State Space Markov Switching Models Using Wavelets

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State Space Markov Switching Models Using Wavelets

Airlane P. Alencar, Pedro A. Morettin and Clelia M. C. Toloi University of S˜ao Paulo, Brazil

Abstract

We propose a state space model with Markov switching, whose regimes are associated with the model parameters and regime transition probabilities are time-dependent. The estimation is based on maximum likelihood method using the EM algorithm. The distribution of the estimators is as- sessed using bootstrap. To evaluate the state variables and regime probabilities, the Kalman filter and a probability filter procedure conditional to each possible regime at each instant are used. This procedure is illustrated with simulated data and the United States monthly industrial production index from January 1960 to January 1995.

Key words:State space model, Markov switching model, Kalman filter, EM algorithm, Wavelets.

Classification: C15, C32

Pedro A. Morettin, corresponding author Department of Statistics

University of S˜ao Paulo C.P. 66281

05311-970- S˜ao Paulo, SP Brazil

telephone: + 55-11-30916125 fax: + 55-11- 30916130 e-mail: [email protected]

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1 Introduction

We observe often in practice several changes in the behavior of time series. These changes may be modeled with unobserved regimes, each one determining a different set of parameters in a proposed model. Examples of such regimes are present in the economic activity indicators that exhibit some patterns of recession or expansion periods. Regime switching models were first addressed in the context of regression models with independent (Quandt, 1972) and Markovian transitions (Goldfeld and Quandt, 1973) and in autoregressive models (Hamilton, 1989). Transition probabilities dependent on explanatory variables using the logistic function are reported in Diebold, Lee and Weinbach (1994), who propose a Gaussian model with mean and variance switching regimes, and in Filardo (1994), with a switching in the parameters of the autoregressive model.

State space models are dynamic models that depend on unobserved variables, called state variables. Models with unobserved trend and cyclical components and ARMA models can be included in the state space approach.

The recursive procedure used to estimate the state variables is the Kalman filter (Kalman, 1960). The method of maximum likelihood to estimate the parameters may be based on Newton-Raphson procedure, BFGS or the EM algorithm.

The EM algorithm is a non-linear optimization algorithm, made popular by Dempster et al. (1977), appropriate for applications in models involving non-observed components or irregularly observed data. It is a method to find the maximum likelihood estimate (MLE) of the observed (incomplete) data by using the likelihood of the complete data in a clever way.

State-space models and switching models have been highly productive areas of research recently, specially the case of Markov switching models.

These systems are characterized by a state (unobserved) variable, which allows for shifts in the parameters of the system. The transition probabilities of the chain may be constant or vary along time.The Kalman filter will provide the estimate of the state variable conditional to the regime and the regime probabilities will be calculated using the probability filter. A comprehensive treatment of state-space and Markov switching models is given by Kim and Nelson (1999).

The aim of this work is to evaluate the maximum likelihood method to state space Markov switching models with time varying transition probabilities using the EM algorithm. In order to implement the algorithm we propose another smoothing probability algorithm to estimate the regime

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probability based on the whole sample information. To evaluate the distribution of the estimators, we propose a bootstrap for state space models developed by Stoffer and Wall (1991).

We illustrate the proposed procedure with simulated data and with real data, namely the US monthly industrial production index from January, 1960 to January 1995.

2 Background

In this section we provide some background information on state space models and wavelets. In most of the cases the estimation methods use maximum likelihood estimators or Bayesian methods. In this paper we restrict our attention to maximum likelihood estimators for state space models. Readers interested in Bayesian methods for these models can look for details in Harrison and Stevens (1976) and West and Harrison (1997).

2.1 State space models

Consider the usual model defined by the equations

yt = Ftxt+Dzt+vt, (1) xt = µt+Gtxt−1+wt, (2) fort = 1, . . . , T, where y_t is an q×1 vector of observations, F_t is an q×p matrix (called the system matrix),xtis ap×1 vector of unknown states,zt

is ak×1 vector of exogenous variables andG_t is ap×pmatrix (called the transition matrix) that describes how the states behave across time. The observation error vt and the error wt associated with the state vector are assumed to be independent Gaussian white noise processes, with zero means and covariance matricesV_t and W_t, respectively. Fort= 0 we also assume that x0 is normal, with mean ν and covariance matrix Σ. In this way, the process y_t is completely specified by the so-called characterization vector ϕ= (F_t, G_t, V_t, W_t), in the notation of Harrison and Stevens (1976). This vector may depend on a set of unknown parameters that will have to be estimated.

The Kalman filter is a recursive procedure used to compute the optimal estimator of the state vector at any instant t of time, having information up to time t. The procedure may be viewed as a two-stage one, for which in the first step we want to obtain the best estimate of observation at time

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tusing the information up to time t−1 (corresponds to obtaining the best prediction through the prediction equations) and in the second step, the knowledge of the new available observation is used to update the prediction obtained in the previous step (this is done through the updating equations).

The parametersϕ, ν and Σ are assumed to be known for allt. The mean of x_tobtained by the Kalman filter, under normality, is an optimal estimator in the sense of minimizing the mean square error. If we do not have normality, the Kalman filter provides the optimal estimator within the class of linear estimators (Harvey, 1989). Even with normality assumption the Kalman filter does not provide robust estimates. Meinhold and Singpurwalla (1989) present a method to robustify the Kalman filter.

The following notation will be used:

x^s_t = E(x_t|y₁, . . . , y_s), (3) P_t^s = Var(x_t|y₁, . . . , y_s), (4) P_t,t−1^s = Cov(x_t, xt−1|y₁, . . . , y_s). (5) Then x^t_t and P_t^t will be the estimators derived from the Kalman filter, x^T_t andP_t^T, t≤T are the smoothed estimators of minimum mean square error, based on all observations y₁, . . . , y_T, and x^T_t, P_t^T, t > T are the predictors ofθ_t.

We do not present here the equations of the usual Kalman filter. Details can be found in Anderson and Moore (1979). For references on non-linear and non-Gaussian state space models see Fahrmeir (1992), Kitagawa and Gersh (1996)and Durbin and Koopman (2001).

2.2 Wavelets

We now turn to some ideas on wavelets. The basic fact about wavelets is that they arelocalized in time (and space), contrary to what happens with the trigonometric functions. This behavior makes them ideal to analyze nonstationary signals and those with transients or singularities. Fourier bases are localized in frequency but not in time: small changes in some of the observations may induce substantial changes in almost all the components of a Fourier expansion, a fact that does not hold for a wavelet expansion.

We start with amotherwaveletψ and afatherwavelet (orscaling function ) φ, such that they generate an orthonormal system of L₂(IR), which we call{φ_j₀_,k(t)} ∪ {ψ_j,k(t)}_j≥j₀_,k, withφ_j₀_,k(t) = 2^j⁰^/2φ(2^j⁰t−k), ψ_j,k(t) = 2^j/2ψ(2^jt−k), for j ≥ j and j is the coarsest scale. Some properties

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may hold for these wavelets, as the admissibility condition R

ψ(t)dt= 0, or that the first (r−1) moments of ψ vanish, for some r ≥ 1. The degree of smoothness ofψ is then given byr.

For anyf ∈L2(IR), we may then consider the expansion f(t) =X

k

α_kφ_j₀_,k(t) +X

j≥j₀

X

k

β_j,kψ_j,k(t), (6) where the true wavelet coeffcients are given by

α_k= Z

f(t)φ_j₀_,k(t)dt, β_j,k = Z

f(t)ψ_j,k(t)dt. (7) An estimate will take the form

fˆ(t) = X

k

ˆ

αkφj0,k(t) +X

j≥j0

X

k

δ( ˆβj,k, λ)ψj,k(t), (8) where the ˆαk, βˆjk are estimates ofαk, βjk respectively, andδis a threshold, with threshold parameterλ.

Several issues of interest here are the choice of the wavelet basis, thresholding policy, parameters appearing in the thresholding scheme and the estimation of the scale parameter (noise level). For details see Morettin (1997).

Concerning the choice of the wavelet basis, some possibilities are the Haar, compactly supported wavelet bases (Daubechies, 1992), complex wavelets (Morlet, or modulated Gaussian), Mexican hat (second derivative of Gaussian), Shannon, Meyer, etc. In this paper we will use the Haar wavelet.

3 Model specification

The idea of the Markov switching state space model is to propose a state space model where the parameters can assume different values accord- ing to a non observed processSt, determining the regimes.

The state space model with Markov regime switching consists of (1)-(2) written as

yt=FStxt+DStzt+vt, (9) xt=µSt +GStxt−1+wt. (10) To complete the model specification, we need to define how the regimes behave along time. Considermpossible regimes,S_t=jwherej= 1, . . . , m.

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First, the probability of regimeStmust depend on the previous regime,St−1. Further, it is very reasonable that the transition probabilities, denoted by π_ij(t) =P(S_t= j|S_t−1 =i), i, j = 1, . . . , m, t = 1, . . . , T, vary with time.

These two properties defines the process (St) as a non-stationary Markov chain. The probability of initial regimeS₀is denoted byπ_j(0) =P(S₀ =j), whereπ_j(0), j= 1, . . . , m−1 are unknown parameters.

A suitable way to model the time-varying transition probabilities is to define them as follows:

πij(t) = exp(β₀^ij +PL

l=1β_l^ijψ_l(t/T)) 1 + exp(β^ij₀ +PL

l=1β_l^ijψ_l(t/T)), i= 1, . . . , m, j = 1, . . . , m−1 1

1 + exp(β^ij₀ +PL

l=1β_l^ijψ_l(t/T)), i= 1, . . . , m, j =m

whereψ_l(·) corresponds to mother Haar wavelets andlrepresents each pair of scale and location indexes.

Advantages of this parametrization is to avoid the parameter estimation under restrictions on the parameter space, what is achieved using the logistic function, and allows transition probabilities to vary along time.

The parameter vector is Θ = (Fi, Di, µi, Gi, Vi, Wi, ν,Σ, πj(0), β_l^ij), l = 0, . . . , L, i = 1, . . . , m e j = 1, . . . , m−1. The iterative estimation process proceeds as follows:

• For fixed parameters values, run the Kalman filter conditional to regimes at timet−1 andt to estimatex_t;

• Run the probability filter to obtain the probabilities of (St=j, St−1 = i) conditional to the information up to timet;

• Obtain the estimate of Θ by the maximum likelihood method using the EM algorithm.

These three steps are repeated until convergence is reached.

4 Conditional Kalman Filter

For usual space state models, the estimators of the unobserved variable xt given the observed data are provided by the Kalman filter and smoothing procedure. Including the Markov switching regimes, the Kalman

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filter and smoothing will be accomplished in the usual way but conditional to the regimes (Kim and Nelson, 1999).

Throughout this work, denote ye_s = (y₁, . . . , y_s), Ses = (S1, . . . , Ss),

x^ij_t|s = IE(xt|yes, St=j, St−1 =i), P_t^ij

1,t2|s = IE[(xt1−x^ij_t

1|s)(xt2 −x^ij_t

2|s)⁰|yes, St=j, St−1 =i]

as in Shumway and Stoffer (2000).

For fixed parameters values and conditional to St−1 = i e S_t = j, the conditional Kalman filter prediction and updating equations provides re- spectivelyx^ij_t|t−1 and x^ij_t|t.

Prediction

x^ij_t|t−1 = µ_j+G_jxⁱ_t−1|t−1, (11)

P_t|t−1^ij = GjP_t−1|t−1ⁱ G⁰_j+Wj. (12) Updating

xîj_t|t = xîj_t|t−1+K_tîj(y_t−F_jxîj_t|t−1−D_jz_t) (13)

P_t|tîj = (I−K_tîjF_j)P_t|t−1îj , (14)

where the Kalman gain is

K_tîj = P_t|t−1îj F_j⁰(FjP_t|t−1îj F_j⁰+Vj)⁻¹.

Note that conditioning on all regimes until time t, Se_t, it is necessary to considerm^t possible cases. Then it is necessary to propose some aprox- imations to implement the Kalman filter. We employ the approximations proposed by Harrison and Stevens (1976) and used by Kim and Nelson (1999). This approximation, properly called collapsing, consists on replac- ing x^ij_t|t and P_t|t^ij by x^j_t|t and P_t|t^j , respectively, conditional only to S_t = j, which is necessary to complete the conditional Kalman filter.

Collapsing

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x^j_t|t =

m

X

i=1

∆^ij_t x^ij_t|t, P_t|t^j =

m

X

i=1

∆^ij_t = P(St−1=i|S_t=j,ye_t) = P(St−1 =i, St=j|yet) P(S_t=j|ye_t) .

The smoothing filter (Kim and Nelson, 1999) predicts xt conditional on eyT, St = j, and St+1 = k, for t = T, . . . ,1. Here it is also necessary to collapse, which means to find x^j_t|T and P_t|T^j based on x^jk_t|T and P_t|T^jk. The following equations summarize the smoothing procedure.

Smoothing

J_t^jk = P_t|t^j G⁰_k(P_t+1|t^jk )⁻¹ x^j_t|T =

Pm

k=1P(St=j, St+1=k|yeT)x^jk_t|T P(S_t=j|ye_T) , P_t|T^j =

Pm

i=1P(S_t=j, S_t+1 =k|ey_T)h

As it will be seen later, to maximize to likelihood function using the EM algorithm, it will be necessary to calculate the covariances (similar to Shumway and Stoffer, 2000).

P_T,T^j _−1|T = (I−K_T^jFj)GjP_T−1|T−1,

P_{t−1,t−2|T}^j = P_t−1|t−1^j J_t−2⁰ +J_t−1^j (Pt,t−1|T −GP_t−1|t−1^j )J_t−2⁰ , t=T, . . . ,2, whereJ_t−1^j =P_t−1|t−1^j G⁰[P_t|t−1^j ]⁻¹.

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5 Probability Filter

Likewise the Kalman filter, the probabilities of{S_t−1 =i, S_t=j}, are obtained conditional toyet−1 and eyt. This is called probability or Hamilton filter and can be implemented following the five next steps fort= 1, . . . , T.

1. Using the prediction error decomposition for state space models, yt|(St−1=i, St=j,eyt−1)∼N(Fjx^ij_t|t−1+Djzt, FjP_t|t−1^ij Fj+Vj);

2. f(yt|eyt−1) =P

St

P

St−1f(yt|S_t, St−1,yet−1)P(St, St−1|yet−1);

3. P(S_t=j, St−1 =i|eyt−1) =P(S_t=j|S_t−1 =i)P(St−1 =i|yet−1);

4. Updating foryt,

P(St=j, S_t−1=i|yet) = f(yt|St=j, S_t−1=i,ye_t−1)P(St=j, S_t−1=i|ey_t−1)

M

X

i=1 M

X

j=1

f(yt|St=j, St−1=i,eyt−1)P(St=j, St−1=i|eyt−1)

;

5. Finally, calculate the marginal probabilityP(S_t=j|ye_t).

In order to obtain the smoothed probabilityP(St=j, St−1 =i|yeT), Kim and Nelson (1999) suggest the approximation

P(St=j, St+1=k|ye_T) =P(St+1 =k|ey_T)×P(St=j|S_t+1 =k,ye_T)

≈P(S_t+1 =k|ey_T)×P(S_t=j|S_t+1 =k,ye_t).

In this work, we use an alternative probability smoothing procedure based on Diebold et al. (1994). For each pair (St, St−1), t = 2, . . . , T, calculate

1. Forτ = 1,

P(S_τ, S_τ−1, S_t, S_t−1|ey_τ) =

m

X

S_τ−2=1

f(y_τ|Sτ, S_τ−1,ey_τ)P(S_τ|Sτ−1)P(S_τ−1, S_τ−2, S_t, S_t−1|ey_τ−1)

f(yτ|yeτ−1) ;

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3. Whenτ =T, the marginal probabilities are P(St−1, S_t|ey_T) =

m

X

ST=1 m

X

ST−1=1

P(S_T, S_T−1, St−1, S_t|ye_T)

and

P(S_t|ye_T) =

m

X

St−1=1

P(St−1, S_t|ye_T). (15)

From now on, letP(St−1 =i, St=j|yeT) =πij(t|T) andP(St=j|yeT) = πj(t|T).

6 Maximum Likelihood Estimation

The log-likelihood function for the Markov switching state space model is

` = ln[f(ye_T)] =

T

X

t=1

lnf(y_t|eyt−1)

=

T

X

t=1

ln



 X

St

X

St−1

f(y_t|S_t, St−1,yet−1)P(S_t, St−1|yet−1)



.

Non-linear optimization methods may be used to find the maximum likelihood estimates. Alternatively, the EM algorithm requires the complete likelihood function, that is the likelihood if we could observe the vector (x0, . . . , xT, y1, . . . , yT, S0, . . . , ST) (Shumway and Stoffer, 2000). For our proposed model, the complete log likelihood is

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` = −1

2ln|Σ| −1

2(x0−ν)⁰Σ⁻¹(x0−ν) +

m

X

j=1

I(S0 =j) ln(πj(0))

+

T

X

t=1 m

X

i,j

I(St=j, St−1 =i)ln(πij(t))

−1 2

T

X

t=1 m

X

j=1

h

ln|W_j|+ (xt−µj−Gjxt−1)⁰W_j⁻¹(xt−µj −Gjxt−1) i

11e(St=j)

−1 2

T

X

t=1 m

X

j=1

h

ln|V_j|+ (y_t−F_jx_t−D_jz_t)⁰V_j⁻¹(y_t−F_jx_t−D_jz_t)i

11_o(S_t=j).

The functions 11_e(S_t=j) and 11_o(S_t=j) indicate respectively the pres- ence of regime switching in the state and observation equations.

Theorem 1. The expected log-likelihood conditional to ye_T considering regime switching only in the observation equation, only in the state equation or in both are respectively given by:

Ho = IE(ln(L)|eyT)

= −1

2ln|Σ| −1

2tr{Σ⁻¹[P_0|T + (x_0|T −ν)(x_0|T −ν)⁰]}+

m

X

j=1

P(S0 =j|yeT) ln(πj(0))

+

T

X

t=1 m

X

i,j

P(St−1=i, St=j|yeT)ln(πij(t))

−1

2tr[W⁻¹(C−BG⁰_t−G_tB⁰+G_tAG⁰_t)]−T 2 ln|W|

−1 2

T

X

t=1 m

X

j=1

πj(t|T) n

ln|V_j|+tr h

V_j⁻¹[(yt−Fjx^j_t|T)(yt−Fjx^j_t|T)⁰+FjP_t|TF_j⁰] io

,

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He = IE(ln(L)|yeT)

= −1

2ln|Σ| − 1

2tr{Σ⁻¹[P_0|T + (x_0|T −ν)(x_0|T −ν)⁰]}+

m

X

j=1

P(S₀=j|ye_T) ln(π_j(0))

+

T

X

t=1 m

X

i,j

P(St−1 =i, St=j|eyT)ln(πij(t))

−1 2

T

X

t=1 m

X

j=1

πj(t|T) n

ln|W_j|+tr[W_j⁻¹(Cj−BjG⁰_j−GjB_j⁰ +GjAG⁰_j)]

o

−1 2tr{

T

X

t=1

V⁻¹[(yt−F xt|T)(yt−F xt|T)⁰+F Pt|TF⁰]} −T 2ln|V|, H = IE(ln(L)|ye_T)

= −1

2ln|Σ| −1

2tr{Σ⁻¹[P0|T + (x0|T −ν)(x0|T −ν)⁰]}+

m

X

j=1

P(S0 =j|eyT) ln(πj(0))

+

T

X

t=1 m

X

i,j

P(St−1 =i, S_t=j|ye_T)ln(π_ij(t))

−1 2

T

X

t=1 m

X

j=1

π_j(t|T)n

ln|W_j|+tr[W_j⁻¹(C_j−B_jG⁰_j−G_jB_j⁰ +G_jAG⁰_j)]o

−1 2

T

X

t=1 m

X

j=1

π_j(t|T)n

ln|V_j|+trh

V_j⁻¹[(y_t−F_jx^j_t|T)(y_t−F_jx^j_t|T)⁰+F_jP_t|TF_j⁰]io ,

where

A=

T

X

t=1

P_t−1|T +x_t−1|Tx⁰_t−1|T,

B =

T

X

t=1

Pt,t−1|T + (xt|T −µ)x⁰_t−1|T,

C=

T

X

t=1

P_t|T + (x_t|T −µ)(x_t|T −µ)⁰,

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Bj =

T

X

t=1

P_t,t−1|T^j + (x^j_t|T −µj)x⁰_t−1|T,

C_j =

T

X

t=1

P_t|T^j + (x^j_t|T −µ_j)(x^j_t|T −µ_j)⁰.

Proof: Some results about quadratic forms (Koch, 1999) are used to find the expected value, IE(ln(L)|ey_T). Let ey and xe be random vectors, n×1, with expected values µ_y and µ_x, respectively, and variance matrix Cov(y,e x) = Σe yx, then

IE(xe⁰Ay) =e tr(AΣ_yx) +µ⁰_xAµ_y.

The estimates of all parameters F_i, G_i, µ_i, V_i, W_i, ν, Σ, π_j(0) and β_l^ij, l = 0, . . . , L, i = 1, . . . , m and j = 1, . . . , m−1, are the values that maximize H. The estimates for β^ij_l must be obtained using a nonlinear optimization procedure like Newton-Raphson or BFGS methods (details in Fletcher, 1987).

7 Bootstrap

The bootstrap algorithm for state space models is presented in Stoffer and Wall (1991). As the regimes St, t= 1, . . . , T are unknown, we propose to simulate them, using the smoothed probabilitiesP(S_t=j|ye_T) =π_j(t|T), and implement the bootstrap, proposed by Stoffer and Wall, considering the switching in the parameters. From the Kalman filter equations, the innovations, their variance and the Kalman gain are given by

t = yt−F_S_tx^S_t|t−1^t −D_S_tzt, Σ_t = F_S_tP_t|t−1^S^t F_S⁰_t +V_S_t, K_t = (G_S_t+1P_t|t−1^S^t F_t⁰)Σ⁻¹_t .

In order to obtain a set ofBbootstrap estimates, the bootstrap algorithm consists in:

1. SimulateS_t, t= 1, . . . T, usingπ_j(t|T).

2. Obtain standardized innovationse_t( ˆΘ) = Σ^−1/2_t ( ˆΘ)_t( ˆΘ).

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3. From the standardized innovations {e₁( ˆΘ), . . . , eT( ˆΘ)}, sample with replacement{e^∗₁( ˆΘ), . . . , e^∗_T( ˆΘ)}.

4. Construct the bootstrap datay_t^∗, t= 1, . . . , T, using y_t^∗ = F_S_tx^∗_t|t−1+D_S_tz_t+ Σ^1/2_t e^∗_t, x^∗_t+1|t = µSt+1+GSt+1x^∗_t|t−1+KtΣ^1/2_t e^∗_t.

The initial values, x_1|0 and P_1|0, remain fixed at the values obtained in the Kalman filter and the parameter values are replaced by ˆΘ.

5. For{y₁^∗( ˆΘ), . . . , y^∗_T( ˆΘ)}, implement the conditional Kalman filter and the probability filter and obtain the maximum likelihood estimates.

6. After repeatingB times the steps 2 to 5, obtain the bootstrapped set of parameter estimates {Θˆ^∗_b, b= 1, . . . , B}.

As noted in Stoffer and Wall (1991), the initial variance values Σ(t) may assume higher values, so the initial innovations must be discarded in the sample procedure. A complete residual diagnostic analysis must be done to guarantee permutable residuals.

8 Simulation

In this section we present two simulation studies to illustrate the proposed methodology. Maximum likelihood estimates (MLE), mean and standard deviation for 1000 bootstrap estimates were obtained for two simulated series. As in Stoffer and Wall (1991), 1000 data series were simulated using the true parameter values and for each one we obtained the maximum likelihood estimates. We present the mean and standard deviation for these 1000 estimates to compare with the bootstrap statistics. Transition probabilities are estimated from MLE, means of simulation and bootstrap estimates for β_i,l. All program routines were developed using Ox language (Doornik (1996)).

8.1 Simulation 1 - Regime switching in observation equation The first simulated model is

yt=FStxt+vt,

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where

1. t= 1, . . . ,512, G=1.0, V= 0.1, W=0.2,P(S0 = 0) =P(S0 = 1) = 0.5;

2. Consider two regimesS_t= 0 or 1 respectively associated toF₀=M₀ = 0 andF1 =M1 = 1. The value M1 is considered as a known value to guarantee the identificability of this model;

3. Take L = 3 Haar mother wavelets with coefficients βil, i = 0,1, l = 1,2,3: β₀₀ = 0.5, β₀₁ = 1.0, β₀₂ = 1.0, β₀₃ = 0.5, β₁₀ = −1.0, β₁₁= 0.0,β₁₂=−1.0 andβ₁₃=−1.5.

The MLE, mean and standard deviation for bootstrap and simulation estimates are presented in Table 1. The MLE and the means of bootstrap and simulation estimates are very close to the true values forM0, G, V and W. For these parameters, the standard deviations of bootstrap and simulation estimates are very similar, indicating that the variability of MLE can be evaluated using bootstrap. For transition probability parameters, bootstrap standard deviations underestimate the variability, as we can see compar- ing the bootstrap and simulation results. Moreover, MLEs and bootstrap sample means underestimateβ00 and β02, but simulation sample means indicate that MLE is an unbiased estimator. In Figure 1, we can observe the similarities of transition probability estimates. Standardized residuals in Figure 2 seem to be uncorrelated and normally distributed (p=0.892 for Kolmogorov-Smirnov test).

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Table 1: Parameter value, maximum likelihood estimate (MLE), mean and standard deviation for the 1000 estimates obtained in the simulation and bootstrap - Simulation 1.

True Simulation Bootstrap

Parameter value MLE Mean SD Mean SD

M0 0.0 -0.001 0.000 0.005 0.001 0.005 G 1.0 1.000 0.997 0.006 0.999 0.005 V 0.1 0.093 0.106 0.012 0.100 0.012 W 0.2 0.166 0.189 0.030 0.166 0.028 β₀₀ 0.5 0.078 0.455 0.463 -0.125 0.316 β₀₁ 1.0 0.889 1.049 0.484 0.780 0.337 β02 1.0 0.450 1.008 0.522 0.112 0.229 β₀₃ 0.5 0.266 0.416 0.778 0.171 0.601 β₁₀ -1.0 -0.760 -0.968 0.343 -0.604 0.098 β11 0.0 -0.097 0.029 0.312 0.034 0.095 β₁₂ -1.0 -0.970 -0.959 0.573 -0.679 0.177 β₁₃ -1.5 -1.622 -1.513 0.346 -1.601 0.095

8.2 Simulation 2 - Regime switching in state equation The second simulation was based on the model

y_t = x_t+v_t,

xt = µSt+Gxt−1+wt, where

1. t= 1, . . . ,512, G=1.0, V= 0.1, W=0.1,P(S₀ = 0) =P(S₀ = 1) = 0.5;

2. Consider two regimesS_t= 0 or 1 respectively associated toµ₀=M₀ =

−2 and µ₁=M₁ = 2;

3. Take L = 3 Haar mother wavelets with coefficients β_il, i = 0,1, l = 1,2,3: β00 = 1.5, β01 = 0.5, β02 = −0.5, β03 = 0.0, β10 = −1.0, β11=−0.4,β12= 0.5 andβ13= 0.2.

In Table 2, the MLE and the mean of simulation and bootstrap estimates are close to the true values and the standard deviation of bootstrap and simulation estimates are very close for M , M , G, V and W. For the

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Figure 1: y, x, x_t|T, P_t|T, true and estimated transition probabilities, and

“smoothed” probabilitiesπ_0|T,π_1|T - Simulation 1.

t

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0 100 200 300 400 500

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0 100 200 300 400 500

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true value MLE simulation bootstrap

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0 100 200 300 400 500

0.00.20.40.60.81.0

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Figure 2: et( ˆΘ) and Σ^−1/2_t ( ˆΘ) time series and histogram, Q-Q plot and autocorrelation function of e_t( ˆΘ) - Simulation 1.

t et(Θ^)

0 100 200 300 400 500

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0 100 200 300 400 500

0.40.60.81.01.2

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parametersβij, the MLE and the mean of bootstrap estimates are far from the true values, but we also observe that the mean of simulation estimates are close to the true values. Nevertheless, maximum likelihood, simulation and bootstrap transition probability estimates presented in Figure 3 are close to the true ones. The variability of bootstrap estimates is smaller than that obtained for the simulation estimates.

Standardized residuals in Figure 4 seem to be uncorrelated and normally distributed (p=0.701 for Kolmogorov-Smirnov test).

Table 2: Parameter value, maximum likelihood estimate (MLE), mean and standard deviation for the 1000 estimates obtained in the simulation and bootstrap - Simulation 2.

True Simulation Bootstrap

Parameter value MLE Mean SD Mean SD

M0 -2.0 -1.9955 -2.0022 0.0301 -1.9933 0.0390 M1 2.0 2.0162 1.9991 0.0358 2.0184 0.0289 G 1.0 1.0001 1.0000 0.0003 1.0000 0.0001 V 0.1 0.1025 0.1004 0.0131 0.0989 0.0119 W 0.1 0.0913 0.0983 0.0150 0.0865 0.0135 β₀₀ 1.5 1.0141 1.5001 0.1675 1.0053 0.0042 β01 0.5 0.5748 0.5117 0.1671 0.5598 0.0050 β02 -0.5 -0.3437 -0.5065 0.2725 -0.3147 0.0066 β₀₃ 0.0 -0.0892 0.0087 0.1911 -0.0565 0.0021 β10 -1.0 -1.8441 -0.9656 0.1709 -1.8492 0.0040 β11 -0.4 -0.7486 -0.4020 0.1733 -0.7496 0.0048 β₁₂ 0.5 0.6100 0.4855 0.2922 0.6129 0.0066 β₁₃ 0.2 -0.1710 0.2061 0.2079 -0.1541 0.0018

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Figure 3: y, x, xt|T, Pt|T, true and estimated transition probabilities, and

“smoothed” probabilitiesπ_0|T,π_1|T - Simulation 2.

0 50 100 150 200 250 300 350 400

0 100 200 300 400 500

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0100200300

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0100200300

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0 100 200 300 400 500

0.0400.0500.060

t π00

0 100 200 300 400 500

0.00.20.40.60.81.0

t π10

0 100 200 300 400 500

0.00.20.40.60.81.0

t π0|T

0 100 200 300 400 500

0.00.20.40.60.81.0

t π1|T

0 100 200 300 400 500

0.00.20.40.60.81.0