A System of Simultaneous Equations (SEM) for the Study of the Effectiveness of the Japanese Monetary Policy

In this paper, the authors study the effectiveness of the Japanese monetary policy set by the Bank of Japan (BOJ) to contrast the three major crises that the country has experienced since the second half of the 90s: that of the lost decade, that of 2008 and that of the Covid-19 pandemic. To this end, they use a particular type of mathematical-statistical model that is widely applied today in the economic field, namely a simultaneous equation model (SEM). This simultaneous equation model is estimated through an Iteratively reweighted least squares (IRLS) using quarterly historical series in the sample period Q1 1994 - Q2 2020. All data are in real terms. The results, appropriately compared with those of other authors, suggest that the monetary policy has a (limited) impact only on the interbank market. The fiscal policy, instead, has a greater ability to influence the money supply, the private consumption and the inflation expectations.


Introduction
The debate regarding the effectiveness of the Japanese monetary policy has developed since the end of the 1990s, together with that relating to the causes of the structural change that the national economy was experiencing in those years.
In fact, precisely between the end of the 90s and the early 2000s, the very long period of GDP growth that began after World War II finished and started a new

Research Aims
The aim of this paper is to offer a generalized analysis of the effectiveness of the More precisely, the SEMs represent a multivariate statistical analysis technique that allows to examine the linear relationships between one or more independent variables and one or more dependent variables, which may be, in either case, measurable (namely directly observable) or latent (not directly observable and, therefore, indirectly measured through two or more detectable indicators or proxies). This simultaneous equations model (SEM) is estimated through an iteratively reweighted least squares (IRLS), using quarterly historical series in the sample period Q1 1994 -Q2 2020 and at constant prices. Among the independent variables of the model, the government expenditure and the government debt were added, in order to control for the fiscal policy.
The main innovation of this manuscript consists in demonstrating that the monetary policy has no effect on the GDP, on the consumption, on the investments and on the inflation expectations, instead it has a limited impact on the interbank call money market rate.
Another important result is that the fiscal policy is able to influence the consumption, the money supply, the inflation expectations (even if slightly) and the interbank call money market rate.
In other words, the importance of this contribution lies in the fact that it addresses a controversial topic, widely discussed and which has aroused considerable interest in the academic community for a long time. The purpose of the authors, in fact, is to conduct a rigorous and impartial normative analysis of the Japanese economic policy, which stimulates the debate on the reasons of the decline of the Japanese economy.
The paper is organized as follows. In Section 1, an introduction relating to the subject of the manuscript is placed. In Section 2 the aims of the research are explained and the main results are summarized. In Section 3 the theoretical SEM model is presented and, after having appropriately described the data relating to each variable included in it, the coefficients of the system estimated through the IRLS are exposed. In Section 4 the results of the estimation of the previous paragraph are discussed, while in Section 5, finally, the conclusions are developed, remarking the innovative component of the article. The last part contains a detailed bibliography and an appendix divided into two sections.

Model and Data
As is well known, the constitutive unit of structural equation models is the structural equation, namely a regression equation that expresses the causal relationship existing between a dependent variable and several independent variables, while the causal links between the variables are formalized, as a whole, through a system of equations The coefficients of such system may be calculated using the iteratively reweighted least squares (IRLS) estimator, a particular type of weighted least squares (WLS).
Compared to the traditional Least squares, which provides the solutions of a function defined in the space 2 l , the IRLS gives the solutions of a function defined in the space p l with 1 2 p ≤ ≤ and it is robust, as the weights are calculated as a function of leverage and residuals [12].
A brief description of the IRLS algorithm is presented by Li [13] (Cfr. Section I of the Appendix).
The dynamic system of simultaneous equations written by the authors is the following: 6,1 6,2 3 6,3 6,4 6,5 6,6 6,7 1 i is the Basic discount and loan rate of the BOJ; t r is the rate of return of 10-year Japanese government bonds; t s is the Japanese interbank call money market rate; t n is the immediate rate of the BOJ (namely the interest rate applied on loans with a maturity of less than twenty-four hours); t I are the investments; t C is the private consumption; e t π is the expected inflation at time t for the period 1 t + ; t π is the actual inflation; The (1) and (2) are, respectively, the dynamic money multiplier and the mon- The two variables government expenditure and government debt have been inserted into each of the six equations in order to control for the fiscal policy.
Note in particular that these variables in (3) and (4) capture the crowding out effect on the private demand, an effect discussed in the Japanese case by recent contributions in the literature [14].
According to previous studies [15], the crowding out, during the 90s, did not have any effect on the interest rates in Japan; instead it implied a loss in the physical capital. This thesis is tested by Equation (6)  In the Japanese case, Kamada and Nakajima [16] construct a particular measure of the expected inflation based on Purchasing power parity (PPP) and find that, in the long run, it coincides with the break even inflation. This is precisely The government final consumption expenditure is considered for the government expenditure.
Regarding the Basic discount and loan rate, its value in each quarter is calculated as the average of the four corresponding monthly rates.
The model was identified using the method of instrumental variables. More precisely, the following thirteen exogenous variables were used as instruments: It is easy to see that for the i-th equation of the system is verified: where K is the total number of exogenous variables, k is the number of exogenous variables in the i-th equation and v is the number of endogenous variables present in the i-th equation.
In other words, each equation of the system is correctly identified.
After presenting the SEM model and after explaining how its variables were measured and how it was identified, the estimation of the coefficients through the IRLS was carried out.
The output is shown below in Table 1.

Discussion of the Estimated Results
In (1), the estimated coefficients 1,1 α and 1,3 α are statistically insignificant and this implies that in the period considered the dynamic money multiplier is zero, namely the real GDP is insensitive to changes in the monetary aggregate 3 M . It is interesting to note that in the same equation 1,4 α and 1,5 α are also null, so the dynamic fiscal multiplier in the same period is also zero.
In (2), 2,1 α and 2,2 α are statistically not significant, demonstrating the fact that an increase in the real GDP does not involve an increase in the money supply.
The interpretation of this result is that the monetary expansion programs undertaken by the BOJ, such as the Quantitative easing of 2001 and the Quantitative and qualitative easing of 2013, were, on the whole, useless. According to some economists, the lack of transmission of the monetary impulses to the economy is mainly attributable to the presence of "zombie banks", namely those banks that record a large amount of Non-performing loans (NPLs) in their balance sheets.
However, this explanation cannot be considered exhaustive. Indeed, as observed by Gandrud and Hallerberg [17], many efforts have been made by Japanese authorities since the second half of the 1990s to address this problem and to recapitalize distressed banks. As a result, the Japanese non-performing loan ratio fell Instead, there is strong evidence ( 2,6 0 α > ) that an increase in the government debt at the current time of 1% causes the money supply to contract strongly (−4.47%). This outcome is not surprising. In fact, Japanese households have a high marginal propensity to save (that according to the authors is equal to α is less significant and so it may be neglected. In (3), the real GDP has a considerable effect on the private consumption ( 3,1 0 α > ). The coefficient 3,5 α is very significant, but its dimension is very small and therefore may be ignored. The primary liquidity ( 1 t M e In (4), the values of the estimated coefficients suggest that private investments are rigid with respect to all dependent variables. In fact, although they are respectively significant at 1%, at 5% and at 10%, 4,1 α , 4,11 α and 4,13 α are close to zero. The relevant result is that a change in the interest rates has no impact on the investments. The thesis supported by Barseghyan, which has been previously mentioned, is not verified.
In (5), the coefficient 5,1 α is very significant, negative and close to zero; instead the coefficient 5,2 α is small and not very significant. All other coefficients are either not significant or very close to zero and, therefore, negligible. This proves that Japanese operators formulate mainly adaptive expectations, that the forecast error they make is limited, that they expect a mild deflation and that interest rates (including those set by the BOJ) and the open markets operations are irrelevant. This evidence is consistent with the fact that, since the beginning of the crisis of the Lost Decade, phases of low inflation and negative inflation alternate in Japan. The coefficient 5,6 α (positive and significant at 1% level) reveals that an increase in the 10-years government bonds real interest rate has a small positive effect on the expected inflation. In some previous contributions, a relationship consistent with the one just described has been found: a decrease in the corporate tax rate and in the deficit-financed consumption tax corresponds to an increase in the real GDP growth rate and in an improvement in the inflation expectations [18]. In (6), the value and the high significance of the coefficient 6,5 α indicate that an increase in the immediate rate by the BOJ cause a growth in the interbank call money market rate. This evidence shows that the BOJ is able to influence the banking sector through its interest rate decisions. The remaining coefficients may be ignored because, again, they are either insignificant or close to zero.

Conclusions
From the results obtained, the authors deduce that monetary policy seems to The government's fiscal policy, on the other hand, proves to be more effective.
In fact, the increases in government expenditure imply a rise in private consumption, a change in the 10-years government bonds real interest rate influences the inflation expectations and the money supply may be regulated by issuing government bonds.
In any case, the fiscal multiplier and the investment multiplier are zero.
It is also important to note that the failure to improve inflation expectations may not be attributed to the limited rationality of the operators, who, as already pointed out, make a very limited error in formulating their forecasts. The responsibility, therefore, must rather be attributed to the inadequate economic policy decisions of policy makers (in particular monetary policy decisions).
The achievements reached by the authors in this paper are consistent both with those of the scholars who have criticized Japanese monetary policy since the early 1990s (already exposed in the first paragraph) and with the empirical evidences relating to the QE of 2001 and to the Abenomics of 2013, evidences which show that these policy interventions have been modest in macroeconomic terms.
In conclusion, it is possible to assert that only the expenditure and the taxation decisions by the government have some relevance (in any case limited) for the Japanese economy. As already mentioned above, the main innovation of this manuscript consists precisely in demonstrating that monetary policy has no effect on the GDP, on the consumption, on the investments and on the inflation expectations, while it has a small impact on the interbank call money market rate.
In light of these results, the expansionary fiscal policy, not the expansionary monetary policy, is probably expected to be the switch that will favor the Japanese recovery after the crisis caused by the Covid-19 pandemic.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.

Ethical Approval
This article does not contain any studies with human participants performed by any of the authors.