## _wp11231

## Source details

**Canonical URL:** [_wp11231](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11231.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11231.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11231.pdf.json)

---

### Introduction: motivation and key empirical observations
- National house prices experienced an unprecedented and synchronized rise across OECD countries in the years preceding the Great Recession; subsequent violent declines include Spain: 20 percent, U.S.: 32 percent, and Ireland: 38 percent (Data as of March 2011).
- Historical U.S. house price movements cited: fell by 7.2 percent from 1980 to 1982; rose by 16.2 percent from 1982 to 1989; fell by 8 percent from 1989 to 1995; rose by 40 percent from 1995 to 2004.
- Long-run historical evidence shows recurring house price expansions and contractions at least since the 17th century.
- Paper focus: duration of housing upturns and downturns across 19 OECD countries over 40 years to inform forecasting and constrain theoretical models of housing cycles.

### Data: sources, coverage, and caveats
- Dataset and coverage:
  - OECD dataset containing nominal and real house prices, price-income ratios, and price-rent ratios.
  - 19 countries with quarterly observations spanning from the first quarter of 1970 to the first quarter of 2010; series are seasonally-adjusted.
  - Countries: Australia, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Korea, Netherlands, Norway, New Zealand, Sweden, Spain, Switzerland, United Kingdom (U.K.), and United States (U.S.).
- Macroeconomic controls:
  - Real GDP, interest and inflation rates, and working age population from the OECD Economic Outlook; credit to the private sector from the IMF International Financial Statistics (IFS).
- Measurement caveats:
  - House heterogeneity, decentralized bilateral negotiations, and low turnover of housing stock in a year—interpretation should account for these challenges.
- Reassurance: dataset used in a number of other cross-country studies.

### Methodology: identifying house price cycles
- Cycle dating method:
  - Harding and Pagan (2002) algorithm (BBQ), a quarterly implementation of Bry and Boschan (1971).
  - Purpose: find local maxima and minima to segment series into expansions (upturns) and contractions (downturns).
- Three-step procedure (summary):
  1. Identify points higher/lower than a window of m quarters on each side.
  2. Enforce alternation: a local maximum must be followed by a local minimum and vice versa; where two consecutive maxima (minima) occur, select the highest (lowest).
  3. (Third step referenced but not included in provided excerpt.)
- Rationale: algorithmic identification is necessary because no official housing turning-point dating (analogous to NBER) exists.

### Censoring rule and dating-algorithm parameters
- Censoring rule: the distance between two turning points has to be at least ݍ quarters, where ݍ is chosen to retrieve only significant series movements and avoid series noise.
  - Harding and Pagan (2002) choose ݍ = 2 for U.S. GDP.
- Asset-price-specific guidance:
  - Borio and McGuire (2004) suggest a rolling window of 13 quarters, which implies j = 6.
  - Girouard et al. (2006) require a minimum phase length (q) of 6 quarters.
- Additional cycle rule:
  - One can impose a minimum cycle duration so that distance between two consecutive maxima (minima) is at least ݇ quarters; Harding and Pagan (2002) choose ݇ = 5.
  - The author does not impose an additional cycle-duration restriction beyond the censoring rule, noting an entire cycle already must last longer than 12 quarters because of the censoring rule.
- Practical notes:
  - The method examines series in level ("classical cycle") to avoid parametric detrending sensitivity.
  - Stata program used is available at http://econpapers.repec.org/software/bocbocode/s457284.htm.

### Characteristics of upturns and downturns (Section III) — sample counts and descriptive statistics
- Definitions:
  - Upturn duration = distance in quarters between a trough and a peak.
  - Downturn duration = distance in quarters between a peak and the trough.
- Dataset censoring status at last observation (2010q1):
  - 49 completed upturns
  - 49 complete downturns
  - 6 right-censored upturns
  - 13 right-censored downturns
- Descriptive statistics (Table 3; values preserved exactly):
  - Complete upturns: Sample 49; Duration mean 24.1; Duration StDev 14.8; Amplitude mean 61.3; Amplitude StDev 56.3
  - Complete + ongoing upturns: Sample 55; Duration mean 28.0; Duration StDev 20.6; Amplitude mean 66.7; Amplitude StDev 60.1
  - Complete downturns: Sample 49; Duration mean 18.2; Duration StDev 8.7; Amplitude mean 30.7; Amplitude StDev 28.4
  - Complete + ongoing downturns: Sample 62; Duration mean 18.4; Duration StDev 12.5; Amplitude mean 28.8; Amplitude StDev 27.5
- Interpretation:
  - On average, upturns last more than downturns and have larger amplitudes.
  - Downturns display less duration variability than upturns, suggesting clustering of downturn durations.

### The role of the last upturn (Section III.B)
- The last upturn (end-20th/early-21st century boom) was exceptional in duration for many OECD countries.
- Selected entries from Table 4 (Complete upturns — Trough-Peak — Duration (quarters) — Amplitude (%)):
  - Denmark 1993q2-2007q1 — 55 — 176.6
  - Finland 2001q4-2007q3 — 23 — 37.6
  - France 1997q1-2007q4 — 43 — 117.8
  - Ireland 1994q4-2006q3 — 47 — 286.4
  - Italy 1997q3-2007q4 — 41 — 59.3
  - Korea 2005q1-2007q1 — 8 — 14.0
  - New Zealand 2000q4-2007q3 — 27 — 98.5
  - Norway 1993q1-2007q3 — 58 — 200.1
  - Spain 1996q3-2007q3 — 44 — 121.6
  - United Kingdom 1996q2-2007q4 — 46 — 160.7
  - United States 1995q1-2006q4 — 47 — 64.3
- Selected ongoing upturns (Trough-Peak — Duration (quarters) — Amplitude (%)):
  - Australia 2005q3- — 18 — 26.3
  - Belgium 1985q2- — 99 — 186.7
  - Canada 1998q3- — 46 — 86.9
  - Netherlands 1985q1- — 100 — 199.8
  - Sweden 1996q1- — 56 — 140.2
  - Switzerland 2000q1- — 40 — 22.5
- Regression evidence on time trend:
  - Complete sample: regression coefficient on time trend = 1.291 (standard error .042).
  - Excluding the last boom: regression coefficient on time trend = 1.094 (standard error .048).
  - Once the last boom is excluded, it is not possible to reject the null hypothesis of no constant time trend.
- Interpretation:
  - The exceptional duration and amplitude of the last upturn largely drive the impression of an upward time trend in house prices.

### Duration dependence: concept, tests, and distributional facts (Section IV)
- Definition:
  - Duration dependence: the probability an expansion (or contraction) ends depends on its elapsed duration.
- Distributional facts (Table 5; values preserved exactly):
  - Minimum duration is 6 for both phases (construction enforces this).
  - Complete upturns (49): Min 6; Pct10 8; Pct25 12; Pct40 16; Median 21; Pct60 24; Pct75 32; Pct90 47; Max 58
  - Complete downturns (49): Min 6; Pct10 7; Pct25 13; Pct40 16; Median 17; Pct60 20; Pct75 23; Pct90 32; Max 41
- Empirical pattern:
  - The 10th, 25th, and 40th percentiles of upturns and downturns are substantially equal.
  - Upturns are longer than downturns only above the 40th percentile.
  - Upturn durations decay slowly; downturn durations cluster in the 10–20 quarter range with very few downturns lasting more than 20 quarters.
- Synchronization caveat:
  - International housing cycles are partially synchronized; regression-based tests with country and year fixed effects control for synchronization.

### Nonparametric and regression-based duration-dependence tests
- Nonparametric setup (Ohn et al., 2004):
  - Define binary ܵ௖௧ = 1 if series is in an upturn at time t, 0 otherwise.
  - For upturns, keep observations where ܵ௖,t-1 = 1 and estimate:
    - ܵ௖௧ = ߚ d௖,௧ିଵ + ߙ  (Equation 1)
    - d௖,௧ିଵ is ongoing duration of the upturn at time t; significant ߚ denotes duration dependence.
  - Full panel specification with fixed effects:
    - ܵ௖௧ = ߙ௖ + ߛ௧ + ݀௖,௧ିଵ ߚ  (Equation 2)
  - Functional forms tested include log and quadratic of d; main specification is linear probability model (LPM) with Year FE = Yes and Country FE = Yes.
- Main estimation results (Table 6 highlights; values preserved exactly):
  - log(݀ ௖,௧ିଵ): -0.0450*** (standard error (0.0079)), -0.0822*** (0.0146), 0.0201 (0.0141)
  - ݀ ௖,௧ିଵ: -0.0025*** (0.0006), -0.0050*** (0.0012), -0.0083*** (0.0020), -0.0138** (0.0044), 0.0020 (0.0013), 0.0052 (0.0028)
  - ݀ ௖,௧ିଵଶ: 0.0000* (0.0000), 0.0001 (0.0001), -0.0001 (0.0000)
  - Fixed effects: Year FE = Yes, Country FE = Yes.
  - Observations and R-squared:
    - Upturns: Observations = 1,273; R-squared = 0.1676, 0.1601, 0.1656
    - Upturns without last boom: Observations = 566; R-squared = 0.2143, 0.2090, 0.2133
    - Downturns: Observations = 825; R-squared = 0.1414, 0.1418, 0.1439
- Findings:
  - Upturn equations indicate a significant duration effect: longer upturns are more likely to terminate (positive duration dependence).
  - Downturns display no significant duration dependence.
  - Robustness: results consistent across specifications and when excluding the last boom.

### Inspecting mechanism: role of macroeconomic fundamentals (Section IV.C and Appendix A2)
- Hypothesis and approach:
  - If positive duration dependence in upturns arises from non-fundamental components, including macroeconomic fundamentals in Equation 2 should not alter the main result.
  - Duration models in discrete time are equivalent to binary limited dependent-variable (LDV) models where duration enters as an independent variable (Appendix A2).
  - Choice is between LPMs and LDVs (not between LPMs and duration models); Table A1 shows logit and LPM give same qualitative results.
- Macroeconomic variables included (real terms; some as HP deviations, some as annualized changes or growth):
  - ܶܰܧܴ (user-cost/rent measure), real mortgage interest rate proxy ݅, inflation ܮܨܰܫ, credit to the private sector ܶܫܦܧܴܥ, national income ܲܦܩ, working-age population growth ܲܣܹ.
- Key coefficient patterns from Table 7 (values preserved exactly):
  - Duration terms:
    - log(݀ ௖,௧ିଵ): -0.0334*** (0.0072), -0.0649** (0.0167), 0.0150 (0.0117)
    - ݀ ௖,௧ିଵ: -0.0020** (0.0006), -0.0041*** (0.0010), -0.0070** (0.0020), -0.0069 (0.0051), 0.0016 (0.0012), 0.0045 (0.0021)
    - ݀ ௖,௧ିଵଶ: 0.0000* (0.0000), -0.0000 (0.0001), -0.0001* (0.0000)
  - Selected macroeconomic coefficients (examples):
    - ܶܰܧܴ ௖,௧ିଵ: -0.4549* (0.1622), -0.5016* (0.1866), -0.4710* (0.1698); other columns show -0.7304 (0.7731), -0.6040 (0.8170), -0.6040 (0.8185), -0.1539 (0.2934), -0.1442 (0.3026), -0.1350 (0.2961)
    - ݅ ௖,௧ିଵ (real mortgage interest rate proxy): -0.7415** (0.2545), -0.7394** (0.2495), -0.7424** (0.2522); other columns show -0.7562* (0.3014), -0.7556* (0.3000), -0.7555* (0.3005), and small positive coefficients 0.0350 (0.1292), 0.0347 (0.1291), 0.0355 (0.1291)
    - ܮܨܰܫ ௖,௧ିଵ (inflation): -1.0049* (0.3530), -0.9933* (0.3493), -1.0029* (0.3530); other columns -1.1175* (0.4765), -1.1089* (0.4819), -1.1087* (0.4852), and small positive 0.0801 (0.2247), 0.0768 (0.2237), 0.0784 (0.2247)
    - ܶܫܦܧܴܥ ௖,௧ିଵ (credit): -0.1471 (0.1035), -0.1466 (0.1082), -0.1479 (0.1087); other columns -1.1855* (0.4455), -1.1784* (0.4770), -1.1784* (0.4776), -0.4356 (0.2292), -0.4440 (0.2293), -0.4243 (0.2222)
    - ܲܦܩ ௖,௧ିଵ (national income): -2.2733* (0.9001), -2.2841* (0.9109), -2.3370* (0.9072); other columns -2.1464 (1.2983), -2.4671 (1.3769), -2.4701 (1.3359), -1.5534* (0.6516), -1.5500* (0.6512), -1.5568* (0.6454)
    - ܲܣܹ ௖,௧ିଵ (working-age population growth): -0.0190 (0.3836), 0.0059 (0.3802), -0.0341 (0.3785); other columns 2.8257 (3.1628), 2.7888 (3.2276), 2.7855 (3.1704), -11.6136 (7.9127), -11.7701 (7.8913), -11.5827 (7.8651)
  - Fixed effects: Year FE = Yes, Country FE = Yes.
  - Observations and R-squared (Table 7):
    - Upturns: Observations = 1,132; R-squared = 0.2130, 0.2101, 0.2149
    - Upturns without last boom: Observations = 444; R-squared = 0.2944, 0.2996, 0.2997
    - Downturns: Observations = 723; R-squared = 0.1268, 0.1277, 0.1293
- Findings:
  - Duration dependence of upturns remains after controlling for macroeconomic fundamentals.
  - Positive changes in interest rates and inflation are associated with an increased probability that an upturn ends (negative coefficients imply higher values increase hazard of termination).
  - Above-trend GDP associated with higher probability of upturn ending in some specifications; above-trend domestic product reduces likelihood of exiting a downturn in some associations.
  - Duration effect captures dynamics not fully explained by included fundamentals.

### Appendix A1 — ARIMA(1,1,0) simulation and implications for duration tests
- Simulated ARIMA(1,1,0) process:
  - ݕ௧ = 0.1 + ݕ௧ిଵ + 0.8 Δݕ௧ిଵ߳௧.
  - 100,000 observations generated.
  - Applied BBQ dating algorithm; left-truncation correction: removed first 5 observations of each phase.
- Empirical counts from simulation:
  - Counted 2248 upturns and 2248 downturns.
- Distributional findings:
  - Frequencies of upturn and downturn durations show a decaying pattern typical of a geometric distribution.
  - Q-Q plot: simulated distribution and theoretical geometric distribution "almost identical."
  - Interpretation: geometric form implies constant probability parameter ߣ across durations (absence of duration dependence).
- Regression test applied to simulated phases (results preserved in text):
  - Upturns and downturns regressions show coefficients indicating near-constant hazard consistent with geometric distribution.
- BBQ algorithm distortions and remedies:
  - BBQ enforces minimum phase length of 6 quarters, creating left-truncation.
  - Remedy: discard observations with duration less than 6 and rescale durations so 6th quarter corresponds to ܦ = 1 (Jenkins, 2005).
  - Conclusion: BBQ algorithm does not create duration dependence endogenously once left-truncation is addressed.

### Appendix A2 — equivalence of discrete-time duration models and binary regressions
- Discrete-time survivor and hazard framework summarized; likelihood contributions for complete and right-censored phases provided.
- Discrete-time log-likelihood shown to be equivalent to log-likelihood of a standard binary model (probit/logit) with dependent variable indicating last quarter of a phase.
- Practical estimation steps:
  - Run logit (or LPM) on quarter-country dataset with dependent variable equal to 1 on the last quarter of expansions or contractions; create ongoing-duration variable d; discard observations with d < 6 and rescale so 6th quarter = 1.
  - Duration functional forms experimented: log(d); linear d; quadratic d^2.
- Mixed proportional hazards and interpretation provided.
- Empirical tables (A1, A2) show logit specifications yield patterns consistent with LPM estimates reported in main text.

### Key empirical findings and policy-relevant interpretations
- Duration dependence:
  - House price upturns are more likely to end as they get longer (robust across specifications and when excluding the last boom).
  - House price downturns do not show signs of duration dependence.
- Macroeconomic associations:
  - Higher real mortgage interest rates and higher inflation are associated with increased probability that an upturn ends.
  - Some evidence that above-trend GDP is associated with higher probability of upturn ending; above-trend domestic product may reduce likelihood of exiting a downturn (associations, not necessarily causal).
- Interpretation:
  - Positive duration dependence in upturns is consistent with booms departing from fundamentals (speculation, overbuilding); longer upturns increase imbalances and the hazard of reversal.
  - Duration dependence implies nonlinear dynamics in aggregate house price indices; duration characterization provides an intuitive description without imposing complex parametric models.
- Policy relevance:
  - Understanding durations and duration dependence helps identify when reversals are more likely and informs policy responses to real estate booms and overheating.

*Source: _wp11231 (excerpt of IMF working paper content provided).*

### References .............................................................................................................

### _wp11231 - References

### Introduction: motivation and key empirical observations
- National house prices experienced an unprecedented and synchronized rise across OECD countries in the years preceding the Great Recession; subsequent violent declines include Spain: 20 percent, U.S.: 32 percent, and Ireland: 38 percent (Data as of March 2011).
- Historical U.S. house price movements cited: fell by 7.2 percent from 1980 to 1982; rose by 16.2 percent from 1982 to 1989; fell by 8 percent from 1989 to 1995; rose by 40 percent from 1995 to 2004.
- Long-run historical evidence shows recurring house price expansions and contractions at least since the 17th century.
- Paper focus: duration of housing upturns and downturns across 19 OECD countries over 40 years; two motivations:
  - Policymakers’ interest in expected lengths of expansions and contractions to inform forecasting.
  - Researchers’ interest in empirical regularities to constrain theoretical models of housing cycles.

### Main empirical findings on durations and duration dependence
- Two sets of results:
  - Average length: upturns are longer than downturns on average; however, this difference disappears once the last house price boom is excluded.
  - Length distribution: upturns (but not downturns) are more likely to end as their duration increases.
- Interpretation: the increasing likelihood that upturns end with duration is consistent with boom-bust theories where rigidities and frictions cause prices to overshoot fundamentals; as expansions lengthen they become progressively unsustainable.
- Terminology: house price expansions referred to as “upturns”, and contractions as “downturns”.

### Duration dependence: concept and context
- Duration dependence defined as the feature that the probability an expansion (or contraction) ends depends on its elapsed duration.
- Noted contrast with traditional linear stochastic processes (e.g., random walks or autoregressive processes) where the likelihood of reversal does not depend on the length of the prior sequence; Appendix A1 discusses this intuition in more detail.
- Literature context:
  - Business cycle literature on duration dependence following Diebold and Rudebusch (1990) and Sichel (1991).
  - Few housing-cycle studies: Claessens et al. (2011) on “financial cycles”; Cunningham and Kolet (2011) on U.S. and Canada metropolitan areas.
- This analysis differs from prior work in (1) the way housing cycles are identified and (2) the test used to detect duration dependence.

### Data: sources, coverage, and caveats
- Dataset: OECD dataset containing nominal and real house prices, price-income ratios, and price-rent ratios.
- Coverage: 19 countries with quarterly observations spanning from the first quarter of 1970 to the first quarter of 2010; series are seasonally-adjusted.
- Countries included: Australia, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Korea, Netherlands, Norway, New Zealand, Sweden, Spain, Switzerland, United Kingdom (U.K.), and United States (U.S.).
- Macroeconomic controls: Real GDP, interest and inflation rates, and working age population from the OECD Economic Outlook; credit to the private sector from the IMF International Financial Statistics (IFS).
- Measurement caveats: house heterogeneity (physical characteristics, location, state of building), decentralized bilateral negotiations, and low turnover of housing stock in a year—interpretation should account for these measurement challenges.
- Reassurance: dataset used in a number of other cross-country studies.

### Methodology: identifying house price cycles
- Cycle dating method: Harding and Pagan (2002) algorithm (BBQ), a quarterly implementation of Bry and Boschan (1971), itself rooted in Burns and Mitchell (1946).
- Purpose: find series of local maxima and minima to segment series into expansions (upturns) and contractions (downturns).
- Three-step procedure applied to a quarterly series y_t (description truncated in provided content but includes):
  1. Identification of points higher or lower than a window of surrounding observations using a window of m quarters on each side; local maxima and minima are defined by comparisons to surrounding observations.
  2. Alternation rule: a local maximum must be followed by a local minimum and vice versa; in case of two consecutive maxima (minima), the highest (lowest) observation is chosen.
  3. (Third step referenced but not included in excerpt.)
- Note: algorithmic identification is necessary because no official dates exist for housing turning points analogous to NBER recession dating.

### Structure of the paper (as outlined)
- Section II: describes the data and the algorithm employed to identify turning points.
- Section III: analyzes average upturn and downturn durations.
- Section IV: discusses the duration distribution and the duration dependence property.
- Section V: concludes.
- Appendixes and supporting materials mentioned include:
  - Appendix A1: The Distribution of Phase Durations for an ARIMA(1,1,0) Process.
  - Appendix A2: Discrete-Time Hazard Models and Binary-Dependent Variable Regressions.
  - Appendix Tables and Appendix Figures (titles given in the content inventory).

*Source: _wp11231 - References (excerpt of main text, including Introduction and Data and Methodology) — IMF PDF.*

### 3. Censoring rule. The distance between two turning points has to be at least ݍ quarters,

### _wp11231 - 3. Censoring rule. The distance between two turning points has to be at least ݍ quarters,

### Censoring rule and dating algorithm parameters
- The censoring rule: the distance between two turning points has to be at least ݍ quarters, where ݍ is chosen by the analyst to retrieve only significant series movements and avoid series noise. Harding and Pagan (2002) choose ݍ ൌ2 for U.S. GDP.
- For series different from GDP (e.g., house prices) the analyst must decide:
  - the dimension of the rolling window (j)
  - the minimum phase duration (q)
- Asset-price-specific guidance cited:
  - Borio and McGuire (2004) suggest a rolling window of 13 quarters, which implies j = 6.
  - Girouard et al. (2006) require a minimum phase length (q) of 6 quarters.
- Additional cycle rule: one can impose a minimum cycle duration so that the distance between two consecutive maxima (minima) is at least ݇ quarters. Harding and Pagan (2002) choose ݇ ൌ5. The author does not impose an additional cycle-duration restriction beyond the censoring rule, noting that an entire cycle already has to last longer than 12 quarters because of the censoring rule.
- The method used examines series in level ("classical cycle") and is preferred here to avoid restrictive parametric assumptions and sensitivity to detrending; the Harding and Pagan (2002) algorithm is used and tested for influence in Section IV.

### Binary outcome and applications
- Outcome variable: binary series where expansion quarters are tagged with “1” and contraction quarters are tagged with “0”.
- Historical and cross-asset uses:
  - Initially confined to business cycles.
  - Pagan and Sossounov (2003): identify bull and bear markets in stock prices.
  - Helbling and Terrones (2003), Borio and McGuire (2004): upturns and downturns in the housing market.
- Practical implementation note: the Stata program used for this paper is available at http://econpapers.repec.org/software/bocbocode/s457284.htm.

### Rationale for larger thresholds in housing cycles
- House price cycles are known to be longer and asset prices more volatile than GDP cycles; therefore threshold parameters (q, j) should be set higher to avoid identification of spurious phases.
- Volatility and potential for spikes in asset prices motivate wider rolling windows.

---

### Characteristics of upturns and downturns (Section III)
- Turning points: Table 1 lists peaks and troughs for all countries; the identified turning points match those in Girouard et al. (2006), Van den Noord (2006), and André (2010), with occasional 1–2 quarter differences due to seasonal adjustment updates or differing choices about reporting only “major” phases.
- Definitions:
  - Upturn duration = distance in quarters between a trough and a peak.
  - Downturn duration = distance in quarters between a peak and the trough.
- Dataset structure and censoring:
  - Every country has an ongoing upturn or downturn at the time of the last observation (2010q1).
  - Descriptive statistics are computed with and without censored phases.

Key dataset counts (preserved exactly):
- 49 completed upturns
- 49 complete downturns
- 6 right-censored upturns
- 13 right-censored downturns

Key descriptive findings (from Table 3; numbers preserved exactly):
- Complete upturns: Sample 49; Duration mean 24.1; Duration StDev 14.8; Amplitude mean 61.3; Amplitude StDev 56.3
- Complete + ongoing upturns: Sample 55; Duration mean 28.0; Duration StDev 20.6; Amplitude mean 66.7; Amplitude StDev 60.1
- Complete downturns: Sample 49; Duration mean 18.2; Duration StDev 8.7; Amplitude mean 30.7; Amplitude StDev 28.4
- Complete + ongoing downturns: Sample 62; Duration mean 18.4; Duration StDev 12.5; Amplitude mean 28.8; Amplitude StDev 27.5

Interpretation:
- On average, upturns last more than downturns and have larger amplitudes.
- Downturns display less duration variability than upturns, hinting at clustering of downturn durations.

---

### The role of the last upturn (Section III.B)
- The last upturn (the house price boom at the end of the 20th century and early 21st century) was exceptional in duration for many OECD countries.
- Table 4 reports dates, durations (quarters), and amplitudes (%) for the last upturn of each country. Selected preserved entries from Table 4:

Complete upturns (Trough-Peak — Duration (quarters) — Amplitude (%)):
- Denmark 1993q2-2007q1 — 55 — 176.6
- Finland 2001q4-2007q3 — 23 — 37.6
- France 1997q1-2007q4 — 43 — 117.8
- Ireland 1994q4-2006q3 — 47 — 286.4
- Italy 1997q3-2007q4 — 41 — 59.3
- Korea 2005q1-2007q1 — 8 — 14.0
- New Zealand 2000q4-2007q3 — 27 — 98.5
- Norway 1993q1-2007q3 — 58 — 200.1
- Spain 1996q3-2007q3 — 44 — 121.6
- United Kingdom 1996q2-2007q4 — 46 — 160.7
- United States 1995q1-2006q4 — 47 — 64.3

Ongoing upturns (Trough-Peak — Duration (quarters) — Amplitude (%)):
- Australia 2005q3- — 18 — 26.3
- Belgium 1985q2- — 99 — 186.7
- Canada 1998q3- — 46 — 86.9
- Netherlands 1985q1- — 100 — 199.8
- Sweden 1996q1- — 56 — 140.2
- Switzerland 2000q1- — 40 — 22.5

- Regression evidence on trend:
  - For the complete sample: regression coefficient on time trend = 1.291 (standard error .042).
  - For the sample excluding the last boom: regression coefficient on time trend = 1.094 (standard error .048).
  - Once the last boom is excluded, it is not possible to reject the null hypothesis of no constant time trend.

Interpretation:
- The exceptional duration and amplitude of the last upturn largely drive the impression of an upward trend in house prices; excluding the last boom removes the significant time trend.

---

### Duration dependence (Section IV)
- Motivation: the same mean duration can arise from different duration-dependence shapes (constant hazard vs. increasing hazard over time). Thus average duration alone is not informative about duration dependence.
- Distributional facts (Table 5; preserved exactly):
  - The minimum duration is 6 for both phases (construction of phases enforces this).
  - Complete upturns (49): Min 6; Pct10 8; Pct25 12; Pct40 16; Median 21; Pct60 24; Pct75 32; Pct90 47; Max 58
  - Complete downturns (49): Min 6; Pct10 7; Pct25 13; Pct40 16; Median 17; Pct60 20; Pct75 23; Pct90 32; Max 41
- Empirical pattern:
  - The 10th, 25th, and 40th percentiles of upturns and downturns are substantially equal.
  - Upturns are longer than downturns only above the 40th percentile.
  - Frequencies: upturn durations decay slowly; downturn durations cluster in the 10–20 quarter range with very few downturns lasting more than 20 quarters.
- Synchronization caveat:
  - International housing cycles are partially synchronized; synchronization can produce spurious clusters of durations. To control for synchronization, regression-based tests with country and year fixed effects are used.

Nonparametric test (Ohn et al., 2004) — preserved equation and procedure:
- Define binary variable ܵ௖௧ = 1 if series y is in an upturn at time t, 0 otherwise.
- For upturns, keep observations where ܵ௖,t-1 = 1 and estimate:
  ܵ௖௧ = ߚ d௖,௧ିଵ + ߙ (Equation 1 in text)
  - where d௖,௧ିଵ is the ongoing duration of the upturn at time t.
  - Significant ߚ denotes duration dependence.
- Full panel specification controlling for country fixed effects (ߙ௖) and year fixed effects (ߛ௧) estimated:
  ܵ௖௧ = ߙ௖ + ߛ௧ + ݀௖,௧ିଵ ߚ + (Equation 2 in text)
- Alternative functional forms tested: logarithmic and quadratic specifications of d, and probit/logit link functions are discussed but the main specification is a linear probability model (LPM).

Findings from estimation (Table 6 summary as described in text):
- Upturn equations (estimated with and without the last boom) indicate a significant and positive effect of duration on the probability that upturns end (positive duration dependence).
- Downturn equation displays no significant duration dependence.
- Robustness and interpretation:
  - Result consistent with Cunningham and Kolet (2011) for U.S. and Canadian cities.
  - Positive duration dependence in upturns is consistent with booms departing from fundamentals; such departures are increasingly difficult to sustain, raising the hazard of reversal as the boom gets longer.
  - Real-world mechanisms suggested: speculation and overbuilding increase reversal probability as booms extend.

Methodological note:
- The author prefers LPM for diagnostic, nonparametric transparency; link functions (probit/logit, Weibull, log-logistic) are acknowledged alternatives commonly used in survival/duration analysis, but LPM is chosen to avoid parametric assumptions.

---

### Inspecting the mechanism: the role of fundamentals (Section IV.C)
- Hypothesis: if positive duration dependence in upturns arises from non-fundamental components of house price expansions, then including macroeconomic fundamentals in Equation 2 should not alter the main result.
- The text introduces this logic and moves to test inclusion of macroeconomic variables; results and details of those regressions are presented in the paper beyond the excerpt provided.

*Source: _wp11231 - 3. Censoring rule. The distance between two turning points has to be at least ݍ quarters,*

### Appendix A2 shows that, once duration models are translated into a discrete-time setting, they are equivalent to

### _wp11231 - Appendix A2 shows that, once duration models are translated into a discrete-time setting, they are equivalent to

### Equivalence of duration models and limited dependent-variable (LDV) models
- Appendix A2: duration models translated into a discrete-time setting are equivalent to binary limited dependent-variable (LDV) models where duration is included as an independent variable.
- Choice is between LPM’s and LDV’s (not between LPM’s and duration models).
- Table A1: results from a logit regression are the same as those from the LPM.
- Note: duration inclusion accounts for time dependencies/serial correlation that simple probit/logit models may not control for (Beck et al., 1996; Harding and Pagan, 2011).

### User-cost formula and macro fundamentals included in regressions
- Starting point: user-cost formula proposed by Poterba (1984): ሻߨെߜ൅ݔܽݐ൅݅ሺܧܥܫܴܲൌ ܶܰܧܴ.
- Formula suggests including a measure of ܶܰܧܴ and interest rate (݅) in Equation 2.
- To control for additional credit availability channels, include credit to the private sector (ܶܫܦܧܴܥ
  ௖௧).
- Include inflation (ܮܨܰܫ
  ௖௧), national income (ܲܦܩ
  ௖௧), and working age population (ܲܣܹ
  ௖௧).
- All variables are expressed in real terms; some variables enter as deviations from trend (Hodrick-Prescott filter), some as annualized changes or growth (see notes to Table 7).

### Duration-dependence tests (Table 6 highlights)
- Upturns show duration dependence: negative coefficient on log(݀ ௖,௧ିଵ) indicates longer upturns are more likely to terminate.
  - log(݀ ௖,௧ିଵ): -0.0450*** (standard error (0.0079))
  - log(݀ ௖,௧ିଵ): -0.0822*** (standard error (0.0146))
  - log(݀ ௖,௧ିଵ): 0.0201 (standard error (0.0141))
- Additional duration terms from Table 6:
  - ݀ ௖,௧ିଵ: -0.0025*** (0.0006), -0.0050*** (0.0012), -0.0083*** (0.0020), -0.0138** (0.0044), 0.0020 (0.0013), 0.0052 (0.0028)
  - ݀ ௖,௧ିଵ ଶ: 0.0000* (0.0000), 0.0001 (0.0001), -0.0001 (0.0000)
- Fixed effects included: Year FE = Yes, Country FE = Yes.
- Observations and R-squared reported in Table 6:
  - Upturns: Observations = 1,273; R-squared = 0.1676, 0.1601, 0.1656 (across columns)
  - Upturns without last boom: Observations = 566; R-squared = 0.2143, 0.2090, 0.2133
  - Downturns: Observations = 825; R-squared = 0.1414, 0.1418, 0.1439

### Duration dependence and fundamentals (Table 7 key coefficients)
- Estimation: ܵ ௖௧ ߙൌ ௖ ߛ൅ ௧ ݀ ߚ൅ ௖,௧ିଵ ܺ ߩ൅ ௖,௧ିଵ, where ܺ ௖,௧ିଵ are macroeconomic fundamentals.
- Dataset described: covers 19 countries with 161 observations each (noted as a relatively small sample given many regressors).
- Duration terms (Table 7):
  - log(݀ ௖,௧ିଵ): -0.0334*** (0.0072), -0.0649** (0.0167), 0.0150 (0.0117) across columns.
  - ݀ ௖,௧ିଵ: -0.0020** (0.0006), -0.0041*** (0.0010), -0.0070** (0.0020), -0.0069 (0.0051), 0.0016 (0.0012), 0.0045 (0.0021)
  - ݀ ௖,௧ିଵ ଶ: 0.0000* (0.0000), -0.0000 (0.0001), -0.0001* (0.0000)
- Selected macroeconomic coefficients (Table 7):
  - ܶܰܧܴ ௖,௧ିଵ: -0.4549* (0.1622), -0.5016* (0.1866), -0.4710* (0.1698); other columns show -0.7304 (0.7731), -0.6040 (0.8170), -0.6040 (0.8185), -0.1539 (0.2934), -0.1442 (0.3026), -0.1350 (0.2961)
  - ݅ ௖,௧ିଵ (real mortgage interest rate proxy): -0.7415** (0.2545), -0.7394** (0.2495), -0.7424** (0.2522); other columns show -0.7562* (0.3014), -0.7556* (0.3000), -0.7555* (0.3005), and small positive coefficients 0.0350 (0.1292), 0.0347 (0.1291), 0.0355 (0.1291)
  - ܮܨܰܫ ௖,௧ିଵ (inflation): -1.0049* (0.3530), -0.9933* (0.3493), -1.0029* (0.3530); other columns -1.1175* (0.4765), -1.1089* (0.4819), -1.1087* (0.4852), and small positive 0.0801 (0.2247), 0.0768 (0.2237), 0.0784 (0.2247)
  - ܶܫܦܧܴܥ ௖,௧ିଵ (credit to private sector): -0.1471 (0.1035), -0.1466 (0.1082), -0.1479 (0.1087); other columns -1.1855* (0.4455), -1.1784* (0.4770), -1.1784* (0.4776), -0.4356 (0.2292), -0.4440 (0.2293), -0.4243 (0.2222)
  - ܲܦܩ ௖,௧ିଵ (national income): -2.2733* (0.9001), -2.2841* (0.9109), -2.3370* (0.9072); other columns -2.1464 (1.2983), -2.4671 (1.3769), -2.4701 (1.3359), -1.5534* (0.6516), -1.5500* (0.6512), -1.5568* (0.6454)
  - ܲܣܹ ௖,௧ିଵ (working-age population growth): -0.0190 (0.3836), 0.0059 (0.3802), -0.0341 (0.3785); other columns 2.8257 (3.1628), 2.7888 (3.2276), 2.7855 (3.1704), -11.6136 (7.9127), -11.7701 (7.8913), -11.5827 (7.8651)
- Fixed effects: Year FE = Yes, Country FE = Yes in Table 7.
- Observations and R-squared reported in Table 7:
  - Upturns: Observations = 1,132; R-squared = 0.2130, 0.2101, 0.2149
  - Upturns without last boom: Observations = 444; R-squared = 0.2944, 0.2996, 0.2997
  - Downturns: Observations = 723; R-squared = 0.1268, 0.1277, 0.1293

### Key empirical findings
- Duration dependence:
  - House price upturns are more likely to end as they get longer (robust across specifications and whether the last boom is included).
  - House price downturns do not show signs of duration dependence.
- Macroeconomic associations:
  - Positive changes in interest rates and inflation are both associated with an increase in the probability that an upturn ends (negative estimated coefficients on interest rate and inflation in upturn regressions imply higher values increase hazard of termination).
  - Above-trend rents and domestic product have associations with upturns: above-trend GDP associated with higher probability of upturn ending; above-trend domestic product reduces the likelihood of exiting a downturn (association, not necessarily causal).
- Duration effect on upturn terminations appears to capture dynamics independent of included macroeconomic fundamentals.

### Interpretation and implications
- Nonlinearity: duration dependence implies aggregate house price indices behave in a nonlinear way; nonlinear models (Markov regime-switching, conditional heteroskedasticity) can account for these features, but duration dependence provides an intuitive characterization without complex modeling.
- Theoretical framing: upturns as "hot" periods with departures from fundamentals; longer upturns increase imbalances and the probability of reversal.
- Policy relevance: since house prices are cyclical in all analyzed countries, understanding where and when reversals are more likely matters for policy responses to real estate booms. An “overheated” economy—via unusually long upturns or above-trend GDP growth—is more likely to initiate a house price downturn.

*Source: _wp11231 - Appendix A2 shows that, once duration models are translated into a discrete-time setting, they are equivalent to (IMF working paper content provided).*

### Appendix A1 – The Distribution of Phase Durations for an ARIMA(1,1,0) Process

### Appendix A1 – The Distribution of Phase Durations for an ARIMA(1,1,0) Process

### Simulation of an ARIMA(1,1,0) process and observed phase durations
- Simulated data-generating process:
  - ݕ௧ൌ0.1൅ݕ௧ిଵ൅0.8 Δݕ௧ిଵ߳௧.
- Simulation details and preprocessing:
  - 100,000 observations generated from the ARIMA(1,1,0) process.
  - Applied the same BBQ dating algorithm used for the national house prices.
  - Left-truncation correction: removing the first 5 observations of each phase.
- Empirical counts from the simulation:
  - Counted 2248 upturns and 2248 downturns.
- Distributional findings:
  - Frequencies of both upturn and downturn durations show a decaying pattern typical of a geometric distribution.
  - Q-Q plot comparison: the simulated distribution and the theoretical geometric distribution are described as "almost identical."
- Interpretation:
  - The geometric distribution form indicates a constant probability parameter ߣ across durations, implying absence of duration dependence (the probability of an upturn/downturn ending is the same regardless of elapsed duration).
- Regression test of duration dependence (Equation 1 applied to simulated phases):
  - For upturns:
    - ܵ௧ൌ     .96159     ሺ.00133ሻ כככ    െ    1.00001     ሺ.00003ሻ݀  ௧ిଵ,                       ሺ1ܣሻ
  - For downturns:
    - ܵ௧ൌ     .08223     ሺ.00322ሻ כככ    െ    1.00003     ሺ.00016ሻ݀  ௧ిଵ.                                ሺ2ܣሻ

### BBQ algorithm, left-truncation, and duration dependence
- Known algorithmic distortion:
  - The BBQ algorithm creates a clear distortion: no phase ends before 6 quarters (minimum phase length of 6 quarters).
  - Remedy applied: ignore observations with duration less than 6 and rescale durations so that the 6th quarter of a phase corresponds to ܦ ൌ1 (Jenkins, 2005).
- Concern addressed:
  - Possible subtle endogenous generation of duration dependence via the dating algorithm (due to nonlinear construction and rolling-window local maxima/minima).
  - Harding and Pagan (2011) note such constructed cycles may display properties of higher-order Markov processes; duration independence implies a first-order Markov process.
- Conclusion from simulations and Equations A1–A2:
  - The BBQ algorithm does not create duration dependence endogenously.
  - Left-truncation is the primary distortion (handled by discarding early durations and rescaling).

### Continuous-time hazard models (definitions and likelihood)
- Survivor function and hazard rate:
  - Survivor function: ሻݔሺܩ  — probability that an upturn or downturn reaches at least durationݔ.
  - Hazard rate: ሻݔሺߣ  defined by ߣሺݔሻ݂ൌሺݔሻሻݔሺܩ/.
  - One-to-one relationship between hazard rate and survivor function.
- Likelihood contributions:
  - Right-censored phase with ongoing duration݀: contribution is ሻ݀ሺܩ .
  - Complete phase with duration ܦ: contribution is ሺܦሻൌߣሺܦሻሻܦሺܩ.
- Log-likelihood for the sample:
  - logLൌ ෍ሼܿ௡ሾlogߣሺܦ௡ሻ൅logܩሺܦ௡ሻሿ൅ሺܿ1െ௡ሻlogܩሺ݀௡ሻሽே௡ీଵ    (Equation (1) in the Appendix)
- Parametric example (Sichel, 1991):
  - Hazard parameterization: ߣሺܦሻܦߙߤൌఈిଵ
  - Interpretation:
    - Parameterߙ > 1: positive duration dependence.
    - Parameterߙ < 1: negative duration dependence.
    - Parameterߙ = 1: duration independence (Weibull reduces to exponential/geometric in discrete time).
  - Claessens et al. (2011) use this Weibull specification on credit, stock, and housing cycles.

### Hazard models in discrete-time and equivalence to binary regressions
- Discrete-time survivor function:
  - Survivor at ongoing duration݀ represented as ∏൫1െߣሺݍሻ൯ௗ௤ ఐ (product across periods at risk).
- Discrete-time log-likelihood (Equation (2)):
  - logLൌ ෍෍ൣݕ௡௤݀൫ߣlog௡௤൯൅൫1െݕ௡௤൯logൣ1െߣ൫݀௡௤൯൧൧ ொ௤ీଵే௡ీଵ
  - Every quarterݍ of phase݊ enters the likelihood;ݕ௡௤ is 1 if the observation is the last quarter of the upturn/downturn, 0 otherwise.
- Equivalence to binary models:
  - The discrete-time likelihood is the log-likelihood of a standard binary model (probit/logit) with dependent variableݕ௡௤.
  - For this paper, the probability takes a logit form:
    - ܦ൫ߣ௡௤൯ൌexpቀ݀൫݄௡௤൯ቁ1൅expቀ݀൫݄௡௤൯ቁ
    - Where݀ሺ݄௡௤ሻ is some function of ongoing duration݀௡௤.
- Practical estimation steps:
  - Run logit regression on quarter-country dataset.
  - Dependent variableݕ௡௤ equals 1 on the last quarter of expansions or contractions.
  - Create variable݀௡௤ that keeps track of ongoing durations.
  - Run models separately for upturns and downturns.
  - Discard observations not at risk (those with݀௡௤ less than minimum allowed quarters, 6) and rescale duration variable so the 6th quarter corresponds to 1.
- Duration functional forms experimented:
  - Specification 1:݀൫݄௡௤ߛ൯ൌ଴log݀௡௤.
  - Specification 2: linear termߛଵ݀௡௤.
  - Specification 3: quadratic polynomialߛଵ݀௡௤ߛ൅ଶ݀௡௤ଶ.
- Mixed proportional hazard model:
  - General form: ߣሺܺ,ܦሻ݄ൌሺܦሻ݁ఉᇲ௑
  -܄(ܦ) is baseline hazard (shape determined by duration dependence specification).
  -ܺ is set of explanatory variables; ߚ interpreted as proportional effect (semi-elasticity) on the hazard.

### Empirical results summarized (figures and tables)
- Figure A1 (Simulated Phase Durations):
  - Reports histograms and Q-Q plots for upturns and downturns from the 100,000-observation simulation, adjusted for left-truncation (first 5 observations discarded).
  - The parameter ߣ of the theoretical geometric distribution corresponds to the unconditional probability of an upturn (downturn) ending and is computed as the ratio between the number of complete upturns (downturns) and the total amount of quarters spent in upturns (downturns).
- Table A1 (Logit Duration Dependence Test) — model highlights (summary of reported coefficients and model fit):
  - Sample sizes and fit:
    - Observations: 834 (columns (1)-(3)); 441 (columns (4)-(6)); 595 (columns (7)-(9)).
    - Loglikelihoods reported: -152.9, -163.2, -155.6, -105.7, -112.4, -109.5, -148.0, -147.7, -144.1.
  - Selected coefficient patterns (as displayed in the table):
    - log(݀௖,௧িଵ) coefficients: -1.727*** (column (1)); -2.099** (column (2)); 0.569 (column (3)).
    - ݀௖,௧ிଵ linear coefficients: -0.077***, -0.242**, -0.139**, -0.301**, 0.067, 0.268* (various columns).
    - ݀௖,௧িଵଶ (squared) coefficients: 0.003* (one specification), 0.004, -0.006 (other specifications).
  - Fixed effects:
    - Year FE: Yes in all reported specifications.
    - Country FE: Yes in all reported specifications.
- Table A2 (Logit Duration Dependence and Fundamentals) — model highlights:
  - Sample sizes and fit:
    - Observations: 592 (columns (1)-(3)); 261 (columns (4)-(6)); 497 (columns (7)-(9)).
    - Loglikelihoods reported: -96.44, -104.0, -95.18, -48.19, -50.74, -49.55, -109.5, -109.0, -106.0.
  - Variables included and coefficient patterns (coefficients reported as in table):
    - log(݀െ1ݐܿ): -2.457***; -4.134***; 0.563 (across different specifications).
    - ݀ݐܿ linear: -0.103**, -0.303***, -0.358***, -0.532***, 0.071, 0.314.
    - ݀െ1ݐܿ2 (squared): 0.003**, 0.004, -0.007 (across specifications).
    - Fundamentals (expressed as deviations from trend; see table notes):
      - ܶܰܧܴ௖௧ିଵ coefficients reported (large negative estimates with large standard errors in some columns, e.g., -14.120, -13.383, -8.447, -17.216, -20.037, -17.822, -10.372, -10.794, -13.497).
      - ݅െ1ݐܿ: -13.489**, -10.987**, -13.385**, -25.774*, -23.750*, -24.931*, -0.503, -0.547, -0.179.
      - ܮܨܰܫ௖௧ିଵ: -22.903**, -19.284*, -23.602**, -34.071**, -34.065**, -35.217**, -0.627, -0.813, -0.301.
      - ܶܫܦܧܴܥ௖௧ిଵ: -7.714, -8.255, -8.564, -52.741*, -40.712*, -43.082*, -11.545*, -11.955*, -11.584*.
      - ܲܦܩ௖௧ిଵ: -72.122**, -62.569**, -86.650***, -98.697***, -127.515***, -130.883***, -43.812*, -43.344*, -44.362**.
      - ܲܣܹെ1ݐܿ: 20.416, 13.061, 31.131, 429.209, 387.202, 450.917, -550.759**, -548.992**, -537.965**.
  - Fixed effects:
    - Year FE: Yes in all reported specifications.
    - Country FE: Yes in all reported specifications.
- Notes on variable transformations (from Table A2 notes):
  - ܶܰܧܴ௖௧ිଵ, ܲܦܩ௖௧ிଵ, and ܶܫܦܧܴܥ ௖௧ିଵ are expressed as deviations from the trend (computed with a Hodrick-Prescott filter).
  - ݅௖௧ିଵ and ܮܨܰܫ௖௧िଵ enter as annualized changes; ܲܣܹ ௖௧ిଵ is expressed as annualized growth.

*Italic: Source — _wp11231 - Appendix A1 – The Distribution of Phase Durations for an ARIMA(1,1,0) Process*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11231.pdf_