# RDP 2020-02: The Distributional Effects of Monetary Policy: Evidence from Local Housing Markets 2. Theoretical Framework

To develop some empirical predictions, we outline a theoretical framework that explains how monetary policy can affect local housing markets differently and why simple estimates of the local effects of monetary policy might be biased. The framework is based on an asset pricing model used extensively in the finance literature. It is typically referred to as the dynamic Gordon growth model, as applied to the housing market (Hiebert and Sydow 2009).[7] Consider the standard asset pricing formula:

$P t = P t+1 + D t R t$

This states that the housing price in the current period (Pt) is equal to the discounted value of the future price of housing (Pt + 1) and current rents (Dt), where the discount rate is equal to R = 1 + r. The discount rate can also be thought of as the ‘required rate of return’, which consists of the risk-free rate plus a risk premium that might vary according to the local housing market. All variables are in real terms. Taking natural logs of this expression:

$log( P t )=log( P t+1 + D t )−log( R t )$

We would like to obtain a (log) linear relationship between housing prices, rents and discount rates (and ultimately interest rates). But the relationship here is nonlinear as it involves the log of the sum of the future housing price and rents. However, Campbell and Shiller (1988) show that the expression can be approximated as.[8]

$log( P t )=k+( 1−ρ )log( P t+1 )+ρlog( D t )−log( R t )$

Here, the log of the price level (log(Pt)) is approximately equal to a constant (k), the weighted average of the log level of the future housing price $\left(\left(1-\rho \right)\mathrm{log}\left({P}_{t+1}\right)\right)$ and the current rent $\left(\rho \mathrm{log}\left({D}_{t}\right)\right)$ less the log level of the discount factor (log(Rt)) which is approximately equal to the discount rate.[9] Taking the difference between two consecutive periods, we obtain:

$Δ p it =−Δ r it +( 1−ρ )Δ p it+1 +ρΔ d it$

where lower-case letters represent the natural log of the upper-case counterparts $\left(\text{e}\text{.g}\text{.}\text{\hspace{0.17em}}p=\mathrm{log}\left(P\right)\right),\text{\hspace{0.17em}}\text{Δ}$ represents (quarterly) changes over time and each local housing market is denoted with subscript i. Here, current housing price growth is a function of the change in the discount rate, expected housing price growth in the next period and current rent growth.

We assume that housing investors (including home owners) use market interest rates to discount the future and that, at least for some investors, the relevant discount rate is the short-term cash rate $\left(\text{Δ}{r}_{t}\right)$ . As such, changes in household discount rates across local housing markets $\left(\text{Δ}{r}_{it}\right)$ are a function of changes in monetary policy $\left(\text{Δ}{r}_{t}\right)$ and shocks to local discount rates $\left({\epsilon }_{it}^{R}\right)$ . The sensitivity of local discount rates to monetary policy varies across local housing markets, as captured by the coefficient ${\theta }_{i}$ .

$Δ r it = θ i Δ r t + ε it R$

We similarly assume that local housing price expectations are a function of market interest rates, and that the sensitivity of expectations to monetary policy varies across local markets, as shown by the coefficient ${\delta }_{i}$ :

$Δ p it+1 = –δ i Δ r t + ε it P$

One possible basis for this variation is that housing supply conditions vary across regions, and both home owners and investors know this, such that they expect local housing price growth to vary across regions in response to aggregate demand shocks, including monetary policy.

Substituting the two expressions, we obtain a regression model that we can explore using local housing market data:

$Δ p it = β i Δ r t +ρΔ d it + ε it$

where the sensitivity of local housing price growth to the cash rate is a function of various structural parameters that vary by local housing market $\left({\beta }_{i}=-\left(\left(1-\rho \right){\delta }_{i}+{\theta }_{i}\right)\right)$ and the error term is a function of both local shocks to discount rates and expectations $\left({\epsilon }_{it}={\epsilon }_{it}^{P}-{\epsilon }_{it}^{R}\right)$ . This simple model indicates that changes in monetary policy can affect current housing prices by influencing either discount rates (through ${\theta }_{i}$ ) or expectations about future housing prices (through ${\delta }_{i}$ ).[10]

The key assumption to pin down the distributional effects of monetary policy on local housing prices $\left({\beta }_{i}\right)$ is that monetary policy does not respond to local housing shocks $\left(E\left(\Delta {r}_{t}*{\epsilon }_{it}\right)=0\right)$ . This strategy assumes that there is no aggregate variable, such as GDP growth or unemployment, to which the monetary policy is responding and which has a heterogeneous effect on local housing markets. Suppose that each market has a different sensitivity to changes in aggregate output growth $\left(\text{Δ}{Y}_{t}\right)$ :

$ε it = μ i Δ Y t + ω it$

where the sensitivity of local housing prices to aggregate output growth is given by the coefficient ${\mu }_{i}$ and ${\omega }_{it}$ is a white noise error. If monetary policy systematically responds to aggregate output growth then the key identification assumption is invalid because $E\left(\text{Δ}{r}_{t}*{\mu }_{i}\text{Δ}{Y}_{t}\right)\ne 0$ . If monetary policy tightens in response to stronger output growth (which drives higher housing prices) this would attentuate the estimates. To account for this bias, we directly control for relevant macroeconomic variables that may have distributional effects on housing prices. We therefore estimate:

1 $Δ p it = β i Δ r t +ρΔ d it + μ i Δ Y t + ω it$

Equation (1) forms the basis of our empirical model and is discussed in more detail in Section 4. In Section 6, a version of the model is also estimated using estimates of monetary policy ‘shocks’ that purge the changes in monetary policy of any systematic response to macroeconomic conditions and forecasts.

## Footnotes

We could instead derive empirical predictions from a user cost of capital model, as used, for example, by Fox and Tulip 2014 (and references therein). This is basically the same framework, with a focus on housing price growth (rather than rental yields) and without the added complexity of introducing depreciation and taxes on housing in deriving the required rate of return. [7]

This implicitly assumes that the housing price-to-rent ratio is approximately constant over time, which is a reasonable assumption over the sample period from 1990 to 2019. [8]

The weight $\left(1-\rho \right)$ is equal to $1/\left(1+\mathrm{exp}\left(\overline{d-p}\right)\right)$ where $\overline{d-p}$ is the sample average of the log ratio of rents to housing prices. [9]

The specification assumes that the correlation between changes in housing prices and current rents does not vary by local housing market, which is essentially the same as assuming a stationary price-to-rent ratio. [10]