Price bubbles and policy interventions in the Chinese housing market

Abstract

The recent dramatic increases in house prices have led to considerable attention being focused on housing markets in China. Using a unique dataset of city-level house prices and rents, this paper investigates the presence of price bubbles in major Chinese city housing markets. Our findings show evidence of price bubbles in most housing markets, even though the bubbles might be, to a large extent, mitigated by strong government intervention. In order to distinguish rational bubbles from irrational bubbles, we extend the existing work to allow a deterministic time trend and break points in the unit root test of house price–rent ratios. As a consequence, our results demonstrate that the price–rent ratios in most of the sample cities are no longer non-stationary, implying the non-existence of rational bubbles in the housing markets.

Appendices

Appendix 1: Present value model

Assuming a two-period model, the house price–rental relation can be written as:

$$ P_{t} = \frac{{E_{t} \left[ {P_{t + 1} + D_{t + 1} } \right]}}{1 + r} $$

(5)

where $P_{t}$ denotes the asset price at time t, $D_{t}$ the housing rental income at time t, and r the constant discount rate. A rational investor may evaluate the price of an asset as the present value of its next period price plus rental income. Recursively solving this equation, we obtain

$$ P_{t} = \theta (1 - \alpha )\sum\limits_{i = 0}^{\infty } {\alpha^{i} E_{t} (D_{t + i} )} $$

(6)

where $\alpha = \frac{1}{1 + r},\theta = \frac{\alpha }{(1 - \alpha )} = r^{ - 1}$. Campbell and Shiller (1987) rearrange this equation as an empirically testable form:

$$ P_{t} - D_{t} \theta = \theta \sum\limits_{i = 1}^{\infty } {\alpha^{i} E_{t} (\varDelta D_{t + i} )} $$

(7)

Since both asset prices and rental incomes usually evolve as nonstationary I(1) processes, a long-run cointegrating vector ($1, - \theta$) exists if the present value model holds. Alternatively, by Campbell and Shiller (1988), because

$$ r_{t + 1} \approx \log \left( {P_{t + 1} + D_{t + 1} } \right) - \log \left( {P_{t} } \right) $$

a logarithm transformation for this equation can generate

$$ p_{t} - d_{t} = - r_{t + 1} + \varDelta d_{t + 1} + \ln (1 + e^{{p_{t + 1} - d_{t + 1} }} ) $$

(8)

where the lower case letters represent the logarithms of the upper case variables. By means of a first-order Taylor expansion, we can linearize model (8) as the following expression:

$$ p_{t} - d_{t} \approx - r_{t + 1} + \Delta d_{t + 1} + k + \rho (p_{t + 1} - d_{t + 1} ) $$

(9)

where $\rho = \frac{{e^{{\bar{p} - \bar{d}}} }}{{1 + e^{{\bar{p} - \bar{d}}} }}$ and $k = \ln (1 + e^{{\bar{p} - \bar{d}}} ) + \rho \left[ {\bar{p} - \bar{d}} \right]$, $\bar{p}$ and $\bar{d}$ are unconditional mean of each variable. Under this log-linear form, a stationary log price-to-rent ratio process implies the nonexistence of a rational bubble in housing prices, while failing to reject a unit root hypothesis indicates the presence of such a bubble.

Appendix 2: Zivot and Andrews (1992) test

The null hypothesis for the test is:

$$ H_{0} :y_{t} = \mu + \beta t + y_{t - 1} + \varepsilon_{t} $$

(10)

The following augmented regression is considered when testing for unit roots:

$$ y_{t} = \mu + \theta DU_{t} (\lambda ) + \beta t + \gamma DT_{t} (\lambda ) + \alpha y_{t - 1} + \sum\limits_{j = 1}^{k} {c_{j} \varDelta y_{t - j} + \varepsilon_{t} } $$

(11)

where $\text{for}\;\lambda \in (0,1),DU_{t} (\lambda ) = 1\quad \text{if}\;{\text{t}} > T\lambda ,0\;\text{otherwise, and}\;DT_{t} (\lambda ) = t - T\lambda \quad \text{if}\;t > T\lambda ,0\;\text{otherwise}.$

Zivot and Andrews (1992) calculate sequentially one-side t-statistics for testing $\alpha^{i} = 1$ and select the minimum value as the test statistic. The break point is selected as the point generating a minimum t-statistic: $t_{{\hat{\alpha }^{i} }} \left[ {\lambda_{\inf }^{i} } \right] = \mathop {\inf }\limits_{\lambda \in \varLambda } t_{{\hat{\alpha }^{i} }} (\lambda )$.

Appendix 3: Club convergence

Panel data, Y _it , we can decompose it into a systematic component (g _it ) and a transitory component (a _it ) as,

$$ Y_{it} = g_{it} + a_{it} $$

(12)

which can in turn be transformed into a time-varying factor representation:

$$ Y_{it} = \left( {\frac{{g_{it} + a_{it} }}{{\mu_{t} }}} \right)\mu_{t} = \delta_{it} \mu_{t} $$

(13)

where $\delta_{it}$ is an idiosyncratic component and $\mu_{t}$ is a common component. The idiosyncratic component measures the distance between $Y_{it}$ and the common component. If the factor $\delta_{it}$ converges to a constant $\delta$, then there will be convergence clubs. Scaling the common factor $\mu_{t}$, the PS method introduces a relative loading or transition coefficient:

$$ h_{it} = \frac{{Y_{it} }}{{\frac{1}{N}\sum\nolimits_{i = 1}^{N} {Y_{it} } }} = \frac{{\delta_{it} }}{{\frac{1}{N}\sum\nolimits_{i = 1}^{N} {\delta_{it} } }} $$

(14)

This relative transition parameter has a cross-sectional mean of unity; and when $\delta_{it}$ converges to a constant, $h_{it}$ converges to unity. The variance of this transition parameter $H_{t} = \frac{1}{N}\sum\nolimits_{i = 1}^{N} {\left( {h_{it} - 1} \right)}^{2}$ converges to zero in the long run (t → ∞).

The PS method also allows for a test of convergence starting from the following regression

$$ \log \left( {{{H_{1} } \mathord{\left/ {\vphantom {{H_{1} } H}} \right. \kern-0pt} H}_{A} } \right) - 2\log L\left( t \right) = \hat{c} + \hat{b}\log \left( t \right) + u_{t} $$

(15)

In (15) $L(t)$ is a slowly varying function, defined as $\log \left( t \right)$ for $t = \left[ {rT} \right]$,…, T, where $\left[ {rT} \right]$ denotes the integer part of $rT$, and $r$ is a positive number. PS also show that the parameter $b$ equals twice the convergence rate $a$. The one-sided $t$-statistics for $\hat{b}$ using HAC standard errors can be constructed, and the (one-sided) null hypothesis of convergence is rejected at the 5 % level if $t_{b} < - 1.65$.

To accommodate any possibility of sub-group convergence, PS introduce a systematic way of clustering based on repeated log-t regressions. It is a four-step process, namely, ordering, forming core group, seizing membership and stopping rule (see Phillips and Sul 2007 for more details).