## 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.

- Standard user cost model extended with heterogeneous beliefs
- Agents boundedly rational
- Agents agree on fundamentals but disagree on speed of convergence to/divergence from fundamentals
**Type 1**: Agents who believe in mean-reversion toward fundamentals**Type 2**: Agents who are trend-followers, believing prices will further diverge from fundamentals- Fractions of belief types updated each period depending on past performance of forecasting strategies

- Estimate a city-level panel regression where fundamental price depends on income per capita, mortgage rates and population

- Use deviations from fundamentals to estimate behavioural parameters of agent-based model
- Estimate non-linear time-varying autoregressive (AR) (1) model using non-linear least squares

- The exuberance indicator is the time-varying AR coefficient
- The housing market is exuberant when time-varying AR coefficient >1

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.

### Chart 1: The HPEI was informative during the most recent house price cycle in the GTA

Note: HPEI is the House Price Exuberance Indicator. GTA is the Greater Toronto Area.

Sources: Canadian Real Estate Association, Canadian Survey of Consumer Expectations, Realosophy Inc. and authors’ calculationsLast observation: January 2021

## An exuberance heat map for Canadian cities

We use a heat map to help visualize our results for a broad set of Canadian cities. The colour-coding depicts the distance of the HPEI from 1. Shades of green indicate when the HPEI is below 0.95.2 Shades of orange show when the HPEI is between 0.95 and 1. When the HPEI exceeds 1, the colours turn to red and deepening shades of red. The most broad-based period of exuberance identified by the HPEI occurs in the late 1980s, corresponding to a period of record house price growth in many parts of the country.

In recent years, the most notable period of exuberance is the 2016–17 episode in the GTA, with similar results observed in the neighbouring city of Hamilton. While the HPEI identifies periods of exuberance in Vancouver in 2015–18, these are not as persistent. This likely reflects various policy interventions by the provincial government. These interventions successfully cooled the housing market, in part by lowering house price expectations (Khan and Verstraete 2019).

Since the start of the COVID‑19 pandemic, the HPEI has identified three cities as exuberant:

- Toronto as of the third quarter of 2020
- Hamilton as of the fourth quarter of 2020
- Montréal as of the first quarter of 2021

In addition, Ottawa is on the verge of being identified as exuberant, with its HPEI reaching 0.99 in the first quarter of 2021. In other cities, the HPEI is not yet detecting exuberant house price dynamics.

### Chart 2: The HPEI has detected exuberance in some housing markets during the COVID-19 pandemic

Chart 2: The HPEI has detected exuberance in some housing markets during the COVID-19 pandemic

Note: HPEI is the House Price Exuberance Indicator.

Source: Bank of Canada calculationsLast observation: 2021Q1

## 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:

$$ X_{t} = \frac{Φ_{1} n_{1,t} + Φ_{2} n_{2,t}}{R+\tilde{\alpha}} \; X_{t-1} + U_{t}, $$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

- 1. Alternative approaches to estimating fundamentals—such as those based on the price-to-rent ratio—generate persistent and implausibly large deviations from fundamentals because of the poor measurement of market rents in the consumer price index (CPI).[←]
- 2. 0.95 corresponds roughly to the average equilibrium value of the HPEI (i.e., the value at which there are equal proportions of mean-reverting and trend-following agents) across cities. The HPEI can also exceed 1 when house prices are persistently undershooting fundamentals. However, these episodes are relatively rare.[←]
- 3. Further technical details, including the estimated behavioural parameters for all cities, will be available in a forthcoming Bank of Canada working paper.[←]
- 4. We combine average resale prices and benchmark prices from the Canadian Real Estate Association (CREA) to construct long time series of house prices. These time series typically begin only in the 2000s.[←]

## References

- 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. - 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.
- Khan, M. and T. Webley. 2019. “Disentangling the Factors Driving Housing Resales.” Bank of Canada Staff Analytical Note No. 2019-12.

## Acknowledgement

We would like to thank Cees Diks for helpful discussions and for providing the code to estimate the behavioural model.

## Disclaimer

Bank of Canada staff analytical notes are short articles that focus on topical issues relevant to the current economic and financial context, produced independently from the Bank’s Governing Council. This work may support or challenge prevailing policy orthodoxy. Therefore, the views expressed in this note are solely those of the authors and may differ from official Bank of Canada views. No responsibility for them should be attributed to the Bank.