The Natural Real Interest Rate and the Output Gap in the Euro Area: A Joint Estimation

Garnier, Julien; Wilhelmsen, Bjørn-Roger Working Paper The Natural Real Interest Rate and the Output Gap in the Euro Area: A Joint Estimation Working Paper, No. 2005/14 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Garnier, Julien; Wilhelmsen, Bjørn-Roger (2005) : The Natural Real Interest Rate and the Output Gap in the Euro Area: A Joint Estimation, Working Paper, No. 2005/14, ISBN 82-7553-334-1, Norges Bank, Oslo, https://hdl.handle.net/11250/2498430 This Version is available at: https://hdl.handle.net/10419/209861 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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The Natural Real Interest Rate and the Output Gap in the Euro Area: A Joint Estimation∗ Julien Garnier†and Bjørn-Roger Wilhelmsen‡ Abstract The notion of a natural real rate of interest, due to Wicksell (1936), is widely used in current central bank research. The idea is that there exists a level at which the real interest rate would be compatible with output at its potential level and stationary inflation. Such a consept is of primary concern for monetary policy because it provides a benchmark for the monetary policy stance. This paper applies the method recently suggested by T. Laubach and J.C.Williams to jointly estimate the natural real interest rate and the output gap in the euro area using data from 1960. Our results suggest that the natural real rate of interest has declined gradually over the past 40 years. They also indicate that monetary policy in the euro area was on average stimulative during the 1960s and the 1970s, while it contributed to dampen the output gap and inflation in the 1980s and 1990s Key words: Real interest rate gap, output gap, Kalman filter, euro area JEL Classification Numbers: C32, E43, E52, O40 ∗Work on the project began while Julien Garnier was doing an intership and Bjørn-Roger Wilhelmsen was on secondment at the European Central Bank. Allviewsexpressedinthepaperareonlythoseofthe authors and are not necessarily those of the European Central Bank (ECB) or the institutions to which the authors are affiliated. We are grateful to Siem Jan Koopman for very helpful suggestions and comments. We also thank P.Cour-Thimann, V.Curdia, F.Drudi, S.McCaw, D.Rodriguez-Palenzuela, R.Pilegaard, H.Pill, L.Stracca, T. Laubach, J.C.Williams and the participants of the ECB workshop on natural interest rates on February 19th 2004. †European University Institute and University of Paris X - Nanterre. [email protected] ‡Central Bank of Norway, Economics Department. b[email protected]. 1 1Introduction In the long run, economists assume that nominal interest rates will tend toward some equilibrium, or ”natural”, real rate of interest plus an adjustment for expected long-run inflation. The natural rate of interest is a central concept in the monetary policy literature since it provides policymakers with a benchmark for monetary policy. In theory, it is important for evaluating the policy stance since rates above (below) the natural rate are expected to lower (raise) inflation. From an empirical point of view, the "natural" real rate of interest is unobservable. The estimation of the natural real interest rate is not straightforward and is associated with a very high degree of uncertainty. In practice, therefore, policymakers cannot rely exclusively on the real interest rate gap, defined as the difference between the real short term interest rate and estimates of the natural real interest, as an indicator of the monetary policy stance. Rather, a comprehensive approach using a wide set of information is required. This notwithstanding, central bank economists have increasingly devoted attention to developing estimation strategies for the natural real interest rate. The methods used range from as simple as calculating the average actual real interest rate over a long period to building dynamic stochastic general equilibrium (DSGE) models with nominal rigidities. A recent contributions to the literature on how to empirically approach the concept of the natural rate is a paper by Laubach and Williams (2003; henceforth LW)1with an application to data for the United States. They suggest to estimate the natural real interest rate and potential output growth simultaneously, using a small-scale macroeconomic model and Kalman filtering techniques. In this model, the natural real interest rate is related to the potential growth rate of the economy. Thus, the estimate of the natural real interest rate is time-varying and related to long-term developments in the real characteristics of the economy, consistent with economic theory. This method has become popular since it strikes a compromise between the theoretically coherent DSGE approach and ad-hoc statistical approaches, as emphasized by Larsen and McKeown (2002). In this paper we employ the technique of LW for the euro area. The present work differs from others in that we use a relatively long (synthetic) dataset starting in the early 1960s. Besides, we estimate the natural real interest rate in Germany and the US for comparison. Such comparison is interesting because, prior to 1999, monetary policies in Europe were considerably influenced by the Bundesbank policy. At the same time, the US is often considered as a useful proxy for global influences that affect monetary policy worldwide. Third, we apply simple statistical tests to investigate the leading indicator properties of the baseline estimate of the euro area real interest rate gap on inflation and economic activity. Thepaperisstructuredasfollows.Insectiontwowetakealookatthedataforthereal interest rate from 1960. Sections three reviews the recent empirical literature and discusses the natural rate concept in the context of different horizons. Section four presents the 1Some papers have already used this framework on European data, including Sevillano and Simon (2004) for Germany, Larsen and McKeown (2002) for the United Kingdom and Crespo-Cuaresma et al (2003) and Mésonnier and Renne (2004) for the euro area. 2 modeling approach and displays the estimation results. Section five briefly discusses the leading indicator properties of the estimated real rate gap. Section six concludes. 2 The data In this section we take a closer look at the data used in this study. The data covers a total of 161 quarterly observations from 1963q1 to 2004q1 for the euro area and Germany and 162 quarterly observations from 1961q1 to 2002q4 for the US. The dataset consists of short-term interest rates, inflation and Gross Domestic Products for the three economies2. The computation of real interest rates is subject to several practical and conceptual difficulties. Ideally, an estimate of the real interest rate should be obtained by substracting ex ante inflation expectations from nominal interest rates. However, the lack of good data for inflation expectations forces us to take a more straightforward approach. In this paper, the real interest rates are calculated from the three-month money market rates and annual consumer price inflation rates3. While we believe that the problem with using current inflation as a proxy for inflation expectations is less severe when assessing developments over longer horizons, the deviations may be stronger in periods with unanticipated inflation, notably in the 1970s. It should be borne in mind that monetary policy regimes differed significantly over time and across countries. Moreover, in many euro area countries, specifically in the 1960s and early 1970s, other instruments than interest rates were important in the conduct of monetary policy. In particular, capital controls prevailed in many euro area countries. Furthermore, inflation, economic growth and interest rates were very volatile in some euro area countries. Finally, the euro area real interest rate has been significantly influenced by other factors than monetary policy, such as tensions within the Exchange Rate Mechanism (ERM) at the turn of the 1990s (Cour-Thimann et al, 2004). The euro area real interest rate fell dramatically in the 1970s, when overheated economies and rising oil prices pushed up inflation at a level that could not be offset by the nominal real interest rates (See Figure 1). Following the trough in the mid-1970s, as European monetary authorities gradually put more emphasis on disinflationary policies, the real interest rate increased slowly over a period of more than 15 years. After peaking in the early 1990s, the real rate declined gradually again, influenced by the monetary authorities’ achievement of more favourable inflation developements. Similar to the euro area aggregate, the real interest rate in the United States was also lower than average in the 1970s and higher thanitsaverageinthe1980s. However,the persistence in the data seems less pronounced, as indicated by the quick rise in the real rate at the turn of the 1980s. In Germany the real interest rate has been more stable around its long-term average, reflecting the achievment of lower and more stable inflation over the whole sample. 2We are grateful to the ECB for providing the data for Germany and the euro area, and Thomas Laubach and John C. Williams for providing the US data. 3For the euro area, national levels for interest rates and consumer prices have been aggregated prior to 1999 using GDP and consumer spending weights respectively at PPP exchange rates, see ECB (2003). 3 1965 1970 1975 1980 1985 1990 1995 2000 2005 0 5 Germany 1965 1970 1975 1980 1985 1990 1995 2000 2005 0 5 10 US 1965 1970 1975 1980 1985 1990 1995 2000 2005 0 5 10 Euro area Figure 1: Real interest rates 3 A review of the recent empirical literature: short vs long-run perspectives The concept of a natural rate of interest was first introduced by Knut Wicksell in the late 19th century (1898, with 1936 translation). Today the concept knows a revival of interest following Woodford’s seminal book, Interest and Prices. According to the recent literature, fluctuations in the real interest rate may be decomposed into two different components: a natural real rate and a real rate gap (Woodford, 2003; Neiss and Nelson, 2003; Cour-Thimann et al, 2004). The natural real rate is related to structural factors and is the real interest rate that in theory would prevail under perfectly flexible prices. This is commonly referred to as the "Wicksellian" definition of the natural rate of interest. The real interest rate gap is related to the business cycle and reflects the existence of nominal rigidities in the economy. The available estimates of historical developments in the euro area natural real interest rate differ considerably from one author to the other. We briefly address these differences below and classify estimates of the natural real rate, taking as cirterium the time horizon at which they should be interpreted. Some papers find that most of the fluctuations in the real interest rate should be attributed to fluctuations in the real interest rate gap rather than the natural real interest rate. This group of papers, which includes Giammarioli and Valla (2003), Mésonnier and Renne (2004), Neiss and Nelson (2003), Sevilliano and Simon (2004) and LW, associate the fluctuations in the natural real interest rate with the evolution of real fundamentals such as determinants of trend GDP growth and preferences. These variables are typically stable in the short to medium term, but may display some variation in the longer run. Consequently, 4 the natural real interest rate is also relatively stable in the short run, and the natural rate in these papers should be considered in a ”long-run” perspective. It refers indeed to the level expected to prevail in, say, the next five to ten years, after any business cycle ”booms” and ”busts ” underway have played out. Note however that the estimated natural real rate of Mésonnier and Renne (2004) is much more volatile that that of LW or Sevilliano and Simon (2004). On the contrary, other papers conclude that fluctuations in the natural real interest rate explain most of the variation in the real interest rate (Basdevant et al, 2004; Cuaresma et al, 2004; Cour-Thimann et al, 2004; Larsen and McKeown, 2002). The papers consistent with this view typically make use of the Kalman filter or other filtering techniques to split the actual real rate into a trend (the natural real rate) and a cyclical component (the real rate gap). However, the models they use do not necessarily contain judgements about the determinants of the natural rate. Rather, the approach they take is closer to a pure statistical measure. Consequently, variations in the natural rate are more pronounced, because the natural rate tends to follow more closely the medium term fluctuations in the actual real rate. The interpretation of the natural real interest rate in this context is therefore likely to be more relevant in a ”shorter” time perspective in that it refers to a neutral monetary policy stance in a situation where the economy has not necessarily settled at its long-run levels. 4 Estimating the natural rate of interest This paper takes a ”long-run” time-perspective and uses economic theory as a benchmark for determining the developments in the natural real interest rate. As recalled by LW, standard growth models imply that the natural real interest rate varies over time in response to shifts in preferences and the trend growth rate of output, themselves unobservable variables. 4.1 The model The empirical framework suggested by LW is to run the Kalman filter on a system of equations to jointly estimate the natural real interest rate, potential output growth and the output gap. They propose a model of a neo-keynesian inspiration, that jointly characterises the behaviour of inflation and the output gap through modified IS and Phillips curves. Neo-keynesian models are not so much interested in the levels of variables composing these curves, but rather the deviations from equilibrium values. The main equations of the model are given by: ay(L)˜yt=ar(L)˜rt+ε1,t (1) bπ(L)πt=by(L)˜yt+ε2,t (2) where ˜rtis the real interest rate gap, ˜ytrepresents the output gap defined as the difference between the (log) GDP ytand (log) potential output y∗ tsuch that ˜yt=100(yt−y∗ t)(3) 5 πtis consumer price inflation, ε1,t and ε2,t are white noise errors, and ay,a r,b πand byare polynomials in the lag operator Lsuch that ay(L)=−Pn i=0 ay,iLi,with ay,0=−1. The laws of motion of unobservable potential output and its trend growth rate are specified as the following: y∗ t=y∗ t−1+gt−1+ε4,t (4) gt=gt−1+ε5,t (5) where ε4,t and ε5,t are white noise errors. The economic theory imposed by LW is represented by the following relationship for the natural real interest rate: r∗ t=cgt+zt(6) where gtis the unobservable trend growth rate of the economy, which is linked to the natural real interest rate r∗ twith the parameter c, capturing the relative risk aversion. ztrepresent other possible determinants of the natural rate of interest, such as households time preferences, variation in public saving and uncertatinty about interest rates. ztis assumed to follow a stochastic process determined by: zt=αzt−1+ε3,t (7) In LW, zt, is either a stationary AR process or a random walk. The measure of the random determinants of the natural real interest rate, zt,is obviously associated with a considerable degree of uncertainty. We face here a technical problem in that r∗ tis an unobserved variable, itself composed of two unobserved components. This difficulty has already been pointed out by Mésonnier & Renne (2004). In most of the specifications that we have tried, the results were highly sensitive to initial conditions and were often not reasonable. This is especially the case when ztfollows a random walk. Indeed, while the first element gtis explicitly linked to the output through (3) and (4), ztis only defined through (7), which makes it more sensitive to small variations in the initial specifications of the model. To overcome this problem, we only consider the case where ztfollows a stationary AR process, and we restrict its variance σ2 3through a signal-to-noise ratio, λz. Weelaboratefurtheronthispointinthenextsection. In addtion, we claim that the possible elements composing ztshould be stationary, thereby making a random walk specification useless. Equation (4), (5), (6) and (7) constitute the state (transitory) equations of our statespace model, and the IS curve (1) and the Phillips curve (2) constitute the observation equations (see Harvey (1989)). On this system, the Kalman filter is run twice: first in order to identify parameters by maximum likelihood, and second in order to estimate the unobserved components r∗ t,y∗ t,gtand zt. The model can be written under its state-space form (see appendix A): yt=Zαt+Bxt+Gεt(8) αt+1 =Tαt+Hεt(9) 6 1965 1970 1975 1980 1985 1990 1995 2000 2005 -5.0 -2.5 0.0 2.5 5.0 ~ rt euro area ~ rt Germany Figure 6: Natural real interest rate gap ˜rt(= rt−r∗ t),euroareaandGermany. 1965 1970 1975 1980 1985 1990 1995 2000 2005 2 3 4 2 3 4 Euro area r* average r* 1965 1970 1975 1980 1985 1990 1995 2000 2005 1 2 3 1 2 3 Germany 1965 1970 1975 1980 1985 1990 1995 2000 3.0 3.5 4.0 3.0 3.5 4.0 US Figure 7: Baseline estimates of r∗ tfor the euro area, Germany and the US 13 5 Statistical properties of the real interest gap in the euro area We now examine some statistical properties of the euro area model, focusing on simple statistics that describe the relationship between the real interest rate gap and inflation. Table 2 and 3 report standard deviations and correlations of selected variables used in this analysis, namely log output yt, log potential output y∗ t, the output gap ˜yt, the actual and the natural real rate and the real rate gap (rt,r ∗ tand ˜rt)and inflation πt. A notable feature of the reported statistics is that the correlation between the actual real interest rate and the real interest rate gap are high and their standard deviations are roughly identical. In other words, the variation in the real interest rate is not primarily related to variation in the natural real interest rate. This is consistent with the results in Giammarioli and Valla (2003) for the euro area, Neiss and Nelson (2003) for the UK and LW for the US, but stands against the results of Cour-Thimann (2004) for the euro area. Table 2: Standard deviations, euro area yt0.35 y∗ t0.30 ˜yt1.72 πt3.50 rt2.55 r∗ t0.80 ˜rt2.93 Table 3 : Correlation coefficients, euro area k=0 k=1 k=2 k=3 k=4 Corr(rt,˜rt−k)0.98 Corr(πt,˜rt−k)-0.47 -0.48 -0.50 -0.53 -0.55 Corr(˜ytt,˜rt−k)-0.55 -0.57 -0.59 -0.60 -0.62 Interestingly, the correlation between inflation and the real rate gap is strongly negative at all lags. This indicates that the developments in euro area inflation since 1960 are, in part,relatedtotheevolutionoftherealinterestrategap. Inlinewiththishypothesiswe find that the real rate gap is also strongly and negatively correlated with the output gap. Next, we investigate the leading indicator properties of the real interest rate gap for inflation. Following Neiss and Nelson (2003), we estimate: πt=a+b1πt−1+b2(rt−k−r∗ t−k)+εt 14 where annual inflation πtis regressed on past inflation and lagged values of the real interest rate gap. The regression on the 1962:1 - 2003:1 sample is summarised in table 4 (numbers in parantheses are standard errors). The results indicate that lagging the real interest rate gap 3 to 5 quarters yields statistically significant parameter estimates when added to an autoregression for inflation. The long lags seem consistent with the common view that monetary policy affects inflation after a significant delay. This simple exercise suggests that the estimated real interest rate gap may contain valuable information about future inflation. Table 4 : Parameter estimates, euro area k=1 k=2 k=3 k=4 k=5 a0.08 0.10 0.13 0.15 0.18 b10.98 0.98 0.97 0.97 0.97 b2−0.02 (−1.53) −0.03 (−1.97) −0.04 (−2.85) −0.05 (−3.07) −0.06 (−3.70) R20.98 0.98 0.98 0.98 0.98 6Conclusion This paper estimates the natural real interest rate for the euro area, considering the currency union as a single entity over the period 1963 to 2004, and for Germany and the US. Following closely the methodology suggested by Laubach and Williams (2003), we apply the Kalman filter to a small scale macroeconomic model, which encompasses a Phillips curve and an IS curve. This allows us to estimate the natural real interest rate, potential output and trend growth rate of the three economies simultaneously. Overall, the results are quite comparable with the original results of Laubach and Williams (2003) using US data. According to our baseline estimate for the euro area, the fluctuations in the natural real interest rate have been relatively low since 1963. The natural rate has declined gradually over the past 40 years, from an estimate of around 4% in the 1960s to slightly less than 2% in 2004. The real interest rate gap is relatively persistent over longer periods, with low short-term fluctuations. Moreover, the real interest rate gap indicated that monetary policy was on average stimulative in the 1960s and 1970s, while it contributed to dampen economic activity and inflation in the 1980s and 1990s. Regarding the output gap, the length of the business cycle’s booms and busts are in line with the consensus view in the business cycle literature. However, this model produces an output gap that is influenced by the real interest rate gap. Its average level is negative in periods of tight monetary policy and positive in periods of monetary laxism. This is interesting since it might be taken as an indicator of the degree at which the central bank policy influences the real economy. In the 1970s when inflation become high. the real interest rate gap was negative and the output gap was positive, on average. Likewise, in the 1980s and 1990s, when inflation fell to lower levels, the real interest rate was positive and the output gap was on negative, on average. Simple empirical tests also suggest that the estimated real interest rate gap is negatively 15 correlatedwiththeoutputgapandinflation. Furthermore, the tests show that the real rate gap may contain valuable information about future inflation in the euro area. The general caveats associated with interpreting estimates of the natural real interest rate, which are highly uncertain, also applies for this paper. References [1] Barro, R.J. and Sala-i-Martin, X. (1995): "Economic Growth.": McGraw-Hill. [2] Basdevant, O.N. Björksten and O. Karagedikli (2004): "Estimating a time varying neutral real interest rate for New Zealand ", Reserve Bank of New Zealand Discussion Paper, DP2004/01 [3] Blanchard, O. and S. Fischer (1989): "Lectures on Macroeconomics.": Cambridge, MA: MIT-press. [4] Cour-Thimann, Philippine, Rasmus Pilegaard and Livio Stracca (2004): "The output gap and the real interest rate gap in the euro area, 1960-2003", paper presented at the Bank of Canada workshop on Neutral Interest Rates, 9-10 September 2004 (available on Bank of Canada’s website). [5] Cuaresma, J.C., Gnan, E . and D. Ritzberger-Grunenwald (2004): "Searching for a natural rate of interest: a euro area perspective", Empirica, vol 31, p. 185 - 204 [6] De Long and J. Bradford (1997): "America’s Peacetime Inflation: the 1970s ", in C.D. Romer and D.H. Romer (eds.), Reducing Inflation: Motivation and Strategy. Chicago: University of Chicago Press. 247-276 [7] Ford, Robert and Douglas Laxton (1999), "World Public Debt and Real Interest Rates", Oxford Review of Economic Policy, Vol. 15 No.2. [8] Griammarioli, Nicola and Natacha Valla (2003), "The Natural Rate of Interest in the Euro area", ECB Working Paper No. 233 [9] Harvey, Andrew C. (1989): "Forecasting, structural time series model and the Kalman filter", Cambridge University Press, Cambridge. [10] Koopman S.J., Shephard N. and Doornik J.A. (1998), "Statistical algorithms for models in state space using SsfPack 2.2", Econometrics Journal, 1, 1-55 and http://www.ssfpack.com/ [11] Larsen, Jens D. J. and Jack McKeown (2002), "The information content of empirical measures of real interest rate and output gaps for the United Kingdom", BIS papers No 19 [12] Laubach, Thomas and John C. Williams (2003), "Measuring the Natural Rate of Interest", Review of Economics and Statistics, vol 85(4), p. 1063-1070 [13] Levin, A., V. Wieland and J.C.Williams (2003). "Robustness of Simple Monetary Policy Rules under Model Uncertainty " in John B. Taylor, ed. Monetary Policy Rules, Chicago. University of Chicago Press. pp, 263-299 [14] Maravall, A. (1995) “Unobserved Components in Economic Time Series”, in M.H.Pesaran and M.R.Wickens (eds.), Handbook of applied econometrics. Volume 1. Macroeconomics. Handbooks in Economics, Blackwell, Oxford and Malden, 12-72. [15] Mésonnier, J.-S, and J.-P. Renne (2004) “A Time-Varying Natural Rate of Interest for 16 theeuroarea”,WorkingpaperNo115,BanquedeFrance. [16] Neiss, K. and Nelson, E (2003): "The real rate gap as an Inflation Indicator ", Macroeconomic Dynamics, 7, pp. 239 -262. [17] Nelson, E. (2004): "The Great Inflation of the Seventies: What really Happened?", Fed working paper, Januar 2004 [18] Nelson, E. and K. Nikolov (2002): "Monetary Policy and Stagflation in the U.K. " Bank of England Working paper No. 155, May [19] Nelson, E. and K. Nikolov (2003): "U.K. Inflation in the 1970s and 1980s: the Role of Output Gap Mismeasurement", Journal of Economics ans Business, Vol. 55 (4), 353-370. [20] Proietti, T. (2001) “Unobserved Components Models for Measuring Output Gaps, Capacity Utilisation and Core Inflation”, mimeo, notes prepared for the course ‘Unobserved Components Models and their Applications in Macroeconomics’, European Central Bank, 11 and 12 June 2001. [21] Sevillano, J. M. M. and M. M. Simon (2004): “An empirical approximation of the natural rate of interest and potential growth”, Working paper No 0416, Banco de Espana. [22] Stock J. and M.Watson (1998), "Median unbiased estimation of coefficient variance in a time-varying parameter model", Journal of the American Statistical Association, March, 93, p.349-358. [23] Stokey, N and S. Rebelo (1995): "Growth Effects of Flat-Rate Taxes. " Journal of Political Economy.Vol.103,No.3. [24] Taylor, John B. (1992): "The great inflation, the great disinflation and policies for future price stability. " In A. Blundell-Wignall (ed), Inflation, Disinflation and Monetary Policy. Sydney. Ambassador press. 9-34. [25] Wicksell, K. (1898) "Interest and prices", London, Macmillan, 1936. Translation of 1898 edition [26] Williams, J.C. (2003) "The natural rate of interest", FRBSF Economic letter 2003-32 [27] Woodford, M. (2003): "Interest and prices: foundations of a theory of monetary policy", Princeton University Press. 17 Appendices AState-spaceform The system to be estimated, eq (1), (2), (4), (5) and (7), has to be put into the general state-space form: αt+1 =Tαt+Hεt(10) yt=Zαt+Bxt+Gεt(11) where Z,B,G,Tand Hare matrices of coefficients12. (10) is the so-called measurement equation and (11) the state equation. ytis a vector of dependent variables, corresponding to ytand πtin our model, αtis a vector corresponding to the unobserved components, here y∗ t,g tand zt.r∗ tis suppressed from the model because it can be fully recovered from gtand zt,provided we obtain c.xtis a vector of deterministic variables. εt∼NID(0,I) and α0∼N(a, P).Note that the representation of (12) is particular in that the vector of parameters Bis treated as a vector of unobserved variables. This is a feature of the library Ssfpack for Ox that we use in this paper. The system can be rewritten ⎛ ⎝ αt+1 B yt ⎞ ⎠=⎛ ⎝ T0 0I Zx t ⎞ ⎠µαt B¶+⎛ ⎝ ut 0 vt ⎞ ⎠(12) =Φµαt B¶+ut,ut∼NID(0,utu0 t) where α0 t=¡y∗ ty∗ t−1··· y∗ t−ngtgt−1ztzt−1¢,B0=¡Ca x··· arx bx··· bsx ¢,whereCequals 0 or ρ, depending on the specificiation of zt(see eq. 7). T= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10··· 010 0 0 1··· 00 0 . . ..... . .. . ..... . . 0··· 10 0 10 . . .10 10 0··· 010 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Z=µ1−ay1··· −ayn 0−c.ar0−ar 0−by1··· 00··· ··· 0¶ xt=µ0x1t··· xrt 0··· 0 00··· 0x0 1t··· x0 st ¶ 12We assume here that they are constant since this is the specification used in our model, but they could just as well be defined as time-varying parameters. 18 and u0 t=¡ε4,t 0··· 0ε5,t 0ε3,t 0··· ··· 0ε1,t ε2,t ¢. Yt=µY1 Y2¶=µyt−ay1yt−1−... −aynyt−n−arrt−1 πt−bπ1πt−1−... −bπpπt−p−byyt−1¶ The specification of the vector of dependent variables allows us to impose constraints on the coefficients on some of the variables (e.g. ytand y∗ t). The other exogenous variables, namely the dummies xt,are estimated as in a classical regression model, i.e. are treated as unobserved variables, upon which no constraint can be imposed. B Sensitivity analysis B.1 Shorter sample 1965 1970 1975 1980 1985 1990 1995 2000 2005 0 1 2 3 4 5 0 1 2 3 4 5 λg restricted c restricted full sample, baseline unrestricted λg and c restricted Figure 8: Shorter sample. r∗ testimates 1973Q2-2004Q2 19 B.2 Sensitivity to signal-to-noise ratios: unrestricted c 1965 1970 1975 1980 1985 1990 1995 2000 2005 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 Sensitivity of r* to λg (λz = 0.064, baseline value) λg = 0.081 (baseline) λg = 0.12 (95th percentile) λg = 0.013 (5th percentile) Figure 9: The natural real interest rate. Sensivity to signal to noise ratios with unrestricted c. 1965 1970 1975 1980 1985 1990 1995 2000 2005 -17.5 -15.0 -12.5 -10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 -17.5 -15.0 -12.5 -10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 Sensitivity of output gap to λg (λz = 0.064, baseline value) λg = 0.081 (baseline) λg = 0.12 (95th percentile) λg = 0.013 (5th percentile) Figure 10: The output gap. Sensivity to signal to noise ratios with unrestricted c. 20 1965 1970 1975 1980 1985 1990 1995 2000 2005 1 2 3 4 1 2 3 4 Sensitivity of rt* to λz (λg = 0.081, baseline value) λz = 0.064 (baseline) λz = 0.076 (95th percentile) λz = 0.047 (5th percentile) Figure 11: The natural real interest rate. Sensivity to signal to noise ratios with unrestricted c. 1965 1970 1975 1980 1985 1990 1995 2000 2005 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 Sensitivity of output gap to λz (λg = 0.081, baseline value) λz = 0.064 (baseline) λz = 0.076 (95th percentile) λz = 0.046 (5th percentile) Figure 12: The output gap. Sensivity to signal to noise ratios with unrestricted c. 21 B.3 Sensitivity to signal-to-noise ratios: crestricted to baseline value 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Sensitivity of the natural interest rate rt* to λg λz = 0.064 and c = 0.88 (baseline values) λg = 0.081 (baseline) λg = 0.12 (95th percentile) λg = 0.013 (5th percentile) Figure 13: The natural real interest rate. Sensivity to signal to noise ratios with restricted c. 1965 1970 1975 1980 1985 1990 1995 2000 2005 -4 -2 0 2 4 -4 -2 0 2 4 Sensitivity of output gap to λg λz = 0.064 and c = 0.88 (baseline values) λg = 0.081 (baseline) λg = 0.12 (95th percentile) λg = 0.013 (5th percentile) Figure 14: The output gap. Sensivity to signal to noise ratios with restricted c. 22