*4.1. Vector Autoregression Model*

This paper aims to empirically examine the impact of the circumstance to financial inclusion by establishing a two-variable VAR model. The traditional econometric method relies only on economic theory to describe the relationship between variables, and cannot provide a strict explanation of their dynamic shock effect. In addition, it also needs to consider modeling the hysteresis function of all endogenous variables for each endogenous variable in the system (Liu and Xia, 2018) [34]. Therefore, it is very complicated to analyze economic problems with time series using traditional econometric methods. The VAR model constructs the model by using each endogenous variable in the system as a function of the hysteresis of all endogenous variables in the system. The model is not based on economic theory and does not distinguish between internal and external variables in advance. In this way, it can effectively control the complexity and difficulty of model estimation and analysis.

The general expression of the VAR model is as follows (Lütkepohl, 2005) [35]:

$$\mathcal{Y}\_t = B\_1 \mathcal{Y}\_{t-1} + B\_2 \mathcal{Y}\_{t-2} + \dots + B\_p \mathcal{Y}\_{t-p} + \epsilon\_{tr}$$

where *Yt* is an n-dimensional variable vector; *Bt* is the relevant coefficient matrix to be evaluated; *p* and *t* are the lag order and time, respectively; and *t* is the random disturbance term.

### *4.2. Impulse Response Function*

To observe the actions of financial inclusion factors to some changes of circumstance, this study makes the impulse response function (IRF) to characterize and verify correlation between the variables. The IRF expresses the influence of the current and future impact of a standard deviation of random

disturbance terms on all endogenous variables of the model. IRF can clearly and intuitively describe the dynamic reaction process of each endogenous variable to its own change or that of other variables.

IRF is shown as (Hamilton, 1994) [36]:

$$I\_Y(\mathfrak{n}, \delta, \omega\_{t-1}) = E[Y\_{t+\mathfrak{n}} | \mathfrak{e}\_t = \delta, \mathfrak{e}\_{t+1} = 0, \dots, \mathfrak{e}\_{t+\mathfrak{n}} = 0, \omega\_{t-1}]$$

$$-E[Y\_{t+\mathfrak{n}} | \mathfrak{e}\_t = 0, \mathfrak{e}\_{t+1} = 0, \dots, \mathfrak{e}\_{t+\mathfrak{n}} = 0, \omega\_{t-1}].$$

where *n* is the impact response period, *δ* refers to the impact from variables, *ωt*−<sup>1</sup> represents all the available information when an impact occurs, *IY* is the impulse response value of the *n*-th period.

#### *4.3. Variable Selection*

In general, monetary policy tools can be summarized into two types: price tools and quantity tools. Price tools are mainly through price guidance to play the role of interest corridor, price leverage, stabilization of expectations and financing costs, and then indirectly regulate economic operation. Quantity tools mainly play the role of monetary instruments in regulating the liquidity of the banking system and expanding the aggregate demand through the adjustment of the quantity of money supply, and then directly affect the economic operation (Liu and Xie, 2016) [37]. Many authors selected the Fed Funds rate and money supply (M2) as the proxy variable of price tools and quantity tools, respectively [38–40]. However, there is not a Fed Funds rate in China, the seven-day weighted interbank lending rate (Inter-Lend) is usually selected as proxy variable of price tools. We follow their study and select the Inter-Lend and M2 as the proxy variable of monetary policy. The aim of financial inclusion is to make vulnerable groups to obtain the service of finance. The rural population is about 540 million in China, which is the main service object of financial inclusion. So, we choose the agriculture related loan (Agri-Loan) and agricultural enterprise loan of all financial institutions (Enter-Loan) in Hunan province as the proxy variable of financial inclusion factors (Hunan is a province which lies in the central China). GDP reflects the overall economic strength of the country and is a good proxy variable of economy. Thus, GDP is selected as the proxy variable of economy in this paper. Many scholars have studied the relationship between oil and economy, which found that there was a strong correlation between them [41,42]. To keep the robustness of results, the oil price is selected as another proxy variable of economy. China's economic development is vulnerable to the world oil price fluctuations. We thus choose the crude oil price of West Texas Intermediate (Oil Price) as the proxy variables of the world oil price. The data of interbank lending rate and M2 comes from the People's Bank of China. The financial inclusion factors data gather from Changsha central sub-branch of People's Bank of China. The data of GDP comes from national bureau of statistics in China. WTI is downloaded in U.S. Energy Information Administration.
