1. INTRODUCTION
The empirical likelihood method is one of the most useful statistical tools that allows us to refer to under some regularity conditions. Initially, this statistical method was introduced by Thomas and Grunkemeier [1] and subsequently further developed by Owen [2,3].
Nowadays, many statisticians use this statistical method in different fields of applications. Owen [2,3] have used the empirical likelihood ratio statistic to test the assumptions assumed by nonparametric statistics. They have shown that this statistic has an asymptotic chi-square distribution. Moreover, they have used this statistic to construct a confidence region and to perform a hypothesis testing of parameters. Asymptotic properties and some necessary corrections of this statistic have been studied by DiCiccio and Romano [4] and Hall and Scala [5]. Furthermore, Qin and Lawless [6] has demonstrated that in the nonparametric scheme, the empirical likelihood, along with estimating equations, fits the data well. For more details, see Newey and Smith [7], Chen and Cui, [8,9]), and Qin and Lowless [6].
In many statistical methods, estimating equations are commonly used to estimate the parameters of assumed model. Among them, the most well-known ones include the maximum likelihood (ML) method, the least square method, and the moment method. In this regard, the empirical likelihood approach, by applying unbiasedness constraint on estimating functions, leads to the maximum empirical likelihood (MEL) estimation, which is obtained by maximizing the empirical likelihood function, under the unbiasedness imposed on estimating equations involving the unknown probability p1,p2,…,pn. The works from Chen et al. [9], Hjort et al. [10], Tang and Leng [11], Leng and Tang [12], and Chang et al. [13] have shown that empirical likelihood estimators perform well when the dimension p of parameters and the number r of estimating equations grow slower than the sample size n, i.e. p=o(n) and r=o(n).
Tang and Leng [11], Leng and Tang [12], and Chang and Chen [13] have investigated the properties of empirical maximal likelihood estimators for high-dimensional data with sparse parameter vector. Practically, this is done by incorporating some suitable penalty functions. However, there are some challenges in employing empirical likelihood for large-scale data. Tsao [14] has shown that for a large enough n and fixed p, the probability of the corresponding confidence region containing true parameter values can be substantially smaller than its nominal value, which leads to an under-coverage problem.
In the meantime, the use of empirical likelihood method for time series data analysis has also been a topic of interest for researchers. For more details, interested readers are referred to Mykland [15], Kitamura [16], Chan and Ling [17], and Chan and Liu [18]. Monti [19] has developed the empirical likelihood method for the analysis of stationary time series. The author has used the M-estimator method, introduced by Whittle [21], for periodogram ordinates in time series models. Later, Yau [21] has extended the results in Monti [19] to long memory time series. Nordman and Lahiri [22] has argued that Monti's results were based on the assumption of normal distribution for the error term, and thus has extended the classical empirical likelihood to the one based on spectral distribution via Fourier transforms. Their results apply to both short and long memory dependencies; see Nordman and Lahiri [24] for details.
To the best of our knowledge, all the studies on empirical likelihood for time series rely on the unbiasedness property of estimating equations. But it seems that the unbiasedness condition alone, when data contains outliers, will produce misleading estimates for the parameters of interest (Bayati et al. [24]). In other words, in the usual empirical likelihood approach, features such as the dispersion of estimator functional, which depends on the dispersion of data itself, are not included in the statistical inference process. Therefore, it is necessary to use a pseudo-empirical likelihood that is a squaring function of the estimating equations. In this paper, for the first time we introduce the restricted empirical likelihood (REL) for the AR(r) time series models and present its Bayesian analysis. To our best knowledge, Bayati et al. [24] is the only one that has also used the idea of REL. Nevertheless, Bayati et al. [24] and this paper are very different in the sense that Bayati et al. [24] has studied the REL for independent observations while this paper investigates the REL for the more sophisticated case of dependent observations specifically from autoregressive models. In this paper, we not only prove the asymptotic normality of REL estimator for general models, but also give the detailed Bayesian analysis of AR(r) time series models using the REL and as well develop efficient algorithm for computing posterior and estimating hyper-parameters.
This article is organized as follows. After a glance at the empirical likelihood approach, we will illustrate the REL methodology in Section 2. The Bayesian analysis using the REL is discussed in Section 3. Bayesian analysis of AR(r) time series models, using the REL, are given in Section 4. A simulation study and an application to a real-world dataset are embedded in Section 5.
2. PRELIMINARIES
In this section, we first review the concept of empirical likelihood and then introduce the REL which is the main method proposed in this paper. The notations and definitions are provided along with other content.
2.1. Empirical Likelihood
Suppose x=(x1,x2,…,xn)′∈Rn are n independent observations obtained from the unknown distribution function f and they relate to the vector of parameters ϕ=(ϕ1,...,ϕr)′. Moreover, we assume that gj(x;ϕ), j=1,…,r, are some estimating functions to estimate ϕ and they satisfy the unbiasedness conditions
EP{gj(x;ϕ)}=∑i=1npigj(xi;ϕ)=0,j=1,…,r,(1)
where
P={p1,…,pn}. Moreover, let
g(x;ϕ)=g1(x;ϕ),...,gr(x;ϕ)′.
In the usual empirical likelihood approach, the estimates of probabilities pi's are obtained by maximizing the empirical likelihood function
L(ϕ)∝∏i=1npi(2)
under conditions
(1) and
∑i=1npi=1. Equivalently,
L(ϕ)=supP∏i=1npi:pi>0,∑i=1npi=1,∑i=1npig(xi;ϕ)=0.(3)
So, it is straightforward to see that
pi∝11+λ(ϕ)′g(xi;ϕ),
where
λ(ϕ)=λ1(ϕ),λ2(ϕ),…,λr(ϕ)′ is the vector of Lagrange multipliers (see Owen [
2] and Qin and Lawless [
6]). So, the empirical likelihood function is given as
L(ϕ)∝∏i=1n11+λ(ϕ)′g(xi;ϕ).(4)
By using (4), the MEL estimator of ϕ is given by
ϕ̂EL=argminϕmaxλ∑i=1nlog1+λ(ϕ)′g(xi;ϕ).
There is a modified two-layer coordinate descent algorithm to compute ϕ̂EL (Tang and Wu [25]).
2.2. Restricted Empirical Likelihood
In previous subsection, the empirical likelihood function is given by (4). Now, by inequality a′b≤a′a+b′b2=∥a∥2+∥b∥22 for any arbitrary vectors a and b, we have
∏i=1n11+∥λ(ϕ)∥2+∥g(xi;ϕ)∥22≤∏i=1n11+λ(ϕ)′g(xi;ϕ).(5)
We hope that the value of ϕ that maximizes the left side of (5) approaches to the maximizing value of ϕ for the right side. The left side of (5) can be rewritten as
∏i=1n11+∥λ(ϕ)∥2+∥g(xi;ϕ)∥22=∏i=1n2w(ϕ)11+w(ϕ)∥g(xi;ϕ)∥2=[2w(ϕ)]n∏i=1n11+w(ϕ)∥g(xi;ϕ)∥2,(6)
where
w(ϕ)=12+∥λ(ϕ)∥2. Therefore, the pseudo-empirical maximum likelihood estimator (MLE) of
ϕ is given by
ϕ̂=argminϕ∈Φmaxw∈ℝ+n∑i=1nlog1+w(ϕ)∥g(xi;ϕ)∥2−nlogw(ϕ).
Indeed, the above equation is equivalent to the case that the function ∑i=1nlog1+w(ϕ)∥g(xi;ϕ)∥2 is optimized under the penalty function −logw(ϕ). It gives us the idea that we can extend the expression (6) to a general form given by
e−nH(w,ϕ)∏i=1n11+w′g∗(xi;ϕ),(7)
where
w=(w1,…,wr)′ and
g∗(x;ϕ)=g12(x;ϕ),…,gr2(x;ϕ)′. Here, we assume
H(w,ϕ) plays the role of a penalty function and define our REL, an alternative to the usual empirical likelihood, as
LR(w,ϕ)=∏i=1n11+w′g∗(xi;ϕ).(8)
Practically, the REL LR(w,ϕ) has two appealing properties in contrast to L(ϕ) given in (4). First, we can assume in LR(w,ϕ) that the elements of vector w are independent of ϕ, while in L(ϕ) dependency of λ on θ is a basic assumption. This means that LR(w,ϕ) can be considered as a semi-parametric version of the usual EL function. Second, by assuming sparsity of vectors ϕ and w (to reduce the number of parameters and the number of the estimating equations respectively) and −logREL being convex, we can use the REL method for high-dimensional data. For more details on the appealing properties of the REL LR(w,ϕ), readers are referred to Bayati et al. [24]. With these appealing properties, we propose the REL estimator of ϕ as given by
(ϕ̂REL,ŵREL)=argminϕ∈ℝr,w∈ℝ+r∑i=1nlog1+wTg∗(xi;ϕ)+nH(w,ϕ).
Under some regularity conditions on g∗ and H, the REL estimator ϕ̂REL has an asymptotically multivariate normal distribution. This result is presented in following theorem with details. Its proof is straightforward and thus omitted to save space.
Theorem 2.1.
Under some well-known regularity conditions on g∗ and H, i.e., ∂2g∗(x;ϕ)∂ϕi∂ϕj<∞ and ∂2H(w,ϕ)∂ϕi∂ϕj<∞, when x1,x2,…,xn are i.i.d., it holds that
n(ϕ̂−ϕ)→Nr0,I−1(ϕ),
where
I(ϕ)=E∂2log1+wTg∗(x;ϕ)+H(w,ϕ)∂ϕi∂ϕji,j=1,…,r
and
w is replaced with its estimate
ŵREL.
Proof.
Since ϕ̂ is the maximizer of ∑i=1nlog1+wTg∗(xi;ϕ)+nH(w,ϕ), so we define
Tn(ϕ)=1n∑i=1nlog1+wTg∗(xi;ϕ)+nH(w,ϕ)=1n∑i=1nlog1+wTg∗(xi;ϕ)+H(w,ϕ).
It is obvious that ϕ̂ is the MLE when Tn(ϕ) is log-likelihood function normalized by 1n. Therefore, the proof is the same as asymptotic normality of MLEs. Moreover, under the given regularity conditions, MLE ϕ̂ is constant, i.e., ϕ̂→ϕ as n→∞.
In the next section, we focus on Bayesian analysis of ϕ, considering the REL as the usual likelihood function.
3. BAYESIAN ANALYSIS WITH REL
Our main purpose in this paper is the Bayesian analysis of ϕ, viewing the REL as playing the role of a regular likelihood function. Based on this idea, we hope that the Bayesian estimation of ϕ can provide an acceptable approximation of the true values. For simplicity, we assume that the penalty function H(w,ϕ) is an additive function, i.e., H(w,ϕ)=H1(w)+H2(ϕ), and moreover assume e−nH(w,ϕ) is proportional to a multivariate density. Consequently, we assume that
π1(w)∝e−nH1(w)andπ2(ϕ)∝e−nH2(ϕ),(9)
are the corresponding prior densities. We have the following two points to make on the densities.
We assume in (9) that the two vectors w and ϕ are independent. This assumption is often considered for simplicity and computational purposes.
Unlike the usual way, the priors π1 and π2 are dependent of n. This is often assumed in Bayesian statistical analysis; for more details see Bhattacharya et al. [26] and Ghoreishi [27]. However, the functions H1 and H2 can also be chosen in such a way that the effect of sample size is ignorable.
Based on the above assumptions, using (8) and (9) along with observations D={x1,⋯,xn}, the joint posterior density of (w,ϕ) is
πw,ϕD∝LR(w,ϕ)π1(w)π2(ϕ).(10)
Given the estimating functions g∗ and the priors π1 and π2, one can estimate the desired parameters using the Bayesian approach. In practice, the marginal posterior distribution of parameters often do not have a closed form, therefore any inference is derived using samples via MCMC methods. However, in many statistical models in which the estimating functions are usually linear of the parameters, the conditional posterior distributions have the closed form and they are easy to sample using Gibbs sampling method. In the next section, we discuss this issue with more details for autoregressive models of order r, i.e., AR(r).
4. REL FOR AR(r) MODELS
4.1. Basic Concepts
Autoregressive models show a random process where the output variable is a linear function of the same variable. Specifically, the AR(r) model is defined as
Xt=ϕ1Xt−1+ϕ2Xt−2+⋯+ϕrXt−r+ϵt,t=r+1,…,n,(11)
where
ϕ1,ϕ2,…,ϕr are the model parameters, and
ϵt's are white noises with mean 0 and constant variance
σ2. In our empirical approach, we assume the distribution of
ϵt's is totally unknown. In practice, there are several methods to estimate the model coefficients. A popular approach is Yule-Walker's system of equations, i.e.,
∑t=r+1nXt−jXt−ϕ1Xt−1−ϕ2Xt−2−⋯−ϕrXt−r=0,j=1,2,…,r,
where
gjt(X,ϕ)=Xt−jXt−ϕ1Xt−1−ϕ2Xt−2−⋯−ϕrXt−r,j=1,2,…,r(12)
are the estimating equations. Later in our Bayesian approach, we will use the weighed Yule-Walker's system of equations as our estimating equations.
As we see, the estimating equations (12) are linear functions of parameters ϕ1,ϕ2,…,ϕr. Therefore, we hope that the posterior conditional distributions have a closed form when we use the REL function instead of a regular likelihood function. Moreover, in the autoregressive model (11), the conditional distribution of (xr+1,…,xn) given (x1,…,xr) is
f(xr+1,…,xn|ϕ,x1,…,xr)=∏t=r+1nf(xt|ϕ,xt−1,…,xt−r)=∏t=r+1npt.
where
pt:=f(xt|ϕ,xt−1,xt−2,…,xt−r). On the other hand, the corresponding likelihood function is
L(ϕ)=∏t=r+1npt,(13)
Since the probabilities pt's are practically unknown, we try to estimate them here using our REL approach. We assume that the observation xt is weighted with its corresponding conditional density function value pt as the weight. This means that up to a normalizing constant, the estimating equations (12) satisfy
EP{gjt(x;ϕ)}=∑t=r+1nptgj(x,ϕ)=∑t=r+1nptxt−jxt−ϕ1xt−1−ϕ2xt−2−⋯−ϕrxt−r=0,(14)
where
P={pr+1,pr+2,…,pn}. Finally, combining
(8),
(13) and
(14) gives the corresponding REL as
LR(ϕ,w)=∏t=r+1n11+∑j=1rwjxt−j2xt−ϕ1xt−1−ϕ2xt−2−⋯−ϕrxt−r2.(15)
4.2. Posterior Conditional Distributions
Without loss of generality, for Bayesian analysis using the REL (15), we assume wj,ϕj and sj are i.i.d. with the following common priors
wj~π1=inv.gamma(n,β0),ϕj~π2=N(0,σ02),sj~f(uj)=Exp(1),(16)
where the additional variables
sr+1,sr+2,…,sn (the argument
ur+1,…,un in their respective distribution functions) appeared in equality
LR(ϕ,w)=∏t=r+1n11+∑j=1rwjxt−j2xt−ϕ1xt−1−ϕ2xt−2−⋯−ϕrxt−r2=∫0∞⋯∫0∞e−∑t=r+1nut1+∑j=1rwjxt−j2xt−∑ν=1rϕνxt−ν2dur+1⋯dun.
Then the conditional posterior distributions have closed forms given by
ut~Exp2+∑j=1rwjxt−j2xt−ϕ1xt−1−⋯−ϕrxt−r2,t=r+1,…,n,wj~gIG(2∑t=r+1nutxt−j2xt−ϕ1xt−1−⋯−ϕrxt−r2+1,2β0,−n),j=1,…,r,ϕl~Nμl,σl2,l=1,…,r,(17)
where gIG stands for the generalized inverse Gaussian distribution and
μl=∑t=r+1n∑j=1rutwjxt−j2xtxt−l−∑t=r+1n∑j=1rutwjxt−j2∑k≠lrxt−lxt−kϕk∑t=r+1n∑j=1rutwjxt−j2xt−l2+12β0,σl=2∑t=r+1n∑j=1rutwjxt−j2xt−l2+1β0−0.5.
4.3. EM Algorithm for Estimating Hyper-parameters
One of the major challenges in sampling from the conditional posterior densities (17) is that the hyper-parameters β0 and σ02 are unknown in practice. However, there are several methods to estimate these quantities. In this subsection, we introduce an EM algorithm for this purpose.
From (15) and (16), the marginal density of the observations, x, is obtained as
f(x|β0,σ02,x1,…,xr)=∫⋯∫e−∑t=r+1nst1+∑j=1rwjxt−j2xt−∑ν=1rϕνxt−ν2×1σ0r2e−12σ02∑j=1rϕj2×∏j=1rβ0nwjn+1e−β0wjdsdwdϕ.(18)
Now consider the logarithm of the integrand in (18) as a function of β0 and σ02 given by
I(β0,σ02)=−∑t=r+1nst1+∑j=1rwjxt−j2xt−∑ν=1rϕνxt−ν2−r2logσ0−12σ02∑j=1rϕj2+nrlogβ0−(n+1)∑j=1rlogwj−∑j=1rβ0wj.(19)
Our proposed EM algorithm for estimating β0 and σ0 has the following steps:
Select the initial estimates β0(0) (>0) and σ02(0) (>0) and start with k=1.
Given β0(k) (>0) and σ02(k) (>0), produce a sample of size N from the conditional distributions (17).
E-step: Calculate the averaged I(β0,σ02) given by (19) using the generated samples in Step 2. Here, it should be noted that it is enough to calculate the mean of that expression in I(β0,σ02) which only depend on β0 and σ02.
M-step: Update β0 and σ02 by maximizing the following two quantities with respect to β0 and σ02 respectively:
Q3(σ02)=r4logσ02+12σ02(k)∑j=1rEσ02,β0(k)ϕj2,Q4(β0)=nrlogβ0−(n+1)∑j=1rEσ02(k),β0(k)logwj−β0∑j=1rEσ02(k),β0(k)1wj.
The updated estimates at time (k+1) are obtained as
σ02(k+1)=1r∑j=1rEσ02(k),β0(k)ϕj2,β0(k+1)=nc∑j=1rEσ02(k),β0(k)1wj.
Repeat Steps 2-4 until the algorithm converges.
4.4. Model Evaluation Criteria
In practice, there are several criteria in literature for evaluating the efficiency of time series models. One of those is the mean squared prediction error defined as MSPE=1n∑t=1n(xt−x̂t)2. In this article, we will use a similar criterion to BIC. This criterion, for AR(r) model, is constructed based on the REL function, abbreviated as EBIC, defined as
EBIC=rlogn+∑t=1nlog1+∑j=1rwjxt−j2xt−ϕ̂1xt−1−ϕ̂2xt−2−⋯−ϕ̂rxt−r2.
In practice, a small EBIC value confirms the adequacy of the model, whereas a large value indicates the inadequacy of the model.
5. SIMULATION STUDY AND APPLICATION
In this section, we first carry out a simulation study to confirm our theoretical results and then apply the proposed methods to a real dataset.
5.1. Simulation Study
Our simulation study focuses on the following autoregressive model of order 2:
Xt=ϕ1Xt−1+ϕ2Xt−2+ϵt,t=3,4,…,n.
For this model, we assume several scenarios for ϕ=(ϕ1,ϕ2) and n. In order for the AR(2) model to look like a nonparametric model, we assume that ϵt's are generated from the following a little more complex distribution (a combination of normal and Cauchy distributions)
ϵi~0.5N(0,2)+0.5C(0,1).
Here, we run the corresponding Gibbs sampling scheme (12) for two sample sizes n=50 and 500. With the generated data under each scenario, we try to fit the following three models:
Model I:
Xt=ϕ1Xt−1+ϵt
with estimating equation g(X,ϕ1)=Xt−1(Xt−ϕ1Xt−1).Model II:
Xt=ϕ2Xt−2+ϵt
with estimating equation g(X,ϕ2)=Xt−2(Xt−ϕ1Xt−2).Model III:
Xt=ϕ1Xt−1+ϕ2Xt−2+ϵt
with estimating equations g1(X,ϕ1,ϕ2)=Xt−1(Xt−ϕ1Xt−1−ϕ2Xt−2) and g2(X,ϕ1,ϕ2)=Xt−2(Xt−ϕ1Xt−1−ϕ2Xt−2).
Moreover, for simplicity, we assume w:=w1=w2. We ran the sampling scheme (17) in order to generate 5000 samples from the marginal distributions of ϕ1 and ϕ2, and each time we fit Models I–III to the data. In addition to the estimates of (ϕ1,ϕ2) and their standard errors, the MSPE and EBIC criteria are also computed. The numerical results are presented in Table 1. As we can see from it, Model III fits the data much better than the other two competitor models.
|
|
(ϕ1,ϕ2) |
Method |
ϕ̂1 |
ϕ̂2 |
MSPE |
EBIC |
|
|
(−0.5, −0.8) |
REL |
−0.319(0.127) |
- |
50.941 |
10.693 |
|
|
|
|
|
|
|
|
|
n = 50 |
|
EL |
−0.221(0.191) |
- |
54.627 |
|
|
|
(0.5, −0.8) |
REL |
−0.355(0.108) |
- |
34.713 |
10.261 |
|
|
|
|
|
|
|
|
| Model I |
|
|
EL |
−0.312(0.123) |
- |
39.101 |
|
|
|
(−0.5, −0.8) |
REL |
−0.302(0.168) |
- |
58.121 |
14.180 |
|
|
|
|
|
|
|
|
|
n = 500 |
|
EL |
−0.223(0.172) |
- |
61.001 |
|
|
|
(0.5, −0.8) |
REL |
0.368(0.143) |
- |
33.975 |
13.990 |
|
|
|
|
|
|
|
|
|
|
|
EL |
0.311(0.144) |
- |
42.214 |
|
|
|
|
|
|
|
|
|
|
|
(−0.5, −0.8) |
REL |
- |
−0.146(0.221) |
55.914 |
8.573 |
|
|
|
|
|
|
|
|
|
n = 50 |
|
EL |
- |
−0.121(0.243) |
67.423 |
|
|
|
(0.5, −0.8) |
REL |
- |
−0.417(0.197) |
39.128 |
7.857 |
|
|
|
|
|
|
|
|
| Model II |
|
|
EL |
- |
0.314(0.202) |
44.121 |
|
|
|
(−0.5, −0.8) |
REL |
- |
−0.586(0.200) |
56.842 |
10.000 |
|
|
|
|
|
|
|
|
|
n = 500 |
|
EL |
- |
−0.372(0.213) |
71.001 |
|
|
|
(0.5, −0.8) |
REL |
- |
−0.222(0.209) |
28.613 |
8.156 |
|
|
|
|
|
|
|
|
|
|
|
EL |
- |
0.199(0.266) |
37.813 |
|
|
|
|
|
|
|
|
|
|
|
(−0.5, −0.8) |
REL |
−0.547(0.032) |
−0.767(0.047) |
20.542 |
4.705 |
|
|
|
|
|
|
|
|
|
n = 50 |
|
EL |
−0.570(0.035) |
−0.726(0.052) |
22.117 |
|
|
|
(0.5, −0.8) |
REL |
0.546(0.041) |
−0.642(0.032) |
21.222 |
4.440 |
|
|
|
|
|
|
|
|
| Model III |
|
|
EL |
0.561(0.043) |
−0.689(0.033) |
22.473 |
|
|
|
(−0.5, −0.8) |
REL |
−0.505(0.012) |
−0.805(0.014) |
17.124 |
6.805 |
|
|
|
|
|
|
|
|
|
n = 500 |
|
EL |
−0.515(0.013) |
−0.821(0.017) |
17.921 |
|
|
|
(0.5, −0.8) |
REL |
0.491(0.011) |
−0.798(0.011) |
16.978 |
6.508 |
|
|
|
|
|
|
|
|
|
|
|
EL |
0.484(0.012) |
−0.811(0.012) |
17.008 |
|
|
|
|
|
|
|
|
|
Table 1The REL and EL estimates of (ϕ1,ϕ2) under various models (and their standard errors inside the parentheses) and the associated MSPE and EBIC.
5.2. Application
In this subsection, we analyze a real dataset on the Total Index of Consumer Goods and Services in urban areas of Iran for the period 1990–2017, extracted by The Central Bank of the Islamic Republic of Iran. The data are shown in Table 2 which demonstrates the presence of time dependency. A primary analysis reveals that an AR(2) model fits the data well after removing the trend. For this purpose, we first estimate the trend using a cubic polynomial given by
xt=α+βt3+ϵt,ϵt=ϕ1ϵt−1+ϕ2ϵt−2+ut,(20)
where the error terms
ut's have an unknown distribution with mean 0 and variance
σ2. With the cubic polynomial, we obtain the estimates
α̂=−0.639 and
β̂=0.004, by fitting a linear regression model. Kolmogorov–Smirnove test statistic for
ϵt is equal to
0.233 with
Sig=0.000. This shows that the residuals do not have a normal distribution. Since the size of the data in not large enough, there is no strong evidence that
ϵ̂t=xt−α̂−β̂t3
follow any particular distribution. Thus, we will use our proposed REL method to fit a time series model to
ϵ̂t's. Using the posterior conditional distributions
(17), the Bayesian estimates of
ϕ1 and
ϕ2 in model
(20) are
ϕ̂1=1.335 and
ϕ̂2=−0.465 respectively. The results indicate that the two-level model
(20) fits the data well with the determination coefficient
R2=0.996,MSPE=3.96, and
EBIC=6.327.
| Year |
1990 |
1991 |
1992 |
1993 |
1994 |
1995 |
1996 |
1997 |
1998 |
1999 |
2000 |
2001 |
2002 |
2003 |
| xt |
1.00 |
1.20 |
1.50 |
1.80 |
2.50 |
3.70 |
4.50 |
5.30 |
6.30 |
7.50 |
8.50 |
9.40 |
10.90 |
12.60 |
| xt̂ |
- |
- |
1.14 |
1.51 |
1.87 |
2.79 |
4.22 |
4.92 |
5.85 |
7.09 |
8.55 |
9.70 |
10.86 |
12.92 |
| Year |
2004 |
2005 |
2006 |
2007 |
2008 |
2009 |
2010 |
2011 |
2012 |
2013 |
2014 |
2015 |
2016 |
2017 |
| xt |
14.50 |
16.10 |
18.00 |
21.30 |
26.70 |
29.50 |
33.20 |
40.30 |
52.60 |
70.90 |
81.90 |
91.70 |
100.00 |
109.6 |
| xt̂ |
15.04 |
17.39 |
19.31 |
21.85 |
26.19 |
32.76 |
34.96 |
39.65 |
48.56 |
62.91 |
82.94 |
90.53 |
100.02 |
108.18 |
Table 2Total Index of Consumer Goods and Services and their fitted values in urban areas of Iran from 1990 to 2017.
