The effects of macroprudential policy on Hong Kong’s housing market: a multivariate ordered probit-augmented vector autoregressive approach

Abstract

This study evaluates the effects of macroprudential policy on Hong Kong’s housing market using a multivariate ordered probit-augmented vector autoregressive model (MOP-VAR). The proposed MOP-VAR extends the conventional dummy policy variable approach by allowing explicit measurement of time-varying policy intensities that underlie policy rules, and thus facilitates analyses of bilateral relationship between house prices and multiple policy instruments when endogeneity exits among the instruments’ intensities and prices. An impulse response analysis suggests that the dampening effect of macroprudential tightening is stronger and more instantaneous on transactions than on prices. The eventual outcome as indicated by conditional forecasts is dominated by a strong and prolonged own price response to house price shocks and other external developments that undermine the policy’s effectiveness. Moreover, over the long haul, a combination of a stamp duty and stress test tends to be more effective than restricting the loan-to-value ratio in triggering a trend reversal in house prices, despite the government’s preference for the latter. The out-of-sample probabilistic forecasts of policy changes are mostly consistent with the observable outcomes.

Appendix: Estimation algorithm

• Stacking the observations and reorganizing the algebra of the VAR in (1) gives

  $$\tilde{Y} = \tilde{X}H + U,$$

  where $\tilde{Y} = \left[ {\tilde{Y}_{1} , \ldots ,\tilde{Y}_{T} } \right]^{'}$ and $U = \left[ {u_{1} , \cdots ,u_{T} } \right]^{'}$ are both $T \times \left( {m + q} \right)$; $\tilde{X}$ is the $T \times \left( {n + \left( {m + q} \right)p} \right)$ matrix of the conforming exogenous and lagged variables; and $H = \left[ {B A_{1} \cdots A_{p} } \right]^{'}$ is the $\left( {n + \left( {m + q} \right)p} \right) \times \left( {m + q} \right)$ coefficient matrix. The Minnesota prior assumes the following distribution on $h = vec\left( H \right)$:

  $$h \sim N\left( {\mu_{h} , V_{h} } \right),$$

  where $\mu_{h}$ is a vector of zeros except for the elements corresponding to the first own lag of $\tilde{Y}$ that equals one. The prior variance $V_{h}$ is a diagonal matrix with the elements

  $$\left( {\frac{{\lambda_{1} }}{{l^{{\lambda_{3} }} }}} \right)^{2} {\text{if}} i = j\quad {\text{and }}\quad \left( {\frac{{\sigma_{i} \lambda_{1} \lambda_{2} }}{{\sigma_{j} l^{{\lambda_{3} }} }}} \right)^{2} {\text{if}}\quad i \ne j$$

  for lag $l$ of the $j^{th}$ variable in the $i^{th}$ equation. $\sigma_{i}$ is the OLS estimate of the standard deviation of the $i^{th}$ error term. The prior variances of other exogenous variables are $\lambda_{4}$. We set $\lambda_{1} = 0.1, \lambda_{2} = 0.25, \lambda_{3} = 2,\, {\text{and}}\, \lambda_{4} = 10,000$. The VAR order is $p = 2$.

• Let $D$ denote data, $\varTheta$ the set of all of the parameters $\left\{ {h,\varSigma , \beta , \rho , Y_{t}^{*} , c_{ik} } \right\}$, and $\varTheta_{\backslash a}$ the subset of $\varTheta$ excluding $a$. Assuming a conjugate inverted-Wishart prior for the VAR covariance matrix ${{\Sigma }}$, the conditionals $f\left( {h|{{\Theta }}_{\backslash h} ,D} \right)$ and $f\left( {{{\Sigma }}|{{\Theta }}_{{\backslash {{\Sigma }}}} ,D} \right)$ are normal and inverted-Wishart, respectively.

• The probit regression coefficient $\beta$ has a normal prior with a mean and variance set to the OLS estimates of the corresponding univariate ordered probit models. Gibbs sampling from the conditional $f\left( {\beta |{{\Theta }}_{\backslash \beta } ,D} \right)$ is straightforward. We assume the variance of $\varepsilon_{t}$ has the form ${{\Omega }} = \left[ {\begin{array}{*{20}c} 1 & \rho \\ \rho & 1 \\ \end{array} } \right]$ with $\left| \rho \right| < 1$ to ensure that the matrix is positive semi-definite. MCMC updates of $\rho$ are calculated from the simulated outcomes of ${\hat{{\varepsilon }}}_{t}$. For consistency, we normalized the coefficient vector $\beta$ correspondingly.

• For each $i = 1, \ldots ,q$, the $K_{i}$ thresholds can be drawn recursively from uniform conditionals given the assumed diffuse priors:

  $$f\left( {c_{ik} |{{\Theta }}_{{\backslash c_{ik} }} ,D} \right) = U\left( {{ \hbox{max} }\left( {\mathop {\hbox{max} }\limits_{{y_{it} = k - 1}} \left( {y_{it}^{*} } \right), c_{ik - 1} } \right), { \hbox{min} }\left( {\mathop {\hbox{min} }\limits_{{y_{it} = k}} \left( {y_{it}^{*} } \right), c_{ik + 1} } \right)} \right).$$
  
  

  We find that the mixing of the chain improves with shrewdly chosen upper and lower threshold bounds. After numerous trials, we set those at $- 19 < c_{1k} < 199$ and $- 9 < c_{2k} < 19,$ which are wide enough to accommodate their respective OLS estimates.

• Sampling of $Y_{t}^{*}$ is more complicated. To begin, we partition the elements of the VAR in the following manner (we assume an order $p = 2$ in the illustration):

  $$\tilde{Y}_{t} = \left[ {\begin{array}{*{20}c} {Z_{t} } \\ {Y_{t}^{*} } \\ \end{array} } \right], \, u_{t} = \left[ {\begin{array}{*{20}c} {u_{zt} } \\ {u_{Yt} } \\ \end{array} } \right], \, {{\Sigma }} = \left[ {\begin{array}{*{20}c} {{{\Sigma }}_{ZZ} } & {{{\Sigma }}_{ZY} } \\ {{{\Sigma }}_{YZ} } & {{{\Sigma }}_{YY} } \\ \end{array} } \right],$$

  with conforming coefficient matrices $B = \left[ {\begin{array}{*{20}c} {B_{1} } \\ {B_{2} } \\ \end{array} } \right]$ and $A_{j} = \left[ {\begin{array}{*{20}c} {A_{j1} } & {A_{j2} } \\ {A_{j3} } & {A_{j4} } \\ \end{array} } \right], \forall j.$ $B_{1}$ and $B_{2}$ have respective dimensions of $m \times n$ and $q \times n$; $A_{j1}$ is $m \times m$; $A_{j4}$ is $q \times q$; and $A_{j2}$ and $A_{j3}$ are $m \times q$ and $q \times m$, respectively. Now,

  $$Z_{t} = B_{1} W_{t} + \left[ {A_{11} A_{12} } \right]\left[ {\begin{array}{*{20}c} {Z_{t - 1} } \\ {Y_{t - 1}^{*} } \\ \end{array} } \right] + \left[ {A_{21} A_{22} } \right]\left[ {\begin{array}{*{20}c} {Z_{t - 2} } \\ {Y_{t - 2}^{*} } \\ \end{array} } \right] + u_{zt}\quad {\text{and}}$$

  (A1)
  $$Y_{t}^{*} = B_{2} W_{t} + \left[ {A_{13} A_{14} } \right]\left[ {\begin{array}{*{20}c} {Z_{t - 1} } \\ {Y_{t - 1}^{*} } \\ \end{array} } \right] + \left[ {A_{23} A_{24} } \right]\left[ {\begin{array}{*{20}c} {Z_{t - 2} } \\ {Y_{t - 2}^{*} } \\ \end{array} } \right] + u_{Yt} .$$

  (A2)

• Let $\mu_{Zt} = B_{1} W_{t} + A_{11} Z_{t - 1} + A_{21} Z_{t - 2}$ and have $\mu_{Yt}$ similarly defined for terms in (A2). Together with Eq. (2) in the main text, we can rewrite the system in state space form:

  $$\left[ {\begin{array}{*{20}c} {Y_{t}^{*} } \\ {Y_{t - 1}^{*} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_{14} } & {A_{24} } \\ {I_{q} } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {Y_{t - 1}^{*} } \\ {Y_{t - 2}^{*} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\mu_{Yt} } \\ 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {u_{Yt} } \\ 0 \\ \end{array} } \right] {\text{and}}$$

  (A3)
  $$\left[ {\begin{array}{*{20}c} {Z_{t} - \mu_{Zt} } \\ {\left( {I_{q} \otimes X_{t - 1}^{'} } \right)\beta } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_{12} } & {A_{22} } \\ {I_{q} } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {Y_{t - 1}^{*} } \\ {Y_{t - 2}^{*} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {u_{Zt} } \\ { - \varepsilon_{t - 1} } \\ \end{array} } \right],$$

  (A4)

  where the 0 s have conforming dimensions.

• The Kalman filter (KF) is then applied to (A3) and (A4) in the forward-filtering stage with the outcome being stored and used in the subsequent stage of backward smoothing/sampling. A few remarks are warranted. First, a normal prior is assumed for the initial state vector $\left[ {Y_{0}^{*'} ,Y_{ - 1}^{*'} } \right]^{'}$. Second, the error terms in the transition equation (A3) and the measurement equation (A4) are not independent, as in standard KF application, because ${{\Sigma }}_{YZ}$ is not a null matrix. So, modifications to the KF recursion (e.g., Ma et al. 2010) have to be used to account for the correlation between measurement and transition noises. Finally, the drawn values of the latent variables have to obey the inequality constraints instigated by the thresholds $c_{ik}$ and the observed categorical data $y_{it}$. Simon (2010) suggests a few ways to resolve the issue, and we accomplish this by adding an extra constrained minimization step after getting the updated state estimates from the unconstrained forward filter and backward smoother.

• In sum, the algorithm iterates through the normal-inverted-Wishart conditionals for the VAR components, a normal conditional for the probit coefficients and the derived computation of the cross-correlation of the latent variables, uniform conditionals for the thresholds, and a constrained KF update and backward sampling of the latent variables. The simulation runs for 8,000 loops with the last 4,000 used to compute the posterior estimates.