An asset-based approach to housing prices

Abstract

From an asset pricing point of view the excess return of housing over the risk free rate is negatively related to the covariance between the gross housing return rate and the stochastic discount factor. Under log-normality regularity conditions, current return premiums would require an extremely volatile stochastic discount factor, as a consequence of the observed variability of real gross housing returns. We analyse the volatility bound noticing, that the logarithmic transformation of the rent/price ratio is, respectively, negatively and positively related to subsequent real rent growth and housing return rates. In predictive regressions the logarithmic transformation of the rent/price ratio is positively and significantly correlated to subsequent excess return rates and to gross housing return volatility, at horizons of 4–16 quarters. These results are robust to the specification of the regressions conditioning variables. Further analysis with a sign identified structural vector autoregression model shows that shocks to the rent/price ratio take several quarters to be completely realized and have persistent effects on real rent growth and interest rates, which last for more than 15 years. Moreover, the logarithmic rent/price ratio forecasts in each time period both forthcoming rent growth and gross housing return rates.

Sign identification method

We describe the method of sign identification with Jacobi rotations for the n-dimensional case.

An n-dimensional Jacobi rotation matrix takes the form:

$$\begin{aligned} Q_{ij}\left( \theta _{ij}\right) =\left[ \begin{array}{ccccccc}1 &{} \cdots &{} 0 &{} \cdots &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \ddots &{} \vdots &{} \, &{} \vdots &{} \, &{} \vdots \\ 0 &{} \cdots &{} \cos \theta _{ij} &{} \cdots &{} -\sin \theta _{ij} &{} \cdots &{} 0 \\ \vdots &{} \, &{} \vdots &{} \ddots &{} \vdots &{} \, &{} \vdots \\ 0 &{} \cdots &{} \sin \theta _{ij} &{} \cdots &{} \cos \theta _{ij} &{} \cdots &{} 0 \\ \vdots &{} \, &{} \vdots &{} \, &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} 0 &{} \cdots &{} 0 &{} \cdots &{} 1\end{array}\right] . \end{aligned}$$

(A.1)

The matrix $Q_{ij}\left( \theta _{ij}\right) \text {}$ in Eq. (A.1) is obtained from the identity matrix replacing elements i, i and j, j with $\cos \theta _{ij}$, element i, j with $-\sin \theta _{ij}$ and element j, i with $\sin \theta _{ij}$, where $0 \le \theta _{ij} \le 2\pi$. Using the prosthaphaeresis formulas it can be shown that multiplying an n-dimensional row vector $x =\left( x_{1} ,\ldots ,x_{n}\right)$ by a matrix $Q_{ij}\left( \theta _{ij}\right)$ determines a clockwise rotation in the plane of its i, j-th components by an angle equal to $\theta _{ij}$. Moreover, it is straightforward to verify that $Q_{ij}\left( \theta _{ij}\right)$ is orthogonal: $Q_{ij}\left( \theta _{ij}\right) Q_{ij}\left( \theta _{ij}\right) ^{ \prime } =Q_{ij}\left( \theta _{ij}\right) ^{ \prime }Q_{ij}\left( \theta _{ij}\right) =I$. This in turn implies, that the matrix:

$$\begin{aligned} Q =\prod \limits _{i =1}^{n}\prod \limits _{j =i +1}^{n}Q_{ij}\left( \theta _{ij}\right) , \end{aligned}$$

(A.2)

is orthogonal.

The space ${\mathcal {Q}} \equiv \left\{ Q :QQ^{ \prime } =Q^{ \prime }Q =I\right\}$ of orthogonal matrices of order n is identified by matrices that take the form (A.2), with $0 \le \theta _{ij} \le \pi$ for $i ,j =1 ,\ldots ,n$, $j >i$.

For the analysis in the present work we performed a grid search over the space of orthogonal matrices of order 3. We used for each $\theta _{ij}$, for $i ,j =1 ,\ldots ,3$, a zero initial value and a step equal to $\pi /50$. The search procedure allowed us to identify 1462 models. The analysis in the main text uses the first 1000 identified models.