2.1. Data

To be consistent with Del Negro and Otrok (Reference Del Negro and Otrok>2007), we use the FHFA (then OFHEO) seasonally adjusted house price indexes for the 50 US states and District of Columbia over the quarterly period of 1975Q1 to 2017Q4, with the start date driven by the availability of the house price data, and the end of the sample corresponding to the latest data at the time of writing this paper. The FHFA house price indexes provide a broad measure of the movement of single-family house prices. The FHFA indexes are weighted, repeat-sales data, that is, they measure average price changes in repeat sales or refinancings on the same properties. This information is obtained by reviewing repeat mortgage transactions on single-family properties whose mortgages have been purchased or securitized by Fannie Mae or Freddie Mac since January 1975. In particular, we use the quarterly “All-Transactions Indexes.”Footnote ⁶ To create a real version of house price, we deflate the indexes by the (seasonally adjusted) Consumer Price Index (CPI) of the USA, derived from the FRED database of the Federal Reserve Bank of St. Louis. We work with the quarter-on-quarter (QoQ) version of the real house price indexes to obtain the national and local factors from the DFM-TV-SV model for both real housing returns and the corresponding stochastic volatilities.

As far as the data used in the change-point VAR are concerned, besides the two national factors of real housing returns and stochastic volatility, we include data on the federal funds rate (FFR), QoQ growth rate of seasonally adjusted real gross domestic product (GDP), QoQ growth of the CPI measuring the inflation rate, and the term spread, which was defined as the difference between 10-year government bond yield and the FFR. Data on FFR, real GDP, and the long-term government bond yield are again sourced from the FRED database. The transformations of the data imply that our effective sample covers the period of 1975Q2 to 2017Q4.

2.2. The generalized DFM

In this section, we present a generalized DFM that is employed to decompose the real housing returns in all states into a common (or national) factor and an idiosyncratic (or state-specific) factor. The DFM is often used to tease out the common movements among multiple time series and has become a standard tool since the work by Stock and Watson (Reference Stock and Watson>1989). We generalize the standard DFM with constant parameters to one that allows for time-varying loading parameters and the stochastic volatility (DFM-TV-SV, henceforth). As such, the generalized DFM-TV-SV captures important time-varying comovements among multiple time series. Formally, our model specification closely follows Del Negro and Otrok (Reference Del Negro and Otrok>2008) and is specified as followsFootnote ⁷ :

(1) \begin{align} r_{i,t}=\beta _{i,t}\cdot f_t + e_{i,t} \end{align}

Here, $r_{i,t}$ is the first difference of the natural log of the real house price for state $i$ at time $t$ . $f_t$ is the national factor that affects all house prices at time $t$ , and $\beta _{i,t}$ is the time-varying loading parameter of this national factor. $e_{i,t}$ is the idiosyncratic factor.

The common factor and the idiosyncratic factors are assumed to be independent from each other. Therefore, the variance decomposition of our model is given by:

(2) \begin{align} Var(r_{i,t})=\beta _{i,t}^2 \cdot Var(f_{t})+ Var(e_{i,t}) \end{align}

Note that either the time-varying loading parameters or the stochastic volatility of the factors enables the factors contributions to the total variations of each variable to vary over time.

Following the standard practice in this literature, we model the common factor $f_t$ using a stationary AR( $p$ ) process:

(3) \begin{align} f_t=\phi ^f_1 f_{t-1} +\phi ^f_2 f_{t-2}+\ldots \ldots \phi ^f_p f_{t-p} + exp\!\left(h^f_t\right)\cdot \varepsilon ^f_t \end{align}

where $\varepsilon ^f_t \sim i.i.d. N\!\left(0,\sigma ^2_f\right)$ . Therefore, the shock to the factor has a stochastic volatility, and its time-varying volatility is governed by $exp\!\left(h^f_t\right)$ .

To keep the model parsimonious, we employ a driftless random walk process to capture the time variation of the volatility:

(4) \begin{align} h^f_t=h^f_{t-1}+\sigma ^h_f \cdot \xi ^f_t, \quad \xi ^f_t \sim i.i.d. N(0,1) \end{align}

The factor loading $\beta _{i,t}$ varies over time and is also assumed to follow a random walk process:

(5) \begin{align} \beta _{i,t}=\beta _{i,t-1}+\sigma ^{\beta }_i \cdot \eta _{i,t} \ ; \ \eta _{i,t} \sim i.i.d.N(0,1) \end{align}

Here, shocks to the loading parameters in different series are assumed to be orthogonal to each other.Footnote ⁸

The idiosyncratic factor follows a stationary AR( $q$ ) process:

(6) \begin{align} e_{i,t}=\phi _{i,1} e_{i,t-1} +\phi _{i,2} e_{i,t-2}+ \cdots + \phi _{i,q} e_{i,t-q}+exp(h_{i,t}) \cdot \varepsilon _{i,t} \end{align}

where $\varepsilon _{i,t} \sim i.i.d. N\!\left(0,\sigma ^2_i\right)$ . The stochastic volatility of the idiosyncratic factor follows a random walk process:

(7) \begin{align} h_{i,t}=h_{i,t-1}+\sigma ^{h}_{i} \cdot \xi _{i,t}, \quad \xi _{i,t} \sim i.i.d.N(0,1) \end{align}

Here, we assume that the shocks to the stochastic volatility in different factors are independent from each other. This assumption simplifies the estimation algorithm.

As usual, some normalizations of the factor rotations are needed before the model can be identified and estimated. The loading parameters and the variance of the shock to the common factor are not separately identifiable. We choose to set $\sigma ^2_f=1$ to achieve the identification. Following Del Negro and Otrok (Reference Del Negro and Otrok>2008), we also restrict that the time-varying volatility all starts from zero for the same identification purpose. We demean each series before the estimation since the means of factors are not separately identifiable. Finally, following works such as Neely and Rapach (Reference Neely and Rapach>2011) and Bhatt et al. (Reference Bhatt, Kishor and Ma>2017), we set $p=q=2$ to keep the model parsimonious.

2.3. Estimation procedure

We estimate this DFM-TV-SV model using the Monte Carlo Markov Chain (MCMC) Bayesian estimation method. Specifically, we employ the well-established Gibbs sampling algorithm by breaking the model into several blocks and sampling sequentially from posterior conditional densities. The idea of the Gibbs sampling algorithm is that when the algorithm converges after the initial burn-in draws, these random draws from the conditional densities altogether constitute a good approximation of the underlying joint densities. Applying the law of large numbers, the numerical integration can be easily taken to obtain the marginal densities of the parameters and the state variables of our interest. Most blocks in the model are linear and Gaussian, and as a result the standard algorithms in Kim and Nelson (Reference Kim and Nelson>1999) are readily applicable. The stochastic volatility introduces a non-Gaussian feature into the model. We apply the procedure proposed in Kim et al. (Reference Kim, Shephard and Chib>1998) that utilizes a mixture of normal densities to approximate the underlying non-Gaussian distribution in order to simulate the stochastic volatility. This procedure has been widely used in the literature, see, for example, Stock et al. (Reference Stock, Watson and to Forecast?>2007) and Primiceri (Reference Primiceri>2005).

We briefly outline the Gibbs sampling estimation algorithm below. Further details are given in Appendix A.2.

1. 1. Cast the model into its state-space form as given in Appendix A.2 and draw the national factor $\{f_t\}_{t=1}^T$ from the conditional density:

   \begin{equation*} f\!\left (\{ f_t \}^{T}_{t=1} \bigg | \left\{\{\beta _{i,t}\}^{T}_{t=1}\right\}^{n}_{i=1},\phi _f, \{\phi _{i,e}\}^{n}_{i=1},\left\{\sigma ^2_{i}\right\}^{n}_{i=1}, \left\{h_{t}^{f}\right\}^{T}_{t=1}, \bigl \{\{ h_{i,t}\}^{T}_{t=1}\bigr \}^{n}_{i=1}, \left\{\{r_{i,t}\}_{t=1}^T\right\}_{i=1}^n \right ), \end{equation*}
   where $\phi _f = \left(\phi ^f_{1},\phi ^f_{2},\ldots \phi ^f_{p} \right)^{\prime }$ , and $\phi _{i,e} = \left(\phi _{i,1},\phi _{i,2},\ldots,\phi _{i,q} \right)^{\prime }$ for $i=1,2,\ldots,n$ .

2. 2. Draw the AR parameters of the common factor from the conditional density:

   \begin{align*} f\!\left (\phi _f \bigg | \{f_t\}^T_{t=1}, \{h^f_t\}^T_{t=1}\right ) \end{align*}
   Draws of the AR parameters outside the unity circle are discarded to ensure the stationarity.

3. 3. Sample the AR and variance parameters for each idiosyncratic factor from the conditional density:

   \begin{align*} f\left( \phi _{i,e},\sigma ^2_i \bigg | \{f_t\}^T_{t=1}, \left\{\beta _{i,t}\right\}^T_{t=1},\left\{h_{i,t}\right\}^T_{t=1}, \left\{\{r_{i,t}\}_{t=1}^T\right\}_{i=1}^n \right) \end{align*}
   Because the idiosyncratic factors are orthogonal to each other, these draws can be made one by one for each $i=1,2,\ldots ..,n$ . Again, to ensure the stationarity any draw of the AR parameters outside the unity circle is discarded.

4. 4. Draw the loading parameters and the shock variance parameters from the conditional density:

   \begin{align*} f\biggl ( \{\beta _{i,t}\}^T_{t=1},(\sigma ^\beta _i)^2 \bigg | \{f_t\}^T_{t=1}, \phi _{i,e},\sigma ^2_i,\{h_{i,t}\}^T_{t=1}\biggr ) \end{align*}
   Again, due to the orthogonality condition, these draws can be made one by one for each $i=1,2,\ldots ..,n$ .

5. 5. Draw the stochastic volatility of the common factor from the conditional density:

   \begin{align*} f\biggl (\{h^f_{t}\}^T_{t=1},\sigma ^h_f \bigg | \{f_t\}^T_{t=1}, \phi _f\biggr ) \end{align*}
   And draw the stochastic volatility of the idiosyncratic factor from the conditional density:
   \begin{align*} f\biggl ( \{h_{i,t}\}^T_{t=1},\sigma ^h_i \bigg | \{f_t\}^T_{t=1},\{\beta _{i,t}\}^T_{t=1},\phi _{i,e}, \{r_{i,t}\}_{t=1}^T\biggr ) \end{align*}

Starting with initial values, we repeat steps (1) through (5) for $(D+S)$ number of times. Here, $D$ is the initial burn-in draws needed for the algorithm to converge, and the results are based on the saved $S$ number of draws. We set $D$ to 2000 and $S$ to 8000.

2.4. Change-point VAR model

This section reviews the empirical model used for structural analysis. It also discusses how the marginal likelihood of the model is applied to determine: (i) the number of regimes and (ii) the order of lags. A similar framework has been applied by Kapetanios et al. (Reference Kapetanios, Mumtaz, Stevens and Theodoridis>2012) and Liu et al. (Reference Liu, Theodoridis, Mumtaz and Zanetti>2019).

To assess whether agents’ responses to macroeconomic shocks vary across regimes, the following VAR model is estimated:

(8) \begin{equation} Z_{t}=c_{S}+\sum \limits _{j=1}^{K}B_{S}Z_{t-j}+\varepsilon _{t}, \end{equation}

where $\varepsilon _{t} \sim N (\mathbf{0},\Omega _{S} )$ and the data matrix $Z_{t}$ contains quarterly data on the FFR, real GDP growth, inflation, term spread, and the national factors of real housing returns and volatility derived from the DFM-TV-SV model. $B_{S}$ and $\Omega _{S}$ denote the VAR coefficient and covariance matrix, respectively, which vary across regimes.

The empirical model permits $M$ breaks to take place at unknown dates, and as in Chib (Reference Chib>1998) the evolution of these breaks is prescribed by the latent state variable, $S_t$ . The latter state variable follows an $M$ state Markov chain with restricted transition probabilities, $p_{ij}=p ( S_{t}=j|S_{t-1}=i )$ , given by

(9) \begin{eqnarray} p_{ij} &\gt &0\text{ if }i=j \\ p_{ij} &\gt &0\text{ if }j=i+1 \nonumber \\ p_{MM} &=&1 \nonumber \\ p_{ij} &=&0\text{ otherwise. } \nonumber \end{eqnarray}

For example, if $M=3$ , the transition matrix is defined as

\begin{equation*} \tilde {P}=\left ( \begin {array}{c@{\quad}c@{\quad}c} p_{11} & 0 & 0 \\[2pt] 1-p_{11} & p_{22} & 0 \\[2pt] 0 & 1-p_{22} & 1 \end {array}\right ). \end{equation*}

Equations (8) and (9) describe a Markov switching VAR with nonrecurrent states where transitions from one regime are restricted to happen sequentially. For example, to move from regime 1 to regime 3, the process has to visit regime 2. The transition matrix also precludes transitions to past regimes. As discussed in Sims et al. (Reference Sims, Waggoner and Zha>2008), this is a Markov switching model where structural breaks are modeled as multiple change points. We believe that this approach is advantageous over standard Markov switching models as it permits the user to associate changes in the macroeconomic dynamics with structural breaks in the economy. For instance, it is shown below that the marginal likelihood metric selects the best “fitting” model as the one with three regimes. Furthermore, these regimes seem to coincide with the Great Inflation, Great Moderation, and Great Recession-ZLB periods.

2.5. Estimation and selection of the number of change points and lags

We follow Chib (Reference Chib>1998) and adopt a Bayesian Gibbs sampling approach to the estimation of the change-point VAR models. Appendix B provides a detailed description of the prior and Appendix C describes the main steps of the algorithm. The only feature that is perhaps important to be mentioned here is that during the last regime, the policy rate does not respond to any variable into the system (to proxy for the ZLB). This characteristic is imposed via tight priors (see the discussion in the Appendix B).

The choice of the number of breakpoints is a crucial specification issue. The number of regimes is selected by comparing the marginal likelihood across models (i.e., different number of regimes or/and lags). The maximum number of regimes has been set equal to three, while the maximum number of lags equals to four. Both choices are driven by concerns regarding the limited number of observations per regime.Footnote ⁹

Given $m$ and $l$ , the marginal likelihood is estimated based on Chib (Reference Chib>1998) and Bauwens and Rombouts (Reference Bauwens and Rombouts>2012):

(10) \begin{equation} \ln G\!\left ( Z_{t}\mid m, k \right ) =\ln f\!\left ( Z_{t}\mid m, k,\Theta,\tilde{P}\right ) +\ln p\!\left ( \Theta,\tilde{P}\mid m, k\right ) -\ln g\!\left ( \Theta,\tilde{P}\mid Z_{t},m, k\right ) \end{equation}

where $\ln G (Z_{t}\mid m, k )$ denotes the marginal likelihood, $\ln f ( Z_{t}\mid m, k,\Theta,\tilde{P} )$ is the likelihood, while $\ln p ( \Theta,\tilde{P}\mid m, k )$ and $\ln g ( \Theta,\tilde{P}\mid Z_{t},m, k )$ are the prior and posterior distribution of the VAR parameter vector, respectively. Note that $\ln G ( Z_{t}\mid m )$ does not depend on the parameters of the model and in theory it can be evaluated at any value of the parameters. Following standard practices, we evaluate the marginal likelihood at the posterior mean. The first two terms on the right-hand side of equation (10) are easily evaluated, whereas the calculation of the normalizing constant $\ln g ( \Theta,\tilde{P}\mid Z_{t},m )$ requires some work. As described in detail in Bauwens and Rombouts (Reference Bauwens and Rombouts>2012), this term can be evaluated by considering reduced Gibbs runs on an appropriate factorization of $g ( \Theta,\tilde{P}\mid Z_{t},m ) .$ We use 10,000 additional Gibbs replications to evaluate $g ( \Theta,\tilde{P}\mid Z_{t},m )$ at the posterior mean.

2.6. Shock identification

This section explains briefly the identification scheme employed in this paper, and it is motivated by the work of Uhlig (Reference Uhlig>2004), Mountford and Uhlig (Reference Mountford and Uhlig>2009), and Barsky and Sims (Reference Barsky and Sims>2011). The identified shocks maximize their contribution on selected variables and also satisfy the sign restrictions described in Table 1, which are imposed for four periods.

Table 1. Sign restrictions

Notes: All sing restrictions have been imposed for $4$ periods. During the ZLB regime, the policy rate does not respond to the demand and supply shock as well (i.e., additional zero restrictions).

The mapping between reduced and structural errors is given by

(11) \begin{eqnarray} \varepsilon _t=A_{0,S} v_t \end{eqnarray}

For any orthogonal matrix $D$ ( $DD^{\prime }=I$ , where $I$ is the identity matrix), the above mapping can be written as

\begin{equation*} \varepsilon _{t}=A_{0,S}D_{S}v_{t} \end{equation*}

Since

\begin{equation*} \Omega _{S}=A_{0,S}D_{S}D_{S}^{\prime }A_{0,S}^{\prime }=A_{0,S}A_{0,S}^{\prime } \end{equation*}

Using the companion form of the VAR(p) model, the impulse of variable $j$ and the impulse of shock $i$ in the period $h$ can be expressed as

(12) \begin{equation} IRF_{i,j}(h)=J_{j}B_{S}^{h-1}A_{0,S}D_{S}J_{i}^{\prime } \end{equation}

where $J_{i}$ and $J_{h}$ are selection matrices of zeros and ones.

In Uhlig (Reference Uhlig>2005), the matrix $D_{S}$ results from the following minimization problem:

(13) \begin{equation} D_{S}^{\ast }=\arg \min \sum _{j\in \mathcal{I}_{+}}\sum _{h_{j}=\tilde{h}_{j}}^{H_{j,+}\in H_{+}}f\!\left ( -\frac{J_{j}B_{S}^{h-1}A_{0,S}D_{S}J_{i}^{\prime }}{\sigma _{j,S}}\right ) +\sum _{j\in \mathcal{I}_{-}}\sum _{h_{j}=\tilde{h}_{j}}^{H_{j,+}\in H_{+}}f\!\left ( \frac{J_{j}B_{S}^{h-1}A_{0,S}D_{S}J_{i}^{\prime }}{\sigma _{j,S}}\right ) \end{equation}

s.t.

\begin{equation*} D_{S}D_{S}^{\prime }=I \end{equation*}

where $\sigma _{j,S}$ is the standard deviation of variable $j$ and $f ( x ) = \left\{ \begin{array}{c} 100x\text{ if }x\geq 0 \\[2pt] x\text{ otherwise}\end{array}\right. $ . Finally, $\mathcal{I}_{+}$ is the index set of variables, for which identification of a given shock restricts the impulse response to be positive and $\mathcal{I}_{-}$ is the index set of variables, for which identification restricts the impulse response to be negative. The use of this scheme is to identify “meaningful” macroeconomic shocks and to study how these disturbances affect house price movements.

Although the identification scheme employed shares many similarities with the one developed by Mountford and Uhlig (Reference Mountford and Uhlig>2009) and Barsky and Sims (Reference Barsky and Sims>2011), two features of the proposed methodology are of worth to be mentioned explicitly. The first point is that the matrix $D_{S}$ is regime dependent, meaning that the above maximization problem (expression 13) needs to be performed for each regime. More importantly, no restriction is placed on the house price level and volatility factors. In other words, our approach about the effects of “standard” macroeconomic shocks on the series that describe the evolution of the common components associated with real housing returns and stochastic volatility is agnostic.

Table 1 reports the sign restrictions employed to identify a demand, supply, monetary policy, and term-spread shock. The restrictions for the first three shocks are uncontroversial, and they have been used in the literature extensively. Although the last shock has been studied in a number of papers Kapetanios et al. (Reference Kapetanios, Mumtaz, Stevens and Theodoridis>2012), Baumeister and Benati (Reference Baumeister and Benati>2013), and Liu et al. (Reference Liu, Theodoridis, Mumtaz and Zanetti>2019) (among others), it is less common and aims to capture movements at the long end of the yield curve that are not induced by variations in the policy rate. During the Great Moderation period, this shock could reflect the foreign capital inflows to the USA, capturing the so-called savings glut phenomenon [Sá et al. (Reference Sá, Towbin and Wieladek>2014), Sá and Wieladek (Reference Sá and Wieladek>2015), Cesa-Bianchi et al. (Reference Cesa-Bianchi, Ferrero and Rebucci>2018)], while during the ZLB period, this shock could proxy the Federal Reserve’s unconventional policies.

3.1. The national factor of the real housing returns

Figure 1 plots the national factor (in a solid line), together with the 90% probability intervals (in dotted lines)Footnote ¹⁰ . One important advantage of this generalized DFM is that it allows exposures of the real housing returns in all states to the national factor to vary over time and thus permits a time-varying integration of the local housing market with the national market. The national factor started to increase in 1975 but then declined in the late 1970s through the early 1980s. Since around the mid-1980s, the national factor had steadily risen for an extended period of time until about 2006, when the national factor plunged leading to a severe financial crisis and recession in 2007–2009. The national real housing returns factor leveled off around 2010 and has rebounded sharply since 2011. The dynamics of the national factor of real housing returns estimated from our DFM-TV-SV model is in general consistent with those findings documented over the common period of the past literature (see, e.g., Del Negro and Otrok (Reference Del Negro and Otrok>2007) and Fairchild and ShuWu (Reference Fairchild and ShuWu>2015)). We explain this pattern below by identifying various shocks in the context of a change-point VAR framework.

Figure 1. The national factor.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Figures 2 and 3 plot the time-varying loading parameters of the national factor, together with the 90% probability intervals, for the growth of real house price in each state. Overall, the exposures of the real housing returns in all states to the national factor vary substantially over time. These loading parameters appear to be positive for all states at all time periods, indicating that there is no identification issue. The dynamic patterns of these time variations also display substantial heterogeneity across states. A number of states, including Alaska, California, Delaware, Florida, Hawaii, Idaho, Illinois, Maryland, Nevada, New York, Oregon, Virginia, Vermont, Washington, and Wisconsin, all witnessed a steady increase in the exposures of their real housing returns from the early 1990s to the pre-crisis period. Interestingly, many other states, including Arkansas, Colorado, Connecticut, DC, Iowa, Indiana, Kansas, Kentucky, Louisiana, Massachusetts, Maine, Mississippi, North Dakota, Nebraska, New Hampshire, Ohio, Oklahoma, Rhode Island, South Dakota, Texas, Utah, West Virginia, and Wyoming, all experienced a steady decline in the exposures of their real house price growth in the same period.

Figure 2. Time-varying loading parameters of the national factor.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Figure 3. Time-varying loading parameters of the national factor—continued.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Figure 4 shows the stochastic volatility of the national factor with its 90% probability interval. There is a rapid and substantial increase in the stochastic volatility of the national factor from around 1998 to around 2011, followed by a large decline afterwards. Figures 5 and 6 plot the stochastic volatility of each state-specific factor. Overall, there is a substantial time variation in the stochastic volatility for all idiosyncratic factors, and there is a substantial heterogeneity in the dynamic patterns across states.

Figure 4. The stochastic volatility of the national factor.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Figure 5. The stochastic volatility of the individual factor.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Figure 6. The stochastic volatility of the individual factor—continued.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Recall, from equation (2), that both the time-varying loading parameters and the stochastic volatility of the national and the state-specific factors jointly determine the time-varying contributions of the national factor to the total variations of the growth in real house price in the states. Figures 7 and 8 show the dynamic evolution of the relative contribution of the national factor to the real housing returns in all states. The relative contribution of the national factor has increased since the mid-1980s for most states. These include states that have experienced an overall steady increase in its loading of the national factor, such as California and New York, and states that have witnessed a decline in its loading of the national factor, such as Connecticut and Massachusetts. For the latter group of states, it seems that the decline in the loading of the national factor is more than offset by a large increase in the stochastic volatility of the national factor and a large decline in the stochastic volatility of the idiosyncratic factor, resulting in an increasing relative contribution of the national factor. To quantify the relative contribution of the national factor to the total variations, we note that the contribution of the national factor for the full sample period in all states is 44.85% on average. This contribution is only about 28.85% on average during the period from 1975Q1 to 1989Q4, rises to as much as 52.12% during the period from 1990Q1 to 2006Q4, and remains as high as 55.08% during the period from 2007Q1 to 2017Q4. Although the contribution of the national factor has declined somewhat after the financial crisis in some states, we conclude that overall the role of the national factor in explaining the house prices in all states is not only critical but also has been increasing to become more important than the local factors since around 1990. These findings are broadly in line with previous works such as Del Negro and Otrok (Reference Del Negro and Otrok>2007), but recall the sample period of that study ended in 2005 and hence did not cover the most recent periods, including that of the GFC which corresponded with a tumultuous period of the US housing market. Given the dominance of the national factor in explaining state-level house price movements, especially since 1990, identifying the role of various macroeconomic shocks in driving this common real housing returns component in a regime-specific context is clearly of paramount importance, and this is what we focus on in detail below in a short while.

Figure 7. Time-varying variance contributions of the national factor.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

Figure 8. Time-varying variance contributions of the national factor—continued.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

3.2. Cross-state time-varying correlation

The generalized DFM as employed here can capture potentially time-varying comovements among multiple series. To this end, we compute the implied correlation for each pair of states and present the average of these pairwise correlations at each time point in Figure 9. The average cross-state correlations increased from the mid-1980s till around 2011 and then declined until the end of the sample. The increase in this correlation was more rapid in 1985–1995 than in 1996–2005. This correlation increased more rapidly again between 2005 and 2011, which may have been driven by the financial crisis.

Figure 9. The time-varying average of cross-states correlations.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

3.3. Cross-sectional dispersion in volatility

Another metric that can be computed based on the DFM-TV-SV model to usefully summarize the dynamic patters of these growth in real house prices is the volatility dispersion, which is the standard deviation of the implied volatility of all states, as shown in Del Negro and Otrok (Reference Del Negro and Otrok>2008). In Figure 10, we present this time-varying volatility dispersion and the decomposition of it into the component driven by the national factor and that by the state-specific factor. We find that there is a large increase in the volatility dispersion at the beginning of the sample period till around the mid-1980s, followed by a large decline. The third panel in this figure indicates that this rise and decline in the total volatility dispersion is primarily attributed to the idiosyncratic factor. The total volatility dispersion increased between 2000 and 2009 again and then declined until the end of the sample. The second panel suggests that this is primarily due to the national factor.

Figure 10. The total volatility dispersion and its decomposition into the national and the individual components.

Notes: The solid black line is the median of the posterior distribution, while the dotted lines represent the $5\%-95\%$ percentiles.

3.4. Change-point VAR model specification and evolution of regimes

Next, we focus our attention on the change-point VAR, using which we analyze the impact of various identified macroeconomic shocks on the national factors of real housing returns and stochastic volatility.Footnote ¹¹ We start with the discussion of the dynamic specification of the model. Table 2 reveals that the model that gets the most support from the data is the most flexible one. In words, the data prefer a model with three regimes and four lags.Footnote ¹²

Figure 11 helps us to understand the model selection implied by the marginal likelihood statistic. The evolution of the regimes coincides with the three phases of inflation and, consequently, the policy rate during the time interval considered in this study. Namely, the first regime is associated with high inflation and policy rate (“Great Inflation”). In contrast, the second regime overlaps (mostly) with the Great Moderation (stable low inflation and policy rate) and the final regime with the ZLB period. Interestingly, the second regime more or less corresponds to sample period used by Del Negro and Otrok (Reference Del Negro and Otrok>2007) under the rationale that this sample corresponds to a single monetary policy regime and thus helps in correctly identifying the monetary policy shocks. The “good-luck” versus “good-policy” and ZLB literature document that the dynamics across these different regimes are dramatically different (see Sims and Zha (Reference Sims and Zha>2006), Cogley and Sargent (Reference Cogley and Sargent>2005), and Liu et al. (Reference Liu, Theodoridis, Mumtaz and Zanetti>2019) among others). In addition, when we look at these three regimes, we observe the first regime being characterized by the declining national factor of real housing returns and then recovery of the same with low volatility,Footnote ¹³ followed by the rapid increases with eventual collapse and highly volatile national factors in the second regime, and then finally the recovery of the common component with declining variability in the wake of expansionary unconventional monetary policies. As a result, a model with sufficient flexibility is required to capture accurately the extensive non-linearities in the data. The results displayed in Table 2 are inline with this rationale.

3.5. Shocks

The evident time-varying importance of the national real housing returns factor in explaining movements at the state-level especially post-1990 tends to suggest that the housing market boom before the collapse in 2007, and then again the recovery after that is not necessarily purely driven by local factors (“local bubbles”). Hence, this section is devoted to studying the responses of the economy including the national housing market factors of returns and volatility, to each identified shocks and whether these responses are regime dependent. The importance of the shock across all three regimes is also assessed in this section.

Table 2. Marginal likelihood comparison

Figure 11. Evolution of regimes.

Notes: Observed data (solid blue line), regime 1 (red shaded area), span between 1975Q1 and 1984Q4, regime 2 (blue shaded area) between 1985Q1 and 2008Q4, and regime 3 (yellow shaded area) between 2009Q1 and 2017Q4.

3.5.1. Demand shock

Figure 12 illustrates responses to a demand shock. To remind the reader, the effects of the shock on the term spread, the real housing returns factor, and the house prices volatility factor are left unrestricted. Figure 12 makes apparent that the transmission of the shock to the macroeconomy varies dramatically across the three regimes. The shocks seems to have a larger effect on the economy during the “Great Moderation” (regime 2), and the forecast variance decomposition also confirms this (Figure 13 and Table 3). Although the effect on the real GDP growth is comparable across the three regimes, inflation rises substantially more in the second regime, and this leads to a protracted increase in the interest rate. Despite the long-lasting policy rate increase, the term-spread falls indicating that the long-term interest rate rises by less than the short end of the yield curve, perhaps due to well-anchored inflation expectation during the Great Moderation. The housing factor increases initially, but this is only a short-lasting effect, since as monetary authorities start “fighting” higher inflation and the policy rate increases, the real housing returns factor falls.

Figure 12. Impulse responses: demand shock.

Notes: The solid (black) line represents the (pointwise) median, while the shaded area captures the $16\%-84\%$ percentiles of the posterior distribution. The shock has been normalized to increase GDP growth by 1 percentage point in the second quarter.

Figure 13. Forecast variance decomposition.

During the Great Moderation, the demand shock explains $50\%$ of the variability of the policy rate, indicating that the Federal Reserve’s intense effort to mitigate the inflationary consequences of the shock. This elevated impact is also reflected on GDP ( $20\%$ ), inflation ( $20\%$ ), term spread ( $35\%$ ), real housing returns factor ( $30\%$ ), and real house price volatility factor ( $20\%$ ).

Table 3. Forecast variance contributions

Notes: The table reports the posterior mean forecast variance shares. $R=1$ , $R=2$ , and $R=3$ indicate regimes $1$ , $2$ , and $3$ , respectively. While $H=1Q$ , $H=4Q$ , $H=12Q$ , and $H=40Q$ refer to forecast horizons, $1$ quarter, $4$ quarter, $1$ year, and $10$ years, respectively.

3.5.2. Supply shock

The responses of output growth, inflation, and the policy rate are stronger in the first than in the other two regimes (Figure 14). The stimulative monetary policy needed for the (negative) output gap to be closed leads to the higher housing returns factor in regimes 1 and 2. This effect is supported further by the strong (income) growth in the economy, which also leads to lower volatility of the national factor of housing returns for regimes 1 and 2.

Figure 14. Impulse responses: supply shock.

Notes: The solid (black) line represents the (pointwise) median, while the shaded area captures the $16\%-84\%$ percentiles of the posterior distribution. The shock has been normalized to increase GDP growth by 1 percentage point in the second quarter.

The forecast variance contribution of the shock does not seem to vary across the $1^{st}$ and $2^{nd}$ regimes and it fluctuates between $10\%$ and $20\%$ (Figure 13 and Table 3). The supply shock plays a more important role during the $3^{rd}$ regime, with its contribution to GDP growth, and the factors of real housing returns and stochastic volatility rising above $20\%$ .

3.5.3. Policy shock

A policy “cut” increases output growth persistently during the second regime (Figure 15). Interestingly, the term spread rises approximately by $200$ basis points suggesting that long-term interest rates do not decrease as much as the policy rate. The stimulative environment created by the Federal Reserve leads to higher values of the real housing returns factor during the Great Moderation, while this effect is insignificant in the first regime. This highly accommodative policy results in higher volatility of the national housing returns factor, although this effect is not precisely estimated.

Figure 15. Impulse responses: policy shock.

Notes: The solid (black) line represents the (pointwise) median, while the shaded area captures the $16\%-84\%$ percentiles of the posterior distribution. The shock has been normalized to increase GDP growth by 1 percentage point in the second quarter.

The forecast variance contribution to GDP growth and inflation is admittedly quite small (Figure 13 and Table 3), which is in line with a number of existing studies (see Bernanke et al. (Reference Bernanke, Boivin and Eliasz>2005), Smets and Wouters (Reference Smets and Wouters>2007), and Justiniano et al. (Reference Justiniano, Primiceri and Tambalotti>2010) among others). On the other hand, the contribution of the policy shock on the term spread and national housing returns factor rises to (almost) $30\%$ ,Footnote ¹⁴ indicating the strong link between policy actions and decision of agents to invest in either long-term or/and housing debt.

3.5.4. Slope shock

The slope shock is a perturbation that lowers the long end of the yield curve, while the policy rate remains constant contemporaneously (Figure 16). In the first regime, the shock stimulates demand and inflation. As the authorities start increasing the policy rate to restore price stability, the national house price factor starts falling with (about) a year delay, while the corresponding volatility increases contemporaneously. These effects carry over to the next regime. However, the response of volatility of the national housing returns factor is not precisely estimated. As the economy moves into the ZLB regime, the responses of the level and volatility of common factors change sign. In this regime, real housing returns factor increases and volatility falls (but again, the latter response is not precisely estimated). Recall that, the slope shock in the last regime proxies the Federal Reserve’s unconventional policies adopted to repair macroeconomic stability (something which we will return to below) in the wake of the Great Recession.

Figure 16. Impulse responses: slope shock.

Notes: The solid (black) line represents the (pointwise) median, while the shaded area captures the $16\%-84\%$ percentiles of the posterior distribution. The shock has been normalized to increase GDP growth by 1 percentage point in the second quarter.

The shock seems to have a limited effect on the macroeconomic variables including the term spread (Figure 13 and Table 3). An exception is the contribution of the GDP on growth during the Great Moderation, especially at short horizons such as one quarter and one year, whereby the shock explains $35\%$ and $30\%$ of the variability of growth. The contribution of the shock to the slope of the yield curve rises as the forecast horizon increases. However, the magnitude of this effect is (approximately) $15\%$ . Interestingly, the contribution of the term-spread shocks is much higher for the remaining two variables (between $20\%$ and $30\%$ ). What seems to be more noteworthy is the fact that the contribution of the shock peaks in the third regime.

3.6. Sensitivity analysis

It is illustrated in the Appendix D that the results are robust: i) when the value of the hyperparameter that controls the tightness of the VAR coefficients is increased (looser priors), ii) when the unemployment (instead of the GDP growth) series, derived from the FRED database, is used to proxy the real sector of the economy, and iii) when the dynamic order to the VAR is reduced to $3$ , $2$ , and $1$ lags (per regime).Footnote ¹⁵

3.7. The national housing returns factor and identified shocks

The discussion in this section is concentrated on the national real housing returns factor in relation to not only conventional and unconventional monetary policy shocks but also aggregate demand and aggregate supply innovations. Table 3 illustrates that the identified macroeconomic economic shocks explain between $50\%$ and $70\%$ of the variability of the national housing returns factor (depending on the horizon and the regime). This message is further reinforced by Figure 17, where the identified shocks account for (almost) all the historical evolution of the common component of state-level growth in real house prices during the first and second regimes. The explanatory power of the identified shocks collapses during the Great Recession, while it improves in the period between $2011Q1$ and $2014Q2$ but breaks down again from $2014Q3$ till the end of the sample ( $2017Q4$ ).Footnote ¹⁶

Figure 17. Historical decomposition of real housing returns factor.

Notes: The historical decomposition is calculated for each posterior draw. The (posterior) mean of these calculations is reported here. The quarter-on-quarter contributions have been aggregated to annual changes contributions. The same transformation is applied for the data series.

Several interesting facts emerge from both Table 3 and Figure 17. The first one is that, unlike in the high inflation regime, conventional monetary policy played almost no role in the growth of the national factor related to the housing market during the Great Moderation—a result in line with Del Negro and Otrok (Reference Del Negro and Otrok>2007); if anything, its contribution is rather negative [Nelson et al. (Reference Nelson, Pinter and Theodoridis>2018)]. This finding is consistent with the work of Justiniano et al. (Reference Justiniano, Primiceri and Tambalotti>2017) and Justiniano et al. (Reference Justiniano, Primiceri and Tambalotti>2019) where the authors explain that factors related to the credit supply and demand, and not monetary policy, are behind the increase of house prices (and housing debt/leverage). In this study, credit supply and demand shocks are not identified explicitly, but are probably captured by the demand, supply, and slope shocks in our model and, interestingly, the contribution of all these three types of shocks to the national factor of real housing returns is positive during this period, which, in turn, corroborates the findings of Plakandaras et al. (Reference Plakandaras, Gupta, Katrakilidis and Wohar>2018), especially in terms of the importance of the aggregate supply shocks.

The national real house price growth factor collapsed during the Great Recession, with one-third of this fall not being explained by the shocks identified. During this period, the (conventional) monetary policy is constrained by the ZLB, which started binding at the beginning of the $2009$ . Interestingly, substantial negative policy contribution started cumulating a few quarters before the ZLB, and these adverse effects picked up during the ZLB period (persistence effect). This anecdotal evidence could suggest that the inability of monetary authorities to lower the policy rate, which cannot be modeled as a shock, could explain a large part of the unexplained wedge.

Although quantitative easing (QE) was introduced in the last quarter of $2008$ , the model suggests that the slope (or QE) shock started contributing positively to the economy around $2012$ . This period coincides with the introduction of the open-ended QE3 and the forward guidance (FG) unconventional monetary policy. This does not mean that the first two QE programs had no positive effects on recovering the housing market, as this question cannot be answered from the historical decomposition and in turn requires the knowledge of the counterfactual profile of the national factor in the absence of the QE1 and QE2 programs. However, the assumption that the large positive contributions by the slope shock during the third regime are associated with explicit guidance about future policy rate would be consistent with the point made in the previous paragraph. In other words, what matters most for agents when it comes to house prices is the systematic part of the monetary policy described by the policy rate.

Finally, the real housing returns factor post- $2014$ has decoupled again from the identified macroeconomic shocks. This perhaps reflects the concerns, as also as outlined above in the introduction, from important policy institutions and policy-makers (see, e.g., Borio (Reference Borio>2019) and Carstens (Reference Carstens>2019), among others) that low interest rates led many investors to search for higher returns that are also subject to elevated risks. Moreover, studies like Bork et al. (Reference Bork, Møller and Pedersen>2019), Balcilar et al. (Reference Balcilar, Bouri, Gupta and Kyei>2021), Bouri et al. (Reference Bouri, Gupta, Kyei and Shivambu>2021), and Gupta et al. (Reference Gupta, Marfatia, Pierdzioch and Salisu>2021) have recently highlighted the importance of behavioral factors such as sentiment and uncertainty in playing crucial roles, beyond standard macroeconomic and financial variables, in driving house price movements, that is, both returns and volatility for the overall and regional USA.Footnote ¹⁷ In light of this, our findings should not come as a surprise and carry important implications from the perspective of modeling the US housing market. In particular, such behavioral variables should then be explicitly included, and shocks to them analyzed, while performing structural analyses of macroeconomic and financial shocks on the housing market. From the policy perspective, if these shocks are indeed important relative to conventional shocks for the US housing market, then policy authorities would need to target these behavioral variables to indirectly affect the movements in housing returns and volatility, especially given evidence that sentiment and uncertainty are not necessarily exogenous and are indeed predictable [Marfatia et al. (Reference Marfatia, André and Gupta>2020), André et al. (Reference André, Gabauer and Gupta>2021), Ludvigson et al. (Reference Ludvigson, Ma and Ng>2021), Salisu et al. (Reference Salisu, Gupta and Ogbonna>2021)].

3.8. Identifying conventional and unconventional monetary policy shocks using external instruments

This section scrutinizes the role of both conventional and unconventional monetary policy on the real housing returns factor.Footnote ¹⁸ In the first stage, the monetary policy (conventional and unconventional) shocks are identified from interest rate surprises that take place in a narrow window (30 min) before and after the policy meetings by using the methodology developed by Swanson (Reference Swanson>2021). In the second phase, proxy SVAR techniques (as in Mertens and Ravn (Reference Mertens and Ravn>2013) and Mertens and Ravn (Reference Mertens and Ravn>2014)) identify the effects on house prices.

3.8.1. Proxy SVAR

This section briefly reviews the shock identification methodology proposed by Mertens and Ravn (Reference Mertens and Ravn>2013) and Mertens and Ravn (Reference Mertens and Ravn>2014). Although, the scheme proposed initially was for a fixed-coefficient VAR model, the studies of Mumtaz and Petrova (Reference Mumtaz and Petrova>2018) and Mumtaz and Theodoridis (Reference Mumtaz and Theodoridis>2020) illustrate how the process can be extended to allow for time and regime dependence, respectively.

The fundamental idea of this methodology is that a “proxy” $ ( \phi _{t} )$ that is correlated with the shock of interest $ ( v_{t}^{m} )$ and uncorrelated with the remaining shocks $ ( v_{t}^{\bullet } )$ is used to identify the structural shock. These conditions can be expressed as follows:

(14) \begin{eqnarray} E\!\left ( \phi _{t},v_{t}^{m}\right ) &=&\alpha \neq 0 \\ E\!\left ( \phi _{t},v_{t}^{\bullet }\right ) &=&0 \nonumber \end{eqnarray}

As explained in Mertens and Ravn (Reference Mertens and Ravn>2013) and Mertens and Ravn (Reference Mertens and Ravn>2014), the identification of structural shocks (i.e., the first column of $A_{0,S}$ ) can result as a solution of a generalized method of moments estimation problem that satisfies the moment conditions (14). The authors also illustrate that the structural shocks, $v_{t}^{m}$ , can be simply derived by regression of $\phi _{t}$ on $\varepsilon _{t}$ . For instance, let us assume that the fitted value of the $v_{t}^{m}$ is given by:

(15) \begin{eqnarray} \Pi _{S}\varepsilon _{t} &=&E\!\left ( \phi _{t}\varepsilon _{t}^{\prime }\right ) \Omega _{S}^{-1}\varepsilon _{t} \\ &=&\alpha _{S}A_{0,S}^{m}\left ( A_{0,S}D_{S}D_{S}^{\prime }A_{0,S}^{\prime }\right ) _{t}^{-1}\varepsilon _{t} \nonumber \\ &=&\alpha _{S}\left ( A_{0,S}^{m}\left ( A_{0,S}^{-1}\right ) ^{\prime }\right ) \left ( A_{0,S}^{-1}\varepsilon _{t}\right ) \nonumber \\ &=&\alpha _{S}v_{t}^{m} \nonumber \end{eqnarray}

where $\Omega _{S}$ denotes the regime-dependent variance–covariance matrix of the VAR residuals, while moving from the third to the fourth line we employ the orthonormality properties of the identified matrix and mapping between the structural and reduced-form errors discussed above.

3.8.2. Narrative measures for conventional and unconventional monetary policy shocks

The strength of the identification scheme discussed in the previous section relies on the quality of the instrument used to identify the structural shock.Footnote ¹⁹ As explained by Jarociński and Karadi (Reference Jarociński and Karadi>2020) and Miranda-Agrippino and Ricco (Reference Miranda-Agrippino and Ricco>2021), changes in the forward interest rate contracts defined in a narrow window of 30 min before and after the policy announcements can contain also information that is related to the state of the economy and not, necessarily, to the reaction function of the monetary authority.

Jarociński and Karadi (Reference Jarociński and Karadi>2020) propose a methodology that isolates the monetary policy shocks from the disturbances that contain “information” about the state of the economy. However, their scheme does not disentangle “conventional” monetary policy shocks from “unconventional” QE and FG ones. As discussed in the literature (see Andres et al. (Reference Andres, Lopez-Salido and Nelson>2004), Chen et al. (Reference Chen, Curdia and Ferrero>2012), De Graeve (Reference De Graeve>2016), and Liu et al. (Reference Liu, Theodoridis, Mumtaz and Zanetti>2019) among others), the transmission mechanism of different monetary policy is not necessarily the same, meaning that the effects of conventional and unconventional monetary policy shocks on the real housing returns factor could be different.

Fortunately, Swanson (Reference Swanson>2021) and Altavilla et al. (Reference Altavilla, Brugnolini, Gürkaynak, Motto and Ragusa>2019) extend the methodology proposed by Gürkaynak et al. (Reference Gürkaynak, Sack and Swanson>2005) and achieve the identification of all three types of monetary policy shocks. Figure 18 reports the effects of different monetary policy shocks on the real housing returns factor across different regimes using the proxy SVAR techniques and the narrative measures. Before discussing the results, it is important to mention that the lack of available futures interest rate contracts for the USA prior to 1991, limits the evaluation exercise to regimes 2 and 3 and only.

Figure 18. Responses of real housing returns factor across different types of adverse monetary policy shocks and across regimes.

Notes: The solid (black) line represents the (pointwise) median, while the shaded area captures the $16\%-84\%$ percentiles of the posterior distribution.

Similar to the benchmark sign restrictions identification scheme, Figures 15 and 18 indicate that conventional monetary policy shocks have a sizeable effect on the real housing returns factor (first-row and first-column subplot). The latter effect appears to be more persistent and more precisely estimated when narrative measures are used. QE monetary policy shocks have again a large impact on the factor (first-row and second-column subplot). Interestingly, the QE shocks appear to be less persistent than conventional monetary policy shocks. Finally, FG monetary policy shocks do affect house price significantly.

The lack of a significant impact on the real housing returns factor from FG shocks might seem puzzling at first. However, this consideration might be addressed if the nature of interest rate shocks is understood. FG is nothing more than a communication policy that reveals information about the central bank’s reaction function in the future. As a result, it reduces uncertainty about the future monetary policy actions and, consequently, reinforces the effects of conventional and quantitative monetary policy decided today.