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

• Author: Garino, Gaia

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]

   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...