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:
[1]

N. Apergis and A. Rezitis, Housing prices and macroeconomic factors in Greece: prospects within the EMU, Applied Economics Letters, 10 (2003), 799-804.   Google Scholar

[2]

M. BarariN. SarkarS. Kundu and K. B. Chowdhury, Forecasting house prices in the United States with multiple structural breaks, International Econometric Review (IER), 6 (2014), 1-23.   Google Scholar

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[4]

M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, Journal of Business, (1973), 135-157.  doi: 10.1086/296288.  Google Scholar

[5]

J. P. CohenY. M. Ioannides and W. W. Thanapisitikul, Spatial effects and house price dynamics in the USA, Journal of Housing Economics, 31 (2016), 1-13.  doi: 10.1016/j.jhe.2015.10.006.  Google Scholar

[6]

J. C. CoxJ. E. Ingersoll Jr and S. A. Ross, A theory of the term structure of interest rates, Econometrica: Journal of the Econometric Society, (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar

[7]

J. C. CoxJ. E. Ingersoll Jr and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166.  doi: 10.1016/0304-405X(76)90023-4.  Google Scholar

[8]

G. W. Crawford and M. C. Fratantoni, Assessing the forecasting performance of regime-switching, ARIMA and GARCH models of house prices, Real Estate Economics, 31 (2003), 223-243.  doi: 10.1111/1540-6229.00064.  Google Scholar

[9]

Y. Demyanyk and O. Van Hemert, Understanding the subprime mortgage crisis, Review of Financial Studies, 24 (2011), 1848-1880.   Google Scholar

[10]

W. E. DiewertA. O. Nakamura and L. I. Nakamura, The housing bubble and a new approach to accounting for housing in a CPI, Journal of Housing Economics, 18 (2009), 156-171.  doi: 10.1016/j.jhe.2009.07.008.  Google Scholar

[11]

M. I. Dröes and W. H. Hassink, House price risk and the hedging benefits of home ownership, Journal of Housing Economics, 22 (2013), 92-99.   Google Scholar

[12]

E. Eerola and T. Lyytikäinen, On the role of public price information in housing markets, Regional Science and Urban Economics, 53 (2015), 74-84.  doi: 10.1016/j.regsciurbeco.2015.05.006.  Google Scholar

[13]

M. FletcherJ. Mangan and E. Raeburn, Comparing hedonic models for estimating and forecasting house prices, Property Management, 22 (2004), 189-200.  doi: 10.1108/02637470410544986.  Google Scholar

[14]

J. Gallin, The long-run relationship between house prices and income: Evidence from local housing markets, Real Estate Economics, 34 (2006), 417-438.  doi: 10.1111/j.1540-6229.2006.00172.x.  Google Scholar

[15]

R. Gibson and E. S. Schwartz, Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance, 45 (1990), 959-976.  doi: 10.1111/j.1540-6261.1990.tb05114.x.  Google Scholar

[16]

C. Goodhart and B. Hofmann, House prices, money, credit, and the macroeconomy, Oxford Review of Economic Policy, 24 (2008), 180-205.  doi: 10.1093/oxrep/grn009.  Google Scholar

[17]

H. S. GuirguisC. I. Giannikos and R. I. Anderson, The US housing market: asset pricing forecasts using time varying coefficients, The Journal of real estate finance and economics, 30 (2005), 33-53.  doi: 10.1007/s11146-004-4830-z.  Google Scholar

[18]

K. L. Guntermann and S. C. Norrbin, Empirical tests of real estate market efficiency, The Journal of Real Estate Finance and Economics, 4 (1991), 297-313.  doi: 10.1007/BF00161931.  Google Scholar

[19]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[20]

J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592.  doi: 10.1093/rfs/3.4.573.  Google Scholar

[21]

M. Iacoviello, House prices, borrowing constraints, and monetary policy in the business cycle, The American Economic Review, 95 (2005), 739-764.  doi: 10.1257/0002828054201477.  Google Scholar

[22]

D. IganA. KabundiF. N. De SimoneM. Pinheiro and N. Tamirisa, Housing, credit, and real activity cycles: Characteristics and comovement, Journal of Housing Economics, 20 (2011), 210-231.  doi: 10.1016/j.jhe.2011.07.002.  Google Scholar

[23]

J. B. Kau and D. C. Keenan, An overview of the option-theoretic pricing of mortgages, Journal of Housing Economics, 6 (1995), 217-244.   Google Scholar

[24]

T. Kauko, On current neural network applications involving spatial modelling of property prices, Journal of Housing and the Built Environment, 18 (2003), 159-181.   Google Scholar

[25]

K. Kim and J. Park, Segmentation of the housing market and its determinants: Seoul and its neighboring new towns in Korea, Australian Geographer, 36 (2005), 221-232.  doi: 10.1080/00049180500150019.  Google Scholar

[26]

R. Kouwenberg and R. Zwinkels, Forecasting the US housing market, International Journal of Forecasting, 30 (2014), 415-425.  doi: 10.1016/j.ijforecast.2013.12.010.  Google Scholar

[27]

P. Linneman, An empirical test of the efficiency of the housing market, Journal of Urban Economics, 20 (1986), 140-154.  doi: 10.1016/0094-1190(86)90003-3.  Google Scholar

[28]

S. Malpezzi, A simple error correction model of house prices, Journal of Housing Economics, 8 (1999), 27-62.  doi: 10.1006/jhec.1999.0240.  Google Scholar

[29]

S. Malpezzi and S. Wachter, The role of speculation in real estate cycles, Journal of Real Estate Literature, 13 (2005), 141-164.  doi: 10.2139/ssrn.2585241.  Google Scholar

[30]

S. Malpezzi and S. Wachter, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.   Google Scholar

[31]

W. Miles, Boom-bust cycles and the forecasting performance of linear and non-linear models of house prices, The Journal of Real Estate Finance and Economics, 36 (2008), 249-264.  doi: 10.1007/s11146-007-9067-1.  Google Scholar

[32]

D. E. Rapach and J. K. Strauss, Differences in housing price forecastability across US states, International Journal of Forecasting, 25 (2009), 351-372.  doi: 10.1016/j.ijforecast.2009.01.009.  Google Scholar

[33]

E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973.  doi: 10.1111/j.1540-6261.1997.tb02721.x.  Google Scholar

[34]

D. E. SommervollT.-A. Borgersen and T. Wennemo, The stochastic behavior of commodity prices: Implications for valuation and hedging, Journal of banking & finance, 34 (2010), 557-567.   Google Scholar

[35]

S. Stevenson, New empirical evidence on heteroscedasticity in hedonic housing models, Journal of Housing Economics, 13 (2004), 136-153.  doi: 10.1016/j.jhe.2004.04.004.  Google Scholar

[36]

K. Tsatsaronis and H. Zhu, What drives housing price dynamics: cross-country evidence, BIS Quarterly Review, (2004). Google Scholar

[37]

O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188.  doi: 10.1002/9781119186229.ch6.  Google Scholar

[38]

Z. Yang and S. Wang, Permanent and transitory shocks in owner-occupied housing: A common trend model of price dynamics, Journal of Housing Economics, 21 (2012), 336-346.  doi: 10.1016/j.jhe.2012.08.001.  Google Scholar

[39]

Z. G. Zhou, Forecasting sales and price for existing single-family homes: a VAR model with error correction, Journal of Real Estate Research, 14 (2009), 155-167.   Google Scholar

[40]

Z. G. Zhou, Determinants of house prices: a quantile regression approach, The Journal of Real Estate Finance and Economics, 37 (2008), 317-333.   Google Scholar

show all references

References:
[1]

N. Apergis and A. Rezitis, Housing prices and macroeconomic factors in Greece: prospects within the EMU, Applied Economics Letters, 10 (2003), 799-804.   Google Scholar

[2]

M. BarariN. SarkarS. Kundu and K. B. Chowdhury, Forecasting house prices in the United States with multiple structural breaks, International Econometric Review (IER), 6 (2014), 1-23.   Google Scholar

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[4]

M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, Journal of Business, (1973), 135-157.  doi: 10.1086/296288.  Google Scholar

[5]

J. P. CohenY. M. Ioannides and W. W. Thanapisitikul, Spatial effects and house price dynamics in the USA, Journal of Housing Economics, 31 (2016), 1-13.  doi: 10.1016/j.jhe.2015.10.006.  Google Scholar

[6]

J. C. CoxJ. E. Ingersoll Jr and S. A. Ross, A theory of the term structure of interest rates, Econometrica: Journal of the Econometric Society, (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar

[7]

J. C. CoxJ. E. Ingersoll Jr and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166.  doi: 10.1016/0304-405X(76)90023-4.  Google Scholar

[8]

G. W. Crawford and M. C. Fratantoni, Assessing the forecasting performance of regime-switching, ARIMA and GARCH models of house prices, Real Estate Economics, 31 (2003), 223-243.  doi: 10.1111/1540-6229.00064.  Google Scholar

[9]

Y. Demyanyk and O. Van Hemert, Understanding the subprime mortgage crisis, Review of Financial Studies, 24 (2011), 1848-1880.   Google Scholar

[10]

W. E. DiewertA. O. Nakamura and L. I. Nakamura, The housing bubble and a new approach to accounting for housing in a CPI, Journal of Housing Economics, 18 (2009), 156-171.  doi: 10.1016/j.jhe.2009.07.008.  Google Scholar

[11]

M. I. Dröes and W. H. Hassink, House price risk and the hedging benefits of home ownership, Journal of Housing Economics, 22 (2013), 92-99.   Google Scholar

[12]

E. Eerola and T. Lyytikäinen, On the role of public price information in housing markets, Regional Science and Urban Economics, 53 (2015), 74-84.  doi: 10.1016/j.regsciurbeco.2015.05.006.  Google Scholar

[13]

M. FletcherJ. Mangan and E. Raeburn, Comparing hedonic models for estimating and forecasting house prices, Property Management, 22 (2004), 189-200.  doi: 10.1108/02637470410544986.  Google Scholar

[14]

J. Gallin, The long-run relationship between house prices and income: Evidence from local housing markets, Real Estate Economics, 34 (2006), 417-438.  doi: 10.1111/j.1540-6229.2006.00172.x.  Google Scholar

[15]

R. Gibson and E. S. Schwartz, Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance, 45 (1990), 959-976.  doi: 10.1111/j.1540-6261.1990.tb05114.x.  Google Scholar

[16]

C. Goodhart and B. Hofmann, House prices, money, credit, and the macroeconomy, Oxford Review of Economic Policy, 24 (2008), 180-205.  doi: 10.1093/oxrep/grn009.  Google Scholar

[17]

H. S. GuirguisC. I. Giannikos and R. I. Anderson, The US housing market: asset pricing forecasts using time varying coefficients, The Journal of real estate finance and economics, 30 (2005), 33-53.  doi: 10.1007/s11146-004-4830-z.  Google Scholar

[18]

K. L. Guntermann and S. C. Norrbin, Empirical tests of real estate market efficiency, The Journal of Real Estate Finance and Economics, 4 (1991), 297-313.  doi: 10.1007/BF00161931.  Google Scholar

[19]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[20]

J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592.  doi: 10.1093/rfs/3.4.573.  Google Scholar

[21]

M. Iacoviello, House prices, borrowing constraints, and monetary policy in the business cycle, The American Economic Review, 95 (2005), 739-764.  doi: 10.1257/0002828054201477.  Google Scholar

[22]

D. IganA. KabundiF. N. De SimoneM. Pinheiro and N. Tamirisa, Housing, credit, and real activity cycles: Characteristics and comovement, Journal of Housing Economics, 20 (2011), 210-231.  doi: 10.1016/j.jhe.2011.07.002.  Google Scholar

[23]

J. B. Kau and D. C. Keenan, An overview of the option-theoretic pricing of mortgages, Journal of Housing Economics, 6 (1995), 217-244.   Google Scholar

[24]

T. Kauko, On current neural network applications involving spatial modelling of property prices, Journal of Housing and the Built Environment, 18 (2003), 159-181.   Google Scholar

[25]

K. Kim and J. Park, Segmentation of the housing market and its determinants: Seoul and its neighboring new towns in Korea, Australian Geographer, 36 (2005), 221-232.  doi: 10.1080/00049180500150019.  Google Scholar

[26]

R. Kouwenberg and R. Zwinkels, Forecasting the US housing market, International Journal of Forecasting, 30 (2014), 415-425.  doi: 10.1016/j.ijforecast.2013.12.010.  Google Scholar

[27]

P. Linneman, An empirical test of the efficiency of the housing market, Journal of Urban Economics, 20 (1986), 140-154.  doi: 10.1016/0094-1190(86)90003-3.  Google Scholar

[28]

S. Malpezzi, A simple error correction model of house prices, Journal of Housing Economics, 8 (1999), 27-62.  doi: 10.1006/jhec.1999.0240.  Google Scholar

[29]

S. Malpezzi and S. Wachter, The role of speculation in real estate cycles, Journal of Real Estate Literature, 13 (2005), 141-164.  doi: 10.2139/ssrn.2585241.  Google Scholar

[30]

S. Malpezzi and S. Wachter, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.   Google Scholar

[31]

W. Miles, Boom-bust cycles and the forecasting performance of linear and non-linear models of house prices, The Journal of Real Estate Finance and Economics, 36 (2008), 249-264.  doi: 10.1007/s11146-007-9067-1.  Google Scholar

[32]

D. E. Rapach and J. K. Strauss, Differences in housing price forecastability across US states, International Journal of Forecasting, 25 (2009), 351-372.  doi: 10.1016/j.ijforecast.2009.01.009.  Google Scholar

[33]

E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973.  doi: 10.1111/j.1540-6261.1997.tb02721.x.  Google Scholar

[34]

D. E. SommervollT.-A. Borgersen and T. Wennemo, The stochastic behavior of commodity prices: Implications for valuation and hedging, Journal of banking & finance, 34 (2010), 557-567.   Google Scholar

[35]

S. Stevenson, New empirical evidence on heteroscedasticity in hedonic housing models, Journal of Housing Economics, 13 (2004), 136-153.  doi: 10.1016/j.jhe.2004.04.004.  Google Scholar

[36]

K. Tsatsaronis and H. Zhu, What drives housing price dynamics: cross-country evidence, BIS Quarterly Review, (2004). Google Scholar

[37]

O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188.  doi: 10.1002/9781119186229.ch6.  Google Scholar

[38]

Z. Yang and S. Wang, Permanent and transitory shocks in owner-occupied housing: A common trend model of price dynamics, Journal of Housing Economics, 21 (2012), 336-346.  doi: 10.1016/j.jhe.2012.08.001.  Google Scholar

[39]

Z. G. Zhou, Forecasting sales and price for existing single-family homes: a VAR model with error correction, Journal of Real Estate Research, 14 (2009), 155-167.   Google Scholar

[40]

Z. G. Zhou, Determinants of house prices: a quantile regression approach, The Journal of Real Estate Finance and Economics, 37 (2008), 317-333.   Google Scholar

Figure 1.  Development of US National Home Price Index ($1975-2016$) with respect to the selected financial market indicators
Figure 2.  Simulated SDEs compared with observed S&P Case-Shiller Home Price Indices (1975-2015)
Figure 3.  Observed and predicted S&P Case-Shiller Home Price Indices (2015-2016)
Table 1.  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
Table 2.  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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