The impact of monetary policy on local housing markets: Do regulations matter?

Abstract

This paper shows that monetary policy has uneven impacts on local housing markets, and that the magnitude of the impacts are correlated with housing supply regulations. We apply the linearized present value model, which allows the log rent–price ratio to be decomposed into the expected present values of all future real interests rates, real housing premia, and real rent growth, to the housing markets in 23 US metropolitan statistical areas. Based on the indirect inference bias-corrected VAR estimates, we find that MSAs that are more regulated have (i) a higher variance in the log rent–price ratio, (ii) a larger share of the variance explained by real interest rate, and (iii) a stronger impulse response of house price to the real interest rate shock.

Appendices

Appendix 1: Campbell–Shiller decomposition

Starting from the realized log gross return to housing defined in Eq. (1) and using lowercase letters for logs, we have

$$\begin{aligned} \phi _{t+1}&=\ln (P_{t+1}+R_{t+1})-\ln (P_t),\nonumber \\&=p_{t+1}+\ln \left( 1+\frac{R_{t+1}}{P_{t+1}}\right) -p_t,\nonumber \\&=p_{t+1}-p_t+\ln (1+\exp (r_{t+1}-p_{t+1})). \end{aligned}$$

(21)

Applying a Taylor approximation to the function $f(x)=\ln (1+\exp (x))$ around $x=\overline{r-p}$ yields

$$\begin{aligned} \ln (1+\exp (r_{t+1}-p_{t+1}))=\sum _{n=0}^{\infty }\frac{f^{(n)}(\overline{r-p})}{n!}(r_{t+1}-p_{t+1}-\overline{r-p})^n, \end{aligned}$$

(22)

or equivalently,

$$\begin{aligned}&\ln (1+\exp (r_{t+1}-p_{t+1}))= [-\ln (\rho )-(1-\rho )\ln (1/\rho -1)]\nonumber \\&\quad +\,(1-\rho )(r_{t+1}-p_{t+1}) +O((r_{t+1}-p_{t+1})^2), \end{aligned}$$

(23)

where $\rho =(1+\exp (\overline{r-p}))^{-1}\in (0,1)$. The last term $O((r_{t+1}-p_{t+1})^2)$ is positive, since its sign is determined by the second-order derivative of f(x),

$$\begin{aligned} f^{(2)}(x) = \frac{\exp (x)}{(1+\exp (x))^{2}}>0, \end{aligned}$$

Substituting Eqs. (23) into (21) yields:

$$\begin{aligned} \phi _{t+1}= & {} p_{t+1}-p_t+[-\ln (\rho )-(1-\rho )\ln (1/\rho -1)]\nonumber \\&+(1-\rho )(r_{t+1}-p_{t+1})+O((r_{t+1}-p_{t+1})^2), \end{aligned}$$

(24)

and

$$\begin{aligned} p_t= & {} \rho p_{t+1}+[-\ln (\rho )-(1-\rho )\ln (1/\rho -1)]+(1-\rho )r_{t+1}\nonumber \\&-\,\phi _{t+1}+O((r_{t+1}-p_{t+1})^2). \end{aligned}$$

(25)

Iterating Eq.(25) forward yields:

$$\begin{aligned} p_t= & {} \frac{-\ln (\rho )-(1-\rho )\ln (1/\rho -1)}{1-\rho }+\sum _{j=0}^{\infty }\rho ^j ((1-\rho )r_{t+1+j}-\phi _{t+1+j})\nonumber \\&+\,\frac{O((r_{t+1}-p_{t+1})^2)}{1-\rho }, \end{aligned}$$

(26)

and

$$\begin{aligned} r_t-p_t= & {} \frac{\ln (\rho )+(1-\rho )\ln (1/\rho -1)}{1-\rho }+\sum _{j=0}^{\infty }\rho ^j (\phi _{t+1+j}-\Delta r_{t+1+j})\nonumber \\&-\,\frac{O((r_{t+1}-p_{t+1})^2)}{1-\rho }, \end{aligned}$$

(27)

which gives Eq. (2) with a negative approximation bias under rational expectations.

Appendix 2: Estimation without bias correction

See Table 7 and Figs. 3 and 4.

Table 7 Summary of ML estimation results

Fig. 3

Actual versus predicted series in the US national market, ML estimation results

Fig. 4

Actual versus predicted series in the median MSA market, ML estimation results

Appendix 3: Sources of the pricing error

There are two potential sources of error in $\widehat{e}_t$: (1) The VAR-computed $\widehat{\mathcal {I}}_t$, $\widehat{\varPi }_t$, and $\widehat{\mathcal {G}}_t$ might not be good estimates of $\mathcal {I}_t$, $\varPi _t$, and $\mathcal {G}_t$ under rational expectations and (2) the linearized present value model ignores the second- and higher-order moments. We deal with the first source of error by utilizing the indirect inference estimator, and it turns out that the bias correction procedure considerably reduces the pricing error; see a comparison between Figs. 1 and 3 and between Figs. 2 and 4. In order to investigate the pricing error that comes from the second source, we conduct a Monte Carlo experiment for the US national market by repeating the following steps 1000 times:

• 1. We keep $i_t$ and $\Delta r_t$ as in the original sample and construct a new series of log rent–price ratio with small fluctuations around its sample mean, $rp_t'=\overline{rp}+\nu _t$, where $\nu _t\sim \mathcal {N}(0,\sigma _\nu ^2)$. Then, using $rp_t'$ we back out the relevant housing premium $\pi _t'$.

• 2. We estimate the VAR with $\mathbf {Z}_t=(i_t,\pi _t',\Delta r_t)$ and compute the pricing error, $\widehat{e_t'}=rp_t'-\widehat{rp_t'}$.

Figure 5 compares the average absolute value of the pricing error over 1000 repetitions when $\sigma _\nu$ takes on different values. The black line shows that, when $\sigma _\nu$ equals one tenth of the standard error of the actually observed log rent–price ratio, the pricing error is small and relatively stable. When we increase $\sigma _\nu$ to one half of the observed standard error (the purple line), the pricing error becomes larger in magnitude and the series starts to vary over time. As the log rent–price ratio becomes as volatile as the observed series (the blue line), both the first and the second moments of the pricing error increase dramatically.

Fig. 5

Simulated pricing errors

However, the constant standard error can hardly replicate the model-implied pricing error. Therefore, we assign different standard errors to the simulated log rent–price ratio for the earlier period 1978H2:1999H2 and the later period 2000H1:2014H2. As the red line shows, the pricing error shoots up once the simulated log rent–price ratio becomes more volatile.

This simulation exercise suggests that both the presence and the time variation of the second moment in the log rent–price ratio contribute to a sizeable pricing error. A necessary condition for the Campbell–Shiller decomposition, which utilizes a first-order Taylor approximation around the steady-state log rent–price ratio, to fit the data well, is that the log rent–price ratio is relatively stable over time. In the data, however, the log rent–price ratio experienced large fluctuations between 2000 and 2012, during which period the fitted log rent–price ratio largely deviates from the actual counterpart; see Fig. 1.

Appendix 4: Orthogonal impulse response functions

Given the response functions of state variables to an orthogonal impulse in real interest rate, those for VAR-computed $\widehat{\mathcal {I}}_t$, $\widehat{\varPi }_t$, and $\widehat{\mathcal {G}}_t$ are readily available via a linear combination. According to Eqs. (8)–(10), the responses of $\widehat{\mathcal {I}}_t$, $\widehat{\varPi }_t$, and $\widehat{\mathcal {G}}_t$ to an orthogonal impulse in real interest rate are:

$$\begin{aligned} \widehat{\hbox {OIR}}_{\mathfrak {z}_j,i,t+\tau }=\left\{ \begin{array}{ll} e_j^\prime (\mathbf {I}-\rho \widehat{\varGamma })^{-1}\widehat{\varGamma } \left[ \widehat{\hbox {OIR}}_{\mathbf {Z},i,t+\tau }^\prime \quad \mathbf {0}\right] ^\prime &{} \text { if } \tau =0, \\ e_j^\prime (\mathbf {I}-\rho \widehat{\varGamma })^{-1}\widehat{\varGamma } \left[ \widehat{\hbox {OIR}}_{\mathbf {Z},i,t+\tau }^\prime \quad \widehat{\hbox {OIR}}_{\mathbf {Z},i,t+\tau -1}^\prime \right] ^\prime &{} \text { if } \tau >0, \end{array} \right. \end{aligned}$$

(28)

with $\mathfrak {z}_1=\mathcal {I}$, $\mathfrak {z}_2=\varPi$, and $\mathfrak {z}_3=\mathcal {G}$, which infer the response of the VAR-computed log rent–price ratio to an orthogonal impulse in real interest rate as,

$$\begin{aligned} \widehat{\hbox {OIR}}_{rp,i,t+\tau }=\widehat{\hbox {OIR}}_{\mathcal {I},i,t+\tau }+\widehat{\hbox {OIR}}_{\varPi ,i,t+\tau } -\widehat{\hbox {OIR}}_{\mathcal {G},i,t+\tau }. \end{aligned}$$

(29)

Given the log real rent at time $t-1$, $r_{t-1}$, the $\tau$-period ahead prediction of log real rent with $\tau \ge 0$ can be expressed as:

$$\begin{aligned} \widehat{r}_{t+\tau }=r_{t-1}+\sum _{s=0}^{\tau }\widehat{\Delta r}_{t+s}, \end{aligned}$$

(30)

and $\tau$-period prediction of log real price takes the following form:

$$\begin{aligned} \widehat{p}_{t+\tau }=\widehat{r}_{t+\tau }-\widehat{rp}_{t+\tau }. \end{aligned}$$

(31)

Then, we are able to obtain the $\tau$-period ahead responses of the predicted log real rent and the predicted log real price to an orthogonal interest rate shock at time t,

$$\begin{aligned} \widehat{\hbox {OIR}}_{r,i,t+\tau }&=\sum _{s=0}^{\tau }\widehat{\hbox {OIR}}_{\Delta r,i,t+s},\end{aligned}$$

(32)

$$\begin{aligned} \widehat{\hbox {OIR}}_{p,i,t+\tau }&=\sum _{s=0}^{\tau }\widehat{\hbox {OIR}}_{\Delta r,i,t+s}-\widehat{\hbox {OIR}}_{rp,i,t+\tau }. \end{aligned}$$

(33)