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## 统计代写|金融统计代写Financial Statistics代考|Theil Inequality for Bank Competition, Concentration, and Credit Risk

An inequality index that belongs to the entropy class has the following general form:
$$G E(\alpha)=\frac{1}{n\left(a^{2}-a\right)} \sum_{i=1}^{n}\left[\left(\frac{y_{i}}{\bar{y}}\right)^{a}-1\right]$$
where $n$ is the size of the sample, $y_{i}$ is the $i$-th observation and $\bar{y}$ is the mean value of the sample. The parameter $\alpha$ represents the weight given to distances between values at different parts of the distribution. For smaller values of $\alpha, G E(\alpha)$ is more sensitive to changes in the bottom tail of the distribution. For higher values of $\alpha, G E(\alpha)$ is more sensitive to changes in the upper tail of the distribution. The most commonly used values of $\alpha$ are $-1,0,1$ and 2 . The most well-known member of the entropy class of inequality indices is the $G E(1)$ index, called Theil index $(T)$, by the name of Henri Theil who introduced it in 1967. All the members of the entropy class of inequality indices have the advantage of being perfectly decomposable (i.e., with a zero residual part).

Following Fernandez de Guevara et al. (2007), the Theil index for bank market power can be calculated from Equation (11):
$$T=\sum_{g=1}^{\mathrm{G}} s h_{g} T_{w, g}+T_{b}$$
where $T$ is the total market power inequality, $G$ is the number of observation groups in the sample, shg is the share in the sample (in terms of total assets) of a group $g, T_{w, g}$ is the within-group inequality in group $g$ and $T_{b}$ is the between-group inequality.
The within-group inequality in group $g$ is defined by Equation (12):
$$T_{w, g}=-\sum_{l=1}^{N_{g}}\left(\frac{s h_{i}}{s h_{g}}\right) \ln \left(\frac{x_{1}}{\mu_{g}}\right)$$
where $N_{g}$ is the numher of entities (e.g., countries or hanks) in group $g$, sh is the share in the sample (in terms of total assets) of an entity $i$ belonging to group $g$ and $\mu_{g}$ is the weighted average of the Lerner index of entities belonging to group $g$.

## 统计代写|金融统计代写Financial Statistics代考|Theil Inequality for Concentration

The results from the calculation of the total Theil inequality index for HHI and its within-group and between-group components are presented in Table 5 , as well as in Figures 5 and 6 . As it is shown in Table 5, as well as in Figures 5 and 6, the disparities in HHI values are due, almost exclusively, to differences inside each group (EA-Co or EA-Pe). This evolution suggests that there is a clear convergence between the EA-Co and EA-Pe country groups with respect to concentration, as measured by the HHI structural measure. In contrast to the case of the Lerner index, the lack of availability of the bank-level underlying determinants of HHI did not let us to go deeper in order to investigate whether the above-described differences stem from inequalities between different countries (between-country inequality) or from inequalities between banks in a given country (within-country inequality).

The results from the calculation of the total Theil inequality index for CR5 and its within-group and between-group components are presented in Table 5 , as well as in Figures 5 and 6 . As it is shown in Table 5 , as well as in Figures 5 and 6, the disparities in CR5 values are due, almost exclusively, to differences inside each group (EA-Co or EA-Pe). This evolution suggests that there is a clear convergence between the EA-Co and EA-Pe country groups with respect to concentration, as measured hy the CR5 structural measure. The lack of availahility of the hank-level underlying determinants of CR5 did not lel us to go deeper in order to invesligate whether these differences slem from inequalilies between different countries (between-country inequality) or from inequalities between banks in a given country (within-country inequality).

## 统计代写|金融统计代写Financial Statistics代考|Convergence Analysis

The evolution of the coefficient of variation, presented in Table 1 and in Figure 2, provides useful information about changes of inequality in Lerner index values across countries and/or years. However, the visual inspection of these changes cannot always provide safe results regarding the existence of convergence or divergence in terms of the Lerner index across euro area countries during a given period, especially when the Lerner index has presented significant ups and downs along this period. The same situation holds in the case of the two concentration measures (HHI/CR5) and the NPL ratio.
To overcome these problems, we employ two concepts of convergence, namely $\beta$-convergence and $\sigma$-convergence (Barro and Sala-i-Martin 1991), which have been prevailing for many years in the growth literature. $\beta$-convergence applies if poor countries tend to catch up with rich ones in terms of per capita income or product levels. In the case of competition, $\beta$-convergence would apply if countries with lower levels of competition were found to tend to catch up with countries with higher levels of competition. On the other hand, o-convergence applies if the dispersion of per capita income or product across countries declines over time. The existence of $\beta$-convergence is a necessary but not sufficient condition for $\sigma$-convergence. Regarding competition, $\sigma$-convergence would apply if the dispersion of competition levels across countries showed a tendency to decline over time.

In the case of competition, the $\beta$-convergence test is performed through the estimation of Equation (14).
$$\ln \left(\frac{C_{i t}}{C_{i, t-1}}\right)=\alpha+\beta \ln C_{i, t-1}+\sum \text { Country }{i}+\varepsilon{i t}$$
where $C_{i t}$ is the level of competition, as expressed by the (inverse of) the Lerner index, in country $i$ in year $t, \alpha$ and $\beta$ are parameters to be estimated, Country $i$ are dummy variables to control for possible country effects, and $\varepsilon_{i t}$ is a random error term. There is $\beta$-convergence when the coefficient $\beta$ in (14) is statistically significant and negative. A higher absolute value of the coefficient $\beta$ corresponds to a greater tendency towards $\beta$-convergence.

Following Lapteacru (2018), the o-convergence test is performed through the estimation of Equation (15):
$$D_{i t}=\alpha+\sigma T+\sum \text { Country } y_{i}+\varepsilon_{i t}$$
where $D_{\text {it }}$ is the absolute value of the difference between the competition in country $i$ in year $t$ and the average competition in year $t, T$ is a time trend, $\alpha$ and $\sigma$ are parameters to bestimated, Country $i$ are dummy variables to control for possible country effects, and $\varepsilon_{i t}$ is a random error term. There is $\sigma$-convergence when the coefficient $\sigma$ in (15) is statistically significant and negative. A higher absolute value of the coefficient $\sigma$ corresponds to a greater tendency towards $\sigma$-convergence.

## 统计代写|金融统计代写Financial Statistics代考|Theil Inequality for Bank Competition, Concentration, and Credit Risk

$$G E(\alpha)=\frac{1}{n\left(a^{2}-a\right)} \sum_{i=1}^{n}\left[\left(\frac{y_{i}}{\bar{y}}\right)^{a}-1\right]$$

$$T=\sum_{g=1}^{\mathrm{G}} s h_{g} T_{w, g}+T_{b}$$

$$T_{w, g}=-\sum_{l=1}^{N_{g}}\left(\frac{s h_{i}}{s h_{g}}\right) \ln \left(\frac{x_{1}}{\mu_{g}}\right)$$

## 统计代写|金融统计代写Financial Statistics代考|Convergence Analysis

$$\ln \left(\frac{C_{i t}}{C_{i, t-1}}\right)=\alpha+\beta \ln C_{i, t-1}+\sum \text { Country } i+\text { sit }$$

$$D_{i t}=\alpha+\sigma T+\sum \text { Country } y_{i}+\varepsilon_{i t}$$

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