## 1 Introduction

^{1}An additional explanation that is commonly suggested––and which clearly is related to some of the previously mentioned factors––is that the natural rate of interest has declined; see, for example, Holston et al. (2017), Rachel and Smith (2017) and Benati (2023).

^{2}

## 2 The model

^{3}However, as in Laubach and Williams (2003) there are also deviations from this relationship that are modelled with another non-observable and time-varying series \({z}_{t}\) according to

^{4}But while monetary policy is not explicitly modelled, it nevertheless affects the macro economy and is, accordingly, reflected in its behaviour. While this setting does not invite structural interpretations of monetary policy, it does allow us to estimate the monetary policy stance (which is given by the real interest rate gap). Given the cyclical nature of monetary policy, an AR (2) process seems like a reasonable choice for this process.

^{5}

## 3 Data

^{6}As the nominal interest rate, we use the three-month interbank rate.

Denmark | |||

Interest rate, CIBOR, 3 Month | 1990Q1–2022Q4 | None | DFBF |

Consumer price index (CPI) inflation, SA | 1990Q1–2022Q4 | 400 × log difference of price index | Statistics Denmark |

Core inflation, SA | 1990Q1–2022Q4 | 400 × log difference of price index | OECD Economic Outlook |

Gross domestic product, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Denmark |

Final consumption expenditure, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Denmark |

Gross fixed capital formation, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Denmark |

Exports, goods & services, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Denmark |

Imports, goods & services, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Denmark |

Unemployment rate, percent, SA | 1990Q1–2022Q4 | None | Eurostat |

Sentiment indicators, consumer confidence indicator | 1990Q1–2022Q4 | Standardized | Eurostat |

Current level of capacity utilization, balance, SA | 1996Q1–2022Q4 | Standardized | Eurostat |

Norway | |||

Interest rate, NIBOR, 3 Month | 1990Q1–2022Q4 | None | NoRe |

Consumer price index (CPI) inflation, SA | 1990Q1–2022Q4 | 400 × log difference of price index | Statistics Norway |

Core inflation, SA | 1990Q1–2022Q4 | 400 × log difference of price index | OECD Economic Outlook |

Gross domestic product, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Norway |

Final consumption expenditure, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Norway |

Gross fixed capital formation, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Norway |

Exports, goods & services, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Norway |

Imports, goods & services, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Norway |

Unemployment rate, percent, SA | 1990Q1–2022Q4 | None | Eurostat |

Consumer surveys, expectations barometer | 1992Q4–2022Q4 | Standardized | Kantar DNS, Norway |

Capacity utilization, weighted average, trend adjusted | 1996Q1–2022Q4 | Standardized | Statistics Norway |

Sweden | |||

Interest rate, STIBOR, 3 Month | 1990Q1–2022Q4 | None | SFBF |

Consumer price index (CPIF) inflation, SA | 1990Q1–2022Q4 | 400 × log difference of price index | Statistics Sweden |

Core inflation, SA | 1990Q1–2022Q4 | 400 × log difference of price index | OECD Economic Outlook |

Gross domestic product, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Sweden |

Final consumption expenditure, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Sweden |

Gross fixed capital formation, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Sweden |

Exports, goods & services, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Sweden |

Imports, goods & services, constant prices, SA | 1990Q1–2022Q4 | 100 × log of index | Statistics Sweden |

Unemployment rate, percent, SA | 1990Q1–2022Q4 | None | Eurostat |

Sentiment indicators, consumer confidence indicator | 1995Q4–2022Q4 | Standardized | Eurostat |

Current level of capacity utilization, balance, SA | 1996Q1–2022Q4 | Standardized | Eurostat |

^{7}The last two variables (capacity utilization and consumer confidence indicator) should be mean reverting and have a zero mean by construction; they are hence modelled as not having a trend.

## 4 Results

^{8}Apart from inflation, we find that the economic cycle has a clear effect on the other economic indicators for each country, as shown by the posterior estimates provided in Appendix C.

Denmark | Norway | Sweden | ||||
---|---|---|---|---|---|---|

Parameter | Mean | 90% interval | Mean | 90% interval | Mean | 90% interval |

Phillips curve | ||||||

\(1 - \mathop \sum \limits_{\ell = 2}^{p} \phi_{\pi ,\ell }\)
| 0.36 | [0.17, 0.54] | 0.23 | [0.08, 0.39] | 0.37 | [0.21, 0.53] |

\(\phi_{\pi ,2}\)
| 0.05 | [− 0.06, 0.15] | 0.48 | [0.37, 0.58] | 0.38 | [0.26, 0.49] |

\(\phi_{\pi ,3}\)
| 0.18 | [0.09, 0.27] | 0.29 | [0.18, 0.40] | 0.25 | [0.14, 0.36] |

\(\phi_{\pi ,4}\)
| 0.15 | [0.06, 0.24] | − | − | − | − |

\(\phi_{\pi ,5}\)
| 0.27 | [0.18, 0.35] | − | − | − | − |

\(\gamma_{\pi ,1}\)
| 0.34 | [0.18, 0.49] | − 0.90 | [− 1.26, − 0.52] | 0.06 | [− 0.24, 0.36] |

\(\gamma_{\pi ,2}\)
| − 0.25 | [− 0.40, − 0.09] | 1.19 | [0.78, 1.57] | 0.01 | [− 0.30, 0.32] |

\(\sigma_{\pi }^{2}\)
| 0.45 | [0.36, 0.55] | 0.77 | [0.62, 0.94] | 3.94 | [3.20, 4.81] |

Real rate gap | ||||||

\(\varphi_{1}\)
| 1.44 | [0.68, 1.79] | 1.08 | [0.82, 1.30] | 1.19 | [1.05, 1.35] |

\(\varphi_{2}\)
| − 0.55 | [− 0.82, 0.06] | − 0.40 | [− 0.61, − 0.19] | − 0.47 | [− 0.64, − 0.33] |

\(\sigma_{{r^{G} }}^{2}\)
| 0.04 | [0.01, 0.09] | 0.22 | [0.06, 0.47] | 0.42 | [0.21, 0.62] |

Cyclical factor | ||||||

\(\rho_{f,1}\)
| 0.54 | [0.28, 0.80] | 1.30 | [1.04, 1.53] | 1.50 | [1.37, 1.61] |

\(\rho_{f,2}\)
| 0.18 | [0.02, 0.34] | − 0.41 | [− 0.61, − 0.19] | − 0.61 | [− 0.72, − 0.49] |

\(\rho_{r,0}\)
| 0.81 | [− 0.37, 2.07] | − 0.13 | [− 0.24, − 0.05] | 0.11 | [0.03, 0.20] |

\(\rho_{r,1}\)
| − 1.39 | [− 2.38, − 0.49] | − 0.02 | [− 0.12, 0.07] | − 0.25 | [− 0.37, − 0.15] |

\(\sigma_{f}^{2}\)
| 0.03 | [0.01, 0.06] | 0.01 | [8·10 ^{–3}, 0.02] | 0.10 | [0.05, 0.16] |

Trend | ||||||

\(\sigma_{y}^{2}\)
| 2·10 ^{–4} | [10 ^{–4}, 5·10^{–4}] | 2·10 ^{–4} | [10 ^{–4}, 5·10^{–4}] | 2·10 ^{–4} | [10 ^{–4}, 5·10^{–4}] |

\(\sigma_{{y^{*} }}^{2}\)
| 0.55 | [0.43, 0.68] | 0.78 | [0.63, 0.97] | 0.38 | [0.29, 0.49] |

\(\sigma_{g}^{2}\)
| 3·10 ^{–3} | [5·10 ^{–4}, 7·10^{–3}] | 2·10 ^{–3} | [4·10 ^{–4}, 6·10^{–3}] | 2·10 ^{–3} | [5·10 ^{–4}, 5·10^{–3}] |

\(\sigma_{z}^{2}\)
| 0.77 | [0.63, 0.89] | 0.35 | [0.10, 0.57] | 0.16 | [0.04, 0.36] |