Forecasting sales and price for existing single-family homes: A VAR model with error correction

Abstract. In this paper we forecast demand for existing single-family housing in the United States. We first find that sales volume (sales) and median sales price (price) have unit roots. We then find that sales and price are cointegrated. We develop a vector autoregressive (VAR) model with error correction to further examine the causality between sales and price. We find that there exists a bidirectional causality relationship between sales and price. Price affects sales significantly and sales affects price weakly. With the VAR model we then forecast sales and price for existing single-family housing during the period 1991 to 1994 by using a recursive method. We find that our predictions for sales and price fit the actual data well.

Introduction

In recent years researchers have begun exploring the relationships between the real estate market and other related markets. For example, Case and Shiller (1990) find that price changes and excess returns of single-family housing can be predicted by a number of information variables. Using the Granger equilibrium model Goebel and Ma (1993) find that mortgage rates and general interest rates are cointegrated after 1980. Schnitzel (1986), on the other hand, finds that deposit rates Granger-cause mortgage rates for Savings and Loans (S&L) during the period of 1970-78. Over the period 1978-84, however, he finds that it is mortgage rates that determine deposit rates. Less work has been done in exploring the fundamental relationship between sales and price for existing single-family homes. It is particularly interesting to investigate this housing market as it represents the biggest portion of home sales in the United States. The VAR model with error correction obtained here can help to analyze and predict the demand for existing single-family homes. Moreover, since residential investment has a timing lagged effect, forecasting the housing demand also appears to be important for policy makers.

In this study, we concentrate on the time-series behavior and relation between sales volume and median sales price. The sales and price data are for the existing single-family houses in the United States. We find that the levels of sales and price have unit roots. That is to say, the two real estate series are not stationary. Their first-order differences are, however, stationary. Further, we find that sales and price are cointegrated. That is, they tend to move together and converge in the long run. Following Engle and Granger (1987), therefore, we construct a VAR model with error correction to examine the Granger causality relationship between sales and price. We find that sales affect price significantly and price affects sales weakly. In addition, utilizing the VAR model we forecast sales and price of existing single-family homes in the whole nation by using a recursive method. We find that our predictions fit the actual data well. The goodness of fit of the model, measured by R2s from the regression of the fitted values on the actual values, ranges from 0.77 to 0.86. Therefore, we conclude that the existing single-family housing market is not efficient. Sales volume and median sales price can be well predicted by our model.

The study is organized as follows. Section two describes the data set. The third section discusses the methodology used in this analysis, and section four provides the empirical results and their implications. Finally, section five concludes the paper.

Data Set

The data used in this study include monthly time series of the existing single-family housing market sales volume (sales) and median sales price (price) in the United States from January 1970 to December 1994. The data from January 1970 to December 1990 is used in the VAR modeling procedure and the data from January 1991 to December 1994 is used for testing the model's predictability of sales and price. The sources of housing data are provided by the National Association of Realtors in Washington, D. C. The two time series are plotted in Appendix 1. The sales series has a strong seasonal pattern and a time trend. In particular, sales begin to increase in February and continue to increase until August. Starting from September, however, sales begin to fall and reach a bottom in January of the next year. This pattern is repeated year after year. In the long run, sales have a tendency to go up. The annual peak of sales typically exceeds that of the previous year, indicating a long-term upward trend. However, a major decline in sales occurred in 1980-81 when mortgage rates reached their peak. The price series indicates a clear time trend, suggesting strong autocorrelation in the time series and the possibility of the existence of nonstationarity. The first-order differences of these two series appear stationary.

Methodology

We first examine the possible existence of unit roots in our time-series data to ensure that the model constructed later is stationary in terms of the variables used. If a time series has a unit root, the first-order difference of the series is stationary and should be used. A series that is stationary after being differenced d times is said to be integrated of order d, or I(d) (Granger, 1981). If two time series are both integrated of order d, a linear combination of these two series may result in a stationary time series, I(0). In that case, we say that the two original series are cointegrated of order d (Granger, 1981). Following the stationarity tests, we then look at the cointegration of sales and price. If these two series are cointegrated, an error-correction term should be added to the modeling process as suggested by Engle and Granger (1987), and Phillips (1991). We further develop a VAR model with error correction terms to examine the Granger causality relationship between sales and price. Based on the VAR model, we forecast sales and price for existing single-family homes using a recursive method. Finally, we compare the forecasted sales and prices with the actual data to determine whether the VAR model with error correction provides a goodness of fit of the model.

Forecasting Sales and Price with the VAR Model

The VAR model identified above provides us not only with the Granger causality relationship between sales and price but also the opportunity to forecast sales and price in the existing single-family housing market. Therefore, we forecast sales and price for the period of January 1991 to December 1994 using a recursive method. The recursive procedure works as follows. For example, if we would like to forecast sales and price in January 1991 we first estimate equations (5) and (6) to obtain all coefficients, using the data set from January 1970 to December 1990. We then forecast one-month-ahead sales and price (i.e., sales and price in January 1991). As time advances we reestimate (5) and (b), using the data set from January 1970 to January 1991 to forecast the sales and price in February 1991. Unlike most predictions, our forecast is an out-of-sample forecast because we separate the modeling data set from the testing data set. We obtain forty-eight predictions (from January 1991 to December 1994) for sales and price and run regressions of the fitted values on the actual values for both sales and price to test the goodness of fit of our forecasting model.

Results and Implications

In this section, we first report the results of the unit root tests and then provide results of the cointegration test. We then examine the Granger causality relationship between sales and price using the VAR model with error correction terms that were developed in the third section of this study. With the VAR model, we further forecast sales and price for existing single-family homes for the period 1991-94 by the recursive method. Finally, the implications of the results are discussed.

Results from Unit Root Test

Conclusions

In this paper we examine demand in the existing single-family housing market and the causality relationship between sales volume and median price using a nationwide data set. We find that sales and price have unit roots and are not stationary, but are cointegrated of order one. A VAR model with error correction is developed to examine the causality relationship between sales and price. We find that price significantly Granger-causes sales and sales weakly Granger-causes price. Using the VAR model we then forecast sales and price for existing single-family homes. We find that our VAR model provides a good predictive model as the predictions for sales and price fit the actual data well. Our model is useful for policy makers in planning the residential investments.

Notes

tIff >0 the series will not be stationary. That is why we test,(fi

3We not only apply the Dickey-Fuller test on sales and price but also other test procedures, for example, the Augmented Dickey-Fuller test. The results are similar. Thus, we report only the results from the general Dickey-Fuller test.

40ne can argue that in a large structure prices and sales can be both endogenous variables. To address this issue, we add several other explanatory variables, such as the FHA/VA thirty-year mortgage rates and the New York Stock Exchange (NYSE) value-weighted monthly stock returns in our model. We find that although the mortgage rates have a significant negative relation with sales the overall fitness of the model remains almost the same as the model with only sales and

prices. We cannot find a significant relationship between the mortgage rates and prices. Even though the stock market returns seem to affect sales and price, they do not contribute additional predictability to sales and price. Therefore, we only report the results from the VAR model with only sales and price variables. Results from a more generalized VAR model with sales, price, mortgage rates, and stock index returns are available upon request.

References

Case, K. and R. Shiller, Forecasting Prices and Excess Returns in the Housing Market, AREUEA Journal, 1990,18, 253-73.

Dickey, D. A. and W A. Fuller, Distribution of the Estimators for Autoregressive Time Series with a Unit Root, Journal of the American Statistical Association, 1979, 74, 427-31. , Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrica, 1981, 49, 1057-73.

Engle, R. R. and C. W. J. Granger, Co-integration and Error Correction: Representation, Estimation, and Testing, Econometrica, 1987, 55, 251-76.

Goebel, P R. and C. K. Ma, The Integration of Mortgage Markets and Capital Markets, Journal

of the American Real Estate and Urban Economics Association, 1993, 21, 511-38. Granger, C. W J., Investigating Causal Relations by Econometric Models and Cross-Spectral Methods, Econometrica, 1969, 37, 42438.

-, Some Properties of Time Series Data and Their Use in Econometric Model Specification, Journal of Econometrics, 1981, 28, 121-30.

- and P. Newbold, Spurious Regressions in Econometrics, Journal of Econometrics, 1974, 2, 111-20.

Nelson, C. R. and C. I. Plosser, Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications, Journal of Monetary Economics, 1982, 10, 139-62. Phillips, P C. B., Understanding Spurious Regression in Econometrics, Journal of Econometrics, 1986, 33, 31140.

-, 1991, Optimal Inference in Cointegrated Systems, Econometrica, 1991, 59, 283-306. Schnitzel, P., 1986, Do Deposit Rates Cause Mortgage Loan Rates? AREUEA Journal, 1986, 14, 448-4.

Zhong-guo Zhou*

*Department of Finance, Real Estate, and Insurance, College of Business Administration and Economics, California State University, Northridge, California 91330. Date Revised-July 1996; Accepted-October 1996.

The author would like to thank Peter Chung, Judy Posnikoff, Mark Tengesdal, Aman Ullah, Ko Wang, and an anonymous referee for helpful comments and suggestions on this project. He would also like to thank Heeyoung Kim for providing him with part of the data set.

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