Statistical Modeling of Rent Per Square Meter in Munich City, Germany

This study explores a comprehensive statistical model for analyzing rental apartment prices per square meter in Munich, Germany. The research investigates key quantitative and qualitative variables influencing rent dynamics by leveraging a robust dataset comprising over 2.6 million apartments with 59 variables, sourced from FDZ Ruhr and ImmobilienScout24, for the years 2015 and 2019. Thirty-one key variables (9 quantitative and 22 qualitative) were analyzed, and the study identified significant predictors, such as apartment size, furnishing quality, energy efficiency, and amenity availability, through exploratory data analysis and multiple linear regression with nonlinear covariates. Applying log transformations and polynomial terms improved model performance, with the 2019 model achieving an adjusted R-squared of over 0.54 in the Analysis Of Variance (ANOVA) ratio tests. Model diagnostics, including the Akaike Information Criterion (AIC), residual plots, and Variance Inflation Factor (VIF), were employed to assess model fit and multicollinearity, ensuring the robustness and validity of the regression model. The results indicate a consistent trend where larger apartments and permitting pets command lower rent per square meter, while upscale furnishings, kitchens, and the number of bedrooms are associated with higher prices. This study provides meaningful predictive analytics insights into urban housing and Munich’s evolving rental market. The findings provide valuable insights for real estate planning, sustainable housing policies, urban development strategies, and educators, particularly for university administrators and planners who can advocate for informed housing policies. This research contributes to academic literature on rent modeling and provides a data-driven foundation for evidence-based decision-making in high-demand urban housing markets.

• Statistical Modeling
• Munich Housing Market
• Rent Price Modeling
• Multiple Linear Regression

1. Introduction

1.1. Introduction with Munich Rental Apartment Review

The importance of using statistical methods to develop a mathematical equation that models the relationship between a response variable rent_sqm and a set of explanatory variables can not be overemphasized. The demand for apartment rentals in Germany, especially in Munich and Berlin, is relatively high compared to other cities. Between 2011 and 2016, about 45,000 new apartments were built in Munich for roughly 90,000 people, even as the population in Munich rose from 200,000 to 1.55 million during the same period . Therefore, about 55,000 more apartments were needed to accommodate the new arrivals. By 2030, about 150,000 apartments would be required as the population will increase to more than 1.7 million based on the estimate of . Germany is representative of the situation in many respects compared to other high-income countries like the UK, France, the US, Canada, etc., and therefore, apartment prices and rents are causing serious problems as they have risen significantly in the country’s large cities [2] . In international comparisons, like North America or Southern Europe, Germany has a higher share of renters. For instance, in 2018, the homeownership rate in Germany was 51.5% compared to 65.1%, 72.4%, and 96.4% in the UK, Italy, and Romania, respectively.

1.2. Objective

This paper models rent_sqm in Munich using multiple linear regression to uncover key market trends. It also investigates whether a transformation of the response variable is needed, examines influential covariates, and identifies those significantly influencing rent prices. The findings are intended to inform housing policy and guide educational leadership in housing planning.

1.3. Literature Review

Regression models, often log-linear, help address skewness and variability in housing data and key predictors include furnishing quality, energy efficiency, and modernization status [3] [4] . Cross-national comparisons highlight differences between Germany’s state-supported and the U.S.’s market-driven housing systems [5] [6] . Sustainability concerns, especially the impact of energy-efficient design, have also gained attention [7] . This study builds on existing literature by modeling Munich’s rental market and linking statistical analysis to educational policy, with implications for improving student and faculty housing strategies [8] [9] .

1.4. Research Questions

2.1. Research Design

This study employs a quantitative research design to investigate the rental price per square meter in Munich’s housing market using a multiple linear regression model with nonlinear covariates.

2.2. Data Collection

A secondary source of data collection was used for this study. The data was provided by the FDZ Ruhr at RWI (and ImmobilienScout24) institution. The ImmobilienScout24 GmbH, founded in 1998, deals with real estate properties in Germany. The data set contains 2,651,885 observations and 59 attributes from 2007 to 2020. The data description is done in Section 3.

2.3. Sample Selection and Data Filtering

We first selected the two cities (Munich and Berlin) that have the highest number of rental transactions. Thereafter, we chose the two years (2015 and 2019) based on the significant impact observed in the plotted scatter of years with rent prices. For instance, Munic 2015 was filtered using the R code (dfm15 <-df %>% filter(city = = “Munich”, year = = 2015)). We conducted separate studies of the two cities in two different papers. We focus on the city of Munich (2015 and 2019) for this article and conducted separate studies of Berlin in another article. The number of rental properties contained in each data set (Munich 2015 and Munich 2019) is 14,449 and 17,776, respectively, as shown in Table 3 .

2.4. Data Cleaning and Missing Values

During data cleaning, we changed the variable names from German to English and removed outliers using the Interquartile Range (IQR) method. The recorded missing values and the NAs were part of the labels for most categorical variables, as shown in Table 2 .

2.5. Data Analysis: Multiple Linear Regression with Nonlinear Covariate

Often, a relationship between two (or more) variables is found or suspected. Sometimes, one might be interested in investigating whether there is a relationship or trend between two or more variables, and if they are, how they are related. In regression, we want to model the relationship between the variable of interest (dependent or response variable), and other given variables (covariates or independent variables); see [10] . For instance, we may want to know whether a relationship exists between the number of hours students read in a day (independent variable or covariate) and their performance in the examination (dependent or response variable). The goal of regression analysis is to determine the parameters of the linear function that best describes the joint distribution of the response variable and the covariates [11] . We note that the relationship among variables may be linear, nonlinear (quadratic, cubic, etc.), or non-existent at all, and may involve several independent variables. Thus, we need tools for an exploratory data analysis (EDA), which enables us to suggest useful model formulations before fitting specific regression models. We refer to multiple linear regression when several independent variables are involved and the response variable is continuous. In this study, we want to investigate the relationship between the rent per square meter in Munich charged for an apartment characterized by continuous and discrete covariates.

In a regression analysis with a continuous response variable $Y_i$ and $p$ covariates or predictors $X_{i1},X_{i2},\cdots ,X_{ik}$ which may be continuous or qualitative (ordinal or nominal) with $n$ observations, let $\left(y_i,x_i^{\top }\right):={\left(y_i,x_{i1},\cdots ,x_{ik}\right)}^{\top }$, $i=1,\cdots ,n$, $k=p-1$, be a pair of the $i\text{th}$ observation $\left(y_i,x_i^{\top }\right)$ of the random vector $\left(Y_i,x_i^{\top }\right)$, where $x_i={\left(x_{i1},x_{i2},\cdots ,x_{ik}\right)}^{\top }$, then our objective is to analyze the effects of the covariates on the mean value of the response variable ( ${\mu }_i\equiv E\left[Y_i\right]$). The linear model models the response as a linear function of the predictors together with an error term, i.e.

$Y_i={\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+\cdots +{\beta }_k{x}_{ik}+{ϵ}_i={\beta }_0+\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{\beta }_j{x}_j+{ϵ}_i$(2.1)

with mean $E\left[Y_i\right]={\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+\cdots +{\beta }_k{x}_{ik}$.

Definition 2.1. The multiple linear regression model is defined as

$Y_i={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik}+{\epsilon }_i,i=1,\cdots ,n,$(2.2)

where ${\epsilon }_i$ is the random error variable, ${\beta }_0$ is the intercept, and the $k$ parameters ${\beta }_1,\cdots ,{\beta }_k$ are the unknown regression parameters to be estimated from $n$ observations $\left(y_i,x_{i1},\cdots ,x_{ik}\right)$, for $i=1,\cdots ,n$.

Polynomial regression is often appropriate when a relationship exists between the response and the covariates. Given a continuous covariate $V_i$ with observations $v_i$ that has a polynomial effect of degree $d$ on the response, then the model $Y_i={\beta }_0+{\beta }_1{V}_i+{\beta }_2{V}_i^2+\cdots +{\beta }_d{V}_i^d+\cdots +{\epsilon }_i$ can be used. Note, it is a linear regression model of the form (2.2) with $x_{ij}=v_i^j,j=1,\cdots ,d$ [12] and [13] .

In order to increase numerical stability, we orthonormalize the corresponding design matrix $X=\left(\begin{array}{ccc}1& v_1& v_1^d\\ ⋮& ⋮& ⋮\\ 1& v_n& v_n^d\end{array}\right)$ to $X^{\text{*}}$, where all columns have unit norms and are orthogonal. In $R$, this is achieved by $\text{poly}\left(v,d\right)$, see [14] .

Sometimes, the transformation of the response variable is appropriate when non-normality and/or unequal error variances are present in the data. Let $Y_i^{ln}:=\text{ln}\left(Y_i\right)$, then the formulated model $Y_i=\text{exp}\left({\beta }_0+{\beta }_1{x}_{i1},\cdots ,{\beta }_k{x}_{ik}+{\epsilon }_i\right)$ can be expressed in the form of the linear regression model (2.2) as

$Y_i^{ln}={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik}+{\epsilon }_i,i=1,\cdots ,n$(2.3)

2.6. Estimation of Model Parameters

In this section, we will consider the methods of estimating the unknown parameters in the linear regression model of Definition (2.2). Our goal is to determine estimates

$\hat{\beta }={\left({\hat{\beta }}_0,\cdots ,{\hat{\beta }}_k\right)}^{\top }\in {ℝ}^p$(2.4)

and the error variance $\sigma$ based on $n$ observations. Here $\beta$ is the unknown regression parameter vector.

Note that parameter estimators, which are random quantities are different from their realizations called estimates, which are determined by the values of the observations. We will consider two approaches: Least Squares (LS) estimation, and Maximum Likelihood (ML) estimation. These two estimation methods yield the same estimator if the assumptions of independence, homoscedasticity, and normality of errors are satisfied.

Let the fitted values of the Model (2.2) be given as

$\begin{array}{c}{\hat{Y}}_i={\hat{\beta }}_0+{\hat{\beta }}_1{x}_{i1}+\cdots +{\hat{\beta }}_k{x}_{ik},i=1,\cdots ,n\\ ={x^{\prime }}_i\hat{\beta }\end{array}$(2.5)

Also, let the residual be denoted by $\hat{\epsilon }={\left({\hat{\epsilon }}_1,\cdots ,{\hat{\epsilon }}_n\right)}^{\prime }\in {ℝ}^n$, which is the difference between the observed response values $y_i$ and the corresponding fitted values of (2.11), be given as

$\hat{\epsilon }=y-\hat{y}=Y-X\hat{\beta },$(2.6)

where $\hat{y}={\left({\hat{y}}_1,\cdots ,{\hat{y}}_n\right)}^{\top }\in {ℝ}^n$ in the vector notation. Then, least squares minimizes the residual sum of squares (the sum of the squared deviations) of Equation (2.12).

Definition 2.2. (Sum of squared deviations) Given the data $\left(y_i,x_i\right),i=1,2,\cdots ,n$, the sum of the squared deviations which is used in obtaining the estimates $\hat{\beta }$ of Equation (2.10) for the unknown regression parameters $\beta$ is given as

$Q_{LS}\left(\beta \right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-x_i^{\text{T}}\beta \right)}^2=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\hat{\epsilon }}_i^2={\hat{\epsilon }}^{\text{T}}\hat{\epsilon }$(2.7)

In order to minimize $Q_{LS}\left(\beta \right)$ (2.13), we take the partial derivative of $Q_{LS}\left(\beta \right)$ with respect to $\beta$ and set the result to zero. Then, it follows

$\frac{\partial \left(Q_{LS}\left(\beta \right)\right)}{\partial \beta }=0\iff -2X^{\text{T}}y+2X^{\text{T}}X\beta =0\iff X^{\top }X\beta =X^{\top }y$(2.8)

We are now interested in solving the least squares normal equations given in (2.14). If the matrix $X$ has a full rank $p$, then $X^{\text{T}}X$ will be positive definite and will have a unique solution. Thus, the minimum of $Q_{LS}\left(\beta \right)$ is attained at

${\hat{\beta }}_{LS}={\left(X^{\top }X\right)}^{-1}{X}^{\top }y$(2.9)

which is the least squares estimate from the normal equations.

The method of maximum likelihood estimation is based on specifying the distribution we are sampling from and writing the joint density of our sample, unlike in the least squares method where we do not specify the distribution of the response variable $Y_i$. Considering the assumptions of our linear model, we assumed in Equation (2.4) that the random variables $Y_i$ are normally distributed ( $Y~N_n\left(X\beta ,{\sigma }^2{I}_n\right)$). Thus, it follows that the likelihood of the vector $\left(\beta ,\sigma \right)$ given the data values $y$ is

$L\left(\beta ,\sigma |y\right)=\frac{1}{{\left(2\pi {\sigma }^2\right)}^{\frac{n}{2}}}\text{exp}\left(-\frac{1}{2{\sigma }^2}{\left(y-X\beta \right)}^{\text{T}}\left(y-X\beta \right)\right)$(2.10)

Therefore, the corresponding log likelihood is given by

$l\left(\beta ,\sigma |y\right)=-\frac{n}{2}\text{log}\left(2\pi \right)-\frac{n}{2}\text{log}\left({\sigma }^2\right)-\frac{1}{2{\sigma }^2}{\left(y-X\beta \right)}^{\text{T}}\left(y-X\beta \right)$(2.11)

To maximize this log-likelihood (2.17) with respect to $\beta$, we differentiate Equation (2.17) with respect to $\beta$ and set it equal to zero [15] . Thus, we have

$\frac{\partial \left(l\left(\beta ,\sigma |y\right)\right)}{\partial \beta }=0\iff -\frac{1}{2{\sigma }^2}\left(-2X^{\text{T}}y+2X^{\text{T}}X\beta \right)=0\iff X^{\top }X\beta =X^{\top }y$ (2.12)

This shows that ${\hat{\beta }}_{ML}={\hat{\beta }}_{LS}$.

Also, differentiating Equation (2.17) with respect to ${\sigma }^2$ and maximizing over ${\sigma }^2$, we have

${\hat{\sigma }}^2:=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-{\hat{y}}_i\right)}^2=\frac{1}{n}{\Vert \hat{\epsilon }\Vert }^2$(2.13)

and an unbiased estimator $s^2$ of ${\sigma }^2$ is given by

$s^2:=\frac{1}{n-p}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(Y_i-{\hat{Y}}_i\right)}^2=\frac{n}{n-p}{\hat{\sigma }}^2=\frac{1}{n-p}{\Vert \hat{\epsilon }\Vert }^2.$(2.14)

It is of great importance to know the goodness of the fitted model after estimating the parameters of the linear regression model of (2.2). Thus, we need suitable measures of the goodness of fit. Therefore, we will introduce one of the appropriate measures of the goodness of fit called the coefficient of determination (R²), which determines the proportion of variation of the response variable that is explained by the covariates.

Definition 2.3. (Sum of squares) We define the sum of squares SST (total sum of squares), SSR (regression sum of squares) and SSE (error sum of squares) to quantify the amount of variability explained by the regression model as follows

$\begin{array}{l}\text{SST}:=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-\bar{y}\right)}^2\iff \left(\text{total sum of squares}\right)\\ \text{SSR}:=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left({\hat{y}}_i-\bar{y}\right)}^2\iff \left(\text{regression sum of squares}\right)\\ \text{SSE}:=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-{\hat{y}}_i\right)}^2\iff \left(\text{error sum of squares}\right)\end{array}$(2.15)

where $\bar{y}=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{y}_i}$. Thus, we can have the decomposition as

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-\bar{y}\right)}^2=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left({\hat{y}}_i-\bar{y}\right)}^2+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-{\hat{y}}_i\right)}^2$(2.16)

and using the fact that ${\displaystyle {\sum }_{i=1}^n\left({\hat{y}}_i-\bar{y}\right)\left(y_i-{\hat{y}}_i\right)}=0$, it follows from (2.26) that

$\text{SST}=\text{SSR}+\text{SSE}$(2.17)

The multiple coefficient of determination R² is a measure of goodness of fit. It measures how well the covariates in the model explain the variance in the response variable, see [16] .

Definition 2.4. (Multiple coefficient of determination) We define the multiple coefficient of determination R² as

$R^2:=\frac{\text{SSR}}{\text{SST}}=1-\frac{\text{SSE}}{\text{SST}}$(2.18)

We also define the adjusted multiple coefficient of determination $R_{adj}^2$ as

$R_{adj}^2:=1-\frac{\text{SSE}/\left(n-p\right)}{\text{SST}/\left(n-1\right)}$(2.19)

The values of the multiple coefficient of determination range from zero to one ( $0\le R^2\le 1$). Our model accounts for a larger variation of the response when the R² is closer to 1. However, the weakness of R² is that, it always increases when we add more covariates to our model, and therefore cannot be used to compare the goodness of fit for models with different numbers of covariates, see [17] . Thus, there is a need to establish an appropriate measure $R_{adj}^2$ that compares models with different numbers of covariates. We will therefore make use of the adjusted multiple coefficient of determination ( $R_{adj}^2$) as a measure of our model selection in this paper.

To measure the strength and direction of the linear relationship between two continuous variables, we use the correlation analysis. The most commonly used metric is the Pearson correlation coefficient, denoted by $\rho$ for the population and $r$ for the sample. It ranges from −1 to 1, where values close to 1 or −1 indicate strong positive or negative linear relationships, respectively, and values near 0 suggest no linear relationship.

The sample Pearson correlation coefficient between two variables $X$ and $Y$ is given by:

$r=\frac{{{\displaystyle \sum }}_{i=1}^n\left(X_i-\bar{X}\right)\left(Y_i-\bar{Y}\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(X_i-\bar{X}\right)}^2}\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(Y_i-\bar{Y}\right)}^2}},$

where $\bar{X}$ and $\bar{Y}$ are the sample means of $X$ and $Y$, respectively. This metric provides a preliminary indication of potential multicollinearity when applied to predictor variables.

2.7. Hypothesis Testing

A statistical hypothesis is an assumption about the form of a population, which based on sample information from the population, seeks to support or reject this assumption. If there is evidence that the null hypothesis (hypothesis of no difference) denoted by $H_0$ is not true, then it is rejected and its alternative denoted by $H_1$ is accepted. Thus, a test of hypothesis is a rule or a procedure used for deciding whether to accept or reject $H_0$ or to determine whether the observed sample differs significantly from expected results under $H_0$ [18] . This concept can be extended in statistical inference for the model parameters of linear regression [19] . For instance, we may want to know if the response variable is significantly influenced by a particular set of covariate variables, which can be expressed in terms of linear combinations of the unknown regression parameters $\beta ={\left({\beta }_0,\cdots ,{\beta }_k\right)}^{\top }$. We will use the chi-square, F and the univariate t-distribution since the t-test and the F-test rely on quantities of these distributions.

Definition 2.5. (Chi-square distribution) A continuous random variable X is said to have a Chi-square distribution with parameter, $\nu$, if its probability density function is given by

$f_X\left(x|\nu \right)=\frac{2^{-\nu /2}}{\text{Γ}\left(\nu /2\right)}{x}^{\nu /2-1}{\text{e}}^{-x/2},\nu >0,x>0$

Here, $\nu$ is the degree of freedom, $\text{E}\left(X\right)=\nu ,\text{Var}\left(X\right)=2\nu$. Thus, we say that X follows a Chi-square distribution with $\nu$ degree of freedom ( $X~{\chi }_{\nu }^2$).

Definition 2.6. (F-distribution) A continuous random variable X is said to have an F-distribution with degrees of freedom (df) ${\nu }_1$ and ${\nu }_2$, if its pdf is given by

$f\left(x\right)=\frac{\text{Γ}\left(\frac{{\nu }_1+{\nu }_2}{2}\right){\left(\frac{{\nu }_1}{{\nu }_2}\right)}^{\frac{{\nu }_1}{2}}{x}^{\frac{{\nu }_1}{2}-1}}{\text{Γ}\left(\frac{{\nu }_1}{2}\right)\text{Γ}\left(\frac{{\nu }_2}{2}\right){\left(1+\frac{{\nu }_{1}x}{{\nu }_2}\right)}^{\frac{{\nu }_1+{\nu }_2}{2}}},x\ge 0.$(2.20)

If $X_1~{\chi }_{v_1}^2$ and $X_2~{\chi }_{{\nu }_2}^2$ and are independent, it follows in (2.30) that X is F-distributed with ${\nu }_1$ and ${\nu }_1$ df.

$X=\frac{X_1/{\nu }_1}{X_2/{\nu }_2}~F_{{\nu }_1,{\nu }_2}$(2.21)

Definition 2.7. (Univariate t-distribution) A continuous random variable X is said to have a Univariate t-distribution with degree of freedom df $\nu$, if its pdf is given by

$f_{\nu }\left(x;\mu ,{\sigma }^2\right):=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)\sqrt{\pi \nu }\sigma }{\left\{1+{\left(\frac{x-\mu }{\sigma }\right)}^2\frac{1}{\nu }\right\}}^{-\frac{\nu +1}{2}},\nu \ge 1$(2.22)

$\text{E}\left(X\right)=\mu \text{and}\text{Var}\left(X\right)=\frac{\nu }{\nu -2}{\sigma }^2.$

If $X_1~N\left(0,1\right)$ and $X_2~{\chi }_n^2$ and are independent, it can be shown in (2.32) that T has a t-distribution with $\nu$ df.

$T=\frac{X_1}{\sqrt{\frac{X_2}{\nu }}}~t_{\nu }.$(2.23)

2.8. T-Test

Definition 2.8. (t-test) We define the t-test procedure for our model (2.2) as follows, since in a t-test, the test statistic is computed for each ${\beta }_j$, see [20] .

Hypotheses:

$H_0:{\beta }_j=0\text{versus}{H}_1:{\beta }_j\ne 0$

Test statistic:

$T_j=\frac{{\hat{\beta }}_j}{\widehat{se}\left({\hat{\beta }}_j\right)}~t_{n-p},\text{under}{H}_0$(2.24)

Here, $\widehat{se}\left({\hat{\beta }}_j\right):=s\sqrt{{\left({\left(X^{\top }X\right)}^{-1}\right)}_{jj}}$ is the estimated standard error of ${\hat{\beta }}_j$ and $s=\sqrt{s^2}$ defined in Equation (2.22)

Rejection Rule: Reject $H_0$ at level $\alpha$, if $\left|T_j\right|>t_{n-p,1-\alpha /2}$

2.9. Analysis of Variance (ANOVA)

Definition 2.9. ANOVA is mostly used to summarize the hypothesis tests results in linear models in a tabular form. Given two models $M_{\text{reduced}}$ and $M_{\text{full}}$ which are nested: $M_{\text{reduced}}\subset M_{\text{full}}$, that is, all covariates of the reduced model are contained in the full model, we define the ANOVA-test ratio for the comparison of $M_{\text{reduced}}$ and $M_{\text{full}}$ as follows

$F=\frac{\left({\text{SSE}}_{\text{reduced}}-{\text{SSE}}_{\text{full}}\right)/\left(n-p_{\text{full}}\right)}{{\text{SSE}}_{\text{reduced}}/\left(p_{\text{full}}-p_{\text{reduced}}\right)}\sim F_{n-p_{\text{full}},p_{\text{full}}-p_{\text{reduced}}}$(2.25)

Hypotheses

$H_0:{\beta }_i=0\text{versus}{H}_1:{\beta }_i\ne 0$

Test statistic: $F$, defined in Equation (2.25)

Rejection Rule: Reject $H_0$ at level $\alpha$, if $F>F_{\left(1-\alpha \right),n-p_{\text{full}},p_{\text{full}}-p_{\text{reduced}}}$

2.10. Analysis of Residuals

After estimating the model parameters, the credibility of the assumptions of linearity, normality of errors, and homoscedasticity for the given data can be assessed using residuals. It is therefore important to study the residuals to examine the extent to which our model assumptions may be violated. Hence, investigating the patterns in the residual plots can help us determine if our model assumptions are violated or not. This is referred to as the analysis of residuals. Residual plots can help us decide whether to transform any of the covariates that we may want to include in the model or not.

2.11. Statistical Checks for the Plausibility of the Linear Model Assumptions

The check we are going to use is the residuals versus the fitted values plot. If this plot has no trend, then we assume the linearity assumption as plausible [21] .

We are interested in checking if $\text{Var}\left[Y_i\right]=\text{Var}\left[{\epsilon }_i\right]={\sigma }^2$ holds. To check this, we again used the standardized residual versus the residual plots. If the standardized residuals are not spread equally along the range of the fitted values, then we interpret the homoscedasticity assumption as not plausible, see [22] .

To check if $\text{Cov}\left({\epsilon }_j,{{\epsilon }^{\prime }}_j\right)=\rho =0$ holds, we plot the residuals versus the covariates to see if the residuals are randomly and symmetrically distributed around zero. If this is true, we assume that the independence assumption is plausible [21] .

To check for ${\epsilon }_i~N_n\left(0,{\sigma }^2{I}_n\right)$, we use the Quantile versus Quantile plot (QQPlot). If we do not have a straight line on the QQ plots of our variable versus the theoretical normal quantile, then we assume that the normality assumption is not plausible [23] .

To check for multicollinearity among explanatory variables $X_1,X_2,\cdots ,X_p$, we assess whether there is a strong linear relationship between them, which can inflate the standard errors of the estimated coefficients ${\hat{\beta }}_j$. This is commonly evaluated using the Variance Inflation Factor (VIF), defined as

${\text{VIF}}_j=\frac{1}{1-R_j^2},$

where $R_j^2$ is the coefficient of determination from regressing $X_j$ on the remaining predictors. A ${\text{VIF}}_j>5$ suggests a potentially problematic level of multicollinearity. Variables exceeding this threshold must be examined and removed if necessary to enhance model stability and interpretability [24] [25] .

3. Data Description and Management

The data has both quantitative and qualitative covariates with rent per square meter (rent_sqm) as the response variable. We focus on the most relevant 31 variables such as “the additional cost”, “heat cost”, “construction year”, etc. The quantitative covariates are summarized as follows: Min = Minimum, 25% = 1st quartile, 50% = Median, $\bar{X}$ = Mean, 75% = 3rd quartile, Max = Maximum and Not available = NA. On the other hand, the qualitative covariates are summarized with their respective categories. Note that costs are expressed in EUR and rounded to two decimal digits and the following data summaries in Table 1 and Table 2 represent the whole data set.

Table 1

Table 1. Description of quantitative variables.

Variables	Description
rent_sqm	Calculated rent per sqm by rent and size of apartment. Min = 3, 25% = 7, 50% = 9, $\bar{X}$ = 9.39, 75% = 12, Max = 28
Addcost	The extra monthly costs that need to be paid for other bills on top of the base rent excluding electricity. Min = 0, 25% = 100, 50% = 140, $\bar{X}$ = 153.8, 75% = 196, Max = 599, NA = 97,186
Heatcost	The monthly heating cost. Min = 0, 25% = 50, 50% = 70, $\bar{X}$ = 75.2, 75% = 94, Max = 300, NA = 898,984
Conyear	The year in which the object was built Min = 1851, 25% = 1930, 50% = 1970, $\bar{X}$ = 1964, 75% = 1996, Max = 2020, NA = 447,372
Lmod	The year of the last modernization Min = 1981, 25% = 2009, 50% = 2012, $\bar{X}$ = 2011, 75% = 2015, Max = 2018, NA = 1,113,056
Lspace	Living space in square meters Min = 19, 25% = 53, 50% = 68, $\bar{X}$ = 71.15, 75% = 85, Max = 165
Fspace	The usable floor space in square meters Min = 0, 25% = 16, 50% = 57, $\bar{X}$ = 54.8, 75% = 79, Max = 250, NA = 1,053,922
Energycon	The energy consumption per year and square meter in kWh Min = 0, 25% = 82, 50% = 117, $\bar{X}$ = 120.4, 75% = 152, Max = 350, NA = 977,343
Adlength	The difference between edat and adat. Min = 0, 25% = 0, 50% = 0, $\bar{X}$ = 0.71, 75% = 1, Max = 20

Table 2

Table 2. Description of qualitative variables.

Variables	Description
afloor	Apartment-specific variable indicates the floor the apartment is located on. afloorg is used to group afloor as follows: (−1) - 0, 1 - 2, 3 - 9, >9, NA
bfloor	This indicates the number of floors in the building. bfloorg is used to group bfloor as follows: 0 - 2, 3, 4, 5, >5, NA
nrooms	Number of rooms, excluding kitchen, bath or corridors. nroomsg is used to group nrooms as follows: 1 - 1.5, 2 - 2.5, 3 - 3.5, >3.5, NA
nbed	Number of bedrooms of the property. nbedg is used to group nbed as follows: 0 - 1, 2, >2, NA
nbath	Number of bathrooms in the property nbathg is used to group nbath as follows: 0 - 1, >1, NA
elevator	This variable indicates if a property has an elevator. elevatorg is used to group elevators as follows: Yes, No, NA
balcony	This variable indicates the presence of a balcony. balconyg is used to group balcony as follows: Yes, No, NA
kitchen	This variable indicates the presence of a fitted kitchen. kitcheng is used to group kitchen as follows: Yes, No, NA
eww	If the warm water consumption was included in the energy consumption value calculation. ewwg variable is used to group eww as follows: Yes, No, NA
subh	It indicates whether a certificate of eligibility to public housing is needed to rent the apartment. subhg is used to group subh as follows: Yes, No, NA
gtoilet	This indicates the presence of a guest toilet. gtoiletg is used to group gtoilet as follows: Yes, No, NA
garden	This indicates the presence of a garden. gardeng is used to group garden as follows: Yes, No, NA
hww	If the warm water consumption was included in the heating cost value calculation. hwwg is used to group hww as follows: Yes, No, NA
cellar	This indicates whether a property has a cellar room cellarg is used to group cellar as follows: Yes, No, NA
parking	This variable indicates whether a parking space is available. parking is used to group parking as follows:: Yes, No, NA
furnishing	This is an artificial category number indicating the property’s facilities. furnishingg is used to group furnishing as follows: (Upscale, Luxury) = Upscale, (Normal, Simple) = Normal, no specification = NA
eeff	This indicates the energy efficiency rating. eeffg is used to group eeff as follows: (A, APLUS, B) = High, (C, D, E) = Medium, (F, G, H) = Low, no specification = NA
ecert	The type of energy performance certificate that the customer has for the object ecertg is used to group ecert as follows: Final energy demand = building, Energy consumption characteristic = consumption, NA
pets	This indicates whether pets are allowed in the property. petsg is used to group pets as follows: (Yes, by Agreement) = Yes, No = No, no specification = NA
heat	This indicates the type of heating. heatg is used to group heat as follows: Central Heating (CH), Non Central Heating (NCH), NA
apcat	This variable categorizes the property into different classes. apcatg is used to group apcat as follows: (Penthouse, Maisonette, Attic Apartment) = top, Apartment = middle, (Mezzanine, Terrace apartment) = low, Basement = below, NA
pcon	This indicates the condition of a property. pcong is used to group pcon as follows: (First occupancy, First occupancy after renovation) = First, (Maintained, as good as new) = Mt, In need of renovation = Inr, (Modernized, Renovated, Fully Renovated) = Md, NA

3.1. Data Sets

We split the date set described in Table 1 and Table 2 into two sub-data sets: Munich 2015 and Munich 2019. The number of rental properties contained in each data set is given in Table 3 . The summaries of the response variable and the quantitative covariates are given in Table 4 while in Table 5 , we give the summary of each qualitative variable followed by their percentages.

Table 3

Table 3. Number of rental properties in the two data sets.

City	2015	2019
Munich	14,449	14,776

Table 4

Table 4. Univariate data summaries of quantitative covariates: first row = Munich 2015, second row = Munich 2019.

Variable	Summary
	Min	25%	50%	Mean	75%	Max	NA
rent_sqm 2015	3.00	12.00	13.00	12.91	15.00	17.00	0
rent_sqm 2019	4.00	16.00	18.00	18.32	21.00	28.00	0
addcost
	0.00	107.00	153.00	164.04	210.00	540.00	1355
	0.00	120.00	170.00	175.47	220.00	550.00	533
heatcost
	0.00	60.00	85.00	89.35	110.00	288.00	10,075
	0.00	55.00	80.00	84.84	109.00	300.00	11,155
conyear
	1860	1962	1976	1976	1999	2017	3522
	1858	1965	1985	1982	2014	2020	3068
lmod
	1981	2011	2014	2012	2015	2016	9313
	1983	2013	2015	2014	2017	2018	11,386
lspace
	23.00	55.00	71.00	73.79	90.00	161.00	0
	19.00	51.00	67.00	68.54	84.00	157.00	0
fspace
	0.00	10.00	55.00	53.40	81.00	234.00	9276
	0.00	11.00	55.00	53.14	82.00	249.00	11,483
energycon
	0.00	85.00	122.00	122.53	155.00	338.00	5975
	0.00	64.00	103.00	104.11	137.00	339.00	7379
adlength
	0.00	0.00	0.00	0.58	1.00	20.00	0
	0.00	0.00	0.00	0.53	1.00	20.00	0

Table 5

Table 5. Univariate data summaries of qualitative covariates: first row = Munich 2015, second row = Munich 2019.

Variable	Categories
afloorg	(−1) - 0	1 - 2	3 - 9	>9	NA
	1762 (0.12%)	6732 (0.47%)	3848 (0.27%)	63 (0%)	2044 (0.14%)
	1648 (0.11%)	6428 (0.44%)	4687 (0.32%)	97 (0.01%)	1916 (0.13%)
bfloorg	0 - 2	3	4	5	>5	NA
	2744 (0.19%)	2226 (0.15%)	2770 (0.19%)	1833 (0.13%)	1418 (0.1%)	3458 (0.24%)
	2741 (0.19%)	2187 (0.15%)	2517 (0.17%)	2160 (0.15%)	1950 (0.13%)	3221 (0.22%)
nroomsg	1 - 1.5	2 - 2.5	3 - 3.5	>3.5
	2157 (0.15%)	5710 (0.4%)	4841 (0.34%)	1741 (0.12%)
	2836 (0.19%)	5949 (0.4%)	4768 (0.32%)	1223 (0.08%)
nbedg	0 - 1	2	>2	NA
	5636 (0.39%)	3562 (0.25%)	1240 (0.09%)	4011 (0.28%)
	3884 (0.26%)	2329 (0.16%)	684 (0.05%)	7879 (0.53%)
nbathg	0 - 1	>1	NA
	10,690 (0.74%)	1669 (0.12%)	2090 (0.14%)
	11,310 (0.77%)	1657 (0.11%)	1809 (0.12%)
elevatorg	Yes	No	NA
	6125 (0.42%)	8108 (0.56%)	216 (0.01%)
	7929 (0.54%)	6847 (0.46%)	0 (0%)
balconyg	Yes	No	NA
	10,863 (0.75%)	3406 (0.24%)	180 (0.01%)
	11,554 (0.78%)	3222 (0.22%)	0 (0%)
kitcheng	Yes	No	NA
	8756 (0.61%)	5438 (0.38%)	255 (0.02%)
	9878 (0.67%)	4898 (0.33%)	0 (0%)
ewwg	Yes	No	NA
	3775 (0.26%)	10,454 (0.72%)	220 (0.02%)
	1419 (0.1%)	723 (0.05%)	12,634 (0.86%)
subhg	Yes	No	NA
	30 (0.00%)	12,534 (0.87%)	1885 (0.13%)
	162 (0.01%)	14,614 (0.99%)	0 (0%)
gtoiletg	Yes	No	NA
	3186 (0.22%)	11,254 (0.78%)	9 (0.00%)
	2948 (0.20%)	11,828 (0.80%)	0 (0%)
gardeng	Yes	No	NA
	2726 (0.19%)	11,173 (0.77%)	550 (0.04%)
	3074 (0.21%)	11,702 (0.79%)	0 (0%)
hwwg	Yes	No	NA
	8856 (0.61%)	4320 (0.3%)	1273 (0.09%)
	10,161 (0.69%)	4088 (0.28%)	527 (0.04%)
cellarg	Yes	No	NA
	11,315 (0.78%)	3036 (0.21%)	98 (0.01%)
	11,533 (0.78%)	3243 (0.22%)	0 (0%)
parkingg	Yes	No	NA
	59 (0.00%)	0 (0%)	14,390 (1.00%)
	7911 (0.54%)	228 (0.02%)	6637 (0.45%)
furnishingg	Upscale	Normal	NA
	5699 (0.39%)	3591 (0.25%)	5159 (0.36%)
	7156 (0.48%)	2726 (0.18%)	4894 (0.33%)
eeffg	High	Medium	Low	NA
	314 (0.02%)	257 (0.02%)	63 (0%)	13,815 (0.96%)
	474 (0.03%)	388 (0.03%)	50 (0%)	13,864 (0.94%)
ecertg	building	consumption	NA
	2898 (0.20%)	6027 (0.42%)	5524 (0.38%)
	3228 (0.22%)	4393 (0.30%)	7155 (0.48%)
petsg	Yes	No	NA
	947 (0.07%)	4123 (0.29%)	9379 (0.65%)
	3460 (0.23%)	5629 (0.38%)	5687 (0.38%)
heatg	CH	NCH	NA
	8056 (0.56%)	3744 (0.26%)	2649 (0.18%)
	6589 (0.45%)	5560 (0.38%)	2627 (0.18%)
apcatg	top	middle	low	below	NA
	2011 (0.14%)	7627 (0.53%)	515 (0.04%)	80 (0.01%)	4216 (0.29%)
	2066 (0.14%)	7977 (0.54%)	1158 (0.08%)	130 (0.01%)	3445 (0.23%)
pcong	First	Mt	Md	Inr	NA
	1781 (0.12%)	5525 (0.38%)	3280 (0.23%)	17 (0%)	3846 (0.27%)
	2682 (0.18%)	5537 (0.37%)	3175 (0.21%)	11 (0%)	3371 (0.23%)

3.2. Exploratory Data Analysis (EDA)

See Figures 1-3 .

Figure 1

Figure 1. Histograms of response variable—rent_sqm: first column = counts, second column = percentage.--Figure 1. Histograms of response variable—rent_sqm: first column = counts, second column = percentage.

Figure 1

Figure 1. Histograms of response variable—rent_sqm: first column = counts, second column = percentage.--Figure 1. Histograms of response variable—rent_sqm: first column = counts, second column = percentage.

Figure 2

Figure 2. Scatter plots of quantitative covariates versus response (rent_sqm) with Linear Smooth (LS) and Non Linear Smooth (NLS): first column = (rent_sqm) and second column = log(rent_sqm). (first row) = Munich 2015 with LS, (second row) = Munich 2019 with LS, (third row) = Munich 2015 with NLS, (fourth row) = Munich 2019 with NLS.

Figure 2

Figure 2. Scatter plots of quantitative covariates versus response (rent_sqm) with Linear Smooth (LS) and Non Linear Smooth (NLS): first column = (rent_sqm) and second column = log(rent_sqm). (first row) = Munich 2015 with LS, (second row) = Munich 2019 with LS, (third row) = Munich 2015 with NLS, (fourth row) = Munich 2019 with NLS.

Figure 2

Figure 2. Scatter plots of quantitative covariates versus response (rent_sqm) with Linear Smooth (LS) and Non Linear Smooth (NLS): first column = (rent_sqm) and second column = log(rent_sqm). (first row) = Munich 2015 with LS, (second row) = Munich 2019 with LS, (third row) = Munich 2015 with NLS, (fourth row) = Munich 2019 with NLS.

Figure 3

Figure 3. Box plots of qualitative covariates versus response (rent_sqm): first column = Munich 2015, second column = Munich 2019.--Figure 3. Box plots of qualitative covariates versus response (rent_sqm): first column = Munich 2015, second column = Munich 2019.

Figure 3

Figure 3. Box plots of qualitative covariates versus response (rent_sqm): first column = Munich 2015, second column = Munich 2019.--Figure 3. Box plots of qualitative covariates versus response (rent_sqm): first column = Munich 2015, second column = Munich 2019.

3.3. Interpretation of Main Effects for the Quantitative and Qualitative Covariates

Looking at the above transformations on rent_sqm in Table 6 and Table 7 , we may likely go with the log transformation for linear and non-linear covariates based on its suitability with respect to constant variance discussed in Section 2 and the effects of the covariates on rent_sqm.

Table 6

Table 6. Interpretation of main effects for the quantitative covariates on rent_sqm and log(rent_sqm) in Figure 2 : first block = Linear smooth, second block = Nonlinear smooth.

Variables	Munich 2015 (rent_sqm)	Munich 2019 (rent_sqm)	Munich 2015 (log(rent_sqm))	Munich 2019 (log(rent_sqm))
Addcost	Linear (increasing)	Nearly constant	Linear (increasing)	Nearly constant
Heatcost	Nearly constant	Linear (decreasing)	Nearly constant	Linear (decreasing)
Conyear	Constant	Constant	Constant	Constant
Lmod	Nearly constant	Linear (increasing)	Linear (decreasing)	Linear (increasing)
Lspace	Linear (decreasing)	Linear (decreasing)	Nearly constant	Linear (decreasing)
Fspace	Nearly constant	Nearly constant	Nearly constant	Constant
Energycon	Nearly constant	Linear (decreasing)	Nearly constant	Nearly constant
Adlength	Linear (increasing)	Linear (increasing)	Linear (increasing)	Nearly constant
Addcost	Quadratic	Quadratic	Quadratic	Quadratic
Heatcost	Quadratic	Quadratic	Nearly linear	Nearly linear
Conyear	Cubic	Quadratic	Quadratic	Nearly linear
Lmod	Nearly linear	Nearly constant	Nearly linear	Nearly constant
lspace	Cubic	Nearly linear	Cubic	Linear (decreasing)
fspace	Cubic	Quadratic	Cubic	Quadratic
Energycon	Quadratic	Quadratic	Nearly constant	Quadratic
Adlength	Quadratic	Nearly constant	Quadratic	Constant

Table 7

Table 7. Interpretation of main effects for the qualitative covariates on rent_sqm in Munich 2015 and Munich 2019 in Figure 3 .

Variables	Munich 2015	Munich 2019
afloorg	No	Yes
bfloorg	Yes	Yes
nroomsg	Yes	Yes
nbedg	No	Yes
nbathg	No	No
elevatorg	Yes	Yes
balconyg	No	No
kitcheng	Yes	Yes
ewwg	No	No
subhg	Yes	Yes
gtoiletg	No	No
gardeng	No	No
hwwg	No	Yes
cellarg	Yes	Yes
parkingg	No	Yes
furnishingg	Yes	Yes
eeffgg	Yes	Yes
ecertg	No	No
petsg	Yes	No
heatg	Yes	Yes
apcatg	No	No
pcong	Yes	No

4. Model Fittings and Predictions

We discuss how we select the type of model we use to fit the rent_sqm for Munich rental properties in 2015 and 2019. To refine the regression model for the rent per square meter in Munich, a stepwise backward regression was applied using the step() function in R. This method began with a full model containing all relevant predictors and iteratively removed nonsignificant variables based on the Akaike Information Criterion (AIC). The backward selection process ensured a more parsimonious model by retaining only the most influential variables, enhancing interpretability while maintaining predictive strength and minimizing model complexity. We first fit four models for the response variable in Munich 2015 in the following cases:

We also do similar model fitting (the 4 cases) for Munich 2019. The summaries are found in Table 8 .

Table 8

Table 8. Model fitting summary with only main effect.

Munich 2015	Case 1	Case 2	Case 3	Case 4
Adjusted R-square	0.2762	0.2652	0.3101	0.2879
Number of parameters (p)	38	38	39	33
Munich 2019
Adjusted R-square	0.5139	0.5145	0.3078	0.5468
Number of parameters (p)	25	22	41	27

Looking at the model fitting summary in Table 8 , we decided to go with case 4, which is the log transformation on rent_sqm (log(rent_sqm)) for the non-linear covariates as it relatively satisfied most of the listed assumptions with a larger R-square, compared to the others in the four data sets.

4.1. Model Fitting of Log(Rent_Sqm) on Non-Linear Covariates for Munich 2015 and Munich 2019

See Table 9 and Table 10 .

Table 9

Table 9. Munich 2015.

	Estimate	Std. Error	t value	Pr (>|t|)
(Intercept)	2.5838	0.0740	34.90	0.0000
poly (conyear, 2) 1	−0.5052	0.1259	−4.01	0.0001
poly (conyear, 2) 2	0.5825	0.1307	4.46	0.0000
poly (lspace, 3) 1	−1.4841	0.2413	−6.15	0.0000
poly (lspace, 3) 2	0.2820	0.1631	1.73	0.0842
poly (lspace, 3) 3	−0.3785	0.1423	−2.66	0.0080
adlength	0.0066	0.0030	2.19	0.0290
nroomsg 1 - 1.5	−0.0973	0.0317	−3.06	0.0023
nroomsg 2 - 2.5	−0.0877	0.0227	−3.87	0.0001
nroomsg 3 - 3.5	−0.0542	0.0182	−2.97	0.0031
nbedg 0 - 1	0.0638	0.0211	3.02	0.0026
nbedg 2	0.0478	0.0197	2.42	0.0157
nbedgNA	0.0977	0.0279	3.50	0.0005
elevatorgYes	0.0378	0.0096	3.93	0.0001
kitchengNo	−0.0606	0.0383	−1.58	0.1141
kitchengYes	−0.0272	0.0395	−0.69	0.4909
ewwgNo	−0.0808	0.0449	−1.80	0.0721
ewwgYes	−0.0976	0.0450	−2.17	0.0304
subhgNo	0.0422	0.0219	1.93	0.0545
gtoiletgYes	0.0268	0.0138	1.94	0.0528
hwwgYes	0.0205	0.0104	1.98	0.0483
furnishinggNormal	−0.0034	0.0185	−0.18	0.8562
furnishinggUpscale	0.0768	0.0182	4.23	0.0000
eeffgLow	0.1780	0.0658	2.70	0.0070
eeffgMedium	0.1156	0.0478	2.42	0.0158
eeffgNA	0.0725	0.0447	1.62	0.1055
petsgNo	−0.0098	0.0103	−0.95	0.3441
petsgYes	−0.0651	0.0316	−2.06	0.0399
heatgNA	0.0706	0.0213	3.31	0.0010
heatgNCH	−0.0052	0.0121	−0.43	0.6675
pcongInr	0.0700	0.1188	0.59	0.5561
pcongMd	−0.0529	0.0156	−3.40	0.0007
pcongMt	−0.0530	0.0161	−3.30	0.0010
pcongNA	0.0270	0.0262	1.03	0.3031
Observations	711
R2	0.321
Adj. R2	0.288
Residual Std. Error	0.116 (df = 677)
F Statistic	9.698*** (df = 33; 677)
p-value	<2.2e−16

*p < 0.1; **p < 0.05; ***p < 0.01.

Table 10

Table 10. Munich 2019.

	Estimate	Std. Error	t value	Pr (>|t|)
(Intercept)	−8.9984	4.7335	−1.90	0.0586
heatcost	0.0004	0.0003	1.47	0.1425
lmod	0.0060	0.0023	2.54	0.0117
poly (lspace, 2) 1	−1.2984	0.2104	−6.17	0.0000
poly (lspace, 2) 2	0.4809	0.1481	3.25	0.0013
poly (fspace, 2) 1	−0.1854	0.1688	−1.10	0.2733
poly (fspace, 2) 2	−0.4037	0.1690	−2.39	0.0178
poly (energycon, 2) 1	0.0216	0.1557	0.14	0.8901
poly (energycon, 2) 2	0.4404	0.1571	2.80	0.0055
bfloorg 0 - 2	−0.0579	0.0340	−1.70	0.0903
bfloorg 3	−0.0029	0.0335	−0.09	0.9322
bfloorg 4	−0.0181	0.0314	−0.58	0.5648
bfloorg 5	0.0491	0.0320	1.53	0.1266
bfloorgNA	−0.0580	0.1068	−0.54	0.5877
kitchengYes	0.0822	0.0230	3.57	0.0004
hwwgYes	0.0551	0.0216	2.55	0.0116
parkinggNo	−0.1027	0.0561	−1.83	0.0686
parkinggYes	−0.0336	0.0198	−1.69	0.0918
furnishinggNormal	−0.0400	0.0398	−1.01	0.3159
furnishinggUpscale	0.1132	0.0385	2.94	0.0036
ecertgconsumption	−0.0471	0.0214	−2.20	0.0288
apcatglow	−0.1004	0.1636	−0.61	0.5400
apcatgmiddle	−0.1517	0.1564	−0.97	0.3332
apcatgNA	−0.2236	0.1587	−1.41	0.1604
apcatgtop	−0.0951	0.1565	−0.61	0.5441
pcongMd	−0.1170	0.0445	−2.63	0.0091
pcongMt	−0.1001	0.0460	−2.18	0.0305
pcongNA	−0.0194	0.0576	−0.34	0.7362
Observations	244
R2	0.597
Adj. R2	0.547
Residual Std. Error	0.137 (df = 216)
F Statistic	11.859*** (df = 27; 216)
p-value	<2.2e−16

p < 0.1; ** p < 0.05; *** p < 0.01.

4.2. Residual Plots of Model Fittings

We plot the residuals versus the fitted values to see if there is a trend to check for the plausibility of the linearity assumption discussed in Section 2. Also, we plot the QQ plots of the covariates versus the theoretical normal quantile to see if it is a straight line to check for the plausibility of the normality assumption, which was discussed in Section 2.

From the plots in Table 11 , we find that the fitted models do not relatively violate the linear regression assumptions in Section 2.19.

Table 11

Table 11. Residual plots of model fittings for Munich 2015 and Munich 2019.

city	Munich 2015	Munich 2019

4.3. Model Predictions of Rent_Sqm for the Main Effect Models

In this section, we will predict the values of rent_sqm for the main effect models given in Table 9 and Table 10 using the most influential variables from the pairwise selection as shown in Table 12 and Table 13 . We will use the median of the continuous covariates and the mode of the qualitative covariates for our prediction. We consider the mode for the qualitative covariates and the median for the remaining continuous variables while we take 50 values between the 5th and 95th quantile/percentile of the variable we are plotting. We also consider the different categories of each qualitative covariate which we are using for the prediction of rent_sqm while other qualitative covariates remain in their mode and the continuous covariates in their medians respectively. We also computed the Variance Inflation Factor (VIF) for all predictors. The GVIF and adjusted GVIF^{1/(2 ∙ Df)} values were all below 2, as shown in Table 14 , indicating no significant multicollinearity issues. This implies that the predictors were sufficiently independent of each other. Thus, no predictor variables were removed on this basis, and the model structure remains statistically robust.

5. Findings

5.1. Summary of Findings

In Figure 1 , there is a significant shift in the histogram plots of rent_sqm for Munich 2015 and Munich 2019. For instance, in Munich 2015, we can see that the rent_sqm is below 20 Euros, but in 2019, the rent_sqm is over 20 Euros. This shows that the rent price increases with time, which is also confirmed in our prediction. For instance, the predicted rent_sqm increased in Munich from 2015 to 2019 by 31.17%, 31.17%, and 39.86% with apartments that have a kitchen, Upscale furnishing, and First occupancy condition.

Table 12

Table 12. Model predictions of rent_sqm for the influential quantitative covariates.

Munich 2015 prediction plots	Munich 2019 prediction plots

Table 13

Table 13. Munich 2015.

Variables	categories	Munich 2015	Munich 2019
afloorg	(−1) - 0
	1 - 2
	3 - 9
	>9
	NA
bfloorg	0 - 2		16.50
	3		17.44
	4		17.17 (mode = 4)
	5		18.37
	>5		17.49
	NA		16.50
nroomsg	1 - 1.5	12.96
	2 - 2.5	13.09 (mode = 2 - 2.5)
	3 - 3.5	13.53
	>3.5	14.29
nbedg	0 - 1	13.09 (mode = 0 - 1)
	2	12.88
	>2	12.28
	NA	13.54
nbathg	0 - 1
	>1
	NA
elevatorg	Yes	13.59
	No	13.09 (mode = No)
	NA
balconyg
kitcheng	Yes	13.09 (mode = Yes)	17.17 (mode = Yes)
	No	12.66	15.82
	NA	13.45
ewwg	Yes	12.87
	No	13.09 (mode = No)
	NA	14.19
subhg	Yes
	No	13.09 (mode = No)
	NA	12.55
gtoiletg	Yes	13.44
	No	13.09 (mode = No)
gardeng	Yes
	No
	NA
hwwg	Yes	13.36	18.14
	No	13.09 (mode = No)	17.17 (mode = No)
cellarg	Yes
	No
parkingg	Yes		17.17 (mode = Yes)
	No		16.02
	NA		17.77
furnishingg	Upscale	13.09 (mode = Upscale)	17.17 (mode = Upscale)
	Normal	12.08	14.73
	NA	12.12	15.33
eeffgg	High	12.17
	Meduim	13.66
	Low	14.54
	NA	13.087 (mode = NA)
ecertg	consumption		17.17 (mode = consumption)
	building		18.00
petsg	Yes	12.26
	No	12.96
	NA	13.09 (mode = NA)
heatg	CH	13.09 (mode = CH)
	NCH	13.02
	NA	14.04
apcatg	top		18.17
	middle		17.17 (mode = middle)
	low		18.08
	below		19.99
	NA		15.98
pcong	Md	13.09 (mode = Md)	17.17 (mode = Md)
	Mt	13.09	17.46
	First	13.80	19.30
	Inr	14.80
	NA	14.18	18.93

Table 14

Table 14. VIF Munich 2019.

	GVIF	Df	GVIF^(1/(2 * Df))
heatcost	1.68	1.00	1.30
lmod	1.37	1.00	1.17
poly (lspace, 2)	2.76	2.00	1.29
poly (fspace, 2)	2.25	2.00	1.23
poly (energycon, 2)	1.69	2.00	1.14
bfloorg	2.18	5.00	1.08
kitcheng	1.41	1.00	1.19
hwwg	1.25	1.00	1.12
parkingg	1.39	2.00	1.09
furnishingg	1.50	2.00	1.11
ecertg	1.24	1.00	1.11
apcatg	2.03	4.00	1.09
pcong	1.81	3.00	1.10

From Table 12 , we can summarise the behaviour of the predicted rent_sqm for the influential quantitative covariates as follows:

In Table 13 , we can summarise the behaviour of the predicted rent_sqm for the influential qualitative covariates as follows:

5.2. Discussion on Research Questions

The analysis indicates significant relationships between rent per square meter and various predictors. In both the Munich 2015 and 2019 datasets, all examined variables influenced rent per sqm. For instance, in Munich 2015, the advertisement length showed a linear and increasing relationship with rent per sqm, while construction year and living space exhibited nonlinear relationships. Similarly, in Munich 2019, heat cost and last modernization had linear increasing trends, whereas other variables, including living space, displayed nonlinear associations.

These findings align with broader market trends. According to JLL’s Housing Market Overview for H2 2023, Munich remains Germany’s most expensive housing market, with asking rents rising by 5.1% year-on-year to €22.50/sqm per month. This suggests that various factors, including those studied, contribute to rent variations [26] .

A log transformation was applied to the rent per sqm variable to address potential non-linear relationships and meet linear regression assumptions. This transformation improved the model’s fit, as evidenced by a higher R-squared value, indicating a better explanation of variance in the response variable. Such transformations are commonly employed in housing market analyses to stabilize variance and normalize distributions [4] . This approach is consistent with standard econometric modeling practices in real estate studies, where log-linear models account for skewness and heteroscedasticity in rent and housing price distributions [3] . This transformation approach also supports previous findings in [27] - [29] .

The study identified several key predictors impacting rental prices:

Also, the study observed that energy efficiency ratings inversely affected rental prices, with higher efficiency ratings correlating with lower rents. This counterintuitive finding suggests that tenants may not fully value energy efficiency in their rental decisions, a phenomenon also noted in previous research [4] . Furthermore, the scatter plot of the continuous variable (energy condition) versus the rent in both the raw and predicted models, as shown in Figure 2 , Table 6 and Table 12 , demonstrates a consistent decreasing trend in both the raw data and model predictions, supporting the validity of this result. This behavior may reflect market dynamics where energy-efficient features are undervalued or not effectively communicated during the rental process.

5.3. Contribution of the Study

The contribution of this study can be summarized in the following themes:

6. Conclusion and Implications

This study examined the dynamics of rental apartment prices per square meter in Munich using a robust statistical framework grounded in multiple linear regression with nonlinear covariates. Drawing from an extensive dataset provided by FDZ Ruhr in collaboration with ImmobilienScout24, the research analyzed over 29,000 rental listings across two critical periods, 2015 and 2019. Our findings revealed influential factors influencing rental price variations, including apartment size, furnishing quality, energy efficiency ratings, and availability of amenities such as elevators and parking spaces.

The analysis identified a consistent inverse relationship between apartment size and rent per square meter, affirming that larger apartments tend to command lower per-unit rents. Upscale furnishings and energy-efficient features were strongly associated with higher rental values, emphasizing the market’s shift toward sustainable and modern living preferences. Applying log transformation and polynomial terms improved the model’s performance and revealed nuanced nonlinear patterns across years, supported by adjusted R-squared values above 0.5 for the 2019 model.

This study provides valuable insights for stakeholders in the real estate, educational leadership, and urban planning sectors, particularly as Munich grapples with housing shortages and rising rent inflation. Policymakers can use these results to identify leverage points for regulating rental markets and implementing incentive structures for energy-efficient housing. Moreover, the educational implications of the modeling approach underscore the importance of integrating data science and urban economics in curriculum development for future housing strategists.

Future research could extend this model across multiple German cities or apply time-series forecasting techniques to predict rent trends beyond 2019. Incorporating spatial econometrics and GIS-based analysis could also enhance understanding of geographic rental disparities within the city. Also, a comparative study of major U.S. and European university cities could be considered for further research to deepen the knowledge of rental price trends and inform global policy practice.

6.1. Recommendations

Based on the findings of this study, the following recommendations are proposed:

Acknowledgements

This paper builds on the Master’s thesis of Ugochukwu Onumadu, supervised by Prof. Ph.D. Claudia Czado, at the Technical University of Munich, Germany. It refines the original findings, explicitly focusing on the Munich rental market.

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