The Influence Of Financing On The Dynamics Of Housing Prices

Abstract Real estate market can be thought of as an open, dynamic system. It means that it is able to exchange stimuli with other open systems, and that its state evolves in a way that might be described mathematically. It turns out that two main processes contribute to the overall evolution of the real estate market: long-term, predictable evolution, interrupted by sharp changes of catastrophic origin. In this picture, national housing funds play an important role in supporting the housing finance: on one hand they could either stimulate or suppress the real estate market influencing the availability of the mortgage credit, but on the other hand, they could also help to stabilize prices. In this study, an attempt was made to determine the degree of relationship between the volume of mortgage financing from national housing funds and the dynamics of real estate prices.


Introduction
Housing market is widely recognized a vital element of a market-driven economy and a factor influencing society's prosperity and satisfaction. According to Foryś 1 , the property market together with its economic and social environment form a system of communicating vessels, in which stagnation in any segment and lacking communication between the segments have detrimental effect on the whole system. Everyone, either directly or indirectly, is related to this market by taking up various activities concerning: buying, owning or renting a housing property. Owning a place to live is one of the fundamental human needs. According to Maslov's hierarchy of needs, housing satisfies basic physical requirements for safety and survival, as well as the needs for social contact, esteem and self-actualization. In addition, the housing fulfills a broad spectrum of the needs of higher order, which implies that the property market, where housing resources are transferred, affects the behavior of individuals as well as the entire society. As such, housing markets interact in a variety of ways with the rest of the economy. The factors responsible for driving house prices include, for example: income, the housing stock, demography, credit availability, interest rates and others. This is the reason, why the housing market is defined as the space, where housing services are allocated by the mechanisms of supply and demand. In Selim 2 opinion, the characteristics of the housing market differs from the goods and services markets in terms of the inelasticity of housing supply. Housing services belong to the most expensive household expenditures. Variations in housing prices attract great attention of individuals and governments due to their influence on the socio-economic conditions, and further on the overall national economic condition. Expectations of capital gains from housing investments affect housing prices by increasing the demand for housing, which in turn causes high volatility in housing prices.
Behaviors on the housing market are subject to change due to agglomerate fluctuations of internal and external determinants, thus leading to changes in the individual elements of the system. These changes, as well as the environment itself, constitute a multidimensional space in which uncertainty occurs. Even the smallest disturbance may evolve with elapsed time into significant change in other elements of the system, completely altering the relations and dependences occurring in the market 3 .
In the light of the above facts, the housing market is composed of elements (objects and entities) connected with different relations (buying, selling, renting). All processes occurring in this set take place in well-defined periods of time, and their dynamics depend on the space-time Unauthentifiziert | Heruntergeladen 12.02. 20 15:27 UTC scale of perturbations in the system's environment. Hence, the housing market can exchange signals with other systems (open system), and its evolution can be described using mathematical equations (dynamical system).
The major goal of this study is to determine the degree of relationship between the volume of mortgage financing from the government's subsidiary program "Family on its own" (in Polish: "Rodzina na swoim") national housing funds and the dynamics of real estate prices.
This research makes use of models originally developed for description of physical phenomena, that is the model of critically-damped harmonic oscillator (CDHO), and the measures of concurrency.
This work presents results of an interdisciplinary research. According to Czaja 4 , physical laws form a solid base for other scientific disciplines. In recent years, a number of papers has been published, which contribute to the so-called econophysics, for example: Mantegna and

Financing resources on the real estate market and the variability of real estate prices
Mortgage-subsidy programs, through which the government pays some percent of the interest on qualifying mortgages, contribute to popular schemes of financing of the housing market. Such programs are meant to enhance customer demand in the residential sector.
Announced in 2007, the government's subsidiary program "Family on its own" aimed to assist young Poles in the purchase of their first housing property. Relationships between housing prices, and the volume of the government's subsidy (including the program "Family on its own") were deeply analyzed by various authors, among others: Foryś and Batóg 11 , Gołąbeska 12 , Kokot 13 ,Kubów 14 ,Majorek 15 ,Mastalerz 16 ,Wodniak 17 .
In this study, an attempt is made towards an explanation of whether and how the discussed program affects the dynamics of housing prices on local real estate markets. Three cities,  where it was referred to as discontinuous transition 18 , phase transition 19 , or simply catastrophe 20 .
Beyond this point the housing prices asymptotically relax to the equilibrium level depending on their volatility. In general, increasing numbers of loans in local housing markets go hand in hand with decreasing housing prices.

Delay analysis of time series of housing prices in local markets
Time series of the housing prices on local real estate markets can be successfully described using the equation of critically damped harmonic oscillator (CDHO), which serves as an example of simple dynamic system in physics. This model approximates a large variety of systems that vibrate or oscillate, such as a mass attached to a spring, a pendulum, or an electric current in an impedance circuit. The analogy between housing market and harmonic oscillator is drawn from observation that a steep rise in housing prices (housing bubble) is in fact a transient disturbance to otherwise stable price evolution, and is followed by an asymptotic return to a long-term where: t -denotes time, t 0 -absolute time delay, τ -relaxation time, y EQ -long-term equilibrium level, e -Euler's number (e ≈ 2.72), A and B -model constants depending on the initial conditions. These parameters are directly related to a time-dependent evolution of the studied system. Incorporated from physical sciences, CDHO model allows us to derive useful data on the behavior of the housing market using average housing prices.
In general, absolute time delay can be thought of as a difference between arbitrary initial time and the onset of a system evolution. The delay is necessarily introduced in the model, Relaxation time τ establishes a period of time required for an observable to decrease to 1/e (0.368) of its initial value. The higher τ value, the slower system return to the equilibrium state. On the other hand, reciprocal relaxation time is associated with the system inertia and is responsible for its sensitivity to fluctuations in independent variables. In case of the housing market, relaxation time describes specific dynamics of time series of local housing prices. The markets characterized by lower τ values are found to be more sensitive to external stimuli, that is they exhibit larger price variations due to higher volatility. In turn, long-term equilibrium level y EQ defines the asymptotic limit, to which the housing prices are approaching in a series of damped oscillations.
In Figure 2 results of a curve fitting procedure of CDHO model are plotted together with housing prices in local markets: Warsaw (WAW), Gdansk (GDN) and Szczecin (SZZ). The details of the model parameters, such as: the coefficient of determination R 2 , absolute as well as specific time delays, t 0 , and t spec , respectively, and long-term equilibrium level y EQ are all specified in Table 1.

Concurrence analysis of housing prices and price limits for "Family on its own"
Another issue addressed in the work concerns analysis of whether or not the time series of housing prices, and price limits set by the government's program "Family on its own" are concurrent. Through this mortgage-subsidy program for first-time home-buyers, about 50 percent of the interest on qualifying mortgages are co-financed over the first eight years. The price limits determine, which housing properties are available for the program. Detailed methodology for constructing price limits in local housing markets is described elsewhere 21 . Figure 3 shows the time-dependent plots of transaction prices in local housing markets compared with price limits set for the mortgage-subsidy program "Family on its own". At first, large time shifts are seen between corresponding series, which suggest that the price limits are significantly delayed with respect to the housing prices. Such delays exhibit institutional inertia of the government in decision-making process, which might have detrimental influence on the overall housing market. Figure 3 shows that the government gradually increased the price limits in the program until the middle of 2011, probably because it faced increasing pressure from developers.
Unfortunately, this process failed to meet the most important events in the markets in terms of both steep rises, and subsequent relaxation of the housing prices. In addition, it is interesting to note that right after the program was closed at the end of 2012, observed housing prices started to go up simultaneously. It is therefore suggested that the price limits establish "the glass ceiling", i.e. hardly visible barrier that keeps prices from rising to the upper levels.
To study to what degree the time series are concurrent, the following normalized correlation coefficient W 12 is proposed: where: t 0 -is the time shift between time series (y 1 taken as the reference), σ 2 -normalizing constant (variance), while <> denotes arithmetic mean. This coefficient is computed using circular correlation algorithm that makes use of periodic extension of the sequence followed by its circular wrapping. If the W 12 coefficient approaches its maximum value, then the two series completely overlap, and are highly proportional (change in the same direction), otherwise such a linear dependency vanishes.
However, some attention must be paid when asymmetric sequences are taken into consideration, as in the studied problem. At its maximum, the correlation coefficient does not necessarily imply the best coincidence in characteristic features of the series (steep rises and deep falls). Hence, we propose to modify a standard measure of a delay between concurrent sequences and suggest using the difference between time shifts corresponding to W 12max and W 12min instead of that corresponding to W 12max alone.
Analysis of concurrency, despite difficulties in accurate evaluation of asymmetric sequences, exhibits significant delays of the price limits for the subsidiary program in reference to the housing prices equal to, respectively: around 4 years (WAW -17 quarters), 3 years (GDN -12 quarters), and about 2 years (SZZ -7 quarters). Such large delays imply that corresponding time series are hardly concurrent, which is shown in Figure 5. Most importantly, the limits itself are found to influence the prices of new housing properties offered by developers, and hence these time series appear no longer independent.   5. Plots of differences between price limits set in the program "Family on its own" and transaction prices in local housing markets Source: own study. Figure 5 shows that in the beginning the price limits were largely underestimated with respect to the housing prices, then, after few modifications made to the program, they became overestimated, while at the end of the program, the limits were deeply reduced falling below the housing prices. Beyond that point sudden rise in the housing prices occurred, leaving behind long-term levels of equilibrium prices. In our opinion, the price limits for the subsidiary program turn out to be a key factor in a decision-making process of calculating of the housing prices by developers in the primary market. After its expiry in 2012, "the glass ceiling" disappeared, and the prices started to change. Such a suggestion is supported by lacking observation of similar increase in housing prices in the secondary market, excluded from the program.

Conclusions
This paper presents results of interdisciplinary research, which explores the benefits of applying the methods originally derived in Physics, and Mathematics into Economic Science.
Obtained results shed new light on the evolution of the housing markets, regardless of classic analytical methods used so far.
According to our findings, housing markets turn out to have higher inertia than other economic systems, i.e. they are slow to change. Due to exogenous factors (aggregate macroeconomic fluctuations), housing prices increase by similar amounts after specific time delay in reference to dominant market in Warsaw. In turn, endogenous factors differentiate between relaxation time-constants and long-term equilibrium levels of the prices. As a result, the overall housing market can be thought of as a large system of communicating vessels in the form of a network, in which local housing markets are represented by critically damped harmonic oscillators, coupled on one end to external stimuli and to their neighbors on the other.
In accordance with physical laws obtained results suggest that the housing markets best reveal their inertial properties when they change suddenly. In short intervals of steep rises or sudden falls in housing prices, the markets are subjected to deep changes of structural origin, whereas otherwise quasi-stationary long-term evolution keeps the markets in equilibrium.
On the other hand, analysis of time delays between sequences of housing prices in local markets leads to the conclusion that the market in Warsaw is in a dominant position, and plays a role of a trigger of sudden changes in the model depicted above. Because local housing markets are coupled to each other, they evolve in a concurrent manner. Apart from that, however, the price limits for the program are found to be significantly delayed with respect to housing prices due to