# American Institute of Mathematical Sciences

December  2018, 8(4): 481-492. doi: 10.3934/naco.2018030

## A stochastic approach to model housing markets: The US housing market case

 Institute of Applied Mathematics, Middle East Technical University, 6800 Ankara, Turkey

* Corresponding author: Bilgi Yilmaz

Received  September 2017 Revised  December 2017 Published  September 2018

This study aims to estimate the price changes in housing markets using a stochastic process, which is defined in the form of stochastic differential equations (SDEs). It proposes a general SDEs system on the price structure in terms of house price index and mortgage rate to establish an effective process. As an empirical analysis, it applies a calibration procedure to an SDE on monthly S&P/Case-Shiller US National Home Price Index (HPI) and 30-year fixed mortgage rate to estimate parameters of differentiable functions defined in SDEs. The prediction power of the proposed stochastic model is justified through a Monte Carlo algorithm for one-year ahead monthly forecasts of the HPI returns. The results of the study show that the stochastic processes are flexible in terms of the choice of structure, compact with respect to the number of exogenous variables involved, and it is a literal method. Furthermore, this approach has a relatively high estimation power in forecasting the national house prices.

Citation: Bilgi Yilmaz, A. Sevtap Selcuk-Kestel. A stochastic approach to model housing markets: The US housing market case. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 481-492. doi: 10.3934/naco.2018030
##### References:

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##### References:
Development of US National Home Price Index ($1975-2016$) with respect to the selected financial market indicators
Simulated SDEs compared with observed S&P Case-Shiller Home Price Indices (1975-2015)
Observed and predicted S&P Case-Shiller Home Price Indices (2015-2016)
Descriptives of house price index, $h$, and mortgage rate, $r$, (1975-2015).
 Min Max Mean Std Skewness Kurtosis $h$ 25.2 184.62 96.26 47.55 0.36 1.84 log-$h$ -0.023 0.02 0.004 0.006 -0.76 4.84 $r$(%) 3.32 18.44 8.38 3.24 0.79 3.33
 Min Max Mean Std Skewness Kurtosis $h$ 25.2 184.62 96.26 47.55 0.36 1.84 log-$h$ -0.023 0.02 0.004 0.006 -0.76 4.84 $r$(%) 3.32 18.44 8.38 3.24 0.79 3.33
Estimates of the parameters using calibration
 $\hat{\lambda}$ $\hat{\mu}_h (\%)$ $\hat{\sigma}_h (\%)$ $\hat{\kappa}$ $\hat{\mu}_r$ (%) $\hat{\sigma}_r$ (%) $\rho$ 16.30 5.23 6.23 7.74 -0.01 0.31 -0.77
 $\hat{\lambda}$ $\hat{\mu}_h (\%)$ $\hat{\sigma}_h (\%)$ $\hat{\kappa}$ $\hat{\mu}_r$ (%) $\hat{\sigma}_r$ (%) $\rho$ 16.30 5.23 6.23 7.74 -0.01 0.31 -0.77
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