# An Empirical Study on Relationship between Residents’ Disposable Income and CPI Based on TVECM: Application of Bootstrap SupLM Testing for Threshold Cointegration

Authors: Jungang Du
Journal: Journal of Mathematics
 (2022)

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## Abstract

Based on the fluctuation characteristics of residents’ disposable income and CPI (Consumer Price Index) in different periods, this article introduces a nonlinear threshold cointegration theory, establishes a TVECM (Threshold Vector Error Correction Model) of the residents’ disposable income and CPI. We propose an algorithm to obtain maximum likelihood estimation under the condition of cointegration vector and threshold value which are unknown and then propose a SupLM test for the presence of a threshold. The asymptotic distribution of the SupLM statistic is analyzed, and it appears to depend on the moment functionals, so tabulated critical values are unavailable. We discuss how the residual bootstrap can be used to calculate asymptotic critical values andp values, investigate the size and power of the SupLM test using Monte Carlo simulation, and find that the test works quite well. In the empirical section, we apply our methods to test and estimate the TVECM of residents’ disposable income and CPI. According to the experimental results, the causal relationship between residents’ disposable income and CPI under different mechanisms is tested and compared with the test results under the linear cointegration hypothesis. The empirical results show that the disposable income of residents and CPI belong to a two-mechanism nonlinear threshold cointegration system. When the deviation from the equilibrium state exceeds the threshold value, the system may adjust to the equilibrium state, and the adjustment speed of CPI shall be faster than that of the residents’ disposable income.


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## Body

## 1. Introduction

The social price level is a very important indicator in macroeconomic management and an important basis for each economy to formulate the relevant economic policies. Due to the different stages of economic development and economic structure, the change trend of CPI in different economies is significantly different in the recent years. For some developed economies, the change of CPI has been basically stable since 2000. For example, by 2021, the cumulative increase of CPI in the United States was 1.57 times, which was very close to that in the United Kingdom, with an average annual increase of about 2.1%; the cumulative increase of CPI in Japan was 1.08 times, with an average annual growth of about 0.4%. For some developing countries, CPI fluctuates greatly, and shows a large increase with economic development. For example, during the period 2000–2021, the average annual growth of CPI in India reached 6.37%. (The above data are calculated from the public data of the governments of the relevant countries). For China, since the reform and opening up in 1978, with the continuous improvement of the economic level, the CPI has also shown a fluctuating upward trend. According to the analysis of relevant data from the National Bureau of Statistics, as of 2021, the CPI has risen by 6.94 times in the past 44 years, with an average annual increase of nearly 5% (the calculation results are obtained through the data provided by the China National Bureau of Statistics websitehttps://www.stats.gov.cn/). The factors that affect the price level mainly include the issue of currency, the level of actual social demand, social wage level, labor efficiency, and international input-based cost-pulling [1, 2]. Among them, the actual social demand level, social wage level, and other factors are closely related to the disposable income of residents. Residents’ disposable income covers the resident’s wage income, which directly determines the level of consumers’ purchasing power, which in turn affects the price index. On the other hand, changes in the price index will also affect the level of residents’ disposable income to offset the reduction in the actual purchasing power of residents due to rising prices. Therefore, the disposable income of residents and the commodity price level are interrelated, and these two together determine the living standard of residents.Regarding the relationship between the residents’ disposable income and the commodity price level, Li [3] focused on analyzing the factors that affect the rise of China’s CPI, and assessed the degree of their contribution. The increase of wage level is an important force leading to the rise of CPI, and the moderate controlling rate of wage increase is an important means to curb the price increases. However, it is necessary to deal with the relationship between curbing inflation and improving residents’ lives. Li [4] analyzed the impact of inflation on residents’ income and its channels, and used the method of income compensation to measure the loss caused by inflation on residents’ income. Wang [5] focused on the interaction between the CPI and the average annual wages of employees, and holds the opinion that price changes would be directly affected by the rising wages of workers, and rising prices would also raise workers’ wages as well. Based on the data of Guangxi area, Liao [6] made an empirical analysis of the relationship between price changes and residents’ income level. By establishing the VAR model and conducting Granger causality test, Liao Bin found that there was a cointegration relationship between residents’ income level and price index, and the changes of residents’ income level lagged behind the changes of price level. Lin [7]established a VAR model of the consumption level of Chinese urban residents, the consumption price index of urban residents, and the per capita disposable income of urban households based on the data from 1978 to 2015, and found that there was a long-term cointegration relationship between them. In order to explore the relationship between economic growth, price level, and residents’ disposable income, Yang [8] established the VECM model based on the data from 1978 to 2019, and found that there was a negative relationship between price level and residents’ disposable income. Based on the panel data of 31 provinces (cities) of China from 2004 to 2017, Tian [9]used the quantile regression model to analyze the impact of the combination of housing prices and residents’ income on residents’ consumption upgrading, and found that the consumption attribute was the main part of the housing market in China, and the “crowding-out effect” rather than “wealth effect” was brought about by the rising house prices.The researches of the above have a significant effect in understanding the relationship between price index and residents’ disposable income in China. However, most of these documentations are qualitative research in the selection of research methods, and the quantitative analysis is also carried out under the premise of a priori assumption of linear relationship. Whether there is a linear relationship between the price index and residents’ disposable income lacks a strict econometric test. Moreover, for the time series used in modeling, the stability and cointegration factors are not considered, which may lead to the appearance of pseudo-regression phenomenon [10], thus affecting the scientific rationality of the conclusions. Judging from the actual situation, since the reform and opening up, the CPI has shown a fluctuating upward trend, but deflation has once been a situation for a period of time. However, the disposable income of residents showed a significant rigid characteristic, showing a significant upward trend. And this difference trends can be seen in Figure 1. The adjustment of residents’ disposable income shows asymmetric behavior relative to the changes of the CPI, and there may be a certain nonlinear relationship between the residents’ disposable income and the price index. In recent years, the nonlinear threshold cointegration theory [11] that developed based on linear cointegration theory has become an important direction in study of the relationship between nonstationary time series, which has been widely used in the field of economic analysis [12, 13].Figure 1
Sequential change trend of SR and CPI.Therefore, based on the analysis of sequence stability and cointegration, and according to the fluctuation characteristics of price index and residents’ disposable income in different periods, this paper proposes the nonlinear threshold cointegration theory, establishes the TVECM, and analyzes the model estimation and testing methods. At the same time, in view of the small sample nature of the research objects, a bootstrap simulation experiment based on parameterized residuals was designed to test and estimate the threshold cointegration model to test whether there is an asymmetric adjustment relationship between residents’ disposable income and the price index. According to the model results, the causal relationship between residents’ disposable income and price index under different mechanisms is tested. The above results are analyzed and compared with the test results under the linear cointegration hypothesis. This not only enriches the model methods for empirical research on the relationship between the price index and residents’ disposable income, but also makes up the shortcomings or deficiencies of the methods used in the previous literature to study the relationship between them, and some conclusions that cannot be obtained by linear models can be obtained. At the same time, the further research on the method of threshold cointegration in small sample nonstationary time series relationship shall be expanded as well.


## 2. TVECM Specification, Estimation, and Testing Method

The linear cointegration theory proposed by Engle and Granger is a powerful tool for nonstationary time series analysis, which has become the main method of macroeconomic time series data research. Cointegration reflects the long-term equilibrium relationship between variables within the economic system, and when the system is subjected to external shocks, the system repairs itself to tend to the original equilibrium state. Linear cointegration implies three important assumptions for this repair [14]. Firstly, the adjustment from short-term deviation to long-term equilibrium is continuous; secondly, there is a same adjustment speed in each period; thirdly, the adjustment is symmetrical. However, these assumptions are too strict, and inconsistent with many practical economic phenomena. Studies have shown that due to the influence of institutional obstacles, transaction costs, government intervention, and other factors, the adjustment of the macroeconomic system often manifests as asymmetric behavior [15]. Only when the external shock reaches a certain degree, the system will have some adjustment to reach the equilibrium state again. For example, the rigid phenomenon of wage level adjustment and decline, and many price transmission phenomena also have this adjustment mechanism. From this, to describe the special adjustment behavior of this nonlinearity, Balke and Fomby introduced nonlinearity into cointegration system and proposed the concept of threshold cointegration [11], which pioneered the theoretical study of threshold cointegration theory. The research object of this paper is residents’ disposable income and price index. The TVECM form is set based on the actual fluctuation of the two, and the estimation and testing method of this model are analyzed under the condition of unknown cointegration vector and threshold parameters.


### 2.1. Model Specification

In this paper, the CPI, which is closely related to people’s lives, is used as an economic indicator to reflect the fluctuation of social price levels. It is represented by the variable CPI, and the disposable income of residents is represented by the variable SR. The annual sequential index is used for both variables, and the data are from the data regularly released by the National Bureau of Statistics. The sample time range is from 1978 to 2021. The annual sequential change trend of SR and CPI is shown in Figure1.From Figure1, we find that the trends of SR and CPI are basically the same, and their trends have strong similarities. According to the figure, the preliminary conclusion is that SR and CPI may have a cointegration relationship. At the same time, this paper found that during the deflationary period around 2000 and the financial crisis in 2008, CPI fluctuated greatly and was once in a downward trend, while SR was stable and rising. It can be seen that although SR and CPI have a stable equilibrium relationship in the long period, under different inflation levels, the short-term adjustment of SR and CPI relative to the equilibrium state may be asymmetric. Therefore, this paper sets the TVECM of SR and CPI as follows:(1)SRt=β0+β1×CPIt+εt,(2)ΔSRt=b10+b11εt−1β+∑i=13b1iΔSRt−i+∑j=13b1jΔCPIt−id1t+c10+c11εt−1β+∑i=13c1iΔSRt−i+∑j=13c1jΔCPIt−id2t+u1tΔCPIt=b20+b21εt−1β+∑i=13b2iΔSRt−i+∑j=13b2jΔCPIt−id1t+c20+c21εt−1β+∑i=13c1iΔSRt−i+∑j=13c2jΔCPIt−id2t+u2td1tβ,γ=1εt−1β≤γd2tβ,γ=1εt−1β>γ,where β = (β0, β1) is the cointegration vector, reflecting the long-term equilibrium relationship between SR and CPI in the model. εt−1(β) is the error correction term, that is, the deviation of the early value of SR from the equilibrium state, which is used as the transfer variable to divide the TVECM system [16]. As γ is threshold value; 1(.) is indicative function (εt−1(β) ≤ γ, d1t is 1, d2t is 0; Otherwise, d1t shall be 0 and d2t shall be 1). Based on Akaike’s Information Criterion (AIC) and practical economic experience, this paper sets the maximum lag order of TVECM as 3. According to the different values of the error correction term εt−1(β), the short-term regulation of SR can be divided into two mechanisms in model (2), that is, the deviation from the equilibrium state in the previous period determines which mechanism SR is in the current period, and thus determines the regulation of SR in the current period. If the error correction term is εt-1(β) ≤ γ, the established TVECM is the first mechanism, and the regulating effect is described by B1 = (b11, b21). If εt-1(β) > γ, TVECM belongs to the second mechanism, and the regulating effect is described by B2 = (c11, c21). In particular, when B1 and B2 have significant statistical differences, it indicates that SR and CPI have significant differences in short-term adjustment under different levels of economic inflation, that is, there is an asymmetric adjustment effect, which can be described by adjusting parameters bi1 and ci1 (i = 1.2) in model (2). Therefore, the test of SR and CPI threshold cointegration is a test of the significance of statistical differences between B1 and B2.


### 2.2. Analysis of the Estimation Method of the Model

The estimation of cointegration vectorβ and threshold γ is the key and difficulty of threshold cointegration estimation and verification. Assuming that the cointegration vector is known, Kapetanios et al. estimated the cointegration vector by the linear method [17], which was used as the estimation of the threshold cointegration vector, and confused the linear cointegration vector and threshold cointegration vector. Under the condition that both cointegration vector β and threshold γ are unknown, how to obtain consistent cointegration vector of nonlinear model (2), the threshold cointegration statistics with high potential and their critical value, realize the estimation and test of the model are the focus of threshold cointegration theory research. In this paper, the threshold cointegration vector is extended to the unknown vector, and the threshold cointegration vector β of model (2) is estimated based on Hansen and Seo’s method [18], to realize the verification of threshold cointegration. In the case of unknown cointegration vector, model (2) can be extended as(3)Δxt=A1′Xt−1βd1tβ,γ+A2′Xt−1βd2tβ,γ+ut.Here, d1t and d2t are set the same as in model (2); xt = (SRt, CPIt)′ is the q × T dimension (q = 2, T = 44) Matrix; Xt-1(β) = (1, εt-1(β), Δxt-1)′.The maximum likelihood function of model (3) is(4)LA1,A2,Σ,β,γ=﹣lnΣ﹣12∑t=1TutA1,A2,β,γ′Σ-1utA1,A2,β,γ,where ut = Δxt﹣A′1Xt-1(β)d1t(βγ)﹣A′2Xt-1(β)d2t(βγ); Σ = E(utu′t). Therefore, the test of threshold cointegration is transformed into the test of the significance of statistical differences between A1 and A2. In this case, the maximum likelihood estimate MLE (A∧1A∧2Σ∧β∧γ∧) of model (2) is the maximum of the maximum likelihood function L(A1, A2, Σ, β, γ) of Model (3) [16]. Due to the parameters β and γ cannot be identified in the case of unknown cointegration vector, this paper uses the lattice search method to obtain MLE (A∧1A∧2Σ∧β∧γ∧), the specific method is as follows:(1)
Johnsen cointegration relationship test method [19] was used to estimate model (1), and the standard deviation of β^, ε^t−1β^ and β^ was obtained. Then, ε^t−1β^ shall be arranged in ascending order to obtainε^t−1∗β^, and the first 25% and the last 25% elements of E are removed (in this paper, 5%∼30% of the previous and the latter are discarded, and 25% of the previous and the latter are found to be the best values), and the middle value is taken as the search lattice interval [γl, γu]. The 95% confidence interval of β^ is taken as the value of β in the search lattice interval [βl, βu].(2)
For each group of values in the search lattice interval, the least square regression method is used to estimateA^1 (βγ), A^2 (βγ) and Σ^ (βγ). Among them,(5)A^1β,γ=∑t=1TXt−1βXt−1β′•d1tβ,γ−1∑t=1TXt−1βΔxt′d1tβ,γA^2β,γ=∑t=1TXt−1βXt−1β′•d2tβ,γ−1∑t=1TXt−1βΔxt′d2tβ,γΣ^β,γ=1T∑t=1Tu^tβ,γu^tβ,γ′.(3)
Assumingut is an independent homogeneity distribution, the maximum likelihood estimate is equivalent to the least squares estimate.(6)Lβ,γ=LA^1β,γ,A^1β,γ,β,γ=−T2lnβ,γ−12np.Thus, minimizeln| Σ^ (βγ)| corresponding (βγ) shall be (β^γ^).


### 2.3. Method Analysis of the Model Test

For the set threshold cointegration models (1) and (2), to test the difference of the adjustment between the two mechanisms in the model, the following tests shall be performed:If the original hypothesis is H0: A1 = A2 (namely, linear cointegration), and the alternative hypothesis is H0: A1≠A2 (namely, threshold cointegration).IfX1(βγ) and X2(βγ) denote the column stack matrix of Xt-1(βγ) d1t and Xt-1(βγ))d2t, respectively; ξ1(βγ) and ξ2(βγ) denote column stack matrix of u˜⊗Xt-1(β) d1t(βγ) and u˜⊗Xt-1(β)d2t(βγ). In the column stack matrix expression, u˜ is the residual of model (3) under H0. We have(7)M1β,γ=I2⊗X1β,γ′X1β,γ,M2β,γ=I2⊗X2β,γ′X2β,γ,Ω1β,γ=ξ1β,γ′ξ1β,γ,Ω2β,γ=ξ2β,γ′ξ2β,γ,V1∧β,γ=M1β,γ−1Ω1β,γM1β,γ−1,V2∧β,γ=M2β,γ−1Ω2β,γM2β,γ−1.Here, I2 is a 2 × 2 identity matrix; V^1 (βγ) and V^2 (βγ) are the estimation value of (βγ) and A^2 (βγ) covariance matrices. The Lagrange Multiplier (LM) statistic can be expressed as:(8)LMβ,γ=vecA^1β,γ﹣A^2β,γ′•V^1β,γ+V^2β,γ−1•A^1β,γ﹣A^2β,γ.In formula (8), vec is a matrix quantization operator. When (βγ) is unknown, β˜ under H0 shall be as the estimator of β, we can have:(9)SupLM=supγl≤γ≤γuLMβ˜,γ.Among them, the search lattice interval [γl, γu] of γ is unchanged.Except thatβ is fixed at the known value β0, the test statistic of equation (9) is simplified as follows:(10)SupLM0=supγl≤γ≤γuLMβ0,γ.


## 2.1. Model Specification

In this paper, the CPI, which is closely related to people’s lives, is used as an economic indicator to reflect the fluctuation of social price levels. It is represented by the variable CPI, and the disposable income of residents is represented by the variable SR. The annual sequential index is used for both variables, and the data are from the data regularly released by the National Bureau of Statistics. The sample time range is from 1978 to 2021. The annual sequential change trend of SR and CPI is shown in Figure1.From Figure1, we find that the trends of SR and CPI are basically the same, and their trends have strong similarities. According to the figure, the preliminary conclusion is that SR and CPI may have a cointegration relationship. At the same time, this paper found that during the deflationary period around 2000 and the financial crisis in 2008, CPI fluctuated greatly and was once in a downward trend, while SR was stable and rising. It can be seen that although SR and CPI have a stable equilibrium relationship in the long period, under different inflation levels, the short-term adjustment of SR and CPI relative to the equilibrium state may be asymmetric. Therefore, this paper sets the TVECM of SR and CPI as follows:(1)SRt=β0+β1×CPIt+εt,(2)ΔSRt=b10+b11εt−1β+∑i=13b1iΔSRt−i+∑j=13b1jΔCPIt−id1t+c10+c11εt−1β+∑i=13c1iΔSRt−i+∑j=13c1jΔCPIt−id2t+u1tΔCPIt=b20+b21εt−1β+∑i=13b2iΔSRt−i+∑j=13b2jΔCPIt−id1t+c20+c21εt−1β+∑i=13c1iΔSRt−i+∑j=13c2jΔCPIt−id2t+u2td1tβ,γ=1εt−1β≤γd2tβ,γ=1εt−1β>γ,where β = (β0, β1) is the cointegration vector, reflecting the long-term equilibrium relationship between SR and CPI in the model. εt−1(β) is the error correction term, that is, the deviation of the early value of SR from the equilibrium state, which is used as the transfer variable to divide the TVECM system [16]. As γ is threshold value; 1(.) is indicative function (εt−1(β) ≤ γ, d1t is 1, d2t is 0; Otherwise, d1t shall be 0 and d2t shall be 1). Based on Akaike’s Information Criterion (AIC) and practical economic experience, this paper sets the maximum lag order of TVECM as 3. According to the different values of the error correction term εt−1(β), the short-term regulation of SR can be divided into two mechanisms in model (2), that is, the deviation from the equilibrium state in the previous period determines which mechanism SR is in the current period, and thus determines the regulation of SR in the current period. If the error correction term is εt-1(β) ≤ γ, the established TVECM is the first mechanism, and the regulating effect is described by B1 = (b11, b21). If εt-1(β) > γ, TVECM belongs to the second mechanism, and the regulating effect is described by B2 = (c11, c21). In particular, when B1 and B2 have significant statistical differences, it indicates that SR and CPI have significant differences in short-term adjustment under different levels of economic inflation, that is, there is an asymmetric adjustment effect, which can be described by adjusting parameters bi1 and ci1 (i = 1.2) in model (2). Therefore, the test of SR and CPI threshold cointegration is a test of the significance of statistical differences between B1 and B2.


## 2.2. Analysis of the Estimation Method of the Model

The estimation of cointegration vectorβ and threshold γ is the key and difficulty of threshold cointegration estimation and verification. Assuming that the cointegration vector is known, Kapetanios et al. estimated the cointegration vector by the linear method [17], which was used as the estimation of the threshold cointegration vector, and confused the linear cointegration vector and threshold cointegration vector. Under the condition that both cointegration vector β and threshold γ are unknown, how to obtain consistent cointegration vector of nonlinear model (2), the threshold cointegration statistics with high potential and their critical value, realize the estimation and test of the model are the focus of threshold cointegration theory research. In this paper, the threshold cointegration vector is extended to the unknown vector, and the threshold cointegration vector β of model (2) is estimated based on Hansen and Seo’s method [18], to realize the verification of threshold cointegration. In the case of unknown cointegration vector, model (2) can be extended as(3)Δxt=A1′Xt−1βd1tβ,γ+A2′Xt−1βd2tβ,γ+ut.Here, d1t and d2t are set the same as in model (2); xt = (SRt, CPIt)′ is the q × T dimension (q = 2, T = 44) Matrix; Xt-1(β) = (1, εt-1(β), Δxt-1)′.The maximum likelihood function of model (3) is(4)LA1,A2,Σ,β,γ=﹣lnΣ﹣12∑t=1TutA1,A2,β,γ′Σ-1utA1,A2,β,γ,where ut = Δxt﹣A′1Xt-1(β)d1t(βγ)﹣A′2Xt-1(β)d2t(βγ); Σ = E(utu′t). Therefore, the test of threshold cointegration is transformed into the test of the significance of statistical differences between A1 and A2. In this case, the maximum likelihood estimate MLE (A∧1A∧2Σ∧β∧γ∧) of model (2) is the maximum of the maximum likelihood function L(A1, A2, Σ, β, γ) of Model (3) [16]. Due to the parameters β and γ cannot be identified in the case of unknown cointegration vector, this paper uses the lattice search method to obtain MLE (A∧1A∧2Σ∧β∧γ∧), the specific method is as follows:(1)
Johnsen cointegration relationship test method [19] was used to estimate model (1), and the standard deviation of β^, ε^t−1β^ and β^ was obtained. Then, ε^t−1β^ shall be arranged in ascending order to obtainε^t−1∗β^, and the first 25% and the last 25% elements of E are removed (in this paper, 5%∼30% of the previous and the latter are discarded, and 25% of the previous and the latter are found to be the best values), and the middle value is taken as the search lattice interval [γl, γu]. The 95% confidence interval of β^ is taken as the value of β in the search lattice interval [βl, βu].(2)
For each group of values in the search lattice interval, the least square regression method is used to estimateA^1 (βγ), A^2 (βγ) and Σ^ (βγ). Among them,(5)A^1β,γ=∑t=1TXt−1βXt−1β′•d1tβ,γ−1∑t=1TXt−1βΔxt′d1tβ,γA^2β,γ=∑t=1TXt−1βXt−1β′•d2tβ,γ−1∑t=1TXt−1βΔxt′d2tβ,γΣ^β,γ=1T∑t=1Tu^tβ,γu^tβ,γ′.(3)
Assumingut is an independent homogeneity distribution, the maximum likelihood estimate is equivalent to the least squares estimate.(6)Lβ,γ=LA^1β,γ,A^1β,γ,β,γ=−T2lnβ,γ−12np.Thus, minimizeln| Σ^ (βγ)| corresponding (βγ) shall be (β^γ^).


## 2.3. Method Analysis of the Model Test

For the set threshold cointegration models (1) and (2), to test the difference of the adjustment between the two mechanisms in the model, the following tests shall be performed:If the original hypothesis is H0: A1 = A2 (namely, linear cointegration), and the alternative hypothesis is H0: A1≠A2 (namely, threshold cointegration).IfX1(βγ) and X2(βγ) denote the column stack matrix of Xt-1(βγ) d1t and Xt-1(βγ))d2t, respectively; ξ1(βγ) and ξ2(βγ) denote column stack matrix of u˜⊗Xt-1(β) d1t(βγ) and u˜⊗Xt-1(β)d2t(βγ). In the column stack matrix expression, u˜ is the residual of model (3) under H0. We have(7)M1β,γ=I2⊗X1β,γ′X1β,γ,M2β,γ=I2⊗X2β,γ′X2β,γ,Ω1β,γ=ξ1β,γ′ξ1β,γ,Ω2β,γ=ξ2β,γ′ξ2β,γ,V1∧β,γ=M1β,γ−1Ω1β,γM1β,γ−1,V2∧β,γ=M2β,γ−1Ω2β,γM2β,γ−1.Here, I2 is a 2 × 2 identity matrix; V^1 (βγ) and V^2 (βγ) are the estimation value of (βγ) and A^2 (βγ) covariance matrices. The Lagrange Multiplier (LM) statistic can be expressed as:(8)LMβ,γ=vecA^1β,γ﹣A^2β,γ′•V^1β,γ+V^2β,γ−1•A^1β,γ﹣A^2β,γ.In formula (8), vec is a matrix quantization operator. When (βγ) is unknown, β˜ under H0 shall be as the estimator of β, we can have:(9)SupLM=supγl≤γ≤γuLMβ˜,γ.Among them, the search lattice interval [γl, γu] of γ is unchanged.Except thatβ is fixed at the known value β0, the test statistic of equation (9) is simplified as follows:(10)SupLM0=supγl≤γ≤γuLMβ0,γ.


## 3. Asymptotic Distribution, Size, and Power ofSupLM

### 3.1. Asymptotic Distribution

In order to give the asymptotic theory of SupLM, we require the following dependence conditions:Assumption 1.
{β′xt,Δxt} is L4r-bounded, strictly stationary, and absolutely regular. Furthermore, the error {ut} is an MDS, and the error-correction β′xt has a bounded density function.
Let “⇒” denote weak convergence, B(r) is the dimensional l vector, and it is the standard brown motion on [0,1]. For convenience, the B(r), ∫01Brdr and ∫01BrdB′r are denoted as B, ∫01B and ∫01BdB′. And the U is the brown motion with covariance matrix Σ, W=β′φ1−1U.
Firstly, the asymptotic distribution of SupLM0 is analyzed as follows:(11)SupLM0⇒tr∫01B1dB′′∫01B1B1′′∫01B1dB′.Here, B1=B11B1≤0−∫01B11B1≤0B11B1>0−∫01B11B1>0.
The study in [18] proves that under the null hypothesis and Assumption 1, the SupLM and SupLM0 have the same asymptotic distribution, namely, T. This would imply that the use of the estimate β˜, rather than the true value β0, does not alter the asymptotic null distribution of the LM test.
Unfortunately, the distribution of statistic SupLM is determined by unknown functions, which are mainly composed of some nonstationary variables. As these functionals may take a broad range of shapes, critical values forT cannot in general be tabulated.


### 3.2. AsymptoticP values: The Residual Bootstrap

With the exception discussed at the end of Section3.1, the asymptotic distribution SupLM appears to depend upon the moment functionals, so tabulated critical values are unavailable.We discuss in this section how the residual bootstrap can be used to calculate asymptotic critical values andp values, and hence achieve first-order asymptotically correct inference.Becausext is a nonstationary time series, in our applications, we assume ut is i.i.d from an unknown distribution G, and the initial conditions are fixed (other choices are possible), ut is used for bootstrap resampling to generate utb. Then, the utb is used for generatingxtb by recursion given model (3). In this way, bootstrap resampling is not required for xt itself. And this method is called residual bootstrap.Becauseut is unobservable, the value of ut obtained in the parameter estimation algorithm in section 2.2 is used to replace it. The specific steps of the residual bootstrap method are as follows:First step: the value ofut obtained in the parameter estimation algorithm in section 2.2 is used for bootstrap resampling to generate utb.Second step: theutb is used for generating new samples through replacing the original model, and thextb is obtained by residual bootstrap.Third step: Use the new sample data {xtb} to reconstruct statistics:(12)SupLMb=supγl≤γ≤γuvecA^1β,γ﹣A^2β,γ′•V^1β,γ+V^2β,γ−1•A^1β,γ﹣A^2β,γ.Fourth step: repeat the above stepsn times, and the statistic SupLMb is calculated on each simulated sample and stored. The bootstrap pvalue is the percentage of simulated statistics which exceed the actual statistic.


### 3.3. Monte Carlo Experiments: Size and Power of SupLM

Monte Carlo experiments are performed to find out the small sample performance of the test. The experiments are based on a bivariate error-correction model with two lags. Lettingxt = (x1t, x2t), the single-regime model H0 is(13)Δxt=μx1t−1−βx2t−1+ϕΔxt−1+utThe two-regime modelH1 is the generalization of (13) as in (3), allowing all coefficients to differ depending if x1t−1−x2t−1≤γ or x1t−1−x2t−1≤γ.Our tests are based on model (13), allowing all coefficients to switch between regimes under the alternative. The tests are calculated using 40 gridpoints on [γl, γu] for calculation of (9), and using 200 bootstrap replications for each replication. Our results are calculated from 1000 simulation replications.We fixμ=μ1,μ2′=0,0′, and α1 = −1. We vary α2 among (0, −0.5, 0.5), and the value of ɸ is fixed as: ϕ=0000ϕ1=−0.20−0.1−0.2, ϕ2=−0.2−0.1−0.1−0.2.We consider two sample sizes,n = 100 and 250. We generated the errors μ=μ1t,μ2t′ under homoscedastic and heteroskedastic specifications. For a homoscedastic error, we generate μ1 as independent N (0, 1) variates. Then, perform iterative calculation according to the residual bootstrap provided in Section 2.3.We first explored the size of the SupLM and SupLM0 statistics under the null hypothesis H0 of a single regime. This involved generating data from the linear model (13). For each simulated sample, the statistics and p values were calculated using the residual bootstrap. In Table 1, we report rejection frequencies from nominal 5% and 10% tests for the SupLM statistic (β unknown) and the SupLM0 statistic (β known). The results of the SupLM statistic and SupLM0 statistic were very similar. On the whole, the size of the two statistics is consistent with the traditional ADF test.Table 1
Size of SupLM tests.
nStatistic5%10%ɸ0ɸ1ɸ2ɸ0ɸ1ɸ2100ADF0.0350.0540.0430.0890.0980.100SupLM0.0380.0390.0460.0880.0950.084SupLM00.0340.0460.0410.0900.0970.092250ADF0.0490.0480.0450.0880.1020.120SupLM0.0490.0410.0460.0930.1060.110SupLM00.0460.0390.0490.0870.1040.100Using the residual bootstrap, when the sample size is 100, the SupLM test with rejection rates ranges from 0.038 to 0.046, and the SupLM0 test with rejection rates ranges from 0.034 to 0.046. If the sample size is increased to n = 250, then the size has been slightly improved, the SupLM test with rejection rates ranges from 0.041 to 0.049, and the SupLM0 test with rejection rates ranges from 0.039 to 0.049.We next explored the power of the tests against the two-regime alternativeH1. The data generation process is changed to the following form:(14)Δxt=α1εt−1εt−1≤γ+α2εt−1εt−1≤γ+utwhere γ > 0, and take values of 5, 8 and 10 respectively; the parameter α1−a,0α20,a′, and set a = 0.1 or 0.8, respectively. The power of SupLM test with corresponding values is shown in Table 2.Table 2
Power of SupLM and SupLM0 tests and 5% nominal size against two-regime alternatives.
nStatistica = 0.1a = 0.8γ=5810γ=5810100ADF0.3010.2890.2010.7020.6910.390SupLM0.3250.2800.2360.7130.6470.517SupLM00.3230.2760.2400.7100.6240.500250ADF0.8900.6090.3910.9840.7980.509SupLM0.9120.8900.7201.0000.9860.953SupLM00.9010.9010.7121.0000.9870.933Table2 reports the rejection frequency of the SupLM and SupLM0 tests at the 5% size for several values of γ. As expected, the power of SupLM statistic (β unknown) and the SupLM0 statistic (β known) is similar, but they are significantly higher than that of the traditional ADF test, which shows that SupLM test is suitable for testing threshold cointegration. The SupLM0 test has slightly higher power than the SupLM test, but the difference is surprisingly small. At least in these settings, there is little power loss due to estimation of β. When the value of γ decreases, the power of the two tests shows an increasing trend. Another feature is that when a increases, the power of the two tests shows an increasing trend.Based on the analysis of size and power, Monte Carlo simulation experiments show that SupLM test is effective in testing the threshold cointegration relationship, and SupLM test has the similar size and power as SupLM0 test. So, the SupLM test can be applied to the empirical study of this paper.


## 3.1. Asymptotic Distribution

In order to give the asymptotic theory of SupLM, we require the following dependence conditions:Assumption 1.
{β′xt,Δxt} is L4r-bounded, strictly stationary, and absolutely regular. Furthermore, the error {ut} is an MDS, and the error-correction β′xt has a bounded density function.
Let “⇒” denote weak convergence, B(r) is the dimensional l vector, and it is the standard brown motion on [0,1]. For convenience, the B(r), ∫01Brdr and ∫01BrdB′r are denoted as B, ∫01B and ∫01BdB′. And the U is the brown motion with covariance matrix Σ, W=β′φ1−1U.
Firstly, the asymptotic distribution of SupLM0 is analyzed as follows:(11)SupLM0⇒tr∫01B1dB′′∫01B1B1′′∫01B1dB′.Here, B1=B11B1≤0−∫01B11B1≤0B11B1>0−∫01B11B1>0.
The study in [18] proves that under the null hypothesis and Assumption 1, the SupLM and SupLM0 have the same asymptotic distribution, namely, T. This would imply that the use of the estimate β˜, rather than the true value β0, does not alter the asymptotic null distribution of the LM test.
Unfortunately, the distribution of statistic SupLM is determined by unknown functions, which are mainly composed of some nonstationary variables. As these functionals may take a broad range of shapes, critical values forT cannot in general be tabulated.


## 3.2. AsymptoticP values: The Residual Bootstrap

With the exception discussed at the end of Section3.1, the asymptotic distribution SupLM appears to depend upon the moment functionals, so tabulated critical values are unavailable.We discuss in this section how the residual bootstrap can be used to calculate asymptotic critical values andp values, and hence achieve first-order asymptotically correct inference.Becausext is a nonstationary time series, in our applications, we assume ut is i.i.d from an unknown distribution G, and the initial conditions are fixed (other choices are possible), ut is used for bootstrap resampling to generate utb. Then, the utb is used for generatingxtb by recursion given model (3). In this way, bootstrap resampling is not required for xt itself. And this method is called residual bootstrap.Becauseut is unobservable, the value of ut obtained in the parameter estimation algorithm in section 2.2 is used to replace it. The specific steps of the residual bootstrap method are as follows:First step: the value ofut obtained in the parameter estimation algorithm in section 2.2 is used for bootstrap resampling to generate utb.Second step: theutb is used for generating new samples through replacing the original model, and thextb is obtained by residual bootstrap.Third step: Use the new sample data {xtb} to reconstruct statistics:(12)SupLMb=supγl≤γ≤γuvecA^1β,γ﹣A^2β,γ′•V^1β,γ+V^2β,γ−1•A^1β,γ﹣A^2β,γ.Fourth step: repeat the above stepsn times, and the statistic SupLMb is calculated on each simulated sample and stored. The bootstrap pvalue is the percentage of simulated statistics which exceed the actual statistic.


## 3.3. Monte Carlo Experiments: Size and Power of SupLM

Monte Carlo experiments are performed to find out the small sample performance of the test. The experiments are based on a bivariate error-correction model with two lags. Lettingxt = (x1t, x2t), the single-regime model H0 is(13)Δxt=μx1t−1−βx2t−1+ϕΔxt−1+utThe two-regime modelH1 is the generalization of (13) as in (3), allowing all coefficients to differ depending if x1t−1−x2t−1≤γ or x1t−1−x2t−1≤γ.Our tests are based on model (13), allowing all coefficients to switch between regimes under the alternative. The tests are calculated using 40 gridpoints on [γl, γu] for calculation of (9), and using 200 bootstrap replications for each replication. Our results are calculated from 1000 simulation replications.We fixμ=μ1,μ2′=0,0′, and α1 = −1. We vary α2 among (0, −0.5, 0.5), and the value of ɸ is fixed as: ϕ=0000ϕ1=−0.20−0.1−0.2, ϕ2=−0.2−0.1−0.1−0.2.We consider two sample sizes,n = 100 and 250. We generated the errors μ=μ1t,μ2t′ under homoscedastic and heteroskedastic specifications. For a homoscedastic error, we generate μ1 as independent N (0, 1) variates. Then, perform iterative calculation according to the residual bootstrap provided in Section 2.3.We first explored the size of the SupLM and SupLM0 statistics under the null hypothesis H0 of a single regime. This involved generating data from the linear model (13). For each simulated sample, the statistics and p values were calculated using the residual bootstrap. In Table 1, we report rejection frequencies from nominal 5% and 10% tests for the SupLM statistic (β unknown) and the SupLM0 statistic (β known). The results of the SupLM statistic and SupLM0 statistic were very similar. On the whole, the size of the two statistics is consistent with the traditional ADF test.Table 1
Size of SupLM tests.
nStatistic5%10%ɸ0ɸ1ɸ2ɸ0ɸ1ɸ2100ADF0.0350.0540.0430.0890.0980.100SupLM0.0380.0390.0460.0880.0950.084SupLM00.0340.0460.0410.0900.0970.092250ADF0.0490.0480.0450.0880.1020.120SupLM0.0490.0410.0460.0930.1060.110SupLM00.0460.0390.0490.0870.1040.100Using the residual bootstrap, when the sample size is 100, the SupLM test with rejection rates ranges from 0.038 to 0.046, and the SupLM0 test with rejection rates ranges from 0.034 to 0.046. If the sample size is increased to n = 250, then the size has been slightly improved, the SupLM test with rejection rates ranges from 0.041 to 0.049, and the SupLM0 test with rejection rates ranges from 0.039 to 0.049.We next explored the power of the tests against the two-regime alternativeH1. The data generation process is changed to the following form:(14)Δxt=α1εt−1εt−1≤γ+α2εt−1εt−1≤γ+utwhere γ > 0, and take values of 5, 8 and 10 respectively; the parameter α1−a,0α20,a′, and set a = 0.1 or 0.8, respectively. The power of SupLM test with corresponding values is shown in Table 2.Table 2
Power of SupLM and SupLM0 tests and 5% nominal size against two-regime alternatives.
nStatistica = 0.1a = 0.8γ=5810γ=5810100ADF0.3010.2890.2010.7020.6910.390SupLM0.3250.2800.2360.7130.6470.517SupLM00.3230.2760.2400.7100.6240.500250ADF0.8900.6090.3910.9840.7980.509SupLM0.9120.8900.7201.0000.9860.953SupLM00.9010.9010.7121.0000.9870.933Table2 reports the rejection frequency of the SupLM and SupLM0 tests at the 5% size for several values of γ. As expected, the power of SupLM statistic (β unknown) and the SupLM0 statistic (β known) is similar, but they are significantly higher than that of the traditional ADF test, which shows that SupLM test is suitable for testing threshold cointegration. The SupLM0 test has slightly higher power than the SupLM test, but the difference is surprisingly small. At least in these settings, there is little power loss due to estimation of β. When the value of γ decreases, the power of the two tests shows an increasing trend. Another feature is that when a increases, the power of the two tests shows an increasing trend.Based on the analysis of size and power, Monte Carlo simulation experiments show that SupLM test is effective in testing the threshold cointegration relationship, and SupLM test has the similar size and power as SupLM0 test. So, the SupLM test can be applied to the empirical study of this paper.


## 4. Test and Analysis of TVECM Estimation for SR and CPI

### 4.1. Existence Test of Cointegration

To avoid pseudo regression, the stability test of time series shall be carried out before time series analysis [20]. In this paper, Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test were used for the unit root test of variables. AIC and Schwarz Information Criterion (SIC) were used to select the lag order of ADF test and the truncated lag factor of PP test, to ensure the non-autocorrelation of the residuals. The test results are shown in Table 3.Table 3
Unit root test results of variables.
VariableADF testPP testTest value5% critical valueTest value5% critical valueSR−0.146−2.9330.293−2.934D(SR)−4.629−1.949−13.339−1.949CPI−1.505−2.9390.011−2.931D(CPI)−5.626−1.950−8.117−1.949From Table3, we found that, at the significance level of 5%, the horizontal values of SR and CPI cannot reject the original hypothesis containing a unit root, and their first-order difference sequences pass the stability test. Therefore, SR and CPI are I(1) process. On this basis, the Johansen cointegration test may be implemented.In this paper, Log Likelihood (LogL), Likelihood Ratio (LR), Final Prediction Error (FPE), AIC, SIC, and Hannan–Quinn (HQ) test criteria are considered comprehensively, to select the lag order of VAR model constructed in Johansen test. Due to the limitation of sample size, this paper starts with the lag order of 4 and determines the optimal lag order of 3 according to the above criteria. The results are shown in Table4.Table 4
Test of lag order.
LagLogLLRFPEAICSICHQ0−233.9068NA454.685111.7953411.8797811.825871−215.301934.41902219.182511.0650911.31843∗11.156692−208.108912.58768∗187.226910.9054511.3276711.058113−202.52629.211432173.7884∗10.82631∗11.4174211.04004∗4−199.53524.636131184.319710.8767611.6367611.15155The Johansen test is implemented with the optimal lag order 3, and the results are shown in Table5.Table 5
Johansen cointegration test results.
Hypothesized no. of CE(s)EigenvalueMax-eigen statistic0.05 critical valueProb.∗∗None0.23683010.8109714.264600.1639At most 1∗0.1234295.2694843.8414660.0217The test results in Table5 show that the maximum characteristic root statistic of the original hypothesis “0 cointegration vectors” is less than the critical value of 5%, indicating that the original hypothesis is rejected at least at the 95% confidence level; while in the original hypothesis “there is at most one cointegration vector” is greater than the critical value of 5%, and the original hypothesis is accepted, indicating that there is a cointegration relationship between SR and CPI. Therefore, this paper believes that there is a cointegration relationship between SR and CPI in the long period.


### 4.2. Threshold Cointegration Test of SR and CPI

Through the cointegration existence test, SR and CPI have a cointegration relationship in a long period. However, judging from the previous analysis of the changing trends of the two, this relationship may be nonlinear at different levels of economic inflation. Therefore, this paper adopts the threshold cointegration test and estimation method to estimate and test the threshold cointegration models (1) and (2) between SR and CPI established above. Since SupLM converges to a complex stochastic functional [21], and the research object belongs to a small sample, this paper designs a bootstrap test [22] based on the parametric residual (PR) of the TVECM. The threshold cointegration test statistic SupLM between SR and CPI was tested. The simulation test method and steps are as follows:(1)
The data from the first period to the 1 +k period (k is the lag order, k = 3 in this paper.) of the sample of the research object shall be used as the initial value, that is, xt = (SRt, CPIt)′, t = 1,2,3,4.(2)
Using thext = (SRt, CPIt)′ sample constructed in this paper, under the original linear hypothesis, it is applicable to estimate the models (1) and (2), and calculate u˜t.(3)
Simulation data can be generated by the following Data Generating Processing (DGP): Firstly, a random disturbance sequencee with a length of n−4 is established, where n is the sample length of the research object. In this paper, if n is 44, the mean value of e is 0, the variance is 1,e∈G, and G is an independent identically distributed unknown function. Then, a random sequence of pointers can be obtained by rounding e×n [15, 21]. A new random perturbation is obtained by calculating the value of u˜t corresponding to the random pointer sequence. Finally, the new random perturbation sequence is added to the initial value x(:,4), to obtain a new random sequence with length n-4, and finally generate a new 2 × n sequence x∗ with the initial value x(:,4).(4)
Under the original linear hypothesis,x∗ is used to estimate models (1) and (2), and the relevant parameters are substituted into equations (5)∼(9) to generate an SupLM∗ estimate of the sequence x∗ corresponding to 2 × n.(5)
Step (3) and step (4) shall be repeated 5000 times to obtain 5000SupLM∗ values corresponding to x∗, and 5000 SupLM∗ values are arranged in ascending order to obtain an ascending sequence with a length of 5000. The 4750th value of the sequence was taken as the critical value of the SupLM statistic, and the proportion of the estimated value exceeding the critical value was taken as the pvalue of the experiment. The test results are shown in Table 6.Table 6
Test results of SR and CPI threshold cointegration.
StatisticsEstimate valueCritical valuep valueConclusionSupLM1171.34544.0680.001RejectionThe results in Table4 show that the SupLM statistical value calculated in this paper is 1171.345, which is much higher than the right tail critical value of 44.068, and the corresponding pvalue is 0.001. The test results significantly reject the original linear hypothesis, indicating that under different economic inflation levels, the short-term adjustment of SR and CPI is an asymmetric adjustment. The conclusion shows that although SR and CPI constitute an equilibrium system in the long period, the adjustment process of SR and CPI from short-term to long-term equilibrium is asymmetric, that is, the adjustment of SR and CPI in the current period depends not only on the degree of deviation from the equilibrium state in the previous period, but also on the mechanism of SR and CPI in the current period. The rate of adjustment to equilibrium varies significantly among the different mechanisms.


### 4.3. Analysis of Estimation Results of SR and CPI Threshold Cointegration Model

Based on the above simulation test results, the parameters of the long-term cointegration relationship model (1) are shown in equation (15). Threshold γ = −1.722.(15)SRt=﹣8.485+1.816×CPIt+εt,where SRt ≤ −8.485 + 1.816 × CPIt−1.722, the cointegration system is in the first mechanism, that is, SR is lower than the equilibrium value under the current CPI level, and the deviation is greater than 1.722. The samples falling into the first mechanism account for 30% of the total samples in this paper. Where SRt > −8.485 + 1.816 × CPIt −1.722, it belongs to the second mechanism, that is, SR is higher than the equilibrium value under the current CPI level, and the deviation is greater than −1.722. In this paper, the samples falling into the second mechanism account for 70% of the total samples. The first mechanism is when CPI is high, while the second mechanism is when CPI is low. The estimated results of submechanism threshold cointegration model (2) are shown in Table 7. In this paper, Eicker–White standard deviation is applied as the reference of parameter estimation test.Table 7
Estimation results of model (2).
VariableThe first mechanismThe second mechanismΔSRt parameter estimationΔCPIt parameter estimationΔSRt parameter estimationΔCPIt parameter estimationεt−10.053 (0.114)0.558 (0.065)0.281 (0.288)0.674 (0.250)Constant term1.968 (1.400)2.643 (0.708)−1.226 (0.673)−1.345 (0.561)ΔSRt−10.813 (0.299)1.042 (0.145)−0.086 (0.173)0.317 (0.139)ΔSRt−2−0.718 (0.204)−0.148 (0.098)−0.165 (0.168)−0.423 (0.127)ΔSRt−30.659 (0.168)−0.795 (0.076)0.238 (0.339)0.124 (0.285)ΔCPIt−1−0.413 (0.075)−0.087 (0.036)−0.430 (0.560)−0.436 (0.412)ΔCPIt−2−0.781 (0.372)−1.167 (0.162)0.247 (0.221)0.170 (0.165)ΔCPIt−30.829 (0.142)1.635 (0.045)−0.739 (0.127)−0.549 (0.123)The results in Table7 show that the influence of the error correction term εt−1 in the second mechanism is relatively obvious. In the equation of SR, the coefficient of εt−1 is 0.281, indicating that the deviation of 0.281 times of the previous period is repaired in the current period. In the equation of CPI, the coefficient of εt−1 is 0.674, which also adjusts for deviation. In the first mechanism, SR adjusts to the previous deviation at a rate of 0.053 and CPI at a rate of 0.558, both significantly lower than in the second mechanism. If the value of SR and CPI does not exceed the threshold, the system does not adjust the value. The results show that SR and CPI adjust the deviation asymmetrically under different mechanisms, that is, the speed of adjustment from short-term deviation to long-term deviation is different. The system determines whether to adjust the system based on the threshold. At the same time, the adjustment coefficient of CPI in both mechanisms is greater than that of SR, indicating that the adjustment of CPI from short-term deviation to long-term equilibrium state is faster than that of SR. Therefore, the system composed of SR and CPI is a nonlinear threshold cointegration system.


## 4.1. Existence Test of Cointegration

To avoid pseudo regression, the stability test of time series shall be carried out before time series analysis [20]. In this paper, Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test were used for the unit root test of variables. AIC and Schwarz Information Criterion (SIC) were used to select the lag order of ADF test and the truncated lag factor of PP test, to ensure the non-autocorrelation of the residuals. The test results are shown in Table 3.Table 3
Unit root test results of variables.
VariableADF testPP testTest value5% critical valueTest value5% critical valueSR−0.146−2.9330.293−2.934D(SR)−4.629−1.949−13.339−1.949CPI−1.505−2.9390.011−2.931D(CPI)−5.626−1.950−8.117−1.949From Table3, we found that, at the significance level of 5%, the horizontal values of SR and CPI cannot reject the original hypothesis containing a unit root, and their first-order difference sequences pass the stability test. Therefore, SR and CPI are I(1) process. On this basis, the Johansen cointegration test may be implemented.In this paper, Log Likelihood (LogL), Likelihood Ratio (LR), Final Prediction Error (FPE), AIC, SIC, and Hannan–Quinn (HQ) test criteria are considered comprehensively, to select the lag order of VAR model constructed in Johansen test. Due to the limitation of sample size, this paper starts with the lag order of 4 and determines the optimal lag order of 3 according to the above criteria. The results are shown in Table4.Table 4
Test of lag order.
LagLogLLRFPEAICSICHQ0−233.9068NA454.685111.7953411.8797811.825871−215.301934.41902219.182511.0650911.31843∗11.156692−208.108912.58768∗187.226910.9054511.3276711.058113−202.52629.211432173.7884∗10.82631∗11.4174211.04004∗4−199.53524.636131184.319710.8767611.6367611.15155The Johansen test is implemented with the optimal lag order 3, and the results are shown in Table5.Table 5
Johansen cointegration test results.
Hypothesized no. of CE(s)EigenvalueMax-eigen statistic0.05 critical valueProb.∗∗None0.23683010.8109714.264600.1639At most 1∗0.1234295.2694843.8414660.0217The test results in Table5 show that the maximum characteristic root statistic of the original hypothesis “0 cointegration vectors” is less than the critical value of 5%, indicating that the original hypothesis is rejected at least at the 95% confidence level; while in the original hypothesis “there is at most one cointegration vector” is greater than the critical value of 5%, and the original hypothesis is accepted, indicating that there is a cointegration relationship between SR and CPI. Therefore, this paper believes that there is a cointegration relationship between SR and CPI in the long period.


## 4.2. Threshold Cointegration Test of SR and CPI

Through the cointegration existence test, SR and CPI have a cointegration relationship in a long period. However, judging from the previous analysis of the changing trends of the two, this relationship may be nonlinear at different levels of economic inflation. Therefore, this paper adopts the threshold cointegration test and estimation method to estimate and test the threshold cointegration models (1) and (2) between SR and CPI established above. Since SupLM converges to a complex stochastic functional [21], and the research object belongs to a small sample, this paper designs a bootstrap test [22] based on the parametric residual (PR) of the TVECM. The threshold cointegration test statistic SupLM between SR and CPI was tested. The simulation test method and steps are as follows:(1)
The data from the first period to the 1 +k period (k is the lag order, k = 3 in this paper.) of the sample of the research object shall be used as the initial value, that is, xt = (SRt, CPIt)′, t = 1,2,3,4.(2)
Using thext = (SRt, CPIt)′ sample constructed in this paper, under the original linear hypothesis, it is applicable to estimate the models (1) and (2), and calculate u˜t.(3)
Simulation data can be generated by the following Data Generating Processing (DGP): Firstly, a random disturbance sequencee with a length of n−4 is established, where n is the sample length of the research object. In this paper, if n is 44, the mean value of e is 0, the variance is 1,e∈G, and G is an independent identically distributed unknown function. Then, a random sequence of pointers can be obtained by rounding e×n [15, 21]. A new random perturbation is obtained by calculating the value of u˜t corresponding to the random pointer sequence. Finally, the new random perturbation sequence is added to the initial value x(:,4), to obtain a new random sequence with length n-4, and finally generate a new 2 × n sequence x∗ with the initial value x(:,4).(4)
Under the original linear hypothesis,x∗ is used to estimate models (1) and (2), and the relevant parameters are substituted into equations (5)∼(9) to generate an SupLM∗ estimate of the sequence x∗ corresponding to 2 × n.(5)
Step (3) and step (4) shall be repeated 5000 times to obtain 5000SupLM∗ values corresponding to x∗, and 5000 SupLM∗ values are arranged in ascending order to obtain an ascending sequence with a length of 5000. The 4750th value of the sequence was taken as the critical value of the SupLM statistic, and the proportion of the estimated value exceeding the critical value was taken as the pvalue of the experiment. The test results are shown in Table 6.Table 6
Test results of SR and CPI threshold cointegration.
StatisticsEstimate valueCritical valuep valueConclusionSupLM1171.34544.0680.001RejectionThe results in Table4 show that the SupLM statistical value calculated in this paper is 1171.345, which is much higher than the right tail critical value of 44.068, and the corresponding pvalue is 0.001. The test results significantly reject the original linear hypothesis, indicating that under different economic inflation levels, the short-term adjustment of SR and CPI is an asymmetric adjustment. The conclusion shows that although SR and CPI constitute an equilibrium system in the long period, the adjustment process of SR and CPI from short-term to long-term equilibrium is asymmetric, that is, the adjustment of SR and CPI in the current period depends not only on the degree of deviation from the equilibrium state in the previous period, but also on the mechanism of SR and CPI in the current period. The rate of adjustment to equilibrium varies significantly among the different mechanisms.


## 4.3. Analysis of Estimation Results of SR and CPI Threshold Cointegration Model

Based on the above simulation test results, the parameters of the long-term cointegration relationship model (1) are shown in equation (15). Threshold γ = −1.722.(15)SRt=﹣8.485+1.816×CPIt+εt,where SRt ≤ −8.485 + 1.816 × CPIt−1.722, the cointegration system is in the first mechanism, that is, SR is lower than the equilibrium value under the current CPI level, and the deviation is greater than 1.722. The samples falling into the first mechanism account for 30% of the total samples in this paper. Where SRt > −8.485 + 1.816 × CPIt −1.722, it belongs to the second mechanism, that is, SR is higher than the equilibrium value under the current CPI level, and the deviation is greater than −1.722. In this paper, the samples falling into the second mechanism account for 70% of the total samples. The first mechanism is when CPI is high, while the second mechanism is when CPI is low. The estimated results of submechanism threshold cointegration model (2) are shown in Table 7. In this paper, Eicker–White standard deviation is applied as the reference of parameter estimation test.Table 7
Estimation results of model (2).
VariableThe first mechanismThe second mechanismΔSRt parameter estimationΔCPIt parameter estimationΔSRt parameter estimationΔCPIt parameter estimationεt−10.053 (0.114)0.558 (0.065)0.281 (0.288)0.674 (0.250)Constant term1.968 (1.400)2.643 (0.708)−1.226 (0.673)−1.345 (0.561)ΔSRt−10.813 (0.299)1.042 (0.145)−0.086 (0.173)0.317 (0.139)ΔSRt−2−0.718 (0.204)−0.148 (0.098)−0.165 (0.168)−0.423 (0.127)ΔSRt−30.659 (0.168)−0.795 (0.076)0.238 (0.339)0.124 (0.285)ΔCPIt−1−0.413 (0.075)−0.087 (0.036)−0.430 (0.560)−0.436 (0.412)ΔCPIt−2−0.781 (0.372)−1.167 (0.162)0.247 (0.221)0.170 (0.165)ΔCPIt−30.829 (0.142)1.635 (0.045)−0.739 (0.127)−0.549 (0.123)The results in Table7 show that the influence of the error correction term εt−1 in the second mechanism is relatively obvious. In the equation of SR, the coefficient of εt−1 is 0.281, indicating that the deviation of 0.281 times of the previous period is repaired in the current period. In the equation of CPI, the coefficient of εt−1 is 0.674, which also adjusts for deviation. In the first mechanism, SR adjusts to the previous deviation at a rate of 0.053 and CPI at a rate of 0.558, both significantly lower than in the second mechanism. If the value of SR and CPI does not exceed the threshold, the system does not adjust the value. The results show that SR and CPI adjust the deviation asymmetrically under different mechanisms, that is, the speed of adjustment from short-term deviation to long-term deviation is different. The system determines whether to adjust the system based on the threshold. At the same time, the adjustment coefficient of CPI in both mechanisms is greater than that of SR, indicating that the adjustment of CPI from short-term deviation to long-term equilibrium state is faster than that of SR. Therefore, the system composed of SR and CPI is a nonlinear threshold cointegration system.


## 5. Causal Relationship Analyses between SR and CPI

The model results show that when the SR is lower than the long-term equilibrium to a certain extent (bounded by the threshold), there will be a faster adjustment to the long-term equilibrium. However, when SR is at a high level, it adjusts slowly to the long-term equilibrium state. The empirical results show that SR is easy to rise and not to fall, which is consistent with the reality that SR has rigid characteristics. In different mechanisms, the relationship between SR and CPI is different. Next, this paper uses the Granger causality test to analyze the conduction relationship between the two different mechanisms. Considering that the SR and CPI belong to the nonlinear threshold cointegration relationship, and the two have the characteristics of asymmetric regulation of deviation from the long-term equilibrium state, this paper conducts the Granger causality test for the samples in the two mechanisms, respectively. At the same time, the Granger causality test is carried out on the conduction relationship between the two based on the assumption of linear cointegration relationship without considering the nonlinear cointegration relationship between the two. Finally, a comparative analysis of the two cases is carried out.


### 5.1. Causality Test of Threshold Cointegration Mechanism

Based on the two-mechanism TVECM of SR and CPI established in this paper, the Granger causality test method is used to test the causal relationship between SR and CPI under the two mechanisms. The lag order of the first mechanism is 3. Due to the small sample size in the second mechanism, the lag order is selected as 1. The test results are shown in Table8.Table 8
Granger test results of threshold cointegration mechanism.
Original hypothesisThe first mechanismThe second mechanismp valuesConclusionp valuesConclusionCPI is not the Granger cause of SR0.026Rejected0.190RejectedSR is not the Granger cause of CPI0.106Accepted0.035AcceptedThe smaller thep values in Table 8, the stronger the ability of the independent variable to predict the dependent variable. In the first mechanism, for the original hypothesis that CPI is not the Granger cause of SR, the maximum probability of it making the first error is 0.026, so the original hypothesis can be rejected at the 95% confidence level. Similarly, we accept the original hypothesis that SR is not the Granger cause of CPI. In the second mechanism, the test results are the opposite. The Granger causality test of the threshold cointegration mechanism shows that there is different one-way Granger causality between SR and CPI in different mechanisms. In the first mechanism, CPI is the Granger cause of SR, while in the second mechanism SR is the Granger cause of CPI.


### 5.2. Causality Test under Linear Cointegration Hypothesis

Regardless of the nonlinear adjustment relationship between SR and CPI, the causal relationship between the two is tested based on the linear cointegration hypothesis. Through the unit root test, it is found that both SR and CPI are single integral sequences, and the Johansen cointegration test verifies that the lag order of both is 3. The Granger causality between SR and CPI is tested in the VAR (3) model, and the test results are shown in Table 9.Table 9
Granger test results under linear cointegration hypothesis.
Original hypothesisp valuesConclusionCPI is not the Granger cause of SR0.318RejectedSR is not the Granger cause of CPI0.050AcceptedThe results in Table9 show that, without considering the nonlinear adjustment relationship between SR and CPI, the original hypothesis that SR is not the Granger cause of CPI is rejected, while the original hypothesis that CPI is not SR is accepted. The results show that there is a causal relationship between SR and CPI.


### 5.3. Comparative Analysis of Causality Test Results

The results in Table8 show that there is different one-way causality between SR and CPI under different mechanisms. Under the first mechanism, the SR is lower than the equilibrium state, that is, the SR is lower than the equilibrium value at the current CPI level. When the deviation exceeds the threshold, the CPI becomes the Granger cause of the SR change, and a higher CPI level drives the SR to a larger growth, and the adjustment speed is 0.053 times of the previous deviation degree. To ensure the stability and improvement of residents’ living standards, when the SR is lower than the equilibrium value under the corresponding CPI level and exceeds the threshold, the SR will be adjusted upwards.In the second mechanism, SR is higher than the equilibrium state, that is, SR is higher than the equilibrium value at the current CPI level. When the deviation exceeds the threshold, SR becomes the Granger cause of CPI changes, while CPI is not the Granger cause of SR. The Granger causality under the second mechanism shows that when the CPI is at a low level, that is, in the case of low inflation, the SR at a higher level prompts the CPI to adjust upwards, and the adjustment speed is 0.674 times the degree of deviation in the previous period, that is, a higher level of SR. The wage level, through consumption and other activities, drives the CPI moderately to a correction.Under the linear cointegration assumption, there is a causal relationship between SR and CPI, which is a one-way transmission from CPI to SR. SR rises as CPI rises and falls as CPI falls, with no threshold effect. The test results under the linear cointegration assumption cannot explain the fact that there is a nonlinear adjustment between SR and CPI and the typical rigid characteristics of SR.


## 5.1. Causality Test of Threshold Cointegration Mechanism

Based on the two-mechanism TVECM of SR and CPI established in this paper, the Granger causality test method is used to test the causal relationship between SR and CPI under the two mechanisms. The lag order of the first mechanism is 3. Due to the small sample size in the second mechanism, the lag order is selected as 1. The test results are shown in Table8.Table 8
Granger test results of threshold cointegration mechanism.
Original hypothesisThe first mechanismThe second mechanismp valuesConclusionp valuesConclusionCPI is not the Granger cause of SR0.026Rejected0.190RejectedSR is not the Granger cause of CPI0.106Accepted0.035AcceptedThe smaller thep values in Table 8, the stronger the ability of the independent variable to predict the dependent variable. In the first mechanism, for the original hypothesis that CPI is not the Granger cause of SR, the maximum probability of it making the first error is 0.026, so the original hypothesis can be rejected at the 95% confidence level. Similarly, we accept the original hypothesis that SR is not the Granger cause of CPI. In the second mechanism, the test results are the opposite. The Granger causality test of the threshold cointegration mechanism shows that there is different one-way Granger causality between SR and CPI in different mechanisms. In the first mechanism, CPI is the Granger cause of SR, while in the second mechanism SR is the Granger cause of CPI.


## 5.2. Causality Test under Linear Cointegration Hypothesis

Regardless of the nonlinear adjustment relationship between SR and CPI, the causal relationship between the two is tested based on the linear cointegration hypothesis. Through the unit root test, it is found that both SR and CPI are single integral sequences, and the Johansen cointegration test verifies that the lag order of both is 3. The Granger causality between SR and CPI is tested in the VAR (3) model, and the test results are shown in Table 9.Table 9
Granger test results under linear cointegration hypothesis.
Original hypothesisp valuesConclusionCPI is not the Granger cause of SR0.318RejectedSR is not the Granger cause of CPI0.050AcceptedThe results in Table9 show that, without considering the nonlinear adjustment relationship between SR and CPI, the original hypothesis that SR is not the Granger cause of CPI is rejected, while the original hypothesis that CPI is not SR is accepted. The results show that there is a causal relationship between SR and CPI.


## 5.3. Comparative Analysis of Causality Test Results

The results in Table8 show that there is different one-way causality between SR and CPI under different mechanisms. Under the first mechanism, the SR is lower than the equilibrium state, that is, the SR is lower than the equilibrium value at the current CPI level. When the deviation exceeds the threshold, the CPI becomes the Granger cause of the SR change, and a higher CPI level drives the SR to a larger growth, and the adjustment speed is 0.053 times of the previous deviation degree. To ensure the stability and improvement of residents’ living standards, when the SR is lower than the equilibrium value under the corresponding CPI level and exceeds the threshold, the SR will be adjusted upwards.In the second mechanism, SR is higher than the equilibrium state, that is, SR is higher than the equilibrium value at the current CPI level. When the deviation exceeds the threshold, SR becomes the Granger cause of CPI changes, while CPI is not the Granger cause of SR. The Granger causality under the second mechanism shows that when the CPI is at a low level, that is, in the case of low inflation, the SR at a higher level prompts the CPI to adjust upwards, and the adjustment speed is 0.674 times the degree of deviation in the previous period, that is, a higher level of SR. The wage level, through consumption and other activities, drives the CPI moderately to a correction.Under the linear cointegration assumption, there is a causal relationship between SR and CPI, which is a one-way transmission from CPI to SR. SR rises as CPI rises and falls as CPI falls, with no threshold effect. The test results under the linear cointegration assumption cannot explain the fact that there is a nonlinear adjustment between SR and CPI and the typical rigid characteristics of SR.


## 6. Conclusions

In this article, we have presented a threshold cointegration SupLM statistic for the linear non-cointegration hypothesis of TVECM. We derived an asymptotic distribution for this statistic, and developed methods to calculate the asymptotic distribution by Monte Carlo simulation, and how to calculate a bootstrap approximation. Then, the relationship between the disposable income of residents and CPI is tested by this method. Based on the above research, the main empirical study conclusions are as follows:(1)
There is a long-term equilibrium relationship between residents’ disposable income and the CPI, but it belongs to a nonlinear threshold cointegration system. Under different economic inflation levels, the short-term adjustment of residents’ disposable income and CPI is an asymmetric adjustment. That is, the adjustment speed of the short-term deviation to its long-term equilibrium state is different. And take the threshold as the limit to decide whether the system will adjust accordingly. At the same time, the adjustment coefficient of the CPI is greater than the disposable income of residents in both mechanisms. Therefore, the cointegration system composed of residents’ disposable income and CPI is a nonlinear threshold cointegration system.(2)
According to the Granger test results, there are different one-way causal relationships between residents’ disposable income and CPI in different mechanisms. In the first mechanism, the CPI is the Granger cause of residents’ disposable income, and in the second mechanism, residents’ disposable income is the Granger cause of the CPI. The results show that in the first mechanism, in the period of high CPI level, to ensure the stability and improvement of workers’ living standards, when the residents’ disposable income is lower than the corresponding CPI level and the equilibrium value exceeds the threshold, the promotion of the CPI will increase. In the following period, the disposable income of residents will be adjusted to the equilibrium state. In the second mechanism, in a period when the CPI is low, when the disposable income of residents is higher than the equilibrium value under the corresponding CPI level and exceeds the threshold, the disposable income of residents will be adjusted to the equilibrium state, and the adjustment speed is significantly higher than the first mechanism. Moreover, under the second mechanism, the disposable income of residents becomes the Granger cause of the CPI, that is, the higher wage level will pull the CPI down moderately through consumption and other activities. Compared with the Granger test based on the linear cointegration assumption, the submechanism Granger test can better explain the fact that there is a nonlinear adjustment between the residents’ disposable income and the CPI.By establishing the TVECM of residents’ disposable income and CPI, this article analyzes the nonlinear adjustment relationship between residents’ disposable income and CPI, and obtains a conclusion that cannot be drawn under the linear hypothesis. In the case of different mechanisms, the adjustment relationship between residents’ disposable income and the CPI is very different. When establishing the correlation model between residents’ disposable income and CPI, and dealing with the problem of predicting the trend of residents’ disposable income, this should be considered. Due to the existence of these changes, the conclusions of this article are of great help in dealing with these problems.



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*Source: 1691686-2022-09-13.xml*