# Housing price–volume correlations and boom–bust cycles

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## Abstract

This paper investigates the housing price–volume nexus based on different levels of liquidity for the US new one-family housing market over the period between January 1963 and November 2009. We mainly analyze the differential responses of trading volumes to housing price changes (denoted as price elasticity), employing a quantile cointegrating approach that allows us to capture housing boom–bust cycles earlier. In addition, we also explore the long-term impacts of monetary policies on housing sales. According to our findings, the price elasticities perform differently across the booms and busts of housing markets. Shifts in price elasticities can transmit signals of oncoming upturns, recovery, and downturns of housing cycles, which possibly relate to the business cycle. On the other hand, a contractionary monetary policy exerts a much effective control over an overheated housing market, but an expansionary monetary policy has a relatively small influence on stimulating a depressed housing market. Our findings offer some important suggestions and policy implications.

### Keywords

Housing price Housing trading volume Nonlinear relationship Quantile cointegration Momentum cycle### JEL Classification

C32 G12 E32## 1 Introduction

A large amount of studies have attempted to explain the puzzle with respect to the positive price–volume relation in housing markets (e.g., Stein 1995; Genesove and Mayer 2001; Yiu et al. 2008). Some findings indicate a state-dependent result or the linkage of investors’ irrational behavior to the positive price–volume relation (e.g., Case and Shiller 1988; Genesove and Mayer 2001). Our paper demonstrates the long-run price–volume nexus for the US single-family housing market by applying a quantile cointegrating (QC hereafter) approach. This method allows us to consider the impact of market liquidity on the housing price–volume nexus and to further understand the changes of price elasticities across a housing boom–bust cycle.^{1}

Liquidity is one of the primary risks in housing markets, as it relates to housing price, trading volume, and the time cost in markets (Krainer 2001).^{2} Illiquidity makes it difficult for asset prices to achieve their fundamental values (Lin and Vandell 2007; Lee et al. 2013). Liquidity also conveys the price–volume correlation. For example, Fisher et al. (2003) support the positive price–volume nexus while controlling the liquidity of housing markets. Suppliers of housing markets are very much concerned about housing liquidity, because of its great influence on housing developments. In order to realize the effect of housing liquidity on the long-run price–volume correlation, this paper applies the quantile cointegrating method (Xiao 2009), through which we can explore how important market liquidity is to this correlation. Such an investigation gives market analysts or investors a much needed earlier sign of an overheated housing market before a bubble forms, which could harm the overall economy (Winarso and Firman 2002; Harris and Arku 2006; Leamer 2015).

The main motivation and targets of this study are twofold. First, we analyze the housing price–volume nexus based on different liquidity levels via a quantile cointegration model. Traditional cointegration models limit this kind of analysis since they are based on the perspective of an average liquidity (or trading volume) level, thus neglecting possible asymmetric market behaviors between boom and bust markets due to investors’ overreaction or under-reaction (e.g., DeBondt and Richard 1985; Hong and Stein 1999; Lee and Zeng 2011). Investors’ sentiment sometimes dominates their buying or selling decisions. Genesove and Mayer (2001) suggest that people have a relatively low willingness to sell their houses and realize a loss when housing prices are in a decline (i.e., the so-called loss-aversion effect). As such, it is meaningful to look at the price–volume nexus of a housing market based on various liquidity levels. For example, a 1 % increase in the housing price may incur less growth in transaction volume under a depressed market (lower liquidity) than it does under a booming market (higher liquidity) due to some effects such as market momentum. Housing price elasticity could therefore be differentiated between booms and busts.

We may therefore conclude that: (1) during busts, the loss-aversion effect leads to a weaker response of housing trading volume to the decreasing housing prices; (2) in booms, market momentum and investors’ overreaction may drive high price elasticity, while houses at the same time could become a short-term speculative commodity. Hence, analyzing the asymmetric price elasticities between booms and busts allows one to realize whether a housing market is overheated or underreacted. Leamer (2015) suggests that housing transaction volume is a better proxy for capturing housing cycles rather than housing prices. Accordingly, quantile cointegration seems to be more appropriate to handle the asymmetric price elasticities across various liquidity levels. This paper also analyzes the long-term impacts of monetary policies on housing sales, although many studies have already discussed this issue (e.g., Kearl and Mishkin 1977; Hamilton 2008). Compared to the literature, this paper highlights the pass-through effect of monetary policy to housing sales conditioned on various liquidity levels. This analysis entails testing whether loose monetary policies efficiently mitigate the loss-aversion phenomenon and raise trading volume when housing markets are under depressed conditions.

Second, this research explains price–volume behaviors over housing cycles and offers a rough comparison between shifts in housing price and price elasticity that are aimed at downward and upward housing markets. The results indicate that shifts in price elasticities are much better at predicting a looming recession or recovery than housing price changes. Such an early alert should be helpful at finding out the symptoms of oncoming business recessions or recoveries.^{3}

The liquidity of real estate markets exhibits dramatic changes over time and presents different states of nature. Krainer (2001) indicates that state-varying liquidity implies that shocks to the fundamental value of houses are not reflected solely by market prices, but through market liquidity as well. Moreover, he states that “*we should not, in general, expect changes in fundamental values to be accompanied by equal changes in market prices*.” Prominently, it is better to gather information of housing prices and market liquidity simultaneously when gauging or predicting the fundamental values of the housing market. In addition, conditions of an imperfect market, such as asymmetric information, may lead to price rigidity (see, for example, Greenwald and Stiglitz 1993); investors’ behavior toward housing markets, such as loss aversion, also affects price elasticity. In sum, price elasticity varies with market cycles (e.g., Field and Pagoulatos 1997; Parker and Neelamegham 1997).

Housing prices intuitively should be driven by a rising market momentum that possibly results in deviations from their fundamental values (e.g., Black et al. 2006). Nevertheless, some central symptoms of housing markets, such as slowly growing turnover volume in a hot market in comparison with a booming price trend, can signal a convincing upcoming end to the boom. Accordingly, this paper analyzes the long-run price–volume nexus based on different liquidity levels of housing markets so that market participants can recognize the shifts in price elasticities of various market states and prepare relative strategies for the ends of booms/busts. Leamer (2015) indicates that residential investment substantially contributes to weakness before business recessions, offering an earlier sign of a recovery from recessions. Therefore, understanding housing cycles helps at predicting an economy’s business cycles.

Our findings and contributions are as follows. First, we find differential price elasticities across various liquidity innovations—that is, the long-run price–volume relation is liquidity-dependent, suggesting that liquidity risk is significantly associated with the price–volume relation. Second, price elasticity is larger at the median liquidity level than at the low and high liquidity levels, implying that an inverted V-shape can alert market participants to the end of a boom or a bust. The results offer some suggestions for property builders and policymakers.

The remainder of this paper is organized as follows. Section 2 presents the literature review. Section 3 describes the data and the methodology. Section 4 reports empirical analyses and suggestions. The final section is the conclusion.

## 2 Literature review

Most markets have cyclical patterns that make them perform differently across various market states. The well-known business cycle is the most important and notable one, because it impacts nearly all industries. For example, Fama and French (1989) employ US data over the period 1928–1978 and find that expected returns of stocks and bonds are high when business conditions are poor. Pena and Rodriguez (2006) note a systematical linkage between the states of a business cycle and the levels of interest rates and expected stock returns. Akinboade and Makina (2009) analyze the relationship between bank lending and business cycles in South Africa and show that bank lending and lending rates move in tandem with business cycles.

Sometimes there are asymmetric responses/strategies among the downturns and upturns during a business cycle. For instance, Mascarenhas and Aaker (1989) find that firms adjust their strategies significantly and asymmetrically over business cycle stages. Similar evidence is also found in financial markets (e.g. Perez-Quiros and Timmermann 2001). Acemoglu and Scott (1994) strongly support that the nonlinearity of the UK labor market is constructed successfully by a cyclical asymmetric model. Smith et al. (2010) examine the relationship between US stock returns and business cycles and propose that downturns in business cycles cause a greater negative influence on stock returns than a positive effect in upturns—that is, the relation varies over a business cycle. Due to this asymmetry, corresponding strategies over a cycle should be made based on different market states (Dissanaike 1997).

In addition to business cycles, cyclical patterns are also found frequently in stock markets. Kim and Startz (1991) explore the mean-reverting behavior in stock prices and find that the pattern of mean reversion is weak before World War II, but is quite strong after the war. They ascribe this to the resolution of economic and geopolitical uncertainties during the 1930s and 1940s. However, most studies (e.g., DeBondt and Richard 1985; Hong and Stein 1999) believe that investors overreact to good news, and thus stock prices become overvalued versus their intrinsic values. Moreover, the momentum of stock markets associated with firm-specific, industrial, or macroeconomic variables (e.g., Chordia and Shivakumar 2002; Figelman 2007) may drive investors, especially individuals, to be sentimental over the assessment of stocks’ intrinsic values. In addition to these channels, there are, of course, several other aspects for explaining a market’s cyclical patterns.

In real estate markets, the non-pervasive phenomenon of a strongly positive price–volume relationship has attracted the attention of many researchers,^{4} with many scholars explaining it from several perspectives, including down-payment effects, liquidity risks, the loss-aversion theory, and so on (see, for example, Stein 1995; Yiu et al. 2008; Genesove and Mayer 2001). The cyclical pattern of real estate markets is also notable due to its close connection with business cycles. Many recent studies have attempted to explore the determinants of housing cycles. For example, Wang et al. (2000) find that over-confidence induces an over-supply condition and prolongs a real estate cycle. Arbel et al. (2009) show that the average price yield is positive during low price conditions (below their long-term mean), but prices start to drop when prices are greater than their long-term mean values. This result implies short-term overreaction. Similarly, Black et al. (2006) construct a time-varying present value model with UK housing data and find that price dynamics are driven by market momentum. Edelstein and Tsang (2007) attribute residential real estate cycles to some fundamentals, such as growth in employment or interest rates. Novy-Marx (2009) demonstrates that market participants magnify the impact of fundamental shocks, which then create a self-reinforcing feedback loop. Therefore, housing markets characterized by high search costs can be more volatile than their fundamentals would imply.

Some studies have tried to explain the asymmetric/different behaviors between housing booms and busts. Crawford and Fratantoni (2003) and Miles (2008) employ nonlinear models to predict housing prices, as they propose that nonlinear models are capable of capturing asymmetric price behaviors between upward and downward markets. Clayton et al. (2010) examine the lead-lag relationship between housing prices and trading volumes and find an asymmetric effect that decreasing prices reduce trading volume, but rising prices have no significant effect on housing sales.

As to a (business) cycle pattern, this paper emphasizes that price elasticity could shift during a market cycle, and we may thereby get more information about its future market trend, rather than looking at only changes in prices.^{5} Some studies hypothesize this same argument. Harrod (1936) theoretically analyzes the relative elasticities of demand and suggests that imperfect competition can result in smaller elasticities during business cycle booms and greater elasticities in a recession, thus leading to a countercyclical pattern. Field and Pagoulatos (1997) empirically analyze the cyclical behavior of the price elasticity of demand for the US food manufacturing industry. Regarding business cycles, demand elasticity is negatively associated with unemployment and therefore exhibits pro-cyclical behavior—that is, price elasticity is less elastic under expansionary business conditions. Parker and Neelamegham (1997) explore the dynamics of price elasticity over the product life cycle for consumer durables. They find that elasticities are high in the earliest phase of a cycle when uncertainties are very high and repeat purchases are on the decline, and the elasticities are low when repeat purchases are on the increase.

In real estate markets, housing transaction volume is more useful than housing prices for capturing housing cycles (Leamer 2015). However, shifts in housing price elasticity seem to transmit more information and may perform better at recognizing housing market cycles. For example, in an upward market, market momentum can push housing prices higher. The appearance of a reversion in price elasticity will then illustrate the weak response of trading volume to increasing prices, implying the case of market overreaction. On the other hand, in relation to the loss-aversion effect at the beginning of a downward market, a 1 % decrease in housing prices stimulates very little change in the volume of housing sales. Therefore, housing price elasticity may illustrate an upward trend following decreasing prices, even as there is less decline in trading volume at the beginning of a downward market. If housing trading volume, Leamer (2015) suggests, leads housing prices, then elasticity should be decreasing at the end of a boom and smooth out at the end of a bust.

This paper not only stresses the importance of housing liquidity (e.g., Zhou 1997; Leung et al. 2002; Leamer 2010, 2015; Shi et al. 2010), but also deduces the shifts of price elasticities over a housing cycle. As in previous studies (e.g., Kearl and Mishkin 1977; Hamilton 2008) concerning the link between monetary policies and housing markets, this paper also examines the impact of monetary policies on housing sales restricted to different liquidity levels. We are concerned about whether monetary policy can effectively stimulate depressed housing transactions and/or mitigate overheated housing markets.

## 3 Data and methodology

### 3.1 Data

This paper examines the long-run price–volume relation for the US new one-family housing market.^{6} The monthly data we use are the time series of the median prices of new one-family houses and the sales volumes of new houses sold in the US between January 1963 and November 2009. The data are gathered from the US Census Bureau.^{7} In general, the market share for existing homes is larger than the market share for new homes.

Before we begin, we wish to make the following points. First, the variation of new home prices is relatively low, and the sales of new homes present less dramatic jumps. In contrast, both the prices and sales of existing homes can be affected by sellers’ own factors, such as coping with a sudden terrible event or cash flow needs. Second, supply elasticity may be related to housing bubbles or the length of a boom due to overbuilding (e.g., Glaeser et al. 2008), and thus it is more relevant to new homes than to existing homes. According to the findings of Blackley (1999), the range of the long-run supply elasticity of new US homes is much higher than range of their short-run supply elasticity, because property builders must try to accurately predict long-term housing prices. Third and finally, the sales or the number of new homes built relate very much to business conditions, and they may be more useful for predicting the development of countries or local regions. An examination of the price elasticities of new homes can thereby benefit builders or developers in order to more accurately forecast trading volume and possibly alert investors to the end of a boom or a bust.

When calculating real housing prices, their median prices are deflated by the consumer price index (CPI).^{8} Moreover, the variables are taken in natural logarithm, by which the estimated coefficients can be regarded as price elasticities. In this paper we consider the effect of monetary policy, measured by changes in the federal funds rate, on housing markets. Examining the possibility of a nonlinear (or stochastic) cointegrating relationship is of special interest (Lee 2013), particularly when an estimated linear cointegrating relationship deviates irrationally from what is predicted by economic or financial theories. Therefore, a quantile cointegration model is more appropriate herein than traditional linear models.

### 3.2 Methodology

Our main empirical works include the following steps. First, we test the stationarity of the variables based on several unit-root tests. Second, we apply a linear cointegrating model to examine whether a linear cointegration relationship exists or not. Third, we test the null hypothesis of constant cointegrating coefficients based on bootstrapped critical values and Xiao’s (2009) test. Fourth, we analyze price elasticities across various quantiles (or innovations) to housing liquidity. Fifth and finally, we check the robustness of our empirical results with some specific parameters.

#### 3.2.1 Unit-root tests

In testing unit roots, we apply the Augmented Dickey-Fuller (ADF 1979), Phillips-Perron (PP, 1988), and Kwiatkowski et al. (KPSS, 1992) tests.^{9} In testing the cointegration, we apply the cointegrating methodology developed by Johansen (1991, (1995), which is a linear cointegration estimated by a vector auto-regression (VAR) model.

The concept of cointegration applies to non-stationary time series variables, which give rise to stationary residuals in the models. In contrast to traditional linear cointegration models such as the Engle and Granger (1987) cointegration approach, many applications in financial and economic fields suggest that the cointegrating vector coefficients might not be constant (Lee and Zeng 2011; Lee 2013). A few methods may be applied here such as time-varying cointegration (Park and Hahn 1999) and the quantile cointegration model (Xiao 2009).

#### 3.2.2 Quantile cointegration model

*k*-dimensional vector consisting of integrated regressors, and \(y_t \) is also integrated in some order; generally, it is integrated of order 1 expressed as

*I*(1). More importantly, the error term \(u_t \) is stationary with zero mean. The linear cointegrating vector \(\beta \) is restricted to be constant.

In comparison with traditional linear cointegration models, Xiao (2009) attempts to extend linear cointegration models so as to cope with some problems. First, Xiao (2009) expects that the cointegrating coefficients are allowed to vary over time or be affected by some types of shocks since many central economic variables are found to exhibit asymmetric adjustment paths in recent years (e.g., Beaudry and Koop 1993; Enders and Granger 1998). Second, Xiao (2009) follows the idea of Saikkonen (1991) to use leads and lags to mitigate the endogeneity problem existing in traditional cointegration models. Note that the price–volume relation has a larger likelihood for bi-directional causality, which can cause the endogeneity problem. Based on these two points, Xiao (2009) proposes a quantile cointegration model.

*K*with zero mean, where \(u_t=\sum \nolimits _{i=-K}^K v_{t-i}^{\prime } \theta _i +\varepsilon _t \), satisfying that \(\hbox {E}\left( {v_{t-i} \varepsilon _t } \right) =0\) for any

*i*, and \(\varepsilon _t \) is a stationary process. Equation (1) can then be rewritten as:

^{10}Note that Xiao (2009) assumes that the cointegrating coefficients are monotonic functions of the innovation process \(\varepsilon _t \).

*LnV*is the trading volume in log,

*LnHp*is real housing prices in log,

*R*is the federal funds rate, \(\Delta \) denotes the first difference operator,

*K*and

*L*are, respectively, the lengths of the lead and lag terms \(({\hbox {herein we set}}\,K=L=3)\), and \(\varepsilon \) indicates an innovation.

^{11}This paper examines Eq. (7) by employing a quantile cointegrating approach, which allows us to examine the price–volume nexus based on any location of the conditional distribution of trading volume, rather than only on the conditional mean as estimated in linear cointegration models.

Xiao (2009) mentions quantile cointegration with several features. First, it captures systematic influences of the conditioning variables on the location, scale, and shape of the conditional distribution of the response. Second, it allows for additional volatility of the dependent variables, in addition to the regressors, and provides an interesting class of cointegration models with conditional heteroskedasticity. Third, the estimated cointegrating coefficients may be influenced by the shocks received in each period and thus may alter across various innovations or quantiles. Fourth and finally, formal tests for the varying-coefficient cointegrating relationship between variables are conducted by employing a bootstrap-based test, which does not need too many restrictions such as the assumption of normality.^{12}

## 4 Empirical analyses

### 4.1 Results of unit-root tests and cointegration

Unit-root tests of the median price and trading volume for the full sample

Variable | ADF test | PP-test | KPSS test | |||
---|---|---|---|---|---|---|

Level | 1st diff. | Level | 1st diff. | Level | 1st diff. | |

Housing prices | \(-1.73\) | \(-23.86^\mathrm{b}\) | \(-2.06\) | \(-47.39^\mathrm{b}\) | \(2.63^\mathrm{b}\) | 0.08 |

Sales volumes | \(-2.22\) | \(-28.03^\mathrm{b}\) | \(-2.37\) | \(-28.03^\mathrm{b}\) | \(1.04^\mathrm{b}\) | 0.14 |

Fed. funds rate | \(-2.37\) | \(-7.57^\mathrm{b}\) | \(-2.18\) | \(-14.51^\mathrm{b}\) | \(0.80^\mathrm{b}\) | 0.12 |

Johansen’s cointegration test between the price–volume relationship for the full sample

Maximum eigenvalue statistics | Trace eigenvalue statistics | |
---|---|---|

Sales volumes versus housing prices of new homes sold | ||

\(\hbox {H}_\mathrm{0}:r=0\) | \(38.72^\mathrm{b}\) | \(48.11^\mathrm{b}\) |

\(\hbox {H}_\mathrm{0}:r\le 1\) | 7.13 | 9.38 |

\(\hbox {H}_\mathrm{0}:r\le 2\) | 2.26 | 2.26 |

Test for the nonlinearity of the cointegration relationship between prices and volumes for the full sample

\({\hbox {Sup}}{\vert }V(\tau ){\vert }\) | CV10 | CV05 | CV01 | |
---|---|---|---|---|

Null: a constant cointegrating coefficient | 78.70\(^\mathrm{b}\) | 3.24 | 4.19 | 6.46 |

Coefficients of quantile cointegration and linear cointegration

Linear model | Quantile cointegration (nonlinear model) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Low liquidity | Median liquidity | High liquidity | ||||||||

DOLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |

Beta 1 ( | 0.890 | 0.697 | 0.749 | 0.777 | 0.918 | 0.962 | 0.933 | 0.917 | 0.856 | 0.830 |

Beta 2 ( | \(-\)0.015 | \(-\)0.013 | \(-\)0.020 | \(-\)0.022 | \(-\)0.030 | \(-\)0.033 | \(-\)0.033 | \(-\)0.032 | \(-\)0.034 | \(-\)0.037 |

\(\hbox {Null Hyp.}\,\beta 1_{QC} =\beta 1_{DOLS}\,(\hbox {p-value})\) | – | \(-\)2164.7 (0.00) | \(-\)1760.4 (0.00) | \(-\)1441.9 (0.00) | \(-\)392.3 (0.00) | \(-\)988.5 (0.00) | \(-\)564.5 (0.00) | \(-\)417.3 (0.00) | \(-\)386.9 (0.00) | \(-\)725.6 (0.00) |

\(\hbox {Null Hyp.}\,\beta 2_{QC} =\beta 2_{DOLS}\,(\hbox {p-value})\) | – | 1058.5 (0.00) | 1059.6 (0.00) | 1043.1 (0.00) | 844.2 (0.00) | 808.2 (0.00) | 806.2 (0.00) | 818.3 (0.00) | 700.0 (0.00) | 567.0 (0.00) |

### 4.2 Results of the quantile cointegration model

After confirming the existence of the linear cointegrated price–volume relation, we further examine whether their cointegrated relation is nonlinear. Table 3 indicates that the cointegrating coefficient is not constant based on the bootstrapping method. The finite sample critical values reported are computed by means of Monte Carlo simulations using 3000 replications. The results indeed support the long-run nonlinear price–volume relation, and the adjustments toward long-run equilibrium do vary depending on market innovations. Such a nonlinear price–volume relation may make raise a bias in the forecast of traditional linear models (Crawford and Fratantoni 2003; Miles 2008).

The next step is to estimate those varying cointegrating coefficients across different liquidity levels (quantiles). Table 4 presents the results based on the specific nine quantiles, which are categorized into three states: low liquidity, median liquidity, and high liquidity. When computing the statistical significance of coefficients, we utilize both the bootstrap method and the Wald test to confirm whether the cointegrating coefficients (price elasticity) differ from 0 at the 5 % significance level.

We note some economic interpretations for the varying price elasticities across different liquidity levels. We indicate the phase of low liquidity at quantile 0.1 as a trough in a cycle and the state of high liquidity at quantile 0.9 as a peak. Thus, we regard the market from a trough to a following peak as being an upward market, and the market from a peak to a following trough is regarded as a downward market. In the following, we offer economic interpretations with respect to both markets phase-by-phase based on the quantile cointegration results.

**Upward market:**Phase 1 (upturn or recovery phase): At the beginning of this phase, a 1 % increase in price only causes a small rise in trading volume (from quantile 0.1 to quantile 0.3). It presents a slow recovery from a bust.

Phase 2 (normal phase): It exhibits rapidly increasing price elasticity or higher price elasticity than the average level from quantile 0.3 to quantile 0.7. This phase contains the stage of market momentum that may result in overreaction. We may specify this as the first half of a boom.

Phase 3 (boom): Price elasticity appears with a reverse trend or lower price elasticity than the average level from quantile 0.7 to quantile 0.9. In this phase, the growth of trading volume from a 1 % increase in price is lower than the average level. We may regard this as the second half of a boom, and it also spells the end of a boom. A reverse trend may contribute to overreaction. According to the discussion above, we divide an upward market, in turn, into three phases: an upturn, a normal phase, and a boom.

**Downward market:**Phase 1 (downturn phase): It starts from a peak—that is, the beginning of a downward market. In this phase, price elasticity is increasing (from quantile 0.9 to quantile 0.5). In fact, the reasoning for why we specify this range is due to the loss-aversion effect—that is, the decreasing housing price makes house suppliers unwilling to sell their homes, thereby leading to a relatively large cut in housing sales.

Phase 2 (normal phase): From quantile 0.5 to quantile 0.3, the price elasticity starts to go down. In this phase, the housing market seems to stay on a normal state since it gets rid of the effect due to the sentimental factor of loss aversion.

Phase 3 (bust phase): In this phase, market liquidity is low, but the downward trend of price elasticity is somewhat mitigated from quantile 0.3 to quantile 0.1, implying that the decreasing housing price causes a relatively less decline in housing sales. It may indicate the end of a bust and a following recovery. Accordingly, a downward market herein consists of a downturn, a normal phase, and a bust.

### 4.3 Robustness analysis

*Unemp*), and two economic factors (a consumer price index in log (

*LnCPI*) and the ratio of personal disposable income to personal total income (

*RI*)).

^{13}

Table 5 reports the results of robustness. First, the trend of the quantile cointegrating coefficients for *LnHP* does not change, but the trend for *R* is somewhat different in comparison with the results of Table 4. However, the monetary policies essentially do have a relatively large influence on housing sales when the market liquidity is at a high level. On the other hand, financial markets and the unemployment rate have no influence on housing sales when housing market liquidity is at a high level, but they do impact housing sales during levels of low and median market liquidities. In a boom or even an overheated housing market, stock and labor markets are irrelative to housing sales. Inflation does not affect housing sales when market liquidity is at low and high levels. Market states seem to be the primary factor driving housing sales. Finally, the ratio of disposable income to total income presents differential effects on housing sales across various quantiles. Since personal wealth can be related to the performance of stock markets and the housing price, the result needs to be further confirmed in future research. Nevertheless, the influences of the real housing price and the federal funds rate on housing trading volume are mostly unchanged after adding these structural parameters.

### 4.4 Illustrations of price elasticities for upward and downward markets

We have already observed the shifts of price elasticities across low, median, and high liquidity states, and according to our empirical results, price elasticity is the lowest in a bust (quantiles 0.1–0.3), second lowest in a boom market (quantiles 0.7–0.9), and greatest in a normal state (quantiles 0.4–0.6). Accordingly, some symptoms due to shifts in price elasticity can alert markets early for an oncoming end to an upward or a downward market. For example, in an upward market, the price elasticity at the beginning is lower, and thus a dramatic reversion in price elasticity may spell the end to an upward market or a boom.

This section exhibits practical price elasticities over housing cycles and links those illustrations with the results of quantile cointegration in order to provide economic interpretations. Mainly, we intend to observe the shifts of price elasticity over an upward market (UM) or a downward market (DM), where the period of those markets can span over 1 year. Before doing so, we pick out prominent upward and downward markets by employing the methods developed by Pagan and Sossounov (2003) to real housing prices (*RHP*). The Appendix section describes the clear steps and results. Note that we do not consider the data of sub-periods if their changes of the real housing price do not transcend our specified criterion.

*APE*track presents a clear inverted V-shape, we should notice an oncoming reversion of the current market state.

^{14}

Test for robustness

Linear model | Quantile cointegration (nonlinear model) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Low liquidity | Median liquidity | High liquidity | ||||||||

DOLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |

Beta 1 ( | \(1.097^\mathrm{b}\) | \(0.370^\mathrm{b}\) | \(0.531^\mathrm{b}\) | \(0.640^\mathrm{b}\) | \(0.947^\mathrm{b}\) | \(1.233^\mathrm{b}\) | \(1.328^\mathrm{b}\) | \(1.149^\mathrm{b}\) | \(1.108^\mathrm{b}\) | \(1.042^\mathrm{b}\) |

Beta 2 ( | \(-0.006^\mathrm{b}\) | 0.003 | \(0.013^\mathrm{b}\) | \(0.007^\mathrm{b}\) | 0.001 | \(-0.018^\mathrm{b}\) | \(-0.029^\mathrm{b}\) | \(-0.030^\mathrm{b}\) | \(-0.030^\mathrm{b}\) | \(-0.028^\mathrm{b}\) |

Beta 3 ( | \(0.158^\mathrm{b}\) | \(0.122^\mathrm{b}\) | \(0.289^\mathrm{b}\) | \(0.279^\mathrm{b}\) | \(0.228^\mathrm{b}\) | \(0.163^\mathrm{b}\) | \(0.057^\mathrm{a}\) | 0.010 | 0.001 | 0.001 |

Beta 4 ( | \(0.020^\mathrm{b}\) | \(-0.008\) | \(0.018^\mathrm{b}\) | \(0.016^\mathrm{b}\) | \(0.020^\mathrm{b}\) | \(0.033^\mathrm{b}\) | \(0.020^\mathrm{b}\) | 0.005 | \(-0.004\) | \(-0.019^\mathrm{a}\) |

Beta 5 ( | \(-0.142^\mathrm{b}\) | 0.102 | \(-0.015\) | \(-0.047\) | \(-0.116^\mathrm{a}\) | \(-0.180^\mathrm{b}\) | \(-0.170^\mathrm{b}\) | \(-0.080\) | \(-0.080\) | \(-0.065\) |

Beta 6 ( | \(-1.997^\mathrm{b}\) | \(-4.217^\mathrm{b}\) | \(0.647^\mathrm{b}\) | \(0.673^\mathrm{b}\) | \(-0.162^\mathrm{b}\) | \(-3.217^\mathrm{b}\) | \(-4.388^\mathrm{b}\) | \(-2.884^\mathrm{b}\) | \(-1.845^\mathrm{b}\) | \(-1.074^\mathrm{b}\) |

Figure 3a–i shows the cases of UMs, and Fig. 4a–d shows the cases of DMs.^{15} According to the results of quantile cointegration, the *APE* track indeed illustrates a reverse trend (i.e., an inverted V-shape), which can alert investors to an oncoming end of a boom or a bust. At the same time, the response of trading volume to housing price changes is relatively moderate.

Figure 3a shows the first upward market over the period from January 1964 to December 1964 (denoted as 1964M1–1964M12). In this stage, *APE* does not have a clear inverted V-shape; at the same time, *APE* presents a dramatic decline since 1964M4—that is, an increase in the real housing price causes relatively less growth in housing sales. *APE* indeed transmits information about the end of an upward market. Moreover, RHP has a noticeable rise from 1964M10 to 1964M12, but *APE* instead shifts to a negative value, implying that increasing RHP leads to decreasing housing trading volume. This prominently means that the end of an upward market is coming soon.^{16} By contrast, the *APE* track of the upward market in Fig. 3b shows few signals for the end of this upward market over the period 1967M9–1968M12, for which the reasoning should be explored further in the future research.

In the upward market (1972M2–1973M7) shown in Fig. 3c, there is a clear inverted V-shape in *APE*. In this period, RHP has an obvious upturn. The *APE* over the period 1972M7–1972M9 rises notably—that is, a 1 % increase in RHP causes rising housing sales after 3 months, but APE reverses from 1972M10. Put differently, although the real housing price keeps its upward trend, the response of housing sales to the dramatic upward RHP becomes weaker, even though it shifts into a negative response (1972M12–1973M7). Changes in price elasticities have significantly transmitted a signal to the end of the upward market.

In the other upward markets (Fig. 3d–i), a few *APEs* (e.g., Fig. 3d, g, h) do not present significant signals for the ends of the upward markets. Relatively clearer inverted V signals of Fig. 3e, f, i occur in 1982M9, 1984M5, and 2006M2, respectively. In particular, some parts of the sub-periods relate to dramatic economic events. In the sub-period from February 1972 to July 1973 (Fig. 3c), when the price elasticity reverses in the middle of 1972, the real housing price still keeps its increasing trend. However, housing markets suffered a bust after the middle of 1973, which accompanied the 1973–1974 stock crash and subsequent business recession. Figure 3d (September 1975 to June 1979) is linked to downward productivity since 1979 and a business recession (see, for example, Edge et al. 2007). Figure 3f (January 1985 to January 1988) relates to the real estate bubble that ended in 1989 and the following business recession. Figure 3i (March 2005 to April 2006) relates to the sub-prime mortgage crisis that turned very severely around 2007. In fact, according to the shifts of price elasticity, the signal for the end of the boom appeared in 2006, which is earlier than the housing price has predicted. In these figures, the shift of *APE* does present an advanced alarm for the end of housing booms and oncoming business recessions.

Shifts of *APEs* in Fig. 4a–d similarly declare in advance an end to those depressed housing markets and the subsequent business recovery. In Fig. 4a, RHP has a clear downward trend from 1969M2 to 1970M12, but *APE* signals an end to the downward trend starting in 1970M3. At the same time, the response of housing trading volume to decreasing RHP transmits positive messages for a recovery in housing markets. In Fig. 4b–d, the inverted V signals of *APEs* also forecast an end to those downward markets starting in 1981M4, 1991M12, and 2009M7, respectively, because the decline of housing sales is mitigated, albeit reverses, corresponding to those decreasing RHPs. To sum up, although *APE* sometimes offers little evidence for the reversion of housing markets, the shifts in price elasticities mostly present earlier symptoms for the ends of upward and downward markets versus what the housing price can forecast.

## 5 Conclusions

This paper re-examines the long-run price–volume relationship through monthly data of the US new single-family housing market by employing a quantile cointegrating regression. This approach overcomes some problems in examining the price–volume correlation. For example, Arbel et al. (2009) are obstructed by the problem of unit-root variables in examining the price–volume relation with mean reversion and momentum. In addition, Leamer (2015) argues that the housing price seems to not be accurate in capturing housing cycles when compared to housing trading volume. Therefore, we resort to the advantages of quantile cointegration to explore the shifts of housing price elasticities in a boom–bust cycle, by which housing recessions or booms can be detected earlier. In particular, cyclical patterns present differential actions, such as asymmetric decision-making or market performances. This paper examines whether housing price elasticity is also state-dependent. If so, there are some benefits to coping with oncoming bubbles.

Changes of real housing prices

Change of real housing prices (%) | |
---|---|

Mean | 0.102 |

Standard deviation | 3.665 |

Mean\(+\)1 standard deviation | 3.767 |

Mean−1 standard deviation | \(-\)3.563 |

Mean\(+\)2 standard deviations | 7.432 |

Mean−2 standard deviations | \(-\)7.228 |

Mean\(+\)3 standard deviations | 11.097 |

Mean−3 standard deviations | \(-\)10.893 |

Changes of real housing prices in upward markets, downward markets, and unobvious markets

Downward Market | Upward Market | Unobvious Market | |
---|---|---|---|

Periods | May 1966\(\sim \)Oct. 1966 (-8.50 %) | Nov. 1963\(\sim \)Dec. 1964 (11.70 %) | Jan. 1963\(\sim \)Oct. 1963 (3.07 %) |

Jul. 1967\(\sim \)Aug. 1967 (\(-\)7.41 %) | Oct. 1965\(\sim \)Apr. 1966 (12.51 %) | Jan. 1965\(\sim \)Sep. 1965 (\(-\)6.54 %) | |

Jan. 1969\(\sim \)Dec. 1970 (\(-\)20.69 %) | Nov. 1966\(\sim \)Jun. 1967 (8.39 %) | Jan. 1971\(\sim \)Jan. 1972 (peak, Jun. 1971) (7.31 %, \(-\)3.47 %) | |

Jul. 1979\(\sim \)Feb. 1982 (\(-\)23.09 %) | Sep. 1967\(\sim \)Dec. 1968 (9.26 %) | Aug. 1973\(\sim \)Aug. 1975 (\(-\)4.57 %) | |

May 1990\(\sim \)May 1992 (\(-\)17.98 %) | Feb. 1972\(\sim \)Jul. 1973 (18.96 %) | Oct. 1983\(\sim \)Dec. 1984 (\(-\)1.44 %) | |

Apr. 2007\(\sim \)Nov. 2009 (\(-\)16.10 %) | Sep. 1975\(\sim \)Jun. 1979 (20.12 %) | Feb. 1988\(\sim \)Apr. 1990 (trough, Nov. 1988) (\(-\)3.92 %, 0.60 %) | |

Mar. 1982\(\sim \)Sep. 1983 (12.83 %) | Jun. 1992\(\sim \)Dec. 2000 (peak, Nov. 1992; trough, Jul. 1994; peak, Apr. 1997; trough, Oct. 1997; peak, Feb. 1998; trough, Aug, 1999; peak, Nov. 1999) (2.06 %, \(-\)5.48 %, 4.74 %, \(-\)0.64 %, 7.13 %, \(-\)1.73 %, 4.89 %, \(-\)5.09 %) | ||

Jan. 1985\(\sim \)Jan. 1988 (27.33 %) | Mar. 2002\(\sim \)Dec. 2002 (5.63 %) | ||

Jan. 2001\(\sim \)Feb. 2002 (9.58 %) | Oct. 2004\(\sim \)Feb. 2005 (2.64 %) | ||

Jan. 2003\(\sim \)Sep. 2004 (11.37 %) | May 2006\(\sim \)Mar. 2007 (trough, Sep. 2006) (\(-\)5.69 %, 3.06 %) | ||

Mar. 2005\(\sim \)Apr. 2006 (7.54 %) | |||

Months (percent) | 121 (21.5 %) | 214 (38.0 %) | 228 (40.5 %) |

Second, the shifts of price elasticities can transmit the likelihood of oncoming recessions or recoveries. An early sign of reverse price elasticities can alert investors to the end of a boom, while moderate shifts in price elasticities may spell the end of a bust and then an upcoming recovery. The results can help market participants to predict housing prices and to execute relevant strategies much earlier. For example, the reversion of price elasticity in a hot market informs that the current housing market is overheated; at such a time, investors should not risk entering it, while builders could sell homes at better prices. Similarly, the reversion of price elasticity in a depressed market could help people buy a new home at a relatively low price.

Many studies attempt to explain why housing prices are cyclical or are linked to housing bubbles, by considering factors such as economic fundamentals, suppliers’ over-building, and overvalued housing prices. This paper mainly focuses on the liquidity-dependent price–volume relation. We offer participants of housing markets with relatively better and more useful information about when a boom will end, when a bust will bottom out, and when the loss-average effect can be mitigated. We also explore the shifts of price elasticity in a boom–bust cycle. We hope that crises, such as the one experienced in the US housing market starting in 2007, can be prevented earlier. After all, such crises produce a large negative impact on economies.

## Footnotes

- 1.
Herein, trading volume is regarded as the liquidity of housing markets, although there are different definitions/measures for the liquidity of real estate markets (see, for example, Forgey et al. 1996).

- 2.
- 3.
For the link between housing states and economic cycles, please refer to the review in Consulting and Research (2006).

- 4.
Lucas (1978) posits that there is little correlation between price and volume, whereas Arbel et al. (2009) find a significantly positive price–volume correlation. In contrast, Hort (2000) documents a negative relationship, while Leung and Feng (2005) note no significant relationship in commercial real estate markets.

- 5.
Kau and Sirmans (1979) investigate Chicago’s housing market and find that price elasticity of demand changes from elastic to unitary elasticity of demand over the period 1836–1970.

- 6.
Although the price–volume correlation may alter depending on different types of buildings, a monetary policy should essentially aim at general buyers, not firms or enterprises. Hence, we focus on the data of single-family houses.

- 7.
Since the period the median price spans is much longer than the period the average price spans (starting from January 1975), we therefore consider the use of the median price. In fact, the correlation between the median housing price and the average housing price is very high (0.998). In addition, the data of the median price are gathered directly from the US Census Bureau.

- 8.
The data of the consumer price index, the nominal federal funds rate, and the controlled structural parameters in the robustness test are collected from the Federal Reserve Bank of St. Louis.

- 9.
For details of these unit-root tests, please refer to the review of Maddala and Kim (1998).

- 10.
In our empirical analysis, the critical values for the statistic of Eq. (6) are computed by using the re-sampling method. The software we use is MATLAB, and the code is obtained from Professor Xiao.

- 11.
Because Saikkonen (1991) suggests that it is not feasible to have long leads or lags,

*K*is therefore set advisably. - 12.
Ebru and Eban (2011) also employ a quantile regression model to examine the relationship between the housing price and some of the housing characteristics of Istanbul by real estate agency surveys in 2007 (October–December).

- 13.
Although there are many structural parameters that we need to consider, we use only one parameter to proxy for each kind of influential field in order to avoid the possible co-linearity problem. For example, the federal funds rate is related to either mortgage rates or long-run interest rates.

- 14.
Some may argue for using a time-varying cointegration regression, instead of a quantile cointegration regression. Of course, this is practicable, but one needs to know the symptoms of oncoming market/price reversion in advance. We think quantile cointegration regression can better handle this problem.

- 15.
Due to accumulating the past 12 months of price elasticities, the beginning of Exhibit 3–1 is January 1964 rather than January 1963.

- 16.
According to the results of quantile cointegration, price elasticities are all positive. By contrast,

*APE*, a representation of practical time-varying price elasticities, shows some negative values. This could be on account that the calculation of*APE*contains only a piece of time and cannot represent the perspective of a long-run cointegrated relation completely. Nevertheless, the results of quantile cointegration can still offer some useful information in interpreting the illustrations of*APEs*.

## Notes

### Acknowledgments

The authors are grateful to the comments of the editor and the two anonymous referees on our paper. In addition, we are grateful to National Sun Yat-sen University for financial support through Grant 05C0301051.

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