1) , require that each component of is d-order and monolithic;
2) There is a non-zero vector , make , .
The abbreviation is cointegration, and the vector becomes a cointegration vector.
There are at most n − 1 linearly independent cointegration vectors for the n-dimensional vector time series .
2.3. The Basic Idea of Impulse Response Function 
In practical applications, because the VAR model is a non-theoretical model, it does not have to make any a priori constraints on the variables, so when analyzing the VAR model, it is generally not analyzed that one variable will produce another variable. How to influence, but to analyze the dynamic impact on the system when an error term changes, that is, when the model is subjected to some kind of impact, this analysis method is called the impulse response function.
The impulse response function analysis method can be used to describe the response of the impact of the error term on the endogenous variable, that is, the impact of the impact of one standard deviation on the random error term on the current value and future value of the endogenous variable.
The basic idea of the impulse response function analysis method is introduced below by taking the VAR(2) model with two variables as an example.
In formula (2), Pending parameter , random disturbance. Assume that it is a white noise vector with the following properties:
Assume that it is starting from the 0th period, and set
Also set in the 0th period given the disturbance term , , And other disturbance items are 0. . This is called the impulse response of phase 0 to .
2.4. VEC Model 
Engle and Granger combine the cointegration theory with the error correction model to establish a vector error correction model, the VEC model. As long as there is a cointegration relationship between the variables, the error correction model (ECM) can be derived from the autoregressive distribution lag model. In the VAR model, each equation is an autoregressive distribution lag model. Therefore, we can think that the VEC model is a VAR model with cointegration constraints.
According to the cointegration equation, the expression of the VEC model can be obtained:
among them, , is the stationary value of after differential transformation, is a coefficient vector, is a cointegration vector, and is a random perturbation vector.
Equation (4) can also be expressed as:
Each equation in Equation (5) is an error correction model. Where is the error correction term vector, reflecting the long-term equilibrium relationship between variables. The coefficient vector reflects the adjustment speed of the variable to the equilibrium state when it deviates from the long-term equilibrium state. All coefficients of the difference term as explanatory variables reflect the effect of short-term fluctuations of each variable on short-term changes as explanatory variables.
2.5. Modeling Steps
This paper explores the relationship between CPI and PPI by establishing a VEC model and using co-integration test and impulse response function. The modeling steps are as follows:
1) In order to eliminate seasonal trends and heteroscedasticity and reduce fluctuations, first-order differences between CPI sequences and PPI sequences to conduct DCPI and DPPI sequences;
2) Perform unit root test on sequence CPI, PPI, DCPI, DPPI;
3) Determining the maximum lag order p of the VAR model by the SC criterion;
4) Construct the VAR(p) model with the maximum lag order p, and verify the stability of the VAR(p) model using the AR root chart;
5) Johansen cointegration test on the first-order single-sequence CPI and PPI, and check whether there is a long-term cointegration relationship;
6) Based on the cointegration test, the impulse response function of the VAR model is analyzed;
7) Explore the short-term fluctuations and long-term equilibrium between CPI and PPI through the VEC model.
3. Empirical Research
3.1. Data Source
The data in this paper comes from the Eastern Fortune Network. The monthly data of China’s consumer price index (CPI) and industrial producers’ price index (PPI) from January 2008 to December 2018 are selected. The total data are 132 months. Figure 1 shows time series chart of the Consumer Price Index (CPI) and the Producer Price Index (PPI). It can be seen from Figure 1 that the PPI is always higher than CPI from November 2016 to November 2018. PPI fluctuates sharply compared to CPI.
3.2. Sequence Stability Test
It can be seen from Figure 1 that the CPI sequence and the PPI sequence change with time. The time serieschart shows that the CPI sequence and the PPI sequence have obvious non-stationarity. The volatility of the PPI sequence is
Figure 1. Time series diagram of CPI and PPI.
obviously stronger than that CPI sequence. From the objective point of view, it is necessary to judge the stationarity of the sequence by unit root test. In order to eliminate seasonal trends and heteroscedasticity and reduce fluctuations, the sequence CPI and PPI are subjected to first-order difference respectively, and the generated sequences are recorded as sequence DCPI and DPPI respectively, and the unit root test are performed on the two sequences. The P-value is defined as the probability, under the null hypothesis H, for right tail event, for left tail event, for double tail event.
It can be seen from Table 1 that the ADF test values of the sequence CPI and PPI are greater than the t-statistic thresholds of the test levels of 1%, 5%, and 10%, and the probability P values are all greater than 0.10. The original hypothesis cannot be rejected, the sequence CPI and PPI are non-stationary sequence. The differenced sequence is DCPI and DPPI. The ADF test value is less than the t-statistic threshold of 1%, 5% and 10% of the test level, and the probability P value is less than 0.01, which means the sequence DCPI and DPPI are considered to be stationary sequences. The sequence CPI and PPI are first order single-order sequences, indicating that the two may have a long-term co-integration relationship.
3.3. VAR Model Maximum Lag Order
Before constructing the VAR model, the Schwartz Criterion (SC) was used to determine the maximum lag order p of the model. The results are shown in Table 2.
It can be seen from Table 2 that the minimum of order P corresponding to
Table 1. Unit root test results of two sequences.
Table 2. Maximum lag order selection criteria.
the SC is 2, so the VAR model selects the lag order P = 2 as the maximum lag order.
3.4. Establishment and Verification of VAR Model
To establish a matrix form of the VAR(2) model:
The SC value of the VAR(2) model is 3.479301, and the SC value is the smallest, indicating that the established VAR(2) model works well. In order to verify the smoothness of the VAR model, the AR root chart is used for verification. It can be seen from Figure 2 that the modulus of all unit root reciprocals falls within the unit circle, which means the established VAR(2) model is stable. The X and Y axes of Figure 2 represent the coefficients of the eigenvalues, respectively. The four points in the unit circle are (0.92, 0.00), (0.89, −0.18), (0.89, 0.18), (−0.19, 0.00).
3.5. Johansen Cointegration Test
When the sequence is a non-stationary, pseudo-regression may occur in the
Figure 2. Distribution of AR unit roots.
construction model. The purpose of the cointegration test is to test whether the causal relationship described by the regression equation is a pseudo-regression. That is to say, whether there is a long-term stable relationship between the variables. The cointegration test requires that each sequence must be a non-stationary sequence and single-order single-sequence. From the unit root test results, it can be known that the sequence CPI and PPI meet the requirements of the cointegration test.
It can be seen from Table 3 that at the 5% significance level, the trace statistic test value is greater than the critical value, and the p values of the first row and the second row are 0.0007 and 0.0025, both are less than 0.05, rejecting the null hypothesis that there’s no cointegration relationship. The conclusion is that there is a long-term cointegration relationship between the consumer price index and the industrial producer’s ex-factory price index. After standardizing the cointegration coefficient, the resulting cointegration equation is as follows:
3.6. Analysis of Impulse Response Function
In order to describe the impact of a standard deviation on the random error term on the current and future values of the variable, this paper uses the impulse response function to analyze the response of consumer price index to the producer price index.
From the impulse response results in Figure 3, the impact of consumer price index on itself, reached a maximum of 0.53 at the first period, then began to decline to 0.44 of the second period, then rose to 0.46 in the third period, and then began a slow decline. It can be seen that consumer price index has short-term
Table 3. Johansen cointegration test results.
Figure 3. Synthesis of impulse response function.
interference to itself. In the long run, the impact of the consumer price index on itself cannot be ignored. Consumer price index did not respond immediately to the first phase of the producer price index, and began to rise with the extension of the lag period, the sixth period rose to the highest point of 0.27 and then gradually declined. Hence, whether in the long-term or short-term, although the consumer price index will have a certain impact on the producer price index, but the impact is not significant.
The producer price index began to respond to its own impact in the first phase, the fifth period reached a maximum of 1.63, and then began to decline but not to zero, indicating that the producer price index has a certain long-term impact on itself. The producer price index has a certain impact on consumer price index, in the sixth period, it reached a maximum of 0.66, and then began to slowly decline but not to zero.
To sum up: whether it is the impact of CPI on itself or the impact of PPI on CPI, and reversely, whether it is the impact of PPI on itself or the impact of CPI on PPI, it is always positive. However, this did not affect the long-term deviation between CPI and PPI under the new economic normal.
3.7. VEC Model Establishment and Parameter Estimation
According to Johansen’s cointegration test, there is a long-term cointegration relationship between the consumer price index and the producer price index. Based on the previous VAR(2) model, the VEC model can be established to analyze the long-term stability relationship and short-term fluctuations between the two.
The VEC model is as follows:
where, the cointegration equation :
From the estimation results of the VEC model, the SC value in the model is 1.711131, which indicates that the VEC model is very reasonable. It can be seen from the equation that CPI and PPI of the previous period have a positive effect on the current PPI. For each additional percentage-point increase in CPI of the previous period, the current PPI will increase by 0.188154 percentage-point; for each additional percentage-point increase in PPI of the previous period, the current PPI will increase by 0.784060 percentage-point. The CPI of previous period has a reverse effect on the current CPI, the PPI of previous period has a positive effect on the current CPI. For every one percentage-point increase in the previous period of CPI, the current CPI will decrease by 0.168972 percentage-point; for each additional percentage-point increase in PPI of previous period, the current CPI will increase by 0.224526 percentage points. Vecm is an error correction term, and its coefficient is negative, indicating that the CPI reversely corrects the CPI value of the next period with a value of 0.020951 to reach a long-term equilibrium state, and the PPI reversely corrects the PPI value of the next period with a value of 0.033789 to reach along-term equilibrium state.
Figure 4 is the cointegration curve between the CPI sequence and PPI sequence. From 2008 to the end of 2011, the absolute value of the error correction term deviates greatly, especially in May 2008, August 2009, and August 2011. Short-term fluctuations deviate from long-term equilibrium. After 2012, the fluctuation range of the error correction item began to decrease, and gradually returned to the long-term equilibrium state, which was caused by a series of de-capacity, destocking, and supply-side reforms in China.
This paper uses the VEC model to empirically analyze consumer price index and producer price index, and studies the cointegration relationship between CPI and PPI, summarizing the following four points:
1) The CPI sequence and the PPI sequence are stabilized after the first-order
Figure 4. Cointegration diagram of CPI and PPI.
difference, and both belong to the first-order single-order sequence. Using the two-sequence lag phase 2 to be highly significant, the VAR(2) model was constructed, indicating that the current consumer price index and the producer price index will be affected by the changes in the first two periods.
2) There is a cointegration relationship between CPI and PPI, which indicates that there is a long-term stable relationship between consumer price index and producer price index.
3) The impulse response of CPI and PPI indicates that CPI has long-term effects on its own short-term fluctuations about whether it is from short-term fluctuations or long-term effects; CPI has certain impact on PPI, but the impact is not strong. PPI has a great impact on itself and has a long-term impact on CPI, but the impact is not big.
4) The VEC model analyzed the long-term stable relationship between CPI and PPI and the short-term fluctuations. The current CPI will be affected by the reverse impact of the previous CPI and the positive impact of the previous PPI. The current PPI will be affected by the previous CPI, the positive impact and the positive impact of the previous PPI. The coefficient error correction term is a negative value, which plays a role of inversely correcting the next period of CPI and PPI values to achieve a long-term equilibrium state.
Project supported by the National Natural Science Foundation of China (61703117); National Natural Science Foundation of China (61763008); Guangxi Young and Middle-aged Teachers’ Basic Ability Improvement Project (2018KY0261).