 Original article
 Open Access
Debt, economic growth, and interest rates: an empirical study of the Swiss case, presenting a new longterm dataset: 1894–2014
 Guillaume Guex^{1} and
 Sébastien Guex^{2}Email author
https://doi.org/10.1186/s4193701700076
© The Author(s) 2018
 Received: 5 June 2017
 Accepted: 10 December 2017
 Published: 24 July 2018
Abstract
In this paper, relations between public debt, economic growth, and longterm interest rates in Switzerland from 1894 to 2014 are examined. For this purpose, an original longterm dataset on the general gross public debt in Switzerland, namely the aggregation of the Confederation gross debt, the cantons’ gross debts, and the municipal gross debts, was reconstructed. Three different statistical approaches are performed to study relations between this aggregated debt, economic growth, and interest rates. The first consists of the study of correlations between GDPweighted variables, the second is the study of the correlation between residuals of ARIMA time series models, and the last one studies vector autoregression (VAR) models, allowing us to test Granger causalities between variables. Every approach is performed on the whole time period but also on boom phases and recession phases independently. All the results suggest that the public debt during this period in Switzerland did not have a negative impact on economic growth and did not raise longterm interest rates.
Keywords
 Public debt
 Economic growth
 Interest rates
 Time series
 Switzerland
Background
The issue of defining relations between public debt and economic growth is a highly controversial topic. From a theoretical viewpoint, the dominant or “conventional” idea, according to Elmendorf and Mankiw (1999, p. 1617), whose expression we will use henceforth, states that in the long term, a high public debt slows down economic growth. However, this point of view remains very much debated. From an empirical angle, researches tackling this question were already numerous before exploding subsequent to the worldwide financial crisis of 2008 and the “influential” (Panizza and Presbitero 2013, p. 180) work of Reinhart and Rogoff (2010). Nevertheless, on the one hand, these empirical researches suffer, for now, from several methodological flaws, weakening their conclusions (Panizza and Presbitero 2013). On the other hand, a nonnegligible part of them finds that this debt does not clearly or significantly influence economic growth.^{1}
Top: Pearson correlation coefficients between debttoGDP ratio and GDP growth rate for boom phases, recession phases and all data. Bottom: Pearson correlation coefficients between debttoGDP ratio and longterm interest rates for boom phases, recession phases and all data. Italics: significant results with 5% level
B_{t} = 1  B_{t} = 0  B_{t} = 1 or B_{t} = 0  

c  \( \mathrm{cor}\left({D}_t^p,{G}_{t+c}^p\right) \)  p value  \( \mathrm{cor}\left({D}_t^p,{G}_{t+c}^p\right) \)  p value  \( \mathrm{cor}\left({D}_t^p,{G}_{t+c}^p\right) \)  p value 
− 5  0.269  0.038  0.379  0.164  − 0.112  0.235 
− 4  0.284  0.022  0.373  0.115  − 0.129  0.166 
− 3  0.255  0.033  0.276  0.203  − 0.118  0.206 
− 2  0.227  0.051  0.158  0.431  − 0.105  0.258 
− 1  0.223  0.047  0.130  0.484  − 0.084  0.366 
0  0.255  0.019  0.038  0.829  − 0.006  0.949 
1  0.318  0.004  0.177  0.340  0.145  0.116 
2  0.100  0.392  0.016  0.936  0.131  0.159 
3  0.023  0.849  0.086  0.696  0.123  0.185 
4  0.034  0.791  0.133  0.588  0.116  0.217 
5  0.222  0.088  0.129  0.648  0.153  0.102 
c  \( \mathrm{cor}\left({D}_t^p,{O}_{t+c}\right) \)  p value  \( \mathrm{cor}\left({D}_t^p,{O}_{t+c}\right) \)  p value  \( \mathrm{cor}\left({D}_t^p,{O}_{t+c}\right) \)  p value 
− 5  − 0.185  0.153  − 0.762  0.001  0.025  0.791 
− 4  − 0.145  0.246  − 0.785  0.000  − 0.003  0.977 
− 3  − 0.124  0.305  − 0.768  0.000  − 0.028  0.762 
− 2  − 0.101  0.386  − 0.730  0.000  − 0.050  0.588 
− 1  − 0.109  0.334  − 0.624  0.000  − 0.092  0.315 
0  − 0.128  0.240  − 0.621  0.000  − 0.146  0.110 
1  − 0.145  0.196  − 0.650  0.000  − 0.203  0.027 
2  − 0.143  0.217  − 0.606  0.001  − 0.241  0.008 
3  − 0.169  0.158  − 0.569  0.005  − 0.271  0.003 
4  − 0.223  0.072  − 0.499  0.030  − 0.303  0.001 
5  − 0.256  0.046  − 0.534  0.040  − 0.338  0.000 
Secondly, to our knowledge, there exists only one published empirical study specifically concerning the Swiss case: the one done by Schwab (1996, pp. 134–136 and pp. 162–169).^{5} The author himself underlines that due to the “relatively short” period of time studied (1963–1992), “it is difficult to come to conclusive appraisals about the relations between public finances and economic growth in Switzerland” (Schwab 1996, p. 169). Thus, the second contribution of the present research is to remedy this problem by analyzing the Swiss case over a long period of time, namely 120 years.
A third asset of this paper is the use of various atheoretical methodologies in order to consolidate conclusions. Three analyses are performed: a study of correlations between GDPweighted variables, a study of the correlation between residuals of ARIMA models, and reducedform vector autoregression (VAR) modeling allowing us to test Granger causalities. For all analysis, time series are studied independently on economic growth (boom) phases, recession phases, and all phases together.

Hypothesis 1 Public debt has a negative impact on economic growth.

Hypothesis 2 Public debt raises longterm interest rates.
The first hypothesis forms what we can call the general or basic hypothesis of the conventional theory. Following the large majority of other studies, this hypothesis will be tested by examining relations between the general gross public debt and the gross domestic product (GDP) in Switzerland.
The second hypothesis is more restrictive but plays a central role in the conventional theory. As a matter of fact, this theory often puts the increase of longterm interest rates at the very heart of mechanisms having a negative impact on economic growth. Hypothesis 2 will be tested by analyzing relations between the general gross public debt and the nominal yields on Confederation and/or Swiss Federal Railways bonds, which, in Switzerland, are generally considered to be the benchmarks for longterm interest rates.
This paper is divided as follows: the “Data” section presents data used for our research; the “Analyses and empirical results” section contains statistical analysis descriptions, results, and discussions; and, finally, conclusions are found in the “Conclusions” section.
Data
This section contains descriptions and sources for the five time series used in statistical analyses. Actual data figures, from 1894 to 2014, are found in Table SA.1 in Additional file 1.
General gross public debt: D _{t}
This time series consists of the aggregation of the gross debts of the Confederation, cantons, and municipalities. An important fact is that until 1910, it consists solely of the funded debt held by the public. The floating debts and the internal (or administrative) debts are not considered. However, the resulting bias is negligible because the amounts of these kinds of debts are, in fact, quite low before 1910.
 (1)
Internal debt, i.e., the part of the debt not held by the public but by different state or public institutions (e.g., public pension funds, public corporations such as the post, telegraph, and telephone), constitutes an important and growing part of the general gross public debt in Switzerland. When the Confederation debt is examined over time, its internal debt represents 9.9% of the total in 1920, 18.6% in 1950, and 29.7% in 2000. Thus, we cannot consider the weight of the internal debt as negligible or weak.
 (2)
Not taking into account, this debt when empirically testing the conventional conception seems, from our point of view, to slightly diminish the robustness of the results.
 (a)
From 1894 to 1910:

Confederation: Federal Statistical Office (1934, pp. 338–339);

Canton and municipalities: Steiger (1915) and Steiger (1916). Data were collected for 1900 and 1910 and reconstructed with linear interpolation for other years. Note that results without interpolation are also discussed in the “Results without interpolated years” section.
 (b)
From 1911 to 1989:

Confederation: Swiss National Bank (1944, p. 199) before 1943; Federal Council, Compte d’Etat de la Confédération suisse, Bern, for every year between 1943 and 1989;

Cantons: Steiger and Higy (1934, p. 424); Federal Statistical Office (1917, p. 355); Federal Tax Administration (1967, p. 35); Federal Tax Administration (1973, p. 29); Federal Administration of Finance (1978, p. 26); Kirchgässner (2004).

Municipalities: Steiger and Higy (1934, p. 435); Federal Statistical Office (1932, p. 343, 1939, p. 347, 1947, p. 454, 1947, p. 404); Federal Tax Administration (1965, p. 55); Kirchgässner (2004, p. 22). This consists of estimations based on a sample from the major municipalities of Switzerland. Data from 1914 to 1918 were reconstructed with linear interpolation.
 (c)
From 1990 to 2014:

Confederation, cantons, and municipalities: Swiss Federal Statistical Office, Table “Dettes des administrations publiques” (jef18.04.01.xls) https://www.bfs.admin.ch/bfs/fr/home/statistiques/administrationfinancespubliques.assetdetail.3462296.html (accessed October 28, 2016). The reconstruction of data with new standards leads to having a quantitative leap in 1990. The resulting bias is, however, reduced by the fact that a leap of comparable scale also occurs in the GPD time series in 1990, for similar reasons.
Gross domestic product: G _{t}
 (a)
From 1894 to 1948: Historical Statistics of Switzerland online, Table Q.16a, http://www.fsw.uzh.ch/hstat/nls_rev/ls_files.php?chapter_var=./q (accessed August 20, 2016).
 (b)
From 1949 to 1989: Historical Statistics of Switzerland online, Table Q.16b, http://www.fsw.uzh.ch/hstat/nls_rev/ls_files.php?chapter_var=./q (accessed August 20, 2016).
 (c)
From 1990 to 2014: State Secretariat for Economic Affairs (SECO), Table “GDP, production approach, not adjusted” https://www.seco.admin.ch/seco/en/home/wirtschaftslagewirtschaftspolitik/Wirtschaftslage/bipquartalsschaetzungen/daten.html (accessed August 23, 2016).
Longterm interest rates: O _{t}
In Switzerland, the annual average of yields on Confederation and/or Swiss Federal Railways bonds is traditionally considered the benchmark for longterm interest rates.^{7} From 1899 to 2014, the Swiss National Bank gives an almost complete time series about this annual average. From 1894 to 1898, data were obtained by taking the mean between savings bank deposit rates and rates of the mediumterm notes issued by banks.
 (a)
From 1894 to 1898: Swiss National Bank, Historical Time Series, Tables 4.3a_A and 4.6a_A, https://www.snb.ch/en/iabout/stat/statrep/statpubdis/id/statpub_histz_arch (accessed August 24, 2016).
 (b)
From 1899 to 2001: Ibid., Table 3.1_A; for the year 1915, Guex (1993, p. 453).
 (c)
From 2002 to 2014: Swiss National Bank, Monthly Statistical Bulletin, Table E4, https://www.snb.ch/en/iabout/stat/statrep/statpubdis/id/statpub_statmon_arch (accessed August 24, 2016).
Consumer price index: I _{t}
Data were deflated by using the consumer price index (set to be 100 in 2014).
 (a)
From 1894 to 1913: Historical Statistics of Switzerland online, Table H.17, http://www.fsw.uzh.ch/hstat/nls_rev/ls_files.php?chapter_var=./h (accessed August 20, 2016).
 (b)
From 1914 to 2014: Swiss Federal Statistical Office, Statistical Encyclopedia, Table “ccf05.02.08.xls,” https://www.bfs.admin.ch/bfs/fr/home/statistiques/cataloguesbanquesdonnees/tableaux.assetdetail.4242093.html (accessed August 24, 2016).
Dummy variable for the identification of boom/recession phases: B _{t}
In our analyses, boom and recession phases are also treated independently in order to see if hypotheses are conditionally valid for different economic regimes. Years belonging to boom phases are noted with 1 and years belonging to recession phases with 0.
This division is largely inspired from the one used in Müller and Woitek (2012, p. 114) and is based on the distinction between phases with positive GDP growth rate and phases with negative GDP growth rate. The boom phases are 1894–1913, 1922–1929, 1945–1973, 1980–1990, and 1997–2014. Recession phases are 1914–1921, 1930–1944, 1974–1979, and 1991–1996.
Analyses and empirical results
 1.
In the first part, this trend is suppressed by using variables weighted by GDP. The latter expresses the annual output of the country; it can work as a weight, similar to the use of population in other studies. Note that even if this method is frequently employed, it is still highly questionable, as the GDP is simultaneously a weight and a studied variable. Analogous to many empirical researches on the same subject (see, e.g., Schwab 1996, p. 164; Elmendorf and Mankiw 1999, p. 1636; Panizza and Presbitero 2013, p. 6; Greiner and Fincke 2015), correlations between weighted variables are computed with a time lag going from − 5 to + 5 years.
 2.
In the second part, residuals of series ARIMA models are analyzed, as advocated in, e.g., Granger and Newbold 1974. Building ARIMA models for a time series permits the removal of the variability explained by the past values of the variable. The remaining unknown variability, i.e., the residuals of the model, represents the effects coming from external sources. Thus, by studying the correlation between residuals of different series, we examine if these external effects, or shocks, are linked together. Here again, the study of correlations between variables with a time lag going from − 5 to + 5 years is performed.
 3.
The last part is the most sophisticated one. Granger causality tests (Granger 1969) are performed, using reducedform vector autoregressive (VAR) models, with the help of the methodology developed by Toda and Yamamoto (1995) to deal with integrated time series. Although this methodology is more complex and less intuitive, it is the most suitable one for analyzing such data.
Deflated time series
In Additional file 1: Table SA.1, debt and GDP are expressed in nominal value and do not take into account inflation, expressed by the variable I_{t}. Thus, analyses are performed on debt and GDP is expressed in constant francs of 2014, noted respectively as \( {D}_t^r \) and \( {G}_t^r \), and defined by:
These variables, along with O_{t}, are pictured in Additional file 1 with their linear regression curve.
GDPweighted series
Method
In this section, we are testing Hypothesis 1 by examining relations between debttoGDP ratio, \( {D}_t^p \), and GDP growth rate, \( {G}_t^p \), defined by:
These variables are pictured in Additional file 1. To test Hypothesis 2, debttoGDP ratio \( {D}_t^p \) is compared with O_{t}.
After the weighting of variables, the Pearson correlation coefficient test is performed on \( \left({D}_t^p,{G}_{t+c}^p\right) \) and \( \left({D}_t^p,{O}_{t+c}\right), \) where, as described above, c takes values between − 5 and + 5. These correlation tests are performed independently on boom phases, recession phases, and with all phases taken together.
Results
The results are presented in Table 1.
Discussion
DebttoGDP ratio vs GDP growth rate (Table 1, top)
In the study of relations between debttoGDP ratio and GDP growth rate, a positive significant correlation coefficient is found during boom phases with c = {−5, −4, −3, −1, 0, 1}. This might reflect the fact that during boom phases, both GDP and debttoGDP ratio are generally increasing. Anyhow, these results go against the temporality (except for c = 1) and the sign of the effect advocated by the conventional theory. Yet, this result needs to be put into perspective, notably because the variable \( {G}_t^r \) appears simultaneously in the \( {D}_t^p \) denominator and the \( {G}_t^p \) numerator, dangerously increasing the risk of finding spurious correlations.
DebttoGDP ratio and longterm interest rates (Table 1, Bottom)
Regarding correlations between debttoGDP ratio and longterm interest rates, very significant results are found for recession phases, although the sign of the correlation is the opposite of the one claimed by the conventional theory. The strength of the results is baffling, although finding an explanation to this phenomenon might need very complex schemes and is beyond the scope of this article. Note that these results should also be taken carefully: even if these time series are not constructed using a common variable, we can see in Additional file 1 that they are not stationary, which is a requirement for the results to be true asymptotically.
Residuals of ARIMA models
Method
Another way to tackle the trend problem is to study correlations between residuals of series ARIMA models, as advocated in Granger and Newbold 1974. As a matter of fact, by building an ARIMA model, we try to forecast a series with its own past behavior, with residuals representing unexplained, external effects. By comparing correlations between residuals of different series, we study whether the shocks of these series are related.
ARIMA models are built from the Box–Jenkins methodology (Box et al. 2015). To begin with, we see that \( {D}_t^r \) and \( {G}_t^r \) seem to have an exponential behavior, as shown by their residuals. Knowing the multiplicative nature of these variables, this behavior can be expected, and it is quite standard to work with the logarithms of these quantities to avoid heteroskedasticity in residuals and to transform them into additive models. Let us define:
the natural logarithms of the general gross public debt and the gross domestic product (in lnmillions constant francs). These variables, shown in Additional file 1, seem to behave in a more linear way, as shown by their Rsquared coefficients and residuals. Note that this logarithm transformation allows us to compare nonlinear effects with linear models.
Following this step, the order of integration of the series must be obtained to ensure stationarity. In Additional file 1, autocorrelation and partial autocorrelation functions of the first difference series \( {\Delta D}_t^{\mathrm{In}} \), \( {\Delta G}_t^{\mathrm{In}} \), and ΔO_{t}, defined by:
are displayed. These plots, along with the insignificant Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test (Kwiatkowski et al. 1992) and the significant Augmented Dickey–Fuller (ADF) test (Dickey and Fuller 1979), indicate that all three series have an order of integration of one. Note that there is a doubt about O_{t}, as both KPSS and ADF tests are not significant for this variable, i.e., the time series seems not informative enough to be able to decide. However, its autocorrelation function, as shown in Additional file 1: Figure SE.5, displays a relatively slow decrease, favoring the use of its first differences.
Finally, model orders are determined by optimizing values of AICc and BIC. Note that for ΔO_{t}, the AICc give a model ARMA (0,3) and the BIC a model ARMA (0,0). We opted here for the ARMA (0,0) as it seems more parsimonious. In Additional file 1, \( {\Delta D}_t^{\mathrm{ln}} \), \( {\Delta G}_t^{\mathrm{ln}} \), and ΔO_{t} are displayed. The respective models for \( {\Delta D}_t^{\mathrm{ln}} \), \( {\Delta G}_t^{\mathrm{ln}} \), and ΔO_{t} are ARMA (1,0), ARMA (0,0), and ARMA (0,0). Thus, we only need to compute residuals for \( {\Delta D}_t^{\mathrm{ln}} \), as \( {\Delta G}_t^{\mathrm{ln}} \) and ΔO_{t} residuals are equivalent to their first difference series. The residuals series of \( {\Delta D}_t^{\mathrm{ln}} \) is denoted by \( {\Delta D}_t^{\mathrm{res}} \).
Here again, Pearson correlation coefficient tests on \( \left({\Delta D}_t^{\mathrm{res}},\Delta {G}_{t+c}^{\mathrm{ln}}\right) \) and \( \left({\Delta D}_t^{\mathrm{res}},{\Delta O}_t\right) \) are performed independently on boom phases, recession phases, and all phases, where c ranges from − 5 to 5.
Results
Top: Pearson correlation coefficients between the residuals of first differences lndebt and residuals of first differences lnGDP for boom phases, recession phases and all data. Bottom: Pearson correlation coefficients between first differences lndebt and residuals of first differences interest rates for boom phases, recession phases and all data. Italics: significant results with 5% level
B_{t} = 1  B_{t} = 0  B_{t} = 1 or B_{t} = 0  

c  \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {G}_{t+c}^{\mathrm{ln}}\right) \)  p value  \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {G}_{t+c}^{\mathrm{ln}}\right) \)  p value  \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {G}_{t+c}^{\mathrm{ln}}\right) \)  p value 
− 5  − 0.015  0.896  0.056  0.771  0.056  0.549 
− 4  0.066  0.556  − 0.020  0.916  0.015  0.871 
− 3  − 0.026  0.813  0.016  0.932  − 0.012  0.897 
− 2  0.062  0.576  0.047  0.795  0.125  0.176 
− 1  − 0.447  0.000  − 0.126  0.479  − 0.461  0.000 
0  0.354  0.001  − 0.276  0.109  0.039  0.671 
1  0.110  0.318  0.128  0.469  0.031  0.735 
2  0.111  0.317  − 0.128  0.477  0.021  0.823 
3  0.060  0.591  − 0.118  0.521  0.050  0.596 
4  − 0.175  0.117  0.425  0.017  − 0.077  0.409 
5  0.204  0.070  0.088  0.646  − 0.097  0.305 
c  \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {O}_t\right) \)  p value  \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {O}_t\right) \)  p value  \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},{\Delta O}_t\right) \)  p value 
− 5  − 0.059  0.601  0.054  0.777  0.064  0.495 
− 4  − 0.029  0.799  0.111  0.552  0.027  0.776 
− 3  − 0.046  0.678  − 0.210  0.250  − 0.016  0.861 
− 2  0.201  0.068  0.090  0.617  0.252  0.006 
− 1  0.171  0.120  0.198  0.262  0.106  0.251 
0  − 0.180  0.100  0.004  0.981  − 0.121  0.189 
1  − 0.161  0.142  0.002  0.989  − 0.142  0.123 
2  0.097  0.381  − 0.183  0.308  − 0.011  0.904 
3  0.126  0.260  − 0.350  0.050  − 0.108  0.245 
4  0.113  0.315  − 0.014  0.940  0.100  0.287 
5  − 0.106  0.349  0.163  0.391  0.041  0.665 
Discussion
First differences lndebt vs residuals of first differences lnGDP (Table 2, top)
A strong significant result concerning Hypothesis 1 is between \( {\Delta D}_t^{\mathrm{res}} \) and \( \Delta {G}_{t1}^{\mathrm{ln}} \) for boom phases, with a strong negative correlation coefficient. This result implies that an increase (respectively a decrease) in the (ln)GDP is followed next year by a decrease (respectively an increase) in the (ln)debt residuals. One significant positive correlation coefficient is found for boom phases at c = 0. The positive link for the same year indicates that shocks between these time series, due to external effects affecting them, often take a similar direction. Moreover, there is a link from the debt to the GDP for c = 4 during recession phases. This link goes against the conventional theory, with the sign of the correlation indicating a positive effect of the debt on the GDP after a few years.
First differences lndebt vs residuals of first differences interest rates (Table 2, Bottom)
The results concerning Hypothesis 2 are different from the “Results” section. The only significant value, i.e., the correlation coefficient between \( {\Delta D}_t^{\mathrm{log}} \) and ΔO_{t − 2}, which is equal to 0.252, suggests that an increase (respectively a decrease) in the interest rates is followed by an increase (respectively a decrease) in the debt 2 years later. This reversed causality contradicts the results that are supported by Hypothesis 2.
Vector autoregression models and Granger causality
Method
This section holds the most appropriate analysis to test Hypotheses 1 and 2, namely by using Granger causality tests (Granger 1969) from reducedform vector autoregression (VAR) models. Nevertheless, Park and Phillips (1989) and Sims et al. (1990), among others, show that the conventional asymptotic theory for hypothesis testing in this context is generally not suited when time series are integrated or cointegrated, which is the case here, as shown in Additional file 1. Hopefully, in their article, Toda and Yamamoto (1995) propose a methodology to tackle integration issues by fitting a model VAR (p + d_{max}) on nondifferentiated data, where p is the lag length given by usual information criteria and d_{max} is the maximum order of integration of the series. Note that models are fitted with p lags for coefficients estimations and orthogonal impulse responses, and the lag is increased only for testing Granger causalities. Even there, the Ftest is performed only on the first p coefficients of the VAR (p + d_{max}) model.
Unlike precedent methods performed in this article, the independent studies of boom phases and recession phases require more careful treatment. As a matter of fact, we can consider that coefficients of the VAR model will be different between boom phases and recession phases, as found in a Markov switching vector autoregression (MS–VAR) model, except that the switching years are known. In practice, different VAR models will be fit for boom phases, recession phases, and all phases, by hiding years not belonging to their respective set. However, VAR coefficients are estimated on lagged values that can lie outside the studied set.
As Granger causalities are studied for two pairs of variables, \( \left({D}_t^{\mathrm{ln}},{G}_t^{\mathrm{ln}}\right) \) and \( \left({D}_t^{\mathrm{ln}},{O}_t\right) \), and three different year subsets, there will be six different VAR models altogether. In the following section, we quickly summarize construction, assessment, and main results of every model. Supplementary statistics about these models can be found in Additional file 1.
Results
VAR models for \( \left({\mathrm{D}}_{\mathrm{t}}^{\mathrm{ln}},{\mathrm{G}}_{\mathrm{t}}^{\mathrm{ln}}\right) \)

(a) Models’ assessments

• Boom phases: The Johansen procedure (Johansen 1988) does not give any cointegration. All information criteria give a lag length of p = 2. Residuals show no autocorrelation and no conditional heteroskedasticity, though they do not seem to follow a normal distribution, especially for \( {G}_t^{\mathrm{ln}} \) (this problem is recurrent for several models; we will come back to this in the discussion).

• Recession phases: Here, the Johansen procedure does not give any cointegration. Criteria diverge on the lag length. For parsimony reasons, we opted for a lag length of p = 2. With such a model, residuals have no autocorrelation and no conditional heteroskedasticity and follow a normal distribution.

• All phases: The Johansen procedure does not give any cointegration. Information criteria give a lag length of p = 2. This model has no autocorrelation in residuals, though they are not normally distributed and show conditional heteroskedasticity.

(b) Models’ coefficients

The models’ coefficients are presented in Tables 3, 4, and 5.

(c) Orthogonal impulse response functions

The orthogonal impulse response functions are presented in Figs. 1, 2, and 3.

(d) Granger causalities

• Boom phases: \( {\displaystyle \begin{array}{c}{G}_t^{\mathrm{ln}}\to {D}_t^{\mathrm{ln}}:F=23.932,p\ \mathrm{value}=8.613\mathrm{e}10\\ {}{D}_t^{\mathrm{ln}}\to {G}_t^{\mathrm{ln}}:F=4.535,p\ \mathrm{value}=0.012\end{array}} \)

• Recession phases: \( {\displaystyle \begin{array}{c}{G}_t^{\mathrm{ln}}\to {D}_t^{\mathrm{ln}}:F=0.233,p\ \mathrm{value}=0.793\\ {}{D}_t^{\mathrm{ln}}\to {G}_t^{\mathrm{ln}}:F=1.533,p\ \mathrm{value}=0.224\end{array}} \)

• All phases: \( {\displaystyle \begin{array}{c}{G}_t^{\mathrm{ln}}\to {D}_t^{\mathrm{ln}}:F=16.241,p\ \mathrm{value}=2.569\mathrm{e}07\\ {}{D}_t^{\mathrm{ln}}\to {G}_t^{\mathrm{ln}}:F=1.031,p\ \mathrm{value}=0.359\end{array}} \)
Coefficients’ estimations of a VAR (2) model between the \( {D}_t^{\mathrm{ln}} \) and \( {G}_t^{\mathrm{ln}} \) for boom phases. Top: model for \( {D}_t^{\mathrm{ln}} \). Bottom: model for \( {G}_t^{\mathrm{ln}} \). Italics: significant results with 5% level
\( {D}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{ln} \)  1.366  0.000 
\( {D}_{t2}^{ln} \)  − 0.377  0.000 
\( {G}_{t1}^{ln} \)  − 0.586  0.000 
\( {\mathrm{G}}_{\mathrm{t}2}^{\mathrm{ln}} \)  0.590  0.000 
Const  0.072  0.027 
\( {G}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  0.176  0.056 
\( {D}_{t2}^{\mathrm{ln}} \)  − 0.142  0.128 
\( {G}_{t1}^{ln} \)  0.887  0.000 
\( {G}_{t2}^{\mathrm{ln}} \)  0.077  0.413 
Const  0.090  0.011 
Coefficients’ estimations of a VAR (2) model between the \( {D}_t^{\mathrm{ln}} \) and \( {G}_t^{\mathrm{ln}} \) for recession phases. Top: model for \( {D}_t^{\mathrm{ln}} \). Bottom: model for \( {G}_t^{\mathrm{ln}} \). Italics: significant results with 5% level
\( {D}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{ln} \)  1.515  0.000 
\( {D}_{t2}^{ln} \)  − 0.539  0.003 
\( {G}_{t1}^{\mathrm{ln}} \)  − 0.001  0.997 
\( {G}_{t2}^{\mathrm{ln}} \)  0.019  0.923 
Const  0.051  0.505 
\( {G}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  0.099  0.542 
\( {D}_{t2}^{\mathrm{ln}} \)  0.520  0.759 
\( {G}_{t1}^{ln} \)  0.513  0.007 
\( {G}_{t2}^{ln} \)  0.444  0.020 
Const  − 0.083  0.254 
Coefficients’ estimations of a VAR (2) model between the \( {D}_t^{\mathrm{ln}} \) and \( {G}_t^{\mathrm{ln}} \) for all phases. Top: model for \( {D}_t^{\mathrm{ln}} \). Bottom: model for \( {G}_t^{\mathrm{ln}} \). Italics: significant results with 5% level
\( {D}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  1.422  0.000 
\( {D}_{t2}^{\mathrm{ln}} \)  − 0.441  0.000 
\( {G}_{t1}^{\mathrm{ln}} \)  − 0.432  0.000 
\( {G}_{t2}^{\mathrm{ln}} \)  0.445  0.000 
Const  0.051  0.094 
\( {G}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  0.038  0.664 
\( {D}_{t2}^{\mathrm{ln}} \)  −0.019  0.835 
\( {G}_{t1}^{\mathrm{ln}} \)  0.992  0.000 
\( {G}_{t2}^{\mathrm{ln}} \)  −0.010  0.910 
Const  0.031  0.379 
VAR models for \( \left({\mathrm{D}}_{\mathrm{t}}^{\mathrm{ln}},{\mathrm{O}}_{\mathrm{t}}\right) \)

(a) Models’ assessments

• Boom phases: The Johansen procedure does not give any cointegration. Most information criteria give a lag length of p = 3. Residuals show no autocorrelation and no conditional heteroskedasticity, though they do not seem normally distributed, especially for \( {D}_t^{\mathrm{ln}}. \)

• Recession phases: The Johansen procedure gives a cointegration with 5% level. The majority of information criteria give a lag length of p = 2. There is no autocorrelation in residuals and no conditional heteroskedasticity, though they do not seem normally distributed.

• All phases: The Johansen procedure gives no cointegration. The majority of information criteria give a lag length of p = 2. With this model, there is no autocorrelation in residuals, though they possess conditional heteroskedasticity and are not normally distributed.

(b) Models’ coefficients

The models’ coefficients are presented in Tables 6, 7, and 8.

(c) Orthogonal impulse response functions

The orthogonal impulse response functions are presented in Figs. 4, 5, and 6.

(d) Granger causalities

• Boom phases: \( {\displaystyle \begin{array}{c}{O}_t\to {D}_t^{\mathrm{ln}}:F=4.037,p\ \mathrm{value}=0.009\\ {}{D}_t^{\mathrm{ln}}\to {O}_t:F=0.119,p\ \mathrm{value}=0.949\end{array}} \)

• Recession phases: \( {\displaystyle \begin{array}{c}{O}_t\to {D}_t^{\mathrm{ln}}:F=0.768,p\ \mathrm{value}=0.468\\ {}{D}_t^{\mathrm{ln}}\to {O}_t:F=1.101,p\ \mathrm{value}=0.339\end{array}} \)

• All phases: \( {\displaystyle \begin{array}{c}{O}_t\to {D}_t^{\mathrm{ln}}:F=3.928,p\ \mathrm{value}=0.021\\ {}{D}_t^{\mathrm{ln}}\to {O}_t:F=1.194,p\ \mathrm{value}=0.305\end{array}} \)
Coefficients’ estimations of a VAR (3) model between the \( {D}_t^{\mathrm{ln}} \) and O_{t} for boom phases. Top: model for \( {D}_t^{\mathrm{ln}} \). Bottom: model for O_{t}. Italics: significant results with 5% level
\( {D}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  1.215  0.000 
\( {D}_{t2}^{\mathrm{ln}} \)  − 0.142  0.422 
\( {D}_{t3}^{\mathrm{ln}} \)  − 0.076  0.496 
O_{t − 1}  − 0.005  0.702 
O_{t − 2}  0.043  0.014 
O_{t − 3}  − 0.033  0.009 
Const  0.015  0.726 
O_{t}=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  − 0.427  0.701 
\( {D}_{t2}^{\mathrm{ln}} \)  0.426  0.803 
\( {D}_{t3}^{\mathrm{ln}} \)  − 0.024  0.982 
O_{t − 1}  1.284  0.000 
O_{t − 2}  − 0.631  0.000 
O_{t − 3}  0.367  0.002 
Const  0.103  0.863 
Coefficients’ estimations of a VAR (2) model between the \( {D}_t^{\mathrm{ln}} \) and O_{t} for recession phases. Top: model for \( {D}_t^{\mathrm{ln}} \). Bottom: model for O_{t}. Italics: significant results with 5% level
\( {D}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{ln} \)  1.521  0.000 
\( {D}_{t2}^{ln} \)  − 0.525  0.003 
O_{t − 1}  0.014  0.271 
O_{t − 2}  − 0.008  0.589 
Const  0.018  0.819 
O_{t}=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  − 1.273  0.546 
\( {D}_{t2}^{\mathrm{ln}} \)  1.044  0.621 
O_{t − 1}  0.972  0.000 
O_{t − 2}  − 0.153  0.441 
Const  2.245  0.036 
Coefficients’ estimations of a VAR (2) model between the \( {D}_t^{\mathrm{ln}} \) and O_{t} for all phases. Top: model for \( {D}_t^{\mathrm{ln}} \). Bottom: model for O_{t}. Italics: significant results with 5% level
\( {D}_t^{\mathrm{ln}} \)=  Coefficient  p value 
\( {D}_{t1}^{ln} \)  1.343  0.000 
\( {D}_{t2}^{ln} \)  − 0.347  0.000 
O _{ t − 1}  0.012  0.165 
O _{ t − 2}  − 0.001  0.942 
Const  − 0.001  0.975 
O_{t}=  Coefficient  p value 
\( {D}_{t1}^{\mathrm{ln}} \)  − 0.658  0.503 
\( {D}_{t2}^{\mathrm{ln}} \)  0.582  0.552 
O _{ t − 1}  1.098  0.000 
O _{ t − 2}  − 0.204  0.038 
Const  0.904  0.027 
Discussion
Models’ assessments
First, let us briefly review the models’ assessments. Almost every model (all except the model in the “VAR models for \( \left({\mathrm{D}}_t^{\mathrm{ln}},{\mathrm{G}}_t^{\mathrm{ln}}\right) \)” section for recession phases) possesses residuals that are not normally distributed. Note that this is not surprising, as it is wellknown that when the number of data is large, most normality tests tend to become unnecessarily powerful, thus raising the amount of type I errors. Moreover, as long as the residuals are “reasonably” normal, this is barely an issue in order to build models for exploratory purposes. Models concerning all phases also have some heteroskedasticity in residuals. The latter certainly reflects the fact that these models are built without distinguishing boom phases and recession phases, and variances between these phases are highly different. This observation highlights the importance of the additional, independent studies of boom phases and recession phases. Apart from these remarks, the models seem reasonable: Granger causality and impulse response functions analyses can be performed.
LnDebt vs lnGDP (the “VAR models for \( \left({\mathrm{D}}_t^{\mathrm{ln}},{\mathrm{G}}_t^{\mathrm{ln}}\right) \)” section)
These analyses reveal results close to the ones obtained in the “Residuals of ARIMA models” section. For the relation \( {G}_t^{\mathrm{ln}}\to {D}_t^{\mathrm{ln}} \), there is a negative Granger causality during boom phases and all phases (as indicated by the significant Granger Fvalues, part d, and by negative impulse response functions, Figs. 1 and 3, bottomleft graphs). We also see a weak positive causality for \( {D}_t^{\mathrm{ln}}\to {G}_t^{\mathrm{ln}} \) during boom phases (significant Fvalue, part d, and positive impulse response functions, Fig. 1, topright graph), but we see that \( {G}_t^{\mathrm{ln}} \) is largely explained by its previous values (Tables 3, 4, and 5, right part). Again, these results contradict Hypothesis 1: they show that raising the (ln)GDP decreases future (ln)debt and, with weaker evidence, that (ln)debt has a beneficial impact on future (ln)GDP.
LnDebt vs interest rates (the “VAR models for \( \left({\mathrm{D}}_t^{\mathrm{ln}},{\mathrm{O}}_{\mathrm{t}}\right) \)” section)
Remarks
An important remark should be made about the Granger causality. The reduced form of the VAR models used to construct Granger causality tests leads to a weaker form of negative results. Indeed, even though a positive Granger causality test is a strong argument in favor of a relationship between variables, a negative result is not strict evidence of the exogeneity of one variable regarding another (Cooley and Leroy 1985). A way to solve this issue would be to build structural VAR models (SVAR); however, this would imply a set of restrictions between variables and our approach would lose its atheoretical aspect. Nevertheless, these negative results tend to reinforce the results found in previous sections, and the accumulation of weak evidences works against the viability of Hypothesis 1 and Hypothesis 2.
Results without interpolated years
In this article, we have chosen to present and to work with the complete time series from 1884 to 2014. However, the debts for cantons and municipalities are missing between 1885–1899 and 1901–1909 and have been reconstructed by linear interpolation. Therefore, it is justified to wonder what would be the results without those interpolated years, as these years could lead to biased relationships between variables. This is the reason why we reapplied our three methods with data from 1910 to 2014, thus omitting interpolated years. Results of this study can be found in Additional file 1, and we will shortly discuss the differences from previous results.
First of all, let us note that all interpolated years belong to a boom phase. Thus, in the first and third experiments, results are modified only for boom phases and all phases, leaving recession phases results unchanged. However, ARIMA models are built considering all years, so even recession phases results are slightly altered.
By comparing results with and without interpolated years, we can see that interpretations remain mostly unchanged. In the study of GDPweighted series, the correlation between \( \left({D}_t^p,{G}_{t2}^p\right) \) becomes significant during boom phases (Additional file 1: Table SH.1, left) and the correlation between \( \left({D}_t^p,{O}_{t+5}\right) \) during boom phases is not significant anymore (Additional file 1: Table SH.1, right). These results do not impact on conclusions. For the study of ARIMA models, three coefficient significances are modified: \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {G}_t^{\mathrm{ln}}\right) \) during boom phases (Additional file 1: Table SH.2, left), \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},\Delta {G}_{t+5}^{\mathrm{ln}}\right) \) during all phases (Additional file 1: Table SH.2, left), and \( \mathrm{cor}\left({\Delta D}_t^{\mathrm{res}},{\Delta O}_{t+3}\right) \) during recession phases (Additional file 1: Table SH.2, right). The significant negative correlation coefficient between \( \left({\Delta D}_t^{\mathrm{res}},\Delta {G}_{t+5}^{\mathrm{ln}}\right) \) is the only result which is in favor of Hypothesis 1, but it is not very significant (p value = 0.049), and by itself, it is hardly a strong argument. Finally, for the study of VAR models, the biggest changes are the significant coefficients of \( {D}_{t1}^{\mathrm{ln}} \) and \( {D}_{t2}^{\mathrm{ln}} \) for explaining \( {G}_t^{\mathrm{ln}} \) during boom phases (Additional file 1: Table SH.3, middleleft), and in the insignificant Granger causality from \( {O}_t\to {D}_t^{\mathrm{ln}} \) during all phases (Additional file 1: Table SH.4 legend, middleright). This does not lead to any change in our conclusions, as the Granger causality \( {D}_t^{\mathrm{ln}}\to {G}_t^{\mathrm{ln}} \) is still a positive relationship, and the link \( {O}_t\to {D}_t^{\mathrm{ln}} \) was already weak.
To sum up, the results obtained by omitting interpolated years hardly change the results found with complete time series.
Conclusions

Hypothesis 1 Public debt has a negative impact on economic growth.

Hypothesis 2 Public debt raises longterm interest rates.
Similarly, this hypothesis is not supported by any result, although the results are less convincing. The tests performed in the “Residuals of ARIMA models” and “Vector autoregression models and Granger causality” sections show that longterm interest rates affect public debt, but not the other way around. However, the weak levels of significance in the results lead us to believe that relations between public debt and longterm interest rates are either negligible or subject to factors beyond the scope of this paper. Nevertheless, Hypothesis 2 does not hold empirically or seems, at best, overly simplistic in the Swiss case.
To sum up, our results corroborate conclusions found by the only other two empirical studies using the general gross public debt in Switzerland, namely the one by Schwab (1996) and the one (unpublished) by Gay (2012): the empirical analysis of the Swiss case, addressed over a long period of time (1894–2014), does not confirm the conventional theory about relations between public debt, economic growth, and interest rates.
For a summary and an appraisal of the results of the main empirical researches, see for instance Schwab (1996, pp. 170–173), Panizza and Presbitero (2013) or Greiner and Fincke (2015, pp. 233–235).
Computed from Historical Statistics of Switzerland online, Table U.45, http://www.fsw.uzh.ch/histstat/main.php (accessed August 19, 2016); see also Guex (2012).
This time series improves and extends the one found in Historical Statistics of Switzerland online, Table U.45.
There is another study, but it is unpublished and a little rudimentary from the point of view of the statistical tools: Gay (2012).
We would like to deeply thank Dr. Alessandro Dozio, deputy at the city development and communication department, Lausanne, for his precious help in the gathering of data about the Confederation debt.
Declarations
Authors’ contributions
Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Abbas, SA, Belhocine, N, El Ganainy, A, Horton, M (2010). A historical public debt database. In IMF Working Paper, WP/10/245.Google Scholar
 Blanchard, OJ (2008). Crowding out. In SN Durlauf, LE Blume (Eds.), The new Palgrave dictionary of economics, second edition, (pp. 327–329). Basingstoke: Palgrave Macmillan.Google Scholar
 Box, GE, Jenkins, GM, Reinsel, GC, Ljung, GM (2015). Time series analysis: forecasting and control. Hoboken: John Wiley & Sons.Google Scholar
 Brunner, K. (1986). Deficits, interest rates, and monetary policy. Cato Journal, 5(3), 709–731.Google Scholar
 Cooley, TF, & LeRoy, SF. (1985). Atheoretical macroeconometrics: a critique. Journal of Monetary Economics, 16, 283–308.View ArticleGoogle Scholar
 Dickey, DA, & Fuller, WA. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366a), 427–431.View ArticleGoogle Scholar
 Elmendorf, DW, & Mankiw, NG (1999). Government Debt. In JB Taylor, M Woodford (Eds.), Handbook of macroeconomics, Vol. 1C, (pp. 1615–1669). Amsterdam: ElsevierGoogle Scholar
 Federal Administration of Finance (1978). Finances publiques en Suisse 1976. Bern. Federal Administration of Finance.Google Scholar
 Federal Council. Compte d'Etat de la Confédération suisse 19431989. Bern.Google Scholar
 Federal Statistical Office (1917). Annuaire statistique de la Suisse 1917. Bern: Stämpfi.Google Scholar
 Federal Statistical Office (1932). Annuaire statistique de la Suisse 1932. Bern: Stämpfi.Google Scholar
 Federal Statistical Office (1934). Annuaire statistique de la Suisse 1934. Bern: Stämpfi.Google Scholar
 Federal Statistical Office (1939). Annuaire statistique de la Suisse 1939. Bern: Stämpfi.Google Scholar
 Federal Statistical Office (1947). Annuaire statistique de la Suisse 1947. Bern: Stämpfi.Google Scholar
 Federal Tax Administration (1965). Finances et impôts de la Confédération, des cantons et des communes 1963/64. Bern. Federal Administration of Finance.Google Scholar
 Federal Tax Administration (1967). Finances et impôts de la Confédération, des cantons et des communes 1965/66. Bern. Federal Administration of Finance.Google Scholar
 Federal Tax Administration (1973). Finances des cantons 1930–1971. Bern. Federal Administration of Finance.Google Scholar
 Gay, B (2012). Les principes de la doxa libérale à l'épreuve de séries statistiques longues: la dette publique suisse (1894–2000). Lausanne: University of Lausanne, Master Thesis.Google Scholar
 Granger, CWJ. (1969). Investigating causal relations by econometric models and crossspectral methods. Econometrica: Journal of the Econometric Society, 37(3), 424–438.View ArticleGoogle Scholar
 Granger, CWJ, & Newbold, P. (1974). Spurious regressions in econometrics. Journal of Econometrics, 2, 111–120.View ArticleGoogle Scholar
 Greiner, A, & Fincke, B (2015). Public debt, sustainability and economic growth. Theory and empirics. Cham: Springer International Publishing.Google Scholar
 Guex, S (1993). La politique monétaire et financière de la Confédération suisse 1900–1920. Lausanne: Payot.Google Scholar
 Guex, S (2012). Öffentliche Finanzen und Finanzpolitik. In P Halbeisen, M Müller, B Veyrassat (Eds.), Wirtschaftsgeschichte der Schweiz im 20. Jahrhundert, (pp. 1077–1129). Bern: Schwabe.Google Scholar
 Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control, 12(2), 231–254.View ArticleGoogle Scholar
 Kirchgässner, G (2004). Die langfristige Entwicklung der Bundesfinanzen, 1960–2002. St. Gallen: Universität St. Gallen.Google Scholar
 Kwiatkowski, D, Phillips, PC, Schmidt, P, Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics, 54, 159–178.View ArticleGoogle Scholar
 Müller, M, & Woitek, U (2012). Wohlstand, Wachstum und Konjunktur. In P Halbeisen, M Müller, B Veyrassat (Eds.), Wirtschaftsgeschichte der Schweiz im 20. Jahrhundert, (pp. 91–222). Bern: Schwabe.Google Scholar
 Panizza, U, & Presbitero, AF. (2013). Public debt and economic growth in advanced economies: a survey. Swiss Journal of Economics and Statistics, 149(2), 175–204.View ArticleGoogle Scholar
 Park, JY, & Phillips, PCB. (1989). Statistical inference in regressions with integrated processes: part 2. Econometric Theory, 5, 95–l32.View ArticleGoogle Scholar
 Pescatori, A, Sandrim, D, Simon, J (2014). Debt and growth: is there a magic threshold? In IMF Working Paper, WP/14/34.Google Scholar
 Reinhart, CM, & Rogoff, KS. (2010). Growth in a time of debt. American Economic Review, 100(2), 573–578.View ArticleGoogle Scholar
 Schwab, B (1996). Ökonomische Aspekte der Staatsdefizite und Staatsverschuldung in der Schweiz. Bern: Haupt.Google Scholar
 Sims, CA, Stock, JH, Watson, MW. (1990). Inference in linear time series models with some unit roots. Econometrica, 58, 113–144.View ArticleGoogle Scholar
 Steiger, J (1915). Der Finanzhaushalt der Schweiz. Die Kantone (Vol. III). Bern: Stämpfi.Google Scholar
 Steiger, J (1916). Der Finanzhaushalt der Schweiz. Die Gemeinden (Vol. IV). Bern: Stämpfi.Google Scholar
 Steiger, J, & Higy, C (1934). Finanzhaushalt der Schweiz, Bund, Kantone, Gemeinden. Bern, Wyss.Google Scholar
 Swiss National Bank (1944). Manuel statistique du marché financier suisse. Bern: Schulthess.Google Scholar
 Toda, HY, & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1), 225–250.View ArticleGoogle Scholar
 Yule, UG. (1926). Why do we sometimes get nonsensecorrelation between timeseries?—a study in sampling and the nature of timeseries. Journal of the Royal Statistical Society, 89(1), 1–63.View ArticleGoogle Scholar