Skip to main content
SpringerLink
Log in

Is the future really observable? A practical approach to model monetary policy rules

  • Published:
Empirical Economics Aims and scope Submit manuscript

Abstract

We take a practical approach to model the forward-looking monetary policy rule. Unlike existing studies, we recognize that the forward-looking components—future inflation and output growth—are intrinsically unobserved at the time policy formulation. Using the unobserved components framework, we extract the latent components of the policy rule from the short-term and long-term Greenbook forecasts, both individually and in combination, and jointly estimate the policy parameters. We also consider correlations between different components and combine the forecasts from the survey of professional forecasters and the inflation index bonds market. Evidence suggests that the Federal Reserve follows an inflation-tilted policy rule and the long-term state of economy gets a higher weight than the short term. Also, the policy reaction is more aggressive when interconnections between different components of the policy rule are considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Note that the objective of the paper is to develop a new approach to model forward-looking MP function, and not to explicitly test and/or estimate the extent of bias in the estimates.

  2. We would like to highlight that the new approach is remarkably different from the existing studies. Further, the sample period is also different. Hence, one should compare results with a grain of salt.

  3. We consider different sources of forecasts to model MP function in a separate section below.

  4. We acknowledge that Morley et al. (2003) models the cyclical component of inflation as an AR(2) process. Though this is related, it differs slightly in the present context. However, it is important to recognize that the approach used in the paper is novel and in the absence of any preset benchmarks, it is only appropriate to ensure that the model specifications meet the conditions laid by the literature on unobserved component modeling.

  5. Introducing higher-order dynamics is both unnecessary and unwanted. First, it increases the curse of dimensionality, and second, we have tested and found that higher-order lags are statistically insignificant. Thus, it will only lead to losing the important degrees of freedom. We further study interconnections between different components of the model. This also acts as a robustness check of our results.

  6. The only limitation of using Greenbook forecasts is that the data are available exclusively to the Board of Governors staffers, which are publicly available with a 5-year lag. This restricts the use of Greenbook forecasts as a part of the information set of market participants. Nikolsko-Rzhevskyy (2011) provides an alternative approach to overcome the data availability limitation of using Greenbook forecast as a proxy for modeling monetary policy function. The study separately estimates inflation and output forecasts from different model specifications and uses forecasts as close as possible to the Greenbook to estimate the Taylor rule. This strategy overcomes the limitation of the 5-year lag in availability of the Greenbook data.

  7. The results are robust to alternate choice of shadow rates proposed in the literature, for example, Bauer and Rudebusch 2016. Further, one can argue that the estimates of monetary function have an element of uncertainty to the extent the shadow interest rate is seen as a noisy reading of the monetary policy stance. However, the robustness testing of the model provides evidence against the argument that uncertainty in estimates of monetary function is due to the use of a particular shadow rate data. Nevertheless, this does not circumvent the issue that the estimates are conditioned on the shadow rate data used for the ZLB period. The robustness test section elaborates on these issues.

  8. See, for example, Orphanides (2001, (2002, (2004).

  9. There is a rich literature on the links between uncertainty and effectiveness of policy (Brainard 1967; Clarida et al. 1999; Rudebusch 2001; Onatski and Stock 2002; Marfatia 2014, 2015; among others).

  10. Since the paper does not aim at providing the best forecast of the monetary policy rule, which of course is important in itself, we keep the forecasting exercise for future explorations. An in-depth out-of-sample forecasting analysis is also warranted because of the operational constraint that the Greenbook data are publicly available only with a 5-year lag.

  11. Interested researcher should, however, note that HM rate is conditioned on the proxy measure of policy stance during the ZLB period. Though we perform robustness tests using alternative proxy measure in the ZLB period, the HM rate should be interpreted keeping this limitation in mind that it does not completely circumvent the issue that the estimates are conditioned on a stance of policy at the ZLB. We acknowledge anonymous referee(s) for pointing out this limitation.

  12. Though this was in the context of decomposing the level of GDP into its trend and cycle, one can relate this in the present case.

  13. Since the investors intent is to earn returns only in real terms, any difference between a nominal bond (which provides nominal returns) and inflation-protected bond (which provides real returns) should, in theory, exclusively be attributed to future inflation expectations.

  14. Break even inflation rates are also used to model inflation expectation in the new Keynesian Phillips curve (Marfatia 2018).

  15. In interpreting the results, caution is warranted because break-even inflation rates are only available after 2003. Hence, the sample period is different for the results presented in columns 9–10 of Table 4.

  16. For complementary evidence on alternative specifications for computing shadow rates, see Bauer and Rudebusch (2016), Krippner (2013), Christensen and Rudebusch (2014), Ichiue and Ueno (2015), and Kim and Singleton (2012).

References

  • Atkeson A, Ohanian LE. Are Phillips curves useful for forecasting inflation? Fed Reserve Bank Minneap Q Rev. 2001;25(1):2–11.

    Google Scholar 

  • Bårdsen G, Jansen ES, Nymoen R. Econometric evaluation of the New Keynesian Phillips curve. Oxf Bull Econ Stat. 2004;66(s1):671–86.

    Article  Google Scholar 

  • Bauer MD, Rudebusch GD. Monetary policy expectations at the zero lower bound. J Money Credit Bank. 2016;48(7):1439–65.

    Article  Google Scholar 

  • Bovin J. Has US monetary policy changed? Evidence from drifting coefficients and real-time data. J Money Credit Bank. 2006;38(5):1149–73.

    Article  Google Scholar 

  • Brainard WC. Uncertainty and the effectiveness of policy. Am Econ Rev. 1967;57(2):411–25.

    Google Scholar 

  • Breedon F, Chadha JS. The information content of the inflation term structure. Bank of England working paper no. 75 1997.

  • Canzoneri M, Cumby R, Diba B. Monetary policy and the natural rate of interest. J Money Credit Bank. 2015;47(2–3):383–414.

    Article  Google Scholar 

  • Chang AC, Hanson TJ. The accuracy of forecasts prepared for the Federal Open Market Committee. J Econ Bus. 2016;83:23–43.

    Article  Google Scholar 

  • Christensen JHE, Rudebusch GD. Estimating shadow-rate term structure models with near-zero yields. J Financ Econom. 2014;13(2):226–59.

    Article  Google Scholar 

  • Christensen JHE, Lopez JA, Rudebusch GD. Inflation expectations and risk premiums in an arbitrage-free model of nominal and real bond yields. J Money Credit Bank. 2010;42(s1):143–78.

    Article  Google Scholar 

  • Clarida R, Galı J, Gertler M. Monetary policy rules in practice: some international evidence. Eur Econ Rev. 1998;42(6):1033–67.

    Article  Google Scholar 

  • Clarida R, Gali J, Gertler M. The science of monetary policy: a new Keynesian perspective. J Econ Lit. 1999;37(4):1661–707.

    Article  Google Scholar 

  • Clarida R, Gali J, Gertler M. Monetary policy rules and macroeconomic stability: evidence and some theory. Q J Econ. 2000;115(1):147–80.

    Article  Google Scholar 

  • Clark PK. The cyclical component of U.S. economic activity. Q J Econ. 1987;102(4):797–814.

    Article  Google Scholar 

  • D’Agostino A, Whelan K. Federal reserve information during the great moderation. J Eur Econ Assoc. 2008;6(2–3):609–20.

    Article  Google Scholar 

  • English WB, Nelson WR, Sack BP. Interpreting the significance of the lagged interest rate in estimated monetary policy rules. Contrib Macroecon. 2003;3:1.

    Article  Google Scholar 

  • Faust J, Rogers J, Swanson E, Wright J. Identifying VARs based on high-frequency futures data. J Monet Econ. 2004;51(6):1107–31.

    Article  Google Scholar 

  • Fuerst TS. Monetary and financial interactions in the business cycle. Jf Money Credit Bank. 1995;27(4):1321–38.

    Article  Google Scholar 

  • Giannoni MP. Does model uncertainty justify caution? Robust optimal monetary policy in a forward-looking model. Macroecon Dyn. 2002;6(1):111–44.

    Article  Google Scholar 

  • Gulyàs E, Startz R. The Phillips curve: forward-looking behavior and the inflation premium. Unpublished paper. University of Washington; 2006.

  • Harvey AC. Trends and cycles in macroeconomic time series. J Bus Econ Stat. 1985;3(3):216–27.

    Google Scholar 

  • Ichiue H, Ueno Y. Monetary policy and the yield curve at zero interest. J Jpn Int Econ. 2015;38:1–12.

    Article  Google Scholar 

  • Judd JP, Rudebusch GD, et al. Taylor’s rule and the Fed: 1970–1997. Econ Rev Federal Reserve Bank San Francisco. 1998;3:3–16.

    Google Scholar 

  • Kim C-J, Nelson CR. Estimation of a forward-looking monetary policy rule: a time-varying parameter model using ex post data. J Monet Econ. 2006;53(8):1949–66.

    Article  Google Scholar 

  • Kim C-J, Kishor K, Nelson CR et al. A time-varying parameter model for a forward-looking monetary policy rule based on real-time data. University of Washington, Department of Economics, Working Papers UWEC-2007–32 2006.

  • Kim DH, Singleton KJ. Term structure models and the zero bound: an empirical investigation of Japanese yields. J Econom. 2012;170(1):32–49.

    Article  Google Scholar 

  • Kleibergen F, Mavroeidis S. Weak instrument robust tests in GMM and the new Keynesian Phillips curve. J Bus Econ Stat. 2009;27(3):293–311.

    Article  Google Scholar 

  • Knotek II ES, Verbrugge R, Garciga C, Treanor C, Zaman S. Federal funds rates based on seven simple monetary policy rules. Econ Comment; 2016

  • Krippner L. Measuring the stance of monetary policy in zero lower bound environments. Econ Lett. 2013;118(1):135–8.

    Article  Google Scholar 

  • Lansing KJ, Trehan B. Forward-looking behavior and the optimality of the Taylor rule. FRB of San Francisco working paper, 2001;(2001-03).

  • Leamer EE. Housing really is the business cycle: what survives the lessons of 2008–09? J Money Credit Bank. 2015;47(S1):43–50.

    Article  Google Scholar 

  • Lee J, Nelson CR. Expectation horizon and the Phillips curve: the solution to an empirical puzzle. J Appl Econom. 2007;22(1):161–78.

    Article  Google Scholar 

  • Ma Y, Zhang J. Financial cycle, business cycle and monetary policy: evidence from four major economies. Int J Finance Econ. 2016;21(4):502–27.

    Article  Google Scholar 

  • Marfatia HA. Impact of uncertainty on high frequency response of the US stock markets to the Fed’s policy surprises. Q Rev Econ Finance. 2014;54(3):382–92.

    Article  Google Scholar 

  • Marfatia HA. Monetary policy’s time-varying impact on the US bond markets: role of financial stress and risks. N Am J Econ Finance. 2015;34:103–23.

    Article  Google Scholar 

  • Marfatia HA. Estimating the New Keynesian Phillips curve for the UK: evidence from the inflation-indexed bonds market. BE J Macroecon. 2018;18:1.

    Google Scholar 

  • Mavroeidis S. Weak identification of forward-looking models in monetary economics. Oxf Bull Econ Stat. 2004;66(s1):609–35.

    Article  Google Scholar 

  • Mavroeidis S. Identification issues in forward-looking models estimated by GMM, with an application to the Phillips curve. J Money Credit Bank. 2005;37(3):421–48.

    Article  Google Scholar 

  • Mavroeidis S. Monetary policy rules and macroeconomic stability: some new evidence. Am Econ Rev. 2010;100(1):491–503.

    Article  Google Scholar 

  • Morley J, Nelson CR, Zivot E. Why are the beveridge-Nelson and unobserved-components decompositions of GDP so different? Rev Econ Stat. 2003;85(2):235–43.

    Article  Google Scholar 

  • Nason JM, Smith GW. Identifying the new Keynesian Phillips curve. J Appl Econom. 2008;23(5):525–51.

    Article  Google Scholar 

  • Nikolsko-Rzhevskyy A. Monetary policy estimation in real time: forward-looking Taylor rules without forward-looking data. J Money Credit Bank. 2011;43(5):871–97.

    Article  Google Scholar 

  • Nikolsko-Rzhevskyy A, Papell D, Prodan R. Policy rules and economic performance. Working paper 2018.

  • Nolan C, Thoenissen C. Financial shocks and the US business cycle. J Monet Econ. 2009;56(4):596–604.

    Article  Google Scholar 

  • Onatski A, Stock JH. Robust monetary policy under model uncertainty in a small model of the US economy. Macroecon Dyn. 2002;6(1):85–110.

    Article  Google Scholar 

  • Orphanides A. Monetary policy rules based on real-time data. Am Econ Rev. 2001;91:964–85.

    Article  Google Scholar 

  • Orphanides A. Monetary-policy rules and the great inflation. Am Econ Rev. 2002;92(2):115–20.

    Article  Google Scholar 

  • Orphanides A. Monetary policy rules, macroeconomic stability, and inflation: a view from the trenches. J Money Credit Bank. 2004;36(2):151–75.

    Article  Google Scholar 

  • Romer CD, Romer DH. Federal reserve information and the behavior of interest rates. Am Econ Rev. 2000;90(3):429–57.

    Article  Google Scholar 

  • Rudebusch GD. Is the Fed too timid? Monetary policy in an uncertain world. Rev Econ Stat. 2001;83(2):203–17.

    Article  Google Scholar 

  • Rudebusch GD. Term structure evidence on interest rate smoothing and monetary policy inertia. J Monet Econ. 2002;49(6):1161–87.

    Article  Google Scholar 

  • Scholtes C. On market-based measures of inflation expectations. Bank Engl Q Bull. 2002;42(1):67–77.

    Google Scholar 

  • Taylor JB. Discretion versus policy rules in practice. In: Carnegie–Rochester conference series on public policy, vol 39. Elsevier 1993; pp. 195–214.

  • Tetlow R. Real-time model uncertainty in the united states: “robust” policies put to the test. Int J Central Bank. 2015;11(1):113–55.

    Google Scholar 

  • Thornton DL, et al. How did we get to inflation targeting and where do we need to go to now? A perspective from the us experience. Federal Reserve Bank of St Louis Rev. 2012;94(1):65–81.

    Google Scholar 

  • Watson MW. Univariate detrending methods with stochastic trends. J Monet Econ. 1986;18(1):49–75.

    Article  Google Scholar 

  • Wu JC, Xia FD. Measuring the macroeconomic impact of monetary policy at the zero lower bound. J Money Credit Bank. 2016;48(2–3):253–91.

    Article  Google Scholar 

Download references

Acknowledgements

We acknowledge valuable inputs from two anonymous referees and the editor.

Author information

Authors and Affiliations

  1. Department of Economics, Northeastern Illinois University, BBH 344G, 5500 N. St. Louis Ave., Chicago, IL, 60625, USA

    Hardik A. Marfatia

Authors
  1. Hardik A. Marfatia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hardik A. Marfatia.

Ethics declarations

Conflict of interest

Hardik Marfatia declares that he has no conflict of interest.

Ethical approval

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

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I would like to thank two anonymous referees and the editor for many helpful comments.

Appendices

Appendix A: State-space representation of the model with SPF forecasts appended

Measurement equation

$$\begin{aligned} \left[ \begin{array}{c} i_{t} \\ \pi ^{gb}_t \\ g^{gb}_t \\ \pi ^{spf}_t \\ g^{spf}_t \end{array} \right] = \left[ \begin{array}{c} \beta _0\\ 0 \\ 0 \\ 0\\ 0\\ \end{array} \right] + \left[ \begin{array}{c} \beta _1 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} \right] i_{t-1} + \left[ \begin{array}{cccccccccccc} 1 &{} \beta _2 &{} \beta _3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0 &{} 0 &{}0\\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{}0 &{} 0 &{}0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{}0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{}0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{}0\\ \end{array} \right] \left[ \begin{array}{c} \epsilon _t \\ \pi ^{e}_{t} \\ g^{e}_{t} \\ V_{1,t} \\ V_{1,t-1} \\ W_{1,t} \\ W_{1,t-1} \\ V_{2,t} \\ V_{2,t-1} \\ W_{2,t} \\ W_{2,t-1} \\ \end{array} \right] \end{aligned}$$
(18)

Transition equation

$$\begin{aligned} \left[ \begin{array}{c} \epsilon _t \\ \pi ^{e}_{t} \\ g^{e}_{t} \\ V_{1,t} \\ V_{1,t-1} \\ W_{1,t} \\ W_{1,t-1} \\ V_{2,t} \\ V_{2,t-1} \\ W_{2,t} \\ W_{2,t-1} \\ \end{array} \right] = \left[ \begin{array}{ccccccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \phi _{11} &{} \phi _{12} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \theta _{11} &{} \theta _{12} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \phi _{21} &{} \phi _{22} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \theta _{21} &{} \theta _{22} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} \epsilon _{t-1} \\ \pi ^{e}_{t-1} \\ g^{e}_{t-1} \\ V_{1,t-1} \\ V_{1,t-2} \\ W_{1,t-1} \\ W_{1,t-2} \\ V_{2,t-1} \\ V_{2,t-2} \\ W_{2,t-1} \\ W_{2,t-2} \\ \end{array} \right] + \left[ \begin{array}{c} \epsilon _t \\ e_{\pi ,t} \\ e_{g,t} \\ v_{1,t} \\ 0 \\ \omega _{1,t} \\ 0 \\ v_{2,t} \\ 0 \\ \omega _{2,t} \\ 0 \\ \end{array} \right] \end{aligned}$$
(19)

Appendix B: State-space representation of the model with inflation index bonds market’s forecasts appended

Measurement equation

$$\begin{aligned} \left[ \begin{array}{c} i_{t} \\ \pi ^{gb}_t \\ g^{gb}_t \\ \pi ^{spf}_t \\ g^{spf}_t \\ \pi ^{bo}_t \end{array} \right] = \left[ \begin{array}{c} \beta _0\\ 0 \\ 0 \\ 0\\ 0\\ 0 \\ \end{array} \right] + \left[ \begin{array}{c} \beta _1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} \right] i_{t-1} + \left[ \begin{array}{ccccccccccccc} 1 &{} \beta _2 &{} \beta _3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} \epsilon _t \\ \pi ^{e}_{t} \\ g^{e}_{t} \\ V_{1,t} \\ V_{1,t-1} \\ W_{1,t} \\ W_{1,t-1} \\ V_{2,t} \\ V_{2,t-1} \\ W_{2,t} \\ W_{2,t-1} \\ V_{3,t} \\ V_{3,t-1} \end{array} \right] \end{aligned}$$
(20)

Transition equation

$$\begin{aligned} \left[ \begin{array}{c} \epsilon _t \\ \pi ^{e}_{t} \\ g^{e}_{t} \\ V_{1,t} \\ V_{1,t-1} \\ W_{1,t} \\ W_{1,t-1} \\ V_{2,t} \\ V_{2,t-1} \\ W_{2,t} \\ W_{2,t-1} \\ V_{3,t} \\ V_{3,t-1} \end{array} \right] = \left[ \begin{array}{ccccccccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \phi _{11} &{} \phi _{12} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \theta _{11} &{} \theta _{12} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \phi _{21} &{} \phi _{22} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \theta _{21} &{} \theta _{22} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \phi _{31} &{} \phi _{32} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} \epsilon _{t-1} \\ \pi ^{e}_{t-1} \\ g^{e}_{t-1} \\ V_{1,t-1} \\ V_{1,t-2} \\ W_{1,t-1} \\ W_{1,t-2} \\ V_{2,t-1} \\ V_{2,t-2} \\ W_{2,t-1} \\ W_{2,t-2} \\ V_{3,t-1} \\ V_{3,t-2} \end{array} \right] + \left[ \begin{array}{c} \epsilon _{t} \\ e_{\pi ,t} \\ e_{g,t} \\ v_{1,t} \\ 0 \\ \omega _{1,t} \\ 0 \\ v_{2,t} \\ 0 \\ \omega _{2,t} \\ 0 \\ v_{3,t} \\ 0\\ \end{array} \right] \end{aligned}$$
(21)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marfatia, H.A. Is the future really observable? A practical approach to model monetary policy rules. Empir Econ 61, 1189–1223 (2021). https://doi.org/10.1007/s00181-020-01910-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00181-020-01910-7

Keywords

JEL Classifications

Search

Navigation