Quantifying the reversibility phenomenon for the repeat-sales index

Keywords: reversibility, quantification, information, Monte Carlo simulations, Markovian process

The reversibility phenomenon for the repeat-sales index (RSI) is a serious obstacle for the derivatives products; it could hinder their introduction or their success. It is also an undesirable characteristic for the management of the real estate risk. This article provides a general solution for this problem, using an informational reformulation of the RSI framework.

We present first a theoretical formula, easy to interpret and easy to handle, before implementing it. Our methodology is robust in the sense that its conclusions are not conditioned by any specific dataset; moreover, the numerical estimations of the reversibility percentages are reliable. For the derivatives our technique has strong implications for the choice of the underlying index. Indeed, even if the reversibility of the RSI is probably higher compared to the hedonic one, this index remains a challenger because of the predictability and the quantifiability of its revisions.

Introduction

With the repeat-sales technique, the past seems to change. But actually it is not the past itself that is changing; it is only its knowledge (its representation). This phenomenon is the consequence of the arrival of the new data in the estimation set that are relevant for the past. This mechanism of revision is an obstacle to the introduction of the derivatives written on a RSI and more generally it is an undesirable characteristic for the management of the real estate risk. Thus, it would be profitable to have at one's disposal an empirical methodology that could allow anticipating the size of the potential fluctuations, as mentioned in Clapham et al. (2005) : "If a futures market requires index stability, it would be useful to know how often revision -either period-by-period or cumulative -exceeds some level. Say, for example, that futures markets could tolerate 0.5 percent revision in any one quarter and 2 percent cumulative revision to the initial estimate". But at the present time, such a methodology does not exist in the RSI literature. This article provides a general solution for this problem, using an informational reformulation of the RSI framework. Our methodology is robust in the sense that its conclusions are not conditioned by a single dataset; indeed in Clapham et al. (2005) one can ask if the empirical results are still valid for a non-Swedish sample. What's more the authors of this article also conclude to the superiority of the hedonic indexes because the reversibility fluctuations are smaller. However they do not provide a methodology which would make the anticipations of these variations conceivable. As we will see, the RSI technique makes possible these estimations. Consequently, even if the reversibility for the RSI is probably higher, this index can still challenge the hedonic approach because of its forecasting feature. The rest of this article is organized as follows. In the second paragraph we present more precisely the reversibility problem, with a literature review. The results of [START_REF] Clapp | Revisions in repeat-sales price indexes: Here today, gone tomorrow?[END_REF] are the subject of a particular attention. The third section presents firstly the theoretical reformulation of the RSI. Then, this new formalism is applied more specifically to the reversibility problem; a simple and easy to handle formula is established.

The fourth part is devoted to the empirical implementation. In this section, a simulation algorithm is presented and we will also answer to the above problem, mentioned in Clapham et al. (2005), establishing the law for the distributions of the reversibility percentages.

The reversibility phenomenon, state of the art

The phenomenon

One of the specificities of the RSI is its time dependence to the estimation horizon; a past value Ind t is not fixed once and for all. When the horizon is extended from T 1 to T 2 (T 2 > T 1 ), the new repeat-sales will not only bring information on the interval [T 1 ,T 2 ] but also 1 on [0,T 1 ].

And unfortunately, there is no reason why the new value Ind t (T 2 ) should be equal to the old one Ind t (T 1 ). This phenomenon of retroactive volatility is called reversibility. Figure 1 is an illustration of this instability for the Los-Angles County [START_REF] Clapp | Revisions in repeat-sales price indexes: Here today, gone tomorrow?[END_REF]) and Figure 2 for Paris [START_REF] Baroni | Prices of single family homes since 1970: new indexes for four cities[END_REF]). As we can see, the magnitude of the variations can be substantial, up to 10% in Clapp and Giaccotto for example.

Literature review

The two seminal articles for the repeat-sales technique are [START_REF] Bailey | A regression method for real estate price index construction[END_REF], in an homoscedastic situation, and Case, Shiller (1987) for the heteroscedasctic context. Since these two papers the repeat-sales approach has become of one most popular index because of its quality and its flexibility. It used not only for residential but also for commercial real estate, cf. [START_REF] Gatzlaff | A repeat-sales transaction-based index of commercial property" A study for the real estate research institute[END_REF]. One can also refer to [START_REF] Chau | Real estate price indices in Hong-Kong[END_REF] for a recent example of a 1 For instance, a data with a purchase at t < T 1 and a resale at t' (T 1 < t' < T 2 ) will be informative for [t,T 1 ]. But, as the resale occurs after T 1 , this repeat-sale cannot be used in the first index estimation. multisectorial application of the RSI and to [START_REF] Baroni | Prices of single family homes since 1970: new indexes for four cities[END_REF] for the French context. The reversibilty phenomenon had been analysed more specifically by [START_REF] Hoesli | Three new real estate price indices for Geneva, Switzerland[END_REF], in a two periods model. This very simplified environment allows studying rigorously the mathematics of the RSI; [START_REF] Meese | The construction of residential housing price indices: A comparison of repeat-sales, hedonic-regression and hybrid approaches[END_REF] choose for instance the same model in their appendixes. But when the number of dates increases, the RSI equations become quickly burdensome. Clapham et al. (2005) tried to compare the sizes of the reversibility phenomenon for the various index methodologies. They conclude that the hedonic one was probably the less affected, but as their article is empirical their conclusion probably depends on the Swedish sample they used. And this is maybe the major concerns because a robust and general conclusion can only comes from theoretical arguments. Generally, in the literature, the theoretic approach is not the most frequent situation. We can mention the very interesting article of [START_REF] Wang | Estimating house price growth with repeat sales data: What's the aim of the game?[END_REF], but others examples are globally scarce. For the reversibility problem there is an exception, namely the article of [START_REF] Clapp | Revisions in repeat-sales price indexes: Here today, gone tomorrow?[END_REF] ; their results are presented in the next paragraph.

The solution of Clapp and Giaccotto (1999)

This paper deals with a BMN2 context, but its formulas can be generalised to a CS3 model. The first step consists in running, for the interval [0,T 1 ], the regression Y(T 1 ) = D(T 1 )LInd(T 1 ) + ε(T 1 ), where the unknown is the vector of the logarithms of the index: LInd(T 1 ). Within Y(T 1 ) we have the log-returns realised for the repeat-sales of the sample.

The lines of the matrix D(T 1 ) correspond to each data. In each line +1 indicates the resale date, -1 the purchase date and the rest is made of zeros 4 . In a second step, the estimation interval is extended to [0,T 2 ] and the regression becomes Y(T 2 ) = D(T 2 ) LInd(T 2 ) + ε(T 2 ). The vector of the log-returns can be written Y(T 2 )' = ( Y(T 1 )' ; Y(T 2 /T 1 )' ) : the old observations Y(T 1 ) completed with the new ones Y(T 2 /T 1 ). The matrix D(T 2 ) is a four blocks matrix:

In the upper left hand side corner we have the old matrix D(T 1 ). The lower part of D(T 2 ) is associated to the new repeat-sales. D 1 (T 2 /T 1 ) is for the transactions realised before T 1 (only purchase in that case) and D 2 (T 2 /T 1 ) for the transactions realised after T 1 (purchase and resale). The new data are of two types : purchase before T 1 and resale after T 1 , or purchase and resale after T 1 . For the first case, the -1 is registered in D 1 (T 2 /T 1 ) and the +1 in D 2 (T 2 /T 1 ), whereas both are in D 2 (T 2 /T 1 ) for the second. We denote ∆(T 2 ) = (D(T 1 )' ; D 1 (T 2 /T 1 )')' the left part of the matrix and F(T 2 ) = ( 0' ; D 2 (T 2 /T 1 )')' its right part. The vector LInd(T 2 ) gives the logarithms of the index values for the second estimation. It can be separated in two pieces, the first one gives the levels of the index on [0,T 1 ] and the second on ]T 1 ,T 2 ] : LInd(T 2 )' = (LInd 1 (T 2 )' ; LInd 2 (T 2 )'). The Clapp and Giaccotto's formula establishes the link between the vectors LInd(T 1 ) and LInd 1 (T 2 ), which both give the index values on the interval [0,T 1 ], but using only the information embedded in Y(T 1 ) for LInd(T 1 ) while LInd 1 (T 2 ) uses the completed dataset Y(T 2 ). This formula requires an auxiliary regression Y(T 2 /T 1 ) = D 1 (T 2 /T 1 )AUX + ε'. But, even if it looks like to the previous regressions, "AUX is not an index of any kind. It's just the vector of coefficients in the artificial regression of Y(T 2 /T 1 ) on D 1 (T 2 /T 1 )" - [START_REF] Clapp | Revisions in repeat-sales price indexes: Here today, gone tomorrow?[END_REF]. We also have to introduce a matrix Ω, quite hard to interpret: Ω = [ D(T 1 )' D(T 1 ) + D 1 (T 2 /T 1 )' D 1 (T 2 /T 1 ) ] -1 D(T 1 )' D(T 1 ). With all these elements the relation for the reversibility is:

LInd 1 (T 2 ) = Ω LInd(T 1 ) + (I-Ω) AUX + [∆(T 2 )'∆(T 2 )] -1 ∆(T 2 )'F(T 2 ) LInd 2 (T 2 ) D(T 1 ) 0 D 1 (T 2 /T 1 ) D 2 (T 2 /T 1 ) D(T 2 ) =

The theoretical solution

An informational reformulation of the RSI

In this paragraph we summarize briefly the theoretical framework established in [START_REF] Simon | An informational reformulation of the repeat-sales index[END_REF].

The classical estimation of the repeat-sales index

In the repeat-sales approach, the price of a property k at time t is decomposed in three parts:

Ln(p k,t ) = ln(Index t ) + G k,t + N k,t
Index t is the true index value, G k,t is a Gaussian random walk representing the asset's own trend and N k,t is a white noise associated to the market imperfections. If we denote Rate = (rate 0 , rate 1 , …, rate T-1 )' the vector of the instantaneous continuous rates for each elementary time interval [t,t+1], we have Index t = exp(rate 0 + rate 1 + … + rate t-1 ), or equivalently rate t = ln(Index t+1 /Index t ). For a repeat-sale we can write at the purchase time t i : Ln(p k,i ) = ln(Index i ) + G k,I + N k,i and at the resale time t j : Ln(p k,j ) = ln(Index j ) + G k,j + N k,j . Thus, subtracting, we get Ln(p k,j/ p k,i ) = Ln(Index j /Index i ) + (G k,j -G k,i ) + (N k,j -N k,i ). The return rate realised for the property k is equal to the index return rate during the same period, plus the random walk and the white noise variations. As each repeat-sales give a relation of that nature, we can express them under a matrix form Y = D*LIndex + ε. Here, Y is the column vector of the log return rates realised in the estimation dataset and LIndex = ( ln(Index 1 ), … , ln(Index T ) )'. ε is the error term and D is a non singular matrix 5 . Moreover, if we remark that there exists an invertible matrix 6 A, such that LIndex = A Rate, we can also write 7 Y = (DA) (A -1 LIndex) + ε = (DA) Rate + ε. In the estimation process, the true values Index and Rate will be replaced with their estimators, respectively denoted Ind = (Ind 1 ,… , Ind T )' and R = (r 0 , r 1 , …, r T-1 )'.

The usual estimation of Y = D*LIndex + ε or Y = (DA) Rate + ε is carried out in three steps because of the heteroscedasticity of ε. Indeed, the specification of the error term leads to the relation Var(ε k ) = 2σ N ²+ σ G ²(j-i) in which the values σ N and σ G are the volatilities associated with G k,t and N k,t , and j-i is the holding period for the k th repeat sales. Thus, the first step consists in running an OLS that produces a residuals series. These residuals are then regressed on a constant and on the length of the holding period to estimate σ N , σ G and the variancecovariance matrix 8 of ε denoted Σ. Finally the last step is an application of the generalised least squares procedure with the estimated matrix Σ. This approach is the traditional one. In [START_REF] Simon | An informational reformulation of the repeat-sales index[END_REF] we established that it was equivalent to an algorithmic decomposition (cf.

Figure 3), where the informational framework becomes explicit. The next paragraphs present briefly the mechanism of this algorithm.

Notations, basic concepts and decomposition of the RSI

Time of noise equality

The variance of the residual ε k measures the quality of the approximation Ln(p k,j/ p k,i ) ≈

Ln(Ind j /Ind i ) for the k th repeat-sales . This quantity 2σ N ² + σ G ²(j-i) can be interpreted as a noise measure for each data. As a repeat-sales is compound of two transactions (a purchase and a resale), the first noise source N k,t appears twice with 2σ N ². The contribution of the second source G k,t depends on the time elapsed between these two transactions : σ G ²(j-i).

Consequently, as time goes by, the above approximation becomes less and less reliable. To make the interpretation easier it is useful to modify slightly the expression of the total noise, 7 The basic rules of linear algebra imply that the matrix DA gets as many lines as the number of repeat sales in the sample, and that the columns correspond to the elementary time intervals. In each line of DA, if the purchase occurs at t i and the resale at t j , we have ( 0 … 0 1(t i ) 1 … 1(t j-1 ) 0 … 0). Therefore, the relation Y = (DA) Rate + ε simply means that Log(return) = rate i + … + rate j-1 + ε 8 ∑ is a diagonal matrix with a dimension equal to the size of the repeat sales sample.

factorising by

σ G ² : 2σ N ²+σ G ²(j-i) = σ G ²[(2σ N ²/σ G ²)+(j-i)] = σ G ²[Θ+(j-i)]. What does Θ = 2σ N ²/σ G ² represent?
The first noise source provides a constant intensity (2σ N ²) whereas the size of the second is time-varying (σ G ²(j-i)). For a short holding period the first one is louder, but as this one is constant and the second is increasing regularly with the length of the holding period, we can find a time where the two sources will reach the same levels. Thereafter, the Gaussian noise G k,t will exceed the white noise. This time is the solution of the equation:

2σ N ² = σ G ² * time time = 2σ N ²/ σ G ² = Θ.
For that reason, Θ will be called the "time of noise equality". In the below formula the function G(x) = x/(x+Θ) will sometimes appear. For an holding period j-i we have G(

j -i ) = (j-i)/(Θ+(j-i)) = σ G ²(j-i)/[2σ N ²+σ G ²(j-i)]. Actually, G(j-i)
will represent the proportion of the time-varying noise in the total noise; these numbers will be used subsequently as a system of weights.

Quantity of information delivered by a repeat-sale

The theoretical reformulation developed in this article brings to the fore the concept of information. As Θ+(j-i) is a noise measure, its inverse can be interpreted as an information measure. Indeed, if the noise is growing, that is if the approximation Ln(p k,j/ p k,i ) ≈

Ln(Ind j /Ind i ) is becoming less reliable, the inverse of Θ + ( j -i ) is decreasing. Consequently, (Θ+(j-i)) -1 is a direct measure9 (for a repeat-sale with a purchase at t i and a resale at t j ) of the quality of the approximation or, equivalently, of the quantity of information delivered. Within the estimation process, the smaller weights associated to the long holding periods make these observations less contributive to the index values.

Subsets and algorithmic decomposition of the RSI

The set of repeat-sales with a purchase at t i and a resale at t j will be denoted by C(i,j). For a time interval [t',t], an observation is relevant only if its holding period includes [t',t] ; in other words, if the purchase occurs at t i ≤ t' and the resale at t j ≥ t. This sub-sample will be denoted Spl [t',t] . For an elementary time-interval [t,t+1], we will also used the simplified notation Spl [t,t+1] = Spl t . If we organize the dataset in an triangular upper table, the sub-set Spl [t',t] will correspond to the cells indicated in Table 1. From the optimization problem associated to the general least squares procedure, we demonstrated in [START_REF] Simon | An informational reformulation of the repeat-sales index[END_REF] that the repeat-sales index estimation could be realised using the algorithmic decomposition presented in Figure 3. The left-hand side is related to the informational concepts (for example the matrix Î), whereas the right-hand side is associated to the price measures (for example the mean of the mean rates ρ t ). The final values of the index come from the confrontation of these two parts.

The real distribution and its informational equivalent

The time is discretized from 0 to T (the present), and divided in T sub-intervals.

0 1 2 … t t + 1 … T -1 T
We assume that the transactions occur only at these moments, and not between two dates (the step can be for example a month or a quarter, depending on the data quality). Each observations give a time couple (t i ;t j ) with 0 ≤ t i < t j ≤ T, thus we have T*(T+1) possibilities

for the holding periods. The number of elements in C(i,j) is n i,j , and we denote N = ∑ i<j n ij the total number of the repeat-sales in the dataset. Table 2a is a representation of the real distribution of the {n ij }. As each element of C(i,j) provides a quantity of information equal to

(Θ+(j-i)) -1 , the total informational contribution of the n i,j observations of C(i,j) is n i,j (Θ+(j-i)) -1 = n i,j /( Θ + ( j -i )) = L i, j .
Therefore, from the real distribution {n i,j } we get directly the informational distribution {L i,j } (cf. Table 2b), just dividing the n i,j by Θ+(j-i). The total quantity of information embedded in the dataset is then I = ∑ i<j L ij .

Averages for the noise proportions, the periods and the frequencies

The number of repeat-sales included in Spl t is n

t = ∑ i ≤ t < j n i,j .
For an element of C(i,j), the length of the holding period is j -i. With the function G, we can define the G-mean 10 ζ t of these lengths in Spl t by

∑ i ≤ t < j ∑ k' G(j-i) = n t G(ζ t ).
The first sum enumerates all the classes C(i,j) that belong to Spl t , the second all the elements in each of these classes. Moreover, as G(j-i) measures the proportion of the time varying-noise G k,t in the total noise for a repeatsales of C(i,j), the quantity G(ζ t ) can also be interpreted as the mean proportion of this Gaussian noise in the global one, for the whole sub-sample Spl t . In the same spirit, we define the arithmetic average F t of the holding frequencies 1/(j-i), weighted by the G(j-i), in Spl t :

F t = ( n t G(ζ t ) ) -1 ∑ i ≤ t < j ∑ k' G(j-i)*( 1 /(j-i) ) = I t / (n t G(ζ t )) . Its inverse τ t = (F t ) -1
is then the harmonic average 11 of the holding periods j-i, weighted by the G(j-i), in Spl t . If at first sight the two averages ζ t and τ t can appear as two different concepts, in fact it is nothing of the sort.

We always have, for each sub-sample Spl t , ζ t = τ t .

Two matrixes

The matrix η is a diagonal one, its T diagonal coefficients are:

n 0 G(ζ 0 ) , … , n T-1 G(ζ
[t',t+1] = ∑ i ≤ t' ≤ t < j
L i,j (for an interval [t,t+1] we 10 We recall here that the concept of average is a very general one. If a function G is strictly increasing or decreasing the G-mean of the numbers {x 1 , x 2 , … , x n }, weighted by the (α 1 , α 2 , … , α n ), is the number X such that:

αG(X) = α 1 G(x 1 ) + α 2 G(x 2 ) +…+ α n G(x n ) with α = ∑ i=1,...,n α i .
An arithmetic mean corresponds to G(x) = x, a geometric one to G(x)= ln(x) and the harmonic average to G

(x) = 1/x 11 We have ( n t G(ζ t ) ) / τ t = ∑ i ≤ t < j ∑ k' G(j-i) * ( 1/(j-i) ) = I t simply denote I t for I [t,t+1]
). As exemplified in Table 3 . For all the intervals included in [0,T] we get this way the quantities of information related. These values are arranged in a symmetric matrix Î.

I [0,1] I [0,2] I [0,3] I [0,T] I [0,2] I [1,2] I [1,3] I [1,T] I [0,3] I [1,3] I [2,3] I [2,T] ¦ I [0,T] I [1,T] I [2,T] I [T-1,T]

The mean prices

Within each repeat-sales class C(i,j), we calculate the geometric and equally weighted averages of the purchase prices h p

(i,j) = (Π k' p k',i ) 1/n i,j
and of the resale prices h f (i,j) = (Π k' p k',j ) 1/n i,j . For an elementary time-interval [t,t+1], the relevant classes C(i,j) are the ones that satisfy to the inequalities i ≤ t < j. With these classes, we calculate the geometric average H p (t) of the h p (i,j)

, weighted by the corresponding L i,j (the total mass of the weights is

I t = ∑ i≤t< j L i,j ): H p (t) = ( Π i ≤ t < j (h p (i,j) ) L i,j ) 1/I t = ( Π i ≤ t < j (Π k' p k',i ) 1 /( Θ + (j-i))
)

1 / I t
As indicated in the second part, H p (t) is also the geometric mean of the purchase prices, weighted by their informational contribution 1/(Θ+(j-i)), for the investors who were owning real estate during at least [t,t+1]. Similarly, we also define the mean resale price H f (t):

H f (t) = ( Π i ≤ t < j (h f (i,j) ) L i,j ) 1/I t = ( Π i ≤ t < j (Π k' p k',j ) 1 /(Θ + ( j -i ))
)

1 / I t Î =
As we can see, H p (t) can be interpreted as a mean purchase price weighted by the informational activity, buy-side, of the market. The interpretation is the same for H f (t), with the informational activity of the market, sell-side.

The mean of the mean rates

For a given repeat-sales k' in C(i,j), with a purchase price p k',i and a resale price p k',j , the mean continuous rate realised during its holding period j-i is r k' (i,j)

= ln(p k',j /p k',i ) / (j-i). In the subset Spl t , we calculate the arithmetic mean of these mean rates r k' (i,j)

, weighted 12 by the G(j-i) :

ρ t = (n t G(ζ t )) -1 ∑ i≤t<j ∑ k' G(j-i)r k' (i,j)
. This value is a measure of the mean profitability of the investment for the people who were owning real estate during [t,t+1], independently of the length of the holding period. The weights in this average depend on the informational contribution of each data. We demonstrate in [START_REF] Simon | An informational reformulation of the repeat-sales index[END_REF] that ρ t can also be written, in a simpler way, with the following formula: = ln(p k',j /p k',i ) / (j-i), with the harmonic mean of the holding periods τ t , the mean purchase price H p (t) and the mean resale price H f (t).

ρ t = ( 1/ τ t )
All these averages are weighted by the informational activity of the market. We denote the vector of these mean rates P = (ρ 0 , ρ 1 , …, ρ T-1 ).

The index and the relation Î R = η P

The global estimation of the RSI can now be realised just solving the equation:

ÎR = ηP R = (Î -1
η) P. The single unknown is the vector R = (r 0 , r 1 , …, r T-1 )' of the monoperiodic growth rates of the index. The three others components of this equation (Î, η and P) are calculated directly from the dataset. The main advantages of this formalism are its interpretability and its flexibility: the matrix Î gives us the informational structure of the dataset, the matrix η counts the relevant repeat-sales for each time interval [t,t+1] and the vector P provides the levels of profitability of the investment, for the people who are owning real estate at the different dates.

The reversibility formulas

Notations

We are now going to study how we can deal with the reversibility phenomenon using the above reformulation of the RSI. In all this section we assume that the initial horizon T 1 is extended to T 2 > T 1 . Table 4 illustrates this extension for the informational distribution. We will keep the same notations, however the horizon will be added as a parameter; for example H p (t) will be denoted H p (t;T 1 ) or H p (t;T 2 ) according to the associated horizon of the estimation. There exists two kinds of new repeat sales: the ones with a purchase before T 1 and a resale after T 1 (i < T 1 < j ≤ T 2 ), delimited by the continuous lines in Table 4, and those with a purchase and a resale realised between T 1 and T 2 (T 1 ≤ i < j ≤ T 2 ), delimited by the dotted lines. In Table 4 the relevant repeat sales for [t,t+1], if the horizon is T 1 , are represented with a light grey. And if the horizon becomes T 2 , the dark grey cells should also be included in this set.

Reversibility for I t , n t , Hp(t) and H f (t)

For an interval [t,t+1], t < T 1 , the quantities of relevant information are I

t (T 1 ) = ∑ i ≤ t < j ≤ T 1 L i,j
for the first horizon and

I t (T 2 ) = ∑ i ≤ t < j ≤ T 2 L i, j = I t (T 1 ) + ∑ i ≤ t < T 1 < j ≤ T 2
L i,j for the second. The sum with i ≤ t < T 1 < j ≤ T 2 corresponds to the additional information (dark grey ). In the following, the notation T 2 \T 1 will refer to the dataset of the new repeat sales that appear when the horizon is extended.

Reversibility for the mean prices H p (t) and H f (t)

We first calculate H p (t) with the purchase prices for the two horizons :

[H p (t,T 1 )] I t (T 1 ) = Π i ≤ t < j ≤T 1 ( Π k' p k',i ) 1/ (Θ + (j -i )) [H p (t,T 2 )] I t (T 2 ) = Π i ≤ t < j ≤T 2 ( Π k' p k',i ) 1/ (Θ + ( j -i ))
Therefore we have : [H p (t,T 2 )]

I t (T 2 ) = [H p (t,T 1 )] I t (T 1 ) Π i ≤ t < T 1 < j ≤ T 2 ( Π k' p k',i ) 1/ (Θ + (j -i ))
Introducing h p (i,j)

, this product becomes :

Π i ≤ t < T 1 < j ≤ T 2 (Π k' p k',i ) 1/(Θ+(j-i)) = Π i ≤ t < T 1 < j ≤ T 2 (h p (i,j) ) L i,j
The total mass of these weights L i,j is equal to I t (T 2 \T 1 ). We denote this geometric average by:

[H p (t, T 2 \ T 1 )]

I t ( T 2 \ T 1 ) = Π i ≤ t < T 1 < j ≤ T 2 ( h p (i,j) ) L i,j = Π i ≤ t < T 1 < j ≤ T 2 ( Π k' p k',i ) 1/ (Θ + (j -i ))
For

I t ( T 2 \T 1 ) = Π i ≤ t < T 1 < j ≤ T 2 (h f (i,j) ) L i,j = Π i ≤ t < T 1 < j ≤ T 2 ( Π k' p k',j ) 1/(Θ+(j-i)) we have: [H f (t,T 2 )] I t (T 2 ) = [H f (t,T 1 )] I t (T 1 ) [H f (t, T 2 \ T 1 )] I t ( T 2 \ T 1 )

Reversibility for τ t

We study in this paragraph the link between the mean holding periods τ t (T 1 ) and τ t (T 2 ). We have:

I t (T 2 \T 1 ) = ∑ i ≤ t < T 1 < j ≤ T 2 L i,j = ∑ i ≤ t < T 1 < j ≤ T 2 ∑ k' G( j -i ) * ( 1 / ( j -i )). Thus I t (T 2 \T 1 ) is
almost the arithmetic average of the 1/(j-i) weighted by the G(j-i). It just lacks in this formula the total mass of the weights, that is ∑ For t < T 1 we have 14 :

i ≤ t < T 1 < j ≤ T 2 ∑ k' G(j-i) = n t (T 2 \T 1 ) G(ζ t (T 2 \T 1 )),
[n t (T 2 )G(ζ t (T 2 ))]/τ t (T 2 ) = [n t (T 1 )G(ζ t (T 1 ))]/τ t (T 1 ) + [n t (T 2 \T 1 )G(ζ t (T 2 \T 1 ))]/
ρ t (T 2 ) = [(1/τ t (T 1 ))*(lnH f (t,T 1 )-lnH p (t,T 1 ))]*[ (I t (T 1 )τ t (T 1 )) /(I t (T 2 )τ t (T 2 ))] + [(1/ τ t (T 2 \T 1 )) * (ln H f (t, T 2 \T 1 ) -ln H p (t, T 2 \T 1 ))]*[(I t (T 2 \T 1 )τ t (T 2 \T 1 )) / (I t (T 2 )τ t (T 2 ))]
In the first square brackets we recognize ρ t (T 1 ). Moreover, we can easily prove that the third brackets are also equal to [n

t (T 2 \T 1 )G(ζ t (T 2 \T 1 ))] -1 ∑ i ≤ t < T 1 < j ≤ T 2 ∑ k' G(j-i)r k' (i,j)
. This expression 13 Here also, the quantity G(ζ t (T 2 \T 1 )) can be interpreted as the mean proportion of the Gaussian noise in the whole noise, for the new data.

14 ρ t (T 2 ) = [I t (T 2 )/( n t (T 2 )G(ζ t (T 2 )))] * ln[H f (t,T 2 )/H p (t,T 2 )] = [ I t (T 2 )/( n t (T 2 )G(ζ t (T 2 )) ) ] * [lnH f (t,T 2 ) -lnH p (t,T 2 )] = [ I t (T 1 )lnH f (t,T 1 )+I t (T 2 \T 1 )lnH f (t,T 2 \T 1 ) ]/[I t (T 2 )τ t (T 2 )] -[ I t (T 1 )lnH p (t,T 1 )+I t (T 2 \T 1 )lnH p (t,T 2 \T 1 ) ]/[I t (T 2 )τ t (T 2 )]
is simply the weighted mean of the mean rates r k' (i,j) for the new repeat sales, and of course it will be denoted ρ t (T 2 \T 1 ). Thus, the reversibility formula for ρ t , t < T 1 , is:

ρ t (T 2 ) = [I t (T 1 ) / I t (T 2 )][τ t (T 1 ) / τ t (T 2 )]ρ t (T 1 ) + [I t (T 2 \T 1 ) / I t (T 2 )][τ t (T 2 \T 1 ) / τ t (T 2 )]ρ t (T 2 \T 1 )
The quantity I

Vectorial formula

The above formulas are valid for t < T 1 . However the expressions that define I t (T 2 \T 1 ), τ t (T 2 \T 1 ), ζ t (T 2 \T 1 ), n t ( T 2 \ T 1 ) and ρ t (T 2 \T 1 ) can be generalized for t ≥ T 1 . Indeed, for these quantities the sums concern the classes C (i,j) such that i ≤ t < T 1 < j ≤ T 2 , that is the new repeat-sales relevant for [t,t+1], with t < T 1 . Now, if we choose t ≥ T 1 , the relevant cells will be the ones satisfying 15 to i ≤ t < j ≤ T 2 . But, what we get this way is not new ; it is just I t (T 2 ),

τ t (T 2 ), ζ t (T 2 ), n t ( T 2 ) and ρ t (T 2 ). For instance I t (T 2 \T 1 ) = ∑ i ≤ t < T 1 < j ≤ T 2 L i,j gives for t ≥ T 1 : ∑ i ≤ t < j ≤ T 2 L i,j = I t ( T 2 ).
We can now write the formulas in a more synthetic manner. We gather the values ρ t (T 2 ), for 0 ≤ t < T 2 , in a T 2 -vector P(T 2 ) and the values ρ t (T 1 ), for 0 ≤ t < T 1 , in a T 1 -vector P(T 1 ). From the vector P(T 1 ), a T 2 -vector is created adding at its end T 2 -T 1 zeros; it will be noted in italics P(T 1 ). The numbers ρ t (T 2 \T 1 ) are gathered in a T 2 -vector P(T 2 \T 1 ), and actually its last T 2 -T 1 coordinates are simply equal to ρ t (T 2 ). Thus, for t < T 1 we have:

τ t (T 2 ) I t (T 2 ) ρ t (T 2 ) = I t (T 1 ) τ t (T 1 ) ρ t (T 1 ) + I t (T 2 \T 1 ) τ t (T 2 \T 1 ) ρ t (T 2 \T 1 )
15 i ≤ T 1 ≤ t < j ≤ T 2 is not correct because it would exclude the repeat-sales with a purchase at i, such that T 1 < i ≤ t. As these couples belong to the new data and are perfectly relevant for [t,t+1], we cannot forget it.

n t (T 2 ) G(ζ t (T 2 )) ρ t (T 2 ) = n t (T 1 ) G(ζ t (T 1 )) ρ t (T 1 ) + n t (T 2 \T 1 ) G(ζ t (T 2 \T 1 )) ρ t (T 2 \T 1 )
And for t ≥ T 1 : n

t (T 2 ) G(ζ t (T 2 )) ρ t (T 2 ) = n t (T 2 \T 1 ) G(ζ t (T 2 \T 1 )) ρ t (T 2 \T 1 )
The diagonal matrix η(T 1 ) can be injected in a T 2 -matrix, completing it with zeros, and denoted in italics by η(T 1 ). η(T 2 ) is the usual T 2 -diagonal matrix and we denote η(T 2 \T 1 ) the T 2 -diagonal matrix built with n 0 (T 2 \T 1 ) G(ζ 0 (T 2 \T 1 ),…,n T 2 -1 (T 2 \T 1 ) G(ζ T 2 -1 (T 2 \T 1 )). We can now write simultaneously these two kinds of equations (for t < T 1 and for t ≥ T 1 ):

η(T 2 ) P(T 2 ) = η(T 1 ) P(T 1 ) + η(T 2 \T 1 ) P(T 2 \T 1 )

Reversibility for the informational matrix Î

For an interval [t i , t j ] the relevant information is denoted I

[t i , t i ] (T 1 ) or I [t i , t i ]
(T 2 ), according to the horizon. The associated informational matrixes are Î(T 1 ) and Î(T 2 ), dimension T 1 and T 2 respectively. A third matrix Î(T 2 \T 1 ), dimension T 2 , is the link between Î(T 1 ) and Î(T 2 ). Its values are computed only with the new L i,j (cf. Table 5), and for each interval [t i , t j ] C [0,T 2 ] they represent the additional quantity of information. Î(T 2 \T 1 ) can be written with three submatrix Î a (T 2 \T 1 ), Î b (T 2 \T 1 ) and Î c (T 2 \T 1 ) (cf. Figure 4). Î a (T 2 \T 1 ) and Î c (T 2 \T 1 ) are two square matrixes of dimension T 1 and T 2 -T 1 , whereas Î b (T 2 \T 1 ) is a T 1 *(T 2 -T 1 ) matrix and its transpose t Î b (T 2 \T 1 ) a (T 2 -T 1 )*T 1 matrix. Î a (T 2 \T 1 ) is symmetric and its diagonal elements correspond to the first column of Î b (T 2 \T 1 ) ; from one of these diagonal elements, the matrix values are the same on the right and below. The matrixes Î b (T 2 \T 1 ) and Î c (T 2 \T 1 ) are simply extracted from Î(T 2 ). Î a and Î c respectively represent the additional information for an interval [t i ,t j ] C [0,T 1 ] and [t i ,t j ] C [T 1 ,T 2 ], whereas Î b is for the intervals [t i ,t j ] C [0,T 2 ] with T 1 C ]t i ,t j [; that is the ones which straddle the first horizon T 1 . If we inject now the matrix Î(T 1 ) in a T 2 *T 2 matrix, completed it with zeros and denoted in italics Î(T 1 ), the reversibility formula for the informational matrix is simply: Î(T 2 ) = Î(T 1 ) + Î(T 2 \T 1 ).

Reversibility for the RSI

The last step consists in establishing the reversibility formula for the index. For an horizon T 1 and t < T 1 , the building blocks I = η(T 1 )P(T 1 ) + η(T 2 \T 1 )P(T 2 \T 1 ), we finally get a very simple reversibility formula for the repeat-sales index : Î(T 2 ) R(T 2 ) = Î(T 1 ) R(T 1 ) + Î(T 2 \T 1 ) R(T 2 \T 1 )

Comments

The above formalism allows summing up the logic of the reversibility phenomenon as follows. First we estimate the RSI with the old data on [0,T 1 ]; we get an informational matrix Î(T 1 ) and a vector R(T 1 ). Then, only with the new data T 2 \T 1 , we estimate the index on [0,T 2 ] ;

it gives Î(T 2 \T 1 ) and R(T 2 \T 1 ). At last, using the whole dataset (old data + new data), we calculate the RSI on [0,T 2 ] with Î(T 2 ) and R(T 2 ). What is expressed in the reversibility formula is simply that the quantity Î R is additive when the horizon is extended from T 1 to T 2 .

In their article of 1999, Clapp and Giaccotto proposed a formula to deal with this problem (cf.

paragraph 2.3). How should we understand these two different approaches? At the theoretical 16 In order to have a T 2 -vector R(T 1 ), we will sometimes complete the T 1 -vector R(T 1 ) with T 2 -T 1 final zeros. 17 We saw above that I t , τ t , ζ t , n t and ρ t are equal for T 2 and T 2 \T 1 when t ≥ T 1 . Unfortunately, for the repeat-sales index, this relation is not true. level they are of course equivalent because they are measuring the same phenomenon. But in an empirical point of view, things are quite different. The Clapp and Giaccotto's formula is rather complex and its financial interpretation is not obvious. For instance, what does the matrix Ω represent? Moreover, as it is pointed in this article of 1999, the auxiliary regression is just an abstract estimation which does not correspond to an index of any kind. On the other hand, the formula Î(T 2 )R(T 2 ) = Î(T 1 )R(T 1 ) + Î(T 2 \T 1 )R(T 2 \T 1 ) is simple, easy to handle and easy to interpret. The basic concepts are simply the informational matrix and the vector of the monoperiodic growth rates of the index; these two notions are strongly intuitive. What is more, the equivalent of the auxiliary regression AUX, namely R(T 2 \T 1 ), can be interpreted as the RSI for the interval [0,T 2 ] that we get if we run the estimation only with the new dataset T 2 \T 1 . Thus, this new formula seems to be very interesting for the empirical applications.

More generally, our relation could be understood as a kind of « equation of energy preservation » for the datasets. Indeed, if we consider that the product ÎR measures the quantity of energy embedded in a dataset, the reversibility formula simply asserts that:

This idea of energy delivered by a sample also allows interpreting the relation ÎR = η P. The left hand side can be understood as the energy of the informational system of the index values, whereas the right hand side can be analysed as the energy provided by the gross (real) dataset

The empirical quantification methodology

Leaning on these theoretical results we are now going to implement a methodology which allows estimating the size of the potential variations due to the reversibility phenomenon.

The exponential benchmark

In order to simulate the behaviour of the repeat-sales between T 1 and T 2 we introduce a simple model based on an exponential distribution of the resale decision. More precisely, we assume that: 1) the quantities of goods traded on the market at each date are constant and denoted K 2) the buy decisions and the sell decisions are independent between the individuals 3) the length of the holding period follows an exponential distribution, with a parameter λ > 0 (the same for all the owners). This last hypothesis means that, conditionally to a purchase at t=0, the probability of not having sold the house at time t is e -λt . This choice is unrealistic because it implies that the probability of selling the house in the next year is not influenced by the length of the holding period 18 . If we introduce the hazard rate 19 which measures the instantaneous probability of a resale: λ(t) = (1/∆t)*Prob( resale > t+∆t | resale ≥ t), we can demonstrate that the choice of an exponential distribution is equivalent to the choice of a constant hazard rate. In the real world things are of course different. For the standard owner (cf. Figure 5) we can reasonably think that the hazard rate is first low (quick resales are scarce). In a second time, it increases progressively to a stationary level, maybe modified by the economical context (residential time). Then, as time goes by, the possibility of a moving associated to the retirement or even the death of the householder would bring the hazard rate to a higher level (ageing). However, even if our assumption is not entirely realistic, we keep it because it generates a simple model in which the resale decision could be compared to a 18 Prob ( resale > t | resale ≥ t ) = Prob ( resale > s | resale ≥ s ) 19 λ(t) is a classical concept for the survival models, cf. Kalbfleisch, Prentice ( 2002). It appears for example in the econometrical studies for the prepayment and the default options embedded in the mortgages, cf. Deng, Quigley, Van Order (2000).

radioactive disintegration of an atom. The aim of this benchmark does not consist in describing precisely the reality; we just try to modelize a basic behaviour. For an interval [0,T], the benchmark dataset is fully determined if the parameters K and α = e -λ are known.

We demonstrated in [START_REF] Simon | Is there a functional relation between the median index and the repeat-sales index[END_REF] that the number of repeat-sales in an exponential sample is 20 N = K T ( 1 -π ) and the total quantity of information embedded in this dataset is 21 I = K' [ (T+Θ+1) u T -T π]. These two expressions will be useful for the calibration step.

An example

For practical reasons, we are working in this article with artificial samples, randomly generated 22 . However, the methodology can be applied directly to the real datasets, without any difficulties. Figure 6 presents the results of the estimation when T 1 = 40 and T 2 = 45. The green curve gives the index values on [0,40], for the old dataset. The yellow curve gives the index values on [0,45], using only the new data T 2 \T 1 . And the red one is for the completed sample. As for the black curve, it gives the percentage of reversibility (Ind t (45)/Ind t (40) -1) for t = 0,…,40. The sample of the new data T 2 \T 1 is smaller than the two others, thus its curve logically presents a higher volatility. For the majority of the dates the difference between the old index and the completed one is negligible; the black curve is near zero. It is only in the last quarter of the interval [0,40] that the two curves can diverge; the spread reach up to 1% with our simulated data. The direction of the variation is given by the new data. For instance, at t = 34 the index T 2 \T 1 is at 110 whereas the old index is around 104. Consequently, the yellow curve brings the old value ( 104) at a higher level (105). As we can see the reversibility phenomenon presents a strong temporal framework. It appears essentially for the nearest dates. But unfortunately, in an investment point of view, these recent past values are in

20 π = d(T) * ( α / T(1-α) ) d(k) = 1 -α k 21 K' = K (1-α)/α Θ = 2σ N ²/ σ G ² u n = α /(Θ + 1) + α² /(Θ + 2) + α 3 /(Θ + 3) + … + α n /(Θ + n)
22 First we first fix the numbers of transactions for each date, for the whole market. Then, the resale rates for each cohort are randomly generated. The estimation sample is made of the repeat-sales with a resale date before T.

general the most important ones. Therefore, it is really crucial to elaborate a methodology able to indicate the level of reliability of the old index values. In other words we are looking for a kind of confidence interval.

The simulation process

We will use for that purpose a Monte Carlo approach; the simulation algorithm is presented in Figure 7. From a repeat-sales sample on [0,T 1 ], we calculate the associated index with R(T 1 )

and Î(T 1 ). These two quantities are fixed during all the process. The present time is T 1 and we are trying to infer what could be the variations of the index when the estimation will be renwed at T 2 . The first step consists in calibrating the exponential benchmark with the old data on [0,T 1 ]. Precisely, we are looking for the values of the parameters K (constant flow on the market) and α (resale speed) such that the total number of repeat-sales N and the quantity of information I be equal between the real dataset and the benchmark sample 23 .

Mathematically speaking, we can estimate the parameter α working with the quantity I/N which does not depend on K (numerical resolution). When α is known, we calculate K with the equation N = K T ( 1 -π ). Once the benchmark is calibrated, we assume that the arrival of the information on the interval [T 1, T 2 ] will occur according to the same rhythm than previously. We get this way an approximation Î bench (T 2 \T 1 ) for the matrix Î(T 2 \T 1 ) which represents the informational distribution of the new repeat-sales 24 . In the same time, we also get the matrix Î(T 2 ) adding Î(T 1 ) and Î bench (T 2 \T 1 ). After the left-hand side of Figure 7, devoted to the informational matrixes, we now focus on the right-hand side dedicated to the growth rates vectors. R(T 1 ) gives the index evolution on the interval [0,T 1 ]. For the rest of the interval [T 1 ,T 2 ], we complete it in a T 2 -vector R hyp = (R(T 1 ), R hyp (T 1 ;T 2 )) making economical 23 Others choices are possible for this calibration step, according to the economical contexts. 24 When K and α are known we demonstrated in [START_REF] Simon | Is there a functional relation between the median index and the repeat-sales index[END_REF] that L i,j = K'α j -i /(Θ + j -i). We first build the informational distribution of the {L i,j } for the benchmark and for the interval [0,T 2 ]. Then, we just keep the columns between T 1 and T 2 (as in Table 5), which represent the new data for the exponential sample. From this partial table, adding its components, we get the matrix Î bench (T 2 \T 1 ).

hypotheses on the future of the real estate prices. In [START_REF] Simon | An informational reformulation of the repeat-sales index[END_REF] we established that the vector R(T 2 \T 1 ), is a Gaussian one. It is centred on the growth rates of the theoretical index values 25 and its variance-covariance matrix26 is σ G ² Î(T 2 \T 1 ) -1

. Because of its unobservability at T 1 we have to generate it randomly as a Gaussian vector N (R hyp ; σ G ² Î(T 2 \T 1 )

-1

). The theoretical expectation is replaced with the best estimator that we have on [0,T 1 ], that is R(T 1 ), completed with the economical assumption on [T 1 ,T 2 ]. For the second parameter we simply use the benchmark matrix as an approximation. At this stage of the process we have Î(T 1 ), Î(T 2 \T 1 ), Î(T 2 ), R(T 1 ) and R(T 2 \T 1 ). The final step consists in calculating the vector R(T 2 ) with the equation Î(T 1 )R(T 1 ) + Î(T 2 \T 1 )R(T 2 \T 1 ) = Î(T 2 )R(T 2 ). Once R(T 2 ) is known we can calculate the index values Int t (T 2 ) on the interval [0,T 1 ] and we can measure the size of the reversibility phenomenon for this simulation. Running many times this procedure we finally get an empirical distribution for the spreads.

Comments

In the above process the randomness just appears in the generation of the Gaussian vector R(T 2 \T 1 ). For its practical implementation we have to use the Cholesky factorization27 .

However, if we are interesting in deepening the simulation, we could introduce two additional random sources: the vector R hyp (T 1 ;T 2 ) and the couple (K, α). Indeed, in order to estimate the expectation of the vector R(T 2 \T 1 ), we completed the vector R(T 1 ) with the economical assumptions associated to R hyp (T 1 ;T 2 ), foreseeing a scenario for the evolution of the real estate prices on [T 1, T 2 ]. However, as the future is uncertain, it could be reasonable to let these 25 rate t = ln(Index t+1 /Index t )

The second generalisation concerns the couple (K, α). The first variable represents a constant level of liquidity for the market and the second the resale speed. With the calibration step on the interval [0,T 1 ] we found a mean couple (K 0 ,α 0 ). However, on the interval [T 1 ,T 2 ] the market conditions could be slightly different. To take this possibility into account we can randomly choose the parameter K in an interval [K 0 -ε ; K 0 + ε], and α 0 in [α 0 -ε' ; α 0 + ε'].

We could even go further with this methodology, considering that the rhythm of the transactions depends on the economical context and especially on the future real estate prices.

Thus, we should firstly calibrate a proportional hazard model on [0,T 1 ], as the one developed by [START_REF] Cheung | The effects of attributes on the repeat sales pattern of residential property in Hong-Kong[END_REF]. And then, according to the scenario simulated on [T 1 ,T 2 ], the rhythm of the repeat-sales could be deduced.

The theoretical law of reversibility in a simplified context

Working in the simplified context with one random source (paragraph 4.3), we can deepen the mathematical analysis. As previously, a repeat-sales sample ω 0 is generated on the interval [0,T 1 ]. Then the benchmark is calibrated on this dataset, and using the corresponding parameters we get an estimation for the matrix Î(T 2 \T 1 ). The quantities Î(T 1 ), R(T 1 ), Î bench (T 2 \T 1 ), Î(T 2 ) are fixed and we just have one random source, that is R(T 2 \T 1 ). The vector R hyp (T 1 ;T 2 ) of the economical assumptions on [T 1 ,T 2 ] is also constant in all this paragraph.

With the formula R(T 2 ) = [Î(T 2 )] -1 [Î(T 1 )R(T 1 )+Î bench (T 2 \T 1 )R(T 2 \T 1 )] it's easy to demonstrate that the vector R(T 2 ) is Gaussian; we have:

E[R(T 2 )] = R(T 1 ) + [Î(T 2 ) -1 Î bench (T 2 \T 1 )] R hyp (T 1; T 2 ) and V [R(T 2 )] = σ G ² [Î(T 2 ) -1 Î bench (T 2 \T 1 )] [Î(T 2 )] -1
The matrix Î(T 2 \T 1 ) represents the new information, Î(T 2 ) the total information. Consequently the product Î(T 2 ) -1 Î bench (T 2 \T 1 ), which appears in these two formulas, can be interpreted as the (vectorial) proportion of the new information in the total one. The first formula simply asserts that the expectation of R(T 2 ) is equal to the old and constant vector R(T 1 ), plus a quantity which represents the influence of the economical hypotheses R hyp (T 1; T 2 ) on [T 1 ,T 2 ]. This influence of R hyp (T 1; T 2 ) is weighted by [Î(T 2 )] -1 Î(T 2 \T 1 ) ; a relative measure of the informational weight of the new data. As for the variance expression, we have to compare it to the formula V [R(T 2 )] = σ G ² [Î(T 2 )] -1 that we would have to apply if we wanted to run the estimation for the index on [0,T 2 ], directly with the whole dataset, without doing an halfway estimation at T 1 . In a reversibility situation we already know a part of the total sample; the resulting index is thus less volatile. What is expressed with the second formula is simply that the attenuation coefficient for the volatility is nothing else than [Î(T 2 )] -1 Î(T 2 \T 1 ), once more.

Now, if we decide that on [T 1 ,T 2 ] the real estate growth is null, in other words R hyp (T 1; T 2 ) = 0, we can demonstrate 28 the following result:

Reversibility law

For t = 1,…,T 1 the ratio Ind t (T 2 ) / Ind t (T 1 ) is log-normally distributed: LN( 0

; v(t)) v(t) is the t th diagonal element of the matrix 29 σ G ²A(T 2 )[[Î(T 2 )] -1 Î(T 2 \T 1 )] [ Î(T 2 )] -1 [A(T 2 )]'
Thus, the reversibility percentage 30 for the date t is a random variable that we can write 100*(Y t -1), with Y t ~ LN(0; v(t)). Figure 8 represents the theoretical deciles, anticipated at T 1 , using the sample ω 0 on [0,T 1 ]. The black curve gives the observed reversibility for this specific sample when the horizon is extended from T 1 to T 2 . As we can see, the theoretical curves are a good approximation of the empirical ones. The size of the potential revisions is small and approximatively constant for the left side of the interval. But for its right side, things are different. As we go closer to T 1 the fluctuations become more and more important, potentially, as testified by the divergence of the theoretical curves in Figure 8. With the 28 For that purpose we just have to use the relations LInd(T 2 ) = A(T 2 ) R(T 2 ) and E[R(T 2 )] = R(T 1 ) 29 The matrix A(T 2 ) is square and its dimension is T 2 . It is composed of 1 on its diagonal and below, 0 elsewhere. 30 100*(Ind t (T 2 ) / Ind t (T 1 ) -1) methodology developed in this paragraph, it now becomes possible to anticipate and to quantify the reversibility effects in a very reliable way.

Conclusion

By means of an informational reformulation of the RSI framework we established first an intuitive and easy to handle formula for the reversibility phenomenon. Then, using an exponential benchmark for the resale decision and Monte-Carlo simulations, we developed a methodology to quantify the size of the potential revisions. In this way we answered to the problem 31 mentioned in Clapham et al. (2005) for the repeat-sales index. For the moment, as we do not have such a similar technique for the hedonic indexes, we cannot assert that the RSI is a bad underlying for the future contracts. Indeed, if its fluctuations are probably higher they are nevertheless predictable, contrary to the hedonic approach. Now, if we want to go further in the derivatives study, the next step would consist in choosing a stochastic dynamic for the RSI in order to price the contingent claims. Unfortunately things are rather complex because of the reversibility. If we consider the basic assumption related to the concept of market efficiency for the stochastic processes in finance, that is their Markovian 32 behaviour, a problem occurs. Is it really possible to describe the dynamic of the RSI with a single Markovian process? The answer is no. We can understand heuristically the problem just rewriting the reversibility formula Î(T 2 )R(T 2 ) -Î(T 1 )R(T 1 ) = Î(T 2 \T 1 )R(T 2 \T 1 ). The left handside measures an increment between the present T 1 and the future T 2 . If the Markovian 31 "If a futures market requires index stability, it would be useful to know how often revision -either period-byperiod or cumulative -exceeds some level. Say, for example, that futures markets could tolerate 0.5 percent revision in any one quarter and 2 percent cumulative revision to the initial estimate -how often do the four indexes violate these criteria?" 32 A process is said Markovian if its future depends on its past only through its present. In others words, the path followed by the process to arrive at the level X s , at the date s, will not influence the probability of realisation of its future X t (t > s). Financially, this mathematical assumption is one of the formulations for the concept of market efficiency. The present value incorporates all the past information; it is useless to study the past in order to get a better level for X s . The market already integrated all the available and relevant information with the fixing of X s .

… t t + 1 … T -2 T -1 T 0 n 0,1 n 0,2 n 0,3 n 0,t n 0,t+1 n 0,T-2 n 0,T-1 n 0,T 1 n 1,2 n 1,3 n 1,t n 1,t+1 n 1,T-2 n 1,T-1 n 1,T 2 n 2,3 n 2,t n 2,t+1 n 2,T-2 n 2,T-1 n 2,T 3 n 3,t n 3,t+1 n 3,T-2 n 3,T-1 n 3,T ¦ t n t,t+1 n t,T-2 n t,T-1 n t,T t + 1 n t+1,T-2 n t+1,T-1 n t+1,T ¦ T -2 n T-2,T-1 n T-2,T T -1 n T-
1 2 3 … t t + 1 … T -2 T -1 T 0 L 0,1 L 0,2 L 0,3 L 0,t L 0,t+1 L 0,T-2 L 0,T-1 L 0,T 1 L 1,2 L 1,3 L 1,t L 1,t+1 L 1,T-2 L 1,T-1 L 1,T 2 L 2,3 L 2,t L 2,t+1 L 2,T-2 L 2,T-1 L 2,T 3 L 3,t L 3,t+1 L 3,T-2 L 3,T-1 L 3,T ¦ t L t,t+1 L t,T-2 L t,T-1 L t,T t + 1 L t+1,T-2 L t+1,T-1 L t+1,T ¦ T -2 L T-2,T-L T-2,T T -1 L T-1,T
1 … t t + 1 … T 1 … T 2 0 L 0,1 … L 0,t L 0,t+1 … L 0,T 1 … L 0,T 2 1 … L 1,t L 1,t+1 … L 1,T 1 … L 1,T 2 ¦ … … … … … … t L t,t+1 … L t,T 1 … L t,T 2 t + 1 … L t+1,T 1 … L t+1,T 2 ¦ … … … T 1 … L T 1 ,T 2 ¦ … T 2
Continuous lines: new repeat sales with a purchase before T 1 and a resale after T 1 (i<T 1 <j≤T 2 ) Dotted lines : new repeat sales with a purchase and a resale between T 1 and T 2 (T 1 ≤ i < j ≤ T 2 )

Relevant repeat sales for [t,t+1] if the horizon is T 1 : light grey.

Relevant repeat sales for [t,t+1] if the horizon is T 2 : light grey + dark grey. Estimation of the volatilities σ N and σ G for the white noise and the random-walk (step 1 and 2 of the Case-Shiller procedure). The time of noise equality is Θ = 2σ N ²/ σ G ² L i,j = n i,j / (Θ + j -i) : Quantity of information delivered by the n i,j repeat-sales of C (i,j). These numbers are also organized in an upper triangular table.

0 1 … T 1 T 1 +1 … T 2 0 0 … 0 L 0,T 1 +1 … L 0,T 2 1 … 0 L 1,T 1 +1 … L 1,T 2 ¦ ¦ ¦ … ¦ T 1 L T 1 ,T 1 +1 … L T 1 ,T 2 T 1 +1 … L T 1 +1 ,T 2 ¦ ¦ T 2
We get the matrix Î from the informational distribution of the {L i,j } summing for each time interval [t,t'] the relevant L i,j , that is the ones whose holding period is including For the subset of the people who were owning real estate during [t,t+1], the mean purchase price H p (t) (the mean resale price H f (t)) is the geometric average of the h p (i,j) (respectively the h f (i,j)

), weighted by the L i,j , for all the relevant repeat-sales classes:

H p (t) = ( Π i ≤ t < j ( h p (i,j) ) L i,j ) The vector R of the monoperiodic growth rates of the index is the solution of the equation: The black curve gives the observed empirical reversibility (at T 2 ) and the coloured ones the theoretical deciles deducted from the reversibility law, just using the information known at T 1 . The two extreme curves are the percentiles at 1% and 99%; the others give the deciles from 10% to 90%. % of reversibility Time

ÎR

  measure the lengths of the holding periods for the old data and for the new ones, relatively to the lengths of the whole sample. If we assume that the average holding periods are all equal, the relation simply becomes: ρ t (T 2 ) = )]ρ t (T 2 \T 1 ).

  system. Here also, we have a kind of equation of preservation:

Figure 1 :Figure 2 :Figure 3 :

 123 Figure1: Reversibility for the Los Angeles County,[START_REF] Clapp | Revisions in repeat-sales price indexes: Here today, gone tomorrow?[END_REF]

  [t,t']. The diagonal elements of the diagonal matrix η are equal to the sums (rows or columns indifferently) of the components of the matrix Î.Dividing the diagonal elements of Î by the diagonal elements of η we obtain directly the mean holding periods τ t .For each repeat-sales class C (i,j), the geometric averages of the purchase prices h

Figure 8 :

 8 Figure 4: Informational matrixes

  with ζ t (T 2 \T 1 ) the G-average of the holding periods for the new repeat sales13 . Therefore, as in the

	basic situation, I	t	(T 2 \T 1 ) / [n t	(T 2 \T 1 ) G( ζ t (T 2 \T 1 ))] is a mean frequency F t	(T 2 \T 1 ), and its
	inverse a mean harmonic holding period τ t	(T 2 \T 1 ), for the new repeat-sales. We can now
	establish the formal link between τ t	(T 1 ) and τ t	(T 2 ) with the relations I	t	(T 2 \T 1 ) =
	[n

t (T 2 \T 1 )G(ζ t (T 2 \T 1 ))] / τ t (T 2 \T 1 ) and I t (T 2 ) = I t (T 1 ) + I t (T 2 \T 1 )

. We get the formula:

  t that we can get if the dataset is restricted only to the new repeat sales, as illustrated in Table5. Thus, it suggests that it is useful to estimate the RSI on the interval [0,T 2 ] just with the sample T 2 \T 1 ; we get this way a T 2 -vector R(T 2 \T 1 )

						t (T 1 ), τ t	(T 1 ), ζ t (T 1 ), n t (T 1 ) and ρ t (T 1 ) give the repeat sales
	index R(T 1 ) 16 . Similarly I	t	(T 2 ), τ t	(T 2 ), ζ t (T 2 ), n t	(T 2 ) and ρ t (T 2 ), calculated for t < T 2 , give the
	repeat sales index R(T 2 ). The link between these two families of intermediate measures is
	known thanks to the quantities I	t	(T 2 \T 1 ), τ t	(T 2 \T 1 ), ζ t (T 2 \T 1 ), n t	(T 2 \T 1 ) and ρ t (T 2 \T 1 ). But if we
	examine precisely their definitions, we can notice that they are corresponding exactly to the
	quantities I	t	, τ t	, ζ t , n

t , ρ 17 . If we now use the general relation ÎR = ηP and the reversibility formula established for the vector P, η(T 2 )P(T 2 )

Table 1 :

 1 Relevant repeat-sales for the time interval[t',t]

							Resale date
		0	…	t'	…	t	…	T
		0
		¦
	Purchase date	t' ¦ t ¦
		T

Table 2b :

 2b Informational distribution for the repeat-sales sample 0

		1,T
	T
	Vertical axis: purchase date	Horizontal axis: resale date

Table 4 :

 4 Informational distribution when the horizon is extended from T 1 to T 2 0

Table 5 :

 5 Informational distribution for the dataset T 2 \T 1

1 / I t H f (t) = ( Π i ≤ t < j ( h f (i,j) ) L i,j ) 1 / I t

  The mean of the mean rates ρ t , realised by the people who were owning real estate during [t,t+1], can be calculated as a return rate with the fictitious prices H p (t) for the purchase H f (t) for the resale, and the fictitious holding period τ t

	ρ t = ( 1 / τ

t ) * ln [ H f (t) / H p (t) ]

Bailey, Muth and Nourse

Case, Shiller

The purchases at t = 0 are not included to avoid a singular matrix in the estimation.

D is a matrix extracted from another matrix D'; the first column has been removed to avoid a singularity in the estimation process. The number of lines of D' is equal to the total number of the repeat-sales in the dataset and its T+1 columns correspond to the different possible times for the trades. In each line -1 indicates the purchase date, +1 the resale date and the rest is completed with zeros.

A is a triangular matrix whose values are equal to 1 on the diagonal and under it, 0 elsewhere.

These measures are relative ones. The matter is the relative sizes and not the absolute levels. They can be defined dividing by a constant in order to standardize the quantities.

This formula is a general one. The variance-covariance matrix of the vector R, whatever be the repeat-sales distribution, is alwaysV (R) = σ G ² Î -1

If Γ is a square matrix of dimension d, symmetric, positive, and with rank r then we can find a matrix B, dimension d x r , rank r such that Γ = B B' (Cholesky factorization) Now, for a vector M of dimension d, and for a square matrix Γ of dimension d, symmetric, positive, rank r with its Cholesky factorization Γ = B B' : If Y ~ N ( 0 , I d ) then M + B Y ~ N (M , Γ)

assumption is satisfied this variation cannot depend on the dates before T 1 . But we know that the right-hand side Î(T 2 \T 1 )R(T 2 \T 1 ), associated to the new data arrived with the time extension, not only brings information on [T 1 ,T 2 ] but also on the interval [0,T 1 ].

Consequently, the RSI do not have a Markovian behaviour. What follows from this result is that we cannot use, at least directly, the usual stochastic dynamics (geometric Brownian motion, Ornstein-Uhlenbeck…) to price a contingent claim. A solution could consist in describing the reversibility risk itself with a dynamic, and then to model the RSI as a noisy asset like in [START_REF] Cheung | The effects of attributes on the repeat sales pattern of residential property in Hong-Kong[END_REF]Childs et al. ( , 2002aChilds et al. ( , 2002b)). Using this approach, we could catch the mechanism of price discovery associated to the reversibility phenomenon. But in spite of everything and even if the technical problems are important, the stakes are real and crucial for the finance industry. It is nothing less than the possibility to price the real estate derivatives written on a RSI. We now conclude this article with two small remarks. In Clapham et al. (2005) we can read: "This suggests that there is a link between the index revision and the sample selectivity of repeat-sales data". This affirmation could clarified because we saw previously that the reversibility phenomenon is inherent and intrinsic to the RSI framework. If the sample selectivity matters it's in a second time, through a specific information content of Î(T 2 \T 1 ) or R(T 2 \T 1 ). We probably cannot reduce the whole phenomenon to a single sample effect. The final remark concerns the title of the article [START_REF] Case | The efficiency of the market for single-family homes[END_REF]: "The efficiency of the market for single-family homes". If the RSI is not Markovian, can we really study the efficiency with this index?


D Pun

S L L Au

S S L Aw

D Q Cao

FORCED VIBRATION ANALYSIS OF A MULTIDEGREE IMPACT VIBRATOR

The dynamics of a multidegree impact vibrator subject to harmonic loading is investigated. The system is represented by a lumped mass model which hits and rebounds from a rigid wall during vibration. The periodic solution to the equations of motion with N forcing cycles and P impacts is formulated. The variational equations and the resulting transition matrix for investigating local stability of the periodic solutions are derived. A two-degree-of-freedom example is analysed, and a variety of motion types are found. Chaotic windows are present between regions of periodic response, and at these boundaries N-P motions are prevalent. Low velocity impacts are evident at exciting frequencies away from the natural frequencies. Two basins of attraction are computed, and the sensitivity to initial conditions is noted. The quality of the N-P motion is discussed from an engineering application perspective.

INTRODUCTION

The vibration of a system subject to displacement restraint is examined. The analysis of impacting systems has been of considerable interest to researchers, for there are many situations in engineering practice where such behaviour occurs. Some examples are cutting tools, jack hammers, compacting equipment, mechanical components which overshoot their designed travel and impact other parts under abnormal conditions. Shaw and Shaw [1] analysed the chaotic behaviour of an oscillator with an inverted impacting pendulum and considered motions with one and two impacts per response cycle. Thompson and Ghaffari [2] observed a cascade of period-doubling bifurcations leading to chaos for an impact oscillator in a study of mooring line dynamics. Shaw and Holmes [3] considered a piecewise unsymmetric oscillator which could be used to represent an impact vibrator. For this case they computed analytically period N orbits with one impact per response cycle and examined their stability. Popplewell et al. [START_REF] Popplewell | [END_REF] formulated the equations for the solution of period N motion with P number of impacts per response cycle (N-P motion) for a single-degree-of-freedom (DOF) impact damper. They presented numerical results for 2-1 to 5-1 motions, and performed experiments to verify their results. They further suggested that the oscillator should be excited at or slightly above its natural frequency for the damper to work best, and that the driving speed should be carefully controlled because response sensitivity is high in this region.

Masri [5] analysed a multidegree system with an impact damper which can be attached to any location on the system. He assumed damper motion of two symmetric impacts per

X l K l M l F l K l-1 C l-1 C l K 2 K 1 F 2 F 1 M 2 M 1 X 2 X 1 B C 2 C 1
load cycle, and obtained good correlations with corresponding experiments. Applications of the analysis to tall buildings demonstrated that the damper can be effective over a wide frequency range. Aidanpaa and Gupta [6] analysed in detail the one-impact motion of a two-DOF impact vibrator. They studied the influences of system parameters and noted behavioral similarities between one-and two-DOF systems. Nigm and Shabana [7] studied the dynamics of a multidegree system with an impact damper placed at one end. The damper is connected to the main system by a viscous damper and a spring. They analysed motion with two unsymmetric impacts per load cycle, and obtained experimental results that generally correlated with computations. Their experiments showed that there are frequency regimes for which steady state solutions apparently do not exist. The motion is such that for some time durations no impact takes place and the vibration amplitude builds up until collisions occur, and the amplitude decreases again until impact no longer materializes, and so on. Moore et al. [8] pointed out in their work on impact dampers in a cryogenic environment that amplitude and frequency of impact per forcing cycle are useful and important characterizations, and noted in their simulation work similar unsteady behaviour as observed in reference [7]. Cao and Shu [9] investigated a multidegree impact vibrator for N-1 motion and obtained robust stability bounds. This paper presents a solution method for periodic N-P motion of a multi-DOF impact vibrator, which has not been analysed before. Such motion is significant because it includes the class of response in which impacts occur occasionally, and may be part of the apparently unsteady behaviour reported in references [7,8]. The transition matrix for the stability analysis of the periodic orbits is obtained through the perturbed motion technique. A two-DOF example is presented, for which regimes of N-P motion and chaotic dynamics are found. The results are discussed with the aid of frequency spectra, phase diagrams, Poincare´maps, and basins of attraction. Finally, some comments with respect to the performance of impact machinery are made.

FORMULATION

equations of motion

Consider a multiple-DOF lumped mass system which experiences impact when the displacement X 1 of mass M 1 reaches the barrier at B, as shown in Figure 1. The following normalized quantities are introduced:

m i = M i /M 1 , k i = K i /K 1 , j i = C i /(2zK 1 M 1 ), v = V/zK 1 /M 1 , t = TzK 1 /M 1 ,
Figure 1. Impact vibrator model.

F 0 = 0 s i = 1 F 2 i 1 1/2 , f i = F i /F 0 , b = BK 1 /F 0 , x i = X i K 1 /F 0 , i = 1, 2, . . . , L, (1)
where M i is the mass, K i the stiffness, C i the damping, F i the force, X i the displacement of the ith DOF, V the excitation frequency, T at time, B the clearance between M 1 and the point of impact, and L the total number of DOF. The equation governing motion between impacts is

Md 2 x/dt 2 + C dx/dt + Kx = f cos (vt + u), (2)
where M, C, K, f, x are matrices or vectors of the normalized quantities, and u is the phase angle of the excitation. The conditions for impact are

x 1 (t 1 ) = b x ˙1(t - i ) q 0, (3)
where t i is the time at the ith impact, and t - i is the moment just before impact, and the dot denotes differentiation with respect to time. The velocities before and after impact are related by the impact rule x˙1(t + i ) = -rx˙1(t - i ), where t + i is the moment just after the ith impact and r is the coefficient of restitution and is an indication of the energy lost during impact. The impact is assumed to be instantaneous as far as the displacement is concerned, hence

x 1 (t + i ) = x 1 (t - i ) = x 1 (t i ) = b.
The impact rule can be expressed as

$ x(t + i ) x˙(t + i ) % = L $ x(t - i ) x˙(t - i ) % , (4)
where L is a diagonal matrix with elements equal to 1, except that the element 1 L + 1 = -r.

The goal here is to seek the periodic solutions of the impacting system. In general each response period may contain N forcing periods and P number of impacts. The first step of the solution process is to cast equation (2) into modal co-ordinates. It is assumed that proportional damping is used, so equation (2) can be reduced to q¨+ Jq˙+ Lq = p cos (vt + u), (5) where

q = [q 1 , q 2 , . . . , q L ] T , x = Uq, (6, 7)
L = diag [v 2 i ], J = diag [2j i v i ], p = U T f = [p 1 , p 2 , . . . , p L ] T , (8, 9)
q is a vector of modal co-ordinates and U is a matrix whose columns are the eigenvectors of the free vibration problem of equation ( 2), normalized with respect to the mass matrix. v i is the ith eigenvalue, j i is the modal damping coefficient, and p is the modal load vector. The solution of equation ( 5) is

q = Ga + Hb + F cos u + c sin u, ( 11
)
where

G = diag [g i ], H = diag [h i ], (12, 13) f = [f 1 , f 2 , . . . , f L ] T , c = [c 1 , c 2 , . . . , c L ] T , (14, 15) v di = v i z1 -j 2 i , ( 16
)
g i = exp (-j i v i t) sin (v di t), h i = exp (-j i v i t) cos (v di t), (17, 18) 2N x 1 b t / 2N / 2 / 1 / f i = a i cos vt + b i sin vt, c i = b i cos vt -a i sin vt, (19, 20) a i = (v 2 i -v 2 )p i (v 2 i -v 2 ) 2 + 2(j i v i v) 2 , b i = 2j i v i vp i (v 2 i -v 2 ) 2 + 2(j i v i v) 2 , (21,22)
and a, b are yet to be determined constant vectors related to the initial conditions. The modal velocity vector can be expressed as

q˙= G a + H b + f cos u + c sin u. ( 23
)
Using equations ( 7), ( 11) and ( 23) to express a, b in terms of the initial conditions, one finally has

$ x x˙% = U DU -1 $ x 0 x˙0 % + U F $ cos u sin u % , (24)
where

U = $ U 0 0 U % , D = $ D 11 D 11 D 12 D 12% , F = $ F F C C % (25-27) f = [f 1 , f 2 , . . . , f L ] T , C = [c 1 , c 2 , . . . , c L ] T , ( 28
, 29)

f i = f i -(h i + j i g i v i /v di )a i -b i g i v/v di , (30)
c i = c i -(h i + j i g i v i /v di )b i + a i g i v/v di , (31)
D 11 = diag [h i + j i g i v i /v di ], D 12 = diag [g i /v di ], (32, 33)
where x 0 , x˙0 are the initial displacement and velocity vectors, respectively. Defining then equation ( 24) can be recast with explicit time dependence, that is

W = U DU -1 , V = U F, (34, 35) y = [x 1 , . . . , x L , x˙1, . . . , x˙L ] T = [x T , x˙T] T , g = [cos u, sin u] T , ( 36
y(t) = W(t)y(0) + V(t)g(u), (38)
where y(0) is a vector of initial displacements and velocities. To find periodic responses of N forcing cycles and P impacts, let the motion at the instant after an impact be used as a reference starting point. The time t is set to zero and the initial conditions are y(0), u, where y 1 (0) = b. Equation (38) governs the motion up to the next impact at time t 1 , which can be solved by considering the equation for the first mass

b = s 2L n = 1 W 1n (t 1 )y n (0) + V 11 (t 1 ) cos u + V 12 (t 1 ) sin u. ( 39
)
The motion after the first impact at t 1 is given by equations ( 4) and (38)

y 1 (0) = LW(t 1 )y(0) + LV(t 1 )g(u), (40)
u 1 = vt 1 + u, ( 41
)
y 1 (t) = W(t)y 1 (0) + V(t)g(u 1 ), (42)
where y 1 (0) is the initial condition after the first impact, y 1 (t) the subsequent motion up to the next impact, t the time which begins from 0 after each impact, and u 1 the corresponding phase angle used to maintain force continuity. Similarly, the initial motion after P impacts

y p (0) = LW(t p )y p -1 (0) + LV(t p )g(u p -1 ). ( 43
)
It is noted that equation (43) contains the impact times t 1 , t 2 , . . . , t p -1 implicitly through y p -1 (0). This relation can be written as where [LW(t i )] and [LV(t j )] are now treated as single matrices, and in the second term [LW(t i )] = I if i q P, and u 0 = u. Using the periodic condition y p (0) = y(0), then y(0) can be solved in terms of t i (i = 1, . . . , P)

y p (0) = t p i = 1 [LW(t i )]y(0) + s p j = 1 t p i = j + 1 [LW(t i )][LV(t j )]g(u j -1 ), (44)
y(0) = - $ t p i = 1 [LW(t i )] -I % -1 $ s p j = 1 t p i = j + 1 [LW(t i )][LV(t j )]g(u j -1 )
% . (45) y(0) can then be substituted into the impact conditions, using the first row of the matrix equation (44), for the kth impact

y k 1 (0) = b = s 2L m = 1 0 t k i = 1 [LW(t i )] 1 1m y m (0) + s 2L m = 1 0 s k j = 1 t k i = j + 1 [LW(t i )][LV(t j )] 1 1m g m (u j -1 ), k = 1, 2, . . . , P, (46)
where subscripted brackets indicate matrix element of the enclosed quantity. As there are P impacts, P equations can be set up on this basis. The variables u i (i = 1, . . . , P -1) can be expressed in terms of t i (i = 1, . . . , P -1) by the relation

u k = vt k + u k -1 , k = 1, 2, . . . , P, (47)
and note that t p can be written as

t p = 2Np v -s p -1 i t i , (48)
so that the final unknowns are t i (i = 1, . . . , P -1) and u. With the initial conditions given, equation ( 46) furnishes a set of simultaneous non-linear equations that can be used to solve for N -P motion. The process is doubly iterative, for in addition to being non-linear in the unknown variables, the values of N and P are not known beforehand and have to be assumed.

stability analysis

The local stability analysis is carried out as illustrated in Figure 2, where the original initial conditions are perturbed to an adjacent state y˜(0), given by y˜(0) = y(0) + Dy(0).

(49) From equation (36), it follows that

Dy(0) = [Dx(0) Dx˙(0)] T . ( 50
)
As the reference state is at the moment just after an impact, it is clear that x 1 (0) = b and Dx 1 (0) = 0. In place of Dx 1 (0), the impact time is subject to variation and this is incorporated by varying the phase angle u. Taking variation of equation (38) gives

Dy(t) = W(t)Dy(0) + V(t)g'(u)Du + [W (t)y(0) + V (t)g(u)]Dt, (51)
where

g'(u) = [-sin u, cos u] T . ( 52
)
The initial deviations after the first impact are obtained by applying equation (4) to equation ( 51) It can be appreciated that [LV(t 1 )]g'(u) and L[W (t 1 )y(0) + V (t 1 )g(u)] are column vectors.

Dy 1 (0) = [LW(t 1 )]Dy(0) + [LV(t 1 )]g'(u)Du + L[W (t 1 )y(0) + V (t 1 )g(u)]Dt 1 . ( 53
The first row of this matrix equation is

Dy 1 1 (0) = s 2L i = 1 [LW(t 1 )] 1i Dy i (0) + [LV(t 1 )g'(u)] 1 Du + {L[W (t 1 )y(0) + V (t 1 )g(u)]} 1 Dt 1 , (54)
noting that Dy 1 1 (0) = Dy 1 (0) = 0, Dt 1 can be found

Dt 1 = -s 2L i = 2 [LW(t 1 )] 1i Dy i (0) + [LV(t 1 )g'(u)] 1 Du {L[W (t 1 )y(0) + V (t 1 )g(u)]} 1 . ( 55
)
The relationship between the time and phase variations can be inferred from equation ( 47) Equation ( 55) can be substituted into equation ( 56) to obtain a relation for Du 1 in terms of Dy i (0) (i = 2, . . . , 2L) and Du. It should also be apparent that by using equations ( 55) and ( 56) and the 2 to 2L rows of equation ( 53), one can obtain the expression where the matrix G 0 gives the variations after the first impact in terms of the initial ones. This procedure can be carried out P times to arrive at the motion deviations after P impact occurrences. Defining

Du 1 = vDt 1 + Du. ( 56
)
Dy 1 2 Dy 2 • • • • • =G 0 • =G 0 z, (57)
G G G G G G G K k G G G G G G G L l G G G G G G G K k G G G G G G G L l • • Dy 1 2L Dy 2L
G = G P -1 • • • G 1 G 0 (58) then z 1 = Gz, (59)
where z 1 is the variation at the end of the one response cycle and G is the transition matrix. The variations after k cycles are

z k = G k z, (60)
and the stability of a periodic solution is determined by the eigenvalues of the transition matrix. The case of interest in this study is the flip or period doubling bifurcation, where an eigenvalue becomes -1 and leaves the unit circle along the real axis.

RESULTS

A system with the following properties is analysed below: The coefficient of restitution is 0•6, the gap b is 0, and the system natural frequencies are v 1 = 0•618 and v 2 = 1•618. The initial conditions are

M = $ 1 0 0 1 % , K = $ 1 -1 -1 2 % , C = $ 0•05 -0•05 -0•05 0•1 % , f = $ 1 0 % . ( 61
)
x = $ -1 -1 % , x˙= $ 1 1 % . ( 62
)
The solutions are obtained from equations ( 38), ( 39) and (43); the first 1800 normalized secs of the responses are ignored, and the motion of the next 100 impacts are recorded. The initial conditions are fixed at the specified values as the solution branches are traced out over the frequency spectrum. Figures 3(a) and (b) show the displacement and impact velocity spectra for the frequency range 0-5. As the system's state space is five-dimensional, the Poincare´sections for the displacement and impact velocity spectra are four-dimensional. The displacement section is taken at once per period of the excitation whereas the impact velocity is sampled when x 1 = 0 and x˙1 q 0, where impact occurs.

Hence, the number of branches in the displacement spectrum corresponds to N forcing cycles, whereas those in the velocity spectrum give P impacts. It can be seen from Figure 3 that relatively narrow windows of chaotic motion are imbedded between regimes of harmonic response. The chaotic windows, labelled D, are near the integer values of frequency, but it is possible to discern that the windows are gradually displaced to higher frequency slots as

, v = 0•50; (b) 1-2 motion, v = 0•60; (c) 1-1 motion, v = 1•00; (d) 2-1 motion, v = 2•50; (c) 3-1 motion, v = 3•50.
the frequency is increased. Furthermore, the width of the chaotic windows seems to increase with the frequency. Outside the chaotic windows, N -1 motions, where N seems to equal the adjacent lower integer frequency, are prevalent and these become chaotic by undergoing a series of period-doubling bifurcations.

Figures 4(a) and (b) show the displacement and impact velocity spectra for frequencies of 0•45 to 1. Here, again, chaotic slots are embedded into broader regions of periodic response. The prevailing periodic responses are of the 1-P type, where P ranges from 1 to 3 and increases with decreasing frequency. The response here is qualitatively the same as that in Figure 3, but the results show that chaotic regions are not necessarily close to integer values of frequency. There is period one motion at the natural frequencies of 0•618 and 1•618, as might be intuitively expected. As the frequency increases, the impact velocity decreases. For excitation frequency larger than one, motions with N generally larger than P are present in the periodic regimes, and the reverse is true for the freuqency region less than one.

The first bifurcation takes place at a frequency of 1•69, according to the time integration results. This confirmed by the stability analysis, which shows that an eigenvalue of the transition matrix G leaves the unit circle at -1•0 within the frequency range of 1 to 1•8, as shown in Figure 5(a). Furthermore, a plot of the spectral radius in Figure 5 velocity excursions. Two more chaotic windows, as shown in Figures 8 and9, exist at higher frequencies. Both the displacement and velocity excursions show decreasing magnitudes with increasing frequency, just as in the case for the periodic regimes. The chaotic windows D 2 and D 3 at higher frequencies have progressively higher bandwidths, contain more periodic-doubling bifurcations, and more chaotic bands. Low velocity or grazing impacts, indicative of movement close to the wall, are present in all the chaotic windows, and the propensity for this type of motion increases with increasing frequency, as is evident in Figure 3. Figures 10 and11 show the impact velocity spectra of the thin chaotic bands at frequencies of 0•9 and 0•5. Using the former as a reference, it can be seen that the number of grazing impacts also increases in the direction of decreasing frequency. Previous studies have concentrated on one impact motion, and these are evident in the major periodic regimes outside of the chaotic bands for frequencies larger than one. Inside the chaotic windows, and for the entire frequency range of less than one, there are motions with more than one impact. In particular, high N, low P motion can be observed in several instances; for example, there are 5-3, 6-2 and 8-2 motions at frequencies of 2•8, 3•25 and 4•47, respectively, as illustrated in Figures 8 and9.

The time histories and phase diagrams of representative N-P motions for the mass m 1 are shown in Figures 121314. For 1-4 motion, there is a high speed impact followed by a lower speed one and then grazing impacts. Then the decaying motion abruptly speeds up to initiate another cycle, showing traits of jerky movement. Grazing impacts are absent from the 1-2 motion, which also features one high speed collision. The 1-1 motion has a slightly lower impact speed, and this trend continues for the 2-1 and 3-1 motions. The mass m 1 undergoes secondary looping responses, as can be seen in Figure 14; for N smaller than P the secondary loops are close to the wall, resulting in more impacts, whereas for N greater than P, the looping motion is away from the barrier, causing fewer impacts.

A picture on the transition to chaos at a frequency of about 1•8 is illustrated by phase diagrams and Poincare´sections in Figures 15 and16. The period one motion undergoes a series of flip bifurcations to become chaotic.

A basin of attraction at a frequency of 2•755 is shown in Figure 17. The shaded areas correspond to 2-1 motion and the blank areas are for 17-10 motion. The initial displacement and velocity for m 2 are -1•0 and 1•0, respectively. The basin is roughly divided into two halves, but there are scattered spots of 2-1 motion in the 17-10 region, and there is a substantial island of 17-10 motion in the 2-1 regime. It follows that in parts of the basin, small changes in initial conditions can lead to significant changes in motion quality. A basin of attraction for motion at frequency of 4•25 is shown in Figure 18. Periodic 12-8 motion exists in the shaded areas, whereas chaotic motion is present in the blank spaces. The overall pattern is largely irregular, as the two motions are mostly mixed with no major distinctive boundaries. Clearly, small deviations in initial conditions can cause a periodic motion to shift to a chaotic one and vice versa.

CONCLUSIONS

The equations of motion for period N motion with P impacts for a multi-DOF impact vibrator subject to harmonic loadings were formulated. The variational equations for determining the stability of such motion were also obtained. The equations were applied to a two-DOF example, and different regimes of N-P response were obtained. Observations concerning the evolution of the dynamics with frequency were made, and in particular it is noted that, as the excitation frequency departs from the natural frequencies, grazing impacts become more numerous. High N, low P motions were found to exist at the transition boundaries between harmonic and chaotic behaviour. Two basins of attraction were computed, and in one case it was shown that two very different types of motion can exist in a mixed pattern. The consequence is dynamics that is sensitive to initial conditions. This aspect was reinforced by the results of the second case, where periodic response is mingled with chaotic motion.

For frequencies larger than one, the major periodic windows contain period one or greater one impact motion. In contrast, for frequencies less than one, similar windows contain period one, one or more impacts motion. That such a wide range of dynamics exists is beneficial for applications, and one can infer from the displacement and velocity time histories useful characteristics for qualifying such motions from an engineering perspective. The 1-4 motion offers a high speed impact followed by hits of lower speed, the 1-1 motion gives high impact velocity and appears to be relatively smooth, whereas the 3-1 motion exhibits low impact velocity.

Figure 2 . 1 D 2 D 3 Figure 3 .

 21233 Figure 2. Local stability analysis.

Figure 4 .

 4 Figure 4. (a) Displacement x1 versus frequency, v Q 1•05. (b) impact velocity x˙1 versus frequency, v Q 1•05.

Figure 5 .

 5 Figure 5. (a) Real and imaginary parts of eigenvalues, 1•0 E v E 1•8; (b) spectral radius versus frequency.

Figure 6 .

 6 Figure 6. (a) Displacement x1 versus frequency, D1 window; (b) impact velocity x˙1 versus frequency, D1 window.

Figure 7 .Figure 8 .

 78 Figure 7. (a) Displacement x1 versus frequency, D11 window; (b) impact velocity x˙1 versus frequency, D11 window.

Figure 9 .Figure 10 .

 910 Figure 9. (a) Displacement x1 versus frequency, D3 window; (b) impact velocity x˙1 versus frequency, D3 window.

Du 1 Figure 11 .

 111 Figure 11. Impact velocity x˙1 versus frequency, v 1 0•5.

Figure 12 .

 12 Figure 12. Displacement x1 versus time: (a) 1-4 motion, v = 0•50; (b) 1-2 motion, v = 0•60; (c) 1-1 motion, v = 1•00; (d) 2-1 motion, v = 2•50; (c) 3-1 motion, v = 3•50.

Figure 13 .

 13 Figure 13. Velocity x˙1 versus time: (a) 1-4 motion, v = 0•50; (b) 1-2 motion, v = 0•60; (c) 1-1 motion, v = 1•00; (d) 2-1 motion, v = 2•50; (c) 3-1 motion, v = 3•50.

Figure 14 .

 14 Figure 14. Phase diagrams, x˙1 versus x1: (a) 1-4 motion, v = 0•50; (b) 1-2 motion, v = 0•60; (c) 1-1 motion, v = 1•00; (d) 2-1 motion, v = 2•50; (c) 3-1 motion, v = 3•50.

  Figure 15. Phase diagrams showing route to chaos, x˙1 versus x1: (a) v = 1•500; (b) v = 1•750; (c) v = 1•780; (d) v = 1•794; (e) v = 1•796; (f) v = 1•800.

Figure 16 .

 16 Figure 16. Poincare´maps showing route to chaos, x˙1 versus x1:(a) v = 1•500; (b) v = 1•750; (c) v = 1•780; (d) v = 1•794; (e) v = 1•796; (f) v = 1•800.

Figure 17 .Figure 18 .

 1718 Figure 17. Basin of attraction for 2-1 (shaded), 17-10 (blank) motions, x˙1(0) versus x1(0).

  )

	0.2
		(a)
	-0.2
	-0.6
	-1.0 3.94	4.04	4.14	4.24	4.34	4.44	4.54
	0.6	(b)
	0.4
	0.2
	0.0 3.94	4.04	4.14	4.24	4.34	4.44	4.54


Waeibrorheem Waemustafa
email: waeibrorheem@uum.edu.my

Azrul Abdullah

Mode of Islamic Bank Financing: Does Effectiveness of Shariah Supervisory Board Matter?

Keywords: Shariah Supervisory Board, Mode of Financing, Islamic Bank

This paper examines the relationships between the effectiveness of Shariah supervisory board (SSB), their remuneration and mode of financing Islamic bank. The SSB effectiveness is evaluated by an index based on 9 attributes score. This study comprises 18 Islamic banks in which operating in Malaysia from the year 2012 to 2013 as a sample. Our regression analysis shows that the effectiveness of SSB does not concern with the mode of Islamic bank financing. However, we found that SSB remuneration and bank's financial growth shown a positive and significant relationship with mode of financing. The implications of these findings are discussed.

Introduction

There is a strong debut on the Islamic mode of financing which do not underlie with the banner of Islamic banking in which constitutes the elimination of injustices of the interest-based system as ordained by Holy Qur'an 1 . According to [START_REF] Rosly | Risk-Based Pricing in Al Bai Bithaman Ajil (BBA)/Murabahah Sales: fiscal Liability with Business and Credit risk exposure[END_REF], [START_REF] Anwar | Islamicity of banking and modes of Islamic banking[END_REF] and [START_REF] Usmani | An Introduction to Islamic Finance[END_REF] the current mode of financing still cannot be expected to completely free from the injustices of the interest-based system or to contribute to the achievement of socioeconomic objectives which Islam seeks to achieve. Although there are many views can be discussed towards the mode of financing adopted by the Islamic banks, this study aims to explain whether Shariah supervisory boards (SSB) and their remuneration have any significant influence towards the choices of Islamic banks mode of financing. Based on the work of [START_REF] Ahmed | Islamic Banking and Shari'ah Compliance: A Product Development Perspective[END_REF] who suggested that, SSB plays an important role, particularly in harmonizing the Islamic rules and guidelines in Islamic banks (i.e. Shariah-compliance gatekeeper), this study argues that the choices of style of financing by Islamic banks should be tempted by this board as to avoid any injustices as what has been proposed in Shariah Law. In other words, the SSB should put highly concern with regard to any mode of financing by the Islamic bank in which are not aligned with the profit-sharing principles.

Literature Review

Islamic mode of financing drawbacks

According to [START_REF] Rosly | Shariah parameters reconsidered[END_REF] there are five drawbacks of Murabahah mode of financing. First, on the matter of ownership and ownership whereby bank has been claimed not actually purchase the asset prior to sale. Second, is on the issue of Murabahah's pricing mechanism. Third, is on the extent profit determination with consideration of time value of money with a fixed compensation since the Murabahah deals with instalment sale. Fourth, the issue of customer defaulted toward Murabahah facility. Fifth, refer to the liability of an asset that found defective upon delivery and lastly is on the principle of al-kharaj bil daman that being applied in the contract.

Shariah Governance

According to [START_REF] Alman | shari'ah supervisory board composition effects on Islamic banks' risk-taking behavior[END_REF], Shariah Supervisory Board (SSB) is one of internal governance mechanisms which involves the monitoring of Islamic banks' activities in accordance to the Sharia (i.e. Islamic law) specifically on its implementation and compliance. It is a part of the big corporate governance framework and what makes it different is only by the existence of Shariah Supervisory Board or Shariah Supervisory Committee or Shariah unit [START_REF] Rahajeng | The effectiveness of shari'ah supervisory board roles in Islamic Banks[END_REF]. Since Shariah governance is part of corporate governance, thus, good and effective SSB should reflect the issue of independence, transparent, accountable, responsible, and fair [START_REF] Bhatti | Toward understanding Islamic corporate governance issues in Islamic finance[END_REF]. In this context, studies by [START_REF] Grais | Corporate governance in institutions offering Islamic financial services: issues and options[END_REF] and [START_REF] Farook | Determinants of corporate social responsibility disclosure: The case of Islamic banks[END_REF], argued that the unique attributes of SSB must be clarified as to incorporating SSB as part of corporate governance mechanisms framework. Although the existence of SSB seemed able to monitor Islamic banking activities, there are still Islamic banks have experienced failure similar to that of conventional in the recent year. Supposedly, by the attaining of Shariah and good corporate governance, Islamic bank basically able to achieve Shariah objectives (i.e., Maqasid Shariah). However, having this kind of committee is sort not like providing reliable Shariah Governance in which utmost important and beneficial to all parties inside and outside the Islamic Banks.

Hypotheses Development

This study uses agency theory as underpinning theory to develop the hypotheses. The following hypotheses have been developed in this study: H1 There is positive relationships between the effectiveness of SSB and mode of financing adopted by the Islamic bank. H2 There is no significant relationship between SSB remuneration and mode of financing adopted by the Islamic bank.

Methodology

The sample used in this study comprises 18 Islamic banks in which operating in Malaysia from the year 2012 till 2014. To extract the mode of financing and information about SSB (i.e., effectiveness and remuneration), the annual reports of Islamic bank were scrutinized. The dependent variable of this study denoted as MODE is proxy by Bai Bhithaman Ajil to total Finance. This study measured SSB effectiveness based on 9 attributes scored in which aggregately measured. The attributes is captured based on four elements of effectiveness which is proposed in conventional corporate governance model [START_REF] Ika | Audit committee effectiveness and timeliness of reporting: Indonesian evidence[END_REF][START_REF] Dezoort | Audit committee effectiveness: a synthesis of the empirical audit committee literature[END_REF]. These elements are 1) composition, 2) adequate authority, 3) resources and 4) diligence. Since Shariah governance is part of corporate governance model [START_REF] Rahajeng | The effectiveness of shari'ah supervisory board roles in Islamic Banks[END_REF], the use of this elements can be assumed valid. To examine the relationship between SSB effectiveness, remuneration, control variables and the mode of financing, a multiple linear regression is performed. The regression model is built as follows:

MODEi = αi+β1SSBEFFi +β2SSBREMUi +β3FGROWTHi+β4PROFi+β5MSUPPi+ β6GDPi +εi

Results and Discussion

The results of descriptive statistics for the variables are shown in Table 2. The average of MODE is 53 percent indicating that more than half of the total finance is dominated by Bai bhithaman Ajil.

The mean score of 0.7857 for SSBEFF indicates that on average, SSB in Malaysia quite effective.

On average, all the SSB members are being paid off RM518, 346 annually, however, there are some bank that did not disclose the SSB remuneration clearly. percent of the variation towards the mode of financing, which is relatively low. Although the adjusted R 2 is low, the finding is consistent with those of the previous governance studies, which pointed out that the low R 2 is common in corporate governance researches [see [START_REF] Farook | Determinants of corporate social responsibility disclosure: The case of Islamic banks[END_REF][START_REF] Rahman | The influence of the shariah supervision board on corporate social responsibility disclosure by Islamic banks of gulf co-operation council countries[END_REF]. Our analysis show that SSBEFF do not have a significant relationship with mode of financing, but there are significant relationship between SSBREMU and FGROWTH. Table 3 shows that SSBREMU (t value = -0.029) has a significant and positive relationship with the mode of Islamic bank financing, at p < 0.1 in which indicates that the increase in BBA moves in tandem with the increase in SSB's remuneration.

In this respect H2 is rejected. Thus, we support the notion that there is no serious effort was taken to implement the profit and loss sharing mode of financing among Islamic bank [START_REF] Rosly | Risk-Based Pricing in Al Bai Bithaman Ajil (BBA)/Murabahah Sales: fiscal Liability with Business and Credit risk exposure[END_REF][START_REF] Anwar | Islamicity of banking and modes of Islamic banking[END_REF][START_REF] Ahmed | Islamic Banking and Shari'ah Compliance: A Product Development Perspective[END_REF]]. It's not a matter of SSB effectiveness, but it seems that their remuneration may determine the preferences toward BBA and Murabahah mode of financing. This might hold a truth since we find that there is an insignificant portion of these asset portfolios is mainly dominated by murabahah, BBA and other than profit and loss mode of financing. In addition, judging from the FGROWTH that is positively significant to MODE as reported, we also perceive that the trade-based mode of financing that must be applied temporarily by the Islamic bank as for their transition phase is not much concern after 25 years, it sort like they are maximizing the shareholder wealth. Another possible explanation is that the SSB committee members may have a relationship with the management as to remain their long period as committee members. As a result to this, we perceive that their assessment may have been influenced by the management since their appointment as committee member is merely due to their relationship with the company.

Table 3. Regression Results

Note: ***Significant at 0.01 level, **significant at 0.05 level, *significant at 0.1 level.

Conclusion

The objective of this study is to examine the influence of SSB effectiveness and their remuneration to the choices of Islamic mode of financing by Malaysian Islamic bank. The analysis shows that an effective SSB does not have significant bearing towards the choice of Islamic mode of financing in Malaysia but their remuneration have. Intrinsically, this finding might give some insight towards the notion of 'cosmetic reason' [START_REF] Zaman | Turnbull-generating undue expectations of the corporate governance, role of audit committee[END_REF] for regulator, policy maker and another researcher on the effectiveness and the existence of SSB in Malaysia especially when incorporating SSBs as part of the corporate governance mechanism.

Table 1 .

 1 Measurement of Explanatory Variables

	Variable	Composition of variables
	MODE	: Bai Bhithaman Ajil to Total Finance
	SSBEFF	: SSB effectiveness Index = Company's actual score on RMC characteristics/
		company total possible score of RMC characteristics
	SSBREMU : Log of Total Remuneration
	FGROWT	: Log of total finance
	H
	PROF	: Number of RMC meetings during the financial year
	MSUPP	: M2 plus financial assets
	GDP	: Growth rate of real GDP
	ε	: Error term

Table 2 .

 2 Descriptive ResultsTable3presents the outcomes of the regression analysis between the mode of financing, SSB The Variance Inflation Factor (VIF) statistics indicate that multicollinearity does not exist in the model as the values are less than 10. The model explains 22.3

	N=42
		Mean	Standard Deviation
	MODE	.5295	.23113
	SSBEFF	.7857	.06639
	SSBREMU	12.7312	.78972
	FGROWTH	16.1884	1.25395
	PROF	.0864	.02689
	MSUPP	2.300	.5959
	GDP	8.300	1.0054

The Holy Qur'an versus

2:275 and 3:130.