Speculative bubbles in U.K. house prices: some new evidence.

1. Introduction

A widely held view among academics and practitioners is that the time-series behavior of house prices of several major industrialized countries may have been characterized by explosive bubbles, which are not present in the underlying fundamentals and which, therefore, may drive an explosive wedge between house prices and their fundamental determinants. In particular, with regard to the United Kingdom, a large empirical literature has studied the determinants of the two booms experienced by house prices in the early 1970s and late 1980s, providing some tentative evidence in support of the view that speculation on expected future house prices might have been an important force driving actual house prices (see, inter alia, Hendry 1984; Muellbauer and Murphy 1997; Renaud 1997; Terasvirta 1998). (1) More recently, concerns of an ongoing house price bubble in the United Kingdom have been repeatedly raised by the press and policymakers. These concerns are well summarized in the following quote:

   It will be different this time. Those are the most dangerous words in
   investment and in economic policy. The U.K. is, yet again, in an
   unbalanced expansion. Behind it lies explosive household consumption
   and a surge in house prices. The last time this happened, in the
   early 1970s and late 1980s, the British cried buckets of tears. Will
   it be different this time? Unlikely, is the answer. (Martin Wolf,
   'An Economy Heading for a Crash,' Financial Times, November 10,
   2002, p. 3)

In this article, we contribute to the relevant literature on several fronts. We empirically examine the conjecture that rational bubbles have been present in U.K. house prices by applying two recently developed econometric testing procedures for indirectly testing for the presence of rational bubbles using quarterly U.K. house price data for the last 20 years. These econometric testing procedures are particularly appropriate in the present context in that they alleviate several problems typically encountered in testing for rational bubbles. Using both procedures, we find strong evidence of explosive, bubble-type behavior in U.K. house prices during the late 1980s and during the late 1990s and early 2000s, consistent with the house price bubbles hypothesis. Indeed, the second bubble we detect appears to be still ongoing at the end of our sample period in 2002.

Our empirical analysis is also motivated by using a stylized model designed to provide a theoretical rationale of the house price bubbles hypothesis. We build a simple three-period overlapping-generations model of house price determination, which produces a housing demand function of the form assumed by a large literature in this context. (2) We then show how a rational bubble may arise as a solution to the house price determination equation in this illustrative model.

The remainder of the article is set out as follows. In section 2, we describe our model and show how a housing demand function of the type assumed by the relevant literature can be derived from first principles and how a rational bubble may arise as a solution to the resulting house price determination equation. Section 3 describes the data, while in section 4, we discuss the econometric techniques employed in testing for bubbles. In section 5, we report the empirical results from formally testing for bubbles in U.K. house prices over the sample period 1983-2002. A final section concludes.

2. The Model

Assumptions

At any date t, a competitive firm produces a stock [X.sub.t] of indivisible and homogeneous houses, which provides a continuously divisible flow of housing services [x.sub.t] at unit price [[PI].sub.t] > 0. The relationship between stocks and flows is exogenous of the form [x.sub.t] = [f.sub.t]([X.sub.t]), where, for simplicity, we restrict [f.sub.t](*) to be linear. There are no transactions costs, and in each period, physical depreciation occurs at a constant rate [delta]. We define [H.sub.t] = (1 - [delta])[H.sub.t-1] + [X.sub.t] as the stock of all houses existing in period t, so that the flow of services provided by [H.sub.t] is [h.sub.t] = [f.sub.t]([H.sub.t]) = [f.sub.t]([H.sub.t]) + (1 - [delta])[H.sub.t-1]). (3) Without loss of generality, we also assume that [delta] = 0. So, at any date t,

(1) [h.sub.t] = [f.sub.t]([X.sub.t]) + [f.sub.t](H.sub.t-1]) = [x.sub.t] + [h.sub.t-1].

The representative consumer's objective is to maximize lifetime utility subject to his budget constraint. Each consumer lives for three periods, t = 0, 1, 2, and in each period, there are three overlapping generations of consumers: young, middle aged, and old. Each consumer has identical lifetime preferences [U.sub.0]([c.sub.0], [h.sub.0], [c.sub.1], [bar.l], [c.sub.2]) defined over the composite consumption good [c.sub.t] (priced [p.sub.t]), the housing services [h.sub.t] (priced [[PI].sub.t]), and labor supply [bar.l] (which is strictly positive and fixed). (4) There is no utility of housing and no labor supplied in old age.

We assume intertemporal and intraperiod additivity and time stationarity with a constant time preference parameter, say [eta], implying that lifetime utility may be written as u([c.sub.0]) + u([h.sub.0]) + [gamma][u([c.sub.1]) + u([h.sub.0])] + (1 + [gamma])u([bar.l]) + [[gamma].sup.2]u([c.sub.2]), where [gamma] = 1/(1 + [eta]); a discount rate of this form excludes dynamically inconsistent choices (Strotz 1956). We assume the simplest possible form of log linear subutilities, u([c.sub.t]) + u([h.sub.t]) + u([bar.l]) = log([c.sub.t]) + log([h.sub.t]) + log([bar.l]) for all t = 0, 1, 2. (5) Note that this satisfies u(*) being twice continuously differentiable and strictly concave as well as u'(0) [right arrow] [infinity], so that the optimally chosen quantities of housing and period consumption are expected to be strictly positive. (6)

We also assume that, in each period, a finite and constant number of identical consumers are born/die; there are no bequests; all consumers have perfect foresight of future prices, wages, and interest rates.

Next, we give details of the consumer's period budget constraints and then we solve the resulting optimization problem. In particular, we show below the case of the young generation (t = 0), which contains all the relevant information for our purposes.

Solving the Model

The intertemporal utility of a young consumer is maximized subject to the opportunities available, comprising the exogenous nominal wages (say [W.sub.0] and [W.sub.1] in young and middle age, respectively) times the consumer's labor supply. Also, young consumers buy [h.sub.0] units of divisible housing services at price [[PI].sub.0] from the newly old consumers; we assume that each unit requires its own separate (one-period) mortgage and that consumers are allowed to take out more than one mortgage. The mortgage size is [[PI].sub.0][h.sub.0] - [k.sub.0], where [k.sub.0] is the amount of deposit optimally chosen by the young consumer--so, for example, if [k.sub.0] = 0, a 100% mortgage is taken out, whereas no mortgage is taken out if [k.sub.0] = [[PI].sub.0][h.sub.0].

In principle, there should be no constraints on the sign of [k.sub.0], which may be either saving or borrowing; also, because, in this model, optimal consumption in each period is always strictly positive, it follows that [k.sub.0] < [W.sub.0][bar.l]. The mortgage repayment in middle age is (1 + [P.sub.1])([[PI].sub.0][H.sub.0] - [k.sub.0]), where [P.sub.1] is the nominal mortgage interest rate. So housing not only gives direct utility to the consumer, but also affects the desirability of consumption between young and middle age via [k.sub.0].

We assume that, (i) in young age, there is no other asset available except [k.sub.0]; (ii) between middle and old age, consumers cannot take out any new mortgage to buy new housing services, but they can borrow or save in a one-period bond [s.sub.1] paying a nominal interest rate [R.sub.2] in the next period; (iii) middle-age wealth is always high enough to allow the mortgage repayment, that is, [(1 + [R.sub.1])[W.sub.0] + [W.sub.1][bar.l] + [[PI].sub.2][h.sub.0]/(1 + [R.sub.1]) - (1 + [P.sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0]) > 0. Assumptions (i), (ii), and (iii) imply that we have a segregated asset market, in the sense that mortgages can only be taken out in young age. Also note that housing in old age does not give direct utility. (7)

Our discussion implies the following young consumer's period budget constraints:

[p.sub.0][c.sub.0] + [k.sub.0] = [W.sub.0][bar.l] in young age at t = 0, [p.sub.1][c.sub.1] + (1 + [P.sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0]) + [s.sub.1] = [W.sub.1][bar.l] in middle age at t = 1, [p.sub.2][c.sub.2] = (1 + [R.sub.2])[s.sub.1] + [[PI].sub.2][h.sub.0] in old age at t = 2.

Intertemporal choice of consumers is determined by backward induction (the derivation is given in Appendix A), and yields the final demand for housing by the young generation,

(2) [h.sub.0] = [[w.sub.0] + [w.sub.1]/(1 + [[rho].sub.1])][bar.l](1 + [gamma])/[[pi].sub.0] - [[pi].sub.2]/(1 + [r.sub.2])(1 + [[rho].sub.1])][1 + [(1 + [gamma]).sup.2]],

where [w.sub.t] = [W.sub.t]/[[p.sub.t], [[pi].sub.t] = [[PI].sub.t]/[p.sub.t], 1 + [[rho].sub.t] = [p.sub.t-1](1 + [P.sub.t])/[p.sub.t], 1 + [r.sub.t] = [p.sub.t-1](1 + [R.sub.t])/[p.sub.t] for t = 0, 1, 2 denote the real wage, the real house price, the real mortgage rate, and the real risk-free interest rate, respectively.

Bubbles in House Prices

If consumers hold perfect foresight or rational expectations, (8) the expression for the demand for housing services, Equation 2, may be generalized as follows:

(3) [h.sup.d.sub.t] = [[w.sub.t] + [E.sub.t][w.sub.t+1]/(1 + [E.sub.t][[rho].sub.t+1])](1 + [gamma])/[[pi].sub.t] - [E.sub.t][[pi].sub.t+2]/(1 + [E.sub.t][r.sub.t+2])(1 + [E.sub.t][[rho].sub.t+1])][1 + [(1 + [gamma]).sup.2]],

where [E.sub.t][[pi].sub.t+2] = [E.sub.t]([[pi].sub.t+2]|[I.sub.t]) and [E.sub.t][w.sub.t+1] = [E.sub.t]([w.sub.t+1]|[I.sub.t]) are conditional expectations of the variable in question on the basis of the information set available at time t, [I.sub.t] = {[[pi].sub.t-i], [w.sub.t-i], [p.sub.t-i], [[rho].sub.t-i], [r.sub.t-i]} for i = 0, 1, ..., [infinity]; the superscript d is introduced to distinguish the demand for housing from the supply of housing, say [h.sup.s.sub.t]; and [bar.l] is normalized to unity. Because the number of young consumers is exogenously fixed and the physical depreciation rate [delta] = 0, the overall supply of houses to the young generation is also exogenously fixed at a level [[bar.l].sup.s.sub.t]. Then in equilibrium,

(4) [h.sup.d.sub.t] = [[bar.l].sup.s.sub.t].

Substituting Equation 3 for [h.sup.d.sub.t] in Equation 4 and rearranging gives

(5) [[pi].sub.t] = [a.sub.t][E.sub.t][[pi].sub.t+2] + [b.sub.1][E.sub.t][w.sub.t+1] + [cw.sub.t],

where [a.sub.t] = 1/(1 + [E.sub.t][[rho].sub.t+1])(1 + [E.sub.t][r.sub.t+2]), [b.sub.t] = (1 + [gamma])/[[bar.l].sup.s.sub.t][[(1 + [gamma]).sup.2] + 1](1 + [E.sub.t][[rho].sub.t+1]), and c = (1 + [gamma])/[[bar.l].sup.s.sub.t][[(1 + [gamma]).sup.2] + 1].

Assuming the transversality condition [lim.sub.t[right arrow][infinity]] [[PI].sup.T.sub.i=0] [E.sub.t][a.sub.t+2][[pi].sub.t+2T] = 0, where T is the time horizon of the consumer, (9) Equation 5 has a general solution of the form

(6) [[pi].sub.t] = [[pi].sup.*.sub.t] + [B.sub.t],

where [[pi].sup.*.sub.t] denotes the static solution and [B.sub.t] = [[zeta].sub1][b.sub.1t] + [[zeta].sub.2][b.sub.2], with [[zeta].sub.1] + [[zeta].sup.2] = 1, that is, we take a weighted average of the two complementary solutions. (10) Two-period-ahead expectations of the general solution may then be taken as follows:

(7) [E.sub.t][[pi].sub.t+2] = [E.sub.t][[pi].sup.*.sub.1+2] + [E.sub.t][B.sub.t+2],

and using Equation 7 in Equation 5, we obtain

(8) [[pi].sup.*.sub.t] + [B.sub.t] = [a.sub.t][E.sub.t][[pi].sup.*.sub.1+2] + [a.sub.t][E.sub.t][B.sub.t+2] + [b.sub.t][E.sub.t][w.sub.t+1] + [cw.sub.t].

By the definition of the particular solution [[pi].sup.*.sub.t], however, Equation 8 reduces to the bubble expression,

(9) [B.sub.t] = [a.sub.t][E.sub.t][B.sub.1+2] [??] [E.sub.t][B.sub.t+2] = [B.sub.t]/[a.sub.t],

which, assuming [absolute value of [a.sub.t]] < 1, (11) verifies

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

In particular, one could consider a bursting bubble of the form

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [E.sub.t[e.sub.t+2] = 0. It is easy to check that a bursting bubble of the type in Equation 11 is compatible with Equation 3. Given Equation 3 then, our model predicts the following behavior of house prices:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

provided [B.sub.t] > 0, which cannot usually be ruled out. However, [B.sub.t] < 0 can be ruled out in a free-disposal world. Individuals cannot distinguish between a house price rise that is due solely to fundamentals from a rise due to fundamentals plus the bubble. Nevertheless, individuals are willing to pay a price over the fundamentals price as long as the bubble yields them the required rate of return next period and is expected to persist. Thus, one implication of rational bubbles is that [B.sub.t] < 0 cannot occur because the bubble element falls at a faster rate than the house price, and therefore, a negative rational bubble ultimately ends in a zero house price. Rational agents realize this and hence know that the bubble will eventually burst (see also Blanchard 1979; Blanchard and Fisher 1989). (12,13)

The bubble bursts with probability 1 - q each period and continues with probability q. If it bursts, it returns in expected value to zero, behaving as a periodically collapsing bubble; the disturbance term [e.sub.t] allows bubbles to have an additional noise term and permits new bubbles to form after a bubble has collapsed (see Evans 1991).

Overall, therefore, our stylized representative overlapping-generations model illustrates how it is possible to derive analytically a parametric function of the demand for housing of the form assumed by a large literature. Most tellingly, our simple theoretical model shows how a rational bubble may arise as a solution to the house price determination equation.

One may make the model richer by including additional fundamentals, including demographic variables (e.g., Mankiw and Well 1989), taxes (e.g., Poterba 1991), employment, and returns on equities. However, given the illustrative nature of the theoretical component of the article and our intention to keep the model tractable, we decided to include only the least controversial fundamentals for housing prices, namely income and borrowing rates. Nevertheless, at the empirical level, we investigated the robustness of our results using further fundamentals in estimation, as discussed below. (14)

Another feature missing in the model relates to the mechanism through which house prices respond to employment shocks, because the model effectively assumes full employment. Employment shocks may be important because the response of house prices to employment shocks may be quite substantial. (15) However, for employment shocks to appear explicitly in our theoretical representation, we would have to model firms and their labor demand as well as consumers' choice of leisure. Effectively, this would lead to a general equilibrium setting with production. We have used a simple fixed labor supply condition in order to keep to a partial equilibrium setting, for pure reasons of tractability, implicitly assuming that flexible wages clear the labor market.

We now turn to the empirical counterpart of our theoretical model, which relies on the housing demand function derived above to construct a test for bubbles in U.K. house prices.

3. Data

Our data set comprises seasonally unadjusted quarterly time series for the U.K. constant-quality house price, real personal disposable income per capita, treasury bill (TB) interest rate, average mortgage rate, and consumption expenditure deflator (CED) over the sample period from 1983:I through 2002:IV. The beginning of the sample period is dictated by the fact that the constant-quality house price series is available since 1983. Data for real personal disposable income per capita, TB interest rate, and CED were taken from the Central Statistical Office's (CSO) Economic Trends; data for the average mortgage rate were obtained from the CSO's Housing Finance. (16)

The time series for the constant-quality house price is the Halifax house price index, provided by HBOS plc. (17) This index is based on detailed records of the prices, physical characteristics, and the regional location of houses. The methodology used to construct the index is based on the hedonic approach to price measurement characterized by valuing goods for the attributes they possess. In the case of housing, prices will reflect the valuation placed by a purchaser on the particular set of physical and locational attributes possessed by the property they wish to buy. Prices are disaggregated into their constituent parts using multivariate regression analysis. This permits the estimation of the change in average price from one period to another on a standardized basis. It is inappropriate to make simple comparisons of average prices of groups of properties traded in different periods because the composition of property types, sizes, locations, etc. will differ from one period to the next. Any rise in average price between two periods, for example, could be attributed to more expensive properties (e.g., large detached houses) being sold in the later period and not to any intrinsic rise in average property value. A reliable comparison of house price movements over time will only be achieved if the difference in the mix of houses traded over different periods is accommodated. The Halifax house price index allows for differences in the mix of properties between different periods by producing a standardized or mix-adjusted measure for house price movements. The average contribution made by each individual property characteristic is calculated, holding all other characteristics constant. By weighting together this average contribution by the specific mix of characteristics in a base period (for the Halifax house price index, this is 1983), a standardized or mix-adjusted index can be obtained. This will be the ratio between the standardized price in the relevant period and the average price in the base period. (18)

The time series of interest in the empirical analysis are the natural logarithm of the real constant-quality house price (p), the natural logarithm of real personal disposable income per capita (y), the real TB interest rate ([r.sub.f], which we use as a proxy for the risk-free rate), and the real average mortgage rate ([r.sub.m]). (19)

4. Testing for Explosive Bubbles in House Prices

In this section, we briefly describe the two econometric testing procedures that we employ for testing for explosive bubbles in U.K. house prices. Both testing procedures are indirect tests for bubbles in the sense that they are not based on the formulation and estimation of a parametric specification of the bubble process. (20) The first test is based on testing for cointegration in a fundamentals-based model of house prices using a robust estimator that has been shown to be particularly appropriate in this context, while the second test is based on a generalization of the Dickey-Fuller (hereafter DF) (Dickey and Fuller 1981) test statistic, which makes use of the class of Markov regime-switching models.

A Robust Test for Bubbles

A house price bubble may be thought of as an explosive component of the house price that is not present in the underlying fundamentals and that, therefore, drives an explosive wedge between the house price and the fundamentals. This is essentially the definition of bubble captured by Equation 6, where the fundamental solution [[pi].sup.*] may be thought of as empirically determined by real personal disposable income, the real risk-free rate, and the real mortgage rate. If the house price and the fundamentals are realizations of I(1) (difference stationary) stochastic processes, then in the absence of bubbles, our model of house prices implies cointegration between the house price and the fundamentals, that is, a linear combination of p, y, [r.sub.f], and [r.sub.m] must exist that defines a stationary process. If the house price contains an explosive bubble term, however, which is not by definition in the fundamentals series, then this will drive a wedge between house prices and the fundamental determinants so that cointegration between p, y, [r.sub.f], and [r.sub.m] cannot be established. (21, 22)

The above discussion logically suggests a simple procedure for testing for bubbles in house prices. We first apply unit root tests to p, y, [r.sub.f], and [r.sub.m], and then we test for cointegration in a static regression of p on y, [r.sub.f], and [r.sub.m]. If cointegration can be established, this would indicate rejection of the hypothesis of house price bubbles.

The interpretation of cointegration tests as tests for rational bubbles has, however, been shown to be potentially misleading in the presence of bubbles that collapse from time to time over the sample period, that is, periodically collapsing bubbles (Evans 1991). (23) Specifically, Evans (1991) shows that cointegration tests will tend to reject the null hypothesis of bubbles more often than suggested by the chosen significance level when there are periodically collapsing bubbles.

The essence of the problem lies in the fact that periodically collapsing house price bubbles will tend to generate skewness and excess kurtosis in time series for house prices over and above that that may be present in the fundamentals series. (24) Because the maintained hypothesis in many standard unit root and cointegration tests is a linear autoregressive model in which the error term is assumed to be Ganssian, a test based on the estimated autoregressive coefficients will tend to average out the exploding part of the bubble and its collapse in the estimated coefficients, generating bias toward rejection of nonstationarity (Evans 1991). The bias should be reduced, however, if a test for nonstationarity is used that allows the collapse in the bubble to be attributed largely to a sudden movement in a nonnormal error term rather than the estimated coefficients of the autoregressive model.

Taylor and Peel (1998) employ a test statistic based on the work of Im (1996) and Im and Schmidt (2000), which explicitly exploits the skewness and excess kurtosis that bubbles engender in the data and alleviates some of the econometric problems encountered in the presence of periodically collapsing bubbles. The essential features of the procedure may be briefly described as follows. Consider the simple linear regression model,

(13) [s.sub.t] = [PSI]'[z.sub.t] + [u.sub.t],

where [z.sub.t] = (1 [x'.sub.t]), [x.sub.t] is a (k - 1) x 1 vector of time series observed at time t, while [PSI] = ([??] [beta]') is the parameter vector, where [??] is the intercept and [beta] is the (k - 1) x 1 vector of parameters of interest. Greater efficiency can be obtained by including in a regression variables that covary with the errors but that are uncorrelated with the regressors (see inter alia Wooldridge 1990). Im and Schmidt (2000) design covariates that have these properties when the residuals are characterized by the presence of skewness and excess kurtosis and that can therefore be used to obtain a more efficient estimator. In particular, skewness and excess kurtosis in the residuals beyond what would be expected given Gaussian errors imply that the standardized third central moment of the series is nonzero and the standardized fourth central moment exceeds three. Hence,

(14) E([u.sup.3.sub.t] - [[sigma].sup.3]) = E[[u.sub.t]([u.sup.2.sub.t] - [[sigma].sup.2])] [not equal to] 0,

(15) E([u.sup.4.sub.t] - 3[[sigma].sup.3]) = E[[u.sub.t]([u.sup.3.sub.t] - 3[[sigma].sup.2][u.sub.t])] [not equal to] 0,

in turn implying that both ([u.sup.2.sub.t] - [[sigma].sup.2]) and ([u.sup.3.sub.t] - 3[[sigma].sup.2][u.sub.t]) are correlated with [u.sub.t] but not with the regressors (because [x.sub.t] and [u.sub.t] are, by assumption, independent). Im and Schmidt (2000) derive a two-step estimator that can be computed from ordinary least squares (OLS) applied to Equation 13 augmented by the term [[??].sub.t] = [([[??].sup.3.sub.t] - 3[[??].sup.2][[??].sub.t]) x ([[??].sup.2.sub.t] - [[??].sup.2])]', where [??], denotes the residuals and [[??].sup.2] the standard residual variance estimate obtained from OLS applied to Equation 13. The resulting estimator is the residuals-augmented least squares (RALS) estimator of [beta], say [[beta].sup.*], and Im and Schmidt (2000) derive analytically its asymptotic distribution and show how the covariance matrix for [[beta].sup.*] can be consistently estimated. Im and Schmidt (2000) also provide a measure of the asymptotic efficiency gain from employing RALS as opposed to OLS through the statistic [[rho].sup.2], which is constructed as the ratio of the OLS and RALS residual variance terms ([[rho].sup.2] is small for large efficiency gains). Im and Schmidt (2000) suggest that the decision as to whether to employ the RALS estimator may be based on the results of tests for normality of the error distribution, such as that suggested by Jarque and Bera (1987).

The unit root and cointegration test statistics we employ in our empirical analysis are Dickey-Fuller test statistics constructed using the RALS estimator--the RALS Dickey-Fuller and RALS cointegrating regression Dickey-Fuller test statistics.

A Markov-Switching Unit Root Test

The second test for bubbles we consider in this article is based on a generalization of the Dickey-Fuller procedure, which makes use of the class of Markov regime-switching models and was recently proposed by Hall, Psaradakis, and Sola (HPS) (1999).

The conventional augmented Dickey-Fuller (ADF) unit root test statistic is based on the regression

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [DELTA] is the first-difference operator and the lag k is chosen such that [[omega].sub.1t] is approximately white noise. For stationarity, we require [phi] < 0, while, if [v.sub.t] is a realization of a unit root process, one should expect to find [phi] = 0. The ADF test statistic is then the standard t-ratin test statistic for the null hypothesis that [phi] = 0, and the rejection consists of large negative values. As is well known, this statistic does not, however, follow the standard Student's t-distribution under the null hypothesis because of the theoretically infinite variance of [v.sub.t], and finite-sample critical values have been computed using Monte Carlo methods by Fuller (1976) and MacKinnon (1991). This statistic is termed DF statistic if k = 0 and ADF statistic if k > 0.

The test proposed by HPS (1999) is essentially a generalization of the ADF procedure to allow for the possibility that the dynamic behavior of [v.sub.t] may be different for different parts of the sample. In particular, HPS consider the possibility where the parameters in the ADF regression, Equation 16, change according to the state or regime that the process is in. This seems plausible in the context of periodically collapsing bubbles, where the two regimes essentially correspond to the expanding and collapsing phases of the bubble. Hence, assuming that the parameters of the ADF regression are time varying and change with an unobserved indicator [s.sub.t] [member of] {0, 1}, HPS suggest considering a regression of the form

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [[omega].sub.2t] ~ IIN(0, 1). Consistent with a large literature (notably, see Goldfeld and Quandt 1973; Hamilton 1989, 1990), the random process st is assumed to be a homogeneous first-order Markov chain on the state space {0, 1} with transition probabilities

(18) Pr([s.sub.t] = 1 | [s.sub.t-1] = 1) = [q.sub.1], Pr([s.sub.t] = 0 | [s.sub.t-1] = 1) = 1 - [q.sub.1], Pr([s.sub.t] = 0 | [s.sub.t-1] = 0) = [q.sub.2], Pr([s.sub.t] = 1 | [s.sub.t-1] = 0) = 1 - [q.sub.2],

and the innovation [[omega].sub.2t] is assumed to be independent of the state variables [s.sub.m] for all t and m. The test for a unit root in either regime ([[phi].sub.0] = 0 and/or [[phi].sub.1] = 0) may then be constructed using a recursive, nonlinear filter algorithm in order to produce probabilistic inferences of the form Pr([s.sub.t] = 1 |[I.sub.t], [xi]), where [I.sub.t] = {[v.sub.1], [v.sub.2], ..., [v.sub.T]} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and the likelihood of the observed data. The value [xi] of [xi] that maximizes this likelihood can then be found using a numerical optimization algorithm. The test of the null hypothesis of a unit root in [v.sub.t] may be based on the t-ratio associated with the maximum likelihood estimates of [[phi].sub.0]] and [[phi].sub.1]]. Note that, within the Markov-switching ADF framework, the existence of explosive rational bubbles is consistent with [[phi].sub.0]] > 0 or [[phi].sub.1]] > 0, indicating that one of the regimes governing the underlying process is characterized by the presence of an explosive component. Evidence that [[phi].sub.0]] = [[phi].sub.1]] = 0 is inconsistent, however, with the existence of rational bubbles in [v.sub.t]. (25) Then, once [xi] is obtained, probabilistic inferences about the unobserved regimes [s.sub.t] may be made on the basis of the filter probabilities Pr([s.sub.t] = 1|[I.sub.t,[xi]) or the smoothed probabilities Pr([s.sub.t] = 1|[I.sub.T],[xi]).

5. Empirical Analysis

Robust Tests for Bubbles

Hansen (1995) shows that, in general, the asymptotic critical values for unit root tests generated from auxiliary regressions with covariates may lead to substantial small-sample size distortion. Accordingly, we generated the appropriate finite-sample critical values for the RALS Dickey-Fuller statistic, [[tau].sub.A], from 5000 draws across an assumed random walk data-generating process with Gaussian errors, with the drift and error variance calibrated on the basis of the OLS estimates of a random walk model for the quarterly log house price data with 80 data points. (26) corresponding to the number of observations available in practice in the present application. In addition to calculating the empirical distribution of [[tau].sub.A], we also used the Monte Carlo experiments to calculate the empirical distribution of the standard Dickey-Fuller test statistic, [tau].

We also generated the empirical distribution of the RALS cointegrating regression Dickey-Fuller statistic, CR[[tau].sub.A], applied to the residuals from a regression of the artificial log house price onto artificial log real income, real TB rate, and real mortgage rate under the null hypothesis of no cointegration, using the same artificial data-generating process for the log house price and a different drifting random walk process for each of log real income, real TB rate, and real mortgage rate, calibrated from OLS estimates obtained using the actual data over the sample period 1983-2002. The critical values were again generated using 5000 draws. In addition to calculating the empirical distribution of CR[[tau].sub.A], we also calculated the empirical distribution of the standard cointegrating regression Dickey-Fuller statistic CR[tau].

In Table 1, we report the resulting empirical critical values at the 1, 5, and 10% nominal significance level for each of [tau], [[tau].sub.A], CR[tau], CR[[tau].sub.A]. The empirical critical values of [tau] and CR[tau] are close to those implied by the response surface results of MacKinnon (1991, table 1). Although the asymptotic distribution of [[tau].sub.A] and CR[[tau].sub.A] are identical to those of [tau] and CR[tau], respectively, for [[rho].sup.2] = 1, their small-sample distributions appear to differ from one another, so that the critical values for [[tau].sub.A] and CR[[tau].sub.A] deviate more from the corresponding values of the normal distribution compared with [tau] and CR[tau], respectively.

Next, we applied the [tau], [[tau].sub.A], CR[tau], and CR[[tau].sub.A] test statistics to our quarterly data for p, y, [r.sub.f], and [r.sub.m]. The results, reported in Panel A of Table 2, suggest that strong nonnormality characterizes the residual series of the log house price and the real mortgage rate (as evidenced by the Jarque and Bera statistics), suggesting that the [[tau].sub.A] statistic is the more appropriate unit root test for these two series, with the [[rho].sup.2] statistic indicating fairly large efficiency gains from using the RALS estimator. Nevertheless, using both the [tau] and [[tau].sub.A] statistics, the unit root null hypothesis cannot be rejected at standard significance levels for each of p, y, [r.sub.f], and [r.sub.m] over the sample period. Tests of nonstationarity of the residuals from a regression of p onto y, [r.sub.f], and [r.sub.m], based on either the CR[tau] or CR[[tau].sub.A] statistics did not allow us, however, to reject the null hypothesis of no cointegration at conventional significance levels (Table 2, Panel B), hence providing evidence in favor of the existence of bubbles in U.K. house prices. (27) Corroborating evidence of the presence of bubbles in house prices can be adduced by noticing that the residuals from the static regression of p, y, [r.sub.f], and [r.sub.m] display strong evidence of nonnormality, as evidenced by the high Jarque-Bera statistics, suggesting that the CR[[tau].sub.A] is more appropriate than the CR[tau] statistic, with the [[rho].sup.2] statistic indicating a rather large efficiency gain from using the RALS estimator. If a periodically collapsing house price bubble is present, this will impart skewness and excess kurtosis into the house price series over and above that that is present in the fundamentals series. If, however, movements in house prices are in fact largely reflecting movements in the fundamentals and hence bubbles are not the cause of the nonnormality, then any skewness and excess kurtosis in the fundamentals series will be a common feature of both the fundamentals and the house price series and should disappear in the residuals from the cointegrating regression (Engle and Kozicki 1993). (28)

In Figure 1, we graph the RALS residuals from the static regression of p onto y, [r.sub.f] and [r.sub.m]. The graph clearly shows two sharp rises in the residuals during the late 1980s, with a peak in 1989, and during the late 1990s and early 2000s, consistent with our discussion in section 1 and the references therein. Indeed, the results suggest that the second bubble has lasted longer than the bubble of the 1980s and it may be still ongoing at the end of our sample period. (29)

[FIGURE 1 OMITTED]

Markov-Switching Unit Root Tests

Our final exercise was to investigate further whether the nonstationarity of U.K. house prices over the sample may be attributable to explosive rational bubbles using the Markov-switching unit root test described in section 4. The results from fitting the Markov-switching model, Equation 17, to each of p, y, [r.sub.f], and [r.sub.m] are reported in Table 3, where the figures in braces are marginal significance levels of t-type tests for the null hypotheses [[theta].sub.0] = [[theta].sub.1] and [[theta].sub.1] = 0 against the alternative hypotheses [[theta].sub.0] < 0 and [[theta].sub.1] > 0, respectively. (30) Because the distribution of such statistics is unknown, however, the p-values (or marginal significance levels) were obtained by bootstrap; as suggested by HPS (1999), the number of bootstrap replications is 1000.

Interestingly, the estimation results suggest that, for each of the time series under investigation, the estimated value of [[theta].sub.0] is negative in the regime [s.sub.t] = 0, although it is not statistically significant and the unit root hypothesis cannot be rejected. However, the regime represented by [s.sub.t] = 1 is characterized by positive values of [[theta].sub.1], but the unit root hypothesis can be rejected in favor of an explosive alternative hypothesis only for p.

In the bottom graph of Figure 2, we plot the inferred probabilities of p being in the explosive regime ([s.sub.t] = 1) over the sample period. Unsurprisingly, there are clear switches to the explosive regime in the late 1980s and the late 1990s, where Pr([s.sub.t] = 1 | [I.sub.t], [xi]) > 0.5, consistent with the pattern of p shown in the upper graph of Figure 2 as well as the timing of the two sharp rises in the RALS residuals from the static regression of p onto y, [r.sub.f], and [r.sub.m] given in Figure 1. Consistent with the RALS residuals discussed in the previous section, the inferred probabilities also suggest that the current bubble may still be ongoing.

[FIGURE 2 OMITTED]

Finally, we also examined the possibility that the house-price crash of the late 1980s has made homebuyers wary in the 1990s. One simple way to examine this possibility involves investigating whether the parameters [[theta].sub.0] and [[theta].sub.1], which determine whether or not there is explosive behavior in house prices, have changed over time. Thus, we carried out the Markov-switching unit root tests over the two equally large subsamples from 1983:I to 1992:IV and from 1993:I to 2002:IV; respectively. The results suggest values of [[theta.sub.0] and [[theta].sub.1] that are qualitatively identical over the two subsamples and also qualitatively identical to the estimated values [[theta].sub.0] and [[theta].sub.1] obtained for the full samples and reported in Table 3. In turn, this suggests that the stochastic properties of U.K. house prices have been quite similar over the 20 years of data investigated. (31)

Summing Up the Empirical Results

Overall, our results from employing robust tests for bubbles indicate that the U.K. house price series does not cointegrate with the fundamentals series suggested by a stylized model of the form constructed in section 2. This suggests that a bubble term that is not present in the underlying fundamentals may drive an explosive wedge between the house price series and its fundamental determinants. These results are corroborated by the empirical evidence from fitting ADF Markov-switching models to each of p, y, [r.sub.f], and [r.sub.m] because the estimated models are consistent with the hypothesis that, while each of y, [r.sub.f], and [r.sub.m] appears to be a realization of an I(1) process, the house price p appears to be well characterized by a two-regime model, where in one regime, p is a realization of a unit root process, while in the other regime, p is a realization of an explosive process.

5. Conclusions

In this article, we have re-examined the view that rational bubbles characterized the behavior of U.K. house prices during the last two decades or so. Using a stylized model that generates the conventional housing demand function assumed by a large literature in this context, we rationalize the house price bubbles hypothesis and motivate our empirical analysis.

We then formally test the house price bubbles hypothesis using quarterly U.K. house price data spanning from 1983 to 2002 and employing two recently developed econometric techniques that alleviate several problems typically encountered in this context. Our empirical results suggest that two explosive bubbles were present in U.K. house prices over the sample period examined, in the late 1980s and the late 1990s, consistent with the house price bubbles hypothesis. The second bubble identified in U.K. house prices appears to be still ongoing at the end of our sample period in 2002, adding to the widespread concerns repeatedly raised by the press and policymakers in recent times.

Overall, this article adds analytical rigor and empirical support in favor of the conjecture, widely held among both academics and practitioners, that speculative bubbles may be present in house prices, the behavior of which cannot be fully explained empirically by conventional fundamentals.

These results raise several issues for further empirical and theoretical research in the relevant literature. While we report some tentative evidence on a stylized fact concerning the behavior of house prices, further empirical work might be addressed toward establishing the robustness of our conclusions, and theoretical work may incorporate them in investigating the implications of the evolution of house prices on living standards, borrowing and saving patterns, wealth accumulation, commuting decisions, housing finance, and the impact of public policy on housing markets. These issues remain immediate avenues for future research.

Appendix: The Demand for Housing

This appendix outlines the procedure we use to solve the model described in section 2 in order to derive the demand for housing. Throughout, the notation is the same as in section 2.

Consumption at t = 2 comes, as a function of past saving/borrowing choice at t = 1 and of past housing choice at t = 0, directly from the consumer's budget constraint (hereafter expressed in real terms),

(A1) [c.sub.2] = (1/[p.sub.1])[[[pi].sub.2][p.sub.1][h.sub.0] + (1 + [r.sub.2])[[s.sub.1] = [c.sub.2]([h.sib.0], [[s.sub.1]).

When old, consumers sell their housing services and receive the prevailing interest rate on their middle-age savings.

Next, maximize the value function at t = 1,

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

subject to

(A3) [p.sub.0][c.sub.1] + (1 + [[rho].sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0]) + [s.sub.1][p.sub.0]/[p.sub.1] = [w.sub.1][p.sub.0][bax.l],

where housing and mortgage deposit choice is fixed from t = 0. Substitution of [c.sub.1] from the budget constraint gives us the unconstrained problem,

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By our assumption of log linear subutilities, we have u([c.sub.1], [h.sub.0], [bar.l]) = log([c.sub.1]) + log([h.sub.0]) + log([bax.l]) and u([c.sub.2]) = log([c.sub.2]), so that the first-order condition of (A4) for an interior solution in [s.sub.1] is

(A5) [[gamma](1 + [r.sub.2]]/[[[pi].sub.2][p.sub.1][h.sub.0] + (1 + [r.sub.2])[s.sub.1]] - [[p.sub.0]/[p.sub.1]]/[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1])([[pi].sub.0][h.sub.0] - [k.sub.0]) - [s.sub.1][p.sub.0]/[p.sub.1]],

implying

(A6) [s.sub.1] = [gamma](1 + [r.sub.2])[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0])] - [[pi].sub.2][p.sub.0][h.sub.0]/(1 + [gamma])(1 + [r.sub.2])[p.sub.0]/[p.sub.1] = [s.sub.1]([h.sub.0], [k.sub.0]).

Equation A6 suggests that there are no restrictions on the sign of the optimal [s.sub.1]: Given past optimal choice of [h.sub.0] and [k.sub.0], [s.sub.1] can either be positive (saving) or negative (borrowing), depending on the values of prices, wages, interest rates, and time preference. Also, substituting Equation A6 for [s.sub.1] in the budget constraints, Equations A1 and A3, gives us optimal consumption in middle and old age,

(A7) [c.sub.1] = (1 + [r.sub.2])[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0])] + [[pi].sub.2][p.sub.0][h.sub.0]/(1 + [gamma])(1 + [r.sub.2])[p.sub.0] = [c.sub.1]([h.sub.0],[k.sub.0]),

(A8) [c.sub.2] = (1 + [r.sub.2])[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0])] + [[gamma][[pi].sub.2][p.sub.0][h.sub.0]/(1 + [gamma])[p.sub.0] = [c.sub.2]([h.sub.0],[k.sub.0]).

This is a strict regular maximum from the second-order condition in [s.sub.1],

(A9) - [[p.sub.0]/[p.sub.1]]/[[[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1])([PI].sub.0][h.sub.0] - [k.sub.0]) - [s.sub.1][p.sub.0]/[p.sub.1]].sup.2]] - [gamma] [[[[gamma](1 + [r.sub.2])/[[pi].sub.2][p.sub.1][h.sub.0] + (1 + [r.sub.2])[s.sub.1][p.sub.0]/[p.sub.1]].sup.2]] <0.

Thus, the expressions of optimal housing and consumption choice at t = 0 are given by the third and final step, namely,

(A10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

subject to

(A11) [p.sub.0][c.sub.0] + [k.sub.0] = [W.sub.0][bar.l].

Using Equations A7 and A8 and the budget constraint in Equation A11 yields

(A12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Given log linear subutilities, the first-order condition for an interior solution in [h.sub.0] is then

(A13) 1/[h.sub.0] + [[gamma][[pi].sub.2][p.sub.0] - (1 + [r.sub.2])(1 + [[rho].sub.1])[[PI].sub.0]]]/[(1 + [r.sub.2])[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1][[PI].sub.0][h.sub.0] - [k.sub.0]] + [[pi].sub.2][p.sub.0][h.sub.0]] = 0.

Similarly, the first-order condition for an interior solution in [k.sub.0] is

(A14) [[gamma](1 + [gamma])(1 + [r.sub.2])(1 + [[rho].sub.1])]/[(1 + [r.sub.2])[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1])[[PI].sub.0][h.sub.0] - [k.sub.0])] + [[pi].sub.0][p.sub.0][h.sub.0]] - [1]/[[W.sub.0][bar.l] - [k.sub.0]] = 0.

Now substitute Equation A14 into Equation A13 and solve for [k.sub.0],

(A15) [k.sub.0] = [[(1 + [gamma]]).sup.2](1 + [[rho].sub.1]([W.sub.0] - [w.sub.1][p.sub.0]][bar.l]/[1 + [(1 + [gamma]).sup.2]](1 + [[rho].sub.1]).

Once again, the sign of the optimal [k.sub.0] does not appear to be restricted (the optimal [k.sub.0] will only be less than [w.sub.0][bar.l] in order to have positive first-period consumption). Substituting Equation A15 into Equation A14 and solving for [h.sub.0] gives us the optimal demand for housing,

(A16) [h.sub.0] = (1 + [r.sub.2])[[w.sub.1] + (1 + [[rho].sub.1])[w.sub.0]][bar.l](1 + [gamma])/[[(1 + [r.sub.2])(1 + [[rho].sub.1]) [[pi].sub.0] - [[pi].sub.2][1 + [(1 + [gamma]).sup.2]].

Finally Equations A15 and A16 and the budget constraint can be used to obtain optimal consumption in young age,

(A17) [c.sub.0] = [(1 + [[rho].sub.1])[w.sub.0] - [w.sub.1]][bar.l]/[1 + [(1 + [gamma]).sup.2](1 + [[rho].sub.1]).

Note that the commodity demand functions, Equations A16 and A17, are homogeneous of degree zero in current and discounted prices, while asset choice in Equation A15 is homogeneous of degree one in the same variables. In Equation A16, the real user cost or rental equivalence price of housing (1 + [r.sub.2])(1 + [rho].sub.1])[[pi].sub.0] - [[rho].sub.2] appears; we assume that its foresight terms are such that it is always strictly positive, in order to avoid consumers wishing to hold an infinite amount of housing to realize infinite capital gains. (32)

Finally, we can substitute Equations A15 and A16 into Equations A6, A7, and A8 in order to obtain saving/borrowing in middle age,

(A18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where A = [bar.l]/[1 + [(1 + [gamma]).sup.2]], and middle and old age consumption are

(A19) [c.sub.1] = [bar.l][[[w.sub.1]/1 + [gamma]] - [(1 + [[rho].sub.1])[w.sub.0] + [w.sub.1]]/[1 + [(1 + [gamma]).sup.2]]],

(A20) [c.sub.2] = [gamma]([1 + [r.sub.1])[bar.l]/[[w.sub.1]/1 + [gamma]] - [(1 + [[rho].sub.1])[w.sub.0] + [w.sub.1]]/[1 + [(1 + [gamma]).sup.2]]].

Notice again that the second-order conditions ensure that the solution to the problem in Equation A12 is a strict regular maximum,

(A21) - 1/[h.sup.2.sub.0] - [gamma][([[pi].sub.2][p.sub.0] - (1 + [r.sub.2])(1 + [[rho].sub.1])[[PI].sub.0]/(1 + [r.sub.2])[[w.sub.1][bar.l] - (1 + [[rho].sub.1]([[pi].sub.0][h.sub.0] - [k.sub.0])] + [[pi].sub.2][h.sub.0]).sup.2] <0

(A22) - 1/[([W.sub.0][bar.l] - [k.sub.0]).sup.2] - [([gamma](1 + [gamma])(1 + [r.sub.2](1 + [[rho].sub.1])/(1 + [r.sub.2])[[w.sub.1][p.sub.0][bar.l] - (1 + [[rho].sub.1]([[PI].sub.0][h.sub.0] - [k.sub.0])] + [[pi].sub.2][p.sub.0][h.sub.0]) <0.

To sum up, Expressions A15, A16, and A17 give us the optimal present mortgage, housing, and consumption choice of consumers who are young at t = 0, while Expressions A18, A19, and A20 give us their optimal future saving and consumption plans for middle and old age. Because agents have homogeneous expectations, if we sum up Expressions A17, A19, and A20 at a common (arbitrary) point in time, we obtain total consumption of all consumers of our model--young, middle aged, and old--at that point in time. By the same token, we can sum up the fixed labor supply of [bar.l] of both currently young and middle-aged consumers in order to obtain total labor supply. Of course, Expressions A15 and A18 evaluated at a common point in time give us the asset choices (mortgage and savings) of all consumers.

Table 1. Empirical Critical Values of the [tau], [[tau].sub.A],
C[R.sub.[tau]], C[R.sub.[[tau] Statistics at the
1, 5, and 10% Nominal Significance Level

                  1%           5%           10%

[tau]             -3.5207      -2.9042      -2.5896
                  [-3.5130]    [-2.8976]    [-2.5858]
[[tau].sub.A]     -3.7615      -3.1123      -2.7526
C[R.sub.[tau]]    '-4.9015     '-4.2559     '-3.9401
                  [-4.8734]    [-4.2377]    [-3.9212]
C[R.sub.[tau]]    -5.2911      -4.5219      -4.2106

[tau] denotes the standard DF test statistic, while [[tau].sub.A]
is the residuals-augmented DF statistic; the empirical critical
values for [tau] and [[tau].sub.A] are based on 5000 draws from
the data generating process of a random walk with drift with an
independently and identically entically distributed error term,
as described in the text, with 80 artificial data points.
C[R.sub.[[tau] denotes the DF statistic and C[R.sub.[tau]A]
the residuals-augmented DF statistic, each applied to the
residuals from a cointegrating regression under the null hypothesis
of no cointegration, as described in the text, again based on
5000 draws (Fuller 1976: Dickey and Fuller 1979, 1981).
Figures in brackets for [tau] and C[R.sub.[[tau] are critical
values calculated using the response surfaces results of MacKinnon
(1991, Table 1).

Table 2. Unit Root and Cointegration Tests

Panel A: [tau] and [[tau].sub.A] unit root tests on p, y, [r.sub.f],
and [r.sub.m]

                 [tau]     [[tau].sub.A]  [[rho].sup.2]    JB

p               -2.0143       -1.8312       0.7815       {0.0003}
y               -2.2013       -2.1369       0.9217       {0.1922}
[r.sub.f]       -2.2147       -2.0417       0.9279       {0.1302}
[r.sub.m]       -1.9502       -1.7355       0.8853       {0.0263}

Panel B: CR[tau] and CR[[tau].sub.A] tests--cointegration between p, y,
[r.sub.f], and [r.sub.m]

CR[tau]     CR[[tau].sub.A]   [[rho].sup.2]     JB

-2.4910         -1.8322          0.8019       {0.0021}

1, 5, and 10% critical values for [tau], [[tau].sub.A], CR[tau], and
CR[[tau].sub.A] are as given in Table 1. For the Jarque-Bera test
statistic (JB) we report in braces the p-values from referring the
value of JB to the [chi square](2) distribution.

Table 3. Maximum Likelihood Estimation of Markov-Switching ADF Models

                       p                  y

[[phi].sub.0]   -0.0263 {0.2079}   -0.0122 {0.2115}
[[phi].sub.1]    0.0109 {0.0004}    0.0003 {0.5832}
[q.sub.2]    0.8047 (0.2262)    0.5139 (0.1029)
[q.sub.3]    0.9322 (0.0321)    0.9351 (0.0532)

                   [r.sub.f]          [r.sub.m]

[[phi].sub.0]   -0.0053 {0.3303}   -0.0042 {0.3135}
[[phi].sub.1]    0.0020 {0.3591}    0.0018 {0.2530}
[q.sub.2]    0.6637 (0.1291)    0.5731 (0.1637)
[q.sub.3]    0.9577 (0.0402)    0.9574 (0.0394)

These results were obtained from estimating a fourth-order
Markov-switching regression of the form of Equations 17 and 18.
Figures in braces denote p-values of tests of the unit root
hypothesis against the alternative of a stationary process in the
case of [[phi].sub.0] and against the alternative of an explosive
process in the case of [[phi].sub.1]; the p-values were computed by
bootstrap as discussed in the text. [q.sub.1] and [q.sub.2] are the
estimated transition probabilities defined in Equation 18; figures
in parentheses denote estimated standard errors.

This article was partly written while Lucio Sarno was a Visiting Scholar at the International Monetary Fund, the Federal Reserve Bank of St. Louis, and Washington University. The authors are grateful to Kent Kimbrough (editor) and two anonymous referees for constructive comments and suggestions on a previous version of the paper. They are also thankful to Jagjit Chadha, Ben Lockwood, and Peter Simmons for comments or useful conversations. The authors remain responsible for any errors that may remain and for the views expressed in the paper.

(1) See also Case and Shiller (1989) for evidence on U.S. data, Noguchi (1994) for evidence on Japanese data, and Eppli and Shilling (1995) for a broader cross-country investigation. For a review of the literature on house price bubbles and their link to the globalization of financial markets, see Renaud (1997).

(2) In fact, most estimated house price equations in the relevant literature are essentially inverted housing demand equations of the form derived here, with the demand for housing assumed to depend on real disposable income and the real user cost of housing services, for example, see Poterba (1984) and Mankiw and Well (1989).

(3) Indivisibility forces us to assume that the discrete choice is met, that is, that in each period all agents are able to exchange at least one indivisible stock unit (house). However, in oar model, this assumption turns out to be innocuous because consumers purchasing a house can rent to other consumers pan of the housing services provided at the unit price [[PI].sub.t]. Also, note that relaxing the assumption of no transactions costs makes our model slightly less tractable, but does not change any of the main results.

(4) Note that, given Equation 1, we have [h.sub.0] = [h.sub.1] = [h.sub.2], so that [h.sub.0] appears in the utility function.

(5) This is the case except if t represents old age, when [u.sub.t]([c.sub.t]) = log([c.sub.t]).

(6) As it is well known, the log utility function belongs to the class of hyperbolic absolute risk aversion (HARA) utility functions and, in particular, to the class of constant relative risk aversion (CRRA) utility functions. For a discussion of the relative advantages of the log utility function see, for example, the overview of Blanchard and Fisher (1989, Chapter 6).

(7) Nevertheless, our assumption of a constant population does not restrict housing sales across generations and time because housing services are perfectly divisible.

(8) At any t, each consumer has a common distribution of future real wages, house prices, and interest rates. Thus, denoting the means of these distributions as [z.sub.t+i] = {[w.sub.t+i], [[pi].sub.t+i], [[rho].sub.t+i], [r.sub.t+1]} for any i > 0, these coincide with the observed per capita values averaged across all individuals. Under rational expectations, given the available information set, each individual forecasts exactly the mean of the distribution, so that [E.sub.t][z.sub.t+i] = [z.sub.t+i] for any i > 0.

(9) Leading Equation 5 two periods, taking expectations twice, and using the law of iterated expectations yields

[E.sub.t][[pi].sub.t+2] = [E.sub.t][a.sub.t+2][E.sub.t][[pi].sub.t+4] + [E.sub.t][b.sub.t+2][E.sub.t][w.sub.t+3] + c[E.sub.t][w.sub.t+2].

Then, substitute the above expression into Equation 5 to get

[[pi].sub.t+2] = [a.sub.t][E.sub.t][a.sub.t+2][E.sub.t][[pi].sub.t+4] + [a.sub.t][E.sub.t][b.sub.t+2][E.sub.t][w.sub.t+3] + [a.sub.t]c[E.sub.t][w.sub.t+2] + [b.sub.t][E.sub.t][w.sub.t+1] + c[w.sub.t],

which, solved recursively, gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where T is the time horizon of the consumer.

(10) We deliberately rule out particular cases that might occur with second-order difference equations, such as repeated roots of the form [B.sub.t] = [b.sub.1t] + [b.sub.2t]t, or a particular solution changing at a constant rate over time such as, for example, [[pi].sub.t] = [[pi].sup.*.sub.t].

(11) The assumption [absolute value of [a.sub.t]] < 1 is equivalent to assuming that (1 + [i.sub.t+1])(1 + [i.sub.t+2]) < (1 + [P.sub.t+1])(1 + [R.sub.t+2]), where [i.sub.t] [equivalent to] ([p.sub.t] - [p.sub.t-1])/[p.sub.t-1] is the inflation rate. This seems tenable if one thinks that, at least for major industrialized countries, both the nominal interest rate and the nominal mortgage rate are typically above the inflation rate. Also, in the present application, the above condition is easily satisfied throughout the sample period on our data (described in section 3).

(12) This also implies that if the actual house price is below the house price implied by the fundamentals, it cannot be because of a rational bubble. If negative bubbles are not possible, then if a bubble is ever zero, it cannot restart; this is the case because the innovation [B.sub.t+1] - [E.sub.t][B.sub.t+1] in a rational bubble must have a zero mean and, if the bubble started again, the innovation could not be zero mean because the bubble would have to go in one direction only (increase) in order to start again.

(13) Note that, in a model with infinitely lived agents who have perfect foresight of future prices, wages, and interest rates, and where wading profits are not possible, [B.sub.t] ought to be equal to zero. However, the present model is different, in that our representative consumer is not infinitely lived. There are three overlapping generations of consumers and no bequests; each consumer lives three periods and cannot die in debt. This is because consumers are assumed to buy [h.sub.0] housing services when young on a mortgage at the prevailing market prices and interest rates; the mortgage is repaid in middle age, and finally, consumers must sell their housing in old age at the market price prevailing at that period in order to live off the proceeds of the sale and any savings. With a fixed housing supply, this setup gives rise to a rational bubble. See also Blanchard (1979).

(14) Please note that, in our model, lenders lend at market prices and there is no cost to speculative lending in that we have assumed that wealth in middle age is always high enough to repay the mortgage. Effectively, this implies that there is no defaulting state. We should like to point out that we have assumed this only for reasons of tractability. However, our model could be written such that there are, for example, two states of nature in middle age: a good state, where there is enough wealth to repay the mortgage (in our notation [(1 + [R.sub.1])[W.sub.0] + [W.sub.1]][bar.l] + [[PI].sub.2][h.sub.0] - (1 + [P.sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0]) > 0) and a default state, where the above inequality is reversed, so that the lender would cancel the mortgage repayment and seize the house of value [[PI].sub.1][h.sub.0] in middle age. The mortgage interest rate [P.sub.1] would then be assumed to be high enough to contain a default premium and one could generalize our assumptions so that, on average, housing prices in middle age would cover mortgage repayments (i.e., [(1 + [R.sub.1])[W.sub.0] + [W.sub.1]][bar.l] + [E.sub.0][[PI].sub.2][h.sub.0] - (1 + [P.sub.1])([[PI].sub.0][h.sub.0] - [k.sub.0]) > 0) in order to preserve the incentive to lend in the first place. We have not explicitly done this because our use of log utility and our aim to find a simple, closed-form solution require us to minimize the number of states in each period.

(15) At the empirical level, this issue is likely to be relevant on U.K. data, which is characterized by the very large London housing market where employment shocks in industries such as business services, financial services, and tourism can have a substantial impact on London house prices and, generally, on the whole U.K. economy.

(16) The housing finance data are recorded by the CSO directly from the reports of the Chief Registrar of Friendly Societies and the reports of the Building Societies Commission. The average mortgage rate is the tax-adjusted rate charged in the quarter.

(17) The HBOS Group has two direct subsidiaries, namely Bank of Scotland and Halifax, the largest providers of mortgage services in the United Kingdom.

(18) We are very grateful to two referees for pointing out to us the importance of using constant-quality house prices rather than average house prices and for leading us to locate the Halifax House Price Index. The constant-quality house price data may be obtained from the website of HBOS at http://www.hbosplc.com/help/faqs/housepriceindex.asp.

(19) Note that, while we use both the house price and real personal disposable income in log levels (consistent with the relevant empirical literature, e.g., see Hendry 1984; Muellbauer and Murphy 1997), the empirical results reported below were qualitatively identical when using these time series in levels. Also, using the gross mortgage rate (in place of the tax-adjusted rate) as well as using alternative deflators in constructing the two real interest rates employed in the empirical analysis did not induce any qualitative change in our results.

(20) Notable examples of direct tests for bubbles are due to, inter alia, Flood and Garber (1984) and Flood, Garber, and Scott (1984). Direct tests are susceptible to the criticism that they are unable to detect bubbles other than those belonging to the specific parametric class under consideration, so failure to detect a bubble does not necessarily imply the absence of unspecified types of bubbles different from the particular bubble under consideration. It is mainly for this reason that the recent relevant literature appears to prefer the use of indirect tests for bubbles (e.g., see Taylor and Peel 1998; Hall, Psaradakis, and Sola 1999; Sarno and Taylor 1999).

(21) Essentially, this approach is the analogue of the testing procedure originally proposed by Diba and Grossman (1988) in the context of testing for bubbles in stock prices. Diba and Grossman also argued that testing for nonstationarity of the differenced series in which a bubble is suspected may be a useful test for bubbles because, if the series in question contains a bubble component, differencing should not remove it. Subsequent studies showed, however, that this procedure may not always be reliable because, in finite samples, series with bubble components may look very much like stable processes when differenced a sufficient number of times (e.g., see Hall, Psaradakis, and Sola 1999).

(22) In the spirit of the Fisher hypothesis, one may argue that the real interest rate is expected to be stationary. In practice, however, the real interest rate appears to be a persistent process such that it may be hard to distinguish it from an I(1) process (Campbell, Lo, and MacKinlay 1997), hence introducing complications in cointegration tests of the type considered here.

(23) Periodically collapsing bubbles are particularly interesting in analyzing the behavior of house prices because they collapse almost surely in finite time. This is an attractive property because bubbles do not seem to be empirically plausible unless they collapse after reaching high levels.

(24) Note, however, that nonnormality is a necessary but not sufficient condition for the existence of rational bubbles.

(25) For reasons analogous to the case of the Dickey-Fuller testing procedure, the null distribution of these test statistics is, however, unknown and, therefore, has to be calculated by simulation.

(26) Critical values calculated with an artificial data-generating process calibrated using the log real disposable income, the real TB rate, or the real mortage rate led to quantitatively similar and qualitative identical results.

(27) Also, note that allowing for more lags (up to four) and a time trend in testing for unit roots in the series in question as well as in testing for cointegration did not induce any qualitative change in the outcome of the tests (details available from the authors upon request).

(28) We investigated the robustness of this finding by including additional potential fundamentals in the cointegrating regression, namely population, taxes, and the log of the PTSE 100 stock price index. The results were, however, qualitatively unchanged in that we were unable to reject the null of no cointegration, and the residuals from the static regression tested for cointegration displayed significant nonnormality.

(29) In general, the return offered on alternative investments, such as equities, might be expected to be important in determining the behavior of house prices. When the volatility of share prices is very high, for example, the returns on property become more attractive, possibly encouraging households to shift some of their money out of equities into real estate. This suggests that the RALS residuals may depend on the volatility of share prices. However, in regressing the RALS residuals into current and lagged values of the returns from the FTSE 100 stock index, we find no evidence of a statistically significant relationship. As correctly pointed out by an anonymous referee, this may not be surprising because, in the United Kingdom (and in Europe in general), few households invest in the stock market. Most households typically have most of their wealth in housing, with the remaining small fraction of their wealth invested in fixed-income securities. Hence, the impact of the volatility of stock prices on house prices may not be as relevant in the United Kingdom relative to, for example, the United States.

(30) We set the order of the autoregression k = 4 in each case, which proved sufficient to yield approximately white noise residuals for all estimated regressions. In Table 3, we do not report all of the estimated autoregressive parameters as our focus is on testing the unit root null hypothesis associated with the statistical significance of [[theta].sub.0] and [[theta.sub.1]

(31) We also tested for cointegration between the house price and the fundamentals in each of the above two subsamples, but in each case, we could not find any evidence of cointegration. These results are not reported to conserve space but they are available upon request.

(32) A sufficient condition for this is to assume that the present discounted value of lifetime wealth is strictly positive (i.e., bounded).

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Gaia Garino * and Lucio Sarno ([dagger]) ([double dagger])

* Department of Economics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom; E-mail [e-mail].

([dagger]) University of Warwick, International Monetary Fund, Federal Reserve Bank of St. Louis, and Centre for Economic Policy Research (CEPR).

([double dagger]) Finance Group, Warwick Business School, Coventry CV4 7AL, United Kingdom; E-mail [e-mail]; corresponding author.

Received August 2001; accepted July 2003.