviation of the independent innovation is larger (for example, σ = 0.9) the magnitudeof PMSE improvement by the bagging does not vary much across different Markov transition probabilities
as in Table 12. Table 13 shows the results in smaller sample, T = 35. Table 10 compares the benchmark andfive bagging methods for different Markov transition probabilities of states.
6
In Figure 1, plots are at sample sizes T = 35, 40, 50, 60, . . . , 140, 150, 200, 300, . . . , 1000. Sample sizes forforecast evaluation M are set to 100. I use smoothed probability of states to randomly draw states for boththe parametric and the residual bootstrap, and to construct original residuals for the residual bootstrap.
In Tables 4, 5 and 6, I set long horizons and compare the performance of the benchmark and five bagging
methods for sample sizes T = 40, 150, and 500, respectively.

10

nonlinearity exists regardless of sample size; hence, even in large sample. It implies thatthe nonlinear forecasts are unstable even in large sample. The bagging reduces variancesof forecasts that stem from stochastic nonlinearity. Hence, as long as a Markov process forprimitive states generates nonlinearity, the bagging improves forecasts.

5

Application: Postwar Real GNP in the U.S.

I apply the above methods to the real GNP quarterly data in the United States. The data
come from Business Conditions Digest. The sample period is from the second quarter ofyear 1951 to the fourth quarter of year 1984. The level of GNP is measured at an annual
rate in 1982 dollars. I let xt be the real GNP. ỹt = log (xt ). For computational convenience,
I multiply 4ỹt by 100 in the estimation: yt = 100 × 4ỹt . The variable of yt is 100 times
the first difference in the log of real GNP. I set r = 4 as in Hamilton (1989) and study
out-of-sample one-step-ahead point forecasts.
One-step-ahead point forecasts: T=55
3.5

2.5

1.5

1982q3

1980q1

Benchmark1977q3

1972q3y

1975q1

1970q1

1967q3-0.5

1965q1

0.5

-1.5

-2.5

Parametric (s)Residual (s,s)

Figure 2: Out-of-sample one-step-ahead forecast estimates: T = 55
Figure 2 compares (1) forecasts by the benchmark estimator, (2) bagged forecasts using
the parametric bootstrap with smoothed state probability for random draws, and (3) bagged
forecasts using the residual bootstrap with smoothed probability for random draws andoriginal residuals. Table 14, Table 15, Table 16, Table 17, and Table 18 in the appendix
11

show the results of means, biases, variances, and PMSE for T = 45, 55, 65, 75, and 85,
respectively. Due to the variance reduction, the bagging improves forecasts in the sense of
the prediction mean squared error. Overall, the magnitude of improvement in the prediction
mean squared error by the bagging is larger in smaller sample. However, note that sample
sizes for forecast evaluation are not identical across these tables.

6

Conclusion

This paper proposes new variants of point forecast estimators in Markov switching models
utilizing bagging. To construct the bagged forecast estimators, I implement the parametricbootstrap and the residual bootstrap to nonlinear dynamic models in which the nonlinearity
comes from the changes in non-independent stochastic components.
I conduct Monte Carlo experiments to compare performance of the bagged forecast estimators with that of the benchmark forecast estimator in the sense of the PMSE. The
Monte Carlo simulation results show that interactions between a Markov process for primitive states and independent innovations affect the relative performance of the bagged forecast
estimators. First, the advantage to using the bagging becomes small when uncertainty of
the independent innovations is small. Second, if all primitive states are less persistent,
magnitude of the PMSE improvement by the bagging becomes small. Third, if uncertainty
from the independent innovations is smaller, relative performance of the bagged forecasts
depends more on the Markov transition probabilities. Fourth, the Monte Carlo simulations
present that the bagged forecast estimators dominate the benchmark forecast estimator evenin large sample. A possible reason is that the nonlinearity which comes from regime shifts,
is stochastic in Markov switching models. The nonlinear forecasts are unstable regardless ofsample size as long as the Markov process produces nonlinearity. Hence, the bagging reduces
forecast variances that stem from stochastic nonlinearity.
I also apply the bagged forecast estimators to study nonstationary time series of postwar
U.S. real GNP as in Hamilton (1989), where the first difference obeys a nonlinear stationary
process. The empirical evidence on out-of-sample forecasting presents that the bagged forecast estimators outperform the benchmark forecast estimators by Hamilton (1989) in thesense of the prediction mean squared error, due to the variance reduction.

12

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Heidelberg, Tokyo: Springer-Verlag.

A

Appendix: ModelTwo primitive states are assumed for simplicity. The state at date t, st , depends on theprimitive states at date from t to rth lags (s∗t ,...,s∗t−r ) and is independent of past observations
on ỹt . The constant term in Equation (1) becomes cst = α1 s∗t + α0 − φ1 (α1 s∗t−1 + α0 ) − · · · −φr (α1 s∗t−r + α0 ). To normalize, I set α1 > 0. In the estimation, I assume normality of theinnovation term t . Let φ(L) = 1 − φ1 L − φ2 L2 − · · · − φr Lr . I assume the stability condition
that all the roots of φ(z) = 0 are greater than 1 in absolute value.

14

A.1

Discussion of the Modelz̃t is a driftless ARIMA(r,1,0) process whose first difference contains i.i.d.(0, σ 2 ) innovations. In contrast, nt is an ARIMA(1,1,0) process whose first difference contains the nonindependent white noise innovations.
A.1.1

Primitive State Process – Serial Dependence, Non-normality, and Strict
Stationarity

By the assumption of the first-order Markov process, the primitive state process is strictly
stationary, and is an AR(1) process with unusual probability distribution of the innovation
sequence.
s∗t = (1 − q) + λs∗t−1 + υt ,(15)

where λ = −1 + p + q. Thus, conditional on s∗t−1 = 1, υt = 1 − p with probability p,
and υt = −p with probability 1 − p. Conversely, conditional on s∗t−1 = 0, υt = q − 1 withprobability q, and υt = q with probability 1 − q.
The innovation term, υt , is a white noise. That is, it is zero mean and serially uncorrelated: E(υt |s∗t−j ) = 0, hence, E(υt ) = 0, E(υt s∗t−j ) = 0 and E(υt υt−j ) = 0 for
j = 1, 2, . . . . However, the innovation term, υt , is not independent of lagged value of theprimitive state. For instance, E(υt2 | s∗t−1 ) depends on s∗t−1 : E(υt2 | s∗t−1 = 1) = p(1 − p) andE(υt2 | s∗t−1 = 0) = q(1 − q).
First, the innovation term, υt , cannot be normally distributed, whereas I suppose normality of the innovations in the other AR component, t ’s, in the estimation. Second, the serial
dependence of the system ỹt comes only from the innovation term in the Markov process,υt . It is because the other innovation term of t is i.i.d. and is independent of nt+j for all
j. The serial dependence of υt generates a nonlinear process for the observed series ỹt . The
best forecast is nonlinear because of this serial dependence.
Since the innovation term, υt , is a white noise, the primitive state process is covariancestationary when |p + q − 1| < 1. Moreover, the primitive state process is strictly stationary
because the joint distribution of the processes depends only on their relative distance. The
primitive state is strictly stationary when |p + q − 1| = 1, too. But, in this case, the primitive
state is degenerated to be a constant, that is, a non-stochastic process, and the process of ỹt
becomes a usual ARIMA process whose first difference has i.i.d. innovations. The condition
such that |p + q − 1| > 1 cannot be the case, because 0 ≤ p, q ≤ 1 hold.

15

A.1.2

I(1) Process

Although the process of ỹt (= log(xt )) contains the special non-independent innovations, it
has the same property as the usual I(1) process. Information about the economy at an initialpoint has no effect on the long run growth rate, and a permanent effect on the level. To see
it, let’s consider Beveridge-Nelson decomposition.
∆ỹt =
α0 + α1 s∗t + zt

(16)

1−q
1
1
= α0 + α1 1−λ+ ( 1−λLυt + φ(L)t ),where φ(L) = 1−φ1 L−φ2 L2 −· · ·−φr Lr . There exist ψ(L) = 1+ψ1 L+ψ2 L2 +· · · and a white11
υt + φ(L)t ) = ψ(L)ut . The ut is a white noise process, because thenoise ut such that ( 1−λLinnovation terms t and υt are white noises and are independent of each other. I assume oneP∞

j=0 j|ψj | < ∞. Using the following identity ψ(L) = ψ(1) + ∆β(L),Pjwhere ∆ = 1 − L, β(L) = ∞j=0 βj L , βj = −(ψj+1 + ψj+2 + · · ·), the process of ∆ỹt can besummability condition
rewritten as(17)∆ỹt = δ + ψ(1)ut + ηt − ηt−1 ,1−qwhere δ = α0 + α1 1−λ. The process of ỹt is obtained asỹt = δ · t +

Pt

(18)

i=1 ui + ηt + (ỹ0 − η0 ).

By the one summability assumption, β(L) is absolutely summable. Therefore, the thirdterm, ηt , is zero-mean covariance stationary. The first component, δ · t, is linear time trend.
The second component,

Pt

i=1 ui , is stochastic trend—that is, driftless random walk. The lastcomponent, ỹ0 − η0 , is the initial condition. Here, implicitly, I assume that ut is a stationary
martingale difference sequence such that 0 < E(u2t ) < ∞ and T1

PT2
t=1 E(ut |ut−1 , ut−2 , · · ·)converges in probability to E(u2t ).7
Both components, nt and z̃t , contain stochastic trend, stationary process and the initialcondition. The nt term contains the linear time trend. While information about the economy
at date 0 has no effect on the long run growth rate of ỹt , it does exert a permanent effect on
the level. The permanent effect of the information differences between the states comes only
from the Markov trend term. Hence, when I compare certain knowledge about the initial
7

Because the two innovation terms, t and υt , are independent, I can analyze them separately. Since t

is i.i.d., the assumptions for the property of the above stationary martingale difference sequence hold for its
1−qcomponent. As for υt , as t goes to infinity E(υt2 ) = πp(1 − p) + (1 − π)q(1 − q), where π = 2−p−q, and theassumptions for υt also seem to hold.

16

states, I can focus on the effect of nt . The effect of the difference between these knowledge
E0 xt |π0 =1
E0 xt |π0 =0λsets is E[ỹt |s∗0 = 1] − E[ỹt |s∗0 = 0] −→ α1 1−λas t −→ ∞.

κ−(−1+p+q)−→ κ−exp(α1 )(−1+p+q)

as t −→ ∞, where κ is a solution of κ(q + p exp(α1 ) − κ) = exp(α1 )(−1 + p + q).

BAppendix: PMSE

The population prediction mean squared error is population variance plus squared bias of
the forecast estimator. Let M denote a sample size for forecast evaluation. To simplify thenotation, E[.|yt+h ] denotes E[.|yt+h , Yt ; θ], and E[.] denotes E[.|Yt ; θ] in this section. The
PMSE conditional on the target variable of yt+h is
E[(yt+h − ŷt+h )2 |yt+h ]= E[{ŷt+h − E(ŷt+h |yt+h )}2 |yt+h ] + [E(ŷt+h − yt+h |yt+h )]2= E[{ŷt+h − E(ŷt+h )}2 ] + [E(ŷt+h ) − yt+h ]2(19)= V ar(ŷt+h ) + [Bias(ŷt+h |yt+h )]2 .
The derivation from the second line to the third line in Equation (19) comes from the factthat the forecast of ŷt+h is a function of the current and past data of Yt , and does not dependon the target variable. The evaluation method removes randomness of the target variable
and accesses the PMSE from the forecasting errors only.
To obtain the sample PMSE corresponding to Equation (19), I calculate the sample
prediction variance byV ˆar(ŷt+h )= M1

PT +M −1t=T[ŷt+h − M1

PT +M −1t=Tŷt+h ]2 .

(20)

The sample prediction bias is obtained byˆBias(ŷt+h |yt+h )= M1

CPT +M −1t=T(ŷt+h − yt+h ).

(21)

Appendix: Monte Carlo Simulation

A few notes on the Monte Carlo simulation are as follows. First, in the Monte Carlo procedure, I use the same random draws of standard normal and uniform distributions acrossdifferent parameter settings. Given presumed parameter values, I obtain simulated data of
the longest sample size once, and use a part of them in the analysis for a smaller sample

17

size. In other words, across different total sample sizes of simulated data, simulated data at
the overlapping dates are the same in an identical parameter setting.
Second, the first 499 simulated data are discarded. In other words, for estimation, forecasting, and the forecast evaluation, I use observations from the 500th simulated data.
Third, due to computational burden, for each combination of a sample size of estimation,a sample size of forecast evaluation, and parameter setting, I conduct Monte Carlo simulation
once. Instead of many simulations, I let a time horizon be very long—that is, the sample
size of forecast evaluation is large, in some simulation experiments.
In the following tables, ‘Variance’ is sample prediction variance of a forecast from Equation (20). ‘Bias’ is sample prediction bias of a forecast from Equation (21). ‘Mean’ is sample
prediction mean of a forecast: M1

PT +M −1t=Tŷt+h . ‘Difference in PMSE’ denotes the percent-age difference between the prediction mean squared error using the bagging and that in thebenchmark: 100 ×C.1

P M SEbagging −P M SEbenchmark.
P M SEbenchmark

Hamilton Parameter Setting
Table 1: Monte Carlo Simulation

T=40, M=1000, r=4. Parameter values are the same as in Hamilton (1989).
Bagging

BenchmarkType
yp of Bootstrapp

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.97333

0.99554

0.99361

0.992640.990140.99966

Bias-0 04650-0.04650-0 02430-0.02430-0 02622-0.02622-0 02720-0.02720-0 02969-0.02969-0 02018-0.02018

Variance

1.90515

0.09806

0.224700.437960.41203

0.57132

PMSE

1.907310.098650.225390.438700.41291

0.57173-94.83-88.18-77.00-78.35-70.02Difference in PMSE, %18

Table 2: Monte Carlo Simulation
T=150, M=1000, r=4. Parameter values are the same as in Hamilton (1989).
Benchmark

BaggingType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

1.00589

1.02557

1.02134

1.023641.03167

1.03876Bias-0.01389

0.005800.00157

0.00386

0.011900.01899

Variance

1.65749

0.039420.35748

0.33684

0.33735

0.66216PMSE

1.65768

0.03946

0.35748

0.33686

0.33749

0.66252-97.62-78.44-79.68-79.64-60.03Difference in PMSE, %

19

20

Table 3: Monte Carlo Simulation

Benchmark
Bagging
Benchmark
Bagging
Benchmark
Bagging
Benchmark
Bagging

Mean

Bias

Variance

PMSE

0.8
0.8

0.4
0.6
0.29130.28850.01220.0093
2.99630.1603
2.99640.1604-94.650.64670.64660.04780.04783.35330.10673.3556
0.1090-96.75

0.8
0.6

0.2
0.6

-73.22

1.7476
0.46801.74440.46790.0571-0.0133-97.833.45340.07483.45260.07480.02760.0035

0.8
0.4

0.4
0.4

-92.752.99690.21742.99670.21640.01540.03170.30980.3260-96.372.5232
0.09162.52220.08780.0325
0.06160.90540.93450.5
0.5
0.44380.4197-96.262.51430.09422.51300.09110.03620.05570.90900.92860.0470-0.0234-96.402.48210.08942.48130.08590.02840.05910.90130.93200.6
0.2

Percent difference in PMSEpq

Markov transition
probability
0.59190.5790

0.6
0.8

0.4
0.2

-93.63

2.15430.13732.15250.13700.04160.01570.2
0.4-59.091.77040.7243

1.76980.7224-0.0240-0.04380.22470.2049-97.953.2081
0.06573.20800.06570.0083-0.00460.59480.5689-96.332.56430.09412.56380.09050.02150.06020.89440.9331

0.8
0.2

0.2
0.8

0.3
0.3

-63.461.65140.60341.65130.60310.00790.01810.58910.5599

0.6
0.6

0.2
0.2

-33.272.69511.79862.69471.7945-0.0201-0.0636-97.693.21440.0743

3.21440.07380.0055-0.0238

0.28950.2460-94.451.59930.08881.59790.08850.03720.0167-0.0034-0.02390.40890.4190-94.702.99640.15872.99630.15850.01220.01280.29130.29190.4
0.8

Parametric Bootstrap with Normally Distributed Error
Smoothed state probability is used for random draws. T=150, M=100, r=4.
Parameter values except Markov transition probability of primitive states are the same as in Hamilton (1989).

-97.523.20580.07953.20570.07810.0130-0.0366

0.59670.5470

0.6
0.4

C.2

Base Parameter Setting
Table 4: Monte Carlo Simulation

T=40, M=1000, r=1. Parameter values are the same as in the base parameter set.
Benchmark

BaggingType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

1.00206

0.95953

0.84674

0.96408

1.078410.96204

Bias

0.03957

0.00296-0.00296

0.11575-0.11575

0.00159

0.115930.00044-0.00044

Variance

0.10253

0.01660

0.040210.01494

0.01769

0.02446

PMSE

0.104100.01660

0.053610.01494

0.031130.02446-84.05-48.50-85.65-70.10-76.50Difference in PMSE, %21

Table 5: Monte Carlo Simulation
T=150, M=1000, r=1. Parameter values are the same as in the base parameter set.
Benchmark

BaggingType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.99503

0.95755

0.85249

0.96150

1.06749

0.95948Bias

0.03601-0.00146-0.10653

0.002480.108480.00047

Variance

0.08557

0.00767

0.027100.00433

0.00499

0.01328

PMSE

0.08687

0.00767

0.038450.00434

0.01676

0.01328-91.17-55.73-95.01-80.71-84.71

Difference in PMSE, %Table 6: Monte Carlo Simulation
T=500, M=1000, r=1. Parameter values are the same as in the base parameter set.
Benchmark

BaggingType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.98926

0.952650.846170.95525

1.06519

0.95389

Bias

0.03412-0.00250-0.10897

0.000100.11004-0.00126

Variance

0.079540.005770.02506

0.002420.00125

0.01155

PMSE

0.080700.00577

0.03694

0.002420.01336

0.01155-92.84-54.23-97.00-83.45-85.69Difference in PMSE, %22

230.969540.985560.00080
0.016820.281130.04803
0.281130.04831-82.820.968170.982740.000220.01480
0.21215
0.04556
0.21215
0.04578-78.42

Benchmark
Bagging

PMSE

1.1

Benchmark
Bagging

Variance

1

Benchmark
Bagging

Bias

Percent difference in PMSE

Benchmark
Bagging

Mean

Standard deviation

Table 7: Monte Carlo Simulation-85.570.35383
0.051070.353830.050760.00169
0.01773

0.97123
0.98727

1.2

-3.66

0.01158
0.011160.01156
0.011140.003990.004350.96479
0.965150.1-86.200.43161
0.05957

0.43160
0.059130.003200.02088

0.97353
0.991221.3-26.150.015700.01159

0.01567
0.01156

0.005700.00582

0.967290.967410.2-84.870.51769
0.07834

0.51767
0.07774

0.00476
0.02450

0.97589
0.995631.4-44.150.02094
0.011700.020780.01162

0.01258
0.00871

0.974960.971090.3-84.100.61725
0.098140.617190.097220.007770.030310.97969
1.00224

1.5-70.350.038230.01134

0.03782
0.011220.020350.01060

0.98353
0.973780.4-82.860.72935
0.12499

0.72927
0.124220.009020.02764

0.98173
1.00036

1.6-84.61

0.048390.007450.04743
0.00730

0.03094
0.01226

0.99491
0.976240.5-82.480.864770.151530.864730.15037

0.006020.03406

0.97953
1.00757

1.7-93.400.06172
0.004070.05985
0.003960.043260.010541.00803
0.97531

0.6

Parametric Bootstrap with Normally Distributed Error
Smoothed state probability is used for random draws. T=150, M=100, r=1.
Parameter values except standard diviation of innovation terms are the same as the base parameter set.

-81.720.996070.18209

0.99604
0.180830.00591
0.03558

0.980221.009881.8-96.200.074810.00284

0.07250
0.002770.04803
0.00846

1.013590.974020.7-80.830.097200.01863

0.096880.01859

0.01775
0.006200.98411
0.972560.8-74.790.149110.03759

0.149110.037440.002360.01229

0.96952
0.97945

0.9

24Table 8: Monte Carlo Simulation

Benchmark
Bagging
Benchmark
Bagging
Benchmark
Bagging

Bias

Variance

PMSE

Percent difference in PMSE

Benchmark
Bagging

Mean

Coefficient of first lagged
dependent variable-96.200.074810.00284

0.07250
0.002770.04803
0.00846

1.013590.974020.1-96.310.10397
0.00383

0.10157
0.003780.04908
0.00747

1.01391
0.972300.2-96.460.15035
0.00532

0.14775
0.00523

0.051030.009411.014700.973090.3-96.560.22604
0.007770.22327
0.00755

0.052680.01479

1.01465
0.976760.4-96.530.34951
0.012120.34661
0.01161

0.053800.02247

1.013280.98195

0.5-96.500.554930.019420.55203
0.01835

0.05383
0.03274

1.009410.988320.6

0.7-96.320.91068
0.033540.90801
0.030480.05174
0.05537

1.000191.00382

Parametric Bootstrap with Normally Distributed Error
Smoothed state probability is used for random draws. T=150, M=100, r=1.
Parameter values except the coefficient of the first lagged dependent variable are the same as the base parameter set.

-95.501.54456
0.06945

1.542450.059420.04591
0.100170.97755
1.031800.8-92.362.547140.19470

2.545530.140450.040130.232910.917121.109910.9

25

Benchmark
Bagging
Benchmark
Bagging
Benchmark
Bagging

Bias

Variance

PMSE

Percent difference in PMSE

Benchmark
Bagging

0.4
0.6
0.25690.24640.01130.00080.11640.05530.1166
0.0553-52.590.6
0.2
0.5916
0.59760.01010.01600.11020.04150.11030.0418-62.15-85.730.06420.00920.06210.00890.04620.01600.91580.8855-87.620.11270.01400.11000.01370.05150.0151-0.0069-0.04330.6

0.2-85.81

0.06420.0091

0.06210.00880.04620.01670.91580.8862-55.290.10120.04530.10100.04520.01590.00220.27750.26380.4

0.4-85.600.06420.00920.06210.00890.04620.01720.91580.88680.4-85.200.08010.01190.07720.0116

0.05390.01450.58750.54800.2

0.4-85.61

0.06420.00920.06210.00890.04620.01730.91580.88680.2-48.510.29620.15250.29580.15220.02030.01730.23390.23080.4

0.2-54.240.15020.06870.15020.0687-0.00360.00560.56200.57120.8-39.030.3891
0.23720.3862
0.23600.05390.03530.33150.31290.2

0.2-52.360.1166
0.05550.11640.05550.01130.00050.25690.24610.8-91.540.09930.0084

0.09690.00800.04910.0206-0.0413-0.06980.8-53.650.15020.06960.15020.0696-0.00360.00430.56200.56990.6-53.760.15020.06940.15020.0694-0.00360.00400.56200.5695

0.4

0.6

q

0.8

0.6

Mean

Markov transition
probability

Table 9: Monte Carlo Simulation
Parametric Bootstrap with Normally Distributed Error
Smoothed state probability is used for random draws. T=150, M=100, r=1.
Parameter values except Markov transition probability of primitive states are the same as in the base parameter set.
p
0.8
0.8
0.8
0.8
0.6
0.4
0.2
0.6

26
Benchmark
Bagging
Benchmark
Bagging
Benchmark
Bagging

Bias

Variance

PMSE

Percent difference in PMSE

Benchmark
Bagging

Mean

Type of Bootstrap
state probability for draws
primitive state probability for residuals

Table 10: Monte Carlo Simulation-54.240.15020.06870.15020.0687-0.00360.00560.56200.5712-51.970.15020.07210.15020.0713-0.00360.02930.56200.5949-55.190.15020.06730.15020.0673-0.00360.00370.56200.5692-55.450.1502
0.06690.1502
0.0666-0.0036-0.0188

0.56200.5468-54.290.15020.06860.15020.0686-0.00360.00560.56200.5712-85.81

0.06420.0091

0.06210.00880.04620.01670.91580.8862-61.590.06420.02470.0621
0.02200.0462-0.05180.91580.8178-92.81

0.06420.0046

0.06210.00420.04620.01940.91580.8890-82.080.06420.01150.06210.0031

0.0462
0.09160.91580.9612

Markov transition probability of primitive states
p=0.6, q=0.8
p=0.8,q=0.6
Parametric Parametric Residual Residual Residual
Parametric Parametric Residual Residualsmoothed
inferred smoothed smoothed
inferredsmoothed
inferred smoothed smoothedsmoothed
inferred
inferredsmoothed
inferredParameter values except Markov transition probability of primitive states are the same as the base parameter set.
T=150, M=100, r=1.

-82.200.06420.01140.06210.01090.04620.02240.91580.8919Residual
inferred
inferred27
0.6
0.25340.24410.00950.00010.0747
0.03630.07480.0363-51.420.59620.59070.01620.01070.05700.02030.05720.0204-64.37-89.230.04600.00500.4

Benchmark
Bagging

PMSE

0.04380.00430.2

Benchmark
Bagging

Variance

0.04730.02610.91520.89410.8

0.6

Benchmark
Bagging

Bias

Percent difference in PMSE

Benchmark
Bagging

q

0.6

0.2

-79.000.06920.01450.06790.01420.03640.0174-0.0237-0.0426-89.230.04600.00500.04380.00430.04730.02620.91520.8941

0.6

0.4

0.4

-59.230.06110.02490.06090.02490.01370.00320.2736
0.2632-89.230.04600.00500.04380.00430.04730.02610.91520.8941

0.4

0.2

0.4

-49.540.13840.06980.13650.06940.04300.01980.57500.5517-89.230.04600.00500.04380.00430.04730.02610.91520.8941

0.2

0.4

0.2

-25.720.2674
0.1986

0.26700.19820.01860.01950.23060.2314-44.36

0.0971
0.05400.0971
0.0540-0.0012-0.0033

0.56280.5607

0.8

0.2

0.2

-19.050.37250.30160.37020.29980.04770.04130.32370.3173-51.420.07480.03630.0747
0.03630.00950.00010.25340.2441

0.8-86.430.06500.00880.06340.00830.04030.0220-0.0517-0.07010.8-44.36

0.0971
0.05400.0971
0.0540-0.0012-0.0033

0.56280.56070.6-44.36

0.0971
0.05400.0971
0.0540-0.0012-0.0033

0.56280.5607

0.4

Parameter values except standard deviation of innovation terms and Markov transition probability of primitive states are thesame as in the base parameter set.
p
0.8
0.8
0.8
0.8
0.6
0.4
0.2
0.6
0.6

Mean

Markov transition
probability

Table 11: Monte Carlo Simulation
Parametric Bootstrap with Normally Distributed Error
Standard deviation of innovation terms is 0.5.
Smoothed state probability is used for random draws. T=150, M=100, r=1.

28
0.00880.00410.20370.08640.20380.0864-57.600.20080.07480.20080.0752-62.53-60.660.11890.04680.00510.0202

Benchmark
Bagging

PMSE

0.11830.04640.25590.2512

Benchmark
Bagging

Variance

0.02520.0192

0.58830.6034

Benchmark
Bagging

Bias0.89640.89040.4
0.6

Benchmark
Bagging

Mean

0.8
0.8

0.6
0.2

p
q

0.2
0.6

-92.310.17520.01350.17470.01320.02120.0163-0.0357-0.0406-61.110.11890.04630.11830.04590.02520.01980.89640.8910

0.8
0.6

0.4
0.4

-58.810.19180.07900.19160.07900.01380.00520.27700.2683-61.050.11890.04630.11830.04590.02520.01980.89640.8909

0.8
0.4

0.4
0.2

-79.530.06430.0132

0.06320.01290.03430.01770.56950.5528-61.070.11890.04630.11830.04590.02520.01970.89640.8909

0.8
0.2

0.2
0.4

-64.720.28400.10020.28390.10000.00620.01360.22140.2288-57.660.23550.09970.23550.0996-0.0057
0.01210.56150.5793

0.6
0.8

0.2
0.2

-54.300.37620.17190.37510.17110.03230.02830.31140.3074-57.760.20380.08610.20370.08610.00880.00540.25590.25250.4
0.8-93.86

0.15650.0096

0.15600.0091

0.02270.0234-0.0662-0.06540.2
0.8-56.520.23550.10240.23550.1023-0.0057
0.01100.56150.5781

0.6
0.6

0.6
0.4

-59.520.23550.09530.23550.0952-0.0057
0.01230.56150.5795

Parameter values except standard deviation of innovation terms and Markov transition probability of primitive states are thesame as in the base parameter set.

Markov transition
probability

Percent difference in PMSE

Table 12: Monte Carlo Simulation
Parametric Bootstrap with Normally Distributed Error
Standard deviation of innovation terms is 0.9.
Smoothed state probability is used for random draws. T=150, M=100, r=1.

29
0.28610.2941-0.0436-0.0357
0.11550.05290.11740.0541-53.91-0.0358-0.04230.11400.04330.11520.0451-60.86-73.650.07390.01950.63000.6234

Benchmark
Bagging

PMSE

0.07380.01790.4
0.6

Benchmark
Bagging

Variance

0.0074-0.0391

0.96110.91460.6
0.2

Benchmark
Bagging

Bias

Percent difference in PMSE

Benchmark
Bagging

Mean

Markov transition
probability

Table 13: Monte Carlo Simulation

0.2
0.6

-61.340.12760.04930.12730.0476-0.0185-0.04170.02320.0000-74.930.07410.01860.07410.01710.0069-0.03810.96070.91560.4
0.4-53.92

0.1171
0.05400.11520.0527-0.0433-0.0351

0.28650.2946-75.690.07410.01800.07410.01680.0069-0.0355

0.96070.91830.4
0.2-66.260.08530.02880.08470.02830.0243-0.02140.64210.5964-75.570.07410.01810.07410.01690.0069-0.0349

0.96070.91880.2
0.4-36.600.30400.19280.30050.1892-0.0594-0.05970.23840.2381-49.200.17740.09010.17650.0886-0.0304-0.0394

0.58740.57840.2
0.2-26.320.44090.32490.44090.3238-0.0040-0.0330

0.37380.3448-55.22

0.11740.05260.11550.0514-0.0436-0.0351

0.28610.2947-72.430.09920.0274

0.09780.0243-0.0378-0.0552-0.0281-0.0455-50.240.17740.08830.17650.0867-0.0304-0.0401

0.58740.5776

Parametric Bootstrap with Normally Distributed Error
Smoothed state probability is used for random draws. T=35, M=100, r=1.
Parameter values except Markov transition probability of primitive states are the same as in the base parameter set.
p
0.8
0.8
0.8
0.8
0.6
0.4
0.2
0.6
q
0.8
0.6
0.4
0.2
0.8
0.8
0.8
0.6

-50.460.17740.08790.17650.0864-0.0304-0.0391

0.58740.5787

0.6
0.4

D

Appendix: Application

Table 14: Real GNP in the U.S.
T (sample size of estimation) =45, M (sample size of forecast evaluation) =90, r (# lags) =4.
Bagging

BenchmarkType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.679800.758030.747450.73797

0.75698

0.74212Bias-0.06623

0.01200

0.00143-0.00806

0.01095-0.00391

Variance

0.582870.12983

0.13386

0.161720.14801

0.14678

PMSE

0.587260.129970.13386

0.16179

0.148130.14679-77.87-77.21-72.45-74.78-75.00Difference in PMSE, %30

Table 15: Real GNP in the U.S.
T (sample size of estimation) =55, M (sample size of forecast evaluation) =80, r (# lags) =4.
Bagging

BenchmarkType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.78937

0.79537

0.811820.81094

0.78056

0.81182Bias

0.08034

0.08635

0.102800.10192

0.07153

0.10280

Variance

0.32766

0.07483

0.082420.083530.082960.08242

PMSE

0.334110.08229

0.09299

0.093910.088070.09299-75.37-72.17-71.89-73.64-72.17

Difference in PMSE, %Table 16: Real GNP in the U.S.
T (sample size of estimation) =65, M (sample size of forecast evaluation) =70, r (# lags) =4.
Bagging

BenchmarkType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.723850.75869

0.779200.753700.734080.72989Bias

0.08960

0.124450.144960.119450.09983

0.09564

Variance

0.171700.06269

0.04530

0.068360.08738

0.04970

PMSE

0.17973

0.07818

0.066310.082620.09735

0.05885-56.50-63.11-54.03-45.84-67.26

Difference in PMSE, %31

Table 17: Real GNP in the U.S.
T (sample size of estimation) =75, M (sample size of forecast evaluation) =60, r (# lags) =4.
Bagging

BenchmarkType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.716220.717370.76592

0.71105

0.693250.74687

Bias

0.096220.09738

0.14592

0.09105

0.073250.12687

Variance

0.13394

0.05238

0.053220.063640.064900.04660

PMSE

0.143190.06186

0.07452

0.071930.07027

0.06270-56.80-47.96-49.77-50.93-56.21

Difference in PMSE, %Table 18: Real GNP in the U.S.
T (sample size of estimation) =85, M (sample size of forecast evaluation) =50, r (# lags) =4.
B h
Benchmarkk

B i
BaggingType of Bootstrap

ParametricParametric

Residual

Residual

ResidualState probability for draws

smoothed

inferredsmoothedsmoothed

inferredsmoothed

inferred

inferredState probability for residuals

Mean

0.731610.749220.786270.738700.716120.75162Bias

0.12444

0.14205

0.179100.13153

0.10895

0.14445

Variance

0.15789

0.07382

0.036210.07660

0.08826

0.04156

PMSE

0.17338

0.09400

0.06829

0.093900.100130.06243-45.78-60.61-45.84-42.25-63.99Difference in PMSE, %32