Key determinants of sales price in the residential developments in Prague Open Access

During the observation period from 2005 to 2021, the Prague residential real estate market experienced significant price growth in new residential development, except for a downturn that followed the global financial crisis (GFC) of 2008. In the 1990s and the first decade of the new millennium, real estate prices rapidly caught up to the levels seen in Western Europe, although they still remained below the average (e.g. Mihaljek, 2007).

This paper aims to explore the dependencies between the prices of new apartments in Prague’s market and fundamental variables during a sample period from 2005 to 2021. Specifically, it provides insights into the relationship between the price per square meter of newly developed residential units in the Prague market and the set of macroeconomic and real estate-specific explanatory variables. By utilizing a combination of advanced statistical learning (SL) techniques, our goal is to extract the most significant drivers from the rich dataset, focusing on identifying a parsimonious set of key determinants.

Although the model’s structure could be adapted for future forecasting purposes, the central objective of this study is to explain the underlying mechanism driving price changes, prioritizing the interpretation of relationships between variables rather than direct prediction.

It is worth noting that the observation period at least partially covers the turbulent markets period during the COVID-19 pandemic, during which the price levels of new apartments in Prague remained remarkably resilient and even displayed a steady increase despite severely adverse economic developments. To capture these effects, we introduced a dedicated COVID-19 dummy variable into the model.

The first part of the paper reviews the academic literature on the links between residential real estate prices and selected fundamental variables, while the second part describes the logic and structure of the variables used in the model, followed by a proposed modeling methodology based on lasso regularization combined with Bayesian model averaging (BMA).

Several research papers, such as Kwakye and Chan (2020), Algieri (2013) and Belke and Keil (2018), present clear evidence of a relationship between real estate prices and macroeconomic variables, along with consumer expectations. While supply in the residential real estate market reacts to economic changes with a delay due to long development processes and sticky prices, demand tends to react much more quickly. In case of a market downturn, development costs incurred will often lead to a slowdown in sales with relatively stable prices for an extended time period rather than a pronounced price drop (e.g. Head et al., 2012 or Coskun et al., 2018).

Housing can be viewed from two perspectives (Denton et al., 1993): as an investment asset or as a consumption asset. For most buyers, a home purchase represents a combination of both perspectives, often leaning more toward the investment asset view. Consequently, demand and prices in the housing market are also influenced by conditions in other investment asset markets.

Housing is a significant component of living standards, and expenditures on housing typically increase more than proportionally to household income growth, as shown in Figure 1 across European countries. The higher a household’s disposable income, the greater the percentage of income allocated to housing.

The residential real estate market and housing culture are shaped by local attributes, distinct legal and tax frameworks and market-specific cultures with a minor influence from international demand, apart from occasional short-term shocks. A study by Englund and Ioannides (1997), examining 15 selected OECD countries, found no evidence of an international real estate cycle. However, they did not reject the hypothesis of the existence of real estate cycles within individual nations. Similarly, Tsatsaronis and Zhu (2004) identified long cycles in real estate prices, observing price developments in 17 countries between 1970 and 2003. Long cycles are characteristic of real estate due to extended construction periods, lengthy project timelines and the limited flexibility of the supply side to quickly respond to shifting market trends. This pattern is also evident in the Prague market. Following a significant expansion starting in 2000, the market entered a recession in 2008, but since 2010, residential prices for new developments have been continuously rising.

Common determinants influencing both the supply and demand sides of real estate price developments include GDP and interest rates, as evidenced by several studies (e.g. Cohen and Karpavičiūtė, 2017). Belke and Keil (2018) highlight a division between supply-side and demand-side factors, such as rent, age structure or market size. Cohen and Karpavičiūtė (2017) specifically explore the interactions between house prices and explanatory variables such as GDP, unemployment or inflation in the Lithuanian real estate market. El-Montasser et al. (2016) examined the relationship between policy certainty on economic fundaments and house prices, arriving at the conclusion that the more unstable the policies are the more volatile the housing prices are.

A more contemporary approach involves utilizing housing user costs rather than focusing on housing prices, as seen in studies by Díaz and Luengo-Prado (2008) or Gillingham (1980). Nevertheless, many recent studies continue to focus exclusively on real estate prices. Housing user costs encompass both monetary outlays – such as mortgage instalments or rent, maintenance and repairs expenses, insurance costs or taxes – and non-monetary costs. The non-monetary costs include the opportunity cost of the equity used to purchase the property, depreciation of the property and potential appreciation in its value over time. The challenge with this approach lies in its heavy reliance on detailed data and the complexity of calculations. Additionally, the concept of housing user costs includes the non-monetary component that assumes residents consider their housing purely from a rational perspective. However, this assumption does not always hold, as the majority of apartments in Prague are purchased for personal or family use rather than purely for investment purposes.

One might argue that households owning an apartment could save less, as property ownership may be perceived as a form of savings. However, Calomiris et al. (2009) challenge this hypothesis, finding the changes in consumption behavior are insignificant. Ownership provides households with the security of a permanent residence, often causing them to overlook the opportunity costs of investing their equity in real estate. For many, acquiring a home remains a paramount life goal. Furthermore, homeowners typically focus on direct out-of-pocket expenses, disregarding potential depreciation or appreciation of their property.

Appreciation can indeed influence the decision-making process for first-time homebuyers, who may anticipate that gains from the future sale of their property could help to fund the equity needed to purchase a larger home as their family grows. Some researchers, such as Abelson (2009), attempt to incorporate the wealth effect arising from ownership alongside the out-of-pocket costs into a single equation. For most buyers, the ability to afford the nominal payments for housing from their income is the crucial factor, with the wealth effect playing a relatively secondary role. The wealth effect plays a more prominent role in the context of investment properties, where the key consideration is the comparison between mortgage instalments and the potential rental income.

In our paper, we integrate insights from both bodies of literature, incorporating both macroeconomic and fundamental real estate variables to explain the developments in residential real estate prices in Prague.

Supply-side determinants are driven by the profitability of real estate development and its resources, with the availability and price of land being key factors (Saiz, 2010). The long housing development process makes it challenging to respond promptly to fluctuations in demand. Increased demand is often reflected in new developments only after a significant delay, causing prices to rise as demand increases. Conversely, developers cannot easily reduce prices when demand cools due to commitments to construction costs and budgets, which may be further affected by macroeconomic factors such as interest rates or labor costs.

The real estate market is a typical example of price stickiness, as observed by, e.g. Poterba (1984) or Komarek and Hlavacek (2011). Construction costs, land costs and development financing are also influenced by underlying interest rates and changes in political and legislative environments. The average price per square meter of new apartments is further impacted by the average size of the units, with empirical studies showing that smaller apartments tend to have higher prices per square meter.

Specific supply-side variables for the Czech market during the observed period include legislative changes, such as adjustments to VAT on newly built apartments, and changes in mortgage regulation, especially the maximum loan-to-value (LTV) requirements.

From one perspective, an individual buyer’s decision to buy or rent is influenced by two primary factors: the desire for homeownership (housing as a consumption asset) and the view of housing as an investment. This decision is also shaped by the development of the financial market (e.g. Komarek and Hlavacek, 2011) and the performance of other investment classes beyond real estate (e.g. Coskun et al., 2018; Chiang et al., 2020 or Gokmenoglu and Hesami, 2019). Additionally, rental price trends affect the homeownership market since buyers weigh the option of renting versus buying. Positive rental returns can attract new investors to the market (e.g. Ronald, 2008; Doling and Ronald, 2010 or Lux, 2020).

Another perspective considers the mobility of the housing occupants – ownership tends to limit flexibility when it comes to moving for work to another city or country while renting offers greater freedom. Mazacek (2023) highlights a positive correlation between the number of major business cities in a country and the percentage of the population living in rental housing, as it provides more flexibility for relocation. In the Czech Republic, there are only two major business cities, Prague and Brno, with Prague being dominant. Consequently, people working in Prague are less likely to move out of the city for job-related reasons.

The demand-side fundaments can be divided into two groups. The first group includes variables that affect wealth and housing affordability, such as mortgage affordability and its regulation, household disposable income, rental levels and returns on apartment investments (Komarek and Hlavacek, 2011; Doling and Ronald, 2010). It also includes demand-driving factors that are partly independent of economic conditions, such as natural population growth across age groups, cultural shifts in living arrangements that influence household numbers or migration trends.

The second group consists of softer variables that indirectly influence the demand for homeownership. These include unemployment expectations (Dogan and Topuz, 2020), mortgage riskiness, expectations regarding future mortgage interest rates (Komarek and Hlavacek, 2011), returns on other investment classes, expectations for future residential price developments and the need for flexibility in housing and mobility options.

The demand for new apartments in Prague remained strong in the real estate market from 2005 to 2021, despite the long-term rise in prices. The significance of local price drivers differs between mature developed markets and emerging markets. According to Ciarlone (2015), the relationship between real estate prices and fundamentals in emerging markets is characterized by slow adjustments to economic shocks. During the observation period, the Czech Republic transitioned from an emerging market to a more established and developed market.

Vizek (2010) describes how changes in interest rates and construction activity explain long-term price developments in both developed and emerging markets. Their empirical study shows that income changes were relevant only for Western European countries, while housing loan changes were more significant for Eastern European countries, which were considered emerging markets. This distinction is tied to the stability of housing lending and political stability.

The demand for apartments in Prague is driven not only by the natural desire to own a home but also by the increasing number of people relocating to the city for work. Unlike workers in other large cities in the European Union or the USA, Czech workers are traditionally less willing to travel long distances to work. Commuting for one or more hours each day is uncommon, due to the country’s small size and the fact that it has essentially just two major business centers, Prague and Brno. In 2020, less than 6% of Prague’s population commuted to work outside the city, while more than 16% of the population in Central Bohemian Region commuted to Prague (Eurostat – Commuting between regions 2021). In addition, 81% of Prague’s population believes it is easy to find a job in the city, the highest percentage in the EU (Bolsi et al., 2020). About 20% of Prague’s working population resides in the same city district in which they work.

As per the EU publication People in the EU – statistics on Geographic Mobility (2015), the Czech Republic also has one of the lowest percentages of inhabitants moving for work in Europe, with Prague’s average commuting time of 48 min (Leoncikas and Rende, 2020). While Prague has one of the best inner-city public transportation systems in Europe, the mass transit infrastructure connecting nearby cities is less developed. According to the Population Census 2011 by the Czech Statistical Office, less than 3% of Prague’s population moved out of the city in 2021. Given these factors, mobility for work is not a significant consideration when deciding between renting or buying housing units in Prague. Furthermore, the Czech population has a strong historical preference for homeownership, with almost 80% of the population living in their own residences ^[2].

Another key factor is the rising living standards (purchase price parity increased significantly since 2012 in the Czech Republic according to Eurostat ^[3]), which motivate people to move away from the outdated developments of the 1960s and 1970s. The significance of demographic changes is presented in Figure 2, showing that the number of people aged 30–50 years has increased by almost 30% over the past 17 years. This age group represents the majority of mortgage borrowers and apartment buyers (Mazacek, 2018). Additionally, migration to Prague for work continues to fuel the demand for rental apartments, driving the rents higher, which increases the attractiveness of purchasing apartments as investment properties.

According to a study by the Institute of Strategic Investments, Faculty of Finance and Accounting, University of Economics in Prague (2020), the primary reason for renting an apartment instead of owning one is insufficient funds or low creditworthiness, which prevents individuals from obtaining a mortgage.

Figure 3 illustrates the average monthly mortgage installment as a percentage of household net disposable income in Prague. Although the prices of new residential developments have risen, wage inflation and decreasing mortgage interest rates have mitigated the impact in certain periods. As a result, the percentage of disposable income spent on mortgage installments decreased between 2008 and 2014, however, this trend reversed after 2014.

According to Linhart et al. (2021) Prague’s residential market has been suffering from a long-term low supply of new apartments, with market saturation (number of apartments per 1,000 inhabitants) being one of the lowest in Europe. Figure 4 shows the number of newly finished apartments per 1,000 inhabitants in Prague, while Figure 5 presents the number of apartments sold in the Prague market over the past five years. It is evident that almost all available yearly stock placed on the market was sold within the same year.

According to a World Bank study, the Czech Republic ranked 127th of 189 countries in terms of the length of the permitting process ^[7]. Furthermore, the supply of available plots for development is very limited, similar to the situation in many other European cities. The lack of well-developed traffic and public transport infrastructure for expansion into other cities in the Central Bohemian Region further hampers growth, with necessary improvements likely to take decades. Another topic that resonates with Prague’s population is the issue of foreigners purchasing apartments in the city. While there is international demand in the housing market, it generally results in shorter-term shocks rather than long-term trends.

Short-term increases in demand typically arise from macroeconomic fundaments such as favorable exchange rate movements, comparatively lower local market prices with limited barriers to entry or capital relocation to another jurisdiction. However, these factors rarely have a lasting impact. For example, a mid-term demand shock from Eastern European buyers occurred between 2000 and 2008, but since then, the volume of foreign buyers in Prague has significantly decreased.

Although no official data are available on the number of apartments sold to foreign investors in Prague, the institutional market for rental residences in the city is predominantly driven by local investors and funds, largely due to exchange rate and interest rate risks. Therefore, most foreign buyers in Prague are individual purchasers. According to real estate developers, approximately 20–30% of apartments sold in recent years have been purchased by non-Czech buyers, with Slovakians accounting for about half of this foreign demand. Most of these foreign buyers are either living or working in the Czech Republic.

The dependent variable in our model is defined as the transaction price per square meter of the net saleable area of new apartments in Prague, including VAT, on a quarterly basis from 2005 to 2021. This inclusion of VAT is necessary, because the typical individual buyer is not a VAT payer, meaning the final price for them includes VAT. However, a comprehensive dataset of average residential prices per square meter in Prague is not available for a sufficiently long period. The most relevant source of data on new dwelling prices is the Deloitte Price Map ^[11], which uses data from the cadastral office as the primary input for actual prices, as well as developers’ price lists for current apartment offers.

The Deloitte price map is available from 2016 onwards and is issued every two months. To produce quarterly data, we averaged the two-month figures, assigning a weight of two to the whole two-month period and a weight of one to the remaining month. To address longer time series, we constructed data of new apartment prices per square meter starting in 2004 by combining data from the Deloitte Price Map, the Czech Statistical Office and a private database provided by Professor Dolansky ^[12] of Czech Technical University. The private database was utilized for the early years of the time series, prior to when the Czech Statistical Office began publishing the relevant price data.

The Deloitte Price Map provides data at the district level in Prague, while the dataset from Professor Dolansky is more granular, as it is based on Prague’s cadastral areas. To resolve this, we calculated a weighted average for each district, based on the proportion of each cadastral area within the individual districts ^[13], ^[14].

To calculate an overall average price for Prague, we used a weighted average of prices for the individual districts. The weights were designed from the total number of apartments under construction and the number of apartments finished in each Prague’s district over the past twelve months. These data were sourced from the Czech Statistical Office. This methodology summarizes our approach to constructing the desired time series for our dependent variable.

We cross-checked and verified the average prices of new apartments in Prague by referring to research outputs and reports from real estate development companies Trigema, Central Group and Skanska ^[15], which jointly issued a comprehensive report on the residential market.

As described above, we chose to use the actual transaction prices of new apartments as the dependent variable because new apartments represent a more homogenous product, particularly in a rapidly developing and changing economy. In contrast, older apartments vary significantly in terms of overall quality, equipment, furnishing and general wear and tear. The long-term prices of older apartments are also influenced by the overall quality of the apartment stock, interacting with new apartment prices that represent an alternative option to older apartments.

Figure 6 illustrates the price developments in the individual districts of Prague, with the Prague average represented by the black line in both graphs.

Table 1 presents descriptive statistics on the price per square meter for new developments in Prague’s districts. The variation coefficient in Table 2 reveals that the price differences between the districts have been relatively decreasing over time.

The compound annual growth rate (CAGR) for individual Prague districts over the period between 2004 and 2021 ranges from 5% to 8% p.a. The lowest average growth is observed in Prague 1. Although the prices appear to rise rapidly, the average growth rate is not dramatically higher than that of other investment assets because of steady price developments between 2008 and 2015, during which the changes in prices of new apartments were marginal.

The correlation coefficient, calculated between the price developments of individual districts and the Prague’s average, shows a high correlation across districts – no Prague district is losing popularity, and no district is significantly outperforming the others. The lowest correlation with the Prague’s average is observed in Prague 1, where the prices have always been high due to its central location and the supply of apartments is limited by heritage protection.

Based on the literature review, we have identified several categories of explanatory variables that could potentially impact residential real estate prices, either positively or negatively:

1. Macroeconomic variables – influence both supply and demand side

GDP of the Czech Republic, Prague GDP, Czech National Bank Interest rate, Inflation and Disposable income of Prague households.

1. Demand-side fundamental variables

Population, Salary, Rents, Interest rates on Mortgages, Average area of a new apartment, VAT, Ratio between average rent and average monthly mortgage installment, Ratio between average monthly mortgage installment and average gross salary, Rental return on apartment price, Maximum LTV Requirement, FX-rate, ECB-CNB interest differential, Unemployment in Prague and Unemployment in the Czech Republic.

1. Supply-side fundamental variables

Construction price index, Number of apartments with construction started and Number of apartments finished.

1. Dummy variables

Covid-19 dummy.

While the impact and relationship for many of these variables, especially those in the macroeconomic variables category, are clear and well-documented in the academic literature discussed earlier in this paper, we have provided details and reasoning for the inclusion of a selected set of key variables in  Annex A. Additionally, Table 3 presents the descriptive analysis for the full set of explanatory variables used in our analysis.

Many previous studies on real estate markets employ econometric methods such as vector autoregressive (VAR) models, vector error correction models (VECM) or autoregressive distributed lag (ADL) models (Kwakye and Chan, 2020). However, these approaches often struggle with high-dimensional data and model uncertainty, particularly when considering datasets with numerous potential explanatory variables, including both levels and growth rates, which can lead to a high number of potential predictors in the model. To address this complexity, we adopt a modern, computation-driven approach that leverages lasso regularization for efficient variable selection and BMA to manage model uncertainty. This ensures a sparse, robust model that selects only the most relevant predictors while maintaining accuracy and interpretability. This helps to ensure the model remains efficient and avoids overfitting, allowing us to focus on the variables with the strongest impact on residential real estate prices in Prague.

The core of our analysis relies on the ADL model (Pesaran and Shin, 2013), a special case of a linear regression model that incorporates both multiple explanatory variables and their lagged values, as well as the lagged values of the dependent variable. This structure of the ADL model is described in  Annex B.

The ADL model presents a robust framework for analyzing the dynamic relationships between variables over time. It incorporates lagged values of both the dependent and explanatory variables which allows it to capture the temporal dynamics and potential lag effects typical for real estate markets. For instance, changes in policy rates or demographic shifts might not immediately impact real estate prices but could have significant delayed effects. The ADL framework accommodates this reality, enabling a deeper understanding of how current and past values of various determinants influence real estate prices over time.

One of its pivotal strengths also lies in its flexibility to handle a mix of variables that are stationary (integrated of order zero, $I(0)$⁠) and those that are non-stationary (integrated of order one, $I(1)$⁠), without the need to transform them into stationary variables. This feature is especially valuable in the context of real estate market analysis, where the variables being studied often exhibit different levels of stationarity. By accommodating both types of variables, the ADL model allows for a more accurate representation of the underlying dynamics and relationships between variables without losing critical information through differencing or detrending ^[16].

While the ADL model offers many advantages, it is not without limitations. One key concern is the potential for misspecification. Incorporating $I(1)$ variables directly into the ADL model inherently assume that the variables are cointegrated, i.e. that a long-term equilibrium relationship exists between the dependent variable and its predictors and that this relationship can be modeled without inducing spurious regression issues. If the variables are not cointegrated, the model could falsely detect significant relationships where none actually exist.

Additionally, the ADL model’s flexibility to include a broad range of lagged dependent and independent variables can increase the risk of overfitting, especially if too many lags or variables are included. This can also lead to an ill-conditioned regression problem due to the near-singularity of the regression matrix. Thus, careful model selection techniques are crucial in ensuring that the model is well-specified and avoids overfitting.

To mitigate the risk of misspecified regression, we first remove $I(1)$ variables that fail the Engle-Granger cointegration test. This ensures that only variables with valid long-term relationships remain in the model. To further address the potential for ill-conditioned regression and to reduce the risk of overfitting, we apply a SL approach known as lasso (least absolute shrinkage and selection operator) (Tibshirani, 1996). Lasso helps the model focus on the most relevant determinants by shrinking the coefficients of less important variables toward zero, effectively “letting the data speak” by selecting the most appropriate predictors from the broad set of potential explanatory variables. Additionally, we support this method with BMA (Hoeting et al., 1999), another SL technique that explicitly accounts for model uncertainty when selecting the optimal value for the lasso tuning parameter λ, improving the robustness of the model.

The estimated model results are captured in the form of long-run multipliers (LRMs). LRMs represent a cumulative impact of a change in an independent variable on the dependent variable over time. LRMs thus provide a comprehensive measure of the equilibrium relationship between the variables included in the final model, offering insights into how long-term shifts in underlying factors influence the dependent variable.

To evaluate the model’s properties, we employ bootstrapping, a widely recognized nonparametric method for assessing the statistical properties of complex estimators. Bootstrapping allows to generate reliable confidence intervals and comprehensively evaluates the results.

This proposed multi-step approach ensures that our model remains well-specified and avoids the pitfalls of overfitting while accurately identifying the key determinants of Prague’s real estate prices. Further details regarding the lasso regularization, Bayesian model averaging and LRMs are provided in  Annex B.

Overall, the proposed estimation procedure to identify the key determinants of the new apartment prices per square meter in Prague can be summarized in the following steps:

1. Construction of predictor matrix. A predictor matrix is assembled to include all potential explanatory variables, along with their growth rates or differences (when appropriate), and their lags up to $q,$ as well as the lags of the dependent variable up to $p$ ^[17]. For this analysis, both $p$ and $q$ are set to $4$⁠.

2. Lasso regularization. Lasso regularization is applied with the tuning parameter $λ$ varying in a geometric sequence from 100 to 1,250. The mean squared error (MSE) is estimated for each $λ$ value using $k$-fold cross-validation, in which the dataset is divided into $k$ groups. The model is trained on $k−1$ groups, and the withheld group is used to calculate the associated errors.

3. Bayesian model averaging (BMA). To avoid hand-picking a single value of $λ$ from the sequence, the BMA procedure is used to compute posterior probabilities for each model. These probabilities are calculated using the Hannan–Quinn criterion (HQC) (see also footnote 17 and Equation (4) in  Annex B). Models that fail normality, autocorrelation or heteroskedasticity tests ^[18], or that have negligible posterior probabilities, ^[19] are filtered out by assigning their posterior probabilities a value of 0.

4. Calculation of posterior coefficients. Posterior coefficients are calculated using Equation (5) (see  Annex B). These coefficients are used to generate posterior predictions, calculate errors, estimate LRMs and compute additional relevant statistics.

5. Nonparametric bootstrap analysis. Finally, to further evaluate the model’s robustness, a nonparametric bootstrap is performed by resampling from the residuals, allowing us to examine the distribution of the estimates and validate the model’s properties.

This section demonstrates the implementation of the framework introduced in Section 5, using Prague’s housing market data presented in Section 4. The sample period for the analysis spans from 2005Q1 to 2021Q4 and the summary statistics of the data sample are provided in Table 3. Here, we illustrate the calibration process and present the explanatory variables identified by our algorithm as the primary drivers of new apartment prices in Prague.

Figure 7 provides an overview of our estimation process. The figure shows out-of-sample MSE ^[20] estimates with error bars across a range of models, each based on a different value of the tuning parameter λ. The green circle marks the model with the minimum MSE, while the blue circle identifies the model with the highest λ within one standard deviation of the minimum MSE model. The blue bars (right axis) represent the BMA posterior probabilities for the individual models, while the dashed black line indicates the numbers of non-zero coefficients for each model.

As expected, we observe an inverse relationship between $λ$ and the number of non-zero coefficients as higher values of $λ$ lead to fewer non-zero coefficients. Ultimately, the BMA procedure assigns non-zero posterior probabilities to 37 models, with a noticeable preference for models that balance sufficiently low MSE and a limited number of coefficients. This demonstrates the algorithm’s ability to prioritize model parsimony alongside goodness-of-fit, ensuring an optimal balance between explanatory power, complexity and interpretability.

Figure 8 illustrates the fitted values of new apartment prices per square meter in Prague, accompanied by bootstrapped confidence intervals derived from 10,000 random draws. The model demonstrates a robust fit, with an adjusted $R2$ value exceeding $90%$⁠. Furthermore, the algorithm effectively reduced more than 95% of the considered coefficients to 0, highlighting a substantial degree of variable selection and model parsimony.

Additionally, the posterior errors were tested for normality, autocorrelation and heteroskedasticity. The diagnostic results show that we cannot reject the null hypothesis of normality and there is no significant autocorrelation or heteroskedasticity present in the posterior errors. The corresponding statistics and p-values are reported in Table 4. These findings confirm the model’s assumptions and underscore the reliability of its estimates.

While the model demonstrates strong explanatory power and robust econometric diagnostics, it is important to note that the primary goal of this analysis is not to forecast future prices, but to identify and interpret the key drivers of price variations over time. By emphasizing the interpretation of the most relevant variables, we aim to provide deeper insights into the underlying dynamics of the housing market.

The final model identified five key determinants of the new apartment prices in Prague during the sample period. These variables, along with their corresponding LRM values are presented in Table 4 and discussed in more detail below. The LRM estimations, which are crucial for understanding the overall model dynamics, are calculated using Equation (7).

An analysis in Table 4 reveals a strong positive relationship between new apartment prices and net disposable income per household, as well as a weaker but still positive relationship with GDP per capita in Prague. Conversely, an increase in unemployment and mortgage interest rates is associated with a reduction in apartment prices. The effect of unemployment is strong, while the impact of mortgage interest rates is less pronounced.

Additionally, the Covid-19 dummy variable ^[21] exhibits a substantial positive LRM, highlighting the pandemic’s unique influence in driving up apartment prices, independent of other concurrent factors. This finding underscores the distinct impact of the pandemic on Prague’s real estate market.

To provide a complete picture, the individual estimated posterior coefficients of the final model are reported in Table 5.

Net disposable income per household. This variable serves as a robust indicator of households’ purchasing power. Higher net disposable income links to a higher allocation of funds towards housing. This increase in demand typically drives up real estate prices. The significance of this variable is intuitive, given its direct impact on how much households can afford to spend on housing. As noted by Toporowski (1993), homeownership has increasingly become a luxury good.

Furthermore, an increase in disposable income that outpaces inflation usually results in a more than proportional availability of funds for housing, consistent with the economic definition of luxury goods and their high-income elasticities (Samuelson and Nordhaus, 2009).

Prague GDP per capita. GDP per capita is a key measure of the region’s economic output relative to its population size. Higher GDP per capita in Prague signals a stronger local economy, enhancing the city’s appeal and attracting both residents and investors, which in turn drives up real estate prices. Beyond indicating economic prosperity, GDP per capita reflects the overall economic health of the region and the attractiveness of its real estate market.

Additionally, GDP per capita can influence the supply side, as a stronger economy may lead to an increase in housing supply, though this is often delayed due to the lengthy permit and construction processes. Thus, GDP per capita plays a dual role in both demand and supply dynamics, making it a significant determinant of real estate prices in Prague.

Unemployment rate in Czech Republic. The unemployment rate is inversely related to the health of the real estate market. Higher unemployment rates typically signal economic distress, reducing individuals’ capacity to buy an apartment. The psychological impact of even minor changes in unemployment rates is also important, as shifts in unemployment can significantly influence consumer confidence and, consequently, long-term investment decisions like real estate purchases. This underscores the unemployment rate’s importance as a vital determinant of the real estate market, reflecting both economic conditions and consumer sentiment.

Covid-19 dummy. The Covid-19 dummy variable captures the unique and unprecedented impacts of the pandemic on real estate prices, as also discussed in Section 4. This variable isolates the pandemic’s effects, which include reduced construction activity, shifts in housing preferences and economic support measures, all of which contributed to a surge in real estate prices during the crisis.

Disclaimer on Reported Significance Testing: Due to the use of Lasso regularization and BMA in our methodology, the coefficient estimates are regularized, introducing bias. Consequently, traditional significance testing may not fully reflect the uncertainty associated with these estimates. Thus, the significance values reported in this table are approximated based on the final posterior coefficients and bootstrapped standard deviations.

Figure 9 presents the posterior LRMs from Table 4, alongside their corresponding bootstrapped distributions generated from 10,000 random simulations. This analysis provides valuable insights into the relative importance and reliability of the identified variables in explaining the dynamics of Prague’s real estate prices The LRMs of the posterior model are represented by green dots, while the boxplots illustrate the central tendency and dispersion of the bootstrapped distributions for each identified explanatory variable’s LRM.

The analysis of these bootstrapped distributions confirms that the net disposable income per household, Prague’s GDP per capita, and the unemployment rate in the Czech Republic are the primary drivers of the new apartment prices per square meter in Prague. In contrast, the mortgage interest rates appear less significant, as the respective bootstrapped distribution is heavily clustered around zero, indicating a relatively weak impact. Additionally, the Covid-19 dummy variable is identified as significant, highlighting the idiosyncratic and strong influence of the pandemic on Prague’s housing market. This confirms the unique role the pandemic played in shaping real estate prices, distinct from other economic and fundamental factors.

To investigate the stability of the LRMs for the identified determinants of the new apartment prices per square meter in Prague, we applied the algorithm introduced in Section 5, setting the sample to always start in 2005Q1 while gradually increasing the last observation from 2007Q1 to 2021Q4. This approach systematically assesses how the LRMs evolve as additional data points are incorporated, offering insights into the robustness and consistency of the estimated impacts of the explanatory variables over time.

Figure 10 tracks the changes in the LRMs for the identified determinants. The chart shows that while the magnitude of the impacts has fluctuated, the signs of the LRMs remained stable, as none of the individual lines crosses the zero line. This suggests that although the strength of the relationship may vary, the variables consistently predict prices in terms of their directional influence. This outcome aligns with intuition and the economic theories discussed earlier in the paper.

The variability in the magnitudes is discussed in more detail below, providing further insight into how different factors have shaped real estate prices at different points in time.

The LRMs for net disposable income per household (blue line) and Prague’s GDP per capita (red line) have historically been the two most significant determinants of new apartment prices, with both exhibiting a positive long-run relationship. However, their contributions have displayed volatility and an inverse relationship over time; when the impact of one rises, the other’s impact typically declines. Following the GFC, there is a noticeable downward trend in the LRM for net disposable income, accompanied by an upward trend in the LRM for GDP per capita in Prague. During this period, apartment prices, net disposable income and GDP per capita experienced an initial downturn in the immediate aftermath of the GFC, followed by stabilization and marginal growth throughout the European sovereign debt crisis until around 2014, when all three variables regained a clear upward momentum.

Around this time, the trends in the LRMs of net disposable income and GDP per capita in Prague reversed again, continuing until the onset of the Covid-19 pandemic. After incorporating the Covid-19 period into the sample, both factors gained significance alongside the pandemic dummy variable, when no combination of the considered explanatory variables could fully explain the observed price developments during the pandemic due to its idiosyncratic nature, as discussed in Section 4.

Our observations suggest that during periods of robust economic growth, net disposable income per household gains more significance as a determinant of new apartment prices, likely due to its direct effect on consumer purchasing power and its immediate impact on housing demand. Conversely, during economic downturns or stagnation, GDP per capita in Prague assumes greater importance, acting as a barometer of economic stability and influencing long-term investment decisions in the real estate market. This separation may reflect the differing roles these indicators play in shaping consumer and investor confidence: disposable income drives the ability to purchase, while GDP per capita signals the overall economic health and potential for recovery. The cointegration of both indicators with apartment prices highlights their intrinsic connection to the housing market, though their elasticity varies with the prevailing economic conditions.

On the other hand, the LRMs for the unemployment rate in the Czech Republic (yellow line) and mortgage interest rate (purple line) have been identified as determinants of apartment prices that exhibit negative long-run relationships. The LRM for the unemployment rate has shown the highest stability over time, remaining significant throughout most of the observation period, whereas the mortgage interest rate’s relevance has been less consistent. The mortgage interest rate gained significance primarily during the period of exceptionally low interest rates and record-low unemployment in the second half of the 2010s. This period was characterized by its distinct economic climate with favorable borrowing conditions and robust employment, which jointly bolstered the housing market.

The stable and negative LRM value for the unemployment rate reconfirms the persistent link between labor market health and real estate demand, as higher unemployment typically lowers property values in the long run. The “long memory” effect associated with the unemployment rate implies that reductions in unemployment do not immediately alleviate concerns about job security and even small increases in unemployment can send a negative signal across the market. This sensitivity is particularly pronounced in volatile economies, as psychological factors and expectations of further increases in unemployment amplify the effects on the housing market, as noted by Tortorice (2012).

This analysis largely confirms fundamental economic principles regarding how the individual determinants, as previously identified by our algorithm and discussed earlier in the paper, influence new apartment prices per square meter in Prague. Moreover, it highlights the complex ways these principles materialize over time. Our observations reveal a clear pattern: net disposable income per household tends to drive price increases during periods of strong economic growth, whereas Prague’s GDP per capita becomes more influential during downturns and stagnation. The impact of the unemployment rate in the Czech Republic remains relatively stable over time, while mortgage interest rates tend to matter only when they are unusually low.

These insights underscore the multifaceted relationship between the identified determinants and the real estate market, suggesting that not only hard economic data but also public sentiment and one-off events, like the Covid-19 pandemic, play a crucial role in driving the dynamics of new apartment prices in Prague.

This study presents a modern, data-driven approach to understanding the underlying factors driving the prices of new residential apartments in Prague between 2005 and 2021. By combining lasso regularization and BMA, we propose a comprehensive framework for the highly automated selection of parsimonious models with strong explanatory powers and desirable statistical properties. Lasso efficiently extracts the most significant variables by shrinking less relevant coefficients to zero, making the model more parsimonious, while BMA accounts for model uncertainty by combining multiple regularization specifications to ensure robust results. This innovative combination allows us to handle a large dataset while providing valuable insights into the key determinants of Prague’s real estate market. Moreover, the proposed methodology can be adapted and applied to other regions or countries, allowing for the creation of similarly robust models that account for a wide range of factors impacting real estate prices.

Including levels, differences or growth rates and lags, we considered nearly 300 potential explanatory variables in our model. The proposed algorithm identified five main drivers among these variables: net disposable income per household, Prague GDP per capita, the unemployment rate in the Czech Republic, mortgage interest rates and Covid-19 dummy variable. Our findings indicate that net disposable income per household and Prague GDP per capita positively influence new apartment prices, while the unemployment rate and mortgage interest rates have a negative impact. Additionally, Covid-19 dummy variable captures the unique, idiosyncratic effects of the pandemic, which contributed to a rise in prices. Collectively, these variables explain over 90% of the variance in new apartment prices per square meter in Prague.

While the model demonstrates strong explanatory power, it is important to highlight that our primary focus is on identifying and interpreting the key drivers of price variation, rather than forecasting future price trends. This approach allows us to understand the underlying dynamics of the housing market, offering critical insights into the factors that influence price developments over time.

The relative affordability of monthly mortgage installments rather than the absolute prices of the apartments, is a key determinant of demand for new apartments. As disposable income and economic output in Prague increase, household purchasing power rises, making mortgage payments more affordable and allowing buyers to pay higher prices for new apartments. Conversely, the unemployment rate constraints demand by limiting purchasing power and influencing consumer sentiment. Higher mortgage interest rates increase borrowing costs, placing a greater burden on household budgets.

Looking ahead, we anticipate that mortgage interest rates may become a more significant variable in future updates, particularly with the inclusion of data from 2022 to 2023. The trend of declining mortgage interest rates, which had previously coincided with rising real wages and disposable income, reversed in mid-2022. At that point, interest rates sharply increased in response to soaring inflation, while real wages declined, leading to a reduction in the living standards of many households. This shift is likely to have a more pronounced impact on the housing market in the coming years.

We plan to periodically update the model to assess the evolution and stability of the identified key determinants, especially in light of recent global events not yet reflected in our sample period, such as the Russian war in Ukraine, rising inflation or sharply increasing interest rates. These events may alter the way real estate prices react to certain shocks and provide new insights into Prague’s market. Future research may also highlight the growing importance of demographical variables, such as the impact of Ukrainian refugees on housing demand ^[22].

As housing prices rise across Europe, we believe the model can be replicated in other European cities with similar housing market conditions to Prague. Testing the model in different cities could reveal varying sets of explanatory variables, specific to the local market dynamics. For instance, in Western European cities, variables such as immigration and location quality may become more significant, reflecting the larger intra-city differences in these markets compared to Prague.

Funding: This work was suported by the Faculty of finance and accouting of the Prague University of Economics and Business [IGS F1/27/2025].

Gross domestic product. Real estate represents a significant component of the economy, contributing notably to gross domestic product (GDP). Within this sector, some argue, such as Franses and De Groot (2013) with their example of the Netherlands, that commercial real estate holds more predictive power for GDP than residential real estate. The literature commonly suggests a correlation between GDP and real estate activities, including real estate prices. However, Green (1997) provided evidence from long-term US data showing that residential real estate prices are exogenous to GDP. Conversely, Zhang et al. (2021) highlighted a cross-correlation between real estate investment and GDP in China, underscoring the complex relationship between these variables.

In our analysis, we have chosen to include current and lagged values of GDP among the explanatory variables. This decision is based on the understanding that the pricing of new apartments contributes to the future GDP rather than the current GDP. This aligns with the observation that a significant portion of new apartments in Prague are sold either before construction begins (with 20–30% required by financing banks) or during construction, with these sales activities contributing to the future GDP.

Additionally, previous years' GDP growth influences the real estate sector’s expansion, particularly in the commercial domain, where GDP growth drives demand for new office and production spaces. This GDP growth also spurs demand for apartments; however, its impact on the residential sector primarily influences GDP in subsequent years. Therefore, it is not accurate to assert that the GDP of a given year directly affects apartment prices within that same year. Instead, demand sentiment is shaped by the GDP growth of previous years, while the supply of apartments will contribute to the GDP in the years to come.

Mortgage interest rates play a significant role in influencing the demand for new apartments, as higher rates result in increased monthly installments, directly impacting potential buyers' budgets. In reality, buyers tend to focus more on the amount of their monthly mortgage installment rather than the absolute price of the apartment, since the installment has a more direct impact on their finances. In case the PRIBOR rate decreases, the margins of commercial banks on mortgages increases, as mortgage interest rates tend to decline more slowly or with a delay in response to PRIBOR decreases. This dynamic allows banks to maintain higher margins for longer periods. On the other hand, in 2019, mortgage margins approached zero, indicating that banks were unable to absorb significant increases in PRIBOR effectively. These increases were eventually passed onto the customers in the form of higher mortgage interest rates.

Both the interest rate and the price of the apartment determine the mortgage instalment, which represents a significant portion of the out-of-pocket housing user costs. For this reason, mortgage interest rates were tested as one of the explanatory variables in our analysis. In comparison, other out-of-pocket costs like insurance payments and taxes are relatively small in the Czech economy, especially when compared with mortgage installments. Maintenance and repair costs are also relatively minor, particularly for new apartments and are challenging to quantify accurately without relying on depreciation-based estimates.

Unemployment. Unemployment was selected as a potential explanatory variable not only because rising unemployment might reduce demand for new apartments, as fewer people are able to make a purchase, but also due to its psychological impact. Developers in Prague are building new apartments for less than 15% of its inhabitants, and the majority of people living in Prague are generally unable to afford new apartments. Thus, the primary issue is rather psychological than a direct reduction in wealth with increasing unemployment – at least initially – unless unemployment rises to dangerously high levels, as seen in Spain, as described in Irandoust (2019). As more people lose their jobs and struggle to find new employment, societal anxiety grows. This can lead to postponement of investments and an increase in savings. This psychological response can dampen housing demand even if wealth remains relatively stable in the short term. However, once unemployment reaches a critical level, people may be forced to sell their apartments in large numbers due to their inability to service mortgage payments. This influx of properties onto the market can exert additional downward pressure on prices, as weak demand meets a rapidly increasing supply (e.g. Irandoust 2019).

VAT and LTV. Both the VAT rate and LTV ratio were included in the model as they changed during the observed period. An announcement of a VAT increase often leads to

a surge in demand, as buyers rush to purchase an apartment before the increase takes effect. This phenomenon occurred in the Czech Republic in 2008 when a VAT increase was announced, prompting a rise in demand.

Similarly, changes to LTV regulations or any other income-debt regulatory measure can have a comparable effect on demand. During the observed period, the maximum LTV allowance shifted from 80% to 90%, impacting buyers' ability to finance property purchases with lower down payments. Both VAT changes and LTV adjustments influence buyer behavior and were therefore included in our model.

Demographic variables. Demographic variables were also included in the list of potential explanatory variables (Garcia, 2021). From 2003 to 2020, the total population of Prague increased by 14%, but the number of households grew even more significantly, rising by 17%. This trend is driven by a decreasing average household size, as more people in big cities, including Prague, live in single-person households (Czech Statistical Office n.d.b) ^[23]. This shift results in a higher overall demand for housing, particularly for smaller apartments.

In addition to total population growth, the age structure of the population is also crucial. The most active age group of apartment buyers consists of individuals aged 30–50 years, and this group increased by 32% in Prague from 2003 to 2020, making it a key demographic driver of housing demand (Czech Statistical Office). While the total population size has not changed dramatically, these population waves are passing through different life stages, affecting housing demand patterns.

Migration was also considered in the model, and the number of foreigners living in Prague was included as an explanatory variable. However, the impact of immigration on Prague’s housing market was minimal during the observation period, as the city did not face any significant immigration waves. This dynamic may change in the coming years due to the influx of Ukrainian refugees, which occurred after the period covered by the model. Some of these refugees may eventually become buyers of housing units, potentially affecting future market demand.

Apartments under construction, apartments finished and apartments sold. The number of apartments under construction and the number of apartments finished are both key supply-side factors, as they directly determine the availability of new apartments entering the market. Data for both variables were sourced from a data collection conducted by the Czech Statistical Office ^[24]. These factors capture the pace of new developments, which influences market saturation and supply-demand dynamics.

Additionally, the number of apartments sold was included as a potential explanatory variable to account for market demand and transaction activity. To further refine the analysis, we incorporated the ratio of newly completed apartments to the Prague population for each quarter as a potential explanatory variable. This metric provides insight into the relative supply of new housing units compared to population growth, which can affect housing availability and pricing pressure.

Construction price index. The construction price index was included as a potential explanatory variable, given that rising construction costs directly affect the margins of real estate developers. However, this does not always translate into higher prices for end buyers. Instead, increased construction costs may prompt developers to reduce the number of units they build, especially if they perceive the cost rise as a short-term shock. In such cases, developers may postpone new projects, leading to a constrained supply and potentially driving up prices due to the limited availability of new apartments.

Rent to average mortgage installment ratio. For buyers who purchase apartments as investment vehicles, the rent-to-mortgage ratio is an essential indicator of whether the rent generated by the apartment can cover their mortgage installments. Figure A1 illustrates the development of average rent and average mortgage installments in Prague for a typical new apartment of average size. If mortgage payments significantly exceed the rental income, potential investors may be discouraged from purchasing investment apartments, as they would need to continuously contribute additional funds, creating a strain on their budget.

In the context of investment apartment purchases, the difference between achievable rent and the mortgage installment represents the net housing user costs, excluding the opportunity cost of equity. Czech individual investors tend to have a strong preference for real assets over financial instruments, indicating that the foregone returns on equity invested in property do not significantly impact their investment decisions unless potential returns in the financial markets are exceptionally high. This is particularly true for moderate and smaller investors, who may have a limited access to financial markets. To capture investment ratio in the apartments, the return on investment from apartment purchases was also added as an explanatory variable in our analysis. This allows us to account for the financial viability of apartment investments, particularly in relation to rental income and mortgage costs.

Average mortgage instalment to average salary ratio. The ratio between income and mortgage installments is a critical measure of housing affordability, especially for average buyers, such as families or couples purchasing their primary residence or, at most, a single investment apartment. Even a marginal increase in interest rates that raises mortgage payments, can significantly affect their decision to buy.

Return on apartment investment. The ratio of average apartment rent to its purchase price captures the attractiveness of apartments as investment vehicles and links the pure investment perspective of apartment purchases with the comparison to other investment assets outside of real estate. By including this ratio in our model, we are able to assess how the return on apartment investments influences buyer behavior, particularly for those who weigh real estate investments against alternative asset classes.

International demand. While there is insufficient data to quantify the strength of international demand in Prague’s market, common market knowledge suggests that foreign demand does not play a significant role. To account for potential international demand in the absence of direct data, we incorporated underlying factors that may impact it. Specifically, we included the exchange rates of CZK to EUR and USD, as most foreign investments are made in these two currencies, even if they originate from non-EUR or non-USD countries. Additionally, we included the interest rates differential between the CNB’s and the ECB’s monetary policy rates, as well as foreign investment flows into the Czech Republic, as reported by the Czech National Bank (non-resident property trade subsection).

Net disposable income. The growth of net disposable income of Prague’s households has played a crucial role in offsetting the rising apartment prices and mitigating the impact of higher mortgage installments on household budgets. As disposable income increases, households are better able to afford housing, including both mortgage payments and other housing-related expenses. This higher disposable income supports increased spending on housing, contributing to sustained demand despite rising prices. Consequently, net disposable income was included as an explanatory variable in our model to capture its influence on housing affordability and demand.

Covid-19 dummy variable. Including a pandemic dummy in the model is a strategic approach to account for the unique market conditions during the Covid-19 pandemic. The pandemic had a multifaceted impact on global real estate markets, including changing preferences towards larger, more isolated properties, affecting individual’s financial capacity to manage mortgage payments and altering the broader economic landscape. Introducing this dummy variable allows us to isolate the pandemic’s effects on the real estate market, which encompasses both adverse factors driving downward pressure on prices (e.g. economic downturn or reduced purchasing power) and factors contributing to price increases (e.g. shifts in housing preferences, flight to safety, supply limitations or fiscal and monetary policies).

Contrary to expectations that real estate prices would decline during a severely adverse economic shock like the Covid-19 pandemic, the Czech real estate market experienced a paradoxical price increase. Due to pandemic mitigation measures, the construction of new apartments in Prague significantly decreased in Q3 and Q4 of 2020, with only 800 new units initiated. At the same time, public authorities introduced various economic subsidies, while household expenditures on social activities and travel were greatly reduced. This often resulted in households maintaining or even temporarily increasing their net disposable incomes.

Moreover, in many cases, the pandemic underscored the need for better housing as individuals spent more time at home, either with family or in confined spaces, increasing the urgency for more suitable living conditions. The limited availability of new apartments, combined with this heightened demand, led to a surge in the real estate market. This demand was not only from potential homeowners but also from investment buyers, who were driven by uncertainty and increased volatility in financial markets, as noted by Ullah et al. (2023). In this context, real assets like apartments became more attractive as safer investment options.

Given the wide range of factors that influenced real estate prices during the pandemic, the inclusion of a Covid-19-specific dummy variable in the model is crucial. It captures the complex and unique dynamics of the period including both the negative and positive pressures on housing prices.

Let us define the dependent variable $Y$ with a lag limit $p$ and $K$ explanatory variables $X(1),…,X(K)$ with a lag limit $q$⁠. The ADL model equation can then be expressed as

where:

• $α$ is the intercept,

• $ρ1,…,ρp$ are the coefficients for the lagged values of the dependent variable $Y$⁠,

• $β0(1),…,βq(1),…,β0(K),…,βq(K)$ are the coefficients for the lagged values of the explanatory variables $X(1),…,X(k)$⁠,

• $ε$ is the error term,

• and $β=[α,ρ1,…,ρp,β0(1),…,βq(1),…,β0(K),…,βq(K)]$ is the full vector of model parameters.

Assume a vector of observations for the dependent variable $Y=(Y1,Y2,…,YT)$ of length $T$ and a $T×K$ matrix of predictors $X=(X1(1),…,XT(1),…,X1(K),…,XT(K))$⁠. Additionally, let

$β=(β(1),…,β(K))$ represent a vector of $K$ unknown regression coefficients. The ordinary least squares (OLS) estimation of a linear regression model minimizes the following term:

Typically, most of the coefficients’ estimates will be non-zero, complicating the interpretation of the model, especially when $K$ is large. In fact, when $K>T$⁠, the OLS solution is not unique and often overfits the data. Thus, to prevent overfitting, it is necessary to constrain the estimation process. In the lasso approach, this is achieved by adding $l1$-norm regularization term to the loss function. Therefore, problem (2) can be re-written as:

When solving the minimization problem (3), some of the $β(k)$ coefficients are shrunk exactly to 0, resulting in a more parsimonious regression model that is less prone to overfitting. The lasso approach can thus be informally described as following the “bet-on sparsity principle,” providing a natural way to encourage or enforce sparsity and simplicity in the solution.

The tuning parameter $λ$ controls the strength of the $l1$-norm penalty, effectively regulating the amount of shrinkage applied to the model coefficients. In the extreme case where $λ=0$⁠, no shrinkage is applied, and the method reduces to the OLS estimate (as shown in Equation (2)). As λ increases, more coefficients are gradually eliminated, with the extreme case of $λ=∞$ resulting in all coefficients being eliminated. Clearly, the choice of the tuning parameter λ has a crucial impact on the final form of the model, as different values of $λ$ produce different models. Several criteria can be used to select the optimal $λ$⁠, including the minimum MSE rule, the one-standard-error rule, the modified Bayesian information criterion (BIC) or Cohen’s kappa coefficient. In this paper, however, we adopt an alternative approach based on BMA.

The BMA model selection technique explicitly accounts for model uncertainty by working with a large pool of model equations simultaneously. This approach helps to find a more reliable and robust model while avoiding the risks associated with hand-picking a single model, as is common when using other selection rules. In the BMA, weights in the form of posterior probabilities are assigned to individual models, reflecting their explanatory power, predictive performance and other relevant features. These weights are then used to combine the models into a single posterior equation.

Let $F$ represent the set of potential models and assume a non-informative prior where the probability of each model $Fi$ being the correct one is uniformly distributed. Under the assumption of normality, the posterior probabilities can be estimated as follows:

where:

• $IC$ is a selected information criterion ^[25] computed using $MSEFi$ and a number of non-zero parameters $dFi$ for a model $Fi$⁠,

• $πFi$ is the posterior probability of model $Fi$⁠,

• $|F|$ is the total number of candidate models.

In addition, further economic or econometric constraints may be introduced to automatically filter out unsuitable models. This process emulates the work of an econometric modeler by refining the selection of models based on their underlying characteristics.

Using the posterior probabilities of all suitable models, the posterior vector of regression coefficients can be calculated by weighting the estimated coefficients of the individual models $βˆFi$ by their associated posterior probabilities $πFi$⁠:

The final BMA estimates of $Y$ can then be computed in the usual way as:

LRMs can be calculated for each $X(k)$ as follows:

where $σX(k)$ and $σY$ are standard deviations of $X(k)$ and $Y$⁠, respectively.

The value of the LRM for each $k$ can be interpreted as the change in the expected value of $Y$⁠, expressed in terms of the number of standard deviations, resulting from a permanent shift of $X(k)$ by one standard deviation. This formulation provides a standardized metric to evaluate the impact of variations in explanatory variables on the dependent variable within the model.