The out-of-sample forecasting performance of nonlinear models of regional housing prices in the US

Abstract

This article provides out-of-sample forecasts of linear and nonlinear models of US and four Census subregions’ housing prices. The forecasts include the traditional point forecasts, but also include interval and density forecasts, of the housing price distributions. The nonlinear smooth-transition autoregressive model outperforms the linear autoregressive model in point forecasts at longer horizons, but the linear autoregressive and nonlinear smooth-transition autoregressive models perform equally at short horizons. In addition, we generally do not find major differences in performance for the interval and density forecasts between the linear and nonlinear models. Finally, in a dynamic 25-step ex-ante and interval forecasting design, we, once again, do not find major differences between the linear and nonlinear models. In sum, we conclude that when forecasting regional housing prices in the United States, generally the additional costs associated with nonlinear forecasts outweigh the benefits for forecasts only a few months into the future.

Keywords:

• forecasting
• linear and nonlinear models
• US and Census housing price indexes

JEL Classification:

• C32
• R31

Acknowledgement

We acknowledge the helpful comments of two anonymous referees.

Notes

¹ They also consider combined forecasts that produce even better forecasts as well as adjust the tests for data snooping (White, 2000).

² Gupta (2013) and Plakandaras et al. (forthcoming) provide detailed literature reviews.

³ Nonlinear estimation, just like linear estimation, requires stationary variables to avoid spurious estimates. Hence, we convert house prices in the United States and the four Census subregions into annual growth rates. We confirm stationarity of the series, in turn, by the Augmented–Dickey–Fuller (ADF), the Dickey–Fuller with GLS Detrending (DF-GLS), the Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) and the Phillips–Perron (PP) tests. The results are available from the authors.

⁴ This discussion relies heavily on the presentation in Kim and Bhattacharya (2009) and Balcilar et al. (2011). We retain their symbolic representation of the equations.

⁵ A longer version of the current article provides details on the testing procedure for choosing between the ESTAR and LSTAR models. See http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2138980.

⁶ We follow the existing literature in treating the parameters of the linear and nonlinear AR models as known in forming forecasts. Hansen (2006) describes how to include parameter estimation uncertainty into interval forecasts for linear models.

⁷ The four Census subregions and the included states are described as follows: Northeast: Connecticut, Maine, Massachusetts, New Hampshire, New Jersey, New York, Pennsylvania, Rhode Island, and Vermont; Midwest: Illinois, Indiana, Iowa, Kansas, Michigan, Minnesota, Missouri, Nebraska, North Dakota, Ohio, South Dakota, and Wisconsin; South: Alabama, Arkansas, Delaware, District of Columbia, Florida, Georgia, Kentucky, Louisiana, Maryland, Mississippi, North Carolina, Oklahoma, South Carolina, Tennessee, Texas, Virginia, and West Virginia; and West: Alaska, Arizona, California, Colorado, Hawaii, Idaho, Montana, Nevada, New Mexico, Oregon, Utah, Washington, and Wyoming.

⁸ In this case, however, the delay lag changes for the Midwest to 3. We can still reject linearity for the Midwest at the 5% level.

⁹ We test regressions estimated using outlier robust M-estimation. We also select ESTAR models for all Census subregions, using the sample period and estimation method (OLS) in Kim and Bhattacharya (2009). Thus, differences in findings reflect different estimation techniques and sample periods. The details of these results are available upon request from the authors.

¹⁰ The results from estimating LSTAR and AR models are available on request. Van Dijk et al. (2002) suggest a battery of misspecifications tests – no residual autocorrelation, parameter constancy, no remaining nonlinearity, no autoregressive conditional heteroscedasticity (ARCH), besides the test of normality – for the LSTAR model. Our estimated LSTAR models for the United States and its four Census subregions do not exhibit any misspecification.

¹¹ The Ramsey model specification test provides further evidence of nonlinearity in the housing price growth rates of the United States and the four Census subregions. We reject the null hypothesis for a linear AR model specification, against a nonlinear LSTAR model, at the 1-per cent level of significance for all cases.

¹² The parameter c denotes the value for which G(s_t; γ, c) = 0.5 at s_t = c. Therefore, the process switches monotonically towards Regime 1 as s_t increases. Thus, two regimes exhibit equal weights at the threshold value c and switching occurs exactly at c.

¹³ Tables for the results of point, interval and density forecasts appear in a longer version of this article. See http://papers.ssrn.com/sol3/papers.cfm?abstract_id = 2138980.

¹⁴ The ex-ante forecast provides a case study of the difference between the linear AR and LSTAR models. The results may not generalize to other sample periods.