Christophe Tavéra

Jean-Christophe Poutineau

Jean-Sébastien Pentecôte

Isabelle Cadoret

Arthur Charpentier

Chantal Gueguen

Maryline Huchet

Julien Licheron

Guillaume Loeillet

Nathalie Payelle

Sébastien Pommier

The "Mother of All Puzzles" at Thirty: A Meta-Analysis

Keywords: Feldstein-Horioka coefficient, capital mobility, saving-investment correlation, meta-analysis JEL Classification: F32, F40, C83 Feldstein-Horioka coefficient, capital mobility, saving-investment correlation, meta-analysis JEL Classification: F32, F40, C83

This paper provides a meta-analysis of 1651 point estimates of Feldstein and Horioka saving retention coefficient from 49 peer-reviewed papers published over three decades. We get two main results. First, correcting for publication bias, we find a consistent underlying coefficient lying between 0.56 and 0.67 for studies using the original paper. Second, heterogeneity reported in the estimates of the Feldstein and Horioka can be explained by a few main factors. In particular, we find evidence that the saving retention coefficient is systematically underestimated with models written in first difference, models using the saving ratio or the current account ratio as the dependent variable instead of the investment ratio, and models including indicators of the public deficit or indicators of the country size as additional explanatory variables.

Introduction

In a pioneering paper, [START_REF] Feldstein | Domestic saving and international capital flows[END_REF] investigated the consequences of international financial integration on the correlation between domestic saving and investment.

They argued that a low value of the saving retention coefficient should be observed in the case of perfect capital mobility. Testing this assumption over a cross section sample for 16 OECD countries, they found a value of the saving retention coefficient equal to 0.85 and failed to reject the null hypothesis of a one-to-one saving-investment association. They concluded that zero capital mobility was supported by the data, which appeared to be at odds with most theoretical models assuming perfect capital mobility and with the observed increasing financial integration. The Feldstein and Horioka puzzle was thus born.

More than thirty years after the publication of the original paper, the basic finding that national saving and investment are closely related has generated a huge theoretical and empirical literature. As an example, searching on EconLit for keywords related to the seminal publication of Feldstein and Horioka, we find that 994 articles have been published by December 2008, so that the literature on this topic has averaged 3 monthly papers since 1980.

The Feldstein and Horioka result still remains more or less a stylized fact of international macroeconomics. It has even been crowned as "the mother of all puzzles" in international finance [START_REF] Obstfeld | Perspectives on OECD Capital Market Integration: Implications for U.S. Current Account Adjustment[END_REF]. This impressive literature that spans over thirty years is heterogeneous on several aspects: the saving investment link has been measured extensively for advanced and developing countries, for nations and regions, through various regression specifications, various estimation methods, and with alternative data frequencies. As a consequence, reported results are hardly comparable with one another on a direct basis. The quantitative evaluation of an underlying ϰ consistent value of the saving investment relationship thus requires a more rigorous treatment such as the one provided by meta-analysis techniques.

Meta-analysis provides a set of quantitative methods to evaluate empirical results from different studies with similar characteristics or, alternatively, with different characteristics that can be controlled for. This method is helpful for clarifying controversial issues and has been used in various fields of economics such as international economics, public economics, transport economics... Regarding the field of international macroeconomics, this approach has already been applied to study different questions such as the trade effect of monetary union [START_REF] Rose | A meta-analysis of the effect of common currencies on international trade[END_REF], the impact of exchange rate volatility on trade (ûroriü and [START_REF] Pugh | The effects of exchange rate variability on international trade: a meta-Regression Analysis[END_REF], the correlation of business cycle between countries [START_REF] Fidrmuc | Meta-analysis of the business cycle correlation between the euro area and the CEECs[END_REF] or the analysis of capital controls [START_REF] Magud | Capital controls: Myth and reality. A portfolio balance approach[END_REF].

The aim of this paper is to use the statistical tools of meta-analysis to determine the underlying consistent value of the saving-investment coefficient that can be extracted from this literature. This statistical approach is well suited for the question at hand since most papers have an empirical content and report at least one main estimated value. We do not provide a broad survey on the subject and we refer to [START_REF] Apergis | A survey of the Feldstein-Horioka puzzle: What has been done and where we stand[END_REF] for a recent overview of this literature. Rather, we concentrate on a quantitative analysis of the underlying value of the saving retention coefficient that emerges from a selection of the published articles. Then we evaluate the factors that may explain discrepancies across estimates.

The conduct of our meta-analysis of the Feldstein and Horioka relationship is in line with the common practices of the field (Stanley et al., 2013, Havranek and[START_REF] Havranek | Estimating vertical spillovers from FDI: Why results vary and what the true effect is[END_REF]. In a first step we build a representative sample of the literature -called the meta-sample -that has been published in peer-reviewed journals or working paper series, and we evaluate whether the selected studies are subject to publication biases. In a second step we proceed to a meta-regression analysis (MRA) to determine whether the sample delivers a underlying true value for the saving retention coefficient, and to detect the possible factors that could explain the estimate heterogeneities.

In a previous study, [START_REF] Bineau | Une méta-analyse des études sur la mesure de la mobilité internationale du capital selon la méthode de Feldstein et Horioka[END_REF] provides a meta-analysis of the Feldstein and Horioka coefficient. We depart from this study on various aspects. First, we use a different meta sample to provide a complementary analysis of this question. We use a larger data set containing 1651 point estimates (instead of 1349) which proves worthwhile given the great number of papers published on this topic over the last 30 years. Second, his analysis mainly concentrates on the original way of measuring the puzzle that links the ratio of investment over output to the ratio of savings over output. In our paper we enlarge the analysis to alternative approaches provided in the literature (such as the current account as the dependent variable, the relation in variations rather than in levels…). Third, and mainly, [START_REF] Bineau | Une méta-analyse des études sur la mesure de la mobilité internationale du capital selon la méthode de Feldstein et Horioka[END_REF] finds evidence of a publication bias, but he leaves the correction for publication bias for future research, while the present study explicitly deals with the correction for publication bias.

To build a suitable meta-sample we have selected and codified 49 empirical studies drawn from Econlit with regard to their structure, keywords, statistical reports... This sample provides a total number of 1651 point estimates of the Feldstein and Horioka saving retention parameter. The whole sample mean of this parameter is 0.498, while reported values range from -1.70 to 2.09. Since the selected studies give 1 to 211 point estimates, we complete the OLS meta-regression with a mixed-effect multilevel estimation to correct for a potential dependence of results coming from the same paper or the same author(s). Our meta-sample is not subject to a Type I publication bias (which occurs when editors, referees, and/or researchers have a preference for a particular direction of results). However as it includes mainly peer reviewed articles we find a Type II bias (which arises when editors, referees, and/or researchers have a preference for results that are statistically significant). ϲ To understand the heterogeneity in the saving retention coefficient, we codify each of the 1651 point estimates of the meta-sample with regard to 19 main characteristics related to the data, the specification of the estimated equation, the estimation method, and the publication features. Our contribution to the literature is twofold. First, we get a underlying true Feldstein and Horioka coefficient lying between 0.562 and 0.677 for the sub-sample sharing the main characteristics of the original [START_REF] Feldstein | Domestic saving and international capital flows[END_REF] Thus despite the deepening of financial integration, the domestic saving-investment correlation that emerges from the literature, once corrected for alternative biases in the publication process, is high since it lies between one half and two thirds. There could be two ways of interpreting this main finding. On the one hand, since this abnormally strong figure is clearly at odds with financial integration, it questions the relevance of the Feldstein and Horioka regression as a way to measure capital mobility. On the other hand, one may be interested in identifying the external factors behind such a high correlation in a world of high capital mobility. In this alternative view, our reported value can be used as a benchmark for model calibration to assess the relative importance of the various sources of correlation between domestic savings and investment in a situation of perfect capital mobility.

ϳ The rest of the paper is organized as follows. Section 2 presents the selection process of the meta-sample and reports its main characteristics. Section 3 analyses the sources of the publication bias. Section 4 reports the result of the meta-regression and discusses the possible sources of heterogeneity of the saving retention parameters across studies. Section 5 concludes.

The meta-sample

The database

The first step of the analysis is to select a sample of papers that can be handled with the standard econometric tools of meta-analysis. The meta-sample should be both representative of this strand of literature and of a manageable size. This last aspect is critical for the literature on the Feldstein and Horioka puzzle, as this topic appears to have remained popular over the last thirty years. Searching in the EconLit database with the keywords related to the seminal publication we get 994 published articles as of December 2008, thus averaging 3 monthly papers over the period. This is also a safe way to avoid additional biases from more recent studies including "atypical" values for macroeconomic series such as Gross Domestic Product, investment and saving as a result of the recent financial crisis.

To proceed to a homogenous treatment of this literature, we have selected papers according to the following criteria: (i) The title of the paper should contain terms directly related to "Feldstein and Horioka", "Saving", "Investment", "Current Account" or "Capital Mobility";

(ii) The paper should provide quantitative results (we have neglected fully theoretical papers).

(iii) An abstract should be presented so that the presence of econometric estimations of the Feldstein and Horioka coefficients can be checked; (iv) The paper should be written in English; (v) The paper should have been published in peer-reviewed journals or working paper series (in particular we have neglected book chapters and PhD dissertations). Studies that failed to satisfy at least one of the criteria (i)-(v) have been excluded from the metasample.

After having examined the remaining 159 articles, we then skipped studies that do not include any original econometric estimation of the Feldstein and Horioka model. As previously noticed in the study of [START_REF] Bineau | Une méta-analyse des études sur la mesure de la mobilité internationale du capital selon la méthode de Feldstein et Horioka[END_REF], many studies do not include any original econometric estimation of the Feldstein and Horioka model. Furthermore, most papers have been sorted out of the meta sample as they do not give enough precision concerning the type of estimated model (in particular, endogenous/exogenous variables should be clearly defined), the database (in particular, initial/final dates and periodicity should be clearly presented) or the empirical results (in particular, the coefficient of correlation, the estimated parameters, and the standard errors should be clearly reported). Each time one of these rather stringent criteria is not met, the study has been excluded from the meta sample. This stringent selection leaves us with only 49 papers. Noticeably, despite this selection, we have kept a total number of 1651 Feldstein and Horioka coefficients, each corresponding to one regression. Thus, our dataset is quite large with regard to the standard meta-analyses that have already been performed in international macroeconomics.

Table 1 presents (by chronological order of publication) the papers that have been selected in the meta-sample with regard to the number of provided estimates and the average value of the Feldstein and Horioka coefficient. Although our dataset spans from 1980 to 2008 two parts are worth noting: in the first 15 years, we only get 12 studies that conforms to our criteria while from 1996 to 2008, we have 41 papers. Thus the Feldstein and Horioka puzzle has remained a lively area of research in international macroeconomics during the last decade. ϵ Furthermore, given our selection criteria, the dataset contains at least one paper per year since 1995, while we do not get any paper in 1982in , 1985[START_REF] Galbraith | A note on graphical presentation of estimated odds ratios from several clinical trials[END_REF]in , 1993[START_REF] Yamori | The relationship between domestic saving and investment: The Feldstein-Horioka test using Japanese regional data[END_REF]in , and 1997. However, as . However, as shown below, these publication characteristics do not significantly affect the "underlying true value" found for the Feldstein and Horioka coefficient in the meta-regression. The number of estimated Feldstein and Horioka parameters per paper ranks from 1 in [START_REF] Kim | The saving-investment correlation puzzle is still a puzzle[END_REF] to 211 in [START_REF] Obstfeld | International capital mobility in the 1990s[END_REF]. The reported average value of the coefficient varies from 0.07 in [START_REF] Sinha | The mother of all puzzles would not go away[END_REF] to 1.01 in [START_REF] Yamori | The relationship between domestic saving and investment: The Feldstein-Horioka test using Japanese regional data[END_REF].

The econometric practice

Besides heterogeneity in the (per paper) number and reported values of the saving retention coefficient, the meta-sample reveals the variety of practices adopted in the literature.

First it affects the choice of countries. As shown in figure 1a, 72% of the parameters have been estimated in papers with OECD data, 23% in papers with developing or emerging data while only 5% of the parameters have been estimated in papers with a database that combines developed, developing, and emerging countries. Second, as reported in Figure 1b, 45% of the parameters have been estimated in papers with cross section data, 35% in papers with time series data, while 20% of the parameters have been estimated with panel data.

/ / i i i I Y S Y u α β = + + , 1, 2,3, . i N = " (1)
In this relation, I is national investment, S national saving, Y national income and the subscript i refers to countries. Parameter β is the Feldstein and Horioka coefficient. Under the null hypothesis of perfect capital mobility, β is expected to be equal to zero.

While many papers have checked for the empirical validity of this standard version of the model with cross section models or panel data models, some other papers have proposed an alternative version focusing on current account developments. Since, the current account (CA ) is defined as the saving-investment gap, CA S I = -, equation ( 1) can be rewritten as,

( ) ( )( ) / 1 / i i i CA Y S Y u α β = -+ - + , 1, 2,3, . i N = " (2)
According to this modified version, changes in saving are fully reflected in changes in investment and have no impact on the current account in a situation of capital market fragmentation. In this case, 1 β = . However, as shown in Figure 1c, almost 90% of the Feldstein and Horioka coefficients of the meta-sample have been estimated by treating the investment ratio as the dependent variable, in line with Feldstein and Horioka's original paper.

As shown in Figure 1d, 75% of papers in the meta-sample take into account the dynamic aspects of the investment-saving relationship by estimating modified versions of equations ( 1) and ( 2) including lags of both the endogenous and/or exogenous variable. In this case, the model is estimated with time series data for a given country according to, ( ) ( ) ( )

1 1 / / / q p k k t t t k t k k k I Y S Y I Y α β γ ε - - = = = + + + ¦ ¦ , 1, , , t T = " (3)
in the case of a dynamic version of equation ( 1) with t ε a white noise process.

The former specification allows one to distinguish short-term from long-run full capital mobility. In the short-run, the null hypothesis corresponding to complete capital market integration is given by: 0, 1, , .

k k q β = ∀ = " Over a longer horizon, the null hypothesis is based on the moving average representation of equation ( 3). The resulting ratio of lag polynomials must equal zero if capital is perfectly mobile across countries:

1 1 1 1 0 p q k k k k k k L L γ β - = = § • § • - = ¨¸¨© ¹© ¹ ¦ ¦
, setting the lag operator L equal to one.

Finally, some authors have selected a first difference version of the original regression (with or without co-integration relationships). Using time-variations instead of levels accounts for possible non-stationarity of the investment and saving variables and is more consistent with macro models based on intertemporal tradeoffs [START_REF] Apergis | A survey of the Feldstein-Horioka puzzle: What has been done and where we stand[END_REF]. In the case of no co-integration, the first-difference version of the relationship may be written as,

( ) ( ) ( ) 0 1 / / / q p k k t t t k t k k k I Y S Y I Y e α β γ - - = = Δ = + Δ + Δ + ¦ ¦ , 1, , t T = " . ( 4
)
As shown in Figure 1, it should be noted that a majority of papers selected in our sample clearly adopt the main characteristics of the original analysis of Feldstein and Horioka.

Besides the heterogeneity in the econometric practice encountered in the literature, they report estimations based upon cross section and OECD countries and treat investment as the endogenous variable. there is a large variation in the reported parameters, as revealed by the comparison of the 10th and 90th deciles. This feature is in line with the high degree of heterogeneity in the estimates of the Feldstein and Horioka coefficient frequently encountered in the literature for individual ϯ Like Bineau (2010), we have found that some published papers provide some relatively high values of FH parameters. We have dismissed studies with beta values higher than 2.5. Actually, it concerns only one study reporting a beta value of 4.06. Such values are difficult to interpret since the beta coefficient is expected to lie somewhere between 0 and 1 to reflect the degree of international capital market integration. Keeping the corresponding study in our sample would have required a specific treatment to avoid the potential bias of this outlier. countries. However, as the median and the mean values do not differ widely, the Jarque-Bera test statistics leads to rejection of the null hypothesis that the Feldstein and Horioka coefficient is normally distributed (P. value = 0.000).

We provide a preliminary inspection of the basic influence of sample details and equation characteristics on the Feldstein and Horioka coefficient estimates, by reporting descriptive statistics obtained for alternative sub-samples. As can be seen, the mean value of the coefficient is sensitive to the characteristics of the sample and to the specification of the model. The largest mean is obtained with cross section samples while the lowest is obtained for the panel data sub-sample. Moreover, the mean value of the Feldstein and Horioka coefficient is lower with models in variations than with models in levels. These results might indicate that while models estimated with cross section samples capture some form of longrun equilibrium relationship between saving and investment, the Feldstein and Horioka coefficients estimated with panel data are affected by both the long-run equilibrium and the short-run temporary co-variation between saving and investment.

The publication bias

As underlined by [START_REF] Havranek | Estimating vertical spillovers from FDI: Why results vary and what the true effect is[END_REF], the average value of the estimated parameter is a biased estimate of its underlying true value. The meta-sample may be subject to some bias as some coefficient values may be over represented in our selection. Before proceeding to the meta-regression we thus investigate whether the meta-sample at hand is subject to publication bias. It is also important to know whether that sample reveals a "true" underlying value for the Feldstein and Horioka coefficient that is unaffected by the publication process. In what follows we distinguish between these Type I and Type II biases.

Type I publication bias

Type I bias occurs when editors, referees, and/or researchers favor a particular direction of results. Following the underlying theory that links saving, investment, and the current account, estimates of the Feldstein and Horioka coefficient are expected to lie between 0 and 1. Thus, negative estimates of the Feldstein and Horioka coefficient, for instance, might be ignored as an increase in national saving that reduces national investment could be puzzling.

The detection of a Type I publication bias commonly starts with the so called "funnel plot" [START_REF] Egger | Bias in meta-analysis detected by a simple, graphical test[END_REF] u is the regression error term. In this equation, δ denotes the true Feldstein and Horioka coefficient and θ is the size of the publication bias. However the stochastic term in equation ( 5) is heteroskedastic because i SE is a sample estimate of the standard deviation of the dependent variable i FHC . Stanley (2008) suggests dividing equation ( 5) by the standard error of the Feldstein and Horioka coefficient, to perform a weighted least squares estimation. This is achieved by running a simple OLS estimation on the transformed bivariate meta-regression,

1 i i i t v SE θ δ § • = + + ¨© ¹ , 1, ,1651 i = " , ( 6
)
where i t is the t-statistic measuring the significance of the i th Feldstein and Horioka coefficient. Equation ( 6) represents a regression line through the funnel graph, rotated by 90 degrees and adjusted for heteroskedasticity. The Funnel Asymmetry Test (FAT) for publication bias is a simple t-test on the intercept of equation ( 6): a value of β significantly different from 0 indicates a publication bias in the sample. Moreover, values of β close to 2 may also be taken as a signal of extreme publication bias and may be consistent with a situation such that all studies display positive and significant estimated Feldstein and Horioka coefficients while the underlying true one is zero. Moreover, if β is significantly positive (or negative), then the effect size is subject to an upward (or downward) bias.

However, as 96% of the studies selected in the meta-sample report more than one value for the Feldstein and Horioka coefficient, estimates from the same study are likely to be dependent. By so, according to [START_REF] Disdier | The puzzling persistence of the distance effect on bilateral trade[END_REF], equation ( 6) is likely to be missspecified. Thus we must run a mixed-effects multilevel model and account for within-study dependence [START_REF] Doucouliagos | Publication selection bias in minimum-wage research? A meta-Regression Analysis[END_REF]) by incorporating a random individual effect for each study, so as to capture between-study heterogeneity. According to the mixed-effects multilevel model, the regression for the t-statistic of coefficient i in study j ( ) ij t should be,

1 ij j ij ij t SE θ δ λ η § • = + + + ¨© ¹ , 1, ,1651 i = " , ( 7
)
where j

λ is the study level random effect and ij η is a disturbance term. Moreover, as the process of selecting estimates from the literature makes the meta-analysis highly vulnerable to data contamination, the robustness of the funnel asymmetry test is checked by re-estimating equations ( 6) and ( 7) with the robust Least Absolute Deviations (LAD) estimation procedure.

Results are reported in Table 3. The θ coefficient (intercept term) is statistically non significant in nearly all cases. It appears as marginally significant at the 10% confidence level only in the case of the Least Absolute Deviations estimation of the basic model that does not correct for within study dependence.

Thus we can sensibly accept the null hypothesis of no Type I publication bias.

Type II publication bias

Type II publication bias occurs when editors, referees, and/or researchers promote results that are statistically significant. This kind of bias may be induced by the self-censoring behavior of authors or by the decision process of peer-reviewed journal editors which favors papers with high significance levels for the main estimates. Smaller samples and limited degrees of freedom reduce the probability of finding significant results. Type II publication bias may thus appear when researchers using small samples are inclined to search across econometric "tools" (proxies, estimators, specifications) to produce more significant results. Type II bias can thus make empirical effects appear larger than the true effect [START_REF] Stanley | Meta-regression analysis as the socioeconomics of economics research[END_REF]).

Type II bias can be detected with the [START_REF] Galbraith | A note on graphical presentation of estimated odds ratios from several clinical trials[END_REF] plot that reports the estimated precision of the Feldstein and Horioka coefficient against the corresponding t-statistics and the assumed true effect. In case of Type II selection bias, large values (in absolute terms) will be over reported and there will be an excessive likelihood of reporting significant results. In case of no Type II publication bias and of the true effect (labeled TE) was really true, the statistics As shown in Figure 4, the reported t-statistic exhibits both a wide variation and seems to increase with rising precision. Assuming that there is no underlying true effect (TE = 0), only 5% of the studies should report t-statistics larger than 2. However, we find that about 58% of ).

While the Feldstein and Horioka coefficient seems unaffected by Type I publication bias, the Type II bias is clearly present among the studies at hand, according to the Galbraith plot. This result is in line with the nature of the meta-sample that mostly incorporates peerreviewed journal articles. This result is consistent with [START_REF] Bineau | Une méta-analyse des études sur la mesure de la mobilité internationale du capital selon la méthode de Feldstein et Horioka[END_REF] who also finds an evidence of publication bias with a funnel graph analysis. However, while he leaves the correction for publication bias for future research, the present paper uses a multivariate empirical framework which explicitly deals with the correction for publication bias.

The underlying true effect

Besides the detection of publication bias, we check whether the meta-sample delivers an underlying true value for the Feldstein and Horioka coefficient. The true effect is a generic term used in meta-analysis. Its computation rests on two assumptions: first, there exists a unique value for the Feldstein-Horioka parameter for a given degree of capital mobility;

second, the two previous publication biases have been cleaned from the estimates.

The method of testing for Type I bias can be used to test the significance of the true effect irrespective of the publication selection process. The Precision Effect Test (PET) is a simple t-test on the slope coefficient θ of equation ( 6). Monte Carlo simulations show that publication bias is quite efficiently filtered out by equation ( 6) so that the true effect seems to be adequately estimated. Empirical results presented in Table 3 show that the null hypothesis of no underlying true effect is always rejected at 5% since the estimated true effects are statistically significant for the whole sample and for both subsamples. These results are insensitive to the estimation procedure. For the whole sample, the 95% confidence intervals reported by the Precision Effect Test are [0.506; 0.562] with OLS estimators and [0.587; 0.644] with Least Absolute Deviations estimators for the simple bivariate model ( 6). For the mixed-effects multilevel model ( 7), the 95% confidence intervals are [0.462; 0.525] with OLS and [0.569; 0.635] with the Least Absolute Deviations method. As can be seen, the sensitivity of the results is rather weak with regard to the estimation procedure. Thus, the underlying true effect obtained for the Feldstein and Horioka coefficient corrected for Type I bias is rather stable.

As a robustness check [START_REF] Stanley | Integrating the empirical tests of the natural rate hypothesis: a metaregression analysis[END_REF], we must take into account that this procedure for testing for the underlying true effect is reliable only if 2 2 v σ ≤ . Misspecification biases can lead to overestimated true effect when 2 2 v σ > . As both the null H0: 2 2 v σ ≤ and the null of no true effect are systematically rejected by the data, we use the Precision Effect Estimate with Standard Error (PEESE) introduced by [START_REF] Stanley | Meta-regression analysis as the socioeconomics of economics research[END_REF] to estimate the size of the underlying true effect. The PEESE assumes that publication bias is related to the variance of the estimate of the TE instead of the standard error (as with the PET). Equations ( 6) and ( 7) thus respectively becomes,

1 i i i i t SE v SE ρ δ § • = + + ¨© ¹ , 1, ,1651 i = " , (8)
and,

1 ij ij j ij ij t SE SE ρ δ λ η § • = + + + ¨© ¹ , 1, ,1651 i = " . ( 9
)
With the PEESE estimates, the δ coefficient is still strongly significant in both ( 8) and ( 9). When estimating (8), the

The meta-regression

The aim of this section is to determine whether the choice of a particular econometric approach affects the estimated value of the Feldstein and Horioka coefficient of the metabase. Besides the bias introduced by the publication process studied in the previous section, we now focus on the impact of modeling and data choices on the value of the estimated parameters. To understand how these factors may affect the estimated value of the saving retention coefficient, we use the general version of the FAT-PET method to estimate the multivariate meta-regression,

1 1 K ki i k i k i i Z t SE SE β α γ ω = § • § • = + + + ¨¸¨© ¹ © ¹ ¦ ² ² , 1, ,1651, i = " ( 10
)
where each variable ki Z , ( 1, , k K = " ) is a meta-independent variable that is assumed to potentially affect the estimate of the saving retention coefficient and where i ω is the meta- regression disturbance term, with standard characteristics. Each of the ki Z variables is weighted by ( )

1/ i SE and the k γ are K coefficients to be estimated, where each one measures the impact of the corresponding variable on the saving retention coefficient.

Selection of explanatory variables

The nature and coding adopted for the explanatory variables ki Z are presented in table 4. As shown, each of the 1651 point estimates of the Feldstein and Horioka coefficient have been coded with regard to K=19 main characteristics related to the data, the specification of the estimated equation, the estimation method, and the publication features.

First, regarding data characteristics, we include dummy variables to account for cross sectional data (CROSS), time series data (TIME), and panel data (PANEL). We also distinguish between samples dealing with country (COUNTRY), and regional (REGIONS) datasets. Finally regarding the nature of the dataset used in papers we distinguish between papers that mix developed, and developing or emerging countries (MIXED), papers centered on OECD countries (DEVED) and papers focusing only on developing and emerging countries (DEVING). In order to control for possible structural changes in the Feldstein and Horioka coefficient, we finally include the average year of the data period (AVGYEAR).

Second, the specification of the estimated relation distinguishes between equations in which investment is the endogenous variable (ENDOINV) and equations with saving (or current account) as the endogenous variable (ENDOSCA). Dummies have also been introduced to account for variables aimed at controlling for the macroeconomic characteristics of the country. With regard to the literature, three variables have been selected: the level of public deficit (PUBDEF), the level of foreign investment (FORINV) and a measure of the country size (COUNTSIZE). Third, regarding estimation characteristics we proceed as follows: we first distinguish between models in level (LEVEL) and models in first differences (DIFF1) to control for the dynamic properties of the estimated models. We also create dummies for standard econometric methods such as OLS, IV, GMM, panel fixed effects or random effects (STEM)

and for all other estimation procedures (NSTEM).

Finally, regarding publication characteristics, the publication year is also included in the list of explanatory variables in order to capture a possible publication trend induced by advances in the econometric methodology or the inclusion of new variables with different properties in terms of a deeper integration of financial markets.

Results of the multivariate meta-regression

We report the results of the multivariate meta-regression in Table 5. As it is well known, dummies have to be omitted to avoid linear dependence. In this case, the constant term represents the effects of omitted dummies. Here, the constant term captures the influence of CROSS, COUNTRY, MIXED, ENDOINV, LEVEL and STEM. This choice, that reflects the most standard practices encountered in our dataset, does not affect the whole results of the analysis. We begin the multivariate analysis by including all explanatory variables into the regression and estimating the resulting model with OLS. As a robustness check, random individual effects for each study are added to the meta-regression model (as in the mixedeffects multilevel model).

As indicated by the F-statistics, the estimated coefficients of the meta-regression are jointly significant. The overall quality of fit is good for both models, although it is higher with the mixed-effects multilevel model than with the basic MRA FAT-PET model. In addition to the highly significant coefficient on the PRECISION variable, the estimated coefficients associated with the fifteen explanatory variables are significant at the ten percent confidence level for nine variables in the basic meta-regression FAT-PET model and for eight variables in the mixed-effects multilevel model. Results reported in Table 5 should be interpreted as follows: a positive and significant coefficient on a given variable suggests that this study characteristic increases the reported

Feldstein and Horioka coefficient with regard to its underlying true value. Conversely, a negative and significant coefficient suggests that the corresponding study characteristic reduces the reported Feldstein and Horioka coefficient with regard to its underlying true value. As the inverse of the standard error (precision) affects all the moderator variables in (9), the underlying true effect (or the authentic effect) is captured by a combination of all explanatory variables in the multivariate meta-regression.

However, the coefficient associated with the PRECISION variable cannot be interpreted as the estimated underlying true Feldstein and Horioka coefficient. More precisely, the omitted dummies are CROSS, COUNTRY, MIXED, ENDOINV, LEVEL and STEM. The second and third columns of Table 5 suggest that the underlying true coefficient is between 0.562 and 0.677 in the sub-sample of studies with the following characteristics: the ratio of investment to GDP is used as the dependent variable, the database contains cross section data and only includes country data for both developed and developing countries, the model is specified in levels and is estimated with standard econometric procedures. Though being of the same order of magnitude, the PRECISION estimate from the basic model (column 2) is significantly higher4 than the corresponding estimate from the mixed effect models (column 3).

The effect of the sample structure is measured through the dummies TIME and PANEL (the CROSS variable is excluded from the model). As can be seen, the effect of the TIME dummy is unclear since it is significant in the basic FAT-PET meta-regression model but insignificant in the mixed effect multilevel model. The effect of the PANEL dummy may also be considered as statistically unclear: While it is significant with the mixed effect multilevel model, it is marginally significant with the basic FAT-PET meta-regression model. Moreover, as it is negative in both models, it seems that panel data studies are more likely to report lower

Feldstein and Horioka coefficients.

The REGIONS and DEVING dummies are always non significant, thereby suggesting that the use of regional data instead of country data and/or the use of datasets including only developing countries does not change significantly the underlying true Feldstein and Horioka coefficient. Only the DEVED dummy is significant in both models. As the associated coefficient is systematically positive, studies using only developed countries datasets yield a larger Feldstein and Horioka coefficient. This result which is consistent with Dooley et al.

(1987) may be attributed to the "country size" factor. Samples of developed countries include countries that are able to affect world interest rates. Thus their saving-investment correlation may be biased upwards even under conditions of perfect capital mobility.

The characteristics of the estimated equation also matters. First, the coefficient associated with the ENDOSCA dummy is negative and highly significant so that there is some evidence Turning to the NSTEM variable added to control for the method of estimation, the associated positive and significant coefficient seems to show that studies using no standard estimation procedures report a higher level of the Feldstein and Horioka coefficient than in the baseline case. Finally, the estimates of the saving-investment correlation are not significantly dependent upon the estimation period, as indicated by the non significance of the variable AVGYEAR. This lack of any upward or downward trend in the result is in line with [START_REF] Coakley | The Feldstein-Horioka Puzzle and capital mobility: A review[END_REF]. Moreover, the reported estimates of the Feldstein and Horioka coefficient are not dependent upon the publication year.

In order to evaluate the sensitivity of this result to the choice of the control variable, the model has also been estimated using the first year instead of the average year of the sample 5 .

ϱ tĞ ƚŚĂŶŬ ĂŶ ĂŶŽŶLJŵŽƵƐ ƌĞĨĞƌĞĞ ĨŽƌ ƚŚŝƐ ƐƵŐŐĞƐƚŝŽŶ͘

This alternative choice may be justified by the fact that the average year of the sample is closer to the year of publication (control variable PUBYEAR) than the first year of the sample. While the coefficient associated with this alternative control variable is negative and significant with the basic MRA-FAT PET model (coefficient: -0.0017, t.stat: -3.29) it turns out to be non significant with the mixed effect multilevel model (coefficient: -0.0003, t-stat: -0.53). In contrast, all the remaining coefficients and t-stats are qualitatively and quantitatively unchanged with both estimation procedures. This alternative control variable thus seems to yield mixed results which do not invalidate the conclusion that there is no clear empirical evidence that the Feldstein and Horioka coefficients are dependent upon the sample years.

Conclusion

This paper has provided a meta-analysis of the empirical literature on the investment-saving correlation. The multivariate meta-regression reports a underlying true coefficient lying between 0.562 and 0.677 for studies with the following characteristics: i) the ratio of investment to GDP is the dependent variable, ii) the database contains only cross section data, iii) the database includes only country data (no regional data), iv) the database includes both developed and developing countries, v) the model is specified in levels, and vi) it is estimated with standard econometric procedures.

With regard to this benchmark value, the heterogeneity reported in the estimates of the Horioka coefficient seems to be systematically underestimated with models written in first differences, models using the saving ratio or the current account ratio as the dependent variable instead of the investment ratio, and models including indicators of the public deficit or indicators of the country size as additional explanatory variables.

There could be two interpretations for the high value of the saving retention parameter found in this study. On the one hand, since this abnormally strong figure is at odds with financial integration, it questions the relevance of the Feldstein and Horioka regression as a way to measure capital mobility. In this case studies should focus on other indicators such as the importance and structure of gross (rather than net) capital inflows and outflows. On the other hand, one may be interested in identifying the external factors behind such a high correlation in a world of high capital mobility. In this alternative view, our reported value can be used as a benchmark for model calibration to assess the relative importance of the various sources of correlation between domestic savings and investment (e.g. [START_REF] Bai | Solving the Feldstein-Horioka Puzzle with financial frictions[END_REF], for a model with financial friction to solve the puzzle).

Although the present meta-analysis was successful in explaining up to 85 per cent of the variations in the estimated values of the Feldstein and Horioka coefficient, a significant part remains unexplained. Given its renewed interest since the recent Great Financial Crisis [START_REF] Bautista | Saving-investment relationship, financial crisis and structural changes in East Asian countries[END_REF], [START_REF] Boubakri | Assessing the financial integration of Central and Eastern European countries with the euro area: Evidence from panel data cointegration tests[END_REF][START_REF] Chu | The Feldstein-Horioka puzzle and spurious ratio correlation[END_REF] to cite a few), subsequent research will be necessary to improve our understanding of the reasons why the empirical estimates of the investment-saving correlation may be so imprecise.

  paper. Second, heterogeneity in the estimates comes mainly from features related to the origin of variables or from the specification of the regression model. In particular, studies taking into account only developed countries in the sample or adopting econometric methods different from ordinary least squares (OLS), instrumental variables (IV), generalized method of moments (GMM), panel with either fixed or random effects (FE or RE respectively), overestimate the Feldstein and Horioka coefficient. In contrast studies adopting the saving rate or the current account as dependent variables, taking into account the public deficit or the country size as exogenous variables or estimating the Feldstein and Horioka relation in first difference tend to get lower estimates of the saving retention coefficient.

Figure 1 .

 1 Figure 1. Some characteristics of the meta-sample.

Figure

  Figure 1a. Countries Figure 1b. Data pooling or ordering

Figure 2 .

 2 Figure 2 reports the distribution of the 1651 estimated values of the Feldstein and Horioka

  which compares the value of the Feldstein and Horioka coefficient with the inverse of its standard error. The meta-sample is bias free, if the plot represents an inverted funnel: with rising sampling error, estimated Feldstein and Horioka coefficients are expected to scatter more widely. Indeed, observations with high precision should be concentrated closely around the true effect, while those with lower precision should spread at the bottom of the plot. In the absence of Type I publication bias, the funnel plot is thus symmetric around the genuine empirical effect.

Figure 3 .FHC

 3 Figure 3. Funnel plot

  2 more than 5% of the time and the cloud should be randomly distributed around 0, with no systematic relation to precision.

Figure 4 .

 4 Figure 4. Galbraith plot for the whole sample.

  that estimated Feldstein and Horioka coefficients are lower when the estimated model uses saving or the current account as the endogenous variable instead of investment. Second, the dynamic structure of the model affects the estimated value of the saving retention coefficient as shown by the significance of the DIFF1 dummy. Models in first differences produce lower estimated values of the Feldstein and Horioka coefficient. As models in first differences are generally designed to capture the short run temporal characteristics of the data, this result might indicate that these models only capture the short run co-movement between investment and saving and tend to underestimate the underlying true Feldstein and Horioka coefficient.These two effects are rather large in absolute value so that the size of the estimated investment-saving correlation is significantly influenced by the structure and the specification of the selected empirical model. The point estimate of the ENDOSCA dummy implies that the investment-saving correlation is reduced by an amount close to 0.3 in the subsample of studies using saving or the current account as the endogenous variable. More precisely, the choice of the saving ratio or of the current account ratio instead of the investment ratio as the dependent variable in the regression model lowers the estimate from 0.677 to 0.356 (with the basic FAT-PET model) or from 0.562 to 0.353 (with the mixed-effects multilevel model).Because the point estimate of the DIFF1 dummy is also close to 0.3, the estimated Feldstein and Horioka coefficient is lowered by a similar amount in the subsample of studies using a regression model including first differences of the variables instead of levels.While the inclusion of an indicator of foreign investment (FORINV) has no significant effect on the estimated Feldstein and Horioka parameter, the coefficients of PUBDEF and COUNTSIZE are always significant and negative. Thus specifications in which indicators of public deficit and/or indicators of the country size are used as additional explanatory variables are more likely to report lower Feldstein and Horioka coefficients. Overall, the significance of variables aimed at controlling for different specifications of the regression models (ENDOSCA, PUBDEF, COUNTSIZE, DIFF1) clearly suggests that the reported Feldstein and Horioka coefficients are highly sensitive to the modeling strategy and to the inclusion of additional explanatory variables besides the investment and/or the saving variable.

Feldstein

  and Horioka coefficient can be explained by a limited number of variables. First, characteristics of the retained countries do matter. Using databases including only developed countries leads to coefficients that are larger than those obtained with mixed date bases including both developed and developing or transition countries, by an amount comprised between 0.10 and 0.15. Moreover, the specification of the regression model has sizeable influences on the estimated Feldstein and Horioka coefficient. The authenticFeldstein and

Table 1 .

 1 The meta-sample.

	Average	value of	parameter	0.51	0.39	0.55	0.58	0.63	0.69	0.43
	Number	of	estimates	58	4	4	12	16	1	15
	Authors			Decressin J. & Disyatat P.	Coiteux M., Olivier S.	Obstfeld M., Rogoff K.	Bordo M.D., Flandreau M.	Corbin A.	Kim S. H.	Blanchard O., Giavazzi F.	De Vita G., Abbott A.
	Publication	year		2000	2000	2001	2001	2001	2001	2002	2002
	Average	value of	parameter	0.85	0.56	0.85	0.21	0.56	0.6	0.67	0.62
	Number	of	estimates	20	14	12	24	48	19	24	81
	Authors			Feldstein M., Horioka C.	Sachs J.	Feldstein M.	Penati, A., Dooley M.	Frankel J.A.	Obstfeld M.	Dooley M., Frankel J.A., Mathieson D.	Feldstein M., Bacchetta P.
	Publication	year		1980	1981	1983	1984	1986	1986	1987	1989

ϭϬ

Table 2 .

 2 Summary statistics of the sample

			Std.	10th		90th
		Mean	error	percentile Median	percentile Observations
	Whole sample	0.498	0.412	-0.060	0.563	0.930	1651
	Cross section (a)	0.550	0.391	-0.054	0.660	0.916	375
	Time series (a)	0.489	0.442	-0.076	0.530	1.020	297
	Panel (a)	0.389	0.379	-0.056	0.440	0.760	154
	Level (a)	0.530	0.401	-0.027	0.601	0.933	611
	Variation (a)	0.405	0.429	-0.117	0.430	0.926

Table 3 .

 3 Tests for Type I publication bias.

	Dependent variable = t-statistic on the Feldstein Horioka coefficient
		θ	δ	Sample	R 2
		(bias)	(precision effect)	size
	OLS estimator
	Basic model (Eq. 6)	-0.117 (-0.69)	0.534*** (37.46)	1651	0.47
	Mixed effect multilevel	2.011 (0.99)	0.493*** (30.74)	1651	0.63
	model (Eq. 7)
	Least Absolute Deviations
	estimator	-0.318* (-1.85)	0.615*** (42.38)	1651	0.53
	Basic model (Eq. 6)
	Mixed effect multilevel	0.869 (0.41)	0.602*** (35.81)	1651	0.66
	model (Eq. 7)

Notes: Absolute values of the t-statistics in parentheses. ***, **, and * indicate significance at the level of 1%, 5%, and 10%, respectively.

precision (1/SE) t-statistics (if the true effect is 0.00)

	the studies report t-statistics greater than the associated critical value for a 5% threshold (the
	test that the proportion of significant t-statistic is equal to the expected 5% is strongly rejected
	with	z =	62.02	and	p <	0.0000	). Alternatively, if we use the top ten points of the funnel
	graph to estimate the true effect as 0.35, the null hypothesis that the proportion of significant
	t-statistic is equal to 5% is still clearly rejected by the data (	z =	62.57	with	p <	0.0000
		40
		30
		20
		10
		0
	-10
		0			10		20	30	40	50

Table 4 .

 4 Acronyms and description of regression variables

	Variable	Description of code

Table 5 .

 5 Estimated parameters of the meta-regression.

	Dependent variable: t-statistic of the estimated Feldstein and Horioka coefficient
		Basic MRA	Mixed-effects
		FAT-PET model	multilevel model
	Explanatory variables
	CONSTANT	-0.692 (-4.13)***
	PRECISION	0.677 (8.78)***	0.562 (5.80)***
	TIME	0.09 (3.16)***	0.012 (0.29)
	PANEL	-0.041 (-1.58)	-0.100 (-3.03)***
	REGIONS	-0.019 (-0.70)	-0.009 (-0.24)
	DEVED	0.149 (5.05)***	0.097 (2.73)***
	DEVING	-0.046 (-1.37)	-0.046 (-1.25)
	ENDOSCA	-0.321 (-12.18)***	-0.309 (-10.12)***
	PUBDEF	-0.175 (-4.91)***	-0.316 (-5.99)***
	COUNTSIZE	-0.219 (-4.94)***	-0.37 (-5.12)***
	FORINV	0.064 (1.79)*	0.055 (0.95)
	DIFF1	-0.321 (-13.98)***	-0.352 (-10.86)***
	NSTEM	0.308 (9.87)***	0.464 (10.00)***
	AVGYEAR	-0.001 (-1.11)	0.001 (0.92)
	PUBYEAR	-0.001 (-0.42)	0.000 (0.01)
	R 2	0.653	0.844
	F-test (a)	210.22 (P. value = 0.000)	66.83 (P. value = 0.000)
	Notes: t-statistics in parentheses. ***, **, * denote significance at the level of 1, 5, and 10%,
	respectively. (a) H0: Independent variables are jointly equal to zero.

A comparison test for equal means yields a t-statistic equal to 37.73 that is far above the standard critical value at the

percent level from a standard Gaussian distribution given the (relatively large) sample size.

ZdͲ /EZ ͕ ŐƌŽĐĂŵƉƵƐ KƵĞƐƚ ϯ ŽƌƌĞƐƉŽŶĚŝŶŐ ĂƵƚŚŽƌ͗ ĐŚƌŝƐƚŽƉŚĞ͘ƚĂ|ĞƌĂΛƵŶŝ|ͲƌĞŶŶĞƐϭ͘Ĩƌ͕ Z DͲ EZ^͕ hŶŝ|ĞƌƐŝƚLJ ŽĨ ZĞŶŶĞƐ ϭ͕ ϳ ƉůĂĐĞ ,ŽĐŚĞ͕ ϯϱϬϲϱ ZĞŶŶĞƐ ĐĞĚĞdž͕ &ƌĂŶĐĞ͘ WŚŽŶĞ͗ нϯϯ ϮϮϯ Ϯϯ ϯϱ ϮϮ͕ &Ădž͗ нϯϯ ϮϮϯ ϯϱ ϵϵ͘ Ϯ The "Mother of All Puzzles" at Thirty: A Meta-Analysis


Sebastian Götz
email: sebastian.goetz@acm.org

Ilias Gerostathopoulos
email: iliasg@d3s.mff.cuni.cz

Filip Krikava
email: filip.krikava@inria.fr

Adnan Shahzada
email: adnan.shahzada@polimi.it

Romina Spalazzese
email: romina.spalazzese@mah.se

Adaptive Exchange of Distributed Partial Models@run.time for Highly Dynamic Systems

Future software systems are characterized by their high dynamicity. For example, we are experiencing a world where Cyber-Physical Systems (CPSs) play a more and more crucial role. CPSs integrate computational, physical, and networking elements; they comprise a number of subsystems, or entities, that are connected and work together. The open and highly distributed nature of the resulting system gives rise to unanticipated runtime management issues such as the organization of subsystems and resource optimization.

In this paper, we focus on the problem of knowledge sharing among cooperating entities of a highly distributed and selfadaptive CPS. Specifically, the research question we address is how to minimize the knowledge that needs to be shared among the entities of a CPS. If all entities share all their knowledge with each other, the performance, energy and memory consumption as well as privacy are unnecessarily negatively impacted. To reduce the amount of knowledge to share between CPS entities, we envision a role-based adaptive knowledge exchange technique working on partial runtime models, i.e., models reflecting only part of the state of the CPS. Our approach supports two adaptation dimensions: the runtime type of and conditions over the knowledge. We illustrate the feasibility of our technique by discussing its realization based on two state-of-the-art approaches.

I. INTRODUCTION Currently, we are experiencing an increasingly open world

where more and more distributed, highly dynamic systems, as for example Cyber-Physical Systems (CPSs) [START_REF] Lee | Cyber physical systems: Design challenges[END_REF], [START_REF] Derler | Modeling cyber-physical systems[END_REF], [START_REF] -D | CyberPhysical Systems: A Perspective at the Centennial[END_REF], play a key role. CPSs integrate computational, physical, and networking elements that might perform many different tasks within disparate contexts as for instance smart living spaces. CPSs are composed of multiple subsystems or entities that have different goals and different partial views of the world. Subsystems can communicate and form or participate in temporary ad-hoc collaborations with one another and any subsystem can freely arrive and leave the environment at any time. The open and highly distributed nature of these systems gives rise to unanticipated runtime management issues such as the organization of subsystems and resource optimization.

A promising approach to manage these issues is to develop such systems as self-adaptive systems [START_REF] De Lemos | Software engineering for selfadaptive systems: A second research roadmap[END_REF] by using selforganization and self-optimization techniques. Each subsystem, then, becomes autonomous and makes decisions based on its own knowledge while taking into account the invariants of the whole system and its environment [START_REF] Keznikl | Design of ensemble-based component systems by invariant refinement[END_REF]. Consequently, each subsystem has its own knowledge exchange policy and there is no central point of coordination from which the complete knowledge of the system would be accessible.

This work focuses on a challenging problem for distributed self-adaptive systems: how to minimize the knowledge that needs to be shared and synchronized among the entities of such a system. We do not address the problem of integrating systems not developed to interoperate with each other. Since the subsystems hold only a partial view of the complete system, local decision making requires them to exchange their knowledge to synchronize their partial views, which might be a costly operation. A simplistic and naive solution would be that all entities share all their knowledge with each other. This has several drawbacks, which our approach aims to overcome:

-Performance. The intense, unnecessary knowledge exchange decreases system performance, as it leads to unnecessary computing cycles and network saturation. -Energy Consumption. The more knowledge is to be exchanged, the more energy is required for it, but the capacity of the participating subsystems is typically restricted. -Memory Consumption. CPSs typically comprise small devices, which are limited in terms of the memory. Hence, storing superfluous knowledge is not a viable option. -Privacy. The subsystems do not necessarily belong to the same owner or come from the same vendor. In such cases, they might possess knowledge that cannot be shared without threatening the privacy of the owner.

In this paper, we present a role-based adaptive knowledge exchange technique working on partial runtime models of highly dynamic systems. A partial runtime model reflects the partial view of an entity of such a system representing only a part of the system's state. Our technique supports adaptation with respect to two dimensions: the runtime type of and conditions over the knowledge. Concretely, the contributions of this paper are:

(i) three strategies for knowledge exchange between collaborating subsystems with tradeoffs (cf. Section IV-A), (ii) role-based runtime models to specify the partial views of the subsystems (cf. Section IV-B), (iii) an illustration of the role-based adaptive knowledge exchange for the partial runtime views in ad-hoc collaborations (cf. Section IV-C).

The role-based knowledge exchange allows the collaborating subsystems to reduce the amount of knowledge they exchange.

The overall technique faces a tradeoff between the likelihood of optimal decision making w.r.t. the individual goals and the associated cost, which both vary in terms of the amount of knowledge exchanged. By facilitating the exchange of the minimum pertinent knowledge, our technique helps to improve the overall CPS performance, energy and memory consumption, as well as reducing privacy threats whilst assuring good local decision making. We discuss two possible realizations of our approach through two different state-of-the-art technologies. The remainder of the paper is organized as follows. Section II describes a motivating, running example. Section III outlines the concepts of models@run.time and role-based modeling, which our technique is based on. Our approach to knowledge exchange in distributed partial runtime models is presented in Section IV. Section V discusses related work. To show the feasibility of our approach, two realizations of our approach are outlined in Section VI. Finally, Section VII draws conclusions.

II. THE CLEANING ROBOTS RUNNING EXAMPLE

A cleaning company uses cleaning robots to carry out cleaning jobs on a variety of premises. After office hours, the company deploys groups of cleaning robots on contracted sites with the objective to efficiently clean the space before a certain deadline. A site can have multiple floors with different layouts. As the cleaning robots are often changing cleaning sites, it would be impractical to assume that these sites have smart space capabilities or infrastructure. Lacking the static infrastructure that would play the central point that could provide the global view of the environment and coordinate the cleaning job, the robots need to be autonomous and independent. They are equipped only with a short-range communication device (e.g. Bluetooth) allowing them to exchange data while being in proximity of one another.

There are various types of robots (e.g. specialized in dry or wet cleaning) with different specifications. During its operation, each robot maintains information about its own state (e.g., battery level, bin capacity) and about its environment (e.g., floor plan, the parts of the floor it has cleaned) in the form of runtime models. While the robots only acquire a partial knowledge about their environment, they can communicate together and query each others' runtime models in order to extend their knowledge and therefore improve their decisions. For example, a robot might ask for the parts of the floor that have already been cleaned by others in order to avoid revisiting the same places. However, it shall only require such information in the case it is in the "cleaning" state and not e.g. going for charging. It should also only request it from the right robot type. Furthermore, it should limit the query based on its available resources and skip places unreachable within its current battery level. Finally, knowledge exchange should be reduced as much as possible to lower the energy impact.

Essentially, the described scenario is about the exchange of runtime models that are both distributed and partial. The above example helps to explain that the amount of knowledge exchanged should be adaptable at runtime depending on: (i) which entities communicate, i.e., based on the type of the runtime model; (ii) the state of the requester.

An adaptive knowledge exchange technique for runtime models can therefore be regarded as a means that allows developers to express knowledge exchange between runtime models and automate its adaptation at runtime, based on the runtime type of knowledge and conditions over it.

III. BACKGROUND

In this section we give a brief summary of models@run.time and role-oriented modeling, which our approach is based on. Models@run.time. The change from programming stationary to mobile devices (e.g., CPSs) introduced several new challenges on software engineering techniques. Among them is the necessity to handle contextual changes in applications. For example, the change in brightness when a car drives through a tunnel should lead to a color inversion of the navigation display. In other words, software for mobile devices is required to reflect on itself and its environment.

To enable this self-and context-awareness, the systems have to keep the respective information. The models@run.time paradigm proposes to use modeling techniques to capture and process this information using runtime models, i.e., models representing a view on the current state of the system [START_REF] Blair | Models@ run.time[END_REF], [START_REF] Aßmann | Models@run.time[END_REF].

Our approach leverages models@run.time since each autonomous subsystem keeps and updates its own runtime model. Our approach extends the existing work in the models@run.time community by discussing three adaptive knowledge exchange strategies to synchronize the partial runtime models of subsystems forming ad-hoc collaborations. Role-based Modeling. The modeling of complex, dynamic domains has been a key challenge of software engineering for more than 40 years. Nowadays, a huge amount of software is developed according to the object-oriented paradigm, which is based on concepts introduced already in 1967 [START_REF] Kay | The early history of smalltalk[END_REF].

Role-oriented modeling, originally introduced by Bachman in 1973 [START_REF] Bachman | The programmer as navigator[END_REF], is an alternative, which provides means to capture the contextual and relational nature of objects, i.e., allows one to model self-and context-aware software [START_REF] Kühn | A metamodel family for role-based modeling and programming languages[END_REF].

Roles differ from objects in terms of the properties of their types [START_REF] Guarino | A formal ontology of properties[END_REF]. An object, being an instance of a class, is rigid and non-founded, meaning that the object cannot change its type and is not required to depend on any other object. A role, being an instance of a role type, is anti-rigid and founded, i.e., can change its type at runtime, but has to be connected to an object, which is playing this role. For example, Student is a role type, because a person can stop to be a student without ceasing to exist. It is the person which plays the role of a student. The type Person, on the contrary, is a class. Moreover, the focus in role-based modeling is on collaborations. Models covering classes and role types are called role models and can be seen as objectified collaborations.

Our approach leverages on role-based modeling by superimposing role models on runtime models of autonomous entities, which allows to dynamically type the runtime information.

IV. KNOWLEDGE EXCHANGE IN DISTRIBUTED PARTIAL RUNTIME MODELS

In this section, we present our technique for role-based adaptive knowledge exchange between collaborating CPS. First, we introduce three strategies for knowledge exchange characterized by different tradeoffs (cf. Section IV-A). Then, by exploiting our cleaning robots example, we present our role-based runtime models (cf. Section IV-B). Finally, we illustrate the role-based adaptive knowledge exchange of partial runtime models through a query language (cf. Section IV-C).

A. Knowledge Exchange Strategies

Towards the definition of knowledge exchange strategies, there are two aspects that are relevant: the degree of pairing among subsystems for their communication and the amount of knowledge to be shared among them. The first can be either total, i.e., every subsystem communicates with each other, or partial, i.e., only some subsystems communicate with some others. The amount of knowledge, on the other hand, can be either complete, i.e., a subsystem exchanges all its knowledge, or a subset, i.e., a subsystem exchanges only a subset of its knowledge, Consequently, we identified the following three strategies that each subsystem can use to exchange its knowledge with the other subsystems.

1) Total-Complete. Each subsystem exchanges its complete knowledge with every subsystem. In this case each subsystem maintains a complete view of the whole system. 2) Partial-Complete. Each subsystem exchanges its complete knowledge, but only with the subsystems it directly collaborates with. 3) Partial-Subset. Each subsystem exchanges only part of its knowledge, and only with the subsystems it collaborates with. The knowledge part that is being exchanged pertains to the current collaboration.

V 1# Par'cipant#2# Leader# Run'me# Model##2# Goals##2# Decisions# V 2# V 3# system# knowledge## Par'cipant#3# Run'me# Model##3# Goals##3# Decisions# Member# Par'cipant#1# Run'me# Model##1# Goals##1# Decisions# Member#
Fig. 1. An Ad-hoc Collaboration Among Three Subsystems.

Figure 1 depicts three subsystems (Participant 1, 2 and 3) forming an ad-hoc collaboration. The collaboration is represented by the black filled boxes specifying the role (Member or Leader) of each participant in the collaboration. Leaders directly collaborate with members, but members do not collaborate with each other directly (only through the leader). Each subsystem has: its own knowledge, captured in a Runtime Model (RM), its own Goals, and its own Decision making mechanisms. A RM specifies a set of relevant aspects of the world and they capture only a part of the whole System Knowledge (i.e., V 1 , V 2 , V 3 that are partial views). Since each subsystem has its own RM, this shall be updated as frequently as needed to adequately reflect the current world.

When some subsystems form a collaboration, their RMs will overlap as they will capture some common parts of the system knowledge. Since each subsystem potentially employs different abstraction mechanisms (masking) on the observed knowledge, it is highly improbable for two subsystems to have identical partial RMs. Each RM is the result of applying an abstraction function f i on the part of the world V i covered by the system:

RM i = f i (V i ).
Below, we revisit and elaborate on the three identified strategies for knowledge exchange.

First strategy (Total-Complete). In the first strategy, where all subsystems share all their knowledge, all subsystems will base their decisions on i∈1..n RM i , where n is the number of participants. This, in turn, offers the benefit that all participants are guaranteed to have all required knowledge for their local decision making. The downside is the negative impact on performance, memory and energy as well as privacy, because potentially more knowledge than necessary is exchanged.

Second strategy (Partial-Complete). For the second strategy only directly collaborating subsystems exchange their complete knowledge. Knowing the application-specific collaboration, allows to reduce the knowledge exchange. Participant 1 shares its knowledge with Participant 2, but not with Participant 3. Similarly, Participant 3 only shares its knowledge with Participant 2. Only Participant 2 has the knowledge of all participants as it directly collaborates with all others. Thus, the drawbacks of the first strategy are reduced. On the contrary, the local decisions of the participants can be suboptimal, due to not having the complete knowledge.

Third strategy (Partial-Subset). Finally, in the third strategy, each subsystem exchanges only part of its knowledge and only with the subsystems it directly collaborates with. As subsystems can take part in multiple collaborations, the challenge is to identify which knowledge of which participant is actually required for their decision making. This strategy allows the subsystems to cherry-pick which information is exchanged in order to further decrease the amount of knowledge that is transferred. While it might decrease the optimality of the local decision making, this strategy helps to improve performance, memory and energy consumption as well as privacy.

The third strategy can therefore vary the tradeoffs between optimality of local decision making, which can be seen as the utility of the participants, and the respective cost. For this, the optimal tradeoff in terms of utility and cost for local decision making is to be identified, where both utility and cost, are functions over the amount of knowledge to be exchanged. To reach this goal, we propose to leverage role models, which are objectified collaborations. The binding of roles to players enables the identification of knowledge required for decision making based on the runtime type of that knowledge.

B. Role-based Runtime Models

Figure 2 depicts in the upper-left the role model Cleaning that is comprised of five role types: a cleaner, a spot, a clean and dirty spot and a full cleaner role type. The links between the role types are role constraints as introduced by Riehle [START_REF] Riehle | Role model based framework design and integration[END_REF].

For example, the implication constraint between the clean spot and spot denotes the necessity of an object playing the clean spot role to play the spot role, too (this holds similarly for dirty spot and spot). The prohibition constraint between the clean spot and dirty spot denotes the necessity for an object playing one of the two roles, not to play the other role at the same time. The solid arrow between cleaner and spot specifies the ability of the cleaner to work on the spot (i.e., is a role-use/association constraint). Note that each role type can define its own attributes, as shown for the cleaner role type. Consequently, role instances can have a state. The binding between roles and their players is expressed using the fills constraint between role types and classes. In the example, fields can play spot roles and robots can play cleaner roles. Moreover, the Cleaning role model comprises a nested role model for rooms. Thus, rooms are modeled as a collaboration of spots, which enables the movement of walls in office buildings. The annotation {1+} is a multiplicity constraint on the number of role instances required by an instance of the role model to be valid (in the example capturing the fact that a room consists of at least one spot).

A concrete scenario is depicted on the right side of Figure 2. Here the world consists of three fields, where two play the clean spot role and one the dirty spot role. In addition, one robot exists, which is participating in the collaboration in the role of a cleaner. The specification of when a player starts or stops to play a role is part of the description of the role types. For example, the cleaner role type will initiate the playership of the full cleaner role for its associated player, when it senses that it is full, i.e., whilst performing its cleaning behavior.

To illustrate how to distinguish collaboration-specific knowledge, which is to be identified for the third knowledge exchange strategy, Figure 2 additionally depicts the Disposal role model. This additional role model specifies that robots can play the role of being bin lorries with a filling level and fields can play the role of bins, with their own filling level. The bin role type denotes an empty field where the cleaning robots are supposed to collect all the trash. If a robot is currently seeking for a bin, it only requires the knowledge captured by the Disposal, but not by the Cleaning role model.

In general, the specification of collaboration-specific knowledge is realized by modeling multiple collaborations, where role types can be played by the same entities of the runtime model. If two autonomous systems approach each other and intend to exchange knowledge, they are now enabled to query collaboration-specific knowledge by formulating the queries against the role models instead of the runtime model. For a comparison between our proposed approach of using role models superimposed on runtime models, Figure 3 depicts a possible metamodel for the runtime model of the above example without role models, i.e., comprising all knowledgewhich was extracted to the two role models in Figure 2.

C. Illustration

Imagine the scenario where two cleaning robots approach each other and one is seeking for the next dirty spot to clean. In the case without role models, the query could look as follows:

SELECT f. * FROM Field f WHERE f.isDirty = true
As a result, the robot will get all the field objects that the other robot knows of being dirty. But, these objects contain additional, superfluous knowledge, which in this case is even known in advance: the field cannot be a bin and, hence, the fill level is set to a default value. In contrast, the query against the role model of Figure 2 could look, in RSQL [START_REF] Jäkel | Rsql -a query language for dynamic data types[END_REF], as follows:

SELECT * FROM Cleaning PLAYING DirtySpot
The PLAYING clause refines the selection based on role types. This query will return all dirty spot role instances that the other robots know of in its current cleaning collaboration. These instances provide just the required knowledge to the requesting robot: the property of the field to be dirty and its coordinates. The knowledge about disposal is not shared.

Thus, using role models, collaborating entities can share knowledge based on the runtime type of the data, i.e., a type that can change at runtime.

An important benefit of using role models to reduce the amount of knowledge to be exchanged is increased evolvability. The restriction to receive only the coordinates of dirty spots could be realized by explicitly naming them in the projection of the SQL query. That is for the example above:

SELECT f.x, f.y FROM Field f WHERE f.isDirty = true
The problem of explicitly naming the attributes of interest arises when the system evolves and new attributes are added to the field. For example, adding a third coordinate (z) to the fields. Without role models, all queries restricting the knowledge to the coordinates of the field have to be adjusted, whereas with role models all queries can stay unchanged.

Next to this, it is of course also possible to filter the results by conditions using the SQL language construct WHERE. For example, a robot seeking for the next dirty spots could narrow Consequently, role models allow to reduce the amount of knowledge to be shared by dynamically changing the runtime type of the requested objects and by evaluating conditions.

V. RELATED WORK

The problem of execution and knowledge exchange via partial models across different elements in the system has been studied by many researchers from multi-agent systems, component-based systems, and nature-inspired computing disciplines. The proposed approaches differ in terms of (i) how they organize the system and coordinate the partial models to achieve system-level goals, and (ii) the mechanisms they employ for knowledge exchange and collaboration. Hence, we limit our discussion to these criteria.

From an organizational point of view, we find grouping or clustering of components/agents to be a popular approach. For instance, DEECo [START_REF] Bures | DEECo: An Ensemble-based Component System[END_REF], SeSaMe [START_REF] Baresi | SeSaMe: Towards a Semantic Self Adaptive Middleware for Smart Spaces[END_REF] and ASPECS [START_REF] Cossentino | Aspecs: an agent-oriented software process for engineering complex systems[END_REF], [START_REF] Cassou | Toward a Tool-Based Development Methodology for Pervasive Computing Applications[END_REF], form interactions and collaborations between different entities of a system through ensembles, groups and holarchies, respectively. DEECo is an ensemble-based component system where an ensemble represents dynamic binding of a set of components and thus determines their composition and interaction. Each component has a state and corresponding processes that are independent of the ensembles it is part of. SeSaMe coordinates distributed components in various selforganizing inter-composed groups based on the types of roles they can play. ASPECS is an agent-oriented methodology that employs holonic organization of agents to map system capabilities to user injected goals. A holon is a general term denoting an entity that has its own individuality, but, at the same time, is embedded in larger wholes.

SAPERE [START_REF] Castelli | Pervasive middleware goes social: The sapere approach[END_REF], on the other hand, takes a nature-inspired approach for component organization and binding, based on the concept of chemical tuple spaces. Each component in the SAPERE ecosystem has an associated semantic representation (chemical annotation); this enables dynamic component interactions in an unsupervised way. The components interact with each other according to the so called "eco-laws" to fulfill their own goals as well as the goals of the overall ecology.

For communication, the proposed solutions adopt different paradigms that include direct messaging, ad-hoc collaboration spaces, and the use of mediator models [START_REF] Spalazzese | Towards a formalization of mediating connectors for on the fly interoperability[END_REF], [START_REF] Inverardi | A theory of mediators for eternal connectors[END_REF]. Typically, agent-based systems use direct messaging among the participating partial models to enable negotiation and persuasion between agents. ASPECS conforms with the FIPA protocol [START_REF] Fipa | Fipa acl message structure specification[END_REF]. SeSaMe also incorporates direct messaging among the components. SAPERE, on the other hand, makes use of a distributed shared space, where context and knowledge are provided by a set of semantic representations of agents. Another approach [START_REF] Bencomo | The role of models@ run. time in supporting on-the-fly interoperability[END_REF] generates a mediator model to bind systems represented in a labeled transition system.

The amount of knowledge, exchanged at runtime between components, each having a partial view of the system, is significant and influences the efficiency of the underlying system. In DEECo, the knowledge of every component is proactively shared among all components and only used in components of the same ensemble (the ensemble membership condition is evaluated locally in each component) [START_REF] Bures | Gossiping Components for Cyber-Physical Systems[END_REF]. SeSaMe handles knowledge distribution through dynamic roles that restrict the amount of knowledge to be shared among interacting components. SAPERE and ASPECS handle this by matching service providers (capabilities) and consumers (goals).

VI. DISCUSSION

In this section, we provide a discussion about possible realizations of our approach. We focus on two promising approaches surveyed in the related work, namely DEECo and SeSaMe, to analyze how they can employ different proposed strategies and use the proposed role models to guide the design process while developing highly dynamic systems. a) Realization of First Strategy through DEECo: To deal with extreme dynamicity coming from external uncertainty in CPS environments (intermittent connections, physical mobility of nodes, opportunistic communication schemes, etc.), DEECo employs the first strategy (Total-Complete) from our list of knowledge exchange strategies. DEECo provides the component-based abstractions and corresponding machinery for implementation and deployment of CPS applications which fit our view of autonomous collaborating subsystems. In particular, the illustrative example from Section IV-C corresponds to a DEECo application comprising two DEECo components: Robot and Room. The Room contains a map of several spots. It also comprises a dynamic collaboration group (ensemble in DEECo terms): DirtySpotsExchange. Its goal is to exchange the knowledge of the Room, specifically the set of the remaining dirty spots in the room, with the Robot.

In particular, every node in a DEECo application periodically and proactively gossips both its own and the knowledge of other nodes that it knows of with all other nodes 1 [START_REF] Bures | Gossiping Components for Cyber-Physical Systems[END_REF]. In our example, this means that all the knowledge of the Room is passed to the Robot, and vice versa. Then, ensemble evaluation is performed locally on the Robot, which results into mapping data from its temporary knowledge base (used by the gossip protocol) to its internal knowledge, a step that finally makes the dirty spots (pertinent to this collaboration) accessible from within the Robot's processes.

Thus, the knowledge exchange approach favored by DEECo trades having complete knowledge for local decision-making for higher network utilization and data redundancy.

b) Realization of Second Strategy through SeSaMe: SeSaMe employs the second strategy (Partial-Complete) as it uses group-based coordination to form collaborations among heterogeneous subsystems. The groups are formed on the basis of roles that various components are capable to perform. Therefore, the proposed role model can be used to guide the design process in SeSaMe. A SeSaMe application of the running example comprises two component types: Robot and Field. Each Robot maintains its own state including its location. The Robot component can play the Cleaner or BinLorry role, whereas, the Field component can take one of three possible roles: DirtySpot, CleanSpot, and Bin role. Each of these roles has a state (data) and programmed behaviors. Further, these components are coordinated as three dynamic SeSaMe groups: (i) the Cleaning group which comprises components assigned to the cleaning task, (ii) the Disposal group consisting of components participating in the disposal activity, and the (iii) Room grouping all Fields of a room.

In SeSaMe, a group is comprised of a component with a supervisor role and various components with follower roles. Therefore, in the current context, a robot can play both the supervisor and follower roles (as a Cleaner or BinLorry) with respect to its participation in various groups (Cleaning or Disposal). On the other hand, a Room group may be composed with certain Cleaning group(s) within a proximity. 1 Here, gossiping takes the form of probabilistic broadcasting.

SeSaMe creates these dynamic groups at runtime, based on the defined rules (grouping policies) [START_REF] Baresi | SeSaMe: Towards a Semantic Self Adaptive Middleware for Smart Spaces[END_REF] and, hence, allows for the formation of ad-hoc collaborations to facilitate roleoriented knowledge exchange among member components of a group. In this way, SeSaMe enables various subsystems to take part in multiple collaborations (groups) and with each collaboration their knowledge is extended. This strategy improves performance and memory consumption but potentially results in decreasing the optimality of local decision-making as decision making is done at group-level.

Although SeSaMe has the notion of roles, it does not inherently support dynamic and adaptive knowledge exchange. We believe that an approach that can make use of the adaptive knowledge exchange and implements the third strategy, can significantly improve the performance and memory consumption of highly dynamic systems.

VII. CONCLUSION

In this paper we presented a role-based adaptive knowledge exchange technique for partial runtime models. Our approach supports adaptation w.r.t. the runtime type of knowledge and conditions over it. We showed three strategies for knowledge exchange between collaborating subsystems, introduced rolebased runtime models for partial views and illustrated their use for role-based knowledge exchange. Our technique enables cooperating subsystems to reduce the amount of knowledge exchanged, which consequently helps to improve the overall performance, to lower energy and memory consumption, as well as to reduce privacy threats. Moreover, our approach lowers maintenance efforts for queries between subsystems by capturing collaboration-specific knowledge in role types. We discussed the general technique and two possible realizations using the DEECo and SeSaMe technologies.

As future work, we plan to build a general framework capturing the protocol of interaction for highly dynamic systems as described in the paper and will show the optimality of the third strategy. For this, we will do prototypical implementations in DEECo and SeSaME. Next, we will investigate the correlation of the amount of knowledge K with the corresponding utility U (K) and cost C(K) by comparing the best possible decisions a subsystem made with complete knowledge with those it made with partial knowledge.

Fig. 3 .

 3 Fig. 3. Metamodel of Runtime Model Without Role Models.

Fig. 2 .

 2 Fig. 2. Combination of Role Models and Runtime Models.

ACKNOWLEDGMENT

The authors would like to thank the GI-Dagstuhl Seminar 14433 -Software Engineering for Self-Adaptive Systems that stimulated initial discussions about this work. This work is partially financed by the Knowledge Foundation through the Internet of Things and People research profile (Malmö University, Sweden), by the German Research Foundation within the Collaborative Research Center 912 "Highly Adaptive Energy-Efficient Computing", by the European Union Seventh Framework Programme FP7-PEOPLE-2010-ITN under grant agreement n o 264840, by the Joint Open Lab "S-Cube" -Telecom Italia S.p.A. Innovation division, Italy, and by the Datalyse project www.datalyse.fr.