3.1. Standard quantile regressions

In the linear-regression model (LRM) of the form yi=xi'β+ui for i = 1, …, n, β is a K×1 vector of coefficients, x _i is a column vector of regressors and y _i is the i-th observation of the explained variable, whereas u _i is the i.i.d. error term. It is widely known that the ordinary-least-squares (OLS) estimator is obtained by selecting the vector β that minimizes the sum of squares residuals minβ∈RKyi- x'β2. Although LRM is highly attractive when all its assumptions are duly met, it has several disadvantages (^{Hao and Naiman, 2007}) from a quantile-regression perspective.

First, it usually focuses on modeling the conditional mean of a dependent variable and leaves out its full conditional distributional properties. In fact, it attempts “to describe how the location of the conditional distribution behaves by utilizing the mean of a distribution to represent its central tendency” (^{Hao and Naiman, 2007, p. 24}). Second, another disadvantage arises when the homoscedasticity assumption fails and then the modeler could introduce a simultaneous modeling of the conditional mean and the conditional scale, for instance y _i = β₀ + β₁ x ₁ + є^γє _i , where γ is an additional unknown parameter and var(y|z) = σ²є^γ. Unfortunately, the estimation of the conditional scale is not usually available in most statistical software. Another shortcoming relates to the normality assumption, which is frequently violated because data in real life do not mostly have a normal distribution, and if the assumptions were not met, then the calculated p-values would tend to be biased. Lastly, when there are outliers econometricians are inclined to eliminate them because they modify substantially fitted regression lines. As an example, during the 1980s in Mexico there were various years with hyperinflation well and above 100%; in 1986 the annual inflation reached 106% and 159% in 1987. Therefore, under OLS the common remedy is to remove outliers from the series, which for the present research would seriously undermine our understanding of how two variables determine each other at different quantiles of a distribution.

In this paper, we use quantile regressions to gauge the effects of macroeconomic volatility mainly arising from inflation rates at different quantiles of the distribution. Quantile regression estimates a conditional quantile function where quantiles of the conditional distribution of the dependent variable are a function of the observed independent variables (^{Koenker and Bassett, 1978}; ^{Koenker and Hallock, 2001}). The regression estimator of the τth quantile of y _i , when it is derived from a sample of observations, minimizes the following function with linear programming methods:

[1]

As explained by ^{Isengildina-Massa, Irwin, and Good (2010)}, the quantile-regression model (QRM) is semi-parametric because it does not assume a specific distribution for the random portion of the model, even though it is supposed for its deterministic portion. In addition, we can assume that the τ^th quantile of the conditional distribution of (y _i |z _i ) is linear as in:

[2]

Where Qτ=yi|xi=xi'βτ. The term xi' is a K×1 vector of variables and β is a vector of known parameters, and єi=yi-xi'βτ. Hence regression quantiles can be obtained by changing the parameter τ on the interval (0,1). With respect to results, the interpretation of the coefficient of the kth variable, β _K (τ), corresponds to the partial derivative ∂Q _yi (τ|X = x)/∂x _j ; in other words, the marginal change in the dependent variable with respect to the marginal change in the independent variable that belongs to the τth quantile.

3.2. Quantile regression for panel data: A two-step estimator

In this subsection we adhere closely to ^{Canay (2011)}’s proposition of a simple two-step estimator for fixed-effects quantile regression for panel data. In the model:

[3]

(Y _it ,X _it )∈R×R ^k are known variables and (U _it ,α _i )∈R×R are unknown. Variable X _it is supposed to include a constant term. The quantile function τ→X′θ(τ) is strickly increasing in τ∈(0,1) and the parameter to be estimated is θ(τ).

With some manipulation of Equation [3], it can be shown that θ(τ) is identified with at least two time periods available and α _i has a pure location shift effect (^{Canay, 2011, p. 370}). Moreover, with a transformation of data, it is possible to eliminate the fixed effects of α _i as T→∞, since α _i is a location shift.

The two-step estimator is computed as follows:

Which is consistent and asymptotically normal and where ET≡ T-1∑t=1T⋅ and EnT(⋅)≡nT-1∑t=1T∑i=nn⋅. Also, it is supposed that Y^it∼Yit-αi as T→∞, where ∼ means weak convergence.

Our main econometric analysis is based on the method described by ^{Andini and Andini (2014, p. 7)}. According to the authors, if we discard country fixed effects, the two-stage approach is not useful. Instead, we make use of the standard estimator (i.e., QR) proposed by ^{Koenker and Bassett (1978)} with the assumption that ϕ _i = γ∀i. However, for the FEQR technique the authors recommend running OLS estimation on model δY _it = ϕδX _it + δu _it to obtain a consistent estimate ϕ^ for parameter ϕ and define γ^i=1T∑TYit-ϕ^Xit. The second step consists in constructing a new variable Yit*=Yit-γi and then obtaining the standard QR estimates from model Yit*=ϕθXit-uit that corresponds to Equation [5] with the estimated fixed effects on the left-hand side.

3.3. Empirical model

As mentioned before, in this paper we analyze standard quantile and fixed-effects quantile regressions in order to find solid evidence of the effects of inflation on the performance of financial intermediaries in 84 developed and developing countries. We use the following empirical model:

[5]

Where FINANCE _it is an indicator of financial development as a percentage of GDP, such as private sector bank credit (pvt), liquid liabilities (liq) or deposit money bank assets (ba); INF _j is the yearly average inflation rate; CONTROL _it comprises macroeconomic variables like income per capita (y), government expenditures as a share of GDP (gov), the index of human capital (edu) and events of political instability including revolutions (rev); lastly, γ _i absorbs country-specific effects. For our purposes, we expect a negative sign for the variable INF _j , as specified by our main hypothesis and the preliminary evidence shown in Section 2.

Next, we perform hypotheses testing of slope equality across quantiles, that consists in assessing if the inter-quantile difference is statistically significant or if it is due only to sample size (^{Davino, Furno, and Vistocco, 2014, p. 81}). The statistical significance of the inter-quantile difference for two quantiles θ₁ y θ₂ is given by:

[6]

Where Y(θ₁) y Y(θ₂) are the values of the independent variables at different quantiles; the coefficient γ measures the difference of the intercepts throughout quantiles; and δ calculates the difference among the intercepts of the two selected quantiles. In general, when we reject the null hypothesis, the difference among the estimated slopes is statistically different from zero, the quantiles have slopes statistically different and the data are not i.i.d. (^{Davino, Furno, and Vistocco, 2014, p. 82}).

4.1. Ordinary least squares

Table 2(A) displays the results of the OLS estimation of the effects of inflation on financial variables¹¹. Given that we rejected the null hypothesis of constant variance with the Breusch-Pagan/Cook-Weisberg test for heteroskedasticity¹², we obtained the White’s heteroskedasticity-corrected t-values and the corresponding p-values. In the specifications where pvt, liq, and ba are dependent variables, inflation rates exert a negative influence significant at more than 95%. All control variables have the expected sign, although gov and rev are usually insignificant. To understand the economic importance of the estimates, a basic experiment shows that an increase in inflation rates of 8.3% (median value) can cause a fall of 3.4% in private credit, 4.9% in liquid liabilities and 4.5% in bank assets¹³.

Table 2: OLS, POLS and FE estimation of the effects of inflation on financial intermediary performance 

Note: For the OLS estimation, period averages were used (1980-2010). Variable y is initial income per capita. Standard errors are White’s heteroske-dasticity-corrected. POLS is pooled OLS estimation, whereas fe is fixed-effects estimation. In this case, robust standard errors are provided.

Source: Authors’ calculation.

In Table 2(B) we report the POLS estimators. The variable of interest presents clearly negative and statistically significant effects on private credit, liquid liabilities and bank assets. Nonetheless, with respect to the OLS estimators the coefficients of inflation are smaller. In general, the control variables also maintain the same signs as in the previous estimation. The control variables presenting adequate and statistically significant coefficients are still income per capita and education. Interestingly, rev became significant in private credit, liquid liabilities and bank assets, suggesting that an increase in political instability tends to reduce the performance of financial intermediaries in the long run. Thirdly, Table 2(C) exhibits the fixed effects estimators. In comparison to the other two methods, inflation remains negative but statistically insignificant suggesting no influence on financial development¹⁴. Although income per capita and education are positive and highly significant in most cases, gov and rev switched signs, generally contradicting theoretical and empirical findings.

Finally, for POLS and fixed effects we report the coefficients of inflation by country group. With regard to the first method, it is observed that inflation levels are consistently insignificant for all the financial variables, although they keep the expected negative sign. On the other hand, price increases are statistically significant at the 1% level for the first three financial variables in developing countries. By the same token, a similar pattern is seen when we disaggregate the data and apply the fixed effects method. For developed countries, there is no empirical evidence of the influence of inflation using any of the three financial indicators. Contrarily, the evidence is statistically strong for less-developed economies.

In summary, by utilizing a variety of linear regression models we have initially proved the empirical existence of the inflation-finance nexus mainly in developing countries. Preliminarily, we can infer that the statistically significant relationship between inflation and finance observed in the full sample of countries is due to the behavior of developing countries, given that the relationship is insignificant for developed countries.

4.2. Standard quantile regressions

Until now we have discussed the effects of inflation rates on financial variables based on ordinary least squares, pooled OLS and fixed effects estimators. But the main purpose of this paper is to assess the inflation-finance link with quantile regressions to verify how the negative relationship varies along the conditional finance distribution. Henceforth we only report inflation coefficients.

Table 3 shows the estimation results of the three financial variables using the whole sample of countries. In the case of private credit, it is observed that the effects of inflation are statistically significant in the lower and upper part of the distribution (it is significant in the 70th quantile at the 5% level). The results point out that an increase of 1% in inflation leads to a fall between 1 and 2.3% in private credit depending on the quantile. Second, in the equation for liq inflation is significant up to the 50th quantile at the 1% level, suggesting that the adverse effects of inflation rates are concentrated in the lower parts of the conditional liquid liabilities distribution. The value of coefficients goes from -1.1 to -3.9%. Third, the results regarding bank assets are similar to private credit, in the sense that the statistically significant negative effects of inflation are concentrated in the lower parts of the distribution and in the 70th quantile.

Table 3: Effects of inflation on the conditional finance distribution without controlling for country fixed effects: All countries 

Note: p-values are in parentheses. Due to lack of data, Chile was excluded from liq and ba, and Germany and Venezuela from ba. *, ** and *** indicate statistical significance at 1, 5, and 10%, respectively. Bootstrapped standard errors.

Source: Authors’ calculation.

As shown in Table 4, the estimation for developed countries produces bigger coefficients. Apparently, inflation is very damaging to financial sector performance in developed countries, and possibly it is the case because authorities in developed countries have been able to control inflation for a long period of time, and a sudden or unexpected increase in inflation could provoke a sharp deterioration in financial variables. Another explanation is that the activity of financial intermediaries is strongly related to output, thus permeating the behavior of economic agents. Depending on the level of inflation, private sector bank credit would fall between 66.3 and 91%; liquid liabilities between 56.7 and 116%; bank assets between 46 and 116%. We also observe that the effects of inflation are grouped around the lower parts of the distribution, precisely because inflation in high-income countries has been kept controlled for a long period of time.

Table 4: Effects of inflation on the conditional finance distribution without controlling for country fixed effects: Developed countries 

Note: p-values are in parentheses. Due to lack of data, Germany was excluded from ba. * and *** indicate statistical significance at 1 and 10%, respectively. Bootstrapped standard errors.

Source: Authors’ calculation.

The story for developing countries is rather different, as presented in Table 5. Coefficient estimates are more heterogeneous with respect to the preceding results. In relation to private credit, inflation effects concentrate around the middle part of the conditional distribution, and for liquid liabilities, in the first half of the distribution. For the case of bank assets, they were significant in the 40th and 50th quantiles. An increase of 1% in inflation would provoke a drop between 0.6 and 0.7% in private credit. The same small coefficients are obtained for liquid liabilities with values ranging from -0.6 to -1.8%. Thus, these results indicate evidence on the relationship between the level of inflation and finance in developing countries. Considering that inflation rates have been higher in those countries, the negative effects oscillate around the middle part of the distribution.

Table 5: Effects of inflation on the conditional finance distribution without controlling for country fixed effects: Developing countries 

Note: p-values are in parentheses. Due to lack of data, Chile was excluded from liq and ba, and Venezuela from ba. *, ** and *** indicate statistical significance at 1, 5, and 10%, respectively. Bootstrapped standard errors.

Source: Authors’ calculation.

In a few words, our QR estimations confirm some theoretical predictions. The estimations suggest a negative, nonlinear relationship between finance and inflation, which is stronger in the lower and middle parts of the conditional finance distribution. However, we have to remove country fixed effects to show precisely where the significant effects lie on the conditional distribution. We should expect that at higher inflation rates the adverse impact on finance should be stronger.

4.3. Fixed-effects quantile regressions

In this subsection we discuss the results of the estimation by controlling for country fixed effects. Overall, we found fewer significant quantiles than before, they are less heterogeneous, and the few significant ones were pushed up along the conditional finance distribution, i.e., they moved up to the upper tail of the distribution. ^{Andini and Andini (2014, p. 8)} offer a plausible explanation for this comportment: Certain effects originating in other quantiles of the conditional finance distribution produce the dominant effect of inflation over finance in the lower tail of the distribution. Another central outcome is that, based on our preferred QR method, we did not ascertain strong evidence on the inflation-finance relationship in developed economies, as we will discuss below.

Table 6 exhibits results considering the full sample of countries. Inflation rates impact private sector credit on the upper tail of the distribution at least at the 5% level of statistical significance. For liquid liabilities the effects are meaningful from the 30th quantile onwards and for bank assets since the 60th quantile. Therefore, after eliminating country fixed effects the results look more precise in the sense that at the higher quantiles of the conditional finance distribution inflation exerts stronger negative effects.

Table 6: Effects of inflation on the conditional finance distribution, controlling for country fixed effects: All countries 

Note: p-values are in parentheses. Due to lack of data, Chile was excluded from liq and ba, and Germany and Venezuela from ba. * and ** indicate statistical significance at 1 and 5%, respectively. Bootstrapped standard errors.

Source: Authors’ calculation.

In regard to the subsamples of countries, there are sharp differences in comparison to the QR estimates -or even the fe estimates, as revealed by Tables 7 and 8‒. For developed economies, inflation appears to affect only pvt in the 95th quantile at the 10% significance level (statistically weakly) and liq in the 20th quantile at the 10% significance level. In fact, these results are more consistent with the low levels of inflation rates experienced by developed countries in the past 30 years. That is one reason why one would expect that inflation rates had a minor effect on advanced financial intermediaries. By comparison and consistent with the results shown in Table 2, the impact of inflation in less-developed economies varies widely because we observe more statistically significant quantiles in the regressions. For private credit, inflation rates are highly significant after the 80th quantile; for liquid liabilities, most quantiles have at least 5% significance level after the 20th quantile; and for bank assets, the acceptable quantiles are only the 30th, 50th and 70th. Overall, we can argue that the higher the inflation rates, the lower the level of financial sector performance and those effects are more meaningful in developing economies. It should be noted as well that the estimations for the full sample of countries are mainly driven by the impact of inflation in developing countries, given that the influence in developed countries is rather null.

Table 7: Effects of inflation on the conditional finance distribution, controlling for country fixed effects: Developed countries 

Note: p-values are in parentheses. Due to lack of data, Germany was excluded from ba. *** indicate statistical significance at 10%. Bootstrapped standard errors.

Source: Authors’ calculation.

Table 8: Effects of inflation on the conditional finance distribution, controlling for country fixed effects: Developing countries 

Note: p-values are in parentheses. Due to lack of data, Chile was excluded from liq and ba, and Venezuela from ba. *, ** and *** indicate statistical significance at 1, 5 and 10%, respectively. Bootstrapped standard errors.

Source: Authors’ calculation.

4.4. Inter-quantile tests

Table 8 reports the results of the inter-quantile tests as proposed by ^{Koenker and Bassett (1978)}. We test the null hypothesis that β_0.5 ≠ β_0.10 ≠ β_0.20 ... ≠ β_0.95, which corresponds to the estimated quantiles in the standard and fixed effects quantile regressions. When we reject the null hypotheses at a minimum of 5% significance level, then the coefficients of the estimated quantiles are valid. From Table 9 we can infer that for private credit all the quantiles are different from each other in the QR and FEQR regressions. For liquid liabilities, the standard quantile regressions are invalid in the case of the full sample and developed countries. And with respect to bank assets, the results are mixed. Under the FEQR technique, the estimations for developed countries are invalid.

Table 9: Test of slope equality (Q05-Q95) 

Note: F test. p-values are in parentheses. *, ** and *** indicate statistical significance at 1, 5 and 10%, respectively.

Source: Authors’ calculation.

4.5. Additional results including legal institutions

So far we have explored the influence of macroeconomic variables on the performance of the financial sector. However, it is worth analyzing the inclusion of institutional development in our empirical model. In fact, many studies prove that legal protection of rights and the enforcement of contracts enhance financial development in the long run (see, for instance, ^{La Porta et al., 1998}; ^{Billmeier and Mass, 2009}). ^{Miletkov and Wintoki (2012, p. 651)} explain that risks and uncertainty are lowered when countries have strong legal institutions, and that their societies begin to receive the benefits of property right protection when its maintenance costs have been surpassed.

As in ^{Miletkov and Wintoki (2012)} and the references therein, we use the popular Legal Structure and Security of Property Rights Index from the Economic Freedom of the World. This index ranges from 0 to 10, where 0 is the lowest value and 10 is the highest. The index is an average of the following components: Judicial independence, Impartial courts, Protection of property rights, Military interference in rule of law and politics, Integrity of the legal system, Legal enforcement of contracts, Regulatory costs of the sale of real property, Reliability of police and Business costs of crime. Given that the index does not cover all countries, our sample was reduced to 46 developing countries. Since the available years for the latest index are 1980, 1985, 1990, 1995 and 2000-2014, we used averages for the missing years assuming that institutions do not change significantly in the short run. In order to save space we only report the results for our main indicator of financial development (i.e., bank credit to the private sector) and the fixed-effects quantile regressions¹⁵.

Table 10 presents the estimation results adding the index of legal institutions. In general, we obtain the same results as in the previous subsection: There is a persistent negative effect of inflation on financial sector development even after controlling for legal institutions. For the whole sample of countries and developing countries, the effects are statistically significant in the upper quantiles. In the case of developed countries, we can see that the effects of price increases are statistically insignificant in all quantiles. The lack of evidence for developed countries is confirmed as well by the test of slope inequality, as shown at the bottom of Table 10.

Table 10: Effects of inflation on the conditional finance distribution, controlling for country fixed effects and legal institutions 

Note: p-values are in parentheses. Due to lack of data on legal institutions, the following developing countries were excluded from the sample: Belize, Fiji, Gambia, Lesotho, Nepal, Rwanda, Sudan and Swaziland. * indicates statistical significance at 1%. Bootstrapped standard errors are reported.

Source: Authors’ calculation.

4.5. Discussion

In a stable macroeconomic environment accompanied by adequate supervision and regulation, financial intermediaries can be powerful enough to enhance economic output in the long run and, as a corollary, improve the standard of living of people. With the estimations shown in this section, we have found evidence of an inverse and nonlinear relationship between finance and inflation in 84 developed and developing countries. In all of the statistically significant QR and FEQR estimations, the inflation was persistently negative. Based on our preferred FEQR estimations and with some caution about the causality issue, we observe that inflation rates reduce the level of financial development in the upper quantiles of the distribution, principally in developing countries where the effect is stronger. We did not obtain robust empirical evidence for developed economies.

With respect to lack of evidence in developed economies, one explanation is that in those countries inflation rates have been kept controlled for a long period of time and they have also enjoyed of a relatively stable macroeconomic atmosphere. On the other hand, these results agree with the literature that has revealed weak evidence on the finance-growth and finance-inflation link in high-income countries. ^{De Gregorio and Guidotti (1995)} add bank credit to the private sector in a sample of 98 countries for the 1960-1985 period. Even if they discover a positive effect of the proxy of financial development on growth, it is stronger for developing countries only. ^{Huang and Li (2009)} study the finance-growth association in a sample of 71 high- and low-income countries with cross-section IV threshold regressions. Although they determine that the association is nonlinear, the impact is more prominent for low-income countries. With respect to inflation, ^{Beck, Lundberg, and Majnoni (2006)} inspect a panel of 63 countries during the period 1960-1997 with which they found solid evidence of a positive interaction between finance and inflation volatility in low- and middle-income countries. On the contrary, their econometric analysis did not provide any robust effect in countries with well-developed financial markets. The logic behind such results is that in developed countries most intermediary development happens outside the banking system, which diminishes the efficacy of monetary policy and thereby the effects of shocks of consumer prices.