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k ( j) + ∑ a i( j ) Yt −i + ε t r j −1 < Yt − d ≤ r j Yt = a 0 i =1 q h = α ( v ) + ∑ α ( v ) ε 2 c v −1 < Yt −b ≤ c v t t −i i 0 i =1
(2)
ℜp
Using (5) we can obtain an estimator of g ( Z t )σ . In this case, the bandwidth can ˆ CV (u t )) 2 , where u t = ( X t − m ˆ CV ( Z t )) 2 and be determined minimising ∑ (u t − m
where {εt} is i.i.d.; the threshold values, rj e cv, are such that r0<r1<...<rl, r0= -∞,
3
Non-parametric techniques
The most common way of estimating the functions f(⋅) and g(⋅) non-parametrically is to estimate first f(⋅), using an estimator of the conditional mean, and then g(⋅) through the difference between the conditional mean of X t2 nd the previous estimator. One of the main difficulties is that when we estimate the conditional mean, we obtain an estimate of the function m(⋅) defined in the model: (3) X t = m( X t −1 , ) + ε t and, in general, m(⋅) is different from the function f(⋅) defined in model (1). To overcame this problem we propose a new procedure for the estimation of f(⋅). Assuming that only a finite number of lagged Xt’s enters f(⋅) and g(⋅), let Z t , p = ( X t −1 ,..., X t − p ) . First, we use a non-parametric estimator of m(⋅) to obtain an estimator of g(⋅). From the model (1) we have: Xt f (Z t ) ε t = + (4) g ( Z t )σ g ( Z t )σ σ Let ν(⋅) be the non-parametric estimate of the conditional mean of Yt = X t / g ( Z t )σ ; then the estimate of f(⋅) can be obtained by multiplying ν(⋅) by an estimate of g ( Z t )σ . 3.1 Kernel Estimator We consider the Nadaraya-Watson kernel estimator, with one smoothing parameter: ˆ ( x) = m
2
Non-linear Models
Many new models and tools have been proposed in the literature to capture the non-linear features of a given time series. For modelling non-linearity in the conditional mean, Tong’s threshold models (TAR, Tong 1978) have been found useful and successfully applied. In many real situations can be of interest modelling also the changing conditional variance. In this case we can introduce an ARCH component, obtaining the so called TAR-ARCH model. This model assumes conditional variance dependent on a squared error term. In same case, however, conditional variance depends asymmetrically on previous returns. To introduce the asymmetry, both in the level and in the conditional variance, Li and Li (1996), propose to use the threshold principle, not only in the conditional mean, but also in the ARCH residuals, defining the Double Threshold Autoregressive Conditional Heteroscedastic models. A time series {Yt} follows DTARCH models if:
Parametric And Non-parametric Methods In Non-linear Time Series Analysis: A Critical Evaluation
Alessandra Amendola1, Francesco Giordano1 and Cira Perna1 Dipartimento di Scienze Economiche, Università di Salerno, Via Ponte Don Melillo, 84084, Fisciano (SA) Italy.
) + g ( X
t −1 ,
X t −2 ,
)ε
t
(1)
The generality of this representation makes it very hard to handle, especially when dealing with non-linearity in f(⋅) and/or in g(⋅). In this case, two different approaches can be used for estimation: non-linear parametric modelling (Tong, 1990) and non-parametric techniques (Hardle, 1989). The aim of this paper is to evaluate the performance of the two different approaches when future volatility is of interest in addition to the conditional mean. In particular, we propose a method to deal with the non-parametric estimation of the functions f(⋅) and g(⋅). The paper is organised as follows. In the next section, we briefly review the basic methodology of threshold models. In section 3, we discuss kernel and neural network estimators and present the proposal method. In section 4, we compare the alternative procedures using simulated time series.
1
1
Introduction
Consider a process {Xt} generated by the model : X t = f ( X t −1 , X t − 2 , where {εt} is a sequence of zero-mean i.i.d. random variables, f(⋅) and g(⋅) represent, respectively, the conditional mean and the conditional variance given the past observations, with g(⋅)>0.
t =1 n
ˆ CV ( Z t ) is the leave-one-out kernel estimator of m(.). The conditional mean f(.) m can be estimated using model (4).
3.2 Neural Network Estimator
rl= +∞, and c0<c1<...<cl1, c0= -∞ e cl2= +∞, with j=1,2,...,l1, v=1,2,....,l2,; d and b are the delay parameters for the conditional mean and the conditional variance.
(2)
ℜp
Using (5) we can obtain an estimator of g ( Z t )σ . In this case, the bandwidth can ˆ CV (u t )) 2 , where u t = ( X t − m ˆ CV ( Z t )) 2 and be determined minimising ∑ (u t − m
where {εt} is i.i.d.; the threshold values, rj e cv, are such that r0<r1<...<rl, r0= -∞,
3
Non-parametric techniques
The most common way of estimating the functions f(⋅) and g(⋅) non-parametrically is to estimate first f(⋅), using an estimator of the conditional mean, and then g(⋅) through the difference between the conditional mean of X t2 nd the previous estimator. One of the main difficulties is that when we estimate the conditional mean, we obtain an estimate of the function m(⋅) defined in the model: (3) X t = m( X t −1 , ) + ε t and, in general, m(⋅) is different from the function f(⋅) defined in model (1). To overcame this problem we propose a new procedure for the estimation of f(⋅). Assuming that only a finite number of lagged Xt’s enters f(⋅) and g(⋅), let Z t , p = ( X t −1 ,..., X t − p ) . First, we use a non-parametric estimator of m(⋅) to obtain an estimator of g(⋅). From the model (1) we have: Xt f (Z t ) ε t = + (4) g ( Z t )σ g ( Z t )σ σ Let ν(⋅) be the non-parametric estimate of the conditional mean of Yt = X t / g ( Z t )σ ; then the estimate of f(⋅) can be obtained by multiplying ν(⋅) by an estimate of g ( Z t )σ . 3.1 Kernel Estimator We consider the Nadaraya-Watson kernel estimator, with one smoothing parameter: ˆ ( x) = m
2
Non-linear Models
Many new models and tools have been proposed in the literature to capture the non-linear features of a given time series. For modelling non-linearity in the conditional mean, Tong’s threshold models (TAR, Tong 1978) have been found useful and successfully applied. In many real situations can be of interest modelling also the changing conditional variance. In this case we can introduce an ARCH component, obtaining the so called TAR-ARCH model. This model assumes conditional variance dependent on a squared error term. In same case, however, conditional variance depends asymmetrically on previous returns. To introduce the asymmetry, both in the level and in the conditional variance, Li and Li (1996), propose to use the threshold principle, not only in the conditional mean, but also in the ARCH residuals, defining the Double Threshold Autoregressive Conditional Heteroscedastic models. A time series {Yt} follows DTARCH models if:
Parametric And Non-parametric Methods In Non-linear Time Series Analysis: A Critical Evaluation
Alessandra Amendola1, Francesco Giordano1 and Cira Perna1 Dipartimento di Scienze Economiche, Università di Salerno, Via Ponte Don Melillo, 84084, Fisciano (SA) Italy.
) + g ( X
t −1 ,
X t −2 ,
)ε
t
(1)
The generality of this representation makes it very hard to handle, especially when dealing with non-linearity in f(⋅) and/or in g(⋅). In this case, two different approaches can be used for estimation: non-linear parametric modelling (Tong, 1990) and non-parametric techniques (Hardle, 1989). The aim of this paper is to evaluate the performance of the two different approaches when future volatility is of interest in addition to the conditional mean. In particular, we propose a method to deal with the non-parametric estimation of the functions f(⋅) and g(⋅). The paper is organised as follows. In the next section, we briefly review the basic methodology of threshold models. In section 3, we discuss kernel and neural network estimators and present the proposal method. In section 4, we compare the alternative procedures using simulated time series.
1
1
Introduction
Consider a process {Xt} generated by the model : X t = f ( X t −1 , X t − 2 , where {εt} is a sequence of zero-mean i.i.d. random variables, f(⋅) and g(⋅) represent, respectively, the conditional mean and the conditional variance given the past observations, with g(⋅)>0.
t =1 n
ˆ CV ( Z t ) is the leave-one-out kernel estimator of m(.). The conditional mean f(.) m can be estimated using model (4).
3.2 Neural Network Estimator
rl= +∞, and c0<c1<...<cl1, c0= -∞ e cl2= +∞, with j=1,2,...,l1, v=1,2,....,l2,; d and b are the delay parameters for the conditional mean and the conditional variance.
