### Model specifications

In this section, we provide structural VAR estimates of the dynamic effects of shocks in the RMG on the 10-year nominal interest rate (IL), the output gap (YG), inflation (INF) and the (log) real effective exchange rate (REER). The latter variable is introduced to control for exchange rate movements which are very important for the highly open Swiss economy. The data are displayed in Fig. 3.

There have been two episodes of high RMG of approximately 15–20% during the late 1970s and late 1980s. During those episodes, expansionary monetary policies and bank lending were followed, after a few years’ lag reflecting the monetary policy transmission mechanisms, by a booming economy as well as substantial and persistent increases in inflation reaching over 6% during the early 1980s and early 1990s. Thus, episodes of high RMG have been followed by substantial and persistent increases in inflation, and major increases in inflation have been preceded by high RMG.

The positive RMG episodes of 1996–1998, 2003–2004 and 2012–2018 were also followed by positive output gaps and increasing inflation, although those relationships were temporarily affected by special events like the commodity price spike in 2008 (which pushed inflation higher) and the removal of the exchange rate floor (which pushed inflation lower) in 2015. However, the RMG remained relatively low since the 1990s, so inflation remained low as well. When the RMG was negative, the output gap subsequently became negative and inflation decreased.^{Footnote 6} These major monetary expansion and tightness episodes correspond to the findings of the analysis of Baltensperger and Kugler (2017), who examine Swiss monetary history since the early nineteenth century.^{Footnote 7}

Table 2 shows the Phillips Perron (PP) unit root and Kwiatkowski Phillips Schmidt Shin (KPSS) stationarity tests. For RMG, REER, INF and YG, the results support the stationarity assumption. For IL, the tests indicate non-stationarity. However, this might be the result of a historically limited sample with extraordinary interest rate fluctuations. According to Sims et al. (1990), VAR estimates remain consistent in the cases of some unit roots, and the coefficient estimates of stationary right-hand variables have standard asymptotic distributions. Thus, the Granger causality tests shown below would even be valid with a non-stationary IL, except in the cases of the tests involving the influence of IL on the other variables. Moreover, we can expect that the confidence interval of the impulse responses will only be mildly distorted, as we only have a few IL coefficients involved in the calculations.

We set a lag length in the VAR of two, which is optimal according to the Hannan–Quinn information criterion. First, Table 3 shows the test results of the lagged interactions of the five variables (“Granger causality” test). The table reports a chi-squared statistic for each of the other four variables (with two degrees of freedom) as well as for all of the variables jointly (with 8 degrees of freedom), and the corresponding marginal significance level. We find highly statistically significant influences of the output gap and the interest rate on the RMG. Moreover, the interest rate is dynamically influenced by the RMG and the output gap, at least at the 1% significance level, whereas for the output gap and inflation we find a statistically significant influence of the RMG. Moreover, there is a dynamic influence of the real exchange rate on the output gap and on inflation.

In addition, an examination of the correlation matrix of the VAR residuals (Table 4) shows strong contemporaneous relationship between the five variables. In particular, the VAR residuals of RMG and those of IL, YG and INF (and the real exchange rate) are strongly negatively (positively) correlated. The most plausible cause of this pattern is the reaction of monetary policy to changes in the output gap and inflation as well as to changes in the real exchange rate.

Given this correlation pattern, we use the following structural VAR model to identify reasonable structural shocks (*u*) from reduced form shocks (*e*), (e_{t}=Bu_{t}) with the following zero restrictions:

$$begin{aligned} begin{array}{ccccc} hbox {x} &{}quad hbox {x} &{}quad 0 &{}quad 0 &{}quad hbox {x} \ hbox {x} &{}quad hbox {x} &{}quad 0 &{}quad 0 &{}quad hbox {x} \ 0 &{}quad 0 &{}quad hbox {x} &{}quad 0 &{}quad hbox {x} \ 0 &{}quad 0 &{}quad 0 &{}quad hbox {x} &{}quad hbox {x} \ hbox {x} &{}quad 0 &{}quad 0 &{}quad 0 &{}quad hbox {x} \ end{array} end{aligned}$$

The matrix has the same rows and columns as Table 4, and *x* means that the coefficient is unknown, i.e., could be non-zero, and has to be estimated.

This model allows for a simultaneous interdependence between the RMG, the interest rate and the real exchange rate, whereas the reaction of these variables to the output gap and inflation is lagged. Consistent with the monetary literature on monetary policy effects, we assume that the RMG and the interest rate do not affect inflation and real output contemporaneously. Moreover, the real exchange rate may impact the output gap and inflation immediately.

This model is over-identified and the chi-square test of the corresponding restrictions provides a value of 2.372, which is not statistically significant at the usual significance levels with 3 degrees of freedom (marginal significance level 0.498); thus, the model is validated.

### RMG dynamics

The impulse responses for the u-shocks with two standard error confidence bands are displayed in Fig. 4. We see that most of the impulse responses are statistically significant, and they confirm our a priori expectations.

*u*1 appears as an exogenous change in monetary policy or RMG; it leads to a short-term decrease in the long-term interest rate, which is then reversed by increasing inflation expectations. The output gap and inflation exhibit a hump-shaped adjustment pattern, and this pattern is more delayed and longer lasting for inflation.

These results are consistent with the claims made by Friedman (1968) as well as with the findings of Christiano et al. (2005): after an expansionary monetary policy shock, output and inflation respond with a hump-shaped adjustment pattern. Output peaks approximately one and a half years after the shock and returns to pre-shock levels after approximately 3 years, and inflation peaks approximately 2 years after the shock.

*u*2 is interpreted as a shock to the long rate; however, this shock has no significant effect on output gap, inflation or the real exchange rate. The impulse responses to *u*3 suggest that it is a demand shock that triggers a restrictive monetary policy. *u*4 appears to be a cost-push inflation shock that leads to a restrictive monetary policy and correspondingly to a negative influence on the output gap. Finally, the impulse responses to *u*5 show that an exogenous appreciation of the real exchange rate has a negative influence on the output gap and inflation. An expansionary monetary policy mitigates this real exchange rate change, and we see a decline in the interest rate as a result. Note that all the impulse responses converge to zero within 30 quarters; therefore, we see no sign of non-stationarity in our series.

The variance decomposition is displayed in Fig. 5. This figure shows the percentages of the contributions of all five shocks to the forecasting variance of all the variables for different horizons. In the short term, this variance is mostly dominated by the “own” shock, but the other shocks play an important role in the cases of most of the variables over the long term. This is particularly true for the RMG, as the variance share of the exchange rate shock for this variable increases to nearly 70% with an increasing horizon, while the demand shock reaches 10%. This corresponds to the high importance of the real exchange rate for Swiss monetary policy.

For the output gap and inflation, we observe a long-term variance share of approximately one third for the RMG. This is larger than the percentage variance of inflation and output resulting from U.S. monetary policy (interest rate) shocks of 7% and 14%, respectively, which are estimated by Christiano et al. (2005). Moreover, RMG shocks appear very important for the real exchange rate over the long term, as they have a variance share of nearly 60%.

### Robustness tests

To test the stability of our structural VAR (SVAR) model over time, we estimated this model using a sample split in 2007Q3. Therefore, we have three estimates, namely, the full sample, 1976Q3–2007Q3 and 2007Q4–2018Q3. This allows us to calculate a log likelihood for the model with and without a break.

If the hypothesis of no break is correct, then twice the difference of the log likelihood is distributed with 67 degrees of freedom, i.e., the total number of SVAR parameters estimated. This approach produces a test statistic equal to 80.46, which is not statistically significant, even at the 10% level. Therefore, this result indicates that our model is stable over the past 10 years despite the financial and government debt crises as well as the unconventional monetary policy responses to them.

Table 5 presents stability tests of the inflation VAR equation. First, we consider the standard Chow statistic with a break in 2007Q3. Second, we present results for the Bai Perron sequential multiple break test, testing for the stability of all coefficients and of only the coefficients of lagged RMG, respectively.

According to Table 5, we find no evidence of a structural break in the inflation equation, as the stability hypothesis cannot be rejected at any reasonable significance level. This conclusion is supported by the result of the cumulated sum of recursive residuals (CUSUM) displayed in Fig. 6. This statistic remains always clearly within its 95% confidence band under the stability hypothesis.

These latter tests of the stability of a single equation of a VAR system are however of limited value as such a model has to be evaluated including all its dynamic interactions. Moreover, the coefficient estimates of a VAR system are difficult to interpret quantitatively. For example, a shock to the RMG can affect inflation dynamically via the output gap or a combination of variables. Therefore, we have used the log likelihood test at the beginning of this Sect. 4.3 and we will now present the impulse responses of two sub-samples in order to evaluate the properties of the VAR model.

Figure 7 displays the impulse responses for the reduced sample covering 1976Q3–2007Q3, and the impulse responses for the strongly reduced sample 2007Q4–2018Q3 are provided in Fig. 8. The patterns are very similar in both figures, and similar to the whole sample displayed in Fig. 4. For example, the peak responses of inflation and output to RMG shocks are very similar for all three samples. As expected, we see less statistical significance in the sub-samples, especially in the second one. This is not surprising, as we only have about 10 years of data in the second sub-sample. The impulse responses in Figs. 7 and 8 thus confirm the stability of our VAR model and of the effects of the RMG shocks on inflation and output.

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