Putting the present value model into practice: a comparison of two alternative approaches

ABSTRACT

A key issue around putting the present-value model into practice is how to construct the unobserved future expectations of the fundamental variables related to an asset. One approach is to fit a vector autoregression (VAR) for the fundamental variables and deduce their future expectations from the estimated VAR. An alternative is to directly specify the future expectations as unobserved components (UC) and use the Kalman filter to extract their estimates from the realized data. This article examines whether the predictions of the present-value model are consistent across the two approaches. Constructing the VAR and UC versions of the standard present-value model, we examine how the two versions compare in identifying the main driver of the US and UK housing markets. For the UK, the two approaches consistently attribute most variations in the price–rent ratio to the expected future risk premium for housing investment. For the US, however, the two approaches deliver considerably different results: the VAR version marks the expected risk-free rate of return, whereas the UC version singles out the expected risk premium as the main driver of the ratio. We conclude that the choice between the VAR and UC approaches is not a trivial issue related to utilizing the present-value model.

KEYWORDS:

• Present-value model
• vector autoregression
• unobserved components
• variance decomposition
• price–rent ratio

JEL CLASSIFICATION:

• C53
• G12
• R31

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ We are following previous studies (e.g. Campbell and Ammer (1993), Balke and Wohar (2002) and Campbell et al. (2009)) by setting the log of the gross real return, $h_t$, as the sum of the real risk-free interest rate, $i_t$, and the risk premium, ${\pi }_t$.

² In practice, $var\left({\mathbb{Z}}_t\right)$ is calculated from the estimated $A_i$’s and ${\mathrm{\Sigma }}_{\epsilon }$, and var (η${ }_t^{pr}$) is estimated by the sample variance of the residuals of $p{r}_t$ from Equation (6).

³ We used the AR(1) specifications at the initial stage of this study, but the AR(2) specification turned out to fit the data better.

⁴ The database is available at http://www.zillow.com/research/data.

⁵ The long-term mortgage rate archived by the Bank of England can be used for the UK, but the data series is available only from 1995 on.

⁶ The transition equations of the state-space form consist of Equations (8a)–(8c), and the measurement equations are Equations (9) and (11).

⁷ The price–rent ratio is included in the VAR because it is required for a proper VAR-based decomposition, as noted by Engsted, Pedersen, and Tanggaard (2012).

⁸ When calculating the variance-covariance terms, we abstract away uncertainty about the estimated parameters.

⁹ For example, the positive correlation between the expected rent growth and risk premium can be interpreted as follows: in response to news of higher future rent growth, house prices do not fully increase in the current period, so investors expect further increases in house prices and therefore a higher housing investment premium in the future.

¹⁰ The null of the whole ${\alpha }^{\prime }s$ being 0 is rejected at the 5% critical level for both countries. More details on the estimation results are available from the authors upon request.

¹¹ The likelihood ratio test rejects the null of (${\gamma }_{\mu },{\gamma }_{\lambda },{\delta }_g,{\delta }_{\lambda },{\theta }_g,{\theta }_{\mu }$) being 0 at the 5% critical level for both countries. More details on the estimation results are available from the authors upon request.

¹² In relation to this conjecture, an anonymous referee suggested another possibility that the over-parameterization problem inherent to VAR-based predictions may have appeared in the case of the US. We deeply appreciate this comment.

Additional information

Funding

This work was supported by Hankuk University of Foreign Studies [HUFS Research Fund].