Yongheng Deng
email: ydeng@nus.edu.sg

Eric Girardin
email: eric.girardin@univ-amu.fr

Roselyne Joyeux
email: roselyne.joyeux@mq.edu.au

Hkimr

Fundamentals and the Volatility of Real Estate Prices in China: A Sequential Modelling Strategy*

. However many observers and researchers argue that the fundamentals of the housing sector, both sector-specific and macroeconomic, may have been the driving force behind housing price volatility.

 which enables us to document that fundamentals can indeed account for movements in housing price volatility, as well as transaction volume in first-tier cities such as Beijing and Shanghai.

*

We would like to thank Eric Ghysels and Bumjean Sohn for providing us the GARCH-MIDAS program to extract the long run volatility component used in this paper. The paper benefited from comments by an anonymous referee from HKIMR.

1.

Introduction

The housing market in China has often been portrayed as highly speculative, giving rise to "bubble" concerns. Figure 1 shows a dramatic increase in discussion about "housing bubbles" in the Chinese media, based on a Google search. The search covers the period from 2005Q1 to 2013Q4, starting before the global financial crisis and ending with a recent housing market boom triggered by the huge monetary-fiscal reflation programme in China in 2009. Indeed the sharp rise in real estate prices in China has been similar to that experienced in the United States during the subprime decade [START_REF] Deng | Land and House Price Measurement in China[END_REF]and Wu, Gyourko and Deng, 2012). For example, over the last decade, residential land prices increased on average by approximately 25% per annum in Beijing. A similar pattern of residential market volatility has been observed in other major housing markets, such as Shanghai and Hangzhou (see [START_REF] Deng | Land and House Price Measurement in China[END_REF]. There is a presumption that such accelerated rises in prices are generated by a bubble. However, some observers and researchers argue that fundamentals in the housing sector, both sector-specific and macroeconomic, may have been the driving force behind housing price volatility (see, for example, [START_REF] Fu | Speculative Investors and Tobin's Tax in the Housing Market[END_REF]. This paper tests these alternative hypotheses on original high-frequency hedonic residential housing data, which avoid the bias in official housing prices data. We find no statistical evidence that a bubble was driving the rise of real estate prices in China in the second half of the first decade of the new millennium. Rather price volatility seems to be well-explained by movements in classical fundamentals.

The real estate industry contributes greatly to Chinese economic growth. In 2009 it directly accounted for over one tenth of total national GDP growth, and therefore played a substantial part in the quick recovery of the Chinese economy from the global financial crisis (Deng, Morck, Wu andYeung, 2011, 2015;[START_REF] Wu | Evaluating Conditions in Major Chinese Housing Markets[END_REF][START_REF] Wu | Incentives and Outcomes: China's Environmental Policy[END_REF]. The overall importance of this sector is even larger if indirect effects on the construction material industries are taken into account (possibly up to one third, Renmin, 2014). As reported by the People's Bank of China, at the end of 2010 the outstanding balance of developer plus residential mortgage loans represented around 6.28 trillion yuan RMB, over 18% of total loans.

We contribute to a literature trying to assess empirically sector-specific as well as macro-variables as potential determinants of long run asset market volatility, building on [START_REF] Schwert | Why Does Stock Market Volatility Change Over Time?[END_REF]. As stressed by [START_REF] Schwert | Why Does Stock Market Volatility Change Over Time?[END_REF]Schwert ( , p. 1116)), if macroeconomic data "provide information about the volatility of either future expected cash flows or future discount rates, they can help explain why [asset] return volatility changes over time". Existing empirical work along such lines focuses almost exclusively on the stock market, both for large panels of countries [START_REF] Diebold | Macroeconomic Volatility and Stock Market Volatility, Worldwide[END_REF][START_REF] Engle | The Spline-GARCH Model for Low-Frequency Volatility and its Global Macroeconomic Causes[END_REF] and for China [START_REF] Girardin | Market Fundamentals as a Source of Stock Market Volatility in China: A GATCH-MIDAS Approach[END_REF]. In the case of daily real estate data such an approach cannot be applied directly.

In order to address the challenges presented by the modelling of high-frequency real estate prices in an emerging market like China, we propose a sequential strategy in five steps integrating several techniques previously developed in a piecemeal and scattered way. The first issue concerns the appropriate data to be used. Focusing on repeated-sales housing data has become standard practice for large advanced countries, like the US. By contrast, a lack of sufficient comparable transactions of repeated-sales in the housing market in China, a feature common with some other emerging economies, means that the hedonic method is the most commonly adopted house price index model among nascent housing markets [START_REF] Wu | House Price Index Construction in the Nascent Housing Market: The Case of China[END_REF]. A major difficulty in studying the property market in China lies in the low quality of official price data due in particular to the underestimation of quality changes (Wu, Gyourko and Deng, 2015(b)). Our first contribution is the use of a unique data set including daily hedonic prices as well as transactions for the residential resale housing markets of Beijing and Shanghai between January 2005 and December 2010.

In a second step, while it is traditional in time-series analysis to conduct prior stationarity tests, this is not appropriate for real estate prices potentially subject to bubbles. We thus propose using recently developed tests [START_REF] Phillips | Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500[END_REF] with an explosive root as an alternative to the unit root hypothesis. The third step is generated by the necessity of handling microstructure noise present in daily real estate data. If we reject the explosive root in favour of the unit root, filtering the raw data with the extraction of a random walk component (as suggested recently for daily data by [START_REF] Bollerslev | Daily House Price Indices: Construction, Modeling, and Longer-Run Predictions[END_REF] seems all the more relevant. The fourth step aims at extracting a long run volatility component from the filtered daily hedonic real estate data. The GARCH-MIDAS procedure seems well adapted to this task in as much as it enables us to account for the heteroskedasticity in short run volatility as well as extracting a long-run volatility component via a beta-lag distribution on realized volatility. We apply the MIDAS methodology (developed by [START_REF] Engle | Stock Market Volatility and Macroeconomic Fundamentals[END_REF][START_REF] Engle | The Spline-GARCH Model for Low-Frequency Volatility and its Global Macroeconomic Causes[END_REF] to extract long-run volatility from Chinese residential housing market daily squared returns. The finding that bubbles do not characterize the residential housing market in first tier Chinese cities leads us in a fifth step to examine to what extent fundamentals are able to explain the long-run volatility of Chinese housing prices at a monthly frequency.

We study the influence of three types of control groups: sector-specific, and policy variables, as well as speculation-related variables. We consider the direct effect of fundamentals on real estate volatility and also the indirect effect through the impact of transaction volume on prices. Such an influence is well rationalized (in the non-rational expectations search model of [START_REF] Berkovec | Turnover as a Measure of Demand for Existing Homes[END_REF] the down payment model of [START_REF] Stein | Prices and Trading Volume in the Housing Market, A Model with Down-Payment Effects[END_REF] and documented for the residential property market (e.g. for Hong Kong by [START_REF] Leung | Testing Alternative Theories of the Property Price-Trading Volume Correlation[END_REF] for mainland China since its housing system reform by [START_REF] Deng | An Early Assessment of Residential Mortgage Performance in China[END_REF]and for Singapore by [START_REF] Deng | Private Residential Price Indices in Singapore: A Matching Approach[END_REF]. We find that the three types of variables are present, though in a contrasted way, in Beijing and Beijing and Shanghai combined. Among the first category, rent and land supply play a role; in the second, stock market returns and foreign currency reserve inflows are important; and among the last macroprudential, but not monetary, policy matters. Some of these influences impact prices indirectly via transaction volume.

To sum up, our analysis has three main contributions. First, we assemble the first dataset of highfrequency transactions in the Chinese urban real estate market. This data allow us to correct some important biases affecting existing studies. Second, to analyse this data, we propose a five-step strategy involving state-of-the-art econometric techniques. From a methodological point of view, we provide one of the first analysis using bubble detection techniques and mixed frequency data analysis for the real estate market. Third, we assess the determinants of the recent rise in both volume and volatility in housing prices in China. We show that concerns about a housing bubble are probably unfounded and that this rises are well-explained by changes in fundamentals.

The remainder of the paper is organized as follows. Existing evidence on the presence of bubbles in China's housing market, as well as empirical analyses of the effect of sector-specific, speculative and policy variables on real estate prices will be presented in section two. The bubble-detection, microstructure-noise filtering and MIDAS methodologies, as well as the unique hedonic residential resale-price data for major Chinese cities used in the present paper will be introduced in section three. Section four will discuss the results of the application of these techniques to real estate data in China, and provide an interpretation. Section five offers some conclusions.

Fundamentals, Speculation and the Real Estate Market

The difficulties associated with detecting a possible deviation of real estate prices from their fundamentals, and identifying the associated bubble component, are clearly illustrated by work on the US housing market prior to the subprime crisis (e.g. [START_REF] Mccarthy | Are Home Prices the Next "Bubble"?[END_REF][START_REF] Cutts | Reversion to the Mean Versus Sticking to Fundamentals: Looking to the Next Five Years of Housing Price Growth[END_REF]. The literature attempting in a similar way to detect deviations of China's real estate prices from their fundamentals will be presented first. We will then examine work on fundamentals, transactions and volatility.

Fundamentals and Real Estate Prices in China

Research has examined the determinants of, and evidence for, booms and busts in housing markets.

A rich empirical literature has endeavoured to study the drivers of house prices in China, stressing the role of sector specific as well as speculative or policy factors.

Housing-Price Boom-Busts

In his famous book Irrational Exuberance (2005), Shiller argues that the feedback that creates bubbles has the primary effect of amplifying stories that justify the bubble. He points out that contagion tends to work through word of mouth and through the news media. From a theoretical perspective, the modelling of booms and busts in the housing market has increasingly relied on expectations-driven cycles introduced for example by [START_REF] Lambertini | Leaning against Boom-Bust Cycles in Credit and Housing Prices[END_REF] using the model of the housing market developed by [START_REF] Iacoviello | Housing Market Spill overs: Evidence from an Estimated DSGE Model[END_REF]. News shocks1 generate common booms in house prices and credit to households. A macroeconomic bust takes place when expectations about future economic conditions are not matched by economic outcomes. Thus, unrealized news shocks distort borrowing above its equilibrium level. Their sudden reversal has negative effects on real and financial decisions. Similar cycles are generated in the models of [START_REF] Kermani | Cheap Credit, Collateral and the Boom-Bust Cycle[END_REF] and [START_REF] Kanik | Housing Boom-Bust Cycles, and Monetary Policy[END_REF]. Booms and busts can also occur when agents have heterogeneous expectations about longrun fundamentals but change their views because of social dynamics [START_REF] Burnside | Understanding Booms and Busts in Housing Markets[END_REF]. Agents with tighter priors are more likely to convert others to their beliefs. The booms that are not followed by busts typically occur when optimistic agents happen to be correct 2 .

However, real estate cycles may be driven by factors other than "news shocks". [START_REF] Leung | Error-Correction Dynamics of House prices: Benchmark[END_REF] presents a simple dynamic stochastic general equilibrium model in which the reduced form dynamics are consistent with the error correction model estimated by [START_REF] Malpezzi | A Simple Error Correction Model of House Prices[END_REF][START_REF] Capozza | An Anatomy of Price Dynamics in Illiquid Markets: Analysis and Evidence from Local Housing Markets[END_REF].

In addition, the presence of collateral constraints and large shocks, modelled by [START_REF] Chen | Asset Price Spillover, Collateral and Crises: With an Application to Property Market Policy[END_REF], can generate large scale bankruptcy, leading to regime-switching dynamics in house prices, in line with the evidence presented by [START_REF] Chang | Monetary Policy, Term Structure and Asset Return: Comparing REIT, Housing and Stock[END_REF]. Since agents do not know the market perfectly, [START_REF] Kuang | A Model of Housing and Credit Cycles with Imperfect Market Knowledge[END_REF] shows that the fact that agents learn on the one hand and trade on the other may induce cycles in both house prices and credit. Stressing the expectation formation process when the market displays regime-switching, [START_REF] Chen | Asset Price and Monetary Policy: The Effect of Expectations Formation[END_REF] find that the anticipation of possible regime change may impact agents' behavior, and hence aggregate output and house price dynamics.

The conditions for a bubble in the real estate market differ from those in stock markets in as much as the response of supply plays a major role in the former and not in the latter. [START_REF] Malpezzi | The Role of Speculation in Real Estate Cycles[END_REF] introduce a crucial distinction between short run and long run price-elasticities of supply. When the former is smaller than the latter a demand shock will generate price volatility.

In their empirical modelling of monthly Shanghai and Beijing house prices data from 1997 to 2003, [START_REF] Hui | Housing Price Bubbles in Hong Kong, Beijing and Shanghai: A Comparative Study[END_REF] employ Granger causality tests and generalized impulse response analysis, and find evidence of speculative bubbles for Shanghai house prices in 2003 but not for Beijing. [START_REF] Hui | Price Discovery of Property Markets in Shenzhen and Hong Kong[END_REF], using similar econometric techniques and quarterly data between 1996 and 2007, find some mild evidence of a bubble in Shenzhen house prices. [START_REF] Chen | The Income Elasticity of Housing Prices in China and its Regional Diversity[END_REF] detect some risk of speculative bubble in several regional housing markets, especially in the eastern area, in as much as the convergence rate of the housing price towards the long-term equilibrium price is relatively slow in such regions. However, in more recent work, [START_REF] Fang | Demystifying the Chinese Housing Boom[END_REF] report that appreciation in China's housing markets has been mostly accompanied by equally impressive household income growth. In their study of 120 cities, they find that households from the lower end of 1 According to [START_REF] Gomes | Housing Market Dynamics: Any News?[END_REF] news shocks explain a sizable fraction of the variation in house prices and other macroeconomic variables.

the income distribution are able to access financing and purchase homes, even in cities with high house price appreciation. Also, [START_REF] Ren | House Price Bubbles in China[END_REF] At the aggregate level, [START_REF] Liu | Housing Prices and General Economic Conditions: An Analysis of Chinese New Dwelling Market[END_REF], conclude that market fundamentals, including mostly demand-side variables (unemployment rate, total population, and changes in the consumer price index) and some supply-side ones (changes in construction costs), could explain most of the variations in housing prices between 1986 and 2002. At a local level, [START_REF] Dreger | Is There a Bubble in the Chinese Housing Market?[END_REF] examine house-price developments using panel-cointegration techniques and data on 35 major cities.

A long run relationship is detected between real house prices and a set of macroeconomic determinants. The national bubble component was estimated to be less than 15 per cent of the equilibrium value as implied by the fundamentals at the end of 2010, but the bubble was larger in the cities located in the south-east coastal areas and special economic zones. Using data from 35 metropolitan areas, [START_REF] Zhang | Housing Price Fluctuations across China: An Equilibrium Mechanism Perspective[END_REF] find that equilibrium prices are determined by basic economic conditions in China and greatly affect fluctuations of actual prices, which return to their equilibrium level through self-adjustment.

Putting Chinese developments in an international perspective 3 , Glindro, Subhanij, Szeto, and Zhu (2011) examine house price developments in eight economies in the Asia-Pacific area as well as

China, using both panel data and country-specific regressions using quarterly data between 1993 and 2006. Following previous work on the U.S. [START_REF] Capozza | Determinants of Real House Price Dynamics[END_REF], they define a bubble as the component of price not explained by cyclical and fundamental factors. In Glindro et al.'s analysis, demand-side factors include real GDP, population, the real mortgage rate, and the mortgage credit-to-GDP ratio, while supply-side factors consist of the land supply index and real construction costs. The 3 [START_REF] Ghysels | Forecasting Real Estate Prices[END_REF] present an exhaustive survey of the principal methodologies used in constructing real estate price indices and of the main empirical findings on the determinants of price movements for the US residential and commercial real estate markets.

trend component of the mortgage credit-to-GDP ratio and equity prices is used to proxy long-run house price fundamentals. They conclude that the recent house price growth in China can be explained by macroeconomic fundamentals.

Housing-Market Specific Fundamentals

Some literature 4 has focused more specifically on housing-specific fundamentals. [START_REF] Wu | Evaluating Conditions in Major Chinese Housing Markets[END_REF] note that the price-to-rent ratio in Beijing, and seven other large markets across the country, has increased by 30% to 70% since the beginning of 2007. Current price-to-rent ratios imply a very low user cost of no more than 2%-3% of house value. Very high expected capital gains appear necessary to justify such a low user cost. Their calculations suggest that even modest declines in expected appreciation would lead to large price declines, of over 40% in markets such as Beijing, in the absence of offsetting rent increases or other countervailing factors. Using both national and province-level panel data, [START_REF] Chen | The Income Elasticity of Housing Prices in China and its Regional Diversity[END_REF] find that housing prices growth is closely related to demand-side variables, such as increases in household income. Peng, Tam and Yu (2005) provide evidence that, over 1998-2004, a supply-side variable, land price, has a much larger 'multiplier' effect on property prices in coastal areas than in the interior provinces.

Speculative Forces

In the case of China, it is generally considered that speculative forces acting on the real estate market originate either in the stock market or in the provision of liquidity from hot money inflows.

As a by-product of the very successful reforms initiated in the late 1990s, home ownership in China is extremely widespread (87% of households in urban and 97% in rural areas) and substantially higher than stock ownership (only 6% in the late 2000s). However, households who buy new housing mainly do it (62% of them) for investment or speculative purposes, often in the form of a second or third housing unit. China's stock market is also often portrayed as highly speculative, as testified by the fact that the number of new household stock market accounts created every month often reaches several millions (households represent 80% of investors in China's stock market). Accordingly it is worth considering to what extent portfolio choice and wealth effects may be at work between the real estate and stock markets, particularly for households in China whose only alternative (until the very recent gradual capital account opening) store of wealth is a bank account.

It is noteworthy that developments in the real estate markets may not be independent from what happens in stock markets [START_REF] Cocco | Portfolio Choice in the Presence of Housing[END_REF][START_REF] Campbell | A Model of Mortgage Default[END_REF] or possibly bond markets [START_REF] Yao | Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints[END_REF]. Indeed in as much as the portfolio of investors is composed of both types of assets, 4

In addition, previous studies have mostly discussed the distribution of affordability, in particular for low income families [START_REF] Sato | Housing Inequality and Housing Poverty in Urban China in the Late 1990s[END_REF][START_REF] Wang | Impact of Urban Economic Openness on Real Estate Prices: Evidence from Thirty-five Cities in China[END_REF][START_REF] Gan | The Impact of Housing Reform on Durables Consumption in China[END_REF]. [START_REF] Shen | Are House Prices too High in China?[END_REF], defines housing affordability in terms of permanent income, and finds that housing is more affordable in China than in developed economies, although the ratio of housing prices to current income is much higher than in developed nations.

substitution between the two investments may lead to a negative correlation of their returns [START_REF] Shiller | Speculative Asset Prices[END_REF], while wealth effects may lead to positive correlations [START_REF] Gyourko | What Does the Stock Market tell us about Real Estate Returns?[END_REF]. As distinct other financial assets, housing is both a durable consumption good, generating utility to its owner, and an investment vehicle that allows the investor to hold home equity. This is obviously true only for those households who are owner occupiers, while in China many derive housing services from renting their home and investing in a second home.

Huang and Ge ( 2009), find only weak correlation between stock and real estate markets in China, but their (monthly) sample is very short (26 months starting in April 2006). With a longer sample (2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012) but a lower frequency (quarterly) and with a regional panel, [START_REF] Yuan | House Prices and Stock Prices: Evidence from a Dynamic Heterogenous Panel[END_REF] contrast positive short-run with negative long-run effects of stock prices on housing prices. [START_REF] Guo | Does "Hot Money" Drive China's Real Estate and Stock Markets?[END_REF] find a positive effect of excess financial wealth on property prices as well as price volatility. By contrast Xu and Chen (2012) find such an impact insignificant when controlling for the effect of money supply growth.

The role of hot money inflows in driving housing market dynamics in China is potentially important given the spectacular accumulation of foreign currency reserves, representing an ever expanding source of liquidity if not fully sterilized. Such inflows can also affect stock market investment. Feng, Lin and Wang (2012) find that hot money inflows (immediately) impact stock but not housing returns, while foreign direct investment (FDI) impacts the latter (with a lag) and not the former. [START_REF] Guo | Does "Hot Money" Drive China's Real Estate and Stock Markets?[END_REF], with monthly data over 1997-2008, find that hot money does impact significantly on China's real estate, but not stock markets. Besides, while stock price shocks drive housing price movements, the reverse is not true. [START_REF] Xu | The Effect of Monetary Policy on Real Estate Price Growth in China[END_REF], over July 2005 to February 2010, find that a bullish stock market accelerates subsequent housing price growth, but, when controlling for money-supply growth, hot money inflows do not impact housing-price growth.

Monetary Versus Macroprudential Policy

Movements in bank credit are often responsible for overheating in real estate markets. The authorities' toolkit includes either price and quantity monetary instruments or macroprudential tools.

Households' income, representing one of the major fundamentals of housing prices, would represent a bound on the latter in an economy without financial markets. However, the presence of mortgage loans used to finance housing purchases may disconnect house prices from the household-income fundamental. As modelled by Hillebrand and Kikuchi (2013), with rational expectations, and neither heterogeneity nor financial frictions, a household may buy a house exceeding the discounted present value of its income, if it finances the purchase partly with a loan, which it will repay in the future by selling the house at a higher price. At the macro level, persistent income changes generate parallel large movements in credit volumes and housing prices 5 . [START_REF] Qi | Property Prices and Bank Lending in China[END_REF] Focusing on another key aspect of the transmission channels, [START_REF] Wu | Real Estate Collateral Value and Investment: The Case of China[END_REF]) study the dramatic swings in Chinese land markets in recent years and investigate the evidence of an economically meaningful real estate collateral channel effect on non-real estate listed firm investment. Their study indicates that since China's debt markets are not characterized by the frictions found in the U.S. and other developed countries there is no reason to expect a collateral channel effect as borrowers do not need to pledge property as collateral to fund investment programs.

Fundamentals, Transactions and Asset Price Volatility

In the context of the stock market [START_REF] Schwert | Why Does Stock Market Volatility Change Over Time?[END_REF] suggests explaining asset market volatility by the time varying volatility of macroeconomic variables. More recently [START_REF] Engle | The Spline-GARCH Model for Low-Frequency Volatility and its Global Macroeconomic Causes[END_REF] model high-frequency return volatility as the product of a slow-moving component and a GARCH process.

The slow-moving component is then modelled as a function of macroeconomic and financial variables. process and a MIDAS polynomial as proposed by [START_REF] Ghysels | Predicting Volatility: Getting the Most out of Return Data Sampled at Different Frequencies[END_REF]. Such an approach is applied to China's stock market volatility by [START_REF] Girardin | Market Fundamentals as a Source of Stock Market Volatility in China: A GATCH-MIDAS Approach[END_REF], with both daily stock returns and monthly macroeconomic variables.

The impact of transactions volume on prices has been emphasized in the housing market literature.

Stylized facts for the housing market are well described by [START_REF] Berkovec | Turnover as a Measure of Demand for Existing Homes[END_REF] who show that because housing supply is less than perfectly elastic, changes in turnover rates are positively correlated with changes in house prices (via a wealth effect working through the presence of down payments, [START_REF] Stein | Prices and Trading Volume in the Housing Market, A Model with Down-Payment Effects[END_REF]. In addition, such a relationship between price and turnover holds more strongly at low rather than at high frequencies (due to the time lag between the signing and closing of the contract). Finally the response to housing demand is quicker for turnover rates than for prices, which is in line with the search-theoretic cum informal friction model of [START_REF] Berkovec | Turnover as a Measure of Demand for Existing Homes[END_REF] but inconsistent with the rational expectations model. Subsequent research [START_REF] Leung | Testing Alternative Theories of the Property Price-Trading Volume Correlation[END_REF] has

shown that, on disaggregated data (for 35 residential estates in Hong Kong over June 1991 through November 1998), the medium-run component of transaction volume (extracted with a band-pass filter) leads (and Granger causes) a similar component of the housing price (see also [START_REF] Leung | What Drives the Property Price-Trading Volume Correlation? Evidence from a Commercial Real Estate Market[END_REF][START_REF] Leung | Intra-Metropolitan Price and Trading Volume Dynamics: Evidence from Hong Kong[END_REF], on the Hong Kong commercial property market).

Empirical evidence surveyed by Tu, Ong and Han (2009, p.2), leads them to conclude that "the fact that transactions lead to price changes after a demand shock is established" . Wheaton and Lee (2009), on panel data of US local housing markets showed that a large part of the variation in turnover is due to flows between rental and owner-occupied housing. In a search market context, there should be a connection between price volatility and turnover: the more uncertainty, the higher should be reservation prices and the longer houses should stay on the market before being sold. Consistent with this, Tu, Ong and Han (2009) document a negative correlation between volatility and turnover in their analysis of Singapore condominiums. In China a specific pattern may arise due to the particularities of housing policy [START_REF] Huang | Second Home Ownership in Transitional Urban China[END_REF], in which households living in public rental housing are more likely than homeowners to own a second home (in the mid 2000s around one quarter of renters owned second homes). Bond and Hwang (2001) propose a fundamental measure of volatility for the commercial property market by using a stochastic volatility model (applied to the U.K. commercial property market) to filter out the signal in the different sources of volatility. The different measures of volatility are decomposed into transitory noise and unobserved fundamental volatility. Several recent papers have investigated whether GARCH effects exist in the U.S. housing market [START_REF] Dolde | Temporal and Spatial Information Diffusion in Real Estate Price Changes and Variances[END_REF][START_REF] Crawford | Assessing the Forecasting Performance of Regime-Switching, ARIMA and GARCH Models of House Prices[END_REF][START_REF] Miller | Exploring Metropolitan Housing Price Volatility[END_REF][START_REF] Miles | Volatility Clustering in U.S. Home Prices[END_REF]. Some research in finance indicates that the conditional variance of some assets exhibits far greater persistence, or even "long memory", than is accounted for in standard GARCH models. If house prices do indeed have this very persistent volatility, properly estimating the conditional variance to allow for such persistence is crucial for optimal portfolio management. [START_REF] Miles | Long-Range Dependence in U.S. Home Price Volatility[END_REF] examines housing prices in a number of U.S.

metropolitan areas, and finds that, for those with significant GARCH effects, more than half indeed exhibit the very high persistence found in other assets, such as equities.

Methodology and Data

Δ𝑦 𝑡 = 𝛼 𝑓 1 𝑓 2 + 𝛽 𝑓 1 𝑓 2 𝑦 𝑡-1 + ∑ 𝜙 ℎ,𝑓 1 𝑓 2 Δ𝑦 𝑡-ℎ + 𝜉 𝑡 𝑝 ℎ=1
(1)

where 𝑓 1 is the fractional starting point of the regression sample and 𝑓 2 the fractional ending point. The ADF t-ratio on the estimated 𝛽 ̂𝑓1 𝑓 2 is denoted by 𝐴𝐷𝐹 𝑓 1 𝑓 2 . The hypotheses are 𝐻 0 : 𝛽 𝑓 1 𝑓 2 = 0 (unit root dynamics), and 𝐻 1 : 𝛽 𝑓 1 𝑓 2 > 0 (explosive behaviour). The GSADF statistic is defined for each smallest feasible window size, f 0 , as:

𝐺𝑆𝐴𝐷𝐹(𝑓 0 ) = sup 𝑓 2 ∈[𝑓 0 ,1],𝑓 1 ∈[0,𝑓 2 -𝑓 1 ] {𝐴𝐷𝐹 𝑓 1 𝑓 2 }. (2)
The regression sample size is 𝑇 𝑤 = ⌊𝑇(𝑓 2 -𝑓 1 )⌋, where 𝑇 is the total number of available observations and ⌊. ⌋ is the integer part of the argument.

PSY tabulate the asymptotic critical values of the GSADF statistic for different values of f 0 . If the GSADF test statistic is found to be larger than the critical value then, for each sample point ⌊𝑇𝑓 2 ⌋, a sequence of backward recursive tests is computed and a supremum taken:

𝐵𝑆𝐴𝐷𝐹 𝑓 2 (𝑓 0 ) = sup 𝑟 1 ∈[0,𝑟 2 -𝑟 1 ] {𝐴𝐷𝐹 𝑓 1 𝑓 2 }. (3)
The point ⌊𝑇𝑓 2 ⌋ belongs to a bubble regime if:

𝐵𝑆𝐴𝐷𝐹 𝑓 2 (𝑓 0 ) > 𝐶𝑉(𝑎, 𝑓 0 ) (4)
where 𝐶𝑉(𝑎, 𝑓 0 ) is the critical value for a given significance level 𝑎 and window size range f 0 .

Noise Filtering

Daily house prices are usually noisy, due to the small number of transactions on a daily basis.

P r                 (7) 7
Previously similar methods were used for low frequency real estate data by [START_REF] Engle | A Dymimic Model of Housing Price Determination[END_REF], and [START_REF] Brown | Forecasting UK House Prices: A Time Varying Coefficient Approach[END_REF].

The true returns can be obtained from a Kalman-filter estimation assuming that the error terms are i.i.d. normally distributed. The normal assumption in our case cannot be justified and we can only interpret the filtered price series as best linear approximation.

MIDAS and GARCH-MIDAS

In order to model the volatility of the Chinese real estate (filtered) prices as a combination of sectorspecific or macroeconomic effects and time series dynamics, we follow Engle et al. ( 2013) (EGS) and use a Mixed Data Sampling (MIDAS) methodology, as in [START_REF] Ghysels | There is a Risk-Return Trade-Off After All[END_REF].

As in Engle and Rangel (ER) ( 2008) we assume that unexpected asset returns can be written as:

  , 1, , , , i t i t i t t i t i t r E r g      (8)
where r i,t is the log asset return on day i during month t, 8 E i-1,t () is the conditional expectation given information up to time i-1. The high-frequency return volatility is assumed to be the product of a short run component, g i,t , and a long-run component, t . Macroeconomic and sector-specific variables can affect the low-frequency volatility but not the short-run component. Then, according to (8), the same news can affect returns differently depending on economic conditions.

In (6) we modelled the true daily prices as random walks and therefore it follows that  

1, , i t i t

Er   .

We re-write equation (8) as: , , ,

i t t i t i t rg      (9)
where  i,t |  i-1,t ~ N(0,1) and  i-1,t is the information set up to day (i -1) of period t. Note that  in equation ( 9) should be the same as in equation ( 6).

We assume that g i,t follows a GARCH(1,1) process:

    2 1, , 1, 1               it i t i t t r g g (10)
We can use a rolling window MIDAS filer to smooth realized volatility and obtain the long run component, () rw i  , which can vary daily as: 8

In theory t could be chosen to be monthly or quarterly. However, since our sample period spans only a small number of years we set t to be monthly.

(

( ) ( ) ( ) 12 1 ( , ) K rw rw rw rw k i ik k m RV          (11) )
where

' () 2 1 N rw ij i j RV r    
is the rolling window (rw) realized volatility, and i denotes the day of the period. For a monthly rolling-window N' is set to 30.    can be chosen to be either a Beta or an Exponentially-weighted polynomial:

12 12 11 1 2 1 2 11 1 1 ( / ) (1 / ) ( , ) , + =1, Beta ( / ) (1 / ) ( ) , 0 1, Exponentially weighted k K q k k K q q k K k K q K q K                         (12)
A thorough analysis of the various shapes obtainable with Beta and exponentially weighted lags can be found in [START_REF] Ghysels | MIDAS Regressions: Further Results and New Directions[END_REF].

Equations ( 9)-( 12 . K can be selected to minimise some information criterion such as the Bayesian one (BIC).

Data

Existing house price index construction methods, such as the repeated-sales index model which has been adopted in most of the developed markets, are not necessarily suitable for nascent housing markets. Using the booming market in China as an example, Deng, Gyourko and Wu (2012), [START_REF] Wu | House Price Index Construction in the Nascent Housing Market: The Case of China[END_REF], and Wu and Deng (2015) evaluate and compare the performance of three most common house price measurement methods, including the simple-average method without quality adjustment, the matching approach with the repeated-sales modelling framework, and the hedonic modelling approach. Their theoretical and empirical analyses suggest that the simple-average method and the matching approach both tend to produce a downward bias due to the complex-and unit-level quality changes over time of sales in these immature housing markets. The hedonic method seems to provide better control for these effects.

On the basis of this finding, we apply here the hedonic method to the housing market in two major Chinese cities, Beijing and Shanghai. Such daily detailed micro level housing transaction data were collected from multiple leading residential real estate agencies and local government authorities'

websites. The sample covers the period from January 2005 to December 2010. For example, in the Beijing case, the daily data contain more than 360,000 transactions, which include 77,577 resale housing transactions from which our sample is extracted. Each record contains transaction information such as transaction date, transaction price (sale price), unit size, number of rooms, the floor level where the unit is located, and the total number of floors of the building. We then use the "Soufun Website-Based GIS" system 9 to match each transaction into more than 6,000 Properly measuring house prices requires disentangling observed movements in prices from the impact of quality changes on housing units sold in different periods [START_REF] Wu | House Price Index Construction in the Nascent Housing Market: The Case of China[END_REF][START_REF] Malpezzi | Hedonic Pricing Models: A Selective and Applied Review[END_REF]. In Chinese cities the housing sector is mostly composed of non-landed condominium units. It is usual for a condominium complex to be composed of hundreds or often thousands of units located in multiple high-rise residential buildings on a specific land parcel.

Accordingly two groups of housing attributes should be distinguished: i) the complex-level attributes which include the locational characteristics and neighborhood amenities, plus some physical characteristics (building type and construction quality); ii) the unit-level attributes which refer to physical characteristics, usually unit size, floor level, and specific environmental attributes (for example, noise, view, accessibility to sunshine, etc.). Within the same complex, housing units share the former attributes, but differ in the latter.

Based on these features, the hedonic model in the Chinese resale housing market can be expressed as:

vst vt st t vst vst P U C D         ( 13
)
where P vst is the transaction price of unit v in complex s sold at time t; U vt and C st are sets of unit-level and complex-level housing characteristics, respectively, all in natural logarithm; and  vst is an i.i.d. error term.

9

The system can be accessed at map.soufun.com. 10 These 6,000 plus neighborhoods are located in eight urban districts in Beijing, namely, Chaoyang, Haidian, Fengtai, Xicheng, Xuanwu, Dongcheng, Chongwen and Shijingshan Districts. Among them, the Chaoyang District contains about 43% of the sample.

While other parameters might also be of interest, the key task of house price index construction is the accurate estimation of time dummies' coefficients  t . The hedonic method seeks to incorporate the quality adjustment, directly based on the estimation results of the hedonic model. In its most frequently used, time-dummy, specification, housing transactions from multiple periods are pooled into a single hedonic model to estimate the vector of time dummy coefficients  t , and subsequently the house price index can be computed on the basis of  t [START_REF] Kain | Measuring the Value of Housing Quality[END_REF]Gourieroux and Laferrere, 2010).

A major challenge for the hedonic method is the high data requirement, involving not only the transaction price, but also detailed housing-attribute information. Data on housing attributes has started becoming available recently in China, thanks to the required registration with local authorities of all housing transaction contracts. The electronic recording by municipal housing authorities in major cities of each transaction's key information ("Real Estate Market Information System") was initiated in 2003, and became compulsory in 2006 (as mandated by the Ministry of Housing and Urban-Rural Development of China: MOHURD). In April 2007, the MOHURD went as far as to prescribe the list, as well as to define the nature and format, of variables to be reported. Such official detailed recording is a natural data source for the implementation of the hedonic method, even though such data are unlikely to be exhaustive enough to encompass each and every complex-or unit-level attribute.

However this does not affect the reliability of the hedonic method as long as the unobserved housing characteristics do not change monotonically over time [START_REF] Wu | House Price Index Construction in the Nascent Housing Market: The Case of China[END_REF].

Officially-published monthly data on housing prices suffer from two major weaknesses. First these ignore the fall in complex-level quality over time: rapid urbanization pushes new buildings to the periphery [START_REF] Deng | Land and House Price Measurement in China[END_REF], and permitted floor-area ratios rise. Second, these suffer from a downward bias resulting from developers' opportunistic pricing strategy. The downward bias in both the level and rate of change of official prices is illustrated in Figure 2 which reports the monthly average prices in Beijing and a transaction-weighted average of Beijing and Shanghai (TWABS) both for official prices and our hedonic prices.

Since we are interested in testing whether economic fundamentals and their volatility have an influence on the volatility of Chinese housing prices we include a number of macroeconomic and sector-specific variables as suggested by previous literature (see Section 2.1). The sector-specific variables include the growth rate of the monthly land supply (∆Lland), the real rent (∆Lrent) in Beijing (sourced from multiple leading residential real estate data vendors in China, and local land resources authorities' websites, to collect data on all residential usage land sales to private parties for 2005-2010), the growth rate of the permanent or registered population (∆Lreg) in Beijing and Shanghai (sourced from China Data Online, and interpolated using the cubic match last option in Eviews), and proxies for construction costs (growth rates of cement, ∆Lcem, and copper prices, ∆Lcop). The macroeconomic variables include the (month-on-month) CPI national inflation rate extracted from the OECD's Main Economic Indicators (∆Lcpi), monthly industrial output growth (∆Lpi), and household disposable income (∆Ldisp), from CEIC. Foreign currency reserves 11 in RMB (∆Lfxres) as well as the Shanghai stock market A-share index (∆Lstock) and transactions volume (Lstockvol) are also sourced from CEIC. The monetary variables include the growth rate of bank credit to the private sector (∆Lcredit), sourced from IMF's International Financial Statistics, the growth in broad and narrow (∆LM2) and (∆LM1) monetary aggregates, the one-year lending rate, and PBC credit refinancing (∆Lref) all from CEIC. We construct a five-pronged monthly index of the loan to value ratio (LTV) in the spirit of the quarterly index reported in the appendix of Xu and Chen (2005). A loosening (1, from Oct. 2008) corresponds to an LTV of 20% for a first and second home purchases, a moderate tightening (-0.5 from beginning of sample) is defined as an LTV above 20% for the second home, and a tightening (-1 from June 2006) is when the LTV is over 30% for a first home over 90 square meters (sqm), and a sharp tightening( -1.5, from September 2007 and May 2010) is defined as an LTV of over 20% for a first home below 90 sqm, over 30% for a first home over 90 sqm, and 40 % for a second home.

Our TWABS data for prices is the transaction-weighted average of Beijing and Shanghai. Table A.1 in the Appendix provides summary statistics for the monthly sector-specific, macroeconomic and policy series, as well as daily housing prices and transaction volumes. For the monthly sector-specific, macroeconomic and policy series we fit an AR(4) model to the first difference of the log of the variable and use the estimated squared residuals as proxies for the monthly volatility (as in [START_REF] Schwert | Why Does Stock Market Volatility Change Over Time?[END_REF].

We denote the volatility of these variables by their name prefixed by R (R∆Lland, etc…).

Results

In this section we initially test for the presence of explosive behaviour in daily Beijing and national (transaction-weighted) average (TWABS) hedonic resale price series. Subsequently, we filter out the microstructure noise and extract long run volatility from the daily series. Finally, with the monthly data we relate that long-run volatility to housing sector specific, monetary policy and speculative factors.

Detecting Explosive Behaviour

Figure 3 presents the evolution of the PSY test for explosive behaviour for the log of the "raw" national average and Beijing prices, including a constant and a trend in the ADF regression. As illustrated in the Beijing plot, the BSADF statistics remains well below the 10% significance threshold with a small peak on May 3 rd 2010. For the TWABS data the BSADF statistics are also below the critical values but with a slightly earlier peak on April 29 th 2010.

11

We do not substract the trade surplus and net FDI inflows from currency reserves, since according to some survey data (reported by [START_REF] Martin | China's "Hot Money" Problems[END_REF], the latter two items represent over two-thirds quarters of illicit capital inflows into China. It is thus inappropriate to exclude them from a measure of hot money.

Noise Filtering

Table 1 contains the noise-filter estimates and Figure 4 presents the sample autocorrelation for the raw daily hedonic housing returns (LBeijing and LTWABS). The noise to signal ratio    is estimated to be equal to 6.3408 and 4.8326 for TWABS and Beijing respectively, thus emphasizing the importance of filtering the data. The first order autocorrelation is estimated to be equal to -0.4209 and -0.4172 for TWABS and Beijing respectively, which is in line with comparable evidence for the US market [START_REF] Bollerslev | Daily House Price Indices: Construction, Modeling, and Longer-Run Predictions[END_REF]. According to equation ( 7) the first order auto-correlation should be equal to -

Extracting Long-Run Volatility

We use the GARCH-MIDAS model described in Section 3 to model daily changes in the filtered log of Beijing and TWABS real estate prices between January 6 th 2005 and December 31 st 2010. We choose the Bayesian Information Criterion (BIC) to select the optimal number of lags necessary to smooth rolling-window realized volatility weighted by the beta lag function in equation ( 4). 300 lags were selected for Beijing and 380 for TWABS.

We extract the daily long run component of the conditional volatility for both indices, and aggregate them to a monthly frequency (as plotted with transaction volumes in logs in Figure 5). Generally The estimates of the GARCH-MIDAS parameters are presented in Table 2. Although the Beta weighting function in equation ( 12) includes two parameters ω 1 and ω 2 , the optimal ω 1 is always unity such that the weights are monotonically decreasing over the lags. Therefore, we only report a single ω parameter. Except forall of the coefficients are statistically significant. The coefficient,, of the sum of the weighted rolling-window realized volatility in equation ( 11), is highly significant. The GARCH parameters,  and , sum to 0.89 for TWABS and 0.92 for Beijing. Finally, as expected, the estimated s in Table 2 are within two standard deviations of the estimated s in Table 1.

Fundamentals and Long Run Volatility

We relate the long run volatility of residential prices to housing sector specific, speculative and policy or macroeconomic variables. We include a dummy variable for April 2010 which corresponds to a tightening of regulation policy by the Chinese government. 12 We also include a dummy variable for Chinese New Year and March. We follow [START_REF] Berkovec | Turnover as a Measure of Demand for Existing Homes[END_REF], who recommend using lagged values of independent variables due to the time required by housing search and marketing, so that sales lag behind the determinants of demand 13 . We initially estimated the long run volatility equation together with the equation using maximum likelihood but found no simultaneity. The results presented here are the OLS estimations of the individual equations.

As reported in Table 3, an autoregressive model of order two for the logarithm of long run volatility of real estate returns is necessary to eliminate residual autocorrelation. Among housing sector specific variables, the growth in rents and land supply in Beijing has a positive effect, both with a one-month lag, on Beijing long-run volatility (Table 3, column 2). Among macroeconomic variables, only national industrial output growth significantly affects long run volatility. We tested for the impact of proxies for construction costs, like the (filtered) growth in the Shanghai copper or cement price, but neither proved significant. By contrast the volatility of land supply in Beijing does impact positively on Beijing real estate price volatility. We detect an indirect volatility-enhancing effect from the growth of foreign currency reserves 14 through transaction volume. We did not find any evidence of an impact of monetary policy either in a narrow or broad sense (lending interest rate, broad and narrow monetary aggregates, or bank credit 15

) on either volatility or transaction volume. By contrast, transactions are an indirect channel through which macroprudential policy affects housing prices volatility. In addition Beijing real estate transaction volumes (Table 3, column 3) are sensitive to a Chinese-New-Year and a March effect, and contain a deterministic trend.

We find that, after controlling for macroeconomic effects (industrial output growth), transaction volumes impact negatively on price volatility (with a lag). Such an inverse relationship can be interpreted as follows: for a given supply of resale housing, a fall in demand (possibly caused by a 12

As a consequence of the stimulus measures in 2008, housing prices started to soar in early 2009 and continue to grow dramatically until 2010, which led to another round of regulation known as the strictest tightening regulation policy around April 2010. The key points of this new round of regulation included differentiated housing credit policy and home purchase restriction policy. Under the home purchase restriction policy, only local residents who had one housing unit and nonresidents who had worked and lived in the city for more than one year were eligible for home purchase. Local residents with two or more housing units, or non-residents with one or more housing units, or non-residents who could not provide proof of living in the city for over one year, were all suspended from purchasing house in the city. After central government expressed its unprecedented dedication to curb surging property prices, many cities followed suit by issuing local packages limiting purchases on homes. (See Deng, Ren, and Wu, 2013 for more discussion.)

13 "In addition, most existing home sales are recorded as of the 'closing date', which can lag behind the signing of the contract by a month or two. Terms of the sale are set at contract signing, not closing." [START_REF] Berkovec | Turnover as a Measure of Demand for Existing Homes[END_REF] p. 424).

14

We consider foreign currency reserve changes themselves and do not exclude the trade and FDI balance as is often done when trying to proxy hot money inflows. This choice is motivated by the observation that the majority (possibly three quarters) of underground capital inflows operate via misinvoicing, and FDI misreporting, i.e. precisely in the components which such measures exclude [START_REF] Kar | Illicit Financial Flows from China and the Role of Trade Misinvoicing[END_REF]. rise in the loan to value ratio) leads to a fall in transaction volumes, which trigger a fall in housing prices, leading to a rise in price volatility as predicted by the financial markets literature. A standard result in the literature is indeed that volume and prices are positively correlated (see review in Section 2.2. above).

The volatility of the two-city weighted index (Table 3, column 2) responds positively to industrial growth volatility, and the April 2010 dummy and negatively to lagged volumes, as with the Beijing index though in a much milder (in absolute values) way. This implies that volatility responds less to these three variables in Shanghai than in Beijing. In addition, a direct effect of the growth in foreign currency reserves, as well as the volatility of stock returns and the LTV index, is significant for the two-city volatility, implying that such effects are entirely driven by a sensitivity of Shanghai's volatility to these variables.

The long-run effects are derived using the [START_REF] Beveridge | A New Approach to decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the 'Business Cycle[END_REF] decomposition by reparameterizing our model in error correction form. For Beijing an increase in volumes of 1% results in the long-run in a decrease in volatility of 0.57%, whereas the decrease is only 0.36% for TWABS.

Although significant the macroeconomic variables and their volatility have only a limited impact in magnitude. Except for LTV, they have a positive effect on volatility as expected. An increase in LTV, 5.

Conclusion

The growing concerns of a "bubble" in Chinese real estate markets in the second part of the first decade of the new millennium imply that skyrocketing increases in housing prices in many major Chinese cities may have little relationship with economic fundamentals. We propose using a unique high frequency daily data set on residential housing prices and transactions in the Beijing and Shanghai resale real estate market, covering six years, to address such concerns. These data remedy the bias present in official housing prices. Our use of recently developed explosive versus unit-root tests imply that, contrary to such concerns, while near-explosive behaviour can be detected around June 2007 and February 2010, this is not statistically significant. In line with these results, we filter out microstructure noise by extracting a random walk component from raw hedonic housing prices. In a third step, using a Mixed Frequency Data Analysis (MIDAS) method, previously only applied to stock market data, we extract a long-run volatility component from the filtered daily real estate prices. At a monthly frequency this long-run component appears to relate significantly in China to fundamentals. Three series of factors appear to play a role. Among real estate specific variables, rent growth and land supply impact positively on volatility. Speculative elements seem to be present in the form of an impact of stock returns and foreign-currency-reserve inflows on volatility. Finally monetary policy seems unable to influence long run volatility, in contrast with the effectiveness of macroprudential policy (loan to value ratio regulation along with purchase restriction policy). Some of these effects operate only indirectly on housing prices, through transaction volumes. GARCH-MIDAS models are fitted using QMLE.

The estimated parameters correspond to those of equations ( 10) to (11) in the text.

Thein the table is 2 as the optimal 1, such that the optimal weights are monotonically decreasing over the lags, is 1.

The numbers in the parenthesis are robust t-stats computed with HAC standard errors.

LLF is the optimal log-likelihood function value and BIC is the Bayesian Information Criterion.

** Denotes significance at 5%.

  [START_REF] Engle | Stock Market Volatility and Macroeconomic Fundamentals[END_REF] introduce the GARCH-MIDAS approach to model simultaneously the GARCH component and the functional relationship between the long-run component with the volatility and the mean of macroeconomic variables. The GARCH-MIDAS component model uses a daily GARCH

  ) form a GARCH-MIDAS model for time-varying conditional variance with rollingwindow realized volatility and parameter space

  the floor area ratio (FAR), distance to the city centre (Tiananmen Square), and the distance to the nearest subway station based on the neighbourhoods where the transacted units are located. Similar data are constructed for the Shanghai market. (See Ren, Wu and Deng, 2013, for a more detailed discussion on the compilation of the data set).

  speaking, volumes and long run volatility move in opposite directions during our sample period. The most striking episode of large long-run volatility takes place in Spring 2010 for both the Beijing and TWABS series. A similar, but lower-magnitude, occurrence of high volatility takes place for TWABS in Spring 2007, and for both series in Spring 2008.

15

  This is in line with the results of[START_REF] Huang | Do Bank Loans and Local Amenities Explain Chinese Urban House Prices?[END_REF] for housing prices in 35 Chinese cities after controlling for amenities.

  i.e. a loosening of the loan to value ratio requirements, decreases long run volatility for TWABS. The long run volatility rises three-fold (TWABS) and six-fold (Beijing) in May 2010 implying a milder response in Shanghai than in Beijing.Volumes decrease by around 70% in May 2010 for both Beijing and TWABS. An increase of 1% in foreign currency reserves increases volumes in Beijing by around 0.1% in the long-run. An increase in LTV increases volumes for both Beijing and TWABS but the effect is ten-fold for TWABS. An increase of 1% in Shanghai stock market transaction volumes increases volumes by 0.3% and 0.5% in Beijing and TWABS respectively. The Chinese New Year and the resulting March effects are similar for Beijing and TWABS: a decrease of 40% during Chinese New Year and an increase of over 100% in March.

Figure 1 .

 1 Figure 1. Frequency Count of "China Housing Bubble" Mentioned in the Media

Figure

  Figure 2. Official and Hedonic Housing Prices: Beijing and Shanghai Transaction-Weighted Average (TWABS): Jan 2005-Dec 2010.

  investigate the relationship between property prices and bank lending in China over the period 1999Q1-2006Q2. Using an autoregressive distributed lag framework, they find shortand long-run causality from bank lending to property prices. Using panel data models covering 31 Chinese provinces and major cities,[START_REF] Peng | The Property Market and the Macroeconomy of the Mainland: A Cross Region Study[END_REF]), over 1998[START_REF] Cocco | Portfolio Choice in the Presence of Housing[END_REF], find that credit expansion by the four large state-owned banks does not feed property-price inflation.

	Two studies consider a narrow or wide set of monetary policy instruments. Xu and Chen (2012) use
	quarterly (1998Q1 to 2009Q4) and monthly (July 2005 to February 2010) data, and document that key
	monetary policy variables (bank loan rate, money supply growth, and mortgage credit policy indicator)
	are the main driving forces behind real estate price growth dynamics in China. Zhang, Hua and Zhao
	(2012), using a non-linear modelling approach, enlarge the number of monetary and price variables
	(mortgage rate, producer price, broad money supply and real effective exchange rate) as significant

determinants of housing prices in China over the period January 1999 to June 2010. Within a wider study of the transmission mechanism,

[START_REF] Zhang | Money, Housing, and Inflation in China[END_REF]

finds that monetary growth is transmitted to the real economy via changes in house prices 6 .

Given the role of the state in the Chinese economy and banking system, and the existence of direct controls on credit, Deng, Morck,

Wu and Yeung (2011, 2015)

provide evidence that China's monetary stimulation in 2009 and 2010 led to soaring house prices in major cities. They argue that such speed and efficacy of China's monetary policy derives from state control over its banking system and corporate sector. Beijing ordered state-owned banks to lend, and they lent. Beijing ordered centrallycontrolled state-owned enterprises (SOEs) to invest, and they invested. Much of this investment was highly-leveraged purchases of real estate.

Macroprudential policy aimed at limiting the effect of credit expansion on real estate prices relies, in particular, on changing down-payment requirements, which in the Chinese case are differentiated between first, second and possibly third home purchases. Xu and Chen (2012) develop a fivepronged qualitative index of loan to value requirements for housing purchases of first and second homes. They find that restrictive (expansionary) movements in that index significantly curb (stimulate) house prices with monthly panel data, with a one month lag.

[START_REF] Deng | Effectiveness of Cooling Measures in the Housing Markets: Evidence from China and Singapore[END_REF]

find that a combination of restricted housing credit policy and home purchase restriction policies implemented by the Chinese government around April 2010 effectively cooled down the housing booms through curbing transaction volumes in many markets. 6

  . Using the estimated values of 𝜎 𝜀 and 𝜎 𝜂 from Table1, the first order autocorrelation should be equal to -0.4939 and -0.4895 for TWABS and Beijing respectively, which is close to the estimated values.

	𝜎 𝜀 2
	𝜎 𝜂 2 +2𝜎 𝜀 2

Table 1 . Noise Filter Estimates

 1

		Beijing	TWABS
		0.00075	0.00067
		(0.00045)	(0.00034)
	C1	-4.69967	-4.59643
		(0.01438)	(0.01468)
	C2	-7.85045	-8.29043
		(0.04268)	(0.07043)
	𝜎 𝜀	0.0954	0.1004
	𝜎 𝜂	0.0197	0.0158
	𝜎 𝜀 𝜎 𝜂	4.8326	6.3408
	Notes:	1. Standard errors are in brackets.
		2. 22 12 exp( C ) exp( C )   

Table 2 . Parameter Estimates for GARCH-MIDAS with Realized Variance

 2

	GARCH-MIDAS Parameters	Beijing Estimates	TWABS Estimates
		(t-stat)	(t-stat)
		0.0013**	0.0014**
		(5.127)	(6.478)
		0.1651**	0.1837**
		(8.312)	(7.491)
		0.7618**	0.7087**
		(16.224)	(20.928)
	m	0.0089**	0.0081**
		(5.201)	(7.930)
		0.1726**	0.1503**
		(10.855)	(10.465)
		9.2656**	7.8594**
		(2.483)	(5.166)
	LLF	5232.29	5435.90
	BIC	-5.614	-6.086

Table A .1 Summary Statistics

 A

	Variable	Mean	Median	Std. Dev.	Skewness	Excess	Jarque-Bera:
						Kurtosis	P-value
				Monthly Variables
				(2006:3 to 2010:12)
	Lrent						
	Lreg						
	Lcop						
	Lcem						
	Ldisp						
	Lfxres						
	LM1						
	Lpi						
	Lcredit						
	Lref						
	Lcpi						
	Lstock						
	Lstockvol						
	Beijing Log volume						
	TWABS Log volume						

Daily hedonic returns (Jan-7-2005 to Dec-31-2010)

	LBeijing	0.0012	0.0025	0.1266	-0.1420	7.4669	0.00
	LTWABS	0.0001	0.0012	0.1319	-0.4001	7.4908	0.00
	LBeijing filtered	0.0008	0.0010	0.0207	-1.4458	17.9614	0.00
	LTWABS filtered	0.0007	0.0004	0.0164	0.0067	21.1087	0.00

[START_REF] Ascari | Boom and Bust in Housing Market with Heterogeneous Agents[END_REF] propose a partial disequilibrium dynamic model of the housing market in which the rationalexpectations hypothesis is also relaxed, this time in favour of chartist-fundamentalist mechanism to allow for the endogenous development of bubbles.

See[START_REF] Agnello | Booms and Busts in Housing Markets: Determinants and Implications[END_REF] for a study of a sample of eighteen industrialised countries over the period1980-2007.

Research, its Council of Advisers, or the Board of Directors.


Frédéric Boniol
email: frederic.boniol@onera.fr

Virginie Wiels
email: virginie.wiels@onera.fr

Yamine Aït-Ameur
email: yamine@enseeiht.fr

Klaus-Dieter Schewe
email: klaus-dieter.schewe@scch.at

The landing gear case study: challenges and experiments

Embedded critical systems need to be validated very thoroughly; it usually results in very long and onerous test phases. Formal techniques, in particular formal specification lan-guages and associated proof tools, could be an advantageous alternative, or at least a good complement and allow a signif-icant reduction of test phases. However, for these techniques to be used in practice, one issue to consider is their efficiency and scalability on complex industrial systems.

Case studies have played an essential role in the history of formal methods. They have allowed to illustrate the application of formal techniques for modelling and verification, to compare different methods in terms of expressivity, performance and easiness of use. They have also permitted to enact the progress made by these methods.

Dagstuhl seminar 9523 is about the famous Steam Boiler case study in 1995 had a lot of impact on the formal methods community. This case study allowed the assessment of formal

techniques, the comparison of different formal techniques, the identification of areas for future work [START_REF]Formal Methods for Industrial Applications, Specifying and Programming the Steam Boiler Control (the book grow out of a Dagstuhl Seminar[END_REF][START_REF] Abrial | The steam boiler control specification problem[END_REF].

As formal methods have made a lot of progress since 1995, ABZ'2014 has proposed a complex case study, representative of industrial needs. The proposed case study is a landing gear control system. It is composed of three parts: a pilot interface, a mechanical and hydraulic parts, and a digital part. This system is a representative of critical embedded systems. The action to be done at each time depends on the state of all the physical devices composing the system and on their temporal behaviour. When considering such systems, the challenge is first to model and to program the software part controlling the landing and the retraction sequence, and second to prove safety requirements taking into account the physical behaviour of hydraulic devices.

The case study attracted a lot of interest. 11 selected papers were presented at the ABZ'2014 conference and published in [START_REF] Boniol | ABZ 2014: the landing gear case study[END_REF]. They used different formal techniques: Event-B [START_REF] Abrial | Modeling in Event-B-system and software engineering[END_REF], ASM [START_REF] Börger | Abstract state machines: A method for highlevel system design and analysis[END_REF], Fiacre [START_REF] Farail | FIACRE: an intermediate language for model verification in the TOPCASED environment[END_REF]. They also proposed different kinds of verification: proof, model checking, test generation, run-time monitoring, and simulation. One of the main conclusions of the ABZ'2014 case study track was that formal methods are now powerful enough to model and to verify complex case studies as the landing gear system. The associated tools proved powerful enough to support such modelling and verification activities. Such an assessment is good news both for industrial engineers and for the academic community. However, a lot of difficulties remain.

This special issue presents an extended version of six of the models that were presented at the ABZ'2014 conference.

The next section gives an overview of the case study. For a more detailed presentation, the reader should refer to [START_REF] Boniol | The landing gear system case study[END_REF]. Section 3 discusses the main challenges of this case study. Section 4 introduces then the six experiments presented in

The mechanical and hydraulic parts The architecture of the hydraulic part is described in Fig. 2. Each landing set contains three latching boxes: one for locking the gear in the up position, when the gear is retracted, a second one for locking the gear in the down position, when the gear is extended, and a third one for locking the door in the closed position. Note that there is no latching box for the open position, meaning that the door is not mechanically locked when it is open.

The landing gears and doors motion is performed by a set of actuating cylinders:

-For each door, a cylinder opens and closes the door.

-For each landing gear, a cylinder retracts and extends the landing gear.

Hydraulic power is provided to the cylinders from an external hydraulic circuit through a set of electro-valves:

-One general electro-valve to supply the specific electrovalves with hydraulic power from the aircraft hydraulic circuit. -Four electro-valves to set pressure, respectively, on the portion of the hydraulic circuit related to door opening, door closing, landing gear extending, landing gear retracting.

Each electro-valve is activated by an electrical order coming from the digital part. In the specific case of the general electro-valve, this electrical order goes through an analogical switch to prevent abnormal behaviour of the digital part (e.g. abnormal activation of the general electro-valve). Note that the three doors (resp. gears) are controlled simultaneously by the same electro-valve. It is thus not possible to control the doors (resp. gears) separately.

A set of discrete sensors inform the digital part about the state of the equipments: 2 The landing gear case study: brief overview

The landing system is in charge of manoeuvring landing gears and associated doors. It is composed of 3 landing sets: front, left and right. Each landing set contains a door, a landing-gear and associated hydraulic cylinders. A simplified schema of a landing set is presented in Fig. 1.

The system is controlled digitally. From a high level point of view, a basic landing sequence are: [START_REF] Abrial | Modeling in Event-B-system and software engineering[END_REF] open the doors of the landing gear boxes, (2) extend the landing gears and (3) close the doors. Similarly, after taking off, the corresponding basic retraction sequence to be performed are: [START_REF] Abrial | Modeling in Event-B-system and software engineering[END_REF] open the doors, (2) retract the landing gears and (3) close the doors. From a more concrete point of view, these two basic sequences can be interleaved in an intricate way: the pilot can interrupt each sequence at any time and at any point to start the opposite sequence as often as he/she wishes, leading to an infinite number of possible scenarios. This is one source of complexity of the system. The second source of complexity is failure management.

Architecture of the system

The landing gear system is composed of three parts: (1) a mechanical part which contains all the mechanical devices and the three landing sets, (2) a digital part including the control software, (3) and a pilot interface.

The pilot interface To command the retraction and outgoing of gears, an Up/Down handle is provided to the pilot. When the handle is switched to "Up" the retracting landing gear sequence is executed, when the handle is switched to "Down" the extending landing gear sequence is executed.

Three lights inform the pilot of the current position of the gears and the doors, and of the current health state of the system and its equipments: (1) one green light "gears are locked down", (2) one orange light "gears manoeuvring", (3) one red light "landing gear system failure". No light is on when the gears are locked up and when no failure has been observed. The digital part The digital part is composed of two identical computing modules (see Fig. 3). Each one executes in parallel the same control software. This software is in charge of controlling gears and doors, detecting anomalies, and informing the pilot about the global state of the system and anomalies (if any). It is part of a retroaction loop with the physical system, and produces commands for the distribution elements of the hydraulic system with respect to the sensors values and the pilot orders. The two computing modules receive the same input (sensor values and pilot orders). From these input, each module computes five electrical orders (one for each electro-valve). These corresponding electrical orders outgoing from the two modules are physically produced on the same electrical line. The implicit composition of two output is an electrical "OR" as shown in Fig. 3. As a consequence, if the two different computing modules send two different values (true and false) on the same line (for instance in case of failure of one of the two computing modules), then only the true value is transmitted to the corresponding electro-valve.

Similarly the two modules produce global boolean state variables to the cockpit (one for each cockpit light). These output are synthesised by each module from sensors data and from the situation awareness. Similarly to electrical orders provided to the electro-valves, the boolean state variables

Mechanical and hydraulic equipment

The analogical switch (between the digital part and the general electro-valve) The aim of this switch is to protect the system against abnormal behaviour of the digital part. In the open position, no electrical order is sent to the electro-valve. In that case, the oil pressure in the hydraulic circuit becomes down.

In addition to this normal behaviour, the analogical switch can fail at any time. However, the most likely failures to consider are permanent failures: the switch remains blocked in the closed or in the open position.

Electro-valves All the electro-valves supposed to have the same behaviour. As shown in Fig. 4, an electro-valve is an hydraulic equipment with two hydraulic ports Hin and Hout, and an electrical port E with the following behaviour:

-if E = false (the voltage of the electrical order is down), then Hout = 0 (no pressure on the hydraulic output side, the hydraulic circuit is open); -if E = true (the voltage of the electrical order is high), then Hout = Hin (the hydraulic circuit is closed).

Note that the electrical order must be sustained to true (i.e, at the high voltage) to maintain the electro-valve in the closed position. The electrical order is not a discrete event, but can be seen as an analogical signal. Because of inertial reasons, when E rises from false to true (i.e., the electro-valve switches from open to closed), the pressure grows up continuously from 0 to Hin. One can suppose that the rise in pressure approximatively follows a linear law, and that the total duration of the transition phase is 1 s. In the same way, when E falls to false, the pressure goes down linearly from Hin to 0. The total duration of the pressure drop is 3.6 s.

Cylinders Cylinders are pure hydraulic equipments. As shown in Fig. 5, they begin to move when they receive hydraulic pressure, and they stop to move when the pressure goes down or when they reach the end of their race.

Gear cylinders are locked in high or down position by means of a latching box mechanism (the latching boxes are physically on the gears, one for each position). When a gear cylinder is locked in high (resp. down) position and when it receives pressure from the high (resp. down) hydraulic circuit, first it is unlocked from the high (resp. down) position, then it moves to the down (resp. high) position, and finally it is locked in the down (resp. high) position. Note that it is possible to stop and to inverse the motion of any cylinder at any time.

Software specification

The aim of the software part of the system is twofold: (1) to control the hydraulic devices according to the pilot orders and to the mechanical devices positions; (2) to monitor the system and to inform the pilot in case of anomaly.

Expected scenarios in normal mode

When the command line is working (in normal mode), the landing system reacts to the pilot orders by actioning or inhibiting the electro-valves of the appropriate cylinders. Two basic scenarios are considered: the outgoing sequence, and the retraction sequence.

Outgoing sequence The outgoing of gears is decomposed in a sequence of elementary actions. When the gears are locked in retracted position, and the doors are locked in closed position, if the pilot sets the handle to "Down", then the software should have the following sequence of actions:

1. stimulate the general electro-valve isolating the command unit to send hydraulic pressure to the manoeuvring electro-valves, 2. stimulate the door opening electro-valve, 3. once the three doors are in the open position, stimulate the gear outgoing electro-valve, 4. once the three gears are locked down, stop the stimulation of the gear outgoing electro-valve, 5. stop the stimulation of the door opening electro-valve, 6. stimulate the door closure electro-valve, 7. once the three doors are locked in the closed position, stop the stimulation of the door closure electro-valve, 8. and finally stop stimulating the general electro-valve.

Retraction sequence In the same way, the retraction of gears is decomposed in a sequence of elementary actions. When the gears are locked in down position, and the doors are locked in closed position, if the pilot sets the handle to "Up", then the software should have the following sequence of actions:

1. stimulate the general electro-valve isolating the command unit, to send hydraulic pressure to the manoeuvring electro-valves, 2. stimulate the door opening electro-valve, 3. once the three doors are in the open position, if the three shock absorbers are relaxed, then stimulate the gear retraction electro-valve and go to step 4, else (if one of the three shock absorbers is not relaxed) go to step 5, 4. once the three gears are locked up, stop the stimulation of the gear retraction electro-valve, 5. stop the stimulation of the door opening electro-valve, 6. stimulate the door closure electro-valve, 7. once the three doors are locked in the closed position, stop the stimulation of the door closure electro-valve, 8. and finally stop stimulating the general electro-valve.

The previous sequences can be interrupted by counter orders (e.g., a retraction order during the let down sequence) at any time. In that case, the scenario stops and restarts in the counter-sequence from the point where it was interrupted. For instance, if an outgoing sequence is interrupted in the door closure phase (step 6 of the outgoing sequence) by an "Up" order, then the stimulation of the door closure electro-valve is stopped, and the retraction sequence is executed from step 2: the door opening electro-valve is stimulated and the doors begin opening again. Afterwards, the scenario continues up to the final step or to a new interruption.

Timing constraints Because of inertia of the oil pressure and to prevent shock waves in the hydraulic circuit, the outgoing and retraction sequences have to meet three timing constraints. First, stimulations of the general electro-valve and the manoeuvring electro-valve must be separated by at least 200 ms. Second, orders to stop the stimulation of the general electro-valve and the manoeuvring electro-valve must be separated by at least 1s. And third, two contrary orders (closure vs. opening doors, extension vs. retraction gears) must be separated by at least 100 ms.

Health monitoring

The second objective of the control software is to detect anomalies and to inform the pilot. Anomalies are caused by failures on hydraulic equipment, electrical components, or computing modules. Whenever an anomaly is detected, the system is globally considered as invalid. An anomaly signal is sent to the pilot interface. The effect of this action is to put the red light "landing gear system failure" on.

Failures

Potential failures Regarding the mechanical and hydraulic equipment, the most likely failures to consider are permanent failures:

-each electro-valve can fail and end up blocked either in the closed or in the open state; -each cylinder can fail and be blocked in its last position (down, high, or any where between these two positions); -similarly to electro-valves, the analogical switch is a mechanical device which can fail and be blocked either in the closed or in the open state; -finally an hydraulic leak can happen anywhere in the circuit resulting in permanent pressure drop.

Regarding the sensors, let us recall that each sensor is composed of three redundant single discrete sensors. Each of them can fail in two different ways: (1) the single sensor is permanently blocked on one of its two values (true or false); or (2) it behaves erroneously varying randomly between true and false.

Finally, regarding the computing modules, each of them can fail in two different ways: (1) permanent failure, in that case the module does not send any order to the electro-valves and to the lights; (2) the module behaves erroneously by sending random values to the actuators.

All the faults are supposed independent. Note that we do not consider total loss of the electrical power resulting in the simultaneous loss of the two computing modules. Similarly, we do not consider failures of the lights.

Probabilities For the sake of simplicity, all the failures are supposed to follow exponential distributions:

λ mh = 10 -3 for the mechanical and hydraulic equipment (including the electro-valves, the cylinders and the analogical switch), meaning that the equipment will probably fail around 10 3 flight hours; λ s = 10 -3 for each single sensor; -and λ cm = 10 -4 for each computing module.

Normal mode requirements

-(R 11 (resp. R 12 )) When the command line is working, if the landing gear command handle has been pushed DOWN (resp. UP) and stays DOWN (resp. UP), then the gears will be locked down (resp. retracted) and the doors will be seen closed less than 15 s after the handle has been pushed; -(R 21 (resp. s after stimulating the opening (resp. closure) electrovalve, then the red light "landing gear system failure" is on. -(R 63 (resp. R 64 )) If one of the three gears is still seen locked in the down (resp. up) position more than 7 s after stimulating the retraction (resp. outgoing) electro-valve, then the red light "landing gear system failure" is on. -(R 71 (resp. R 72 )) If one of the three doors is not seen locked in the open (resp. closed) position more than 7 s after stimulating the opening (resp. closure) electrovalve, then the red light "landing gear system failure" is on. -(R 73 (resp. R 74 )) If one of the three gears is not seen locked in the up (resp. down) position more than 10 s after stimulating the retraction (resp. outgoing) electrovalve, then the red light "landing gear system failure" is on. -(R 81 (resp. R 82 )) When at least one computing module is working, if the landing gear command handle has been DOWN (resp. UP) for 15 s, and if the gears are not locked down (resp. retracted) after 15 s, then the red light "landing gear system failure" is on.

Because of the independent hypothesis between faults, the probability of having n failures at the same time is the product of the probability of each fault. For instance, the probability for losing the two computing modules is 10 -8 ; the probability for losing two single sensors of the same sensor if 10 -6 , etc.

Requirements

The landing system is a critical system. It has to meet a set of safety requirements. These requirements are divided into two parts: normal mode requirements, and failure mode requirements.

By normal mode, we mean any scenario involving no failure. By failure mode we mean any scenario involving combinations of failures with probability greater than 10 -7 .

The landing gear case system is a critical cyber physical systems which offers several challenges for formal modelling and verification methods.

Size of the state space A first (classical) challenge is the combinatorial explosion. The system involves more than 70 components (from sensors to computing units). All these components evolve in parallel in an asynchronous way. Each component has about four behavioural modes (including functional and dysfunctional modes). The global behaviour of the system is the result of interleaving the behaviour of all the components, which leads to a huge state space. For instance, let us consider a redundant sensor. This sensor is composed of three single sensors. These three single sensors are asynchronous. They have their own inertia and they can change (from true to false for instance) with a small delay. The number of potential states of the redundant sensor is then 2 3 . There are 18 redundant sensors in the system (3 per gear, 2 per door, 1 for the hydraulic circuit, 1 for the analogical switch, and 1 for the handle). The number of potential states of all the sensors is then 8 18 . Adding the behaviour of the mechanical components (electro-valves, cylinders, analogical switch and handle), the whole state space contains more than 2 70 states. Adding now the potential failures, the state space including dysfunctional scenarios goes over 2 100 states. The first challenge is to overcome this combinatorial explosion: how to explore such a huge state space, or how to abstract it in a safe way according to requirements to be verified.

Handling failure modes

The second challenge is the handling of failure modes. As said above, the landing system involves a great number of digital, mechanical and hydraulic components. Each component can fail in several ways following several probabilistic distributions. As explained previously, requirements to be verified on the system mix functional and dysfunctional behaviours in an intricate way. The verification of the system requires an interleaving of functional and probabilistic dysfunctional modelling.

Handling time A third challenge is about time. 12 of the 19 requirements of the case study involve time. Two of them (R 11 and R 12 ) relate to end-to-end latencies, from the pilot's handle to the lights which notify the situation to the crew through all the digital, mechanical and hydraulic components. The other ones specify a maximal response time for the system to inform the pilot in case of anomalies. Verification of these requirements needs to take into account the real-time behaviour of each component in an accurate way. Knowing that time is another cause of combinatorial explosion, the third challenge is how to best model time.

Experiments

This special issue contains six different modelling and verification of the case study.

The first paper "Aircraft Landing Gear System: Approaches with Event-B to the Modelling of an Industrial System" by Wen Su and Jean-Raymond Abrial [START_REF] Su | Aircraft landing gear system: approaches with event-b to the modelling of an industrial system[END_REF], proposes three Event-B modelling of the case study. Modelling a complex system is not an easy automatable task, the three proposed models follow three different approaches and uses different techniques for verification and validation (simulation, model checking, theorem proving, constraint solving). The paper brings valuable insights on modelling complex systems in general and modelling the landing gear system in particular.

An Event-B modelling and verification of the case study is also proposed by the second paper "Modelling a Landing Gear System in Event-B" by Amel Mammar and Régine Laleau [START_REF] Mammar | Modelling a landing gear system in event-b[END_REF]. The construction of the model is incremental with respect to the three challenges stated before: it first models the system without considering time and potential failures, then adds time and failures step by step.

The third paper "Validation of the ABZ Landing Gear System using ProB" by Dominik Hansen, Lukas Ladenberger, Harald Wiegard, Jens Bendisposto and Michael Leuschel [START_REF] Hansen | Validation of the abz landing gear system using prob[END_REF], is another B modelling which puts a special emphasis on visualisation of the model of the landing gear system. Visualisation tools are used to provide different views of the system, the paper describes the help it provides for the development and validation of the model.

The fourth paper "The Landing Gear System in Multi-Machine Hybrid Event-B" by Richard Banach [START_REF] Banach | The landing gear system in multi-machine hybrid event-b[END_REF] proposes an hybrid modelling of the case study. Hybrid modelling is ideally suited for the landing gear system which combines mechanical, hydraulic and software parts. No verification tool is yet available but the hybrid modelling of the case study is a valuable help to understand how the continuous movement of the mechanical devices interferes with the digital components.

A model checking approach is proposed by the fifth paper "Environment-driven Reachability for Timed Systems" by Ciprian Teodorov et al. [START_REF] Teodorov | Environment-driven reachability for timed systems. Safety verification of an aircraft landing gear system[END_REF]. This paper addresses the statespace explosion issue using a context-aware verification approach. This approach brings a significant reduction of the state-space based on a specific handling of the environment of the system. The decomposition into different contexts allows an exploration of combinations of nominal behaviour and failures.

The sixth and last paper "Rigorous development process of a safety-critical system: from ASM models to Java code" by Paolo Arcaini, Angelo Gargantini and Elvinia Riccobene [START_REF] Arcaini | Rigorous development process of a safety-critical system: from asm models to java code[END_REF], proposes a refinement-based development of the landing gear system using Abstract State Machine. Each refinement step can be proved correct using SMT-based approach. A java implementation is produced, its conformance with respect to the specification is checked using two approaches: modelbased testing and runtime verification.

Conclusion

The case study track at ABZ'2014 was a very interesting and lively session. In this special issue, the authors have led further their modelling and verification of the landing gear system. The six papers present interesting aspects of the system and insights on formal modelling and verification approaches. We hope for many more work to come in the future on this case study, from the different existing formal communities.

-

  Front/right/left gear is locked/not locked in the extended position. -Front/right/left gear is locked/not locked in the retracted position. -Front/right/left gear shock absorber is on ground/in flight. -Front/right/left door is open/not open. -Front/right/left door is locked/not locked in the closed position. -Hydraulic circuit (after the general electro-valve) is pressurised/not pressurised. -The analogical switch between the digital part and the general electro-valve is closed/open.

Fig. 1

 1 Fig. 1 Landing set this special issue. The last section gives few words of conclusion.

Fig. 2

 2 Fig. 2 Architecture of the hydraulic part

Fig. 3

 3 Fig. 3 Digital architecture

Fig. 4

 4 Fig. 4 An electro-valve equipment

  Door cylinders are locked (by means of two latching boxes on each door) only in closed position. Doors remain open by maintaining pressure in extension circuit. When a door cylinder is locked in closed position and when it receives pressure from the extension hydraulic circuit, first it is unlocked from the closed position, then it moves to the open position, and finally it is maintained in the open position as long as the pressure is maintained in the hydraulic extension circuit. All these operations are done automatically with the hydraulic pressure only. No electrical part is involved in cylinders. These operations take a certain amount of time, depending on the position of the cylinder in the aircraft and in the hydraulic circuit. The durations are given in Table1. The values are only mean values. The true durations can vary around these values up to 20 %.

Fig. 5

 5 Fig. 5 Extension and retraction of a cylinder

  R 22 )) When the command line is working, if the landing gear command handle remains in the DOWN (resp. UP) position, then retraction (resp. outgoing) sequence is not observed. -(R 31 ) When the command line is working, the stimulation of the gears outgoing or the retraction electro-valves can only happen when the three doors are locked open. -(R 32 ) When the command line is working, the stimulation of the doors opening or closure electro-valves can only happen when the three gears are locked down or up. -(R 41 (resp. R 42 )) When the command line is working, opening and closing doors electro-valves (resp. outgoing and retraction gears electro-valves) are not stimulated simultaneously. -(R 51 )When the command line is working, it is not possible to stimulate the manoeuvring electro-valve (opening, closure, outgoing or retraction) without stimulating the general electro-valve. Failure mode requirements -(R 61 (resp. R 62 )) If one of the three doors is still seen locked in the closed (resp. open) position more than 7

Table 1

 1 Durations

	operations	of the	Duration (in s) of …	Front gear	Front door	Right/left gear	Right/left door
			Unlock in down position	0.8	-	0.8	-
			From down to high position	1.6	1.2	2	1.6
			Lock in high position	0.4	0.3	0.4	0.3
			Unlock in high position	0.8	0.4	0.8	0.4
			From high to down position	1.2	1.2	1.6	1.5
			Lock in down position	0.4	-	0.4	-