Time-varying role of macroeconomic shocks on house prices in the US and UK: evidence from over 150 years of data

Abstract

In this paper, we study the effect of macroeconomic shocks in the determination of house prices. Focusing on the US and the UK housing market, we employ time-varying vector autoregression models using Bayesian methods covering the periods of 1830–2016 and 1845–2016, respectively. We consider real house prices, output growth, short-term interest rates and inflation as input variables in order to unveil the effect of macroeconomic shocks on house prices. From the examination of the impulse responses of house prices on macroeconomic shocks, we find that technology shocks dominate in the US real estate market, while their effect is unimportant in the UK. In contrast, monetary policy drives most of the evolution of the UK house prices, while transitory house supply shocks are unimportant in either country. These findings are further corroborated with the analysis of conditional volatilities and correlations with macroeconomic shocks. Overall, we are able to unveil the dynamic linkages in the relationship of the macroeconomy and house prices. Over time, we analyze the variations in economic events happening at the imposition of the shock and uncover characteristics missed in the time-invariant approaches of previous studies.

Appendices

Appendix A: Estimation of the TVP-VARs models

Our estimation procedure draws directly from Canova and Gambetti (2010).

• 1. Priors

Let $z^{T}$ denote the sequence of z’s up to time T. Let $\gamma$ be the vector containing the nonzero elements of $F^{ - 1}$ that are different from one and are stacked in rows and $\varXi$ a vector including all the $\varXi_{i}$. The transition density is assumed to be

$$ p\left( {\theta_{t} |\theta_{t - 1} ,\varOmega } \right) \propto I\left( {\theta_{t} } \right)f\left( {\theta_{t} |\theta_{t - 1} ,\varOmega } \right) $$

$$ p\left( {\theta_{t} |\theta_{t - 1} ,\varOmega } \right) = N\left( {\theta_{t - 1} ,\varOmega } \right) $$

where $I\left( {\theta_{t} } \right)$ is an indicator function selecting non-explosive draws of $\theta_{t}$ for $y_{t}$. We assume the hyperparameters and the initial states are independent so that the joint prior is simply the product of the marginal densities. Following Cogley and Sargent (2005) we assume:

$$ P\left( {\theta_{0} } \right) \propto I\left( {\theta_{0} } \right)N\left( {\bar{\theta },\bar{P}} \right) $$

$$ P\left( \varOmega \right) = {\text{IW}}\left( {\bar{\varOmega }^{ - 1} ,T_{0} } \right) $$

$$ P\left( {\log \sigma i_{0} } \right) = N\left( {\log \bar{\sigma }_{i} ,10} \right) $$

$$ P\left( \gamma \right) = N\left( {0,10000 \times I_{4} } \right) $$

$$ P\left( {\varXi_{i} } \right) = {\text{IG}}\left( {\frac{0.01}{2}^{2} ,\frac{1}{2}} \right) $$

where $\bar{\theta },\bar{P}$ are OLS estimates of the VAR coefficients and their variances obtained with the initial sample, $\bar{\varOmega } = \lambda \bar{P}$, $T_{0}$ is the number of observations in the initial sample (40 observations), and $\bar{\sigma }_{i}$ is the estimate of the variance of the residual in equation i obtained using the initial sample. The hyperparameter $\lambda$ is set to 0.0005 for all parameters except for the constant terms of output growth, inflation and interest rate. For these constants, it is set to 0.001.

• 2. Posteriors

To draw realizations from the posterior density, we use the Gibbs sampler. Each iteration is composed of four steps and, under regularity conditions and after a burn-in period, iterations on these steps produce draw from the joint density.

• Step 1$p\left( {\theta^{T} |y^{T} ,\gamma ,\sigma^{T} ,\varXi ,\varOmega } \right)$

Conditional on $\left( {\theta^{T} |y^{T} ,\gamma ,\sigma^{T} ,\varXi ,\varOmega } \right)$ the unrestricted posterior of the states is normal. To draw from the conditional posterior, we employ the algorithm of Carter and Kohn (1994). The conditional mean and variance of the terminal state $\theta^{T}$ is computed using standard Kalman filter recursions, while for all the other states the following backward recursions are employed:

$$ \theta_{{t\left| {t - 1} \right.}} = \theta_{t\left| t \right.} + P_{t\left| t \right.} P_{t\left| t \right.}^{ - 1} \left( {\theta_{t + 1} - \theta_{t\left| t \right.} } \right) $$

$$ P_{{t\left| {t - 1} \right.}} = P_{t\left| t \right.} - P_{t\left| t \right.} P_{t + 1\left| t \right.}^{ - 1} P_{t\left| t \right.} $$

where $p\left( {\theta_{t} |\theta_{t + 1} ,y^{T} ,\gamma ,\sigma^{T} ,\varXi ,\varOmega } \right)\,\sim\,N\left( {\theta_{{t\left| {t + 1} \right.}} ,P_{{t\left| {t + 1} \right.}} } \right)$

• Step 2$p\left( {\gamma |y^{T} ,\theta^{T} ,\sigma^{T} ,\varXi ,\varOmega } \right)$

Given that $\sigma^{T}$ and $y^{T}$ are known $\varepsilon_{t}$ is known and since $u_{t}$ is a standard Gaussian white noise, we have $D_{t}^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} F^{ - 1} \varepsilon_{t} = u_{t}$ or $D_{t}^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \varepsilon_{t} = D_{t}^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \left( {F^{ - 1} - I} \right)\varepsilon_{t} + u_{t}$. We can rewrite the ith equation as $z_{it} = - w_{it} \gamma_{i} + u_{it}$ where $z_{it} = \varepsilon_{it} /\sqrt {\sigma_{it} } , w_{it} = \left[ {\varepsilon_{1t} /\sqrt {\sigma_{1t} } , \ldots ,\varepsilon_{i - 1,t} /\sqrt {\sigma_{i - 1,t} } } \right]$ and $\gamma_{i}$ is the column vector formed by the nonzero elements of the ith row of $F^{ - 1} - I$. Given the normal prior, the posterior is $\gamma_{i} = N\left( {F_{1,i} ,V_{1,i} } \right)$ where $F_{1,i} = V_{0,i} \left( {V_{0,i}^{ - 1} \gamma_{0,i} + w_{i}^{ '} z_{i} } \right)$ and $V_{1,i} = \left( {V_{0,i}^{ - 1} + w_{i}^{ '} w_{i} } \right)$ with $V_{0,i}$ and $\gamma_{0,i}$ the prior variance and mean, respectively. Drawing for $i = 2,3,4$, we obtain a draw for $\gamma$.

• Step 3$p\left( {\sigma^{T} |y^{T} ,\theta^{T} ,\gamma ,\varXi ,\varOmega } \right)$

The elements of $\sigma^{T}$ are drawn using the univariate algorithm by Jacquier et al. (2004) along the lines described in Cogley and Sargent (2005) (see “Appendix B” for details).

• Step 4$p\left( {\varXi_{i} |y^{T} ,\theta^{T} ,\gamma ,\sigma^{T} ,\varOmega } \right)$, $p\left( {\varOmega |y^{T} ,\theta^{T} ,\gamma ,\sigma^{T} ,\varXi } \right)$

Conditional on $y^{T} ,\theta^{T} ,\gamma ,\sigma^{T}$ and under conjugate priors, all the remaining hyperparameters can be sampled in a standard way from Inverted Wishart and Inverted Gamma densities (Gelman et al. 1995). We perform 20,000 repetitions and discard the first 5000 draws and, for inference, we keep one every 10 of the remaining draws to break the autocorrelation of the draws.

Appendix B: TVP-VAR parameters

Figures 13, 14 and 15 present graphically the time-varying coefficients for the output growth, inflation and interest rate equations in the VAR for the US, while Figs. 16, 17 and 18 present the respective coefficients for the UK

Fig. 13

US posterior median coefficients estimates and their 68% intervals for output growth equation

Fig. 14

US posterior median coefficients estimates and their 68% intervals for inflation equation

Fig. 15

US posterior median coefficients estimates and their 68% intervals for interest rates equation

Fig. 16

UK posterior median coefficients estimates and their 68% intervals for output growth equation

Fig. 17

UK posterior median coefficients estimates and their 68% intervals for inflation equation

Fig. 18

UK posterior median coefficients estimates and their 68% intervals for interest rates equation

Appendix C: Volatility evolution

See Tables 3 and 4.

Table 3 US volatility evolution

Table 4 UK volatility evolution

Appendix D: Structural Breaks

See Table 5.

Table 5 Bai–Perron multiple break test results