The Impact of Economic Policies on Housing Prices: Approximations and Predictions in the UK, the US, France, and Switzerland from the 1980s to Today

Abstract

I show that house prices can be modeled using machine learning (kNN and tree-bagging) and a small dataset composed of macroeconomic factors (MEF), including an inflation metric (CPI), US Treasury rates (10-yr), Gross Domestic Product (GDP), and portfolio size of central banks (ECB, FED). This set of parameters covers all the parties involved in a transaction (buyer, seller, and financing facility) while ignoring the intrinsic properties of each asset and encompassing local (inflation) and liquidity issues that may impede each transaction composing a market. The model here takes the point of view of a real estate trader who is interested in both the financing and the price of the transaction. Machine learning allows for the discrimination of two periods within the dataset. First, and up to 2015, I show that, although the US Treasury rates level is the most critical parameter to explain the change of house-price indices, other macroeconomic factors (e.g., consumer price indices) are essential to include in the modeling because they highlight the degree of openness of an economy and the contribution of the economic context to price changes. Second, and for the period from 2015 to today, I show that, to explain the most recent price evolution, it is necessary to include the datasets of the European Central Bank programs, which were designed to support the economy since the beginning of the 2010s. Indeed, unconventional policies of central banks may have allowed some institutional investors to arbitrage between real estate returns and other bond markets (sovereign and corporate). Finally, to assess the models’ relative performances, I performed various sensitivity tests, which tend to constrain the possibilities of each approach for each need. I also show that some models can predict the evolution of prices over the next 4 quarters with uncertainties that outperform existing index uncertainties.

1. Introduction

Assessing the status of the housing market is key in assessing the stability of the financial sector and inferring the level of risk taken by the banks, insurance, private equity funds, and private persons. Therefore, it is crucial for banks, insurance, regulators, and governments to understand the real estate market. Market observation is mostly done through the monitoring of prices, checking vacancy rates, and discussing the right level of a given transaction (“What is the fair price of a flat in Zurich?”). Observed values are used to infer model regulatory capital, financial exposures, and the prices of secondary market securities with mortgages as underlying. However, pricing real estate assets is one of the most challenging tasks financial institutions must complete as price structures are not always transparent to them, and secondary investors are not always aware of the local economic context. Here, I take a chance to model data using machine learning (ML) methods to explore, first, which model variables are the most important to us and, secondly, to assess the performance of the more sophisticated models in explaining the prices observed. ML has already been used to solve similar problems in the past (Glynn 2022; Liu et al. 2005; Wang et al. 2014; Pai and Wang 2020; Pinter et al. 2020; Singh et al. 2020; Rodriguez Gonzalez et al. 2022; Levantesi and Piscopo 2020).

• Macro view and risk to the economy

This study does not aim to solve the real estate pricing problem definitely but rather links the observed house prices to an economic context that is wider than the local markets and single transactions. From a macroeconomic perspective, the status of the real estate market is considered an indicator of an economy’s overall health and one of its drivers (Case 1992; Case et al. 2000; Nneji et al. 2013). Market bubbles (Glaeser and Nathanson 2015) and crashes (Gjerstad and Smith 2009) are known to impact the finances of large parts of the population through rent levels, prices of properties, or changes in available income (She and Tang 2007; Deng et al. 2019). The real estate market also sustains large portions of the economy through construction, building maintenance, legal and planning needs. When a housing price downturn happens (e.g., 1990s in Switzerland for flats; in 2008 globally), the level of economic activity of those actors (e.g., building companies, lumber industry) may be lowered, inducing a stagnation of the economy. Those economic issues could be transmitted to other markets (Nneji et al. 2013; Yunus 2018) and countries’ (Nneji et al. 2015) if the economies are open enough. Those contagion may be enabled through lenders covering multiple housing markets (insurances and banks) through a currency crisis. Such contagion effects may indeed be encouraged through the support programs of central banks to regional economies, which are already known to have cross-border effects (Cao et al. 2023; Cao and Dinger 2022). Therefore, in each country, the economic and political spheres constantly contemplate and debate the status of the real estate market.

• Regulations and safety measures

The conclusions of these reflections result in sets of support measures and regulations. To protect buyers, minimum income levels are required while maximum duration and rates of loans are prescribed. For the financial system, real estate exposures require significant financing needs, and regulators request large amounts of capital to guarantee the stability of the banking and insurance sectors in case a systemic crisis hits the housing market. To estimate capital needs, credit rating models and risk management measures are established to ensure the capital level is sufficient at any time. In this context, the price of purchased assets and their evolution over time become important, even for secondary investors (e.g., mortgage-backed securities owners). Those measures nevertheless rely on the price estimation at all times of each asset. Exposures, Exposure at Default (EAD), Loss Given Default (LGD), and Probability of Default (PD) models are all impacted by a mispricing of assets (e.g., through Loan to Value, collateral value, recovery value) requiring the bank to rely on independent pricing of assets and additional risk measures (liquidity monitoring, margin calls, etc.).

• A short literature review on real estate pricing

Published prices are often based on the assets’ parameters (hedonic models: type of asset, living surface, number of rooms, etc.), but they also depend on financial and economic contexts, which may remain unclear to players. For instance, if mark-to-market prices are often thought to be wrong, it is because the rationale of a transaction is usually overridden by considerations based on different goals. The price of a single asset results from a mix of emotional (“location, location, location”, prestige address, rare properties, etc.), cultural (renting vs. buying), and economic variables, such as the structure of the pension investments, professional needs, and touristic locations (Devi and Perumandla 2023; Coskun 2022; Garboden 2023; Logan 1991; Salzman and Zwinkels 2017; Farlow 2004; Schiller 2005, 2006). As said above, many of the pricing models available today focus on explaining prices from asset features and local contexts: the GDP (Tham et al. 2022; Zhang 2011; Huang et al. 2007; Cunha and Lobão 2021), the quality of the assets on the market, their distance to main economic centers, the volumes available, the occurrence of crime (de La Paz et al. 2022; Pinzon et al. 2008), land quality (Lee and Choi 2017), level of salaries (Gonzalez-Val 2022), environmental standards, industrialization, building norms and standards, political context (Tham et al. 2022), the type of immigration (Gopy-Ramdhany and Seetanah 2022), the COVID-19 pandemic (Del Giudice et al. 2020; Ha 2021; Mattarocci and Roberti 2020), the level of inflation (Taltavull 2021) and the quality of transportation networks which enable workers to reach their job (Wang et al. 2007). Those models, called hedonic (see Fahrländer (2006); Bourassa et al. (2010) for Switzerland), certainly allow the ranking of assets within a local context but tend to neglect other effects, such as the impact on personal taxes and macroeconomic conditions, and lead to the mispricing of many assets (Bond et al. 2022; Chinco and Mayer 2015; Lin and Vandell 2007). Another root of mispricing results of the competition between pricings solutions for a same kind of asset: retail transactions also compete with more significant projects (multiple-family buildings or multi-building) carried out by insurance companies and banks, designed to generate cash flow while securing cash deposits from inflation fluctuations. Pricing may be influenced by the demand for buy-to-let investments of private investors. Older generations are making an arbitrage between real estate, collective investment, and anticipated donations to the youngest family members. From their perspective, a variety of mid-term investments are limited, so the purchase of a real estate asset may be prioritized over the profitability of the investment. Finally, markets may be disrupted by the drafting of regulations made in the contexts of political and fiscal reforms, which are not always aligned with economic cycles. Mispricing has strong consequences and cannot be ignored as it is carried only by buyers. A too-high value would lead to a higher risk of default and potentially a charge in capital too high for banks. An underestimated price may not allow the transaction to happen and lead to markets being gripped.

• Can ML bring something if you adopt a holistic perspective?

In this study, I take a step back from pricing complexity and adopt a holistic view of the problem, focusing on the fact that a market is made up of the sum of realized transactions. After all, regardless of the level of financing rates, some contexts are not favorable to real estate investment (e.g., low liquidity of potential buyers). For instance, if investors worry about their economic future or the level of prices, they may not be inclined to invest their savings into securing housing. Similarly, investors unwilling to lose too much may refuse to finalize a transaction. Such a context was observed in France in 2023 when commercial banks did not immediately charge the cost of rising rates to their customers in an attempt to preserve the market. Nevertheless, this commercial measure, which aimed at preserving the bank’s businesses from rate fluctuations, did not impede the building companies from being impacted by a substantial slowdown in transaction volumes. In short, whatever the market and the type of asset, each transaction needs to be completed by making available three parts: 1/an asset (or a seller), 2/a buyer, and 3/a secured financing facility. Those three parties are convenient as they directly represent the transaction from a front desk perspective. Those parties carry all liquidity, financial constraints (willingness to sell, possibility to buy), economic confidence (financing rates and willingness to buy), and arbitrage (profitability of the transaction, investment into equities rather than real estate) that characterize such a transaction. I test here the assumption that the financing constraints strongly determine the price rather than the specifics of the asset being transferred. With the help of ML, I aim at identifying which parameters are the most important to explain prices and potentially to predict those.

2. Materials and Methods

For modeling the data, I used the caret R package (R Core Team 2021; Kuhn 2008) and tested my approach in 15 countries. I have tested both kNN and tree-bagging strategies in order to detect the presence of overfitting in the model solutions (see

Table 1. Training parameters used for all models in this study.

Model Parameters
tr control	method = “repeatedcv”, repeats = 3
empty data	not allowed

for control options). This paper is linearly organized by quantifying the impact quality of including new variable(s) to the previous model. But first, and based on a wide choice of data available at SNB, I select which indicators are helping the most to explain the evolution of flats in Switzerland. Second, and from the parameter selection, I have built an initial model based on inflation, financing rates, and GDP to explain the price evolution in 15 countries using the data already archived by the FRED database. I then tested the inclusion of central bank rates and ECB portfolio size in order to test whether the addition of a parameter significantly improves the data fit. I have tried to model the house prices in nominal and percentage forms to test whether using the nominal value improves the model’s performance. The complete list of models is available in

Table 2. Characteristics of the data used for each model group. HPI rates are computed over 12 quarters1.

	Model Name	GDP	CPI (Rates)	Treasury 10 y	Unemployment	Rent	Central Bank	ECB	HPI	Figure
1	3-parameter	Y	Y	Y, rates	N	N	N	N	Nominal	Figure 4
2	IR models	Y, rates	Y, rates	Y, rates	N	N	Y, US rates	N	Nominal	Figure 5
3	Local Central bank	Y, rates	Y, rates	Y, rates	N	N	Y, local rates	N	Nominal	Figure 6
4	ECB	Y, rates	Y, rates	Y, rates	N	N	N	Y, nominal	Nominal	Figure 7
5	ECB/FED	Y, rates	Y, rates	Y, rates	N	N	Y, plus FED rates	Y	Nominal	Figure 8
6	Rents	Y, rates	Y, rates	Y, rates	N	Y	N	N	Nominal	Figure 9
7	3-parameter 1 yr	Y, rates	Y, rates	Y, rates	N	N	N	N	Rates	Figure 10
8	ECB 1 yr	Y, rates	Y, rates	Y, rates	N	N	N	Y	Rates	Figure 11
9	Rent 1 yr	Y, rates	Y, rates	Y, rates	N	Y	N	N	Rates	Figure 12
10	Permutations	Y, rates	Y, rates	Y, rates	N	N	N	N	Nominal	Figure 13
11	LOCAL 1 yr	Y, rates	Y, rates	Y, rates	Y	N	N	N	Nominal	Figure 14

.

I have used three different sources for the data used in this study. First, I used the Swiss National Bank (SNB) dataset, which helped me to identify which parameters were critical to include. The dataset available at the SNB is extensive and detailed, but I have failed to find the exact same data for other countries. For instance, price indicators for each canton and each asset type (flat vs single-family house) cannot be found for all countries. As I wanted to expand my study to a large number of countries, I used the data available at the Federal Reserve Bank of St. Louis (“FRED”) for economic and financial data and at the Federal Reserve Bank of Dallas house price index database (Mack and Martínez-García 2011).

Regarding data preparation, pre-processing was kept to a minimum. For all models, end-of-quarter figures were used for all data. When missing data were observed (the treasury rates data were not complete at the daily level), gaps were filled using linear interpolation when end-of-quarter values were missing. The original data and the corrected data are shown in the Supplementary Materials for treasury rates. No modification was applied to house prices when nominal values were used. I show that the Consumer Price Indices (CPI) were utilized in their annualized form. The same data treatment has been used for all the countries considered.

2.1. The Swiss National Bank (SNB) Data Portal Dataset

Many economic data for Switzerland are available at the Swiss National Bank (SNB) data portal (www.snb.ch, accessed on 6 January 2023). Real estate prices are available for various types of assets (flats, single-family houses, etc.) in multiple areas. Other potentially valuable data for this study relate to the macroeconomic trends of the Swiss economy (GDP, GNI, subsidies, wages, etc.). The published real estate prices show the dispersion of estimates for each type of asset (flats, single-family homes, buildings) across the pricing tools (Figure 1 and Figure 2). Values for some assets and some years show a dispersion of ±5% for homes and ±15% for flats. For Wuest and Partner, the spread between “ask” and realized prices can reach 30% for some market segments. Gaps between price time series may reflect differences in the size of datasets, segment of the market, or regional location of the assets observed. Those differences will be discussed later in the light of fit performance and predicted price accuracy in the Results Section. To consistently include more countries in this study, I searched for data that could be found in other databases, such as INSEE (https://www.insee.fr/fr/accueil, accessed on 6 January 2023) in France. Unfortunately, data coverage using national agency datasets cannot provide enough consistency across countries (some series are too short to be exploited, and others do not have a sampling rate similar to other countries).

2.2. The FRED Dataset

The Federal Reserve Bank of St. Louis archives macroeconomic data for many countries in the same format and distributes it freely through its website (https://fred.stlouisfed.org/, accessed on 6 January 2023). From the first analysis of the data for Switzerland, I chose the following parameters: Consumer Prices Index (CPI) values, US Treasury rates (10-year maturity), and Gross Domestic Product (GDP) for 15 countries. To those parameters, I added the interest rates of the central banks of each country and the FED/ECB total asset portfolio size, which are known to impact the level of economic activity and house prices (Lin and Tsai 2021a, 2021b).

2.3. The International House Price Database of the Federal Reserve Bank of Dallas

At last, I opted to use the house prices published by the Dallas Federal Reserve (Mack and Martínez-García 2011). The house price index (HPI) dataset covers many countries (see list of countries included in this study; Table 6). I chose to use the HPI and not its inflation-adjusted version (RHPI), as I preferred using the inflation data (CPI) as an input parameter in the raw form. It is possible that the baskets of goods used to compute CPI differ from one country to another. It is not an issue here, as each CPI dataset is for house prices in the same country. Doing this allows for a better match between the local perception of prices and the confidence of buyer/sellers when entering a transaction. Regarding the choice of countries, I focus on the European countries and some countries of interest (USA, Australia, Canada), because their markets are sizable and/or because they are known to use variables interest for part of the duration of the loan. By doing so, I hope to discover similarities between European countries and between countries with strong economic links (e.g., France–Germany; USA–United Kingdom; United Kingdom–Australia; Finland–Sweden–Norway). I show, in Figure 3, the evolution of some of the indicators used against the data published by the Federal Reserve Bank of Dallas house price index (HPI) for Switzerland.

3. Results

3.1. Two First Models to Map Important Variables for Switzerland

In this section, I complete two models that aim at finding the most important variable and quantifying the contribution of new data to model improvement. I first use a dataset of 49 parameters available at the SNB. The complete list of variables is available in Appendix A.

The data included in this model are unmodified from their source. HPI values are used in their nominal form with a reference value of 100 at Q2 2002.

The R package caret allows for quantifying the relative importance of the parameters contributing to explaining the signal. I list the 20 most essential parameters with their categories in

Table 3. The first 20 most contributing variables selected by the kNN algorithm from the complete dataset of 49 parameters for model 1. Those parameters, in their original form (nominal values and percentages), are needed to explain apartment prices in Switzerland.

	Parameter Name (SNB Denomination)	Category	Relative Importance
1	Compensation of employees paid to the rest of the world	Income	100.00
2	Type of goods - Goods - Durables	CPI	99.55
3	Type of goods - Services - Total	CPI	99.49
4	Compensation of employees	Income	99.24
5	Type of goods - Services - Private	CPI	99.02
6	Origin of goods - Domestic	CPI	98.69
7	Gross domestic product	Income	97.09
8	Gross national income GNI	Income	96.95
9	Subsidies	State	96.78
10	Consumption of fixed capital	Investment	95.95
11	Taxes on production and imports	State	95.34
12	Type of goods - Services - Public	Inflation	91.33
13	United Kingdom - GBP - GBP LIBOR - 3-month	Markets	91.32
14	Labour force	Demography	88.95
15	Compensation of employees received from the rest of the world	Income	88.37
16	Switzerland - CHF - Call money rate Tomorrow next - 1 day	Markets	86.84
17	CPI - Total index	Inflation	84.31
18	Net operating surplus	Production	75.30
19	Type of goods - Goods - Non-durables	Inflation	73.91
20	Property income paid to the rest of the world	Income	72.37

. Most of the parameters listed are either related to inflation or income. Interestingly, employment rates and short-term financing rates play a less important role than expected. For Switzerland specifically, these first findings are explained by the fact that only households with a yearly income larger than 1/5 of the total loan are given access to financing (this is a rule of thumb valid to investments purchased with a maximum of 30% collateral provided).

Overall, the data shown in Table 3 are reassuring, and this model confirms that, because the affordability criteria fixed by financial authorities are strictly enforced, the level of unemployment rates has no impact on prices. Some observations can be additionally made:

• Financing rates (e.g., LIBOR 3M) play a role but a less pronounced one than expected. Most actors agree that the financing rate is the most critical parameter in explaining the evolution of house prices. In the view of many, low financing rates are allowing more transactions to be realized, driving the demand upwards. This is undoubtedly true for most buyers, but the transaction’s performance also plays a role for investors involved in large volumes. For buy-to-rent schemes, real estate investments become more profitable when rates are low. Indeed, as the profitability of a real estate transaction is estimated at 3 to 5%, the arbitrage between a real estate transaction and another investment that is equally cash-intensive (e.g., public or private debt) must be considered in the light of the inflation context for the period covered by the transaction (2, 5 or 10 yrs).
• Secondly, this first model shows that inflation plays a role. Inflation can have multiple effects on the health of a portfolio. Those effects depend on the real estate segment:
  • For private investors, inflation is effective at reducing the interest rate charge. For those sensitive to downward income fluctuations, a change in inflation may become a concern. For those with a fixed interest rate or those who use real estate investments to protect their savings, an increase in inflation could be beneficial, especially if real estate prices follow inflation.
  • For commercial and buy-to-let investors, inflation has primarily adverse effects. Indeed, those investors may have inflation-linked and floating loans. If the business plan at inception was not conservative enough, inflation may impact the investor’s liquidity through an increase in maintenance and operation costs (e.g., heating).
• Finally, I contemplate the impact of GDP on house price structure. In reality, GDP is not correlated with price evolution: the GDP never decreased in Switzerland while the HPI fluctuated. Nevertheless, we include GDP into each national dataset as a local parameter, weighing the overall perception of the regional economic conditions against the other countries. For instance, commercial investors may examine GDP dynamics before choosing the location of an office and commercial space investment. However, it is unlikely that private investors will weigh their purchase decisions in light of GDP forecasts.

As a second step, and to increase the number of quarters available in the dataset, I have reduced the set of parameters (

Table 4. List of parameters chosen for model 2. Here, as some parameters include some collinearity with prices (positive for Compensation of employees and GNI; negative for US LIBOR 3 m) the performance of this model should be considered with caution. However, this model presents the advantage of using raw data. This model also presents the advantage of linking some demographics data such as immigration numbers (more workers may lead to a decrease of salary levels).

	Parameter Name (SNB Denomination)	Category	Relative Importance
1	Compensation of employees	Income	100
2	GNI	Income	96
3	Inflation total index (%)	Inflation	62
4	US LIBOR 3m (%)	Rates	24
5	SNB Core inflation (%)	Rates	0

). Model 2 shows similar performance to model 1, although the number of data points is much smaller. From the comparison of the standard deviation of residual values, models 1 and 2 have similar performances (

Table 5. Performance of the first two models based on Swiss National Bank dataset. The model (“lightdata”), which uses fewer input parameters but spans a more extended period, is more accurate (RMS, SD improve while COR and MAPE stay satisfactory). I consider both models are performing well (RMS < 15; MAPE < 0.2; COR > 0.90).

Model	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
Model 1/49v	0.994	0.001	3.307	0.201	3.320	2.649	0.020	0.157	0.213
Model 2/“lightdata”	0.974	0.001	0.464	0.009	0.466	0.373	0.110	0.095	0.702

).

As expected, these first two models show the power of the k-nearest neighbors machine learning approach in discriminating the importance of various input parameters to explain output data.

However, inferring that other datasets could have explained the data equally well is still impossible. We can, however, conclude from these two models that (1) the number of parameters that explain prices is not necessarily high, and (2) that treasury rates cannot be used as the only financial input parameters (i.e., otherwise it would not have been possible to finance real estate in the 1970s).

For each model, and to rank them by their quality, I have computed various statistical figures (Table 5) such as root mean square (RMS)2, standard deviation of residuals3 (i.e., observed modeled values), Mean Absolute Error (MAE)4 and mean absolute percentage error (MAPE)5. Regarding the MAPE values, I use the thresholds proposed by Lewis (1982)6. Those functions are part of the MLmetrics (https://cran.r-project.org/web/packages/MLmetrics/MLmetrics.pdf, accessed on 6 January 2023) R package. The goal here is not to provide definitive answers to the price construction, but to explore which input parameters are significant enough to contribute to explaining the data. For each model performance testing, I run the models 600 times and build statistics from the fit of each model. I consider models are performing well when mean (RMS) < 15 and mean (MAPE) < 0.20.

3.2. Open Economies?

It is well known that economic contagion may happen between countries and regions of the global economy. Contagions are made possible when regional and international institutions invested in multiple asset classes suddenly meet an unforeseen liquidity need or when the source of financing is unique for many markets. One obvious scenario would describe the difficulties of a commercial real estate investor who may be impacted by a sector recession (an increase of vacancies in rented assets), default of an investor (unfinished buildings), and/or liquidity difficulties of a lender forcing him to realize transactions at a loss (fire sale to reduce exposures) to increase capital ratios.

To test the connection(s) between real estate markets, I computed the correlation coefficients between the house price series of various countries (

Table 6. Correlation coefficients of house price indices between countries for 10 countries. Values larger than 0.80 are highlighted in red. Four datasets (Italy, Finland, Japan, and Spain) correlate less with other price series. UK, Sweden, Australia, and the Netherlands show the strongest correlation with prices in the USA. Russia and Japan only publish yearly data for their Gross Domestic Product statistics. In the rest of the article, those two countries are excluded from the pool of countries modeled. Anti-correlations are indicated in green.

		Nobs	USA	FR	GER	SP	IT	FIN	SWE	AU	UK	CH	NL	RU	JP	HU	NO
1	USA	196	1	0.89	0.91	0.75	−0.01	0.79	0.94	0.96	0.98	0.82	0.95	0.84	−0.32	0.93	0.99
2	France	174		1	0.57	0.83	0.21	0.98	0.88	0.90	0.95	0.87	0.86	0.94	−0.96	0.84	0.90
3	Germany	132			1	0.82	0.11	0.61	0.92	0.94	0.97	0.84	0.85	0.53	−0.48	0.96	0.80
4	Spain	116				1	0.67	0.22	0.64	0.66	0.78	0.54	0.83	0.57	−0.88	−0.13	0.62
5	Italy	116					1	0.53	0.20	0.17	0.01	0.28	0.19	0.94	−0.30	−0.61	−0.19
6	Finland	136						1	0.84	0.82	0.75	0.80	0.66	0.84	−0.95	0.66	0.94
7	Sweden	124							1	0.99	0.97	0.95	0.87	0.84	−0.86	0.93	0.99
8	Australia	195								1	0.98	0.91	0.85	0.83	−0.32	0.87	0.99
9	UK	195									1	0.86	0.92	0.83	−0.27	0.90	0.97
10	CH	176										1	0.74	0.83	−0.53	0.76	0.90
11	NL	112											1	0.64	−0.84	0.91	0.82
12	Russia	21												1	−0.92	0.61	0.86
13	Japan	46													1	0.17	−0.52
14	Hungary	67														1	0.85
15	Norway	184															1

). Some series are correlated, others less so. For instance, Italy, Spain, Russia, and Hungary are the countries least correlated with others. Japan’s price history is anti-correlated with that of most countries. The overall correlation between different markets may be explained by a common source of financing, debt origin, and securitization, which makes the market more connected. Local conditions (demography, volumes available, construction rate, etc.) may play a more prominent role for countries that do not show a correlation. Countries publishing data only once a year (Japan and Russia) or only recently (Hungary) are excluded from further analysis as too few data are available. One can consider, however, that Japan, because of its specific context (demography, domiciliation of its debt, and status of its stock market), may stand out rightly. I build on these first observations and independently model house price time series for various countries by testing the impact of adding new variables on the quality of the data fit.

3.3. Modeling House Prices in 15 Countries

As many of the price time series are well-correlated, and buyers/sellers are not in contact together and likely do not contribute to others local economies, one variable should contribute to all markets to sustain similar price increases. One of the candidates is, of course, LIBOR, but as it has been suggested that LIBOR was manipulated and was indeed discontinued from Q4 2016 in the FRED database, I have chosen to use the US Treasury Rate (10-yr), which matches the maturity of the investment hedge of banks when pricing the transaction. For the sake of clarity, and for the rest of the article, I shall only display model results for four countries that already experienced crises (USA, UK, Switzerland, and France), and which, in my view, represent the widest variety of contractual setups (

Table 7. Broad characteristics of the market. ARMS: Adjustable-Rate Mortgage.

	Amortization	Fixed Rates?	Source
USA	Yes, continuous	Possible (ARM, 10%)	https://shorturl.at/g9bZ3, accessed on 6 January 2023.
France	Yes, continuous	Yes, variable 30%	https://shorturl.at/D4FOW, accessed on 6 January 2023.
UK	Yes	Possible, 26%	https://shorturl.at/FaNlm, accessed on 6 January 2023.
Switzerland	At least 35% after 10–15 yrs, otherwise discontinuous.	Yes, LIBOR for 10–15%, ARM 5%	https://shorturl.at/bFb3b, accessed on 6 January 2023. https://www.moneyland.ch/en/variable-rate-mortgages-comparison, accessed on 6 January 2023.

). All figures from other countries are available in Supplementary Materials. I have completed a series of models, which are listed in

Table 8. Variable importance data for the models presented in this paper for Switzerland. The effect of the ECB dataset is visible in the improvement of the model performances.

	Model Name	TR		CPI	GDP	Central Bank Rate	Asset	Additional Data, Imp.
1	3-param	100		>10	>10	-	-	-
2	3-param 1 yr	100		>10	>10	-	-	-
3	Central bank IR models (US rate)	100		>5	0	75	-	-
4	Local Central bank IR models (LIR)	100		93	0	27.9	-	-
5	ECB	90.9		13.4	0	-	100
6	ECB/FED	92.9		>5	>5	-	100	FED. Rate 24.9
7	ECB 1 yr	100		5	0	-	1.7	-
8	LOCAL	-		100	22.5	-	-	Unemployment, 0
9	LOCAL 1 yr	-		100	43			Unemployment, 0
10	Rents	100		0.8	0	-	-	Rent index, 87.0
11	Rents 1 yr	78.4		>1	0	-	-	Rent index, 100

and

Table 9. Same as Table 8 for the tree-bag approach.

	Model Name	Father	TR	CPI	GDP	Central Bank Rate	Asset	Additional Data, Imp.
1	3-param	-	100	13.4	0	-	-	-
2	3-param 1 yr	-	100	1.3	0	-	-	-
3	Central bank IR models (US rate)	1	100	12.6	0	86.2	-	-
4	Local Central bank IR models	1	80.8	100	0	56.6	-	-
5	ECB	1	90.9	13.4	0	-	100
6	ECB/FED	1, 4	76.8	>5	0	-	100	FED. rate > 5
7	ECB 1 yr	2	100	5	0	-	1.7	-
8	LOCAL	-		100	0	-	-	Unemployment, 22.8
9	LOCAL 1 yr			100	0			17.6
10	Rents		77	>10	0	-	-	Rent index, 100
11	Rents 1 yr		61.5	>5	0	-	-	Rent index, 100

. The figures showing the results for the other countries are available in Appendix A.

3.3.1. A Model with Three Parameters (3-Param; CPI, GDP and TR)

I completed the first series of models based on Consumer Price Index (CPI), Gross Domestic Product (GDP), and treasury rates (10-yr) data only (Figure 4, Table 8). The advantage of this model series is that data is freely available in many countries and some of those data go up until the 1970s. One disadvantage is that the dataset is smaller, which may affect the quality of the results. The modeling strategy is the same as all the other models (previous and future) presented in this study. I have tested two algorithms in parallel: kNN and tree-bagging. Those approaches were chosen as they are popular with many, and they were able to produce regression. The tree-bagging technique was chosen over others because it is less prone to overfitting than kNN, and the models produced are thought to be easier to explain than those built with other approaches. kNN is well-known for performing with small datasets and is easy to implement. Both approach solutions are shown with distinctive colors and curves showing residual values in each figure. When uncertain, I conclude that a model is not sufficiently robust because it is of the same magnitude as the difference between price series of various origins or between prices of different assets (houses vs. flats). In total, five countries’ models show an RMS > 15%: (USA, Sweden, Australia, Netherland and Norway;

Table 10. Statistics of the data fit performance of the 3-parameters models (tree-bag strategy). HPI values are in nominal form. COR stands for Correlation, RMS for root mean square, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.949	0.002	22.405	0.383	31.982	17.575	0.228	0.261	0.000
France	0.959	0.001	11.637	0.167	0.295	7.946	0.102	0.183	0.007
Germany	0.924	0.004	11.087	0.333	3.415	6.945	0.055	0.316	0.000
Spain	0.882	0.005	11.404	0.231	2.683	7.658	0.098	0.173	0.067
Italy	0.865	0.004	9.298	0.126	2.554	6.349	0.070	0.189	0.034
Finland	0.956	0.001	10.143	0.121	2.572	7.552	0.095	0.217	0.004
Sweden	0.956	0.003	19.687	0.661	20.861	15.148	0.125	0.166	0.066
Australia	0.956	0.002	21.044	0.433	14.522	15.866	0.247	0.215	0.000
UK	0.976	0.001	11.617	0.313	12.483	8.950	0.193	0.181	0.004
Switzerland	0.950	0.002	8.638	0.160	6.624	7.068	0.066	0.203	0.002
Netherland	0.865	0.011	16.400	0.514	17.635	12.710	0.123	0.160	0.131
Norway	0.965	0.001	18.465	0.339	NA	14.244	0.178	0.149	0.037

). Two observations can be noted from this set of third models. First, the fit with the data is good, as the long-term features of the price time series are recovered overall. Second, from 2015 the models do not explain the price increase for most countries. I note that the France and UK models performs slightly better than USA and Switzerland, where loans are not always amortized. Amortization allows damping of price changes following sudden changes in fiscal and economic policies. Two explanations may come from the economic perspective: negative interest rates and (mostly for Europe) the non-conventional support program of the European Central Bank (e.g., ECB PSPP7 and ECB PEPP8).

I know those two effects may be interlinked, so I modeled those effects separately. For this model, like for some of the following, I have also computed models after differentiating HPIs over 12 quarters.

3.3.2. Models with Three Parameters (CPI, GDP and TR) Plus Central Bank Rates: IR and LIR Models

Having shown that the prices of real estate could be successfully interlinked between countries (the best example of such a statement being the propagation of the 2008 house price drops from USA to other countries), I have completed two sets of computations. The first set of models comprises those for which US rates applied to all countries, the second set uses the local national central bank rates. A second set of models (Australia, Switzerland, and European countries) use local central bank rates and not the USA ones. HPI values are kept in their nominal form.

Overall, the inclusion of additional data was expected to have a positive impact, and this is well observed here (

Table 11. Statistics of the data fit performance of the IR models (tree-bagging strategy). COR stands for Correlation, RMS for root mean squared, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.963	0.002	18.871	0.443	18.999	14.092	0.204	0.262	0.000
France	0.960	0.001	11.550	0.183	3.310	7.964	0.105	0.186	0.006
Germany	0.917	0.008	11.457	0.456	8.660	7.123	0.055	0.353	0.000
Spain	0.890	0.005	11.144	0.240	3.178	7.255	0.094	0.135	0.255
Italy	0.893	0.005	8.431	0.155	3.264	5.405	0.061	0.198	0.025
Finland	0.954	0.001	10.373	0.131	3.170	7.729	0.097	0.212	0.005
Sweden	0.953	0.002	20.070	0.495	9.510	15.929	0.135	0.167	0.066
Australia	0.965	0.002	18.624	0.430	13.469	13.785	0.233	0.214	0.000
UK	0.968	0.001	13.066	0.210	7.663	10.131	0.240	0.216	0.000
Switzerland	0.961	0.001	7.472	0.126	2.100	5.972	0.057	0.182	0.007
Netherland	0.879	0.008	14.983	0.409	6.807	11.102	0.110	0.166	0.112
Norway	0.994	0.000	7.399	0.269	NA	5.752	0.084	0.134	0.081

). However, the addition of the central bank rates data has a different impact. For France, the impact is minimal (Figure 5). This could be explained by the fact that most of the mortgage loans in France are locked in (fixed rates) and are amortized. We can assume that the number of variable rate loans is not significant to cause a drop in prices (limited number of potential forced sales). A change in the central bank rates is only affecting new investors (the youngest of the borrowers) and older investors if the rates used in their contracts are variable. A change of default rate is only expressed through prices when margin calls cannot be met (prices observed are not a direct function of discrete pockets of liquidity stress). The impact on older borrowers is only visible if their income is decreased or they cannot meet margin calls.

In the USA and UK the improvement is minimal, suggesting either inflation is collinear with central bank rates, or, for the most recent years, arbitrage to enter a real estate transaction is partially driven by central bank policies. In Switzerland, we can find a slight improvement over the most recent years (>2015).

Using national central bank rates leads to the same conclusions: the fit to the data is not significantly improved for most recent observations (Figure 6 and

Table 12. Statistics of the data fit performance of the LIR (local interest rates) models (tree-bagging strategy). COR stands for Correlation, RMS for root mean squared, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2. Models with RMS < 20 and MAPE less than 20% are considered as performing.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.961	0.002	19.323	0.400	18.720	14.538	0.214	0.263	0.000
France	0.897	0.004	12.552	0.191	6.631	9.746	0.098	0.193	0.061
Germany	0.864	0.011	15.170	0.558	5.149	8.765	0.065	0.261	0.003
Spain	0.850	0.010	10.154	0.234	3.019	6.801	0.083	0.163	0.161
Italy	0.857	0.008	7.797	0.150	2.474	5.210	0.058	0.229	0.017
Finland	0.923	0.003	9.365	0.169	8.135	6.541	0.067	0.167	0.139
Sweden	0.900	0.006	24.037	0.733	53.721	16.982	0.125	0.165	0.254
Australia	0.965	0.001	17.924	0.367	32.903	12.277	0.212	0.214	0.001
UK	0.963	0.002	13.596	0.260	3.191	10.558	0.253	0.226	0.001
Switzerland	0.975	0.003	5.885	0.304	16.130	4.332	0.041	0.163	0.031
Netherland	0.873	0.014	13.497	0.464	6.959	9.563	0.086	0.212	0.026
Norway	0.964	0.002	16.851	0.443	NA	12.070	0.123	0.264	0.001

). Finally, those data were redundant with the treasury rate (10-yr), which calibrates the rates worldwide when financing is mobilized or when securitization of the position is done to unload lenders’ banking books.

3.3.3. Models Including ECB/FED Asset Sizes

The previous models including central bank interest rates are not sufficient for explaining the recent rise in prices. However, Quantitative Easing (QE) programs of the FED/ECB, but also by the PEPP of ECB, to support the economy may have provided arbitrage to some investors. Those programs started in 2015 and continued until the end of the COVID-19 pandemic. Those programs are nowadays closed by decreasing the book sizes. ECB and FED support programs aim at providing liquidity to the financial sector by relieving them from low long-term assets and by lowering the coupons of corporate debt securities (by increasing the demand on those assets). Because of the nominal values involved, such purchase programs were suspected to have an impact on other markets. However, it has been shown that those may have impacted the equity market variously over time and differently for each crisis (Olasehinde-Williams et al. 2023; Fiordelisi and Galloppo 2018), likely because of a mismatch in time to maturity. For real estate, however, maturities (2, 5 and up to 30) and return to investments (3–5%) are similar, and some investors may then be provided an investment arbitrage. In other words, some investors may be more interested in investing in real estate securities than in corporate or sovereign debt.

Regarding the impact of PSPP/PEPP, it is interesting to note that announcement of those programs stabilizes markets before transactions are executed and costs realized (Motto and Özen 2022). For those programs, amounts are announced one or two semesters beforehand. Impact is therefore almost immediate. For this reason, I chose to use the nominal of ECB asset and not its monthly purchase. I first tested the effect of the ECB program (PSPP, PEPP; Figure 7 and

Table 13. Statistics of the data fit performance of the ECB models (tree-bagging strategy). COR stands for Correlation, RMS for root mean squared, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.983	0.001	12.937	0.232	3.121	10.908	0.202	0.265	0.001
France	0.993	0.001	5.019	0.176	0.829	3.787	0.059	0.159	0.030
Germany	0.985	0.001	4.982	0.201	3.129	3.739	0.031	0.400	0.001
Spain	0.980	0.002	4.806	0.194	1.286	3.582	0.047	0.181	0.045
Italy	0.978	0.002	3.905	0.136	0.198	3.151	0.036	0.243	0.002
Finland	0.985	0.001	5.923	0.136	2.450	4.831	0.064	0.204	0.008
Sweden	0.992	0.000	8.463	0.211	3.544	6.431	0.055	0.151	0.122
Australia	0.988	0.001	11.212	0.390	7.948	9.038	0.199	0.212	0.000
UK	0.984	0.001	9.399	0.217	1.096	7.665	0.204	0.191	0.002
Switzerland	0.976	0.001	5.995	0.100	0.147	4.861	0.049	0.193	0.003
Netherland	0.981	0.002	6.220	0.348	4.405	5.218	0.054	0.207	0.018
Norway	0.990	0.001	9.631	0.310	NA	7.755	0.133	0.200	0.003

) and then those combined from FED programs (QE, QTs, Figure 8 and

Table 14. Statistics of the data fit performance of the ECB/FED models (tree-bagging strategy). COR stands for Correlation, RMS for root mean squared, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.985	0.001	12.428	0.220	1.929	10.599	0.196	0.262	0.001
France	0.993	0.001	5.009	0.193	0.828	3.758	0.060	0.165	0.025
Germany	0.980	0.002	5.642	0.230	3.827	3.979	0.032	0.381	0.001
Spain	0.983	0.002	4.435	0.187	1.146	3.404	0.044	0.182	0.044
Italy	0.974	0.002	4.216	0.130	0.434	3.500	0.040	0.246	0.002
Finland	0.987	0.001	5.468	0.116	2.450	4.490	0.060	0.232	0.002
Sweden	0.993	0.000	7.707	0.186	3.121	5.585	0.050	0.154	0.110
Australia	0.989	0.001	10.814	0.372	8.358	8.440	0.186	0.204	0.001
UK	0.984	0.001	9.234	0.214	1.346	7.465	0.208	0.196	0.002
Switzerland	0.975	0.001	6.095	0.099	0.301	5.011	0.050	0.198	0.002
Netherland	0.980	0.003	6.371	0.383	2.271	5.420	0.056	0.203	0.023
Norway	0.995	0.000	7.000	0.291	NA	5.523	0.089	0.141	0.058

). In any case, the collinearity with the price level is low as: (1) these programs do not cover the complete period of observation (zeros values are available for when asset size was zero or before ECB existed), and (2) those are being decommissioned.

For most of the countries, one can see the fit quality improved for years > 2020 (Table 14). The effect is displayed for both kNN and tree-bag in Figure 7. This indicates that at least the ECB asset book data are not conflicting with the prices of real estate, suggesting a porosity of some markets. For the sake of completeness, I have created another model which includes the data of the FED assets book. The fit for the US is even further improved, although it is difficult to draw definite conclusions from the addition of this dataset.

3.3.4. The “Rent” Model (Switzerland)

As some real estate objects are included into various types of transactions (the same flat can be rented and owned by a private individual or be part of a buy-to-let transaction), I have completed a model for Switzerland for which the rent levels (source: cc-f-05.06.01.17 (https://www.bfs.admin.ch/bfs/de/home/statistiken/preise/harmonisierte-verbraucherpreise.assetdetail.32407393.html, accessed on 6 January 2023), Office de la Statistique, Suisse) are included. This rent index starts in 1939; therefore, it is not limiting the length of the dataset considered so far.

By doing so I take into account that rent and buy markets are connected, at least for the same kind of assets. When housing unit production becomes more abundant, then vacancies may increase and rents decrease. When the standard of housing improves (insulation, connection to transport, etc.) the rent levels of existing houses may decrease locally. The addition of the rent level (percentage) for each quarter greatly improved the fit of the data (smoother fit). The correlation between modeled and observed prices is >0.99 (Figure 9). The standard deviation of the residual values is 3.23 and the RMS is 3.22. After showing that external influence of IR is limited, at least for Switzerland, this last model shows that the rental market influences the property market to some extents. In Switzerland, the rental market is strongly linked with the development of the economy as companies are looking for skilled workers who tend to reside in the city center where housing is available. For the purchase market, the link is more complicated to establish. Indeed, the new-comers are kept out of the market for the time needed to build a sufficient down-payment and to secure an income level which qualifies them to satisfy the affordability criteria. While prices and rents continue to increase, it becomes more profitable to complete a transaction of the buy-to-let type leading to a concentration of real estate to a limited number of players (because of the affordability criteria). In other words, the buy market only benefits the additional new workers who can afford to exit the rental market.

3.3.5. Year to Year Models: Moving Away from HPI Nominal Values

In all models, the input parameters are presented in the form of rates. In order to address the time dependence concerns of the parameters’ predictions, I have differentiated the HPI values over 12 quarters (3 years). Such a period is representative of the period considered by a buyer before taking the decision to buy an asset (e.g., “is the market going up or down?”, “is it more profitable to rent rather than buying?”). In these models, the collinearity of data is reduced to its maximum.

New versions of the models 3-param, ECB, Local and “Rents” have been completed (Figure 10, Figure 11 and Figure 12. The model performances of these models are displayed in

Table 15. Statistics of the data fit performance of the 3-parameters-1 yr models (tree-bagging strategy). COR stands for Correlation, RMS for root mean squared, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.812	0.008	7.605	0.105	4.338	5.646	1.099	0.191	0.003
France	0.838	0.005	6.019	0.066	2.383	4.712	0.992	0.240	0.000
Germany	0.872	0.004	3.840	0.055	4.925	2.809	2.197	0.240	0.001
Spain	0.873	0.005	8.328	0.135	2.025	6.427	1.233	0.142	0.193
Italy	0.954	0.001	10.382	0.136	2.450	7.648	0.096	0.217	0.004
Finland	0.848	0.006	4.522	0.081	4.528	3.717	0.360	0.204	0.012
Sweden	0.808	0.006	6.471	0.071	1.446	5.200	0.346	0.212	0.000
Australia	0.891	0.004	6.736	0.110	1.070	5.204	1.515	0.147	0.042
UK	0.887	0.004	4.483	0.066	1.508	3.328	0.565	0.280	0.000
Switzerland	0.777	0.013	9.610	0.125	1.429	8.201	4.069	0.255	0.002
Netherland	0.707	0.012	23.067	0.290	3.350	16.908	0.838	0.571	0.001
Norway	NA	0.000	NA	0.000	NA	14.244	0.178	0.149	0.037

,

Table 16. Statistics of the data fit performance of the ECB 1 yr models (tree-bagging strategy). COR stands for Correlation, RMS for root mean squared, MAE for mean absolute error, MAPE for mean absolute percentage error. CHIp and CHIs are the parameter and statistic of the Chi2.

Countries	M_COR	S_COR	M_RMS	S_RMS	SD	M_MAE	M_MAPE	M_CHIp	M_CHIs
USA	0.850	0.007	6.680	0.121	4.901	4.828	1.230	0.159	0.023
France	0.923	0.003	4.255	0.073	1.416	3.300	1.073	0.151	0.051
Germany	0.933	0.003	2.878	0.052	4.986	2.180	4.607	0.262	0.001
Spain	0.956	0.002	5.200	0.112	1.285	3.930	0.303	0.142	0.248
Italy	0.958	0.004	3.647	0.166	1.764	2.758	2.252	0.143	0.250
Finland	0.774	0.010	9.773	0.180	2.622	5.984	1.067	0.209	0.012
Sweden	0.866	0.009	3.824	0.099	3.415	3.166	0.233	0.214	0.019
Australia	0.895	0.005	4.723	0.091	1.657	3.918	0.243	0.191	0.003
UK	0.893	0.004	6.648	0.085	1.101	5.177	0.595	0.156	0.027
Switzerland	0.914	0.003	3.935	0.059	1.401	2.742	0.417	0.212	0.002
Netherland	0.958	0.003	4.203	0.146	1.388	2.888	0.487	0.115	0.532
Norway	0.991	0.001	9.400	0.309	NA	7.492	0.129	0.200	0.003

, Table 17, Table 18 and Table 19. All these models have a good performance with an RMS < 8%. Some of these models do not include ECB data which indicates that the models including ECB asset data can also perform well.

3.4. Performance Tests

Time series used in this study are very smooth. That is, a set of data for a given time is not very different from the previous and the next one. As a consequence, the train/test approach used to estimate the performance of ML models is successful most of the time. A very quick test with the introduction of disturbed data (one record multiplied by five) leads to the conclusion that this approach is not strong enough to determine whether the model is reliable. In the same spirit of what has been done by Dimopoulos and Bakas (2019), I have designed more stringent tests and procedures to evaluate the models. In order to get a deeper insight, I have completed other tests, such as permutation, deletion of data and use of unrelated parameters. First, I completed a permutation exercise which shows that parameters (TR, CPI and GDP) do not have the same influence even when collinearity is not removed. The second test shows that the sole use of local data cannot explain the price increase. At last, I tested the models by quantifying their predictive capabilities and not only from the quality of the data fit.

3.4.1. Stability of the Residual Time Series

In order to confirm the possible time-dependence of the price time series, I have computed for all models the stability of the residual time series computed for the tree-bagging model (which I consider as the less stable ones). It is readily visible that the models for which HPI has not been differentiated and those not using the ECB are the most variable (models in plain font, p > 0.10, Table 17).

Table 17. Results of the Dickey–Fuller test on the residual series for various models for Switzerland, France, United Kingdom and USA. Models indicated with bold fonts are judged as performant. I indicate the performing models using bold font.

n	Model of Residuals Series	Statistics	p Value
1	Switzerland	−0.70	0.97
2	Switzerland 1 yr	−3.65	0.03
3	Switzerland IR	−2.13	0.52
4	Switzerland LIR	−1.45	0.81
5	Switzerland ECB	−3.64	0.03
6	Switzerland ECB_1y	−4.25	0.01
7	France res	−2.26	0.47
8	France 1 yr	−3.58	0.04
9	France res_IR	−2.13	0.52
10	France res_LIR	−1.30	0.86
11	France ECB	−5.73	0.01
12	France ECB_1y	−3.53	0.04
13	UK	−3.62	0.03
14	UK 1 yr	−3.45	0.05
15	UK IR	−3.31	0.07
16	UK LIR	−1.94	0.60
17	UK ECB	−5.04	0.01
18	UK ECB_1y	−4.02	0.01
19	USA	−2.20	0.49
20	USA 1 yr	−3.94	0.01
21	USA IR	−3.19	0.09
22	USA LIR	−1.59	0.75
23	USA ECB	−4.18	0.01
24	USA ECB_1y	−4.41	0.01

Table 17. Results of the Dickey–Fuller test on the residual series for various models for Switzerland, France, United Kingdom and USA. Models indicated with bold fonts are judged as performant. I indicate the performing models using bold font.

n	Model of Residuals Series	Statistics	p Value
1	Switzerland	−0.70	0.97
2	Switzerland 1 yr	−3.65	0.03
3	Switzerland IR	−2.13	0.52
4	Switzerland LIR	−1.45	0.81
5	Switzerland ECB	−3.64	0.03
6	Switzerland ECB_1y	−4.25	0.01
7	France res	−2.26	0.47
8	France 1 yr	−3.58	0.04
9	France res_IR	−2.13	0.52
10	France res_LIR	−1.30	0.86
11	France ECB	−5.73	0.01
12	France ECB_1y	−3.53	0.04
13	UK	−3.62	0.03
14	UK 1 yr	−3.45	0.05
15	UK IR	−3.31	0.07
16	UK LIR	−1.94	0.60
17	UK ECB	−5.04	0.01
18	UK ECB_1y	−4.02	0.01
19	USA	−2.20	0.49
20	USA 1 yr	−3.94	0.01
21	USA IR	−3.19	0.09
22	USA LIR	−1.59	0.75
23	USA ECB	−4.18	0.01
24	USA ECB_1y	−4.41	0.01

3.4.2. Permutations Test (Model with 3-Param)

As the machine learning algorithms are always assumed to be reliable to model any data using any input parameter, I tried to model the prices using another set of values. I chose to permute original data before modeling prices. In order to make sure the permutation of a single parameter is well quantified, I have made three models for which input parameters are permuted independently (Figure 13). As expected, those results shows that the input parameter responsible for financing the transaction (US TR10 yr) is one of the most important parameter for Switzerland. Financing rates impact the pool of people who can afford the transaction. As GDP may be seen as not impacting on the population buying houses, CPI also plays a role, especially to explain short-term variations and amplitude changes. As I consider the data at the national level, I am not able to detect local effects of GDP like (Su et al. 2018) did, nor if the number of performing loans are more numerous when economic crisis hits (Tham et al. 2022). For the latter effect, it is not clear whether the number of NPLs actually has an impact on prices. The conclusion of this test is that the algorithm is not able to model anything using any data and therefore the parameters included in the first models contained relevant information to price evolution.

3.4.3. Sensitivity Test: Testing the Prices Using Local-Only Parameters

Here, I try to model the prices with nationally focused parameters. I only show in this paper the results for Switzerland and France9, as the data for other countries were more difficult to gather. I use only three parameters here: GDP, inflation and unemployment rate.

Results of this approach are displayed for two countries (France and Switzerland) in Figure 14 and Figure 15. Those two countries were selected as (1) most of the loans are not amortized in Switzerland and (2) the criteria of loan allocation are more demanding in Switzerland than in France (e.g., affordability parameter of 5%). I consider those models not successful in modeling the prices, as neither the long-term trends nor the short-term variations are recovered (Figure 14 and Figure 15).

3.4.4. Predicting Price Using Models Trained on Amended Datasets

For this test, I have removed the last four data points from each dataset. The models are trained on this amended dataset, and prices are then predicted using the complete datasets (Figure 16). In the current situation, the last quarter’s data were strongly influenced by the action of the central banks to reduce inflation and tame the increase in wages. We nowadays see a combination of inflation and rates which was so far thought of as impossible (e.g., low rates/high inflation). Now that inflation is decreasing again we are able to sample data combination which were previously seen (e.g., in 2003). For this reason, some associations with the parameters used were unprecedented.

As each model is set by a random seed, I have performed 600 runs in order to build the statistics shown in Table 18 and Table 19. For four countries, the direction of HPI changes (negative and positive) was predicted correctly. This observation is supported by the positive values of r (see mean and standard deviation). For the US, the observed price decrease was much larger than the predicted ones (approx. 8%) for both kNN and tree-bag strategies. For both methods, the RMS are found to be lower than price dispersion between estimates and asset types. For the three other countries, the performance is better. Those results maybe contradict findings which suggested that economic fundamentals cannot fully explain the house price evolution (Quigley 1999; Farlow 2004, 2013; De Bondt 1995), or it simply suggests that the economic data available at the time (before the most recent crises happened) were not sufficiently diverse to be tested against house price indices.

Table 18. Statistics of the data fit performance of the ECB_1y_tbag models (tree-bag strategy). Here we train the model after excluding the last four data points. The predictive values are compared to the original datapoint, and statistics are computed by comparing together the eight points. Tree-bag strategy is used.

Country	M_COR	SD_COR	M_RMS	SD_RMS	SD_RES	M_MAE	SD_MAPE	M_CHIp	M_CHIs
USA	−0.93	0.34	10.24	0.66	4.90	9.32	0.49	1.00	0.03
France	0.84	0.57	3.28	0.47	1.42	3.05	0.25	0.85	0.15
Germany	0.72	0.46	13.17	0.38	4.99	12.44	0.65	1.00	0.03
Spain	0.27	0.57	4.43	0.08	1.29	4.28	0.74	1.00	0.03
Italy	−0.62	0.43	2.90	0.47	1.76	2.46	0.61	0.99	0.04
Finland	0.94	0.28	6.59	0.16	2.62	6.19	1.54	1.00	0.03
Sweden	0.83	0.44	9.36	0.58	3.41	8.88	0.49	1.00	0.03
Australia	0.09	0.19	6.51	1.13	1.66	6.35	0.38	1.00	0.03
UK	0.57	0.26	5.98	0.45	1.10	5.90	0.51	1.00	0.03
Switzerland	0.69	0.15	3.08	0.32	1.40	2.83	0.36	0.83	0.16
Netherland	0.99	0.39	2.79	0.55	1.39	2.51	0.11	0.93	0.09
Norway	0.98	0.33	2.62	0.23	3.01	2.38	0.18	0.50	0.77

Table 18. Statistics of the data fit performance of the ECB_1y_tbag models (tree-bag strategy). Here we train the model after excluding the last four data points. The predictive values are compared to the original datapoint, and statistics are computed by comparing together the eight points. Tree-bag strategy is used.

Country	M_COR	SD_COR	M_RMS	SD_RMS	SD_RES	M_MAE	SD_MAPE	M_CHIp	M_CHIs
USA	−0.93	0.34	10.24	0.66	4.90	9.32	0.49	1.00	0.03
France	0.84	0.57	3.28	0.47	1.42	3.05	0.25	0.85	0.15
Germany	0.72	0.46	13.17	0.38	4.99	12.44	0.65	1.00	0.03
Spain	0.27	0.57	4.43	0.08	1.29	4.28	0.74	1.00	0.03
Italy	−0.62	0.43	2.90	0.47	1.76	2.46	0.61	0.99	0.04
Finland	0.94	0.28	6.59	0.16	2.62	6.19	1.54	1.00	0.03
Sweden	0.83	0.44	9.36	0.58	3.41	8.88	0.49	1.00	0.03
Australia	0.09	0.19	6.51	1.13	1.66	6.35	0.38	1.00	0.03
UK	0.57	0.26	5.98	0.45	1.10	5.90	0.51	1.00	0.03
Switzerland	0.69	0.15	3.08	0.32	1.40	2.83	0.36	0.83	0.16
Netherland	0.99	0.39	2.79	0.55	1.39	2.51	0.11	0.93	0.09
Norway	0.98	0.33	2.62	0.23	3.01	2.38	0.18	0.50	0.77

Table 19. Statistics of the data fit performance of the ECB_1y_kNN models (kNN strategy). Here we train the model after excluding the last four data points. The predictive values are compared to the original datapoint, and statistics are computed by comparing the eight points together. kNN strategy is used.

Country	M_COR	SD_COR	M_RMS	SD_RMS	SD_RES	M_MAE	SD_MAPE	M_CHIp	M_CHIs
USA	0.747	0.000	10.064	0.203	4.901	9.516	0.500	1.000	0.029
France	−0.829	0.581	1.873	0.422	1.416	1.601	0.112	0.500	0.771
Germany	−0.723	0.641	14.887	0.128	4.986	14.190	0.681	1.000	0.029
Spain	NA	0.001	5.312	0.062	1.285	5.131	1.059	1.000	0.029
Italy	NA	0.001	3.539	0.001	1.764	3.069	0.902	0.750	0.229
Finland	−0.744	0.279	7.440	0.069	2.622	7.017	1.425	1.000	0.029
Sweden	0.748	0.001	8.221	0.429	3.415	7.889	0.466	1.000	0.029
Australia	NA	0.010	4.160	1.373	1.657	3.868	0.191	0.500	0.771
UK	−0.178	0.577	4.725	0.070	1.101	4.581	0.353	1.000	0.029
Switzerland	0.690	0.001	2.464	0.386	1.401	2.244	0.278	0.656	0.433
Netherland	0.978	0.001	4.340	0.830	1.388	4.296	0.171	0.852	0.147
Norway	0.557	0.000	3.339	0.078	3.011	2.536	0.171	0.500	0.771

Table 19. Statistics of the data fit performance of the ECB_1y_kNN models (kNN strategy). Here we train the model after excluding the last four data points. The predictive values are compared to the original datapoint, and statistics are computed by comparing the eight points together. kNN strategy is used.

Country	M_COR	SD_COR	M_RMS	SD_RMS	SD_RES	M_MAE	SD_MAPE	M_CHIp	M_CHIs
USA	0.747	0.000	10.064	0.203	4.901	9.516	0.500	1.000	0.029
France	−0.829	0.581	1.873	0.422	1.416	1.601	0.112	0.500	0.771
Germany	−0.723	0.641	14.887	0.128	4.986	14.190	0.681	1.000	0.029
Spain	NA	0.001	5.312	0.062	1.285	5.131	1.059	1.000	0.029
Italy	NA	0.001	3.539	0.001	1.764	3.069	0.902	0.750	0.229
Finland	−0.744	0.279	7.440	0.069	2.622	7.017	1.425	1.000	0.029
Sweden	0.748	0.001	8.221	0.429	3.415	7.889	0.466	1.000	0.029
Australia	NA	0.010	4.160	1.373	1.657	3.868	0.191	0.500	0.771
UK	−0.178	0.577	4.725	0.070	1.101	4.581	0.353	1.000	0.029
Switzerland	0.690	0.001	2.464	0.386	1.401	2.244	0.278	0.656	0.433
Netherland	0.978	0.001	4.340	0.830	1.388	4.296	0.171	0.852	0.147
Norway	0.557	0.000	3.339	0.078	3.011	2.536	0.171	0.500	0.771

3.5. Predicted Prices

Now that confidence has been gained in predicting prices over the four last quarters, I produced four sets of MEF values to predict prices using models trained on complete datasets (Figure 17 and Figure 18). The ranges of values tested represent values which could be relevant for pricing mortgages in a mid-term horizon of four quarters to predict values of HPIs. I have used the model 6 (ECB 1 yr, see Figure 11) with the tree-bag approach. Those predictions could be compared to Pillar 2 scenarios simulation as required by regulators. Unlike for Pillar 2 scenarios, all possible combinations of input parameters over specified ranges (

Table 20. Range of MEF values used to compute the predicted values (160,000 models). For each parameter, 20 values were tested over each range.

N	GDP (%)	Inflation (CPI)	ECB Asset Size (M EUR)	Treasury Rates (%)
160,000	−2 to 2%	−2 to 2%	6.5 × 106 ± 1 × 106	0 to 4%

) have been considered, leading to a total number of 160,000 synthetic prices. From this input dataset, I have computed prices predicted for the four countries (France, UK, USA and Switzerland). With the exception of the USA, none of the models predict a decrease in price over the next 4Q (12Q metrics). This means that, in those four countries, systemic margin calls are not expected to be seen in a mid-term horizon for interest-only mortgages (like those available in Switzerland and USA). It should be kept in mind here that the prices predicted are bound by a maximum diminution of ECB assets of 1600 mEUR (from a start value of 6500 mEUR). Finally, I remind you that those prices are averaged over all kinds of assets, and mean prices may be overly optimistic for some areas and some asset types depending on the local economic conditions.

Those predictive prices do not take into account temporal development or MEF changes. Those estimates are end-of-scenario prices. The evolution of prices over n-quarters can be reconstructed using n prices. As the number of solutions computed is large (160,000), the risk that the future is not included in one of the scenarios is low. Such prediction allows for the computation of many scenarios, some of which may be outside economic standards models, allowing the price estimation resulting from non-conventional economic policies.

4. Conclusions

A recurrent criticism of machine learning approaches is that the results cannot be explained. I show here, by completing a step-by-step modeling approach, which parameters can improve the data fit. The chosen parameters can represent all parties, enabling the transaction to be completed under certain conditions (available buyer, available seller, financing, economic support), and ultimately help understand the price structure. Even if the data quality is imperfect (e.g., GDP may have been revised many times after publication), the data quality may be improved in the future thanks to the work of independent rating agencies and the information-sharing of countries with investors. Ultimately, the models are limited by the sampling rate of the dataset and the length of the period observed. Even in such a context, some models are performing well enough to predict prices over the next four quarters.

In summary, and as expected (Weis et al. 2021), the most critical parameter is the level of interest rates at 10 years, but this study allows us to refine our understanding of the mechanisms in place.

• Inflation plays a role, although its contributions vary from one country to another. As real estate investment is often considered as a hedge to inflation, the interaction between the two variables cannot be ignored (Mahlstedt and Zagst 2016; Amenc et al. 2009; Froot 1995).
• The impact of level of employment on prices is likely stronger when the real estate market is open to many and loans are amortized (otherwise, either buyers would not have access to the loan or we would observe a credit event similar to the 2008 subprime crisis).
• When data from the ECB/FED asset purchase program are introduced, the fit with the data improves from 2015 onwards. The correlation coefficient between prices in Switzerland and ECB portfolio size is higher than 92% (computed over the last 24 quarters using the ECB nominal model; to be compared to r = 46% for inflation and r = 0.58 for rates; see also Table 8 for relative importance).
• The differentiation of the HPI allows for predicting prices for the next four quarters for at least four countries.

If I consider the models collectively, for all countries, it is possible to model prices without including the ECB portfolio size up to 2015. From 2015 onwards, price increases and price decreases tend to correlate with the size of the ECB asset book. For some models, the importance of ECB portfolio size is equally important to the level of treasury rates. As countries outside the European community are impacted by this effect, I suggest that some large-volume investors are involved in arbitrages made between real estate, sovereign, and corporate debts investments (Berg et al. 2023; Charles 2016; Rosenberg 2019; Ryczkowski 2019; Storrie 2015, 2019; Gabriel and Lutz 2014). The reversal of ECB policies designed to support the economy since 2015 may change the risk balance for some institutional investors and may lead to unpredictable consequences, such as a risk of contagion from real estate to other financial sectors. For Switzerland, regarding the 1990 real estate bubble, one can infer that the levels of financing and the inflation rate increase triggered the crisis. In particular, the start of the crisis coincides with the time when the ratio of CPI/TR passed 5% (downwards). The accuracy of the models presented in this study may encourage modelers involved in capital management based on economic scenarios such as ICAAP and Basel “Pillar 2 models” to use similar models to further explore the landscape of possible scenarios. In the future, and to ensure all aspects of the market are well included, I would recommend building on those models more closely investigating the influence of rent level (as the linkage between the buy-to-let, build-to-let, and build-to-sell markets is so far not studied enough) and by adding variables such as building rates, vacancy rates, demographic data and the age distribution of migrants. Finally, ML real estate models would benefit from comparison with models based on the capitalized income value method, where cash flows play a greater role in the pricing.