Regional effect of monetary policy based on GVAR model and machine learning

Abstract

Monetary policy is an important means for a country to regulate macroeconomic operations and achieve established economic goals. Moreover, a reasonable monetary policy improves the efficiency of financial operations on a global scale and effectively resolves the financial crisis. At present, scholars from various countries have begun to pay attention to the issue of differentiated formulation of monetary policy among regions. This paper combines machine learning to construct a monetary policy differentiation effect analysis model based on the GVAR model. Moreover, this paper uses the gray correlation analysis method to obtain the gray correlation matrix between industries, and then introduces the industry’s own characteristics, industry relevance and macroeconomic factors into the macro stress test of credit risk. In addition, this paper constructs a conduction model based on the industry GVAR model, and uses the first-order difference sequence of GDP growth rate, CPI growth rate and M2 growth rate of each economic region to construct a GVAR model to test the impulse response function. The results of the test show that the monetary policy shocks of various economic regions are significantly different. All in all, the research results show that the performance of the model constructed in this paper is good.

Keywords

GVAR monetary policy regional effects machine learning

1 Introduction

With the continuous improvement of my country’s monetary policy, the monetary policy has changed from simple control and adjustment of the overall economy to complex control and adjustment of the economy of specific industries. If different industries have different responses to a unified monetary policy (generally called an asymmetric response), it is necessary to clarify the differences in monetary policy for each industry, including qualitative judgments and quantitative measurements. Monetary policy will respond differently to different industries. On the one hand, it will make it difficult to achieve the established macroeconomic goals. On the other hand, it will have a greater negative impact on some industries and make the economic development of different industries more uneven. Recently, scholars have mainly analyzed the different effects of monetary policy on the industry level from the transmission channel of monetary policy. However, they all viewed the impact of monetary policy on each industry in a single, and did not consider the various industries in the entire national economic system as a whole. Therefore, they only saw the greater impact of monetary policy on a certain industry but did not find the mutual influence and superimposed effects of monetary policy effects between different industries [1].

With the continuous changes in the international and domestic economic and financial situations, the formulation of monetary policy has begun to shift from overall macro-control to micro-control of various industries, and the precision of control continues to improve, so as to avoid adverse effects on other industries in the economy. In the increasingly complex international economic and financial situation, the domestic economic situation has also become more severe. In particular, our country is in the process of industrialization, and the development of various industries is very uneven. This requires our country to clearly understand the impact of monetary policy on various industries when formulating monetary policy, so as to better formulate a monetary policy that adapts to the current national economic situation and realize the expected goals of monetary policy. Relevant research at home and abroad has reached a certain consensus on the different effects of monetary policy on different industries. In other words, the transmission of monetary policy in various industries and the final impact are different, which requires qualitative analysis and quantitative measurement. This brings severe challenges to the formulation of effective monetary policy [2].

This paper changes the mechanism of monetary policy transmission from macro to micro, and establishes the micro mechanism of monetary policy transmission. Moreover, this paper unifies the monetary policy and micro-subjects such as banks, enterprises, and households under a theoretical framework, and gives a complete theoretical model, thus giving an empirical analysis of the monetary policy micro-transmission theory. In addition, this paper provides model support and data support for the formulation of monetary policy, which improves the scientificity and rigor of monetary policy formulation, and also enhances the effectiveness of monetary policy regulation [3].

2 Related work

The literature [4] proposed the GVAR method as a practical method to establish a coherent global model of the world economy. The GVAR method is relatively novel, and it provides a complicated but easy-to-follow space-time structure model for analyzing the global economy. Moreover, the model combines time series, panel data and factor analysis techniques. The literature [5] estimated the global economic model. When estimating the GVAR model, they first estimated the country-specific VAR or vector error correction model (VECM). These models include domestic macroeconomic (circle-specific) variables, such as gross domestic product (GDP), overall price level, short-term interest rate level, stock price and money supply. In addition, they also contain the cross-sectional average of external variables (weighted by transaction volume). Then, the literature [6] stacked and combined the estimated single country models to solve simultaneously as a large global VAR model. Each VECM in the GVAR model is linked by a weight matrix, and the weight matrix is determined by the paired transaction volume. The literature [7] refined the GVAR research by providing a theoretical framework. The research results show that the GVAR model can be used as an approximation of the global unobserved common factor model. Moreover, the model is very effective in dealing with the interdependence of public factors and the international trade market of business cycles. The literature [8] also developed a screen program for the overall simulation of GVAR. The empirical example in the literature [9] increased the geographic coverage and extended the estimation period. The literature [10] studied the stability of economic structure and showed how the GVAR model can be used for structural impulse response analysis. The literature [11] studied the impact of the rise of China’s economy on the Latin American economy. According to the GVAR literature, the author uses trade flows to construct a weight matrix of foreign variables. However, due to the sharp increase in China’s share of global trade, the fixed-weight matrix is unlikely to perfectly demonstrate the spillover effect during the sample period. To solve this problem, time-varying weight provides a solution. The experimental results prove that time-varying weights effectively analyze the interaction between economies in a dynamic environment [12].

Monetary policy is an important means for a country to regulate macroeconomic operations and achieve established economic goals. The literature [13] proposed the Mundell-Fleming model based on the Keynesian analysis framework, and systematically carried out theoretical research on the international transmission mechanism of monetary policy. The research carried out in the literature [14] showed that the adjustment of real output after the monetary policy shock takes a long time. Moreover, the study found that special policy arrangements, including exchange rate flexibility, will continue to affect the sensitivity of developing countries to policy changes and shocks in the central economy. The literature [15] summarized the theory of monetary policy transmission mechanism into three categories: traditional interest rate channels, other asset price channels and credit channels. The literature [16] studied the relative strength of the monetary credit, exchange rate and monetary policy interest rate channels of the European Monetary Union countries by constructing an overall measurement standard for the impact of monetary policy on prices and output.

Regarding the macroeconomic model of the new open economy, the literature [17] pioneered the integration of monopolistic competition and nominal price stickiness into the dynamic general equilibrium model, and established a micro foundation for analyzing economic entities. Since then, the results of studying international macroeconomic issues based on this analysis method are endless, which are collectively referred to as “new open economic macroeconomics". Using this new framework, the literature [18] studied the asymmetric effects of currency shocks on domestic and foreign economies, and focused on the impact of monetary and fiscal policies on welfare. The literature [19] proposed a two-state model connecting Poland and the Eurozone and used it to assess the heterogeneous differences between the two regions. The literature [20] further explored the two explanations under the conflict of monetary policy spillover effects, thus expounding the importance of international monetary policy coordination and the current problems in coordination among countries. Literature [21] proposed a Bayesian vector autoregressive model (BVAR) and added a combination of multiple DSGE models. The DSGE model was measured by comparing the impulse response of the DSGE model with the impulse response of multiple DSGE-VARs. The relative degree of error specified. The literature [22] used structural vector autoregressive model (SVAR) to analyze the determination and estimation of monetary policy rules and monetary policy shocks in the US economy. The literature [23] analyzed the effectiveness of China’s monetary policy by using the conventional VAR system with binary policy announcements. The research results show that although the monetary policy framework is very different, the impact of monetary policy on economic development is similar to the propagation process of output growth and inflation in advanced economies.

3 GVAR model

In Pesaran’s model, N + 1 countries (regions) in the global economic system are considered, denoted as i = 0, 1, ⋯ , N. Among them, the 0th country is used as a reference country. The economic variables of each country (region) are mainly affected by the following three aspects: the influence of its own lag and time trend variables, foreign variables (variables corresponding to other countries), and external macro variables. According to this idea, the k_i-th economic variable of the i-th country (region) has the following vector autoregressive model: $\begin{matrix} x_{it} = \partial_{i 0} + \partial_{i 1} t + Φ_{i} x_{i, t - 1} + Λ_{i 0} x_{it}^{*} + Λ_{i 1} x_{i, t - 1}^{*} + Ψ_{i 0} d_{t} + \\ Ψ_{i 1} d_{t - 1} + ɛ_{it} \\ t = 1, 2, \dots, T; i = 1, \dots, N \end{matrix}$ (1)

Among them,

x_it and x_i,t-1 represent the observed values of the k_i-th economic variable of the i-th country (region) at time t and t - 1, respectively, and the dimension is k_i × 1. Φ_i is the coefficient of x_i,t-1, and its dimension is k_i × k_i;

∂_i0 and ∂_i1 respectively represent the coefficients of the constant term and the time trend term;

$x_{it}^{*}$ and $x_{i, t - 1}^{*}$ represent the influence of the corresponding economic variables of other countries on the economic variables of the i-th country. The coefficients are Λ_i0 and Λ_i1 respectively, and the dimensions are both $k_{i} \times k_{i}^{*}$ ;

d_t and d_t-1 represent S public variables that affect all countries, their coefficients are Ψ_i0 and Ψ_i1, and the dimensions are both k_i × s.

ɛ_it represents the shock term of the i-th country, and the dimension is the mean value of k_i × 1 and ɛ_it.

$\begin{matrix} \sum_{ii} = (σ_{ii, ls}) \\ σ_{ii, ls} = cov (ɛ_{ilt}, ɛ_{ist}) \end{matrix}$ (2)

Or it can be simply written as: $ɛ_{it} \sim i . i . d . (0, \sum_{ii})$ (3)

The lag terms of x_it and $x_{it}^{*}$ in the above formula GVAR model are both 1, so the above model can be referred to as GVAR (1, 1) for short. The important feature of the GVAR model is that it assumes that the various regions influence each other, that is to say, the shock items are related, and the shock to one region will inevitably affect the economic variables of other regions. Therefore, the following assumptions are made here:

1. When t = t′, $E (ɛ_{it} ɛ_{{jt}^{'}}^{'}) = \sum_{ij}$ (4)

2. When t ≠ t′, $E (ɛ_{it} ɛ_{{jt}^{'}}^{'}) = 0$ (5)

In addition, in the GVAR model, the foreign variable $x_{it}^{*}$ is a weighted linear combination of the corresponding variables of each country, that is, $x_{it}^{*} = \sum_{j = 0}^{N} w_{ij} x_{jt}$ . Among them, there is $\sum_{j = 0}^{N} w_{ij} = 1$ and w_ii = 0. Here w_ij represents the proportion of the value of a variable from the j-th region to the i-th region in the value of all this variable in the i-th region (for example, you can use the trade share of the j-th region to the i-th region as a proportion of the total trade of all i regions), and the weight matrix W is known.

According to the above equation transformation, it can be written as follows: $x_{t} = b_{0} + b_{1} t + {Fx}_{t - 1} + γ_{0} d_{t} + γ_{1} d_{t - 1} + u_{t}$ (6)

Among them, there is $x_{t} = {(x_{1 t}^{'}, \dots, x_{Nt}^{'})}^{'}$ . x_t is iterated to obtain the predicted value of multiple periods, or iteratively calculate the predicted value of each subsequent period in the random simulation.

If it is assumed that $x_{it}^{*}$ and d_t satisfy the (weak) exogeneity condition at the same time, a two-dimensional vector z_it is defined, and $z_{it} = {(x_{it}, x_{it}^{'})}^{'}$ is assumed, Equation (6) can be written as follows: $A_{i} z_{it} = \partial_{i 0} + \partial_{i 1} t + B_{i} z_{i, t - 1} + Ψ_{i 0} d_{t} + Ψ_{i 1} d_{t - 1} + ɛ_{it}$ (7)

Among them, there is A_i = (1, - Λ_i0) , B_i = (Φ_i, Λ_i1). We assume that $x_{t} = {(x_{1 t}, \dots, x_{Nt})}^{'}$ , $W_{i} = {[e_{i}, t]}^{'} R_{i}^{'}$ , e_i are N-dimensional unit vectors, t is an N × 1-dimensional vector whose element value is 1, and R_i is a N × 1-dimensional vector formed by the ith row of the incidence matrix R, then there is z_it = W_ix_t, i = 1, 2, ⋯ , N, we get: ${Gx}_{t} = \partial_{0} + \partial_{1} t + {Hx}_{t - 1} + Ψ_{0} d_{t} + Ψ_{1} d_{t - 1} + ɛ_{t}$ (8)

Among them, $\begin{matrix} G = [\begin{matrix} A_{1} W \\ A_{2} W_{2} \\ \dots \\ A_{N} W_{N} \end{matrix}], \partial_{j} = [\begin{matrix} \partial_{1 j} \\ \partial_{2 j} \\ \dots \\ \partial_{Nj} \end{matrix}] \\ H = [\begin{matrix} B_{1} W \\ B_{2} W_{2} \\ \dots \\ B_{N} W_{N} \end{matrix}], Ψ_{j} = [\begin{matrix} Ψ_{1 j} \\ Ψ_{2 j} \\ \dots \\ Ψ_{Nj} \end{matrix}] \\ ɛ_{t} = [\begin{matrix} ɛ_{1 t} \\ ɛ_{2 t} \\ \dots \\ ɛ_{Nt} \end{matrix}], j = 1, 2 \end{matrix}$ (9)

The nature of ɛ_it has been explained in the previous analysis, there is: $\begin{matrix} Cov (ɛ_{t}) = \sum = (\begin{matrix} \sum_{11} & \dots & \sum_{1 N} \\ \dots & \dots & \dots \\ \sum_{N 1} & \dots & \sum_{NN} \end{matrix}) \\ \sum_{ij} = Cov (ɛ_{it}, ɛ_{jt}) \end{matrix}$ (10)

The rank of the matrix G is N and it is a non-zero matrix, that is, $rank (G) = N, | G | \neq 0$ (11)

Therefore, multiplying G^-1 on both sides of formula (8) to obtain the industry default risk factor expression is as follows:

$\begin{matrix} x_{t} = G^{- 1} \partial_{0} + G^{- 1} \partial_{1} t + G^{- 1} {Hx}_{t - 1} \\ + G^{- 1} Ψ_{0} d_{t} + G^{- 1} Ψ_{1} d_{t - 1} + G^{- 1} ɛ_{t} \end{matrix}$ (12)

It is assumed there are S macroeconomic variables, here is the first-order white regression equation for each macroeconomic variable as follows: $d_{it} = k_{i, 0} + k_{i, 1} d_{i, t - 1} + k_{i, 2} d_{i, t - 2} + v_{i, t}$ (13)

Among them, there is v_t = (v_1t, v_2t, ⋯ , v_{st)′∼i.i.d.N(0,∑v}). The error term ɛ_i,t and u of industry i and the error term v_i,t of macroeconomic variables are independent of each other, and there is $u_{t} = (\begin{matrix} ɛ_{t} \\ v_{t} \end{matrix}) \sim i . i . d . N (0, \sum), \sum = (\begin{matrix} \sum_{ii} & 0 \\ 0 & {\sum_{v} \end{matrix})$ .

According to the expression form of industry default risk factors, we get: $x_{t} = b_{0} + \partial_{1} t + {Fx}_{t - 1} + {Md}_{t} + {Nd}_{t - 1} + δ_{t}$ (14)

Among them, $\begin{matrix} b_{0} = G^{- 1} \partial_{0}, b_{1} = G^{- 1} \partial_{1}, Γ = G^{- 1} H, M = G^{- 1} Ψ_{0} \\ N = G^{- 1} Ψ_{1}, δ_{t} = G^{- 1} ɛ_{t} \end{matrix}$

According to the second-order autoregressive equations of macroeconomic variables that have been obtained, the values of each period of d_t and d_t-1 are obtained, and the predicted value of the industry risk factor x_t starting from period T and (T + 1, T + 2, ⋯ , T + n) in subsequent periods can be obtained by formula (13), that is: $\begin{matrix} x_{T + n} = Γ^{n} x_{T} + \sum_{τ = 0}^{n - 1} Γ^{τ} [b_{0} + b_{1} (T + n - τ)] + \\ \sum_{τ = 0}^{n - 1} Γ^{τ} [{Md}_{T + n - τ} {Nd}_{T + n - 1 - τ}] + \sum_{τ = 0}^{n - 1} Γ^{τ} δ_{T + n - τ} \end{matrix}$ (15)

It can be seen from the above formula that x_T+n is mainly composed of four parts. The first part Γⁿx_T represents the influence of the initial value x_T on the system state after n periods. The second part is the embodiment of the constant velocity term and the time trend term in the VAR model. The third part represents the impact of macroeconomic variables on endogenous variables, and the last part represents the random (unpredictable) component of x_T+n. Through the above formula, the value of x_t can be predicted in multiple periods, or the value of x_t can be obtained step by step in a random simulation.

4 The basic theory of industry grey relational analysis

At present, the generalized degree of relevance is widely used. Therefore, the generalized degree of relevance is used in the relevance analysis of this paper. The calculation steps are as follows:

1. The analyzed reference sequence and comparison sequence are determined.

The reference sequence is the dependent variable sequence, which is the mother sequence for analyzing the correlation, that is, analyzing the correlation with the reference sequence, and the reference sequence is set to X₀. The comparison sequence is the independent variable sequence, which is the subsequence to analyze the correlation, that is, analyzing the correlation between the comparison sequence and the reference sequence. The comparison sequence is set to X₁, ⋯ , X_N, t = 1, 2, ⋯ , T is the time variable of the variable, and X_i (t) represents the value of the i-th variable at the t-th time.

2. The dimensionless processing of the index and the zero image of the starting point.

Data dimensionless processing generally includes the following processing methods: average processing, extreme value processing, standardization processing, and initial value processing, etc. The initial value method is used for dimensionless processing, and the data sequence obtained is as follows: $X_{i}^{'} (t) = \frac{X_{i} (t)}{X_{i} (0)}$ . On this basis, the polyline sequence used to calculate the gray relative degree of relevance is $X_{i}^{' 0} (t) = X_{i}' (t) - X_{i}' (1)$ , and the polyline sequence used to calculate the gray absolute relevance is $X_{i}^{0} = X_{i} (t) - X_{i} (1)$ .

3. The absolute gray correlation degree and the gray relative correlation degree are calculated.

If $| S_{i}^{'} | = | \sum_{t = 2}^{T - 1} X_{i}^{' 0} (t) + \frac{1}{2} X_{i}^{' 0} (T) |$ is assumed, the gray relative degree of association is: $γ_{0 i} = \frac{1 + | S_{0}^{'} | + | S_{i}^{'} |}{1 + | S_{0}^{'} | + | S_{i}^{'} | + | S_{i}^{'} - S_{0}^{'} |}$ (16)

If $| S_{i} | = | \sum_{t = 2}^{T - 1} X_{i}^{0} (t) + \frac{1}{2} X_{i}^{0} (T) |$ is assumed, the absolute gray correlation degree is: $δ_{0 i} = \frac{1 + | S_{0} | + | S_{i} |}{1 + | S_{0} | + | S_{i} | + | S_{i} - S_{0} |}$ (17)

4. The gray comprehensive correlation degree is calculated.

According to the calculated gray absolute correlation degree and gray relative correlation degree, the two are combined linearly, which not only reflects the similarity of the two broken lines, but also reflects the closeness of the two broken lines relative to the change rate of the initial point during the movement. If the weight is assumed to be θ, the gray comprehensive correlation degree is: $ρ_{0 i} = θ γ_{0 i} + (1 - θ) δ_{0 i} (θ = 0.5)$ (18)

5 A macro stress test method based on the IGVAR model is proposed

The GVAR model is introduced into the mutual influence between industries, and combined with the correlation between industries, the industry GVAR model (This paper is collectively referred to as the IGVAR model) is constructed. It is assumed that there are N industries, namely i = 1, 2, ⋯ , N. First, the industry default probability after logistic transformation is: $p_{it} = \frac{1}{1 + exp (- y_{it})}$ (19)

Among them, there is i = 1, 2, ⋯ , Nt = 1, 2, ⋯ , T. p_it represents the default probability of the i-th industry at time t, and y_it represents the risk factor corresponding to the change in the logarithm of the default probability.

Secondly, according to the analysis of the correlation between regions in the GVAR model, considering the close correlation between various industries, the risk factor y_it here is mainly affected by the following aspects:

The occurrence of default in other industries has a direct impact on the probability of default in industry i. This is mainly because there is a certain correlation between various industries, and other industries can affect the risk factor value of industry i.

The lag effect of the industry itself. The risk factors of each industry will be affected by the risk factors of the previous period or the previous periods, which is mainly caused by the lag effect of the risk factors themselves, and the industry risk factors will also be affected by the time trend item.

The influence of external macroeconomic factors. All industries will be directly affected by macroeconomic factors.

The impact item ɛ_it of industry i risk factor and the impact item ɛ_jt of other industry risk factors are correlated at the same time. Since the covariance of shock items over the same period is not zero, shock items of other industry risk factors will affect the shock items of industry i risk factors, and thus the value of risk factors of industry i. Based on the above four considerations, this paper believes that the risk factor y_it of the default probability of the i-th industry has the following vector autoregressive model:

$\begin{matrix} y_{it} = \partial_{i 0} + \partial_{i 1} t + Φ_{i} y_{i, t - 1} + Λ_{i 0} y_{it}^{*} + Λ_{i 1} y_{i, t - 1}^{*} + Ψ_{i 0} d_{t} + \\ Ψ_{i 1} d_{t - 1} + ɛ_{it} \\ t = 1, 2, \dots, T; i = 1, \dots, N \end{matrix}$ (20)

In the formula, y_it and y_i,t-1 represent the default risk factors of the i-th industry at time t and t - 1, respectively, and $y_{it}^{*}$ and $y_{i, t - 1}^{*}$ represent the impact of other industries on the i-th industry. Meanwhile, d_t (d_1,t, d_2,t, ⋯ , d_s,t) ′ represents the observed values of s macroeconomic variables at time t, and the dimensions of Ψ_i0 and Ψ_i1 are N × s. Since the impact of other industries will affect the related industries, the definition and nature of the industry impact item ɛ_it is ɛ_it ∼ i . i . d . (0, ∑_ii) and $E (ɛ_{it} ɛ_{{jt}^{'}}^{'}) = \sum_{ij}$ .

In the IGVAR model, the default risk factor $y_{it}^{*}$ of industry i is actually a weighted linear combination of the default risk factors of other industries, $y_{it}^{ast} = \sum_{l j = 1 j \neq i}^{N} ρ_{ij} y_{jt}$ and weight ρ_ij reflect the degree of correlation between the i-th industry and the j-th industry (or called the correlation coefficient). In the GVAR model, the share of the industry is used as the weight, but from the perspective of industry default probability, the degree of correlation between industries is often the main factor that determines the “contagion” of risks among various industries. The greater the correlation coefficient, the higher the degree of correlation between industries. Once a credit crisis occurs in a certain industry, this default risk will be transmitted to other industries through the correlation between industries. Therefore, the industry correlation coefficient is an important manifestation of the correlation between industries. The correlation coefficient ρ_ij generates an incidence matrix in the form of Equation (20), which will be generated by the gray incidence matrix. It should be noted that ρ_ij = 0 considers that its own relevance has been reflected in the industry’s own default risk factors, so this contagion is zero. $R = [\begin{matrix} 0, ρ_{12}, ρ_{13}, \dots, ρ_{1 N} \\ ρ_{21}, 0, ρ_{23}, \dots, ρ_{2 N} \\ \dots \\ ρ_{N 1}, ρ_{N 2}, \dots, 0 \end{matrix}]$ (21)

The macro stress test model constructed is as follows: $p_{it} = \frac{1}{1 + exp (- y_{it})}$ (22) $\begin{matrix} y_{it} = \partial_{i 0} + \partial_{i 1} t + Φ_{i} y_{i, t - 1} + Λ_{i 0} y_{it}^{*} + Λ_{i 1} y_{i, t - 1}^{*} + Ψ_{i 0} d_{t} + \\ Ψ_{i 1} d_{t - 1} + ɛ_{it} \\ t = 1, 2, \dots, T; i = 1, \dots, N \end{matrix}$ (23) $y_{it}^{*} = \sum_{l j = 1 j \neq i}^{N} ρ_{ij} y_{jt}$ (24) $d_{it} = k_{i, 0} + k_{i, 1} d_{i, t - 1} + k_{i, 2} d_{i, t - 2} + v_{i, t}$ (25) $u_{t} = (\begin{matrix} ε_{t} \\ v_{t} \end{matrix}) ~ i . i . d . N (0, \sum), \sum = (\begin{matrix} \sum i i & 0 \\ 0 & \sum v \end{matrix})$ (26)

In the formula, y_it and y_i,t-1 represent the default risk factors of the i-th industry at time t and t - 1, respectively, $y_{it}^{*}$ and $y_{i, t - 1}^{*}$ represent the impact of other industries on the i-th industry, and $y_{it}^{*}$ is actually a weighted linear combination of default risk factors in other industries. Meanwhile, ρ_ij represents the gray correlation coefficient between the i-th industry and the j-th industry, and d_t (d_1,t, d_2,t, ⋯ , d_s,t) ′ represents the value of s macroeconomic variables at time t. ∂ΦΨ and k are all parameters that need to be estimated, and ∑is the equation covariance matrix, which represents the degree of dispersion of u.

6 Ordered multi-class logistic method to measure the industry’s original default probability

The selection of scientific industry classification basis is very important for the research of this paper. Different industry classification basis will affect the final classification results and then the final stress test results. At present, some industry classification standards commonly used internationally mainly include ISIC (United Nations International Standard Industry Classification), MSCI (Global Industry Classification Standard), and NAICS (North American Industry Classification Standard). These industry classification standards have clear definitions and strict industry definition principles for various industries. The differences between various industries are relatively obvious and the same industry has internal quality. At the same time, considering the coverage of the industry, the following 9 industries are selected as samples for analysis: agriculture, forestry, animal husbandry and fishery, extractive industries, manufacturing, electricity, gas and water, construction, transportation, wholesale and retail, real estate, and information technology industry.

Considering that my country has not yet conducted statistics on the default situation of listed companies from the industry perspective, and lacks the credit rating of each listed company as a basis, this paper uses an orderly and multi-category method to measure the original default probability of the industry. The calculation formula is as follows: $In [\frac{p (y ⩽ j)}{1 - p (y ⩽ j)}] = μ_{j} - (α + \sum_{1}^{n} β Z_{i})$ (27)

The value of p (y ⩽ j) is obtained by estimation, and then the probability of default for each category is obtained from p (y = j) = p (y ⩽ j) - p (y ⩽ j - 1). According to the five-level classification of loans, commercial banks calculate the final default probability on the basis of the historical default rates of each level. This paper divides the status of listed companies into three categories: normal and ST, * ST. The historical default probabilities of the three categories in the corresponding Logistic model are assumed to be 0.01, 0.25, and 1. Therefore, the final industry default probability formula measured by the ordered multi-class logistic method is: $P = 0.01 * P_{1} + 0.025 * P_{2} + 1 * P_{3}$ (28)

Among them, P₁, P₂ and P₃ correspond to the probabilities of listed companies being in the normal, ST and *ST categories respectively. In the process of estimating the probability of default, 16 financial indicators of listed companies are selected for factor analysis. In order to reduce the impact of business cycles and contingency factors, a longer sample period is selected and the indicators are time-weighted, so that the estimated default probability is more accurate and closer to reality. After calculating the default probabilities of different listed companies in various industries, the default probabilities of listed companies in the same industry are calculated according to industry standards, and the original default probability values of each industry are obtained.

7 Regional effect analysis

In order to verify whether the regional asymmetric effect of monetary policy exists, and to reasonably explain its empirical results, the empirical model in this paper adopts the GVAR model. The stationarity of the sequence is the premise of establishing the GVAR model. However, most of the time series in the real economic field are not stable. Therefore, this paper conducts stationarity test on the original time series. This paper uses the ADF test method in the unit root test to test the stationarity of the time series. The test results are shown in Table 1. CPI and GDP statistics are shown in Figs. 1 and 2.

Table 1
ADF unit root test results of the original data

Area Variable ADF value P Confidence interval Stationarity

1% 5%

North-east area CPI –3.13 0.04 –3.87 –3.06 North–east area CPI

GDP –2.58 0.12 –3.87 –3.06 GDP

North coast CPI –5.36 0.00 –3.87 –3.06 North coast CPI

GDP –2.92 0.07 –3.87 –3.06 GDP

East coast CPI –4.59 0.00 –3.87 –3.06 East coast CPI

GDP –3.28 0.03 –3.87 –3.06 GDP

South coast CPI –3.08 0.05 –3.87 –3.06 South coast CPI

GDP –3.88 0.01 –3.87 –3.06 GDP

Middle Yellow River CPI –3.09 0.05 –3.87 –3.06 Middle Yellow River CPI

GDP –2.44 0.15 –3.87 –3.06 GDP

Middle reaches of the Yangtze River CPI –3.39 0.03 –3.87 –3.06 Middle reaches of the Yangtze River CPI

GDP –2.89 0.07 –3.87 –3.06 GDP

Southwest Region CPI –3.16 0.04 –3.87 –3.06 Southwest Region CPI

GDP –2.77 0.09 –3.87 –3.06 GDP

North-west region CPI –2.61 0.12 –3.93 –3.08 North–west region CPI

GDP –3.41 0.03 –3.87 –3.06 GDP

M2 –5.59 0.00 –3.87 –3.06 M2

Area	Variable	ADF value	P	Confidence interval	Stationarity
				1%	5%
North-east area	CPI	–3.13	0.04	–3.87	–3.06	North–east area	CPI
	GDP	–2.58	0.12	–3.87	–3.06		GDP
North coast	CPI	–5.36	0.00	–3.87	–3.06	North coast	CPI
	GDP	–2.92	0.07	–3.87	–3.06		GDP
East coast	CPI	–4.59	0.00	–3.87	–3.06	East coast	CPI
	GDP	–3.28	0.03	–3.87	–3.06		GDP
South coast	CPI	–3.08	0.05	–3.87	–3.06	South coast	CPI
	GDP	–3.88	0.01	–3.87	–3.06		GDP
Middle Yellow River	CPI	–3.09	0.05	–3.87	–3.06	Middle Yellow River	CPI
	GDP	–2.44	0.15	–3.87	–3.06		GDP
Middle reaches of the Yangtze River	CPI	–3.39	0.03	–3.87	–3.06	Middle reaches of the Yangtze River	CPI
	GDP	–2.89	0.07	–3.87	–3.06		GDP
Southwest Region	CPI	–3.16	0.04	–3.87	–3.06	Southwest Region	CPI
	GDP	–2.77	0.09	–3.87	–3.06		GDP
North-west region	CPI	–2.61	0.12	–3.93	–3.08	North–west region	CPI
	GDP	–3.41	0.03	–3.87	–3.06		GDP
	M2	–5.59	0.00	–3.87	–3.06		M2

Fig. 1

Statistical diagram of CPI.

Fig. 2

Statistical diagram of GDP.

From the ADF test results in Table 1, we find that the ADF values of the original series of GDP growth rates, CPI growth rates, and M2 growth rates in each economic region are somewhat greater than the critical value level of 5%. However, the ADF value of some series is less than the critical value level of 5%. This shows that the original time series of economic data have not all passed the ADF unit root test. By observing the P value, we found that some sequences have P values greater than 0.05, and some sequences have P values less than 0.05. This result shows that the original sequence cannot satisfy the same order single integer. Therefore, we believe that the original sequence is unstable, and the original data needs to be processed. Therefore, this paper conducts first-order difference on the selected time series of M2 growth rate, GDP growth rate, and CPI growth rate respectively, and performs ADF unit root test on the time series after the difference. The test results are shown in Table 2 below, and the DCPI and DGDP statistics are shown in Figs. 3 and 4.

Table 2

ADF unit root test results of the first-order difference sequence

Area	Variable	ADF value	P	Confidence interval			Stationarity
				1%	5%
North-east area	DCPI	–7.34	0.00	–3.90	–3.07	North-east area	DCPI
	DGDP	–6.05	0.00	–3.90	–3.07		DGDP
North coast	DCPI	–7.79	0.00	–3.90	–3.07	North coast	DCPI
	DGDP	–6.20	0.00	–3.90	–3.07		DGDP
East coast	DCPI	–6.28	0.00	–3.93	–3.08	East coast	DCPI
	DGDP	–4.84	0.00	–3.93	–3.08		DGDP
South coast	DCPI	–6.24	0.00	–3.90	–3.07	South coast	DCPI
	DGDP	–5.65	0.00	–3.93	–3.08		DGDP
Middle Yellow River	DCPI	–6.19	0.00	–3.90	–3.07	Middle Yellow River	DCPI
	DGDP	–5.68	0.00	–3.90	–3.07		DGDP
Middle reaches of the Yangtze River	DCPI	–6.44	0.00	–3.93	–3.08	Middle reaches of the Yangtze River	DCPI
	DGDP	–5.18	0.00	–4.00	–3.11		DGDP
Southwest Region	DCPI	–6.72	0.00	–3.90	–3.07	Southwest Region	DCPI
	DGDP	–4.75	0.00	–3.93	–3.08		DGDP
North-west region	DCPI	–5.07	0.00	–3.93	–3.08	North–west region	DCPI
	DGDP	–6.35	0.00	–3.90	–3.07		DGDP
	DM2	–8.58	0.00	–3.90	–3.07		DM2

Fig. 3

Statistical diagram of DCPI.

Fig. 4

Statistical diagram of DGDP.

From the test results in Table 2, among the results of the first-order difference series of economic variables in the eight major economic regions, the ADF value of the first-order difference series of each variable is less than the critical level of 5%, and the P value is much less than the significant level of 5%. From the analysis of the test results, the null hypothesis that the ADF unit root test has a unit root can be rejected. That is, the DGDP, DCPI, and DM2 of each economic region are all stable sequences. We can get the following conclusion: the first-order difference sequence of the growth rates of the three variables GDP, CPI, and M2 DGDP, DCPI, and DM2 obey the I (1) process.

In order to show the regional differences of China’s banking financial institutions more intuitively, according to the data listed in the table above for the basic situation of banking financial institutions’ business outlets and legal entities in each region, this paper compares the proportions of the basic conditions of banking financial institutions and legal person institutions in various regions of my country in 2019, as shown in Table 3 and Fig. 5 below.

Table 3

Proportion of the basic information of the business outlets and legal entities of banking financial institutions in each region

area	Percentage of institutions	Proportion of employees	Proportion of total assets	Proportion of legal entities
North-east area	9.57%	11.15%	8.29%	24.31%
North coast	15.88%	16.16%	25.33%	14.43%
East coast	13.75%	16.09%	24.83%	9.50%
South coast	11.64%	13.19%	18.87%	10.59%
Middle Yellow River	15.07%	14.17%	10.67%	15.93%
Middle reaches of the Yangtze River	15.08%	13.10%	11.67%	13.94%
Southwest Region	16.37%	14.58%	13.18%	23.12%
North-west region	5.32%	4.73%	4.20%	15.61%
Nationwide	100.00%	100.00%	100.00%	100.00%

Fig. 5

Statistical diagram of proportion of the basic information of the business outlets and legal entities of banking financial institutions in each region.

In addition to banking financial institutions, the financial system also has non-bank financial institutions. Non-banking financial institutions are also an important part of the financial market, mainly including securities companies, fund companies, futures companies, insurance companies, etc. The regional differences of non-banking financial institutions will also affect my country’s monetary policy transmission. The number of institutional developments of securities companies, fund companies, and futures companies in each region in 2019 is shown in Table 4 and Fig. 6.

Table 4

The development of the securities industry in each region

	securities company	fund company	Futures company	Listed company
North-east area	6	64	7	152
North coast	4	1	10	270
East coast	31	46	50	886
South coast	31	34	26	4909
Middle Yellow River	8	0	8	183
Middle reaches of the Yangtze River	9	0	9	310
Southwest Region	8	2	7	214
North-west region	6	1	4	113

Fig. 6

Statistical diagram of the development of the securities industry in each region.

From Table 5, we can see that there are obvious differences between regions in terms of the number of insurance companies established. It shows that my country still has a lot of room for improvement in insurance.

Table 5

The development of the insurance industry in each region

	Insurance company (number)	Headquartered in the provincial branch of the jurisdiction (number)	Premium income (100 million yuan)	Various types of compensation	Insurance density (yuan/person)	Insurance depth
North-east area	8	197	2383	795	2149	4
North coast	56	314	6227	2131	4082	5
East coast	67	4036	6064	2099	4340	4
South coast	34	184	5021	1545	2213	3
Middle Yellow River	3	6465	3492	1175	1856	4
Middle reaches of the Yangtze River	5	237	3457	1289	1433	3
Southwest Region	9	5163	3670	1314	1424	4
North-west region	1	100	978	346	1359	4

Table 6 and Fig. 8 show the development of regional capital markets in 2019.

Table 6

The development of capital markets in each region

	A-share financing (100 million yuan)	H-share financing (100 million yuan)	Domestic bond financing (100 million yuan)
North-east area	458.5	379.8	1991.7
North coast	1076.3	70.9	7513.9
East coast	4915.7	437.3	14267.3
South coast	3400.2	21.4	7330.4
Middle Yellow River	1159.9	0.0	5356.3
Middle reaches of the Yangtze River	1269.6	0.0	9190.0
Southwest Region	821.6	0.0	3489.2
North-west region	571.5	0.0	1236.2

Fig. 7

Statistical diagram of the development of the insurance industry in each region.

Fig. 8

Statistical diagram of the development of capital markets in each region.

We can see that the main financing method for companies in various economic regions in 2019 is domestic bond financing, and there are obvious differences in financing between regions. On the whole, the eastern coastal areas are at the regional leading level in both A-share financing and domestic bond financing, while the stock market financing in the northwest is at the last position in the region. By comparing the three financing forms, we can find that domestic bond financing is the main financing method in my country, followed by A-share financing. Although there is still a gap between A-share financing and bond financing, it is in continuous development. However, H-share financing is still in the initial stage of development in my country, and H-share financing in most regions has not yet developed.

8 Conclusion

This paper constructs a conduction model based on the industry GVAR model. On this basis, this paper carries out an industry-level macro stress test, and analyzes the dynamic impact of industry relevance and various macro factors on the industry’s default probability through impulse response function.

The regional effect of monetary policy is an extremely complicated process, which is affected by various factors. Based on the actual differences between regional economy and regional finance, this paper finds that the economic development level and financial development level of my country’s eight economic regions are not completely positively correlated. On the whole, the coastal areas are relatively developed in terms of the level of economic development and financial deepening, and the economic development level of the northwestern region is at the lowest level in the region. However, with the continuous advancement and development of the western development policy, the degree of financial deepening in the northwest region has continued to deepen, and it can even catch up with the central economic zone. Therefore, we can see that different factors have different effects on monetary policy, and the regional effect of monetary policy is actually the result of the combined influence of these factors.

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