Fundamental Real Estate Prices: An Empirical Estimation with International Data

Abstract

We propose two alternative models to estimate fundamental prices on real estate markets. The first model is based on a no-arbitrage condition between renting and buying. The second model interprets the period costs as the result of market equilibrium between housing demand and supply. We estimate both models for the USA, the UK, Japan, Switzerland, and the Netherlands. We find that observed prices deviate substantially and for long periods from their estimated fundamental values. However, we find some evidence that, in the long-run, actual prices tend to return to their fundamental values progressively. This result is due to both impulse–response functions and forecast analyses. In particular, we find that using the fundamental price significantly increases the accuracy of out-of-sample long-term forecasts of the price.

Appendices

Appendix 1: Data

Table 2 Data description

Appendix 2: Econometric Methodology

To recapitulate the results of “The Fundamental Price Equation,” the computation of the fundamental prices given by the rent model requires: (1) the linearization parameter ρ, (2) the VAR coefficients in A and (3) the constant c. In addition, the S/D model needs (4) the supply series s _t , 5) the linearization parameter θ and (6) and the constant c ^∗. The VAR coefficients can be easily estimated with traditional methods. The next sections describe how to get the other elements.

Linearization Parameter $\protect\rho$

Recall from footnote no. 19 that the parameter ρ is defined as

$$ \rho =\frac{1}{1+\exp \left( -\ln \delta -\bar{x}-\Delta \bar{h}\right) }. $$

where $\bar{x}=\bar{p}-\bar{h}$. If both the price and the imputed rent are observable, it is straightforward to compute ρ. Unfortunately, in our case, the imputed rents and the prices are not expressed in the same units. Thus, we have that $\bar{x}=\bar{p}-\bar{h}-\ln \omega$, where ω is the conversion factor of the imputed rent units in terms of price units. This conversion factor is unobservable. To cope with this problem, we estimate ρ with an OLS regression of Eq. 12. For example, with the Rent model, we estimate the equation:

$$ \hat{x}_{t}=a+b\left( \hat{x}_{t+1}+\Delta h_{t+1}\right) -m_{t}+\varepsilon _{t}. $$

The parameter a gives us an estimation of κ ₁ + ln δ − ln ω − φ and the parameter b is our estimation of ρ.

Constants c and c ^∗

To estimate c and c ^∗, we assume that, on average over time , the observed log price-to-imputed-rent ratio $\hat{x}_{t}$ is equal to the fundamental price $x_{t}^{\bullet }$ for the rent model ($E\left( \hat{x} _{t}\right) =E\left( x_{t}^{\bullet }\right)$) and that $\tilde{x}_{t}$ is equal to $x_{t}^{\ast }$ for the S/D model ($E\left( \tilde{x}_{t}\right) =E\left( x_{t}^{\ast }\right)$).

Taking the unconditional expectation of Eq. 24 and using the vector z _t of fundamentals corresponding to each model implies that:

$$ c=E\left( \hat{x}_{t}\right) -\left( \rho g_{1}\Phi \left( I-\rho \Phi \right) ^{-1}+g_{2}\right) E\left( z_{t}^{\bullet }\right) $$

(26)

and

$$ c^{\ast }=E\left( \tilde{x}_{t}\right) -\left( \rho g_{1}^{\ast }\Phi \left( I-\rho \Phi \right) ^{-1}+g_{2}^{\ast }\right) E\left( z_{t}^{\ast }\right) , $$

(27)

respectively.

Supply and Linearization Parameter $\protect\theta$

As mentioned in footnote no. 26, the computation of the supply-demand model requires to first estimate the housing supply s _t , which is not directly observable. Note that the entire series of s _t can be recovered if the initial s ₀ is known. To see how, first note that Eq. 19 states that s _t is a function³⁴ f ₁ of s ₀ and θ

$$ s_{t}=\theta ^{t}s_{0}+\sum_{j=0}^{t-1}\theta ^{t-j-1}\left( \kappa _{2}\left( \theta \right) +\left( 1-\theta \right) b_{\!t+j}\right) . $$

(28)

Furthermore, we know³⁵ that θ is a function of the average $\bar{ b}-\bar{s}$

$$ \theta =1/\left( 1+\exp \left( \bar{b}-\frac{1}{T}\sum_{t=1}^{T}s_{t}-\ln \delta \right) \right) . $$

(29)

By substitution of s _t , we have that

$$ \theta \!=\!\left( 1\!+\!\frac{1}{\delta }\exp \left( \bar{b}\!-\!\frac{1}{T} s_{0}\sum_{t=1}^{T}\theta ^{t}\!-\!\frac{1}{T}\sum_{t=1}^{T}\sum_{j=0}^{t-1} \theta ^{t-j-1}\left( \kappa _{2}\left( \theta \right) \!+\!\left( 1\!-\!\theta \right) b_{t+j}\right) \right) \right) ^{-1}$$

(30)

This equation can be numerically solved for θ once the initial supply is known. Consequently, the only unknown in this problem is the initial housing supply. Once s ₀ is known, θ and the entire series of s _t can be recovered with Eq. 17.

We can determine s ₀ by imposing one additional condition: we impose that the average one-period imputed rent growth rate must be equal to the average renting price growth rate. This is equivalent to saying that, in the long term, the cost of living in a house must grow at the same rate as the cost of living in a rented flat. If this condition is violated one, of these two alternatives will be infinitely more costly than the other. This condition, together with Eq. 15, yields $\overline{\Delta s_{t}}=\overline{\Delta y_{t}}-\overline{\Delta q_{t}}$ where $\overline{ \Delta q_{t}}$ is the average renting price growth rate. As $\overline{ \Delta s_{t}}$ is also a function of s ₀, it is also possible to compute the s ₀ which gives the adequate initial condition. Then, starting from s ₀, the entire series housing supplies can be recovered and used to compute $x_{t}^{\ast }$.