Ryan S Mattson
email: mattsonr@rhodes.edu

Philippe De Peretti
email: philippe.de-peretti@univ-paris1.fr

March

Investigating the Role of Real Divisia Money in Persistence-Robust Econometric Models

. These two environments allow modeling mixtures of I(0)/I(1) variables with possible cointegration without pretesting for integration nor for the dimension of the cointegration space. Moreover the latter method is also robust to various other forms of persistence such as local-to-unity processes, long memory/fractional integration, or unmodeled breaks-in-mean in the causal variables. By implementing the tests on di¤erent sub-samples identi…ed by standard structural break tests, and using three di¤erent measures of money (DM4, DM4-and DM3), the tests suggest a unidirectional causality from activity to money. Moreover, from one period to another, the whole causal structure of the systems seem to change, as well as the stationarity of the series. At last, the two methodologies return similar results.

Introduction

Economists have long been interested in the role of money in the economy, especially regarding its role as a leading indicator for future activity. Following the early works of [START_REF] Sims | Money, Income and Causality[END_REF][START_REF] Sims | Macroconomics and Reality[END_REF], most researchers have focused on non-causality tests à la [START_REF] Granger | Investigating Causal Relations by Econometric Models and Cross-Spectral Methods[END_REF] in Vector Auto-Regressions (VAR). Among many others, [START_REF] Hayo | Money-Output Granger Causality Revisited: An Empirical Analysis of EU Countries[END_REF] and [START_REF] Walsh | Monetary Theory and Policy 2nd edition[END_REF] survey this vast literature, both emphasizing the contradictory results found by many researchers, showing that it is not possible to draw clear conclusions about the role of money in the economy. A similar conclusion was previously reached by [START_REF] Stock | Interpreting the evidence on Money-Income Causality[END_REF], noticing that "Researchers using only slightly di¤erent speci…cations have reached disconcertingly di¤erent conclusions".

In this paper, we argue that such instability of results may have at least two distinct sources: i) Measurement errors in monetary aggregates due to inappropriate aggregation methods and, ii) Two-stage inference in econometric models.

Concerning the former point, as suggested in a number of seminal publications [START_REF] Barnett | Economic Monetary Aggregates: An Application of Aggregation and Index Number Theory[END_REF]; Barnett, O¤enbacher andSpindt, 1981, 1984; [START_REF] Barnett | The Theory of Monetary Aggregation[END_REF], there exists an internal inconsistency between the microeconomics used to model the private sector, and the aggregator functions used to compute monetary aggregates by central banks. As a corollary, simple-sum aggregates, computed by central banks, are likely to return ‡awed measures of money, being based on the unrealistic assumption of perfect substitutability of assets. This critique, known as the Barnett Critique, de…ned by [START_REF] Chrystal | Empirical evidence on the recent behaviour and usefulness of simple-sum and weighted measures of the money stock[END_REF], strongly a¤ects the results given by macroeconomic models, implying that outcomes of studies using simple-sum aggregates are misleading. [START_REF] Belongia | Measurement Matters: Recent Results From Monetary Economics Reexamined[END_REF] has investigated the empirical validity of this Critique by reexamining several studies on money, replacing the ‡awed simple-sum measures with theoretically-consistent ones. He shows that the original conclusions are then deeply altered. Theoretically-consistent measures of money have been used in a number of studies such as [START_REF] Barnett | Consumer Theory and The Demand for Money[END_REF], [START_REF] Chrystal | Empirical evidence on the recent behaviour and usefulness of simple-sum and weighted measures of the money stock[END_REF] who study the relationship between money and GDP, [START_REF] Schunk | The Relative Forecasting Performance of the Divisia and Simple Sum Monetary Aggregates[END_REF] focusing on the predictive content of money, [START_REF] Serletis | The Demand for Divisia Money in the United States : A Dynamic Flexible Demand System[END_REF] and [START_REF] Serletis | The Role of Money in Canada[END_REF] or [START_REF] Yue | Divisia Monetary Services Indexes for Switzerland : Are They Useful for Monetary Targeting?[END_REF]. Other recent works also include [START_REF] Darrata | Money and Macroeconomic Performance: Revisiting Divisia Money[END_REF], Barnett, Chauvet and Tierney (2009), [START_REF] Belongia | Measurement Matters: Recent Results From Monetary Economics Reexamined[END_REF]Ireland (2012a, 2012b) or [START_REF] Rahman | The Case for Divisia Money Targeting[END_REF].

Concerning the latter point, classical tests of non-causality rely on pre-tests (…rst-stage inference) on the persistence of the series. For instance, if all series are stationary, a simple VAR in levels is estimated, whereas if the series are integrated but not cointegrated, a VAR in …rst di¤erences is used. If additionally, cointegration is present, then a Vector Error Correcting Model (VECM) is estimated (Toda andPhillips 1993, 1994). This simple example emphasizes a general method consisting of: i) Trying to identify the correct underlying Data Generating Process (DGP) according to pre-tests for stationarity/cointegration (…rst-stage inference), and ii) Testing for non-causality in the estimated DGP (second-stage inference). Nevertheless, in empirical work, due to lack of power of stationarity tests, estimating the "true" DGP may be di¢cult, as well as di¤erentiating between the di¤erent kind of persistence as local-to unity processes, [START_REF] Phillips | Towards a Uni…ed Asymptotic Theory for Autoregression[END_REF][START_REF] Chan | The Parameter Inference for Nearly Nonstationary Time Series[END_REF], or long memory/fractional integration; a problem that deeply worsens if structural breaks are present [START_REF] Diebold | Long Memory and Regime Switching[END_REF]. Hence, inappropriate persistence modelling, and thus inaccurate DGP identi…cation may lead to incorrect inference and cascading errors in non-causality tests.

The goal of the paper is to study the causal relationships between real money and activity by taking into account the two above points. To avoid the Barnett Critique, we use theoreticallyconsistent measures of money, i.e. Divisia indices. Concerning the econometric models, we follow [START_REF] Bauer | Persistence-Robust Surplus-Lag Granger Causality Testing[END_REF], suggesting agnostic approaches to the form of persistence, that is suggesting tests that are robust to the various forms of persistence, without any pre-testing. In causality testing, to our knowledge, two approaches can be used, i) The Fully-Modi…ed VAR (FM-VAR) of [START_REF] Phillips | Fully Modi…ed Least Squares and Vector Autoregression[END_REF], extending the Fully Modi…ed ordinary least square of [START_REF] Phillips | Statistical Inference in Instrumental Variables Regression with I(1) Processes[END_REF], ii) The surplus-lag VARX model of [START_REF] Bauer | Persistence-Robust Surplus-Lag Granger Causality Testing[END_REF], based on [START_REF] Toda | Statistical Inference in Vector Autoregressions with Possibly Integrated Processes[END_REF], [START_REF] Dolado | Making Wald Tests Work for Cointegrated VAR Systems[END_REF] or [START_REF] Saikkonen | In…nite-Order Cointegrated Vector Autoregressive Processes[END_REF].

Whereas the FM-VAR is designed to model a mixture of I(0)/I(1) variables with possible but unde…ned cointegration, the Bauer and Maynard (2012) surplus-lag VARX approach possesses basically the same features, but goes further. Indeed, the causality test remains robust, when the causal variable has long memory/fractional integration, can be modeled as a local-to-unity process, or contains a certain amount of unmodeled breaks in mean. In this latter case, the noncausality test is interpreted as a co-breaking test.

The main results of the paper are twofold: i) Using three di¤erent Divisia monetary aggregates, tests support an unilateral causality from real activity to real money, ii) Over the di¤erent sub-periods, the whole causal structure of the systems, as well as the long term relationships seem to change. This paper is organized as follows. Section two focuses on monetary aggregation and presents the theoretically consistent Divisia aggregates. Section three introduces the econometric methodologies. In Section 4, we …rst test for structural breaks on the growth rates of the series, and then implement non-causality tests on di¤erent sub-periods. At last Section 5 concludes and discusses the results.

The theoretical approach to monetary aggregation

In this section, we brie ‡y recall the theoretical approach to monetary aggregation. Let x t = (x 1t ; :::; x kt ) 0 and m t = (m 1t ; :::; m pt ) 0 be respectively two vectors of real consumption goods and real monetary assets in period t; t = 1; :::; T: Let l t be leisure. Assume that data are rationalized by a well-behaved utility function:

U t = U (x t ; m t ; l t ) (1)
i.e. each bundle (x 1t ; :::; x kt ; m 1t ; :::; m pt ; l t ) 0 is the solution of the utility maximization program 1 , where the nominal price of money, for an asset m it is de…ned according to [START_REF] Barnett | The User Cost of Money[END_REF]:

p n it = p t R t r it 1 + R t ; i = 1; :::; k (2)
Where:

R t is a benchmark rate, r it is the asset's own rate, p t is a consumer price index.

To allow for aggregation, further assume that the utility function (1) is weakly separable over the monetary assets, and thus admits a rewriting:

U t = U (x t ; m t ; l t ) = V (x t ; U 1 (m t ); l t ) (3)
Where:

V (:) is a strictly increasing function, known as the macro-function, U 1 (:) is the monetary sub-utility function, known as the micro-function, which, if homothetic, is also the aggregator function.

The above weakly separable utility structure ensures that an aggregate exists over money 2 .

Following [START_REF] Diewert | Exact and Superlative Index Numbers[END_REF][START_REF] Diewert | Superlative Index Numbers and Consistency in Aggregation[END_REF], if the preferences over the monetary assets are homothetic, then 1 For theoretical justi…cations of money in the utility function, see [START_REF] Feenstra | Functional Equivalence Between Liquidity Costs and the Utility of Money[END_REF] and [START_REF] Poterba | Money in the utility function: An empirical implementation[END_REF]. 2 Notice that here weak separability is assumed but not tested. For more formal testing procedures, see for example [START_REF] Varian | Nonparametric Tests of Consumer Behavior[END_REF], Swo¤ord andWithney (1987, 1988), de Peretti (2007), [START_REF] Barnett | Admissible Clustering Of Aggregator Components: A Necessary And Su¢cient Stochastic Seminonparametric Test For Weak Separability[END_REF], or Fleissig andWhitney (2003, 2005).

U 1 (:) is interpreted as the aggregator function. An index number Q(m t+1 ; m t ; p n t+1 ; p n t ), must therefore satisfy:

Q(m t+1 ; m t ; p n t+1 ; p n t ) = U 1 (m t+1 ) U 1 (m t ) (4)
If preferences are non-homothetic, the aggregator function is no longer the sub-utility function, but the distance function. The distance function D(U 1 ; m t ) = max l fl : U (m t =l) U 1 g returns by what proportion one has to in ‡ate or de ‡ate the vector m t in order to reach the utility level

U 1 .
A consistent aggregate is given by:

Q(m t+1 ; m t ; p n t+1 ; p n t ) = D(U 1 ; m t+1 ) D(U 1 ; m t ) (5)
Diewert (1976,1978,1980) proved that in both case cases Q(m t+1 ; m t ; p n t+1 ; p n t ) can be nonparametrically and consistently estimated as:

Q(m t+1 ; m t ; p n t+1 ; p n t ) = p Y i=1 m it+1 m it (sit+1+sit)=2 (6)
Where:

s it is the budget share for the monetary asset i in period t,

U 1 = p U 1 (m t+1 )U 1 (m t ).
Clearly, ( 6) is a discrete approximation of the continuous Divisia index.

In their seminal paper, [START_REF] Barnett | The New Divisia Monetary Aggregates[END_REF] present a very comprehensive survey of the so-called Divisia aggregation. They show that very di¤erent conclusions are drawn when one considers Divisia aggregation, rather than simple-sum aggregates, the latter being valid only if the assets are substitutes, which is an unrealistic assumption. As an illustration, Figure [START_REF] Akaike | A New Look at the Statistical Model Identi…cation[END_REF] plots two monetary aggregates: The DM4 (see below), and the simple-sum built by adding the component levels of the DM4 assets. Both aggregates are divided by a consumer price index and are in log-form. On the same …gure, we plot (right scale) the aggregation bias, which is computed as the di¤erence between the two indices. Clearly, the error is not constant over time and dramatically increases up to the early 90's, according to the explosion of …nancial innovations. After the early 90's, the bias decreases, and then constantly increases at a lower growth rate. Notice that the bias re ‡ects di¤erent trends in the two series. The time-varying nature of the bias implies that simple-sum aggregates can not be used as proxies of Divisia aggregates.* variables without specifying the order of integration of each component, nor the dimension of the cointegration space. The latter framework is more general since it also takes into account other forms of persistence in the causal variable as local-to-unity processes, fractional integration/ long memory and unmodeled structural breaks-in-mean in the causal variable. We …rst describe the FM-VAR method.

The Fully-Modi…ed approach to VAR modeling

Let y t be a k-vector time series generated by the …nite-order VAR of order p, VAR(p):

y t = J(L)y t 1 + " t ; t = 1; 2; ::; T (7)
Table 1: SupF T (l + 1jl) sequential tests for structural breaks, monetary components where:

div4r t , Lag = 2, h = 0:1 l + 1 l SupF T (l + 1jl) Cut of f (5%) 1
J(L) = p P i=1 J i L i 1 ;
" t Niid(0; "" ).

(7) can be equivalently re-written as:

y t = B(L) y t 1 + Ay t 1 + " t (8)
where:

B(L) = p 1 P i=1 B i L i 1 with B i = p P j=i+1 J j ,
and A = J(1):

or as:

y t = B(L) y t 1 + y t 1 + " t (9)
where:

= A I k
This latter representation is the Vector Error Correction form representation of a VAR process.

The A matrix or ( ) contains all relevant information about the long-term relationships. Now, re-write model ( 8) as:

y t = Bz t + Ay t 1 + " t (10) = F x t + " t (11)
Where:

z t = ( y 0 t 1 ; :::; y 0 t p+1 ) 0 : x t = ( y 0 t 1 ; :::; y 0 t p+1 ; y 0 t 1 ) 0 ; B = [B 1 ; B 2 ; ::; B p 1 ]:

Or in matrix form as:

Y 0 = BZ 0 + AY 0 1 + E 0 (12) = F X 0 + E 0 (13)
Where:

F = [B 1 ; :::; B p 1 ; A]:
Since the above model can not be e¢ciently estimated by Ordinary Least Square (OLS) due to a second order (endogeneity) bias, [START_REF] Phillips | Fully Modi…ed Least Squares and Vector Autoregression[END_REF] suggest applying the Fully Modi…ed estimator (Phillips and Hansen, 1990) to the system. De…ne b "y and b yy as estimates of the two-sided long-run covariance matrices of respectively t = (b " t = y t F x t ; y t 1 ) and y t 1 ; where b " t are the OLS residuals of model [START_REF] Barnett | New concepts of aggregated money[END_REF]. Also de…ne b " y and b y y as estimates of the one-sided long-run covariances matrices of respectively t = (b " t = y t F x t ; y t 1 ) and y t 1 . Notice that both types of matrices are kernel estimates. For instance b "y and b " y are computed as:

b "y = T 1 P j= T +1 w(j=k1) b (j) (14)
b " y =

T 1 P j=0 w(j=k1) b (j) (15)
Where:

w(j=k1) is a kernel smoothing function, k1 is the bandwidth or truncation parameter, b (j) is the standard covariance estimator, T 1 T P t=1 t 0 t+j
The FM-VAR estimator is then de…ned as:

b F + = [Y 0 ZjjY 0 Y 1 b "y b 1 yy ( Y 0 1 Y 1 T b y y )](X 0 X) 1 (16)
Where: jj is the horizontal stacking operator,

Y 0 1 = Y 0 1 Y 0 2 :
Compared to the standard FM estimator for OLS, no correction for autocorrelation is made. Thus, the only correction is for endogeneity. 1 as the simple OLS estimator, ( 16) can be re-written in order to make apparent the correction for endogeneity:

De…ning b F = [Y 0 ZjjY 0 Y 1 ](X 0 X)
b F + = b F [ b "y b 1 yy ( Y 0 1 Y 1 T b y y )](X 0 X) 1 (17)
A non-causality test from variable j in equation i, amounts to testing:

B 1 [ij] = ::: = B p 1 [ij] = A[ij] = 0
which can be stated as:

H 0 : Rvec( b F + ) = 0 ( 18
)
where R is a (q 1) suitable selection matrix, with here q = p. Then the corresponding Wald test is given by:

W + = T (Rvec( b F + )) 0 [R( b " T (X 0 X) 1 )R 0 ] 1 (Rvec( b F + )) (19)
Phillips (1995) proved that under the null, W + is asymptotically bounded by a Chi-squared distribution with q degrees of freedom, q being the rank of R.

The surplus-lag VARX model

Alternatively, one may also consider surplus-lag models to test for non-causality between two variables, y 1t the dependent one and y 2t the exogenously modeled forcing (causal) variable, given a set z t = (y 3t ; ::; y kt ) 0 of control variables. We here follow Bauer and Maynard (2012), extending and simplifying Toda and Yamamoto (1995). In our framework, i.e. the causality between two series, write down and estimate the single equation of interest of a surplus-lag VARX(p; p 1 ), here a Auto-Regressive ARX(p; p 1 ) process:

y 1t = p X j=1 1 j y 1t j + 2 j z t j + p1+1 P j=1 3 j y 2t j + " t (20)
and jointly test for3 1 = 3 2 = ::: = 3 p1 = 0 using a standard Wald tests 3 .

The two models are clearly complimentary. On the one hand, if series are I(0)/I(1) and cointegration is present, the FM-VAR is e¢cient, unlike the VARX which forces an extra lag.

Nevertheless, as shown by Bauer and Maynard (2012) based on Monte Carlo simulations, e¢ciency losses have a very limited impact on the power of causality tests. On the other hand, for local-to-unity processes, the causality tests in FM-VAR's may not behave well, as shown by [START_REF] Yamada | A Note on Hypothesis Testing Based on the Fully Mod-i…ed Vector Autoregression[END_REF], which is a case for which the VARX is designed for. Hence, di¤erent informations between the two tests may indicate di¤erent kinds of persistence. For other kinds of persistence, such as long memory, nothing is known for the FM-VAR.

We next turn to implementations 4 .

Implementing the tests

The role of real money balance e¤ects has recently been reconsidered by Favara and Giordani (2009), using a structural VAR, imposing restrictions consistent with the New Keynesian framework. They suggest that real money balances shocks may have a signi…cant e¤ect on output and prices. Dorich (2009) reaches a very similar conclusion using a money-in-the-utility model.

Interestingly, he concludes that such e¤ects may be based on a non-separable utility function, a fundamentally di¤erent framework from ours. Those conclusions are in sharp contrast with [START_REF] Woodford | Interest and Prices[END_REF] and [START_REF] Ireland | Money's Role in the Monetary Business Cycle Model[END_REF], who show the negligible impact of real money. All these articles used inconsistent measures of money. Here, we reconsider their conclusions using three di¤erent theoretically-consistent measures of broad money. We …rst describe the data and analyze their statistical properties in terms of structural breaks, and the implement the described non-causality tests.

Data

Let y 1 t = (div4r t ; indpro t ; irst t ; irlt t ; 12 p t ) 0 , y 2 t = (div4mr t ; indpro t ; irst t ; irlt t ; 12 p t ) 0 and y 3 t = (div3r t ; indpro t ; irst t ; irlt t ; 12 p t ) 0 be k-vectors of variables being integrated of order 1 or 0, where: div4r t , div4mr t and div3r t are respectively the logarithms of the real Divisia DM4, DM4and DM3 indices; indpro t is the logarithm of the Industrial Production Index, used as a proxy of the real activity; irst t and irlt t are respectively short (three-month) and long-term (one year) interest rates and 12 p t is the Consumer Price Index in ‡ation rate. All data are on a monthly basis, and span, for the United States, the period covering January 1968 to December 2012. Data Over such a long period, one can expect structural breaks to occur in either the mean or the variance (or in both) of the series, especially in their growth rates. Such breaks have been reported by a number of authors as Guégan and de Peretti (2013). Moreover, testing for structural breaks in …rst di¤erences is of particular importance when dealing with FM-VAR models. Indeed, recall that such a methodology requires computing long-run covariances matrices, using both residuals and …rst di¤erences of series.

To test for structural breaks, we proceed in two steps. We …rst use the procedure suggested in Bai and Perron (1998) to …nd the optimal number of breaks, by sequentially computing the SupF T (l + 1jl) statistics, where l is the number of breaks under the null. For a given series, we begin by computing SupF T (1j0), to test for no break against a single break. If the null is rejected, we then add an additonal break for one of the two segments, compute SupF T (2j1), and so on, until we fail to reject the null of no additional break. Then, taking l as the number of breaks, in a second step, we compute the breaking dates using a global minimizer. Tables (1, and its subsequent take-o¤ after 1996 9 , it is interesting that the break occurs during the period sweeps become relatively large (1995)(1996) in the sub aggregates, but not the broadest: DM4.

For the in ‡ation rate two breaks are found at the 5 % level and 3 at the 10% one. We thus adopt a 3-break model for the in ‡ation rate model, which is line with Benati and Kapetanios (2003) using a slightly di¤erent methodology. For the growth rates of real activity, tests support a one-break model. Interestingly, activity and broad money exhibit a break close to the same time in late 1981.

For the two interest rates, no breaks in growth rates are observed. For these two series, we also tests for breaks in variance. As a rough analysis, we implement structural break tests on the square residuals of the two models de…ned in Table [START_REF] Anderson | Retail Sweep Programs and Bank Reserves, 1994-1999[END_REF]. Results are reported in Table [START_REF] Andrews | Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation[END_REF]. At the 5% threshold, both series exhibit a clear break, and at the 10% two breaks. In both cases, we conclude here in favor of a two-break model for the two series. Notice that both series exhibit a break in the late 2007, at nearly the same date.

Grouping the results of structural break tests suggests testing for non-causality on di¤erent sub-periods, presented in Table (4).

Non-causality in FM-VAR models

To estimate the various FM-VAR models 10 , we have to jointly choose two parameters: i) The lag p of the VAR model, ii) The bandwidth parameter k1, knowing that for k1 several intervals are given by [START_REF] Phillips | Fully Modi…ed Least Squares and Vector Autoregression[END_REF] that rule out automatic bandwidth selection procedures (see [START_REF] Andrews | Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation[END_REF]. To tackle this problem, we adopt a two-step data-driven procedure. First, using a double loop over p and k1, we estimate various FM-VAR models (8) on p = 2; :::; 10, and for k1 2]1=4; 2=3[ (see assumption BW p. 12 and theorem p 37), and keep the models having disturbances. On a second run, among all these models, we select the one with the minimal AIC criterion [START_REF] Akaike | A New Look at the Statistical Model Identi…cation[END_REF], and implement non-causality tests in this model. We report two kinds information: i) The estimates of the various long-term matrices A in Tables 5, 8 and11, where the d variable signals if deterministic terms are added, where d = 1 means that an intercept is added, and d = 2 that an intercept plus a linear trend are included, and

ii) The whole causal structure of the model. This allows analyzing if the long-term relations appear stable over time, as well as the stationarity of the variables, even if not formally tested.

Speci…cations of the models are also presented. Tables [START_REF] Barnett | The User Cost of Money[END_REF], ( 9) and ( 12) present the results of non-causality tests for the di¤erent de…nitions of money.

Non-causality tests for DM4 Looking …rst at Table ( 5), it appears that the long term relationships change from a period to another, as the stationarity of the series. For instance, the long-term interest rate appears to be non-stationary in sub-period 1 but stationary in subperiods 2 and 3. Picking an other example, Divisia money is also integrated of order one in sub-periods 1, 2 and 3. The sub-periods 4 being hard to interpret due to the non-nullity of the cross-coe¤cicients. Turning to non-causality tests, Table [START_REF] Barnett | The User Cost of Money[END_REF], we fail to reject the null of noncausality from to money to activity in all sub-periods, whereas activity clearly granger-causes money in the last three periods. For sub-period 1, causality is found only at the 10 % threshold.

Concerning the other causal relationships, unexpectedly, the short-term interest rate seems to play no driving role concerning activity, but appears to be a causal variable for money. At last, note that, except for few exceptions, the whole causal structure of the systems change from one period to another.

Non-causality tests for DM4-As previously noticed, here again, the …rst striking point is that the long-term matrices are period-dependent, Table [START_REF] Barnett | New concepts of aggregated money[END_REF]. Turning to non-causality tests, Table [START_REF] Barnett | The New Divisia Monetary Aggregates[END_REF], in all sub-periods, the null of non-causality from money to real activity is not rejected.

In the same time, real activity Granger-causes money, but the results depend on the time-period considered. The null is accepted for sub-periods 1 and 3, and deeply rejected over periods 2, 4 and 5. Over these two latter periods, the short-term interest rate is the driving force of real activity. The relation between the interest rate and money is also not constant over time, and we reach a very similar conclusion as for DM4: the causal structures of the systems change over time.

Non-causality tests for DM3 Focusing now DM3, and only on causality tests as presented in Table [START_REF] Barnett | Measurement Error In Monetary Aggregates: A Markov Switching Factor Approach[END_REF], we have the same patterns: On sub-period 1, we have no causal relationships between money and activity On sub-periods 2, 3 and 4, we reject non-causality from activity to money, while we fail to reject the null from money to activity.

Hence, whatever the de…nition of money used, the causal relationship for all models is clearly from activity to money, and not the converse.

Non-causality in surplus-lag VARX models

We next turn to results of non-causality tests in surplus-lag VARX models. Similarly to what we did for the FM-VAR lag selection, we use a double loop over p and p1, for p; p1 = 1; :::; 10, estimate the various ARX(p,p1) models, and keep those with spherical disturbances. In a second trial, among these models, we choose the one with the minimal AIC. Results are given by Tables [START_REF] Barnett | Economic Monetary Aggregates: An Application of Aggregation and Index Number Theory[END_REF], ( 10) and ( 13). We report the kind of model used, as well as the Wald statistic and the associated p-value, only for money and real activity.

Non-causality tests for DM4 Results using the broad monetary aggregate DM4, are similar with those obtained using the FM-VAR methodology. In sub-periods 2, 3 and 4 real activity is a causal variable of money. Results are less clear for sub-period 1 where causality is found only at the 10 % threshold. Interestingly, only in sub-period 2, the causal relation is bilateral, a result which is also suggested in Table [START_REF] Barnett | The User Cost of Money[END_REF].

Non-causality tests for DM4-Results are presented in Table [START_REF] Barnett | Consumer Theory and The Demand for Money[END_REF]. Concerning the null of non-causality from money to activity, it is accepted in all sup-periods. Focusing now on the causal relationship from activity to money it is clearly period-dependent, as within the FM-VAR framework: The null is rejected for sub-periods 2 and 5 (here at 10%), but accepted for other samples. Clearly, compared with DM4, the relationship is much more unstable.

Non-causality tests for DM3 At last, focusing on non-causality tests for DM3 returns exactly the same information as for tests implemented in FM-VAR models. For sub-period 1, no causal relationships are found between money and activity, whereas over other periods, the causal relationship is from activity to money. Hence, DM3 seems to behave as DM4.

Conclusion and discussion

In this paper, we have used Granger non-causality tests to investigate the empirical relationships between real money and activity. Three di¤erent broad measures of money as de…ned by the CFS have been used, DM4, DM4-and DM3. Tests have been implemented within two distinct frameworks: FM-VAR and surplus-lag VARX models. Results return important features: i) For all three aggregates, and for di¤erent sub-periods, tests suggest that if causality is found, it is from activity to real money, ii) In all models, there is no causal links between money and activity during the period January 1968 to July 1979, iii) For DM4-, the results are period-dependent, and the two aggregates showing a stable relationship with activity are the DM4 and DM3 ones, iv) From one sub-period to another, the whole causal structure of the systems, as well as the stationarity of some series seems to change, maybe also explaining the instability found previously by researchers, v) At last, the two methodologies return very similar information.

Tables to be included in the paper

Figure 1 :

 1 Figure 1: Real DM4, simple-sum over the same components as the DM4, aggregation bias. (shaded area, right sclale).

3

  Non-causality tests in persistence-robust econometric modelsTo analyze the role of Divisia money in the economy, we use standard Granger non-causality tests in VAR models. Two frameworks are considered that do not impose pre-testing for persistence nor for cointegration: The FM-VAR approach of[START_REF] Phillips | Fully Modi…ed Least Squares and Vector Autoregression[END_REF], and the surplus-lag VARX model of[START_REF] Bauer | Persistence-Robust Surplus-Lag Granger Causality Testing[END_REF]. The former approach allows modeling a mixture of I(0)/I[START_REF] Akaike | A New Look at the Statistical Model Identi…cation[END_REF]

2 )

 2 report the results for a pure structural change model for a window h = 0:1. At the 5% threshold, all series, except the two interest rates, exhibit one or several breaks. The broadest monetary aggregate exhibit a clear single break in September 1981, coinciding with a contraction in broad money growth. The sub-aggregates DM3 and DM4-di¤er from the broader DM4 in that they experience a break in early 2008. This speci…c break does not coincide with any major revision in the monetary data collection methodology of the Federal Reserve 7 , but does coincide with the rate cuts enacted by the Federal Reserve in September and October of 2007 as well as a Federal Reserve program to conduct a ($100 billion) series of repurchase agreements over the course of twenty eight days in March of 2008. The monetary sub aggregates show an added break in April 1995 and February 1996, close to the beginning of the inclusion of retail sweeps in calculation of the Divisia 8 . A retail sweep occurs when banks reclassify checking deposits as savings deposits to skirt legal reserve requirements. Checking accounts are then underreported while savings accounts are over reported by a signi…cantly large amount. Given the di¤ering weights associated with each monetary component, the entrance of sweeps into the calculation

5 ;

 5 k1 = 0:64; d = 2 p = 4; k1 = 0:53;

4 ; 4 ;

 44 k1 = 0:66; d = 1; AIC = 39:29; BIC = 35:08 # i 146) 2:569 (0:632) 4:648 (0:325) 12:899 (0:012) ****** Sub-period 4 p = 6; k1 = 0:38; d = 2; AIC = 39:10; BIC = 33:74 # i *** 14:287 (0:026) 11:239 (0:081) 10:350 (0:110) 13:735 (0:032) indprot 7k1 = 0:49; d = 1; AIC = 32:61; BIC = 29

5 ;

 5 -VAR causal structure. Non-causality from variable j in equation i. Main entries are the Wald statistics and the p-values (between parentheses) Sub-period 1 p = 5; k1 = 0:52; d = 0; AIC = 36:96; BIC = 33(0:079) 6:956 (0:223) 19:002 (0:002) ****** Sub-period 2 p = 4; k = 0:32; d = 1; AIC = 36:70; BIC = 33:94 # i k1 = 0:47; d = 0; AIC = 39:57; BIC = 34:85 # i 4 p = 4; k1 = 0:50; d = 0; AIC = 34:31; BIC = 30:03 # i

Table 2 :

 2 SupF T (l + 1jl) sequential tests for structural breaks, Other components ( 12 p t ), Lag = 3, h = 0:1 l + 1 l SupF T (l + 1jl) the Federal Reserve of St Louis 5 , except the Divisia indices computed by the Center for Financial Stability 6 (CFS). The CFS monthly reports statistics for Divisia indices as the broad DM4, DM4-, which is DM4 without short term treasury bills, and DM3, which does not include treasuries nor commercial paper, corresponding to the discontinued M3 aggregate.[START_REF] Barnett | The New CFS Divisia Monetary Aggregates: Design, Construction, and Data Sources[END_REF] describe the construction of those monetary aggregates. The nominal price of money for each asset is computed using a corresponding interest rate and a benchmark rate; interest bearing checking accounts are paired with the national average interest rate on those accounts, for example. The benchmark rate is chosen as a maximum rate among the 'basket' of component rates along with other comparable loan rates. Included in this basket is a short

				Cut of f (5%)
	1	0	43.40	16.76
	2	1	21.92	18.56
	3	2	19.40	19.53
	4	3	10.75	20.24
	Final (global) Estimates of the Three Breaks:
	Nov1982, Oct1991, Oct2007
			indpro t , Lag = 3, h = 0:1
	l + 1 l SupF T (l + 1jl)	Cut of f (5%)
	1	0	34.96	16.76
	2	1	09.42	18.56
	3	2	07.96	19.53
	Final (Global) Estimates of the Single Break:
	Dec1981
			irst t , Lag = 1, h = 0:1
	l + 1 l SupF T (l + 1jl)	Cut of f (5%)
	1	0	07.39	12.25
	2	1	04.25	13.83
	3	2	04.01	14.73
			irlt t , Lag = 2, h = 0:1
	l + 1 l SupF T (l + 1jl)	Cut of f (5%)
	1	0	12.09	14.06
	2	1	08.38	13.83
	3	2	06.50	14.73
	are collected from

Table 3 :

 3 SupF T (l + 1jl) sequential tests for structural breaks in variances for the two interest rates

			irst t , Lag = 0, h = 0:1
	l + 1 l SupF T (l + 1jl) Cut of f (5%)
	1	0	78.12	12.25
	2	1	13.25	13.83
	3	2	12.66	14.73
	Global Estimates of the Two Breaks:
	Sep1982, Nov2007
			irlt t , Lag = 1, h = 0:1
	l + 1 l SupF T (l + 1jl) Cut of f (5%)
	1	0	22.13	12.25
	2	1	13.35	13.83
	3	2	12.71	14.73
	Global Estimates of the Two Breaks:
	Jul1979, Nov2007

Table 4 :

 4 Sub-Periods for the non-causality tests Samples with less than 50 observations are not considered term lending rate to commercial and industrial …rms, a suggested in[START_REF] O¤enbacher | Divisia Monetary Aggregates for Israel: Background Note and Metadata[END_REF], who use the loan rate as the maximum hypothetical interest sacri…ced for the liquidity service of the real assets. The loan rate is commonly the highest and therefore the benchmark rate compared. Depending on the period there are between fourteen and seventeen component asset and interest rate pairs included in the broadest possible aggregate, Divisia M4 (DM4) to account for entry and exit of assets, changes in survey methodology, and innovation in the …nancial markets. Money market demand accounts (MMDA) do not enter into the survey until the early 1980s, and then exit the survey in 1991 as they are folded into the more general savings accounts survey that exists throughout the entire period. Given the above discrete approximation methodology a consistent Divisia aggregate is then produced for econometric analysis.

		Sub-Periods for y 1 t
	Sub-Period Points in the Database	Dates
	1	[1; 139[	[Jan1968,Jul1979[
	2	[179; 286[	[Nov1982,Oct1991[
	3	[286; 478[	[Oct1991,Oct2007[
	4	[479; 539]	[Nov2007,Dec2012]
		Sub-Periods for y 2 t
	Sub-Period Points in the Database	Dates
	1	[1; 139[	[Jan1968,Jul1979[
	2	[179; 286[	[Nov1982,Oct1991[
	3	[286; 338[	[Oct1991,Feb1996[
	4	[338; 478[	[Feb1996,Oct2007[
	5	[479; 539]	[Nov2007,Dec2012]
		Sub-Periods for y 3 t
	Sub-Period Points in the Database	Dates
	1	[1; 139[	[Jan1968,Jul1979[
	2	[179; 286[	[Nov1982,Oct1991[
	3	[328; 478[	[Apr1995,Oct2007[
	4	[479; 539]	[Nov2007,Dec2012]

Note:

5 :

 5 Long-term matrices A for the four sub-periods. Money de…ned as div4r t y 1 t = (div4r t ; indpro t ; irst t ; irlt t ; 12 p t ) 0

			Sub-period 1		Sub-period 2
			p = 4; k1 = 0:42; d = 1	p = 4; k1 = 0:51; d = 1
	2 6 6 6 6 4	1:00 0:01 1:09 0:91	0:01 0:99 0:53 0:97	0:00 0:00 0:84 0:01	0:00 0:00 0:05 0:90	0:05 0:01 4:73 0:38
		0:03	0:03	0:00	0:00	0:95

Table 6 :

 6 FM-VAR causal structure. Non-causality variable j in equation i. Main entries are the Wald statistics and the p-values (between parentheses)

				Sub-period 1
			p = 4; k1 = 0:42; d = 0; AIC = 37:41; BIC = 34:76
	# i	! j	div4rt	indprot	irstt	irltt	12pt
	divrt		******	9:339 (0:053) 2:667 (0:615)	8:001 (0:091) 4:243 (0:374)
	indprot		2:947 (0:566)	******	5:742 (0:219)	0:456 (0:977) 1:840 (0:765)
	irstt		15:712 (0:003)	3:420 (0:490)	******	1:425 (0:839) 5:222 (0:265)
	irltt		3:851 (0:426)	4:570 (0:334) 2:595 (0:628)	******	3:239 (0:519)
	12pt		3:643 (0:456) 14:311 (0:006) 8:770 (0:067) 18:793 (0:000)	******
				Sub-period 2
			p = 4; k = 0:51; d = 0; AIC = 34:57; BIC = 30:59
	# i	! j	div4rt	indprot	irstt	irltt	12pt
	divrt		******	10:150 (0:038) 9:497 (0:049)	5:347 (0:235) 3:853 (0:426)
	indprot		9:143 (0:057)	******	1:084 (0:896)	1:857 (0:762) 7:823 (0:098)
	irstt		4:672 (0:322)	0:935 (0:919)	******	13:708 (0:008) 1:201 (0:878)
	irltt		6:879 (0:142)	7:296 (0:121) 1:106 (0:893)	******	3:534 (0:473)
	12pt		1:820 (0:768)	1:358 (0:851) 1:737 (0:783)	0:682 (0:953)	******
				Sub-period 3
			p = 5; k1 = 0:64; d = 1; AIC = 39:98; BIC = 36:84
	# i	! j	div4rt	indprot	irstt	irltt	12pt
	divrt		******	17:434 (0:004) 12:893 (0:024)	8:559 (0:128) 2:642 (0:754)
	indprot		8:647 (0:124)	******	9:444 (0:093)	4:147 (0:528) 3:362 (0:644)
	irstt		1:803 (0:875) 17:691 (0:003)	******	13:947 (0:016) 2:431 (0:787)
	irltt		3:103 (0:684)	3:613 (0:606) 17:441 (0:004)	******	3:821 (0:575)
	12pt		2:419 (0:788)	7:862 (0:164) 15:322 (0:009)	0:756 (0:981)	******
				Sub-period 4
			p = 4; k1 = 0:53; d = 0; AIC = 34:57; BIC = 30:59
	# i	! j	div4rt	indprot	irstt	irltt	12pt
	divrt		******	33:852 (0:000) 23:921 (0:000) 7:430 (0:115) 17:023 (0:002)
	indprot		0:810 (0:937)	******	6:864 (0:143) 4:300 (0:366)	0:511 (0:972)
	irstt		9:664 (0:046) 32:042 (0:000)	******	7:938 (0:094)	1:932 (0:748)
	irltt		1:572 (0:813)	9:187 (0:056)	7:265 (0:122)	******	5:035 (0:284)
	12pt		11:601 (0:021)	6:851 (0:144)	2:047 (0:727) 4:748 (0:314)	******

See Bauer and Maynard (2012) for computational details.

All the routines, i.e the FM-VAR model, the surplus-lag VARX as well as the Bai and Perron (1998) tests (see below) are programmed using SAS/IML and are available under request at the corresponding author email adress: philippe.de-peretti@univ-paris1.fr

http://research.stlouisfed.org/fred2/

www.centerfor…nancialstability.org

The closest major revision to the data methodology as described inBarnett et al (2013) is the discontinuation of M3 components including overnight repurchase agreements, requiring a new data source for such components. The discontinuation occurs in March of 2006, a little more than a year before the identi…ed break in October 2007.

Unfortunately the retail sweep program for the St. Louis Fed has been discontinued. The data has not been available since March 2012, and those wishing to incorporate sweeps are left to estimate them as needed.

SeeAnderson and Rasche (2001) for a more detailed description of sweeps programs. 1 0 Following Andrews and Monahan (1992), a pre-whitened method is used in all models.

 [START_REF] Andrews | Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation[END_REF]This project has received funding from the European Union's Seventh Framework Programme (FP7-SSH/2007-2013) for research, technological development and demonstration under grant agreement no grant agreement 320270 -SYRTO

1:01 0:00 0:00 0:00 0:05 0:05 1:00 0:00 0:00 0:07 5:71 3:49 0:68 0:32 14:87 2:16 2:19 0:00 0:79 0:91 0:03 0:03 0:00 0:00 0:95 0:98 0:01 0:00 0:00 0:05 0:00 0:99 0:00 0:00 0:05 0:11 0:32 0:99 0:06 0:46 0:62 0:43 0:01 0:84 3:67 0:00 0:01 0:00 0:00 0:88


Is big data enough? A reflection on the changing role of mathematics in applications Domenico Napoletani * , Marco Panza † , and Daniele C. Struppa ‡

The advent of computers, and especially high performance computers, has had a dramatic impact on the way in which mathematics is done, and even more on how mathematics is applied, as demonstrated by the growth of computational mathematics as well as what goes under the name of "experimental mathematics", to which a journal is now devoted. More importantly, computers are now used to perform highly complex computations in order to apply mathematical models to a variety of empirical problems that could never be attacked otherwise. It is in this way that mathematics is often now applied in different branches of biology (think of genomics and proteomics) as well as to social sciences, and in what now goes under the generic term of "big data".

High performance computing has also brought changes in the way we think about mathematics, its power, its methods, and the way in which it can be used to solve problems. For this reason, we would like to do something unusual in a mathematical journal, by bringing the mathematical community into one of the discussions that are taking place among philosophers of mathematics and of science. While many mathematicians are weary of asking broad methodological questions, because the danger of being vague and unclear is all too real, we also think that we must have this discussion concerning the modalities in which mathematics is applied, if we don't want to be trapped by our own assumptions, and miss out on more fruitful approaches to understanding reality. This a discussion about the ways computational mathematics has not only changed our approach to science, but even our way to understand a phenomenon.

We believe it is possible to identify four methodological motifs that are closely related to each other and whose elucidation may make explicit the approach to problem solving in data analysis and statistical learning. By using short and evocative labels, we could say that these motifs concern, respectively, the microarray paradigm, the preeminence of historical phenomena, the conceptualization of developmental processes, and the principle of forcing, which we have introduced and partially expanded upon in [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF][START_REF] Napoletani | Artificial diamonds are still diamonds[END_REF][START_REF] Napoletani | Processes rather than descriptions?[END_REF], also in response to some acute commentaries [START_REF] Humphreys | Data Analysis: Models or Techniques? Foundations of Science[END_REF][START_REF] Lenhard | Coal to Diamonds[END_REF]. Here we would like to simply focus on their rationale and potential significance in clarifying the underlying trends of quantitative, data-driven science.

The first methodological motif, the microarray paradigm, is derived by considerations that have become quite important in modern molecular biology. As it is well known, DNA microarray technology (see for example [START_REF] Baldi | DNA Microarrays and Gene Expression: From Experiments to Data Analysis and Modeling[END_REF]) allows the scientist to capture and visually represent, on an extremely large array of microscopic sites, information on the expression level of short strands of messenger RNA (mRNA), extracted and amplified from a given cell population. When the large amount of data derived from DNA microarrays is coupled with simple clustering techniques, it is often sufficient not only to differentiate cell populations from distinct tumors, but also to determine the outcome of a given therapy, without the need for an understanding, in each tumor population, of the actual role of each of the mRNA strands whose expression level is quantified by the microarrays.

Thus in [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF] we have spoken of the microarray paradigm to indicate how much modern data analysis is supported by the belief that sufficiently large data collected from a phenomenon will allow to answer any question about the phenomenon itself, if treated with appropriate methods, and assisted by powerful enough algorithms. Significant methods in statistical learning theory that embody the microarray paradigm include the aforementioned clustering techniques, in their more traditional hierarchical forms [START_REF] Ward | Hierarchical Grouping to Optimize an Objective Function[END_REF], but especially in their more unstructured versions such as the affinity propagation method recently introduced in [START_REF] Frey | Clustering by Passing Messages Between Data Points[END_REF], that exploits a completely local passage of information among data points to obtain their fast and accurate splitting into clusters, whose best representative, the 'exemplar' is also identified. Boosting [START_REF] Freund | A decision-theoretic generalization of online learning and an application to boosting[END_REF] is another striking technique, perhaps the most transparent embodiment of the microarray paradigm, where a large number of classifiers, slightly better than random guesses, are combined, boosted, to build a new classifier that is much more accurate than its components (cf. [START_REF] Hastie | The Elements of Statistical Learning[END_REF], chapter 10). Finally, we mention here the recent field of 'nonlinear manifold learning', a collection of methods to find low dimensional objects, preserving some type of local, neighborhood structure, in unstructured, high dimensional data (see [START_REF] Tenenbaum | A Global Geometric Framework for Nonlinear Dimensionality Reduction[END_REF][START_REF] Roweis | Nonlinear Dimensionality Reduction by Locally Linear Embedding[END_REF] and [START_REF] Izenman | Modern Multivariate Statistical Techniques[END_REF], chapter 16).

As we can see from these examples, within the microarray paradigm, answers are found through a process of automatic fitting of the data to models that do not carry any structural understanding beyond the actual solution of the problem itself, a distinctive lack of knowledge which we wanted to emphasize in [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF] by speaking of "agnostic science". The somewhat naive belief in the assumptions of the microarray paradigm was popularized in a recent cover article in Scientific American [START_REF] Weinberger | The Machine That Would Predict the Future[END_REF], where the author discusses how the collection of unprecedented amount of data could allow the construction of a computing model to predict the future, and even earlier in the opinion piece of Chris Anderson in Wired magazine [START_REF] Anderson | The End of Theory: The Data Deluge Makes the Scientific Method Obsolete[END_REF] that envisioned a future of atheoretical, automatic science.

Our objective in [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF] was less concerned in stating that such paradigm has gained ground, or why this has happened, than in uncovering its structure, and the assumptions it depends upon. Indeed physicists often work and approach their science from a very different point of view. Regardless of how complex nature may be, there is a belief that a fundamental explanation can be found, and that a theoretical, explicative model can reached. So, for example, Newtonian dynamics is considered a fundamentally successful description of the universe, and an archetype of the way science should look like, irrespective of the fact that, within its framework, a solution of a basic question such as the three-body problem is, effectively, not computable. And yet these shortcomings have not led physicists to abandoning Newtonian dynamics: within non-relativistic velocities, and as long as the bodies we study are not too small, Newtonian mechanics is still (rightly) considered a perfectly good model of reality, indeed one of the most successful and enduring.

If it is not the lack of computability (as in the three-body problem) or the lack of precision (as it happens for the relativistic effects that Newtonian dynamics fails to account for or predict), then what does require the agnostic approach embodied in the microarray paradigm? We suggest that such an approach is required when the phenomena we want to deal with are historical in a sense that needs some explanation: at least in an intuitive sense, we call 'historical' those phenomena whose development can only be constrained locally (in time and/or space) by (potentially multiple) optimization processes acting on subsets of variables, and in such a way that the functions to be optimized change over long periods of time. It appears that many of the sciences, which have been less receptive to the classical process of mathematization, concern this sort of phenomena. Biology, economics, the social sciences in general, all are examples of disciplines that study phe-nomena whose development seems to be historical in the sense we suggested above. Our second broad methodological motif is thus the preeminence of historical phenomena in contemporary science.

Fitness landscapes, introduced by Sewall Wright in [START_REF] Wright | The roles of mutation, inbreeding, crossbreeding, and selection in evolution[END_REF], best exemplify and motivate the definition of historical phenomena: the evolution along such landscapes is partially shaped by local optimization constraints but, crucially, these constraints change dramatically their nature over time, so that such evolution is not likely to be fully described by any single global optimization process. The selective pressure, due to varying environmental conditions, on the genotypes of a given population is an example where a local search for an optimal phenotype is subject to constantly varying constraints. Recalling that alternate versions of a same gene are called 'alleles', the basic idea of fitness landscapes is to represent a population as a point in a high dimensional space determined by its average allele frequencies, and to represent the average fitness of the population by the value of a corresponding function; the graph of the function generates a multidimensional surface (which is just the fitness landscape) and the potential evolution of the population is described by the local maximization of the fitness on the graph. Note that the landscape itself will slowly change when the evolution of several competing populations (not necessarily from the same species) is considered at once, since the fitness of one of them will affect the environment of the other (see [START_REF] Gavrilets | of Encyclopedia of Ecology[END_REF] for a recent overview of these issues).

Coming back to the general definition of historical phenomena, we can assume that the time scale at which there is a switching from one local optimization process to another in the development of such phenomena, is much longer than the time scale of the optimization processes themselves. Again, fitness landscapes could provide a motivation for this assumption, since the environmental conditions, and the shape of the landscape, change slowly with respect to the change of the position of the individual genotype points ( [START_REF] Gavrilets | of Encyclopedia of Ecology[END_REF], page 1613). Note also that not all variables are likely to be subject to selective local optimization at the same time. Now it is possible, indeed it happens often, that specific questions about an historical phenomenon can be reduced to simple models, that require knowledge of only a few key measured quantities. But the models derived to answer each individual question are not only usually data-driven, but often unable to answer other relevant questions on the same phenomenon. And the methods exemplifying the microarray paradigm we mentioned earlier certainly lead to temporary models that have this characteristic. This inability to gain a global understanding is not surprising, if we think about the randomness and relative independence of the selective optimization pro-cesses that shapes individual characteristics of the current state of an historical phenomenon.

It may be that a full, structured understanding that potentially leads to answers for a wide enough class of questions about a typical historical phenomenon would require knowledge of its entire history, or the development of a proxy, a data-driven model that is likely as complex, and opaque, as the phenomenon itself, a perspective reminiscent of incompressibility ideas in Kolmogorov complexity [START_REF] Li | An Introduction to Kolmogorov Complexity and Its Applications[END_REF], but in the context of a whole set of suitable potential questions. The methodological danger is that the flood of data generated by our innumerable measuring devices may convince us that data is enough, that there is nothing beyond the microarray paradigm, and that opaque, enormous, data-driven models are the privileged way to approach phenomena, even though they become so similar to the famous map of Borges [START_REF] Borges | Del Rigor en Sciencia[END_REF], that was useless, since it was as big as the geography it was supposed to describe.

So the problem arises of how we can gain meaningful understanding of historical phenomena, given the tremendous potential variability of their developmental processes. Indeed, several techniques in mathematical modeling already show the usefulness of a partial historical modeling of key variables, evolving in time, describing a phenomenon, we think here of nonlinear time series analysis methods [START_REF] Kantz | Nonlinear Time Series Analysis[END_REF], where modeling of random and/or nonlinear processes includes knowledge of its past states at several time points; or of forecasting methods where ensembles of initial conditions are used to best predict the future state of complex systems, such as the ensemble Kalman filter [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF][START_REF] Evensen | Advanced Data Assimilation for Strongly Nonlinear Dynamics[END_REF], or the even less structured particle filters method [START_REF] Kotecha | Gaussian particle filtering[END_REF]. All these methods show a distinct awareness of the relevance of modeling incidental, historical developmental processes. And we should not forget that many successful heuristic methods for optimization are directly inspired to the idea of fitness landscapes. One example is the field of evolutionary algorithms [START_REF] De | Evolutionary Computation[END_REF], modeling optimization of a function as a form of reproduction of solutions, that allows tentative solutions to the optimization problem to generate new ones through mutation and genetic crossover, and therefore allowing unexpected changes of their fitness, seen as the evaluation of the function to be optimized at the tentative solution. Another example is particle swarm optimization [START_REF] Kennedy | ShiSwarm Intelligence. Morgan Kaufmann[END_REF], where the search for optimal solutions to a problem is seen as a collective process in which individual, tentative solutions are changed in time not only by their tendency to improve their current fitness (in the fitness landscape generated by the function to be optimized), but also by their tendency to retain close contact with the best known tentative solution.

Still, all these methods work more at the operational level, where model classes have already been chosen to solve a certain problem. They do not constitute, taken individually, a coherent, conceptual shift in the way to approach historical phenomena, and their complexity. We can try to imagine more radical ways in which such complexity can be tackled by looking at existing attempts within biology to conceptualize rules of development.

In [START_REF] Minelli | A principle of Developmental Inertia[END_REF] for example a general "principle of biological inertia" is introduced, to give a broad conceptual basis to the multiform expression of default dynamics in developmental biology. Roughly stated, the principle asserts that, without external disturbances or internal (genetic) control, there is a "local self-perpetuation of cell-level dynamics" ( [START_REF] Minelli | A principle of Developmental Inertia[END_REF], page 119). Among the many embodiments of this principle, in the context of embryo development, biological inertia would correspond to stating that embryos have a tendency to reproduce spatially the same basic structure indefinitely. This embodiment of biological inertia could be seen in the context of historical processes as due to local optimization functions that tries to maximize replicates, or spatial distribution, of a basic template. However, in a real system we cannot expect this inertial behavior of developmental systems to be indefinitely exact ( [START_REF] Minelli | A principle of Developmental Inertia[END_REF], page 123), therefore the local optimization process will break down in the timeframe of full development and a complex, inhomogeneous organism may arise. Note the subtle and nonconventional use of ideas from physics where only analogy is at work: the principle of biological inertia parallels the principle of inertia in mechanics, but only at the very general level of establishing the equivalent for biological systems of a state of rest or dynamical invariance. At the same time, what is essential in the idea of physical inertia is retained, and its usefulness is seen in the power of conceptualizing the developmental process itself, rather than in the ability to identify and predict the final outcome of the process. This logically rigorous and yet informal use of ideas from mathematics and physics to define principles that partially govern biological processes might become the standard for a proper structuring of historical sciences, exactly because individual historical phenomena do not seem amenable to a compact explanation of their structure, and only the development that led to their current state may be open to meaningful theorization, a shift advocated by our third methodological motif, the conceptualization of developmental processes.

Indeed, the theory of evolution could be seen as the archetypical example of this motif. Such theory, in itself, is not mathematical, and does not allow quantitative predictions, but it has its internal logical structure, and provides a conceptual scaffolding that has inspired biology, and specific mathematical models, since its inception. As an aside, we believe that our point of view is in line with the recent commentary by Wilson [START_REF] Wilson | Great Scientists = Good at Math, The Wall Street Journal[END_REF], who suggested that profound ideas in science need not to be mathematically profound; in this perspective, mathematization is an overflow of the richness and potential of an idea, not a condition of its power.

What is crucial for historical sciences is that the conceptualization of a developmental process may not even lead to a structural, full understanding of the state of the resulting phenomenon. The phenomenon may forever remain hidden to our understanding, and the microarray paradigm will stand as the only way to find quantitative answers to most problems we will ask about it. Conceptualization, at best, can provide the ground, the language, on which to develop data-driven, agnostic methods to solve problems. This acknowledgment does not hinder however the tremendous potential of finding ways to use mathematical structures to solve problems about historical phenomena.

To reflect on this potential, it may be useful to change our focus. Up to now our discourse moved from the microarray paradigm to an attempt to characterize the types of phenomena that most likely will require its application. But we could also reverse this viewpoint, taking for granted the applicability of the microarray paradigm (i.e. that we have enough data), and trying to understand, operationally, how to apply it.

One way to do that is of course to give free rein to statistical learning, data-driven techniques. But here we want to discuss a more general organizational motif: the principle of forcing1 , which we introduced in [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF]. In that paper, we suggested that several disparate techniques developed to apply sophisticated mathematics to empirical problems can be brought together under a common methodological viewpoint, the idea of forcing mathematical ideas and methods on the data. More precisely, by forcing we mean the application to a problem of powerful techniques -such as multiscale methods as applied to image processing and numerical solutions of differential equations ( [START_REF] Brandt | Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics[END_REF][START_REF] Brandt | Multiscale Scientic Computation: Review[END_REF]), continuity and functional data analysis as applied to regularization and statistical analysis [START_REF] Ramsay | Functional Data Analysis[END_REF][START_REF] Ramsay | Applied Functional Data Analysis[END_REF], or topological graph analysis as applied to classification of molecular structures [START_REF] Bonchev | Chemical Graph Theory: Introduction and Fundamentals[END_REF]-not on the basis of a previous evaluation that these techniques fit with the relevant phenomenon because of its specific nature, but rather on the basis of a priori confidence in the power and flexibility of these techniques [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF], and using the large amount of available measurements to adapt the technique to the phenomenon. This is an approach that contrasts with the usual idealization process in modeling, where there is a progressive stripping away of details from the phenomenon, to reach a simplified image of the same, eventually amenable to analytical treatment by a mathematical technique (whose usefulness is often suggested by the idealization process itself).

The example of functional data analysis is perhaps the simplest instance of forcing, and the most telling: a phenomenon may be discrete, and yet, if there is enough data (microarray paradigm) we can force regularization to be able to treat the data as if they were continuous or even smooth, and therefore access the full machinery of analysis. A more recent example of forcing, along the same general trust of functional data analysis, is diffusion geometry [START_REF] Coifman | Geometry, analysis and signal processing on digital data, emergent structures, and knowledge building[END_REF][START_REF] Szlam | Regularization on graphs with function-adapted diffusion processes[END_REF]. The organizational principle of this theory can be seen, at a very general level, as the belief that, regardless of the whether the data available in the empirical problem are categorical, and/or discrete, it is useful to define a notion of geometrical manifold associated to the data, because in this way the whole apparatus of functional analysis on manifolds can be adapted and used to solve the problem itself. For example, the review [START_REF] Coifman | Geometry, analysis and signal processing on digital data, emergent structures, and knowledge building[END_REF] describes how scientific journals can be seen as points on a low dimensional manifold, built by first associating each article to a vector of the frequencies of preselected words in the article itself, and then by projecting the cloud of points associated to the whole set of articles onto the directions of maximal variance. We refer to [START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF][START_REF] Coifman | Geometry, analysis and signal processing on digital data, emergent structures, and knowledge building[END_REF] and to the relevant primary literature [START_REF] Szlam | Regularization on graphs with function-adapted diffusion processes[END_REF] for more details.

To be sure, forcing may be viewed as a coarse and willful attempt to use mathematics for specific purpose, but are there limits to the applicability of forcing? Is it plausible to think that, for sufficiently large and diverse data sets, any mathematical structure can be forced on them in a computationally efficient way to solve problems?

While we do not have answers to these questions, the weakening of the relation between individual models and phenomena should make us think more deeply about the role of mathematics in science. We suggested above that historical phenomena may not be amenable to uniform, structured reduction to simple models, and in these cases a data analysis approach to each problem instance is most likely the best that can be expected when approaching them, whether by forcing, or by standard statistical learning and classification techniques. And these data-driven applications of mathematics lead us to an intriguing version of the famous question of Wigner about the unreasonable effectiveness of mathematics: how can classification methods, which are essentially function fitting on data, be so successful at predicting phenomena? Our suggestion is that the four methodological motifs we highlighted may give us a context for asking such questions and for a comprehensive reflection on the ways modern science is so effective at problem solving. We are also encouraged by the interdependence of the motifs: the notion of historical phenomena, and the principle of forcing seem pertinent only in the context of large data sets, and therefore are deeply dependent on the microarray paradigm; and defining historical phenomena naturally shifts the emphasis from phenomena's states to the developmental processes that generate them.

Surely science may have become agnostic, and data analysis methods are often incapable of providing understanding of phenomena, they only give answers to our problems. But this does not imply that its methods should not be amenable to understanding. On the contrary, it is exactly the absence of understanding of phenomena that brings urgency to a widely shared methodological and epistemological reflection of the scientific community. Mathematics may not work necessarily for historical sciences the way it did for physics, but that does not mean that it has to reduce itself to blind computations, and principles such as biological inertia show that it is possible to gain deep insight into the rules of this historical development.

And therefore this is, in the end, an invitation to mathematicians, to approach biology and other historical sciences on their own terms, a process that frustrates superficial knowledge of each field, and challenges us, if we want to be relevant, not so much to be interdisciplinary, as to be scientifically bilingual. We may discover that what is essential in a field, and the true linchpin of its conceptualization, is often very different from what we deem profound or interesting in our own mathematical disciplines.

We named this principle 'forcing' in[START_REF] Napoletani | Agnostic science. Towards a philosophy of data analysis[END_REF] because this term is strongly evocative of the general methodological approaches it pertains to. It has however no significant relation to the method introduced by Paul Cohen, now of common use in set theory, that goes by the same name.

Acknowledgments

We thank the three anonymous referees, a mathematician, a philosopher, and a biologist, for their thoughtful and very helpful comments.


Philippe De Peretti

Oren J Tapiero

A GARCH analysis of dark-pool trades

The ability to trade in dark-pools without publicly announcing trading orders, concerns regulators and market participants alike. This paper analyzes the information contribution of dark trades to the intraday volatility process. The analysis is conducted by performing a GARCH estimation framework where errors follow the generalized error distribution (GED) and two different proxies for dark trading activity are separately included in the volatility equation. Results indicate that dark trades convey important information on the intraday volatility process. Furthermore, the results highlight the superiority of the proportion of dark trades relative to the proportion of dark volume in affecting the one-step-ahead density forecast.

Introduction

The ability to trade in dark-pools, without publicly announcing trading orders concerns market participants and regulators alike. In 2009, SEC Chairman and head of division on trading and markets -Mary Schapiro and James Brigagliano -expressed their concern indicating that the trading activity in dark-pools (also known as dark liquidity) may impair the price discovery process. In an article on the New-York Times (March 31 st , 2013), regulators have further expressed their concern that such an impairment of the price discovery process would eventually drive ordinary investors away from the markets. Therefore, regulators suspect that dark-pools may negatively affect trading liquidity. To address these concerns, some countries have taken regulatory measures over dark trading. Canada, for example, heavily regulates this activity by allowing these kinds of trades only if there is a significant price improvement relative to executions on public exchanges. While, in Australia regulators have recently1 proposed to impose a minimum threshold for orders in dark-pools. Another potential concern for regulators is that it may be a potential venue for price manipulations. For example, a trader may push up the price on the public exchange (by issuing multiple buy orders) while simultaneously selling in the dark-pool. Nevertheless, Kratz et Al. (2011) overrules this possibility.

There are several incentives for institutional investors to trade in dark-pools. First they are not obliged to make their intentions public. This implies that an institutional investor is able to execute large orders with fewer trades and without significantly affecting market impact risk. Boni et al. (2011) support this claim by indicating improved execution quality for large trades carried in dark-pools. Thus, combined with mid-quote pricing, overall transaction costs paid by the institutional investor decreases. However, an investor engaging in this activity faces an execution risk because the dark-pool does not guarantee trading executions. This may imply that in moments of high intraday price volatility, the investor will prefer to trade in public exchanges.

Another incentive to trade in dark-pools relates to information asymmetry. Zhu (2011) state that dark-pool allows investors to avoid trading against an informed order-flow. Moreover, both medias (e.g.: the New-York Times and the Financial Times) and regulators assert that dark-pool activity has been on a rise almost in tandem with high frequency trading. In other words, dark-pool activity may reflect institutional investors' distrust of public exchanges due to high frequency trading activity2 . Provided this is true and provided institutional investors are able to detect high frequency trading activity, trading in dark-pools may coincide with the latter trading activity. Hence, the study of dark-pools may (perhaps indirectly) relate to the high frequency trading activity.

While regulators and CEO's of public exchanges3 have expressed their concerns, academic papers indicate some of the potential benefit and problematic of dark-pool trading activity. [START_REF] Buti | Diving into Dark Pools[END_REF] indicate dark-pool trading activity is higher on days with high share volume, low intraday volatility and high depth. Hence, overall market quality improves. O'Hara and [START_REF] Ye | A Glimpse into the Dark: Price Formation, Transaction Cost and Market Share of the Crossing Network[END_REF] find that market fragmentation (in general) does not impair overall market quality.

Moreover they find that while short-term volatility has increased, price dynamics has become closer to the random walk (implying greater market efficiency). At last they find that overall executions are faster and transaction costs are lower. Nevertheless, [START_REF] Ye | A Glimpse into the Dark: Price Formation, Transaction Cost and Market Share of the Crossing Network[END_REF] indicate that introducing a dark-pool does negatively affect price discovery on the public exchange while improving overall liquidity. This improvement is explained by less informed trading on the exchange. Weaver (2011) also finds a negative relationship between increased dark-pool activity and market quality (i.e.: price discovery) by indicating the positive effect it has on the measures of bid-ask spread. On the other hand, Zhu (2011) indicates that while price discovery is improved by the presence of a dark-pool, liquidity is reduced in public exchanges. Ready (2012) analyzes volume in dark-pools to finds that lower stock spreads (in dollars term) coincide with reduced dark-pool activity, which conforms Zhu's (2011) prediction. [START_REF] Nimalendran | Informed trading in dark pools[END_REF] mitigate the "price discovery impairment" argument by indicating the possibility that informed traders may also trade in dark-pools and therefore "spilling" information into the quotes that are seen in public exchanges. Nevertheless, two years later, the same authors (using propriety data) find increased quoted spreads on public exchanges following dark-pool transactions [START_REF] Nimalendran | Informational linkages between dark and lit trading venues[END_REF]. Moreover, they find that "informed traders" may be concurrently trading in the "light" and in the "dark".

To compete with dark-pools public exchanges (e.g.: Euronext-Paris, BATS, NASDAQ, NYSE and others) have started to allow traders to hide some or all of their order size. [START_REF] Bessembinder | Hidden liquidity: an analysis of order exposure strategies in electronic stock markets[END_REF] (using data from Euronext-Paris) find that hidden orders take more time to be executed and that there is some execution risk associated with these orders. However, they also find that allowing hidden orders does not drive away "defensive" investors from the exchange. Buti and Rindi (2013) find that allowing hidden orders on public exchanges benefits large traders, while small traders are beneficial only when the tick size is large. Furthermore, they find that internal spreads widen with presence of hidden orders. Therefore, overall it seems that the effect of hidden orders on trading is to an extent similar to effect of dark-pools.

Using data on Microsoft (MSFT), on a millisecond timestamp and provided by the Trades and Quotes (TAQ) database, we analyze the predictive content that dark-pool trading activity may have on return process. To that end we apply a GARCH model to a microstructure problem, where either of the two proxies for dark pool trading (henceforth, dark-trading) are included as explanatory variables. The first proxy is dark trading volume while the second is the number of dark-trades. Both proxies are set within a pre-specified time intervals of 5 minutes. We find that in predicting future intraday returns, the proxy for proportion of dark-trades (within the prespecified time interval) over-performs the proportion of dark-volume. This over-performance is even more striking when accounting for non-linear effects. Nevertheless, including either of these proxies in the GARCH estimation framework over-performs a simple AR(1)-GARCH(1,1)-GED model in forecasting both the center and tails of the distribution. Our results highlight the informational content that dark trades may bear. They also highlight that for the market, the size of dark trades (in monetary terms) is not as important as the frequency at which they occur. This is especially important when considering extreme one-step ahead realizations of returns for which the number of dark-trades is more predictive. Furthermore, as our results indicate, it becomes even more important when considering the non-linear effects that darktrading may have on the returns process. This paper is divided into four sections. The first section describes the dataset used for this work. The second section describes the empirical methodology used in this paper. The third section presents and discusses the empirical results and the last section concludes this paper.

Data description and analysis

We retrieved data from the Trades and Quotes (TAQ) database. The database contains two distinct files, one indicating quotes and another indicating transactions. The data set is time stamped to the milliseconds and reflect the transactions made within active markets hours, i.e.: 9:30:00:000 to 16:00:00:000. The transaction file contains all the transactions made in the existing trading venues. We choose the period that starts on January 2013 and end on March 2013 as our sample period and we choose Microsoft stock as a reference case.

The information on dark-trading is indicated with the letter 'D' in the 'Exchange' data column. To be precise, the designated letter 'D' indicates all trades reported by the Financial Industry Regulatory Authority (FINRA), which oversees trades executed in other Trade Reporting Facilities (TRF) including dark-pools. We indicate that this variable has been used as a proxy for dark-pool trading in Boni et al. (2011) and [START_REF] Weaver | Internalization and market quality in a fragmented market structure[END_REF]. Furthermore, [START_REF] Weaver | Internalization and market quality in a fragmented market structure[END_REF] indicates that 90% of all TRF trades are executed in dark-pools. Therefore, we assume that the 'Exchange' variable provides an adequate proxy for dark-pools trading activity. The transaction files also contain information on trade size and the condition at which it was executed. Thus, we are able to have an approximation of the proportion of dark-pool trading both in monetary and quantity measures, i.e.: the volume that is traded in the dark (monetary) and the number of dark-trades (quantity).

Using the transactions data we compute 5, 15, 30 and 60 minutes log-returns, = ln ( ⁄ ), using only prices reported on public exchanges. Where the price ( ) used to calculate log-return is the last observed price within a predetermined time interval. Then, using the 'Exchange' variable and the indicating letter 'D' in the TAQ transaction data, we compute our two proxies. The first is the proportion of volume traded in the dark designated by the letter , while the second is the proportion of dark-trades designated by the letter . Where, the two variables (for each time interval) are calculated in the following manner:

= × × (1) = (2)
( ) is the transaction price of the s'th public (dark) trade, ( ) is the quantity traded in the s'th transaction carried in public (dark) exchange and ( ) total number of public (dark) transaction within a pre-specified time interval.

Empirical methodology

Let { } be a series of returns, and assume that the data admit the following data generating process (DGP)4 :

= + ! + " #ℎ ℎ % = & ' + & " % + (ℎ % + )* (3)
Where:

• , &, & ' , &
, and ( are parameters to be estimated.

• & ' > 0, α > 0, ( > 0 and α + ( < 1 • ) is a (1 × 0) vector of parameters associated with the (0 × 1) matrix of exogenous variables * .

• " ∼ 3(. )
Moreover, to capture excess kurtosis in intraday returns, define 3(. ) as the Generalized Error Distribution (GED) law with 5 degrees of freedom [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF]. We shall refer to the above model as the AR(1)-GARCHX(1,1)-GED model ('X' standing for included external explanatory variables), which is reduced to the AR(1)-GARCH(1,1)-GED model if ) = 0.

To analyze the informational contents of dark-trades, on the conditional variance (h t ), and then on returns, we focus on the predictive accuracy of competing AR(1)-GARCHX(1,1)-GED models, each one differing by the variables included in the matrix X t . Especially, we focus on pairwise comparisons based on the accuracy of out-of-sample one-step-ahead density forecasts.

The use of density forecasts for comparing both nested and non-nested models is popular in economics (e.g.: Tay and Wallis, 2000). This approach bears several interesting features. First, since a density forecast is an estimate of the full one-step-ahead probability distribution function of a random variable (conditional on an information set), the comparison takes place over the full distribution (or over some regions of the distribution). Therefore, it enables us to see how dark trading informs us on tail events which is of main concern for the financial regulator. Second, the competing models are allowed to be only an approximation of the true underlying DGP. In other words, they are allowed to have a certain degree of misspecification. Third, tests are designed to deal with heterogeneous data. Fourth, for two nested models, the suggested approach allows to analyze the marginal influence of a given exogenous explanatory variable in terms of predictive content. Thus, providing information that is different from the one provided by the standard Student t-statistics.

Following [START_REF] Amisano | Comparing density forecasts via weighted likelihood ratio tests[END_REF], define 6 = ( , * 7 ) 7 and let ℱ = σ(6 , 6 % , … , 6 ) be the information set at time t. Suppose we have two competing AR(1)-GARCHX(1,1)-GED models, say : (6 , 6 % , … , 6 ;< : ) ) and > ( 6 , 6 % , … , 6 ?< : ) % ) (where ) and ) % are parameters to be estimated) and we want to rank these models according to their out-of-sample one-step-ahead forecast accuracy. We can either analyze point forecasts (e.g. Clark and McCraken, 2009) Define, E ∈ (max(I, J) , ( -1)/ ). Using a rolling scheme, one can estimate the two models on the time period 1: M = int(E ). Then, produce density forecasts and re-estimate the model on 1: M = int(E ) + 1. This procedures is repeatedly carried on, yielding two sets of n log-scores -{ln (B C ( < ))} PQR(λ ) and {ln (B D ( < ))} PQR(λ ) . Note that by using this scheme, we allow the models to capture structural changes in the parameters as well as in the kurtosis of the returns. To test for null of equality of density forecasts, the following statistic is used:

M S = T3U VVVVVVV λ ,S W X/√1 (4)
Where:

-Draft -

13 • T3U VVVVVVV λ ,S = 1 ∑ T3U λ , < PQR(λ ) , • T3U λ , < = [( < ) \logB C ( < ) -logB D ( < )_,
• W X is an heteroskedastic and autocorrelation consistent (HAC) estimator of the standard error of T3U λ , < over the n considered periods.

• [( < ) is a weighting function discussed below.

• < is the observed standardized returns defined as < = ( < -`Ŝ )/W X S , where `Ŝ and W X S are the unconditional mean and standard error of the n realizations of < .

Such a test is known as a Weighted Likelihood Ratio (WLR) test. Under the null, t n is distributed as a standard Normal deviate with unit variance. Notice that a large and significant positive value for M S leads to choose f(.) over g(.). While a negative value of M S will lead to choose g(.) over f(.). The weighting function w(.) is used to set to highlight a particular region of the density forecast. If w(.) is uniform, i.e. taking the value of 1 whatever < is (case 0), then the test highlights the entire distribution. Four other definitions of w(.) are of interest for any variable y with zero mean and unit variance:

•

Empirical results and discussion

We implement the WLR test on intraday returns on Microsoft traded shares where the data is aggregated at a five-minute time interval. As mentioned above, two proxies of dark trading are used in various competing models, and . Figure 9 plots the two different measures, together with their trends estimated using a spline. Cleary, the two series exhibit similar trends, but with different volatilities.

GARCHX(1,1)-GED (e = f) model

We next turn to pair-wise comparisons. Table 3 presents the seven competing models used in this study. The M 0 one is the reference model with ) = 0, whereas models M 1 to M 6 all include various proxies of dark trading. Tables 3 to 7 present the results of WLR tests. Main entries are the t n statistics and the p-values, between parentheses. A significant positive value for t n indicates that model M i (row) is to be preferred to M j (column) and conversely. Clearly, four kinds of information are of interest:

i)

The information content dark trading, relative to a simple AR(1)-GARCH(1,1) model,

ii)

The relative information contribution to future returns of the two proxies, i.e.:

proportion of dark volume ( ) versus proportion of trades ( ).

iii) Linear versus non-linear effects of dark trading, iv) Past versus contemporaneous effects.

V t D N t D -Draft - 16 i.

The information contribution of dark trading

We emphasize the first column of tables 3 to 7 with a special attention on models M 1 and M 3 (rows 2 and 4). The indicate that these models over-perform the simple (or, benchmark)

AR(1)-GARCH(1,1)-GED model (M 0 ) in terms of one-step-ahead density forecast (case 0).

Nevertheless, these two models do not provide the same information regarding returns on onestep-ahead forecast. For instance, using as an explanatory variable does not significantly improve the forecasting performances over the simple AR(1)-GARCH (1,1) model when only the center of the distribution is considered (table 4). However, it returns important information about right and left tails (Table 5). For regulators, evaluating the possible uncertainty associated with dark trading, it is an important result. Conversely, including does improve forecasts for both the center of the distribution and for the tails. In other words including or in the variance equation yields significant information about the likelihood of extreme intraday movements in the price of traded shares. ii.

Proportion of dark volume vs. proportion of dark trades

Including or in the variance equation provides different information when examining their effects on the center of the distribution. Nevertheless, these models significantly overperform the benchmark model (M 0 ) in analyzing distribution tails. Emphasizing the second column in table 4 (case 0) the two models appear to be equivalent. However, tables 5 and 6 reveal a slightly different reality, i.e.: model M 3 is over performs model M 1 when only the center of the distribution is considered. This result is consistent with the results reported earlier.

Moreover, table 6 indicates that model M 1 should be chosen when forecasts of the distribution tails are being emphasized. To summarize, the two proxies provide different information about future returns realizations. However, the proportion of dark-trades ( ) seem to provide superior information regarding the one-step-ahead density.

iii.

Linear vs. non-linear effects

Previously we have indicated that there might be a non-linear effects of dark trading on returns variance (Figures 1 and8). To examine this possibility, we perform a pairwise comparison of M 2 relative to M 1 and of model M 4 relative to M 3 . With regard to the former comparison, results are of a particular interest: M 2 over-performs M 1 (case 0). This result appears to be due to its ability to forecast the tails of the returns, especially the left tail (that corresponds to losses). Thus, it provides crucial information concerning the Value at Risk (VaR) metric. For the center of the distribution, M 1 still over-performs model M 2 .

A similar pattern appears in latter comparison since M 4 over-performs M 3. Especially when considering the tails (right and left). If we compare M 4 relative to M 2 , it seems that the former performs better while considering the center of the distribution. This is also the case when considering the right tail of the distribution. Therefore, the proportion of dark trades ( ) provides superior information while considering non-linear relationships with returns. This result is rather surprising (as as all previously discussed results) because the proportion of dark volume includes information on contemporary and past transactions price. Though not investigated here, it seems that these results provide evidence that intraday price is a martingale.

Conclusion

In this paper we have applied the GARCH estimation framework to a problem of market microstructure. More precisely, we have attempted to answer whether the activity of trading in the dark (trading in dark-pools) conveys any information on the intraday return and volatility process. Our results indicate that indeed dark trading activity conveys relevant information to the process determining one-step-ahead returns. Moreover, not only it conveys information over the one-step-ahead return forecast, it also conveys important information on the entire density forecast of returns. This, with a special emphasis on the tails of this density forecast. Hence, we conclude that dark-trading has an important role in determining intraday returns and the uncertainty that may relate to them.

Furthermore, our results indicate that number of dark trades within a predetermined timeinterval provides more information regarding the (one-step-ahead) point and density forecast of returns. Moreover, for non-linear relationships that affect the volatility process, the proportion of dark trades also provides more information. Though we do not discuss the issue of price discovery, it is obvious that dark trading has a role in the price discovery process. From our results, it seems that it may contribute to the price discovery process in the case of Microsoft stock.

Given highlighted results, dark trading may provide valuable information to regulators and market participants alike. For regulators, dark trading maybe provide information over the -Draft -20 effects of high -frequency trading, provided that dark trading activity coincides with the latter activity. Therefore, an important issue for further research is to empirically determine how trading in the dark coincides with high frequency trading. Determining this may provide an important piece of information for regulators in the activity of overseeing financial markets. Another important outcome of the indicated results is that dark-trading seems to be well integrated in current trading activity. Furthermore, as mentioned already, it seems that traders on public exchanges react to dark trading once it is exposed to the public.

Besides determining the relationship between high-frequency (or more generally, informed trading) and dark trading, further research will require to expand our stocks universe to include more stocks with different trading characteristics as well as different time periods. That is: we pre-assume that dark trading activity had a different few years ago. Thus it is necessary to analyze how dark-pool trading played role in the last ten or more years.

  plots the relationship between lead returns and lead returns absolute value with respect to and . These figures highlight the possibility of a concave relationship between dark trading and lead returns absolute value. That is, up to some threshold value dark trading activity is followed by increased returns absolute value.

Figure

  Figure 1 -Distribution of | |, , (5-minute time interval)

  or density forecasts. Since the latter represent the complete characterization associated with the one-step-ahead forecast, it contains all the relevant information. Furthermore, let B C (. ) and B D (. ) be the two out-of-sample one-step-ahead density forecasts and let ln (B C ( < )) and ln (B D ( < )) be the two log-scores evaluated at the outcome r t+1 . Amisano and Giacomini (2007) suggest a test based on a loss function that uses these logarithmic scoring rules.

  Case 1 (Center of the distribution): [(b) = c(b), where c(. ) is the standard normal density function. • Case 2 (Tails of distribution): [(b) = 1 -c(b) / c(0) , where c(. ) is the standard normal density function. • Case 3 (Right tail of distribution): [(b) = Φ(b), where Φ(. ) is the standard normal distribution function. • Case 4 (Left tail of distribution): [(b) = 1 -Φ(b), where Φ(. ) is the standard normal distribution function.

Figure

  Figure 9 -Time evolution of and (5-minute time interval)

Figure 10 -

 10 Figure 10 -Histogram, residuals, QQ-plots and squared returns for the AR(1)-

  At last, we analyze whether including past information about contributes significant information in volatility equation. By comparing models M 5 and M 6 to M 4 , it appears the latter performs best. This is valid for all considered case (the entire or only section of the distribution).

Table 1 -Summary statistics -transactions data

 1 i.e.: a two dimensional plot) reveals a concave relationship between absolute log-returns and the two proxies (proportion of dark trades and volume). Note that this concave relationship is likely to be related to dark trading occurring when volatility is low. Figures5 -8

		Stocks traded	Trade size (FINRA)	Log-returns (%)
			Time interval: 5 minutes
	Mean	1455	479	0.001
	Std. Dev	1087	331	0.122
	Median	1187	398	0
	Min	14	5	-1.82
	Max	11930	4993	1.37
	Obs.	4740
			Time interval: 15 minutes
	Mean	4258	1403	0.003
	Std. Dev	2929	911	0.2%
	Median	3551	1206	0
	Min	30	18	-1.62
	Max	30980	9761	2.04
	Obs.	1620
			Time interval: 30 minutes
	Mean	8215	2705	0.054
	Std. Dev	5409	1731	0.29
	Median	7012	2389	0
	Min	33	20	-1.53
	Max	53305	15571	3.50
	Obs.	840
			Time interval: 60 minutes
	Mean	14290	4886	0.012
	Std. Dev	9025	3136	0.39
	Median	13321	4546	0
	Min	38	22	-2.23
	Max	56761	20782	3.20
	Obs.	420
	Table 1 provides summary statistics (per pre-specified time interval) for log-returns, trade
	size reported by other reporting trading facilities (FINRA) and all other exchanges. Then, for
	each pre-specified time interval; the relationship between absolute returns, proportion of dark-
	trades ( ) and the proportion dark volume ( ) is plotted (figures 1 -4) in a three dimensional
	figure. A-priori, these figures indicate that the absolute value of log-returns decreases with

respect to the proportion of dark-trades ( ) and the proportion dark volume ( ). However, a 7 closer examination (

Table 2 -FIML Parameter estimates of the AR(1)-GARCHX(1,1)-GED (e

 2 = f) model

	Parameter	Estimate	Std. Err	t-stat	p-value
	! & ' & ( 5 Q-stat(1-6)	-0.08669 3.725E-9 1.629E-7 0.279467 0.616957 1.02131 8.44 (0.2074)	0.0334 1.776E-6 2.423E-8 0.0329 0.0377 0.0264	-2.60 0.00 6.72 8.48 16.35 38.68	0.0095 0.9983 <.0001 <.0001 <.0001 <.0001
	ARCH-LM (1-6)	0.92 (0.9885)

Table 3 -Estimated models

 3

				Variance Equation
	Model g ' g g % g h g i g j g k	Mean Equation AR(1) AR(1) AR(1) AR(1) AR(1) AR(1) AR(1)	GARCH (p,q) p=1,q=1 p=1,q=1 p=1,q=1 p=1,q=1 p=1,q=1 p=1,q=1 p=1,q=1	Exogenous Variables None , ( ) % , ( ) % , ( ) %	Distribution of Errors GED GED GED GED GED GED GED

Table 4 -Weighted Likelihood Ratio tests

 4

		Case 0 -(the entire distribution), λ λ λ λ=0.75.
	Model g ' g g % g h g i g j g k	g ' 3.956 (0) 3.487 (0) 5.851 (0) 4.962 (0) 2.274 (0.022) 2.343 (0.019)	g 2.840 (0.004) 0.169 (0.865) 4.190 (0) -2.167 (0.03) -1.395 (0.16)	g % -1.942 (0.05) 1.817 (0.069) -2.947 (0.00) -2.989 (0.00)	g h 4.224 (0) -2.476 (0.03) -1.391 (0.16)	g i -4.373 (0) -4.258 (0)	g j 0.458 (0.646)	g k

Table 5 -Weighted Likelihood Ratio tests

 5

			Case 1 (Center of distribution), λ λ λ λ=0.75.
	Model g ' g g % g h g i g j g k	g ' -0.568 (0.570) -2.242 (0) 3.792 (0) 1.566 (0.117) 0.496 (0.619) -1.699 (0.089)	g -3.329 (0) 5.852 (0) 2.494 (0.012) 1.121 (0.261) -0.970 (0.331)	g % 4.676 (0) 5.377 (0) 2.799 (0.005) 1.74 (0.080)	g h 0.409 (0.681) -3.099 (0.002) -5.243 (0)	g i -1.462 (0.143) -2.391 (0.016)	g j -2.429 (0.015)	g k

Table 6 -Weighted Likelihood Ratio tests

 6

			Case 2 (Distribution tails), λ λ λ λ=0.75.
	Model g ' g g % g h g i g j g k	g ' 5.525 (0) 5.825 (0) 5.601 (0) 5.688 (0) 3.782 (0) 3.481 (0)	g 5.645 (0) -2.805 (0.005) 4.059 (0) -4.139 (0) -1.286 (0.198)	g % -4.949 (0) -0.678 (0.497) -5.837 (0) -5.851 (0)	g h 5.362 (0) -1.344 (0.178) 0.756 (0.449)	g i -5.279 (0) -4.394 (0)	g j 2.111 (0.034)	g k

Table 7 -Weighted Likelihood Ratio tests

 7

				Case 3 (Right tail), λ λ λ λ=0.75.
	Model g ' g g % g h g i g j g k	g ' 3.046 (0) 2.318 (0) 5.551 (0) 4.915 (0) 2.237 (0.025) 2.212 (0.027)	g 1.607 (0.108) 1.537 (0.124) 4.992 (0) -1.547 (0.121) -0.167 (0.866)	g % -0.661 (0.508) 3.311 (0) -1.850 (0.064) -1.427 (0.153)	g h 4.276 (0) -2.835 (0.004) -0.981 (0.326)	g i -4.642 (0) -4.051 (0)	g j 1.140 (0.253)	g k

Table 8 -Weighted likelihood ratio tests.

 8

				Case 4 (Left Tail), λ λ λ λ=0.75
	Model g ' g g % g h g i g j g k	g ' 2.982 (0) 2.961 (0) 3.474 (0) 2.762 (0.005) 2.275 (0.022) 1.215 (0.224)	g 2.753 (0.005) -1.061 (0.288) 1.713 (0.087) -1.838 (0.065) -2.165 (0.030)	g % -2.255 (0.024) -0.285 (0.775) -2.653 (0.007) -3.254 (0.001)	g h 2.276 (0.022) -0.985 (0.324) -1.122 (0.261)	g i -2.252 (0.024) -2.584 (0.009)	g j -0.473 (0.635)	g k

Reuters web site (April 9 th ,

2013): http://www.reuters.com/article/2013/04/09/us-exchanges-sec-darkpool-idUSBRE93818520130409

[START_REF] Boni | Dark Pool Exclusivity Matters[END_REF] indicate that some dark-pools are specifically designed for institutional investors and discourage over participants such as high frequency traders.

Reuters web site (April 9 th , 2013): http://www.reuters.com/article/2013/04/09/us-exchanges-sec-darkpool-idUSBRE93818520130409

Before choosing the AR(1)-GARCH(1,1) model, we have estimated various models, also with different laws for the residuals (Student, Skew-Student, Skew-GED). Clearly the AR(1)-GARCH(1,1)-GED performs best.