# Locally Most Powerful Test for the Random Coefficient Autoregressive Model

Authors: Li Bi; Feilong Lu; Kai Yang; Dehui Wang
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In this article, we study the problem of testing the constancy of the coefficient in a class of stationary first-order random coefficient autoregressive (RCAR(1)) model. We construct a new test statistic based on the locally most powerful-type (LMP) test. Under the null hypothesis, we derive the limiting distribution of the proposed test statistic. In the simulation, we compare the power between LMP test and empirical likelihood (EL) test and find that the accuracy of using LMP is 6.7%, 28.8%, and 26.1% higher than that of EL test under normal, student’st, and symmetric contamination errors, respectively. A real life data is given to illustrate the practical effectiveness of our test.


---

## Body

## 1. Introduction

In time series analysis, autoregressive and linear processes are widely used due to their mathematical tractability. In fact, the autoregressive model has two prominently advantages: their estimation procedures are well established and the existence of stationary solutions are easily derived. Therefore, this model has been investigated in the field of signal detection and classification, psychometry, and biomedical engineering. See, for example, Hoque [1], Ogawa et al. [2], Subasi et al. [3], and Maleki et al. [4]. The first-order autoregressive model (AR(1)) is defined as(1)Xt=αXt-1+εt,t≥1,where {εt} is an independent and identically distributed (i.i.d.) random error sequence with probability density function fε, E(εt)=0, Var(εt)=σε2, and {εt} independent of X0 for all t. However, it has been found that a variety of data sets cannot be modelled precisely by assuming linearity. Financial data, for instance, present heteroscedasticity, and biological data suffer from random perturbations. To address this problem, Conlisk [5] considered a random coefficient autoregressive model defined as follows:(2)Xt=αtXt-1+εt,t≥1,where {αt} is a sequence of i.i.d random variables with E(αt)=α and Var(αt)=σα2. {εt} is an i.i.d. random error sequence with mean zero and variance σε2. The random variable X0 is assumed to be independent of {αt}, which is independent of {εt}. Note that when σα2=0, the RCAR(1) model reduces to the standard AR(1) model.The RCAR(1) models have been widely applied in mathematical literature due to its strong application value in practice. For instance, Nicholls and Quinn [6–8] derived the least square (LS) estimates of the model parameter and showed that the LS estimates are strongly consistent and obey the central limit theorem. Brandt [9] restated the necessary and sufficient conditions for the existence of a strict stationarity and ergodicity solution in the RCAR(1) model. Wang and Ghosh [10] considered the Basyesian estimation method to estimate RCAR model parameter and explored the frequentist properties of the Bayes estimator. Wang et al. [11] obtained the asymptotic properties of the maximum likelihood estimators of these parameters in RCAR(1) model under the unit root assumption.More recently, the problem of testing traditional AR(1) model against RCAR(1) model is thus of essential issue in this settings. This detection problem was firstly investigated by Nicholls and Quinn [8], where a Gaussian Lagrange multiplier test is derived for the problem. However, this likelihood ratio approach has several weaknesses, as the true value of the parameter under the null hypothesis lies on the boundary of the parameter space, the asymptotics will not be easily established. Ramanathan and Rajarshi [12] considered a non-Gaussian test method to handle the change problem for parameters in RCAR(1) case. Their signed rank test, however, require a symmetry assumption on the innovation density, which is highly unnatural in this context. Still for the RCAR(1) model, Lee [13] proposed the cumulative sum test for parameter change in RCAR(1) model. But, it detects a change only in a function of the parameter rather than the parameter themselves. Recently, Moreno and Romo [14] studied robust unit root test with autoregressive errors. The unit root test shows some size distortions and requires many estimates in the construction of test.To deal with these drawbacks, we construct a locally most powerful-type (LMP) test for testing the constancy of parameters in the RCAR(1) model. The LMP test has been investigated by several authors. Rohatgi et al. [15], Chikkagoudar and Biradar [16], and Manik et al. [17] considered the LMP test for parameter constancy. The test has merit in that its calculation process is less cumbersome and it produces stable sizes especially when parameter nearby the true value. In contrast, the traditional tests mentioned above show some size distortions and require many estimates in the construction of tests, see Ramanathan and Rajarshi [12]. In a general way, our test can conventionally discard correlation effects and enhance the performance of the test. In this paper, we illustrate how our proposed method can be implemented for finite samples under normal, student’st, and symmetric contamination errors (see Huber [18] and Hettmansperger [19]). Through numerical simulation studies, we can see that our test has a stronger power in terms of maintaining the empirical level and power than the empirical likelihood (EL) test suggested by Zhao et al. [20].The outline of this paper is organized as follows. In Section2, we introduce our test statistic and derive its limiting distribution under the null hypothesis. Numerical simulations to evaluate the empirical size and power of our test technique are discussed in Section 3. Section 4 presents a real life data example to illustrate the superior of our methods. We provide brief concluding remarks in Conclusions. Appendix provides the proofs of the main results.


## 2. Methodology and Main Results

In this section, we will construct a test statistic to test whetherαt is a constant. To achieve this task, we set up the null and alternative hypotheses(3)H0:σα2=0vsH1:σα2>0.In what follows, we give our main results. Suppose that the time series data x1,x2,…,xn are generated from (2). Let {αt} be an i.i.d. sequence of random variables with common probability distribution Fα; {εt} is an i.i.d sequence of random variables with density function fε. In this article, we regard X0 as a given number; alternatively, if X0 is a random variable, we shall consider only inferences conditionally on X0 are fixed. Note that X0, {αt}, and {εt} are independent.Next, we assume the following conditions to establish the asymptotic properties of the test statistic:(C1)α2+σα2<1.(C2) The distributionFα of αt is such that Eαt3<∞.(C3) The third and mixed derivatives oflog⁡fε with respect to α and σε2 are uniformly bounded on (α,σε2).(C4) Differentiation thrice with respect to(α,σε2) of fε under the integration is bounded.It is easy to derive thatXt is a Markov chain on {0,1,2,⋯} with the following transition probabilities: (4)fXt∣Xt-1xt∣xt-1=∫fXt∣Xt-1xt∣xt-1,αtdFα=∫fεxt-αtxt-1dFα.The Markovity follows from (2) and the fact that {αt} is an i.i.d sequence. The conditional density fXt∣Xt-1(xt∣xt-1) is obtained by noting that conditional on αt. Conditional on (Xt-1,αt) and {εt} is an i.i.d sequence with the density fε, hence, the probability density function of Xt is given by fε(xt-αtxt-1). So, we get the result in the above equation.Therefore, we can write down the likelihood functionLH1 for RCAR(1) model:(5)LH1x1,…,xt=fX0x0∏t=1n∫fεxt-αtxt-1dFα.Furthermore, employing this spirit of LMP approach introduced by Manik et al. [17], we can obtain our test statistic(6)Qnβ=∑t=1n12fε′′xt-αxt-1fεxt-αxt-1=∑t=1nRtβ,where(7)Rtβ=12fε′′xt-αxt-1fεxt-αxt-1.And the detailed procedures can be found in Appendix A.Letβ=(α,σε2)T denote vector of all unknown parameters of the RCAR(1) process. Hereafter, we use the notation β^=(α^,σ^ε2) to represent the maximum likelihood estimator (MLE) of β. In addition, the actual test statistic can be obtained by replacing β with their MLE β^. It can be easily derived that {Rt(β),FtR} is a zero mean martingale.Before we state our main results, the following assumptions will be made:(A.1)supt⁡ERtβ2+η<∞, for some η>0.Then about the asymptotic distribution ofQn(β^), we have the following theorem.Theorem 1.
Let{xt} be a sequence of strictly stationary, ergodic, and Ft-measurable solutions to equation (2) under conditions of (C1)-(C4) and above (A.1). Then under H0, we have asymptotic normality(8)n-1Qnβ^→dN0,ω^2,here ω^2=σ2-WTΣ-1W.(The expressions for ,σ2, and Σ-1 are derived in Appendix B).


## 3. Simulation

In this section, we carry out some simulation studies to compare performances of the locally most powerful-type test and the empirical likelihood test in terms of empirical size and power. The empirical size and power for the two tests in Tables1–6 are based on 1000 repetitions with the help of R software. Within each study, we set the initial values X0≡1 and employ the significance level at 0.05. Throughout this simulation, we used notation LMP for locally most powerful-type test by our algorithm, EL for empirical likelihood method.Table 1
Empirical sizes of LMP and EL tests at nominal level 0.05 for model N1.
N  ( σ N , α ) ( 0.5,0.3 ) ( 0.5,0.5 ) ( 0.5,0.7 ) ( 0.5,0.9 ) ( 1,0.3 ) ( 1,0.5 ) ( 1,0.7 ) ( 1,0.9 ) 100 MP 0.095 0.105 0.109 0.101 0.116 0.103 0.089 0.098 EL 0.008 0.004 0.002 0.003 0.003 0.002 0.003 0.002 200 LMP 0.089 0.082 0.085 0.086 0.085 0.082 0.083 0.087 EL 0.006 0.007 0.004 0.005 0.004 0.004 0.003 0.003 300 LMP 0.074 0.077 0.082 0.069 0.076 0.071 0.076 0.074 EL 0.006 0.009 0.007 0.006 0.005 0.007 0.008 0.009 400 LMP 0.063 0.070 0.075 0.068 0.067 0.063 0.070 0.072 EL 0.007 0.009 0.006 0.004 0.007 0.008 0.005 0.004 500 LMP 0.066 0.060 0.077 0.060 0.065 0.062 0.060 0.055 EL 0.009 0.005 0.006 0.005 0.006 0.009 0.014 0.008 600 LMP 0.062 0.057 0.067 0.053 0.065 0.060 0.059 0.068 EL 0.009 0.005 0.011 0.007 0.009 0.007 0.008 0.007 700 LMP 0.064 0.059 0.068 0.056 0.057 0.062 0.058 0.063 EL 0.005 0.010 0.008 0.007 0.007 0.008 0.005 0.003 800 LMP 0.060 0.061 0.060 0.056 0.058 0.055 0.055 0.063 EL 0.009 0.007 0.007 0.010 0.011 0.012 0.008 0.009 900 LMP 0.055 0.062 0.054 0.048 0.053 0.053 0.062 0.062 EL 0.015 0.015 0.008 0.010 0.011 0.007 0.010 0.009 1000 LMP 0.063 0.051 0.057 0.052 0.058 0.062 0.057 0.054 EL 0.019 0.009 0.012 0.009 0.011 0.012 0.007 0.009 5000 LMP 0.051 0.051 0.051 0.053 0.050 0.048 0.049 0.053 EL 0.020 0.022 0.018 0.020 0.016 0.021 0.024 0.022Table 2
Empirical sizes of LMP and EL tests at nominal level 0.05 for model N2.
N  ( m , α ) ( 8,0.3 ) ( 8,0.5 ) ( 8,0.7 ) ( 8,0.9 ) ( 10,0.3 ) ( 10,0.5 ) ( 10,0.7 ) ( 10,0.9 ) 100 MP 0.060 0.065 0.066 0.061 0.065 0.068 0.075 0.067 EL 0.003 0.003 0.002 0.003 0.004 0.003 0.004 0.001 200 LMP 0.059 0.061 0.065 0.059 0.066 0.063 0.068 0.061 EL 0.002 0.004 0.002 0.002 0.004 0.004 0.002 0.004 300 LMP 0.058 0.057 0.059 0.055 0.063 0.058 0.063 0.055 EL 0.005 0.004 0.003 0.009 0.006 0.006 0.004 0.002 400 LMP 0.058 0.055 0.058 0.055 0.062 0.054 0.061 0.058 EL 0.004 0.003 0.005 0.003 0.004 0.003 0.005 0.006 500 LMP 0.056 0.055 0.056 0.052 0.057 0.059 0.055 0.054 EL 0.006 0.004 0.005 0.005 0.007 0.005 0.006 0.007 600 LMP 0.060 0.057 0.058 0.060 0.062 0.059 0.056 0.053 EL 0.003 0.005 0.005 0.005 0.008 0.007 0.009 0.006 700 LMP 0.055 0.056 0.058 0.057 0.060 0.060 0.056 0.058 EL 0.006 0.005 0.005 0.008 0.006 0.006 0.004 0.008 800 LMP 0.050 0.054 0.054 0.052 0.056 0.054 0.054 0.054 EL 0.005 0.007 0.008 0.006 0.006 0.008 0.008 0.008 900 LMP 0.051 0.052 0.058 0.056 0.056 0.060 0.049 0.051 EL 0.005 0.006 0.006 0.009 0.006 0.008 0.010 0.005 1000 LMP 0.052 0.056 0.057 0.055 0.050 0.053 0.050 0.054 EL 0.007 0.008 0.005 0.006 0.012 0.009 0.009 0.007 5000 LMP 0.052 0.050 0.050 0.050 0.051 0.049 0.054 0.052 EL 0.015 0.014 0.012 0.016 0.017 0.014 0.012 0.013Table 3
Empirical sizes of LMP and EL tests at nominal level 0.05 for model N3.
N  ( σ 1 , σ 2 , α ) ( 1,3 , 0.1 ) ( 1,3 , 0.2 ) ( 1,3 , 0.3 ) ( 1,3 , 0.4 ) ( 1,3 , 0.5 ) ( 1,3 , 0.6 ) ( 1,3 , 0.7 ) ( 1,3 , 0.8 ) ( 1,3 , 0.9 ) ξ = 0.8 100 LMP 0.064 0.068 0.065 0.072 0.053 0.078 0.071 0.053 0.047 EL 0.000 0.000 0.001 0.000 0.000 0.001 0.000 0.000 0.001 300 LMP 0.045 0.066 0.053 0.038 0.054 0.081 0.051 0.043 0.040 EL 0.000 0.001 0.000 0.001 0.001 0.002 0.000 0.001 0.000 500 LMP 0.058 0.057 0.057 0.049 0.050 0.060 0.043 0.049 0.044 EL 0.001 0.000 0.001 0.000 0.000 0.001 0.000 0.000 0.001 1000 LMP 0.050 0.059 0.052 0.051 0.045 0.054 0.036 0.043 0.036 EL 0.002 0.001 0.001 0.001 0.001 0.001 0.003 0.001 0.001 5000 LMP 0.053 0.050 0.054 0.050 0.048 0.050 0.047 0.050 0.046 EL 0.003 0.004 0.003 0.007 0.007 0.006 0.008 0.006 0.011 ξ = 0.9 100 LMP 0.078 0.081 0.089 0.069 0.089 0.095 0.083 0.091 0.086 EL 0.000 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.001 300 LMP 0.080 0.070 0.074 0.070 0.070 0.056 0.063 0.057 0.055 EL 0.000 0.000 0.001 0.000 0.000 0.001 0.002 0.000 0.001 500 LMP 0.072 0.081 0.068 0.067 0.058 0.059 0.060 0.046 0.053 EL 0.001 0.000 0.001 0.000 0.000 0.001 0.001 0.000 0.004 1000 LMP 0.064 0.061 0.067 0.067 0.055 0.055 0.052 0.051 0.045 EL 0.000 0.000 0.000 0.001 0.002 0.003 0.003 0.000 0.001 5000 LMP 0.055 0.053 0.050 0.050 0.055 0.046 0.050 0.047 0.050 EL 0.004 0.002 0.003 0.002 0.002 0.006 0.003 0.003 0.009Table 4
Empirical powers of LMP and EL tests at nominal level 0.05 for model A1.
N  ( σ N , a , b ) ( 0.5,0.5,0.5 ) ( 0.5,0.5,1 ) ( 0.5,1 , 0.5 ) ( 0.5,1 , 1 ) ( 1,0.5,0.5 ) ( 1,0.5,1 ) ( 1,1 , 0.5 ) ( 1,1 , 1 ) 100 LMP 0.165 0.102 0.171 0.110 0.171 0.107 0.174 0.090 EL 0.055 0.024 0.040 0.023 0.050 0.022 0.041 0.025 200 LMP 0.318 0.143 0.294 0.147 0.320 0.140 0.311 0.160 EL 0.191 0.070 0.156 0.073 0.202 0.049 0.155 0.067 300 LMP 0.513 0.209 0.494 0.253 0.511 0.175 0.449 0.241 EL 0.405 0.144 0.332 0.173 0.397 0.124 0.294 0.168 400 LMP 0.674 0.232 0.561 0.330 0.663 0.264 0.590 0.338 EL 0.578 0.176 0.456 0.272 0.559 0.189 0.450 0.260 500 LMP 0.791 0.377 0.720 0.427 0.770 0.374 0.725 0.438 EL 0.722 0.298 0.629 0.366 0.705 0.303 0.625 0.374 600 LMP 0.842 0.415 0.792 0.521 0.832 0.424 0.790 0.508 EL 0.798 0.352 0.717 0.460 0.793 0.365 0.712 0.458 700 LMP 0.912 0.483 0.862 0.598 0.919 0.478 0.855 0.598 EL 0.882 0.429 0.809 0.541 0.885 0.423 0.794 0.536 800 LMP 0.964 0.560 0.915 0.645 0.947 0.568 0.918 0.643 EL 0.938 0.512 0.877 0.604 0.928 0.516 0.872 0.599 900 LMP 0.967 0.633 0.936 0.727 0.971 0.623 0.955 0.703 EL 0.952 0.586 0.910 0.696 0.959 0.586 0.933 0.687 1000 LMP 0.988 0.712 0.963 0.799 0.978 0.702 0.962 0.776 EL 0.981 0.678 0.942 0.774 0.969 0.676 0.938 0.749 5000 LMP 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 EL 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000Table 5
Empirical powers of LMP and EL tests at nominal level 0.05 for model A2.
N  ( m , a , b ) ( 6,0.5,0.5 ) ( 6,0.5,1 ) ( 6,1 , 0.5 ) ( 6,1 , 1 ) ( 10,0.5,0.5 ) ( 10,0.5,1 ) ( 10,1 , 0.5 ) ( 10,1 , 1 ) 100 LMP 0.289 0.110 0.220 0.149 0.241 0.096 0.183 0.131 EL 0.045 0.016 0.027 0.019 0.047 0.018 0.027 0.029 200 LMP 0.596 0.252 0.519 0.310 0.525 0.199 0.452 0.272 EL 0.161 0.052 0.118 0.055 0.186 0.057 0.146 0.063 300 LMP 0.790 0.400 0.722 0.475 0.762 0.351 0.686 0.429 EL 0.336 0.111 0.276 0.124 0.373 0.124 0.293 0.152 400 LMP 0.899 0.538 0.861 0.644 0.862 0.462 0.815 0.564 EL 0.486 0.147 0.399 0.196 0.520 0.173 0.440 0.223 500 LMP 0.964 0.624 0.933 0.757 0.957 0.596 0.906 0.699 EL 0.622 0.218 0.529 0.285 0.695 0.262 0.590 0.345 600 LMP 0.979 0.701 0.951 0.819 0.965 0.657 0.947 0.754 EL 0.708 0.237 0.606 0.351 0.767 0.301 0.650 0.395 700 LMP 0.995 0.791 0.974 0.862 0.983 0.727 0.978 0.824 EL 0.814 0.351 0.669 0.404 0.839 0.389 0.773 0.483 800 LMP 0.996 0.837 0.990 0.927 0.994 0.809 0.984 0.882 EL 0.845 0.404 0.767 0.512 0.895 0.488 0.839 0.585 900 LMP 0.999 0.874 0.995 0.934 0.996 0.831 0.993 0.924 EL 0.879 0.430 0.801 0.548 0.929 0.514 0.877 0.622 1000 LMP 1.000 0.936 1.000 0.971 1.000 0.889 0.996 0.951 EL 0.913 0.515 0.887 0.626 0.946 0.601 0.930 0.706 5000 LMP 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 EL 0.992 0.967 0.994 0.985 1.000 1.000 1.000 0.999Table 6
Empirical powers of LMP and EL tests at nominal level 0.05 for model A3 with(σ1,σ2)=(1,3).
N  ( a , b ) ( 0.5,0.5 ) ( 0.5,1 ) ( 0.5,1.5 ) ( 1,0.5 ) ( 1,1 ) ( 1,1.5 ) ( 1.5,0.5 ) ( 1.5,1 ) ( 1.5,1.5 ) ξ = 0.8 100 LMP 0.141 0.054 0.034 0.120 0.073 0.038 0.084 0.082 0.046 EL 0.039 0.009 0.006 0.026 0.012 0.008 0.018 0.012 0.009 300 LMP 0.775 0.388 0.183 0.628 0.493 0.299 0.475 0.448 0.331 EL 0.325 0.102 0.030 0.231 0.097 0.052 0.143 0.114 0.061 500 LMP 0.959 0.675 0.381 0.880 0.791 0.594 0.754 0.729 0.583 EL 0.656 0.236 0.070 0.500 0.268 0.143 0.347 0.222 0.131 1000 LMP 1.000 0.967 0.753 0.996 0.992 0.934 0.982 0.982 0.939 EL 0.945 0.599 0.255 0.891 0.673 0.399 0.720 0.609 0.402 5000 LMP 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 EL 1.000 1.000 0.981 1.000 1.000 0.997 0.999 1.000 0.997 ξ = 0.9 100 LMP 0.118 0.043 0.035 0.071 0.041 0.025 0.049 0.052 0.037 EL 0.042 0.013 0.015 0.032 0.021 0.014 0.020 0.013 0.011 300 LMP 0.673 0.278 0.101 0.545 0.372 0.190 0.388 0.313 0.220 EL 0.301 0.077 0.021 0.199 0.101 0.043 0.128 0.093 0.055 500 LMP 0.914 0.558 0.277 0.868 0.681 0.423 0.700 0.622 0.467 EL 0.571 0.183 0.054 0.486 0.239 0.128 0.353 0.212 0.113 1000 LMP 1.000 0.929 0.627 0.995 0.965 0.849 0.970 0.954 0.865 EL 0.927 0.503 0.204 0.852 0.637 0.375 0.702 0.548 0.368 5000 LMP 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 EL 1.000 1.000 0.963 1.000 1.000 0.996 1.000 1.000 0.994


### 3.1. Empirical Size

To calculate empirical sizes, we pay our attention to three kinds of model (1) defined as follows:(N1)αt=α, εt~N(0,σN2).(N2)αt=α, εt~t(m).(N3)αt=α, εt~Fε(x). The distribution function of ξ-contamination is(9)Fεx=ξΦxσ1+1-ξΦxσ2,where ξ is a fixed constant satisfying 0<ξ<1 and Φ(x) is the distribution function of standard normal random variable, σi>0,i=1,2.Let us now describe how the simulated results are obtained. First of all, we use the model N1 to generate data when theσN= 0.5, 1 and α= 0.3, 0.5, 0.7, 0.9. Secondly, we generate data from model N2 with the degrees of freedom m= 8, 10 and α= 0.3, 0.5, 0.7, 0.9. For model N1 and N2, we take the sample size N= 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. Thirdly, we simulate samples from model N3 with the (ξ,σ1,σ2,)=(0.8,1,3), (ξ,σ1,σ2,)=(0.9,1,3), and α= 0.3, 0.5, 0.7, 0.9. For model N3, we set the sample size N=100,300,500,1000 and 5000. The empirical levels of three models are presented in Tables 1–3, respectively. As seen from Tables 1–3, the LMP test give similar results under normal, student’st, and symmetric contamination errors. In addition, the LMP test produces sizes closer to the nominal significance level 0.05 quite satisfactorily, especially for larger sample sizes. However, the EL method has lower levels for each sample size, so one may make wrong decisions in tests based on them. Looking at three models results, we can conclude that our method has a greater effect than the EL test in terms of the empirical level.


### 3.2. Empirical Power

In order to investigate the empirical powers, we consider the alternatives under three versions of model (2):(A1)αt~Beta(a,b), εt~N(0,σN2).(A2)αt~Beta(a,b), εt~t(m).(A3)αt~Beta(a,b), εt~Fε(x). The distribution function of ξ-contamination is(10)Fεx=ξΦxσ1+1-ξΦxσ2,where ξ is a fixed constant satisfying 0<ξ<1 and Φ(x) is the distribution function of standard normal random variable, σi>0, i=1,2.To calculate the empirical powers of the two tests, in the first, we generate samples from the model A1 in which the parametersa= 0.5, 1, b= 0.5, 1, and σN= 0.5, 1. In the second setting, we simulate samples from model A2 with the degrees of freedom m=6,10 the parameters a=0.5,1 and b=0.5,1. For model A1 and A2, we take the sample size N= 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. In the last case, we generate data from the model A3 in which the (ξ,σ1,σ2,)=(0.8,1,3), (ξ,σ1,σ2,)=(0.9,1,3), and the parameters a=0.5,1,1.5, b=0.5,1,1.5. For model A3, we employ the sample size N= 100, 300, 500, 1000, and 5000. The empirical power of the above three models are summed up in Tables 4–6. In three cases, the power of two test statistics increase with the sample size, while our test produces relatively better powers than the EL test. Furthermore, both tests powers are close to 1 at the sample size N=5000. Overall, from these simulations, we conclude that the accuracy of using LMP is 6.7%, 28.8%, 26.1% higher than that of EL test under normal, student’s t, and symmetric contamination errors, respectively. As anticipated, our finding shows that the LMP test is a functional tool to detect a parameter change for RCAR(1) model. Therefore, we recommend the LMP for practical use because its computation is not difficult and the overall performance is better than that EL under normal, student’st, and symmetric contamination errors.


## 3.1. Empirical Size

To calculate empirical sizes, we pay our attention to three kinds of model (1) defined as follows:(N1)αt=α, εt~N(0,σN2).(N2)αt=α, εt~t(m).(N3)αt=α, εt~Fε(x). The distribution function of ξ-contamination is(9)Fεx=ξΦxσ1+1-ξΦxσ2,where ξ is a fixed constant satisfying 0<ξ<1 and Φ(x) is the distribution function of standard normal random variable, σi>0,i=1,2.Let us now describe how the simulated results are obtained. First of all, we use the model N1 to generate data when theσN= 0.5, 1 and α= 0.3, 0.5, 0.7, 0.9. Secondly, we generate data from model N2 with the degrees of freedom m= 8, 10 and α= 0.3, 0.5, 0.7, 0.9. For model N1 and N2, we take the sample size N= 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. Thirdly, we simulate samples from model N3 with the (ξ,σ1,σ2,)=(0.8,1,3), (ξ,σ1,σ2,)=(0.9,1,3), and α= 0.3, 0.5, 0.7, 0.9. For model N3, we set the sample size N=100,300,500,1000 and 5000. The empirical levels of three models are presented in Tables 1–3, respectively. As seen from Tables 1–3, the LMP test give similar results under normal, student’st, and symmetric contamination errors. In addition, the LMP test produces sizes closer to the nominal significance level 0.05 quite satisfactorily, especially for larger sample sizes. However, the EL method has lower levels for each sample size, so one may make wrong decisions in tests based on them. Looking at three models results, we can conclude that our method has a greater effect than the EL test in terms of the empirical level.


## 3.2. Empirical Power

In order to investigate the empirical powers, we consider the alternatives under three versions of model (2):(A1)αt~Beta(a,b), εt~N(0,σN2).(A2)αt~Beta(a,b), εt~t(m).(A3)αt~Beta(a,b), εt~Fε(x). The distribution function of ξ-contamination is(10)Fεx=ξΦxσ1+1-ξΦxσ2,where ξ is a fixed constant satisfying 0<ξ<1 and Φ(x) is the distribution function of standard normal random variable, σi>0, i=1,2.To calculate the empirical powers of the two tests, in the first, we generate samples from the model A1 in which the parametersa= 0.5, 1, b= 0.5, 1, and σN= 0.5, 1. In the second setting, we simulate samples from model A2 with the degrees of freedom m=6,10 the parameters a=0.5,1 and b=0.5,1. For model A1 and A2, we take the sample size N= 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. In the last case, we generate data from the model A3 in which the (ξ,σ1,σ2,)=(0.8,1,3), (ξ,σ1,σ2,)=(0.9,1,3), and the parameters a=0.5,1,1.5, b=0.5,1,1.5. For model A3, we employ the sample size N= 100, 300, 500, 1000, and 5000. The empirical power of the above three models are summed up in Tables 4–6. In three cases, the power of two test statistics increase with the sample size, while our test produces relatively better powers than the EL test. Furthermore, both tests powers are close to 1 at the sample size N=5000. Overall, from these simulations, we conclude that the accuracy of using LMP is 6.7%, 28.8%, 26.1% higher than that of EL test under normal, student’s t, and symmetric contamination errors, respectively. As anticipated, our finding shows that the LMP test is a functional tool to detect a parameter change for RCAR(1) model. Therefore, we recommend the LMP for practical use because its computation is not difficult and the overall performance is better than that EL under normal, student’st, and symmetric contamination errors.


## 4. A Real Life Data Analysis

In this section, we illustrate how the LMP method can be applied to a practical application. This data consist of 78 monthly number of annulus growth rate of the import bill in Australia, starting in December 2010 and ending in May 2017. The data are available online at the CEInet Statistics Database site http://db.cei.cn/page/Default.aspx. The mean and variance of the data are found to be 0.2069 and 15.3347. The data are denoted asy1,y2,…y78. Figure 1 is the sample path plot for the real data yt, t=1,2,…,78.Figure 1
The left one is the sample path of original seriesyt and the right one is the sample plot of centering series Xt for the real data.For such a process, the mean function of{yt} is constant and we may assume that the process mean is subtracted out to produce a process Xt=yt-E(yt) with zero mean. The plot of the sample path, autocorrelation function (ACF), and the partial autocorrelation function (PACF) for series {Xt} are given in Figures 1 and 2, respectively. From Figure 1, we can see Xt may come from a stationary autoregressive time series process. From Figure 2, we conclude that Xt is from first-order autoregressive process. Moreover, we test the normality of the residual data Xt according to the Normal Q-Q plot introduced by Henry C Thode [21]. From Figure 3, we notice that the scatter points on the Normal Q-Q plot are close to the reference line, so the residual data Xt can basically be seen as following the normal distribution. Hence, we derive that the model N1 would be more suitable to model these data.Figure 2
The sample autocorrelation function (ACF) and partial autocorrelation function (PACF) plots of centering seriesXt for the real data.Figure 3
The normal Q-Q plot of centering seriesXt for the real data.From the time series plot given in Figure1, the constancy of the parameter may be suspected, and, therefore, we are interested in testing the hypothesis of constancy of the coefficient parameter. We carry out the test for H0:σα2=0 against H1:σα2>0. The value of the test statistic in (8) turned out to be 2.1151, with p value 0.0344, which indicates the rejection of the null hypothesis at 5% level of significance. Thus, in this case the RCAR(1) model would be much more appropriate as opposed to the AR(1) model.


## 5. Conclusions

In this article, we propose a locally most powerful-type test for testing the constancy of the random coefficient parameter in autoregressive model and derive their limiting null distributions under regularity conditions. It is clear from the applications that the coefficient need not remain constant throughout the time. Therefore, it is essential to have such a test conducted whenever the random variation is suspected. Through our simulation study and a real-data analysis, we demonstrate that our test succeeds and it performs better than the competitor. Finally, we anticipate that our locally most powerful-type test can be extended to other types of time series model.



---

*Source: 6593821-2019-06-27.xml*