As can be seen from Figure 1, the volatility of the entrepreneur confidence index during the sample period is greater than that of the PPI index, but the changing trend of entrepreneur confidence index is basically consistent with PPI index. That is, the entrepreneur confidence index rises and the PPI rises, conversely,

Table 1. Critical values of sup-Wald test (q = 1, 2).

Figure 1. The changing trend of the entrepreneur confidence index and PPI index.

the entrepreneur confidence index falls and so does PPI. The changes between the two series can be mainly divided into three parts: one is from 2005 to the end of 2007, China’s economy is overheating in this period, and both entrepreneur confidence index and PPI index are running at a high level, the entrepreneur confidence index is much higher than 100, indicating that enterprises are positive and optimistic about the current economic situation and future expectation. Macroeconomic investment rises and the economy has entered a rapid growing stage; Two is from 2008 to 2010, influenced by international financial crisis and in the economic downturn, entrepreneurs lack confidence in investment. China’s economic development has received a greater negative impact, especially in the fourth quarter of 2008, the entrepreneur confidence index is always below 100; Three is the post-crisis from 2010 to 2017, the government has issued a series of economic policies to strengthen domestic and foreign investment to stimulate economic growth. The confidence of enterprise investors has gradually recovered. However, due to the lack of technological innovation capacity, structural industry surplus and other problems, the growth momentum of China’s economic development is not strong, and the entrepreneur confidence index has dropped significantly.

Figure 2 and Figure 3 depict the normality test results of the PPI index series. In Figure 2, the histogram shows obvious bimodal characteristics, which indicates that the PPI index series is not normal distribution, and there is a large deviation between its kernel density curve (solid line) and normal density curve (dotted line). The upper tail of the Q-Q diagram deviates significantly from the

Figure 2. Histogram and kernel density curve of PPI.

Figure 3. Q-Q diagram of PPI.

line In Figure 3, we, therefore, reject the assumption that the PPI index series follows normal distribution. This result shows that the precondition for mean regression model based on the classical assumption is no longer true. For this purpose, we need to establish a linear quantile regression model to explore the relationship between entrepreneur confidence index and PPI index.

3.2. Data Source and Processing

To study the causality between the entrepreneur confidence index and PPI index, the quarterly data from 2005 to 2017 are selected as variables. Among them, the entrepreneur confidence index (denoted as QYJ) is obtained from Oriental Fortune Net (http://www.eastmoney.com/), the logarithmic form is expressed by LNQYJ, and the first order difference is DLNQYJ. The PPI index data is obtained from National Bureau of Statistics (http://data.stats.gov.cn/), the seasonal average of PPI is expressed as LNPPI in logarithmic form and DLNPPI in first difference. Besides, in order to eliminate the effect of heteroscedasticity on regression results and reduce volatility, the data used in the empirical part were all taken with the natural logarithm, and the transformation of natural logarithm will not change the characteristics of the original data.

3.3. Unit Root Test

The Augmented Dickey-Fuller (ADF) unit root test is employed to test stationarity of the series. Checking for stationarity of data series is an important prerequisite in most empirical time series analysis, as these methods require stationarity of the variables. Results of unit root test are reported in Table 2. The results show that ADF statistics of LNQYJ and LNPPI are more than the critical values of 1%, 5% and 10%, and the p-values are also more than 0.05, so we cannot reject the null hypothesis of unit roots for both variables in level form. However, the null hypothesis is rejected when ADF unit root test is applied to the first differences of each variable. The first differences of the QYJ and PPI are stationary indicating that these variables are integrated of order one, I (1).

3.4. Cointegration Test

The purpose of cointegration test is to prevent spurious regression. There are two theories to test the cointegration relationship of time series. One is Engle-Granger (E-G) two step method, the other is Johansen test based on VAR model. This paper uses E-G two step method to test the cointegration relationship between LNQYJ and LNPPI series. The E-G two step method is to conduct unit root test of regression residual series based on OLS model. Firstly, we apply OLS to fit equation ${y}_{t}=\alpha +\beta {x}_{t}+{\mu}_{t}$ , and the regression equation is used to calculate the non-equilibrium error ${\stackrel{^}{\mu}}_{t}$ ; then the stationarity of residual series ${\stackrel{^}{\mu}}_{t}$ is tested. If ${\stackrel{^}{\mu}}_{t}$ is a stationary series, it is considered that ${x}_{t}$ and ${y}_{t}$ are cointegration variables, and there is a long-run equilibrium relationship between them. As Engle and Granger (1987) pointed out, only non-stationary variables with the same order of integration could be tested for cointegration. Since QYJ and PPI series are integrated with the same order I (1), cointegration test can be conducted by E-G two step method.

By using OLS estimation, the cointegration regression equation is

$\begin{array}{l}\text{LNPPI}=85.37+0.13\text{LNQYJ}+{\mu}_{t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{21}.\text{5}00\text{62}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{3}.\text{4}0\text{4283}\right)\end{array}$ (13)

The t statistics of each parameter are shown within parentheses in Equation (13), the adjusted R-squared is 0.865107, the F-statistics is 75.35606, the p-values of the constant term C and the explanatory variable LNQYJ are 0.0000 and 0.0014 respectively, both of which are less than 0.05. It can be seen from the regression results that the fitting effect of the cointegration regression equation is very good.

Table 2. Results of ADF unit root test.

And then the unit root test is performed on the estimated residual series ${\stackrel{^}{\mu}}_{t}$ . The critical value applied by ADF test during the cointegration analysis is different from the traditional ADF test, but refers to the table of cointegration critical values provided by Engle-Granger. The equation for calculating critical value is $C\left(\alpha \right)={\varphi}_{\infty}+{\varphi}_{1}{T}^{-1}+{\varphi}_{2}{T}^{-2}$ , Where T represents the sample size and $\alpha $ is the significance level. According to Equation (13), the critical value $C\left(\alpha \right)$ is −3.3410 at the given $\alpha =0.05$ . When conducting ADF test of residual series, the test statistic is −4.531120, which is less than the critical value at $\alpha =0.05$ . The result shows that we can reject the null hypothesis. Therefore, the residual series is stationary. So we have reasons to believe that, there is a cointegration relationship between LNQYJ and LNPPI, and there is a long-run equilibrium relationship between the entrepreneur confidence index and PPI index. In the long-run, every 1% rise in entrepreneur confidence index is associated with increases of 0.13% in PPI index.

Based on the above ADF test and E-G cointegration test, we find that the entrepreneur confidence index and PPI index are both non-stationary time series, and there is a long-run equilibrium relationship between the two series, thus it is suitable for the two economic variables to analyze their relationship by using Granger causality test.

3.5. Mean Granger Causality Test

The Granger causality test is used to analyze the causal relationship between the variables’ conditional mean, which is to test whether the explanatory variable X is favorable for predicting the behavior of the prerequisite is that X and Y are stationary series or non-stationary series with cointegration relations. Granger pointed out that if two integral series with the same order had cointegration relationship, there must be causality to support this long-run equilibrium, and the influence might be one-way or two-way. To explore the relationship between the entrepreneur confidence index and PPI index, the mean Granger causality test is performed on LNQYJ and LNPPI series. Since the optimal lag order of VAR is one, this paper chooses the mean Granger causality test of one lag order. In order to make the test results more reliable, the two lag order and three lag order are also tested. Table 3 presents the results of mean Granger causality test.

Table 3. Results of mean Granger causality test.

From Table 3, at 5% level, the results show that the PPI index is not the Granger cause of the entrepreneur confidence index, while reject the entrepreneur confidence index is not the Granger cause of PPI index. It indicates that the change in entrepreneur confidence index can cause the change in PPI index to some extent.

3.6. Quantile Granger Causality Test

For the quantile Granger causality test, the causality between variable LNPPI_{t} and LNQYJ_{t} can be tested in two ways: one is to test a single coefficient, that is, the significance of the lag parameter of the independent variable is tested respectively. We considered the null hypothesis
${H}_{0}:{\beta}_{j}=0\left(j=1,2,\cdots ,q\right)$ . LNQYJ_{t} is considered as the Granger causal of LNPPI_{t} if the hypothesis is not true at a certain
$\tau $ . The other is the joint test of all parameters, the null hypothesis is
${H}_{0}:{\beta}_{1}=\cdots ={\beta}_{q}=0$ . LNQYJ_{t} is considered as the Granger causal of LNPPI_{t} if the hypothesis is not true at all
$\tau $ .

3.6.1. Single Coefficient Test in Quantile

For each PPI-QYJ relation, we consider the following model:

${y}_{t}=\alpha \left(\tau \right)+{\displaystyle \underset{i=1}{\overset{p}{\sum}}{\alpha}_{i}\left(\tau \right){y}_{t-i}}+{\displaystyle \underset{j=1}{\overset{q}{\sum}}{\beta}_{j}\left(\tau \right){x}_{t-j}}+{\mu}_{t}\left(\tau \right)$ (14)

To determine whether PPI index Granger causes entrepreneur confidence index, y is LNPPI and x is LNQYJ; for reversed causal relations, y is LNQYJ and x is LNPPI. This model specification allows us to investigate whether lagged x delivers information (about y) that is not contained in lagged y.To illustrate, we estimate model (14) with $p=q=1$ according to the optimal lag order of VAR. For each y, 19 quantile regressions (with $\tau =0.05,0.1,\cdots ,0.9,0.95$ ) least-squares regression (OLS) are computed. The results of Granger causality test in quantile are reported in Table 4.

The coefficient estimates ${\stackrel{^}{\beta}}_{1}$ , the test statistics t and the p-values are provided for this test. Table 4 shows the testing of whether PPI index Granger causes entrepreneur confidence index. Since the p-values are greater than 0.05 across entire conditional distribution (when $0.05\le \tau \le 0.95$ ), it is evident that there is no causality from PPI index to entrepreneur confidence index, the result is the same as OLS test and the above mean Granger causality test. We conclude that PPI index changes do not lead entrepreneur confidence index changes in $0.05\le \tau \le 0.95$ . When we test whether entrepreneur confidence index Granger causes PPI index, the p-values are significant and are all less than 0.05 in $0.05\le \tau \le 0.45$ , hence there are strong reasons to accept that entrepreneur confidence index Granger causes PPI index at lower quantiles. However, when $0.5<\tau \le 0.95$ , it is not significant that entrepreneur confidence index Granger causes PPI index, which indicates entrepreneur confidence index has little impact on PPI index when the conditional distribution of PPI is located at higher quantile. The result of the OLS test also show that there is causality from entrepreneur confidence index to

Table 4. Results of quantile Granger causality test.

PPI index, while this test is only conducted at a mean level and does thus not provide an overall picture of the existing causality from entrepreneur confidence index to PPI index. Quantile Granger causality test can provide the influence of entrepreneur confidence at different quantiles. Through this test, we not only look at the causality beyond the mean estimates, but we also account for the structural breaks.

It can be seen form Table 4 that the regression estimates of ${\stackrel{^}{\beta}}_{1}$ vary with quantiles. The values of ${\stackrel{^}{\beta}}_{1}$ at lower quantiles are greater than that at higher quantiles. They are significantly positive for lower quantiles but insignificant at higher quantiles ( $\tau $ in [0.5, 0.95]). This implies that, the effect of entrepreneur confidence on PPI has an obvious difference when PPI is in different periods. For the PPI index at low quantiles, the estimates of ${\stackrel{^}{\beta}}_{1}$ on the previous entrepreneur confidence index are positive and significant, we therefore conclude that improving entrepreneur investment confidence will have a certain positive impact on PPI.

3.6.2. Joint Test in Quantiles

To be sure, this paper applies the sup-Wald test to check joint significance of ${\stackrel{^}{\beta}}_{1}$ on [0.05, 0.95]. We first test the Granger causality from entrepreneur confidence index to PPI index, the sup-Wald statistic is 10.3 and reject the null hypothesis of estimate ${\beta}_{1}=0$ at 5% level, this indicates that entrepreneur confidence Granger cause PPI. We then analyze the causality from PPI index to entrepreneur confidence index, the sup-Wald statistic is 5.6 and cannot reject the null hypothesis of estimate ${\beta}_{1}=0$ at 5% level, thus there is no evidence to believe that the PPI index is the Granger causality for entrepreneur confidence index. The results of the joint test are the same as the single coefficient test, except that the single coefficient test know the causality in which areas of the quantile.

4. Conclusion

In this study, cointegration, and methodology of Granger causality test are employed to empirically investigate causal link between entrepreneur confidence index and PPI index in China. We make use of quarterly data from the first quarter 2005 to the fourth quarter 2017. The cointegration test results indicate that there is a long-run equilibrium relationship between the entrepreneur confidence index and PPI index. And in the long-run, every 1% rise in entrepreneur confidence index is associated with increases of 0.13% in PPI index. Both the mean and quantile causality test indicate a one-way causality between the entrepreneur confidence index and PPI index. The direction of causality is from entrepreneur confidence index to PPI index. The impact of entrepreneur confidence on PPI varies with quantiles. At lower quantiles, the causality from entrepreneur confidence index to PPI index is significant, and plays a positive role in promoting PPI. While entrepreneur confidence index has little impact on PPI at higher quantiles. The estimates of ${\stackrel{^}{\beta}}_{1}$ at low quantiles are positive and significant. The results indicate that improving entrepreneur investment confidence will have a certain positive impact on PPI. When analyzing whether PPI index is the causality for entrepreneur confidence index, the p-values are not significant either in the mean test or in the quantile test. It fails to pass the significant test at the 5% level. The result indicates that PPI index is not the Granger causality for entrepreneur confidence index.

Conflicts of Interest

The authors declare no conflicts of interest.

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