Detecting exuberance in house prices across Canadian cities

Context and motivation

The Canadian housing market has been exceptionally strong during the COVID‑19 pandemic. In March 2021, national resales reached all-time highs and house price growth surpassed its previous peak. Strong demand fundamentals, shifting preferences for more space, and limited supply of single-family homes have together contributed to rapid house price growth.

Sustained periods of rapid growth in house prices can create the expectation that prices will continue to rise, even if economic fundamentals cannot support these increases. Such extrapolative expectations can become self-fulfilling when the prospect of higher prices in the future raises housing demand today.

In this note, we introduce a model aimed at detecting periods of extrapolative expectations in house prices across Canadian cities. The resulting House Price Exuberance Indicator (HPEI) can be updated on a quarterly basis to support the Bank of Canada’s broader assessment of housing market imbalances.

A model-based indicator of housing exuberance

We introduce a non-linear heterogeneous agent model (Bolt et al. 2019) for the housing market in which agents switch between different types of house price expectations. Specifically:

• Some agents expect house prices to converge toward levels consistent with economic fundamentals.
• Others are trend-followers, believing prices will further deviate from fundamentals.

The relative proportions of these two types of agents determine whether house prices are:

• in a mean-reversion regime, where prices are expected to converge back to fundamentals, or
• in a temporary explosive regime, where prices are expected to deviate further from fundamentals.

We estimate the model using house price data for Canadian cities from 1988, as outlined in Figure 1. Further technical details are provided in the appendix.

Since house prices in the model are formulated in deviations from fundamentals, we first estimate the levels of house prices consistent with typical demand-side fundamentals.1 Specifically, we estimate a city-level panel regression in which the real house price depends on:

• disposable income per capita
• population
• the real effective mortgage rate

We then feed the deviations from fundamentals into our behavioural model, from which the HPEI is derived as the time-varying autoregressive (AR) coefficient. This coefficient is a function of:

• the relative fractions of mean-reverting and trend-following agents
• how fast these agents expect prices to converge to, or diverge from, fundamentals

The housing market is deemed to be exuberant when the HPEI exceeds 1, meaning house prices are expected to diverge further from their fundamental level.

The Greater Toronto Area through the lens of the HPEI

The Greater Toronto Area (GTA) provides a good test case for assessing the usefulness of the HPEI. The house price dynamics in the GTA from 2016 to 2017 appear to have been driven, at least in part, by expectations (Khan and Webley 2019).

Chart 1 shows that the HPEI for the GTA started rising in 2015 and exceeded 1 in the first quarter of 2016. This suggests good early warning properties of the HPEI given that year-over-year house price growth was only around 10 percent at the time but peaked at roughly 30 percent about a year later.

Moreover, the HPEI tracked movements of other relevant indicators. For example, the HPEI broadly matched:

• the dynamics of the share of homes sold over their asking price
• expectations of growth in house prices from the Bank’s Canadian Survey of Consumer Expectations

Importantly, these other indicators were not sending a clear signal when the HPEI first exceeded 1, with the former hovering around 40 percent and the latter exhibiting volatility. Therefore, this episode demonstrates how the HPEI can complement other indicators in providing a timely assessment of housing exuberance.

Caveats and areas for future work

Our results suggest that the model introduced in this note provides useful information for detecting periods of extrapolative house price expectations. However, no single model can provide the appropriate signal in all circumstances. Therefore, it will be important to use the HPEI in conjunction with other tools and data to arrive at a full assessment of housing market imbalances.

One important source of uncertainty concerns estimating fundamental house prices. The behavioural model used to detect periods of exuberance is independent of the process used to estimate fundamental prices. However, this initial step has important implications for the estimated parameters of the behavioural model and the resulting HPEI. The approach we use in this note to estimate fundamental prices considers only demand-side factors. This could lead to upward-biased estimates of the deviation of prices from fundamentals. As a result, the HPEI could be higher in cities experiencing increasing constraints on the supply of land or home-building materials and labour. Better modelling of the supply-side of the housing market remains an important avenue for future research.

Appendix

Technical details of the model

Technical details of the model3

The stochastic model we estimate is a non-linear AR(1) model of the form:

where:

• \(X_{t}\) is the real house price,4 expressed in deviations from fundamentals
• \(n_{n,t}\) is the fraction of agents in period \(\mathit{t}\) that hold expectations of type \(h, h \; \epsilon \; \{\mathit{1},\mathit{2}\}\)
• \(Φ_{1} \; (Φ_{2})\) is the speed of convergence to (divergence from) the fundamental house price
• \(R\) and \(\tilde{\alpha}\) are fixed fundamental parameters
• \(u_{t}\) is the error term

The two types of agents have different forecasting strategies for house prices. Endogenous switching between belief types is based on the past performance of these strategies. Specifically, the fractions of belief types are updated in each period according to a discrete choice model:

• fraction of type 1 agents:
  \(n_{1,t} = \delta \; n_{1,t-1} + (1-\delta) \frac{1}{1+e^{β(u_{2,t-1}-u_{1,t-1})}} \)
  
  \(= \delta \; n_{1,t-1} + (1-\delta) \frac{1}{1+e^{-β(X_{t-1}+\tilde{\alpha}-RX_{t-2})(Φ_{1}-Φ_{2})X_{t-3}}} \)
  
• \(n_{2,t} = 1-n_{1,t}, \)

where:

• \(U_{h,t-1}=(X_{t-1}+\tilde{\alpha}-R\,X_{t-2}) (E_{h,t-2}(X_{t-1})+\tilde{\alpha}-R\,X_{t-2})\) is a fitness measure
• \(1-\delta\) is fraction of agents updating beliefs each period
• \(β\) represents the sensitivity of agents to small changes in past performance

Parameters \(Φ_{1},Δ Φ\) (which equals \(Φ_{2} - Φ_{1}\)), \(β\) and \(δ\) are estimated, and \(Φ_{2}\) is then implied.

Endnotes

References

1. Bolt W., M. Demertzis, C. Diks, C. Hommes and M. van der Leij. 2019. “Identifying Booms and Busts in House Prices Under Heterogeneous Expectations.” Journal of Economic Dynamics and Control 103 (C): 234–259.
2. Khan, M. and M. Verstraete. 2018. “Non-Resident Taxes and the Role of House Price Expectations.” Bank of Canada Staff Analytical Note No. 2019-8.
3. Khan, M. and T. Webley. 2019. “Disentangling the Factors Driving Housing Resales.” Bank of Canada Staff Analytical Note No. 2019-12.