House Prices and Bubbles in New Zealand

Abstract

This paper studies actual (real) house prices relative to fundamental (real) house values in New Zealand for the period 1970–2005. Utilizing a dynamic present value model, we find disparities between actual and fundamental house prices in the early 1970s and 1980s and from 2000 to date. We model the bubble component that is related to fundamentals (the intrinsic component), making it possible to highlight whether a bubble still exists after that component is accounted for. We then analyze any remaining bubble to detect any momentum behavior. Much of the overvaluation of the housing market is found to be due to price dynamics rather than an overreaction to fundamentals.

Appendix

The Fundamental Price-Income Model

The model described in the empirical framework section above has the following present value expression for the real value of household property, V _t :

$$ V_{t} = \gamma E_{t} {\sum\limits_{i = 1}^\infty {{\left( {\frac{1} {{{\prod\limits_{j = 1}^i {{\left( {1 + \rho ^{ * }_{{t + j}} } \right)}} }}}} \right)}Y_{{t + i}} } } $$

(8)

where V _t is a constant proportion, γ, of the expected value of future real disposable income, Y _t , discounted at the real discount rate, $\rho ^{ * }_{t}$. Assuming the relationship between the real house price index P and market capitalization V, and the relationship between the value of all income, Y, and income covered by the house price index, are constant, then Eq. 8 is re-written as:

$$ P_{t} = E_{t} {\sum\limits_{i = 1}^\infty {{\left( {\frac{1} {{{\prod\limits_{j = 1}^i {{\left( {1 + \rho ^{ * }_{{t + j}} } \right)}} }}}} \right)}Q_{{t + 1}} } } $$

(9)

where P _t  = β′V _t , and, defining β = β′(γ) and Q _t   =  βY _t .

We define the time stream of realized discount rates, ρ _t , to satisfy:

$$ P_{t} = {\sum\limits_{i = 1}^\infty {{\left( {\frac{1} {{{\prod\limits_{j = 1}^i {{\left( {1 + \rho _{{t + j}} } \right)}} }}}} \right)}Q_{{t + i}} } } $$

(10)

Given the discussion of Eq. 9 above, Eq. 10 is a particular solution to $P_{t} = {{\left( {P_{{t + 1}} + Q_{{t + 1}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {P_{{t + 1}} + Q_{{t + 1}} } \right)}} {{\left( {1 + \rho _{{t + 1}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \rho _{{t + 1}} } \right)}}$, and it follows that:

$$ 1 + \rho _{{t + 1}} = {{\left( {P_{{t + 1}} + Q_{{t + 1}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {P_{{t + 1}} + Q_{{t + 1}} } \right)}} {P_{t} }}} \right. \kern-\nulldelimiterspace} {P_{t} } $$

(11)

where P _t is the real price at the end of period t, and Q _t+1 is real disposable income measured during t + 1. Taking logs and using lower case letters to represent the logs of their upper-case counterparts, we can write:

$${r_{{t + 1}} = \ln {\left( {1 + \exp {\left( {q_{{t + 1}} - p_{{t + 1}} } \right)}} \right)} + p_{{t + 1}} - p_{t} }$$

(12)

where r is defined as ln(1 + ρ) and the term (q − p) can be viewed as the economy-wide income-price ratio. The first term in Eq. 12 can be linearized using a first-order Taylor’s approximation and Eq. 12 can be written as:

$$ r_{{t + 1}} = - {\left( {p_{t} - q_{t} } \right)} + \mu {\left( {p_{{t + 1}} - q_{{t + 1}} } \right)} + \Delta q_{{t + 1}} + k $$

(13)

where k and μ are linearization constants:

$$ \begin{array}{*{20}l} {{k = - {\text{ln}}\mu - {\left( {1\mu } \right)} \cdot \overline{{{\left( {q - p} \right)}}} } \hfill} \\ {{\mu = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp \overline{{{\left( {q - p} \right)}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp \overline{{{\left( {q - p} \right)}}} } \right)}}} \hfill} \\ \end{array} $$

where $\overline{{{\left( {q - p} \right)}}}$ is the sample mean of (q − p) about which the linearization was taken. Clearly, 0 < μ < 1 and in practice is close to 1.

Empirically, it is common that both p and q are I(1) so that the variables are transformed to ensure stationarity. Denote by pq _t the (log) price-income ratio, p _t  − q _t , and rewrite Eq. 13 as:

$$pq_{t} = k + \mu pq_{{_{{t + 1}} }} + \Delta q_{{t + 1}} - r_{{t + 1}} $$

(14)

After repeated substitution for $pq_{{t + 1}} ,pq_{{t + 2}} , \ldots$ on the right-hand side of Eq. 14, we get:

$$pq_{t} = \frac{{k{\left( {1 - \mu ^{i} } \right)}}}{{{\left( {1 - \mu } \right)}}} + {\sum\limits_{j = 0}^{i - 1} {\mu \,^{{j + 1}} \Delta q_{{t + j + 1}} - {\sum\limits_{j = 0}^{i - 1} {\mu \,^{{j + 1}} r_{{t + j + 1}} + \mu ^{i} pq_{{t + 1}} } }} }$$

(15)

Letting $i \to \infty$ and assuming that the limit of the last term is 0, results in the following alternative form of Eq. 15:

$$pq_{t} = \frac{k}{{{\left( {1 - \mu } \right)}}} + {\sum\limits_{j = 0}^\infty {\mu \,^{{j + 1}} \Delta q_{{t + j + 1}} } } - {\sum\limits_{j = 0}^\infty {\mu \,^{{j + 1}} r_{{t + j + 1}} } }$$

(16)

Hence, if q _t  ∼ I(1) then Δq _t  ∼ I(0) and, assuming that r _t  ∼ I(0) (recall that it is the real discount rate), then pq _t will be I(0) and we have the model linearized and expressed in terms of stationary variables. Finally, taking conditional expectations of both sides:

$$pq_{t} = \frac{k}{{{\left( {1 - \mu } \right)}}} + {\sum\limits_{j = 0}^\infty {\mu \,^{{j + 1}} E^{c}_{t} \Delta q_{{t + j + 1}} - {\sum\limits_{j = 0}^\infty {\mu \,^{{j + 1}} E^{c}_{t} r_{{t + j + 1}} } }} }$$

(17)

where $E^{c}_{t}$ are conditional expectations and we interpret $r_{{t + j + 1}}$ as investors’ required return.

In order to use Eq. 17 to generate a series for pq _t *, the price-income ratio implied by the model and from it the implied or fundamental house price, p*, we need to obtain empirical counterparts to the terms on the right-hand side involving expectations. For the first of these, the expectation of disposable income growth, we incorporate disposable income growth into a three-variable VAR model (see below) while for the second we assume a time-varying risk premium, which we also include in the empirical VAR. Here we follow the work of Merton (1973, 1980) on the intertemporal CAPM, and model the time-varying risk premium as the product of the coefficient of relative risk aversion, α, and the expected variance of returns, $E^{c}_{t} \sigma ^{2}_{t}$.¹⁶ The equation for the price-income ratio then becomes:

$$pq_{t} = \frac{{k - f}}{{{\left( {1 - \mu } \right)}}} + {\sum\limits_{j = 0}^\infty {\mu \,^{{j + 1}} E^{c}_{t} \Delta q_{{t + j + 1}} - \alpha {\sum\limits_{j = 0}^\infty {\mu \,^{{j + 1}} E^{c}_{t} \sigma ^{2}_{{t + j + 1}} } }} }$$

(18)

where f is the constant real-risk free component of real required returns. In this case, we forecast both real income growth and the housing return variance using a three-variable VAR in $z_{t} = {\left( {pq_{t} ,\Delta q_{t} ,\sigma ^{2}_{t} } \right)}\prime$. The empirical VAR is written in compact form as:

$$ z_{{t + 1}} = Az_{t} + \varepsilon _{{t + 1}} $$

(19)

where A is a (3 × 3) matrix of coefficients and ɛ is a vector of error terms. We assume a lag length of 1 for ease of exposition. If, in the empirical application, a longer lag length is required, the companion form of the system can be used.

Forecasts of the variables of interest j + 1 periods ahead are achieved by multiplying z _t by the jth + 1 power of the matrix A:

$$ z_{{t + j + 1}} = A^{{j + 1}} z_{t} $$

(20)

The equation from which we compute the fundamental price-income ratio (and hence the fundamental house price) is:

$$ pq^{ * }_{t} = \frac{{k - f}} {{1 - \mu }} + {\left( {e^{\prime }_{2} - \alpha e^{\prime }_{3} } \right)}A{\left( {I - \mu A} \right)}^{{ - 1}} z_{t} $$

(21)

where ${\mathbf{e}}^{\prime }_{2} {\mathbf{A}}^{{j + 1}} {\mathbf{z}}_{t} = E^{c}_{t} \Delta q_{{t + j + 1}}$ and ${\mathbf{e}}^{\prime }_{3} {\mathbf{A}}^{{j + 1}} {\mathbf{z}}_{t} = E^{c}_{t} \sigma ^{2}_{{t + j + 1}}$ where ${\mathbf{e}}^{\prime }_{2}$ and ${\mathbf{e}}^{\prime }_{3}$ are, respectively, the second and third unit vectors. Hence the fundamental value of the price-income ratio is generated by a combination of the present value model and the forecasting assumptions.

Therefore pq _t * provides a measure of the fundamental house price series once we have estimated the VAR coefficients and the constants μ, k, and f. Given that we wish to generate a series for real house prices that is warranted by (predicted) income growth, we generate (the log of) fundamental house prices as:

$$ p^{ * }_{t} = pq^{ * }_{t} + q_{t} $$

(22)

Equation 21 can also be used to derive tests of how far actual house prices deviate from their fundamental value as warranted by real disposable income. This is simply a test of $pq_{t} = pq^{ * }_{t}$ for all t. Since $pq_{t} = {\mathbf{e}}^{\prime }_{1} {\mathbf{z}}_{t}$ where ${\mathbf{e}}^{\prime }_{1}$ is the first unit vector, we can write Eq. 21, after transforming the variables to deviations from their means to remove the constant term, as:

$$ e_{1} \prime {\left( {{\mathbf{I}} - \mu {\mathbf{A}}} \right)} = {\left( {e_{2} \prime - e_{3} \prime } \right)}{\mathbf{A}} $$

(23)

This restriction is linear in the elements of A (denoted a) and in the present case simply amounts to:

$$ \begin{array}{*{20}l} {{\mu a_{{11}} - \alpha a_{{31}} + a_{{21}} = 1;} \hfill} \\ {{\alpha a_{{32}} - a_{{22}} + \mu a_{{12}} = 0} \hfill} \\ {{\alpha a_{{33}} - a_{{22}} + \mu a_{{13}} = 0.} \hfill} \\ \end{array} $$

(24)

and can be tested with a standard Wald test which is asymptotically χ ²-distributed with three degrees of freedom.