Intrinsic bubbles test
In this paper, the intrinsic bubble method is utilized to test the Hong Kong residential property market. Intrinsic bubbles are driven by fundamental value alone in a nonlinear way, thereby entailing a nonlinear relationship between the property price index and the rental price index. The model indicates a simple condition that links the time-series of real property prices to the time-series of real rent payments when the expected rate of return is constant. To test the bubbles in the Hong Kong residential property market, I use rental price index as the fundamental value of property price index, following the Froot and Obstfeld’s (1991) approach, but replacing dividends with rental price index, and stock prices with property price index. In notations, if Pt is the real property price index at the beginning of the period t, Dt is the real rental price index paid at the end of period t, and r is the constant real rate of lending, then the present value (PV) model of property price index can be written as:
$$ {P}_t={e}^{-r}{E}_t\left({D}_t+{P}_{t+1}\right), $$
(1)
where Et(.) is the market’s expectation conditional on information known at the beginning of the period t.
Under this condition, \( {P}_t^{PV} \) is the present value of the property price index in period t can be written as:
$$ {P}_t^{PV}=\sum \limits_{s=t}^{\infty }{e}_t^{-r\left(s-t+1\right)}{E}_t\left({D}_s\right). $$
(2)
Froot and Obstfeld (1991) demonstrate that if a bubble exists, and \( {P}_t={P}_t^{PV}+{B}_t \) is a solution to Eq. (1), then real property price index Pt can be thought of as the sum of present value \( {P}_t^{PV} \) and a bubble Bt which is the difference between the actual price and the fundamental value.
$$ P\left({D}_t\right)={P}_t^{PV}+B\left({D}_t\right) $$
(3)
The intrinsic bubble model states that bubbles are generated by a nonlinear function of rents. Therefore, the intrinsic bubbles are function of rental price index satisfies the following:
$$ B\kern0em \left({D}_t\right)={cD}_t^{\lambda } $$
(4)
where c is an arbitrary and λ is the positive root of a quadratic equation.
When the process of log rental price index dt = ln(Dt) must follow a random walk with a drift μ,
$$ {d}_{t+1}=\mu +{d}_t+{\xi}_{t+1} $$
(5)
And residual of regression ξt + 1~N(0, σ2) is a normal random variable with conditional zero mean and standard deviation σ, the present value of the property price index becomes directly proportional to the rental price index during period t:
$$ {P}_t^{PV}={kD}_t $$
(6)
where \( k={\left({e}^r-{e}^{\mu +{\sigma}^2/2}\right)}^{-1} \). It follows that, if the sum in Eq. (2) converges, the expression μ + σ2/2 will be smaller than r. If there is a bubble present, the observed property price index can be thought of as the sum of present value \( {P}_t^{PV} \) and a bubble Bt:
$$ P\left({D}_t\right)={P}_t^{PV}+B\left({D}_t\right)={kD}_t+{cD}_t^{\lambda } $$
(7)
And
$$ \frac{\sigma^2}{2}{\lambda}^2+\mu \lambda -r=0. $$
(8)
Under this setup, the inequality r > μ + σ2/2 implies λ must be greater than 1, and it is an explosive nonlinear relation between bubbles and the rental price index, so the property price index may overreact to information about the rental price index. Empirically, Froot and Obstfeld (1991) divide Eq. (7) by Dt because of collinearity among the explanatory variables:
$$ \frac{P_t}{D_t}={c}_o+{c}_1{D}_t^{\lambda -1}+{\xi}_t $$
(9)
To test for bubbles, the null hypothesis of no bubble implies that c1 =0, whereas the bubble alternative in Eq. (9) predicts that c0 = k and c1 > 0.
Granger causality test
To interpret the results as to whether changes in the rental price index returns can influence changes in the property price index returns and vice versa, first, co-integration between these variables needs to be tested. If co-integration exists, the Granger causality test can be used to determine whether changes in the rental price index return can predict changes in the property price index returns in the Hong Kong residential property market. Prior work suggests property price movements are closely related to a common set of macroeconomic variables. Iacoviello (2005) uses a structural VAR approach, and identifies that U.S. monetary policy impacted real estate prices during 1974Q1 to 2003Q2. Giuliodori (2005) provides quantitative and qualitative evidence of the interaction between property price and monetary transmission mechanisms across nine European countries. Ahearne et al. (2005) study the property prices in 18 advanced economies and also conclude that there is a link between monetary policy and housing prices. Taylor (2007) provides an early example of a study that ascribes a substantial role to overly-loose monetary policy in the US after the recession, in which an overly low interest rate irritated housing activity. Taylor (2008, 2009) also suggests that loose monetary policy is the primary cause of bubbles in property prices and activity. In the wide selection of empirical papers, the majority of researchers conclude that loose monetary policy is a primary cause of bubbles in the property prices of Western countries. Koivu (2010) studies the wealth effect in China, and uses a VAR model and finds that loose monetary policy actually leads to higher asset prices, especially house prices. Yao et al. (2011) use monthly data from June 2005 to September 2010 to investigate the long-run relationship between monetary policy and asset prices in China. Their empirical results show that monetary policy has little effect on residential prices. Regarding the selection of variables for the factor analysis, the most recent research papers have chosen the past rent price or rental price index changes and changes in interest rates as the explanatory variables. For example, Lai and van Order (2010) use the Gordon growth model to express the housing market in the US with rental income, discount rate and the expected rental income growth; Nneji et al. (2011) use past price changes, changes in interest rates and the unemployment rate to determine the change in property price. Baltagi and Li (2015) use the rental price index and the home purchase index to examine the relationship between them in the Singapore residential property market. In this paper, the following main macroeconomic variables are used: (1) Property Price Index (Pt); (2) Rental Price Index (Dt); (3) Lending Rate (Rt); and (4) Consumer price index (It). The analyses in the paper are then carried out using these monetary and economic variables under a VAR/VECM framework.
The Granger (1969) causality test is a well-known test for causality and is usually conducted in the context VAR. For the causality tests used in this paper, I use the following VAR equations which do not impose common lags across all variables,
$$ \Delta {P}_t={c}_p+\sum \limits_{i=1}^{k_1}{\alpha}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_2}{\alpha}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_3}{\alpha}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_4}{\alpha}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{1t} $$
(10a)
$$ \Delta {D}_t={c}_d+\sum \limits_{i=1}^{k_5}{\beta}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_6}{\beta}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_7}{\beta}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_8}{\beta}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{2t} $$
(10b)
$$ \Delta {R}_t={c}_r+\sum \limits_{i=1}^{k_9}{\chi}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_{10}}{\chi}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_{11}}{\chi}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_{12}}{\chi}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{3t} $$
(10c)
$$ \Delta {I}_t={c}_i+\sum \limits_{i=1}^{k_{13}}{\delta}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_{14}}{\delta}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_{15}}{\delta}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_{16}}{\delta}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{4t} $$
(10d)
The null hypothesis that changes in rental price index returns do not Granger cause changes in the property price index returns are tested by \( {H}_0:{\alpha}_{2{k}_2}=0 \), for all k2 in Eq. (10a). Also the null hypothesis that changes in property price index returns do not Granger cause changes in the rental price index returns are tested by \( {H}_0:{\beta}_{1{k}_5}=0 \), for all k5 in Eq. (11b). If co-integration exists among these variables, an error correction term (ECT) is required in testing Granger causality as shown below,
$$ \Delta {P}_t={c}_p+{\gamma}_p{z}_{t-1}+\sum \limits_{i=1}^{k_1}{\alpha}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_2}{\alpha}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_3}{\alpha}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_4}{\alpha}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{1t} $$
(11a)
$$ \Delta {D}_t={c}_d+{\gamma}_d{z}_{t-1}+\sum \limits_{i=1}^{k_5}{\beta}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_6}{\beta}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_7}{\beta}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_8}{\beta}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{2t} $$
(11b)
$$ \Delta R{}_t={c}_r+{\gamma}_r{z}_{t-1}+\sum \limits_{i=1}^{k_9}{\chi}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_{10}}{\chi}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_{11}}{\chi}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_{12}}{\chi}_{4i}\Delta {I}_{t-1}+{\varepsilon}_{3t} $$
(11c)
$$ \Delta {I}_t={c}_i+{\gamma}_i{z}_{t-1}+\sum \limits_{i=1}^{k_{13}}{\delta}_{1i}\Delta {P}_{t-1}+\sum \limits_{i=1}^{k_{14}}{\delta}_{2i}\Delta {D}_{t-1}+\sum \limits_{i=1}^{k_{15}}{\delta}_{3i}\Delta {R}_{t-1}+\sum \limits_{i=1}^{k_{16}}{\delta}_{4i}\Delta I{}_{t-1}+{\varepsilon}_{4t} $$
(11d)
in which γp, γd, γr, and γi denote speeds of adjustment, zt represents the deviation from the long-run relation among variables.
The null hypothesis that changes in rental price index returns do not Granger cause changes in the property price index returns in this study is tested by H0 : γp = 0 and \( {\alpha}_{21}={\alpha}_{22}=\dots ={\alpha}_{2{k}_2} \) =0. Also the null hypothesis that changes in the property price index returns do not Granger cause changes in the rental price index returns in this study is tested by H0 : γd = 0 and \( {\beta}_{11}={\beta}_{12}=\dots ={\beta}_{1{k}_4}=0 \).