Caroline Jardet
email: caroline.jardet@banque-france.fr

Baptiste Meunier
email: baptiste.meunier@ecb.europa.eu

Claudia Braz

Franziska Ohnsorge

Sebastian Barnes

Matteo Mogliani

Sebastien Laurent

Ewen Gallic

Jean-Charles Bricongne

Nowcasting world GDP growth with highfrequency data

Keywords: Nowcasting, mixed frequency, high frequency, large factor models, variable selection, big data JEL classification: C53, C55, E37

While the Covid-19 crisis has shown how high-frequency data can help tracking the economy in real-time, we investigate whether it can improve the nowcasting accuracy of world GDP growth. To this end, we build a large dataset of 718 monthly and 255 weekly series. Our approach builds on a Factor-Augmented MIxed DAta Sampling (FA-MIDAS) which we extend with a pre-selection of variables. We find that this pre-selection markedly enhances performances. This approach also outperforms a LASSO-MIDAS -another technique for dimension reduction in a mixed-frequency setting. While we find that a FA-MIDAS with weekly data outperform other models relying on monthly or quarterly data, we also point to asymmetries. Models with weekly data have indeed performances similar to other models during "normal" times but can strongly outperform them during "crisis" episodes, above all the Covid-19 period. Finally, we build a nowcasting model for world GDP annual growth incorporating weekly data which give timely (one per week) and accurate forecasts (close to IMF and OECD projections but with 1-3 months lead). Policy-wise, this can provide an alternative benchmark for world GDP growth during crisis episodes when sudden swings in the economy make usual benchmark projections (IMF's or OECD's) quickly outdated.

Introduction

The sudden shock of the Covid-19 crisis -with some economies shutting down almost entirely in a matter of days -has put new emphasis on high-frequency data [START_REF] Bricongne | Tracking the economy during the Covid-19 pandemic: the contribution of high-frequency indicators[END_REF].

Weekly, daily, or even hourly data have been extensively used to assess in real-time the impact of the Great Lockdown. A number of innovative dataset have emerged: for example real-time marine traffic was used to track world trade [START_REF] Cerdeiro | World Seaborne Trade in Real Time: A Proof of Concept for Building AIS-based Nowcasts from Scratch[END_REF], hourly electricity consumption to estimate the loss of industrial activity in Europe [START_REF] Igan | Tracking the Economic Impact of COVID-19 and Mitigation Policies in Europe and the United States[END_REF], daily credit card spending to quantify the consumption shock [START_REF] Carvalho | Tracking the COVID-19 Crisis with High-Resolution Transaction Data[END_REF], or weekly labour market statistics to model changes in US employment [START_REF] Coibion | Labor Markets During the COVID-19 Crisis: A Preliminary View[END_REF].

In the meantime, world GDP forecasts provided by international organizations such as the OECD or the IMF -widely used by economists as "benchmark" projections -have appeared to be lagging behind. These institutions assembled scenarios and released projections, but which could only be updated every two or three months, making them rapidly outdated given large and sudden changes in economic conditions. Thus, OECD projections as of March 2020 still assumed a positive figure (+2.4%) for world GDP growth in 2020. Two weeks later when most Western countries entered a strict lockdown, this scenario was already outdated. Mid-April, IMF's WEO projected world GDP at -2.9% for 2020. Forecasts were then not updated before mid-June, where they stood at -6.0% (OECD) and -4.9% (IMF). Most macroeconomists -"projections-takers" -were then facing a lack of "benchmark" projection for world GDP as those of usual "projection-issuers" (IMF and OECD) became quickly obsolete.

The purpose of this paper is to assess if high-frequency data can enhance the nowcasting of world GDP growth and therefore provide a timely alternative "benchmark" projection. To that aim, we build a large dataset of 718 monthly and 255 weekly indicators. Our approach builds on the Factor-Augmented MIxed DAta Sampling (FA-MIDAS) proposed by [START_REF] Marcellino | Factor-MIDAS for now-and forecasting with ragged-edge data: A model comparison for German GDP[END_REF] which consists in using a principal component analysis to extract the common trends from large datasets (here one monthly and one weekly) and in running a MIDAS regression using the extracted factors. This set-up suits since: (i) we mix multiple frequencies when forecasting annual/quarterly world GDP growth with monthly and weekly series; and (ii) we rely on the aggregation of multiple national variables to make up for the lack of global variables.

We extend the FA-MIDAS approach with a pre-selection of variables. This pre-selection step aims at identifying variables that are the more informative to forecast the target variable -in our case world GDP growth. Principal components are then extracted from the selected subset of variables. Literature has shown that this pre-selection improves the forecasting accuracy of factor models -see [START_REF] Boivin | Are more data always better for factor analysis[END_REF] or [START_REF] Bair | Prediction by supervised principal components[END_REF]'s "supervised PCA". Following [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF], we adopt a soft thresholding procedure under which only the top ranked predictors are kept. We consider three alternative techniques based on: the correlation between the target variable and each predictor (sure independence screening of [START_REF] Fan | Sure independence screening for ultrahigh dimensional feature space[END_REF], the t-statistic associated with the coefficient in the univariate regression of the target variable on each predictor (similar to [START_REF] Jurado | Measuring Uncertainty[END_REF], and LARS ordering [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF]. We also compare the performances of this approach with the sparse-group LASSO-MIDAS [START_REF] Babii | Machine Learning Time Series Regressions with an Application to Nowcasting[END_REF], another dimension reduction technique for mixed-frequency data.

Comparing across different specifications for the FA-MIDAS, we find evidence that highfrequency data can significantly improve nowcasting performances, but only during very specific "crisis" episodes. Models with weekly data indeed show greater predictive accuracy compared with an AR model during "crisis" episodes -but have similar performances during "normal" periods. This is in line with the literature pointing out asymmetries in forecasting performance across expansions and recessions, e.g. [START_REF] Chauvet | Forecasting output[END_REF] or [START_REF] Siliverstovs | State-dependent evaluation of predictive ability[END_REF] who test a wide range of specifications and find that such asymmetries in forecasting performance across the business cycle phases are rather common. Our results notably confirm [START_REF] Siliverstovs | New York FED Staff Nowcasts and Reality: What Can We Learn about the Future, the Present, and the Past?[END_REF] showing that while the New York FED's nowcasting models are at least as good as an AR model during expansions, they entail substantial gains in accuracy during recessions. In our paper, we show that the predictive accuracy for models with weekly data is largely greater during the Covid-19 crisis than for models relying solely on monthly data , but that both exhibit similar performances in other periods: "normal" times and also the Great Financial Crisis.

These findings contribute to the on-going debate on whether high-frequency data enhance forecasting performances. [START_REF] Ferrara | Macroeconomic nowcasting in times of Covid-19 crisis: On the usefulness of alternative data[END_REF] showed that a nowcasting model based on highfrequency data produced more accurate forecasts for US growth as of end-March 2020 in Q1 than models based on standard macroeconomic information, but on the other hand, the INSEE (2020) find no significant accuracy gains when including high-frequency data. Our paper shows that while the timely signal provided by weekly data allows for substantial accuracy gains when the activity experiences dramatic swings, its contribution is only of second order when economic conditions are stable. It should be kept in mind that these findings are obtained in a strict data-driven procedure -with no intervention of the forecaster in the selection of series -and in a set-up very close to real-time, with the entire nowcasting framework (pre-selection, factors, MIDAS model) re-estimated and re-calibrated at each date.

As regards variable selection, we find that pre-selecting fewer but more targeted predictors markedly enhances the forecasting performances of the FA-MIDAS -in line with [START_REF] Boivin | Are more data always better for factor analysis[END_REF], Bai and[START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF][START_REF] Schumacher | Factor forecasting using international targeted predictors: The case of German GDP[END_REF] among others. Upon testing different techniques, we find that the LARS technique yields more substantially gains. This extends the literature on the FA-MIDAS by adding a variable pre-selection, showing also that this step yields significant accuracy gains when using a high-dimensional dataset. More broadly, we contribute to the literature on forecasting in a mixed-frequency set-up by comparing a FA-MIDAS with the LASSO-MIDAS, respectively a dense and a sparse approach for dimension reduction. Our results show that the FA-MIDAS outperforms the latter, even more when the FA-MIDAS is combined with variable pre-selection. This might appear in line with [START_REF] Giannone | Economic Predictions with Big Data: The Illusion of Sparsity[END_REF] that showed that sparse modelling is outperformed by dense modelling when the data generating process is not of very low dimension.

Finally, we build a model using weekly data to nowcast the annual growth of world GDP. A pseudo-real-time exercise during the Covid-19 crisis shows that this model provides timely estimates with a 1-3 months lead on IMF and OECD releases. It might therefore serve as an alternative "benchmark" during "crisis" episodes when institutional projections are rapidly outdated. This extends the literature on forecasting world GDP. In this strand, some papers have used bridge models (e.g. [START_REF] Golinelli | Tracking world trade and GDP in real time[END_REF], but our paper is closer to those based on large datasets such as [START_REF] Matheson | New Indicators for Tracking Growth in Real Time[END_REF] or [START_REF] Ferrara | Nowcasting global economic growth: A factor-augmented mixed-frequency approach[END_REF]. The two main additions to this literature, in particular to the latter paper, relate to the inclusion of weekly data -shown to significantly increase performances during "crisis" episodes -and of variable pre-selection -shown to also yield significant gains in predictive accuracy.

The rest of paper is organised as follows: section 1 presents the data and statistical issues, section 2 presents the FA-MIDAS extended with pre-selection, section 3 details the strategy to compare models in a real-time set-up, and section 4 discusses results.

Section 1: Data

Use of high-frequency data

The purpose of this paper is the nowcasting of global GDP quarterly (or annual) growth rate 𝑦 𝑡 by exploiting the infra-quarterly (or infra-annual) information available through monthly or weekly indicators as represented in Figure 1. The red square figures a given date -around May 10 th in this example -and available information appear in red. Official GDP data for Q1

are not yet available (they are published around 2 months after quarter end); monthly indicators are available only until month 3 (they are published at best around 20 days after month end) but weekly data are available up to the preceding week. This illustrates how timeliness is a strong comparative advantage of weekly data and the main reason why one can consider incorporating it in a nowcasting model.

Facing a lack of "world" variables -or their lack of timeliness when such series exist, our approach rely instead on pooling national statistics across multiple countries. We build a large cross-national dataset from which we can aggregate the information into a few factors by principal component analysis (PCA). To that aim, we gather a dataset of 718 monthly and 255 weekly series covering a wide range of economic activities and countries (accounting for around 90% of world GDP in PPP terms) described in Table A2.1 in Annex 2. released the day after month end -and significant predictive power [START_REF] Harris | Tracking the Economy with the Purchasing Managers' Index[END_REF]d'Agostino and Schnatz, 2012). Notably, [START_REF] Lahiri | Nowcasting US GDP: The role of ISM Business Surveys[END_REF] find evidence that PMIs improve forecast accuracy in factor models. In addition to the "headline" indices, sub-indices for trade ("new export orders") and those capturing tensions in the production apparatus at an early stage ("new orders", "suppliers' delivery time", "output", and "output prices") are included.

As regards weekly data, while the literature generally only considers financial variables at such high frequency (e.g. [START_REF] Andreou | Should Macroeconomic Forecasters Use Daily Financial Data and How?[END_REF], several series in our dataset relate to the "real" economy. We include China housing price indices, commodity prices, trade indices (e.g. Baltic Dry Index), and various indices for the US economy (e.g. new jobless claims, gasoline consumption, steel production, US business condition index of [START_REF] Aruoba | Real-Time Measurement of Business Conditions[END_REF]). While more weekly data from countries besides US and China would have been desirable, no data with sufficient timespan was available to the best of our knowledge. In addition, gains in forecasting accuracy from US-/China-centric data might still be valuable given their major role in global dynamics. In this vein, [START_REF] Kindberg-Hanlon | Gauging the globe: the Bank's approach to nowcasting world GDP[END_REF] have documented a high correlation between US data and world GDP growth. 1 Chiu et al. (2020) have also demonstrated a high correlation between the Baltic dry index and business cycles in BRICS, suggesting that this series can account for dynamics in emerging economies.

As is more standard in the literature, our weekly dataset also includes financial variables: stock market indexes, nominal effective exchange rates, 3-month interbank rates, and 10-year sovereign rates. The term spread, computed as the difference between the latter two, is included as well as other global or aggregate financial indices (e.g. Standard & Poor's Global 1200, VIX). All these variables are retrieved for around 40 countries accounting all together for around 90% of world GDP in PPP terms.

Handling high-frequency data

An issue arising for monthly data with asynchronous publication lags is that the dataset has a "ragged-edge" pattern: monthly indicators can have different missing elements at the end of the sample -making the monthly dataset unbalanced. To address this, we use the "vertical realignment" procedure of [START_REF] Altissimo | New Eurocoin: Tracking Economic Growth in Real Time[END_REF]. For every series, the last available point is taken as the contemporaneous value and the entire series is realigned accordingly. Formally, for a series 𝑥 𝑡 whose last observation at a contemporaneous date 𝑇 is at 𝑇 -𝑘, the series becomes 𝑥̅ 𝑡 = 𝑥 𝑡-𝑘 .2 While several issues may be induced by this method -most notably that the availability of data determines dynamic cross-correlation between variables and can then change over time, [START_REF] Marcellino | Factor-MIDAS for now-and forecasting with ragged-edge data: A model comparison for German GDP[END_REF] empirically test other methods and find no substantial changes on the nowcasting performance across these methods. 3While the "ragged-edge" pattern does not affect our timely weekly dataset, ensuring stationarity and seasonality are concerns given the absence of well-established method for the seasonal adjustment of weekly series. Both issues might be alleviated by taking the annual growth rate of the series as in [START_REF] Lewis | Monitoring Real Activity in Real Time: The Weekly Economic Index[END_REF]. But while statistically correct, this approach introduces a base effect which might be problematic for our objective: as most indicators suffered a dramatic drop in March 2020, the jump in March 2021 will be symmetrically dramatic and might put at risk the viability of the nowcasting. More broadly, [START_REF] Ladiray | Seasonal Adjustment of Daily and Weekly Data[END_REF] discuss the drawbacks of taking annual growth for weekly data and point out that it not only includes a phase shift by design but also can introduce spurious cycles.

To alleviate these concerns, we use a two-step procedure to obtain de-seasonalised and stationary weekly indicators. In the first step, non-stationary series are transformed in their average weekly variation over the last four weeks. 4 This transformation -equivalent to a moving monthly growth -have the double advantage of making the series stationary and correcting for infra-monthly seasonality. Then the transformed series is regressed on monthly dummies; the final series is the residual of this regression. This last step allows us to correct for any monthly (or lower frequency) seasonality. To avoid potential distortions that could be caused by the unusual behaviour of some variables during the Covid-19 crisis (e.g. the peak at 6.6 million initial jobless claims in the US in the last week of March 2020 vs. a maximum of 660,000 over 2001-2019), the regression with monthly dummies is estimated over 2001-2019.

Section 2: Extending the FA-MIDAS model with variable pre-selection

The FA-MIDAS model

Our econometric framework is based on the two-step FA-MIDAS approach proposed by [START_REF] Marcellino | Factor-MIDAS for now-and forecasting with ragged-edge data: A model comparison for German GDP[END_REF]. The first step is a principal component analysis, run on both the monthly and weekly datasets. Doing so, we extract the common trends at both frequencies. Formally, we assume that the dataset 𝑋 𝑇 can be represented according to a factor structure with a 𝑟-dimensional factor vector 𝐹 𝑇 , Λ the loadings matrix and an idiosyncratic components 𝜉 𝑇 not explained by the common factors. The common components (Λ ⋅ 𝐹 𝑇 ) and the idiosyncratic components are mutually orthogonal.

𝑋 𝑇 = Λ ⋅ 𝐹 𝑇 + 𝜉 𝑇

Once monthly and weekly factors have been extracted, the second step is the modelling in a MIDAS specification in which the dependent variable 𝑦 𝑡 is the quarter-on-quarter growth rate of world GDP at quarter t. MIDAS regression builds on the seminal work by [START_REF] Ghysels | The MIDAS Touch: Mixed Data Sampling Regression Models[END_REF] showing that this specification -by allocating different weights to the different lags of high-frequency regressors -performs better than a flat aggregation where high-frequency regressors are averaged at lower-frequency. Explanatory variables are a quarterly constant 𝛽 0 as well as the monthly 𝑓 𝑡 3 ⁄ 𝑚 and weekly 𝑓 𝑡 13 ⁄ 𝑤 factors. 𝐾 represents the number of highfrequency lags and 𝜃 is a vector of parameters of the MIDAS weighting function 𝑔.

𝑦 𝑡 = 𝛽 0 + 𝑔 (𝑓 𝑡 3 ⁄ 𝑚 , 𝜃 𝑚 , 𝐾 𝑚 , … ) + 𝑔 (𝑓 𝑡 13 ⁄ 𝑤 , 𝜃 𝑤 , 𝐾 𝑤 , … ) + 𝜀 𝑡
The MIDAS weighting function (𝑔) used is a "Almon" polynomial of degree 𝑝 = 3. The coefficients for the lags of the high-frequency regressor are modelled through a polynomial function of degree 𝑝 -1. This specification is the most parsimonious since only 𝑝 parameters are estimated. Formally, the weighting function is:

𝑔(𝑓 𝑡 , 𝜃, 𝐾, 𝑝) = ∑ 𝑐(𝑘, 𝜃) ⋅ 𝑓 𝑡-𝑘 𝐾 𝑘=0 𝑤ℎ𝑒𝑟𝑒 𝑐(𝑘, 𝜃) = ∑ 𝑘 𝑗 ⋅ 𝜃 𝑗 𝑝-1 𝑗=0
The optimal number of lags is determined by minimizing the in-sample sum of squared residuals while fixing an upper limit of 4 lags for monthly factor and 8 for weekly factor. The number of high-frequency lags can be different between the monthly and the weekly factors.

Pre-selection of regressors

When forecasting with a high-dimensional dataset, the literature [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF][START_REF] Schumacher | Factor forecasting using international targeted predictors: The case of German GDP[END_REF] concludes that the accuracy of factor models is significantly improved when selecting fewer but more informative predictors. On a more theoretical ground, [START_REF] Boivin | Are more data always better for factor analysis[END_REF] find that larger datasets lead to poorer forecasting performances when idiosyncratic errors are cross-correlated or when the variables with higher predictive power are dominated.

Against this background, we extend the FA-MIDAS approach by adding a preliminary step to select the regressors with the highest predictive power. The initial dataset is 𝑋 𝑡 = (𝑥 1,𝑡 , 𝑥 2,𝑡 , … , 𝑥 𝑁,𝑡 )′ with 𝑡 = 1, … , 𝑇 and 𝑁 variables (𝑁 𝑚 = 718 for the monthly dataset and 𝑁 𝑤 = 255 for the weekly dataset). Both figures largely exceed the number of observations 𝑇 = 81 (2001Q1-2021Q1). The idea underlying pre-selection is to rank the potential regressors 𝑥 𝑖,𝑡 based on a measure of their predictive power with respect to the target variable. In the following, three main techniques are used:

-t-stat ranking (t-stat): each potential regressor 𝑥 𝑖,𝑡 is ranked based on the absolute value of the t-statistic associated with its coefficient estimates in a univariate regression of 𝑥 𝑖,𝑡 on the target variable 𝑦 𝑡 . The univariate regression also includes four lags of the dependent variable to control for the dynamics of the dependent variable. While originating in genetic studies [START_REF] Bair | Prediction by supervised principal components[END_REF], this pre-selection technique has found its way to economics -see for example [START_REF] Jurado | Measuring Uncertainty[END_REF].

-"Sure Independence Screening" (SIS) of [START_REF] Fan | Sure independence screening for ultrahigh dimensional feature space[END_REF]: regressors are ranked based on their marginal correlation with the target predictor. In their theorem 1, [START_REF] Fan | Sure independence screening for ultrahigh dimensional feature space[END_REF] provide theoretical ground for this approach by demonstrating that it has the sure screening property that "all important variables survive after applying a variable screening procedure with probability tending to 1". This approach has been used for nowcasting in [START_REF] Ferrara | When are Google data useful to nowcast GDP? An approach via pre-selection and shrinkage[END_REF] or [START_REF] Proietti | Nowcasting monthly GDP with big data: A model averaging approach[END_REF].

-Least-Angle Regression (LARS) algorithm [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF]: while the two methods above are based on univariate relationships of the regressors with the target variable, this one provides a ranking of the predictors when the presence of the other predictors is taken into account. Essentially, the LARS -developed by [START_REF] Efron | Least angle regression[END_REF] -is a computationally efficient iterative forward selection algorithm, less aggressive than peer techniques in eliminating too many predictors correlated with the ones included.

Another key advantage of LARS is its generality as [START_REF] Efron | Least angle regression[END_REF] showed that the LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] is in fact a special case of the LARS. By extension, the LARS algorithm can also be used to solve the optimization criterion of the Elastic Net [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]. The algorithm is formally defined in Annex 3 but briefly, starting from no predictors, it adds one at each step by proceeding equiangularly between the variables in the most correlated set. The algorithm adds one regressor at each step, meaning that if we end after 𝑘 steps, it provides an active set of 𝑘 predictors. By continuing until the max number of predictors, the algorithm then provides an ordering of potential regressors 𝑥 𝑖,𝑡 according to the iteration at which they join the active set. This approach had been used in several nowcasting studies such as [START_REF] Schumacher | Factor forecasting using international targeted predictors: The case of German GDP[END_REF], [START_REF] Bulligan | Forecasting economic activity with targeted predictors[END_REF] or Sousa and Falagiardia (2015).

Once a ranking of the potential regressors 𝑥 𝑖,𝑡 is obtained, pre-selection is about defining the optimal number 𝑘 * of regressors to include into the regression. In practice, 𝑘 * is chosen by testing over different values and electing the one optimizing the criterion set by the forecaster.

In our FA-MIDAS approach, this means estimating different factor models using different subsets 𝑋 𝑡 𝑘 = (𝑥 1,𝑡 * , 𝑥 2,𝑡 * , … , 𝑥 𝑘,𝑡 * ) consisting of the 𝑘 predictors with the highest scores. In our case with distinct monthly and weekly factors, the optimal numbers of monthly and weekly regressors (respectively 𝑘 𝑚, * and 𝑘 𝑤, * ) are obtained simultaneously by testing over different combinations of (𝑘 𝑚 , 𝑘 𝑤 ).

Section 3: Comparing performances across models

Set of FA-MIDAS models

To test whether high-frequency data improves predictive accuracy of the nowcasting, we compare performances for three different FA-MIDAS models described below. In addition, we also test an AR model with the latest available data for quarterly world GDP.

-Model 1 includes the monthly factor (𝑓 𝑡 𝑚 ) and the weekly factor (𝑓 𝑡 𝑤 ).

-Model 2 includes a factor at monthly frequency, but which incorporates also weekly series averaged over the month (𝑓 𝑡 𝑚,𝑤 ). 5 to 2005Q1 and the first forecast is 2005Q2; and so on with the estimation sample extended by one quarter following an "expanding window" procedure.

Real-time exercise

To reflect the fact that the nowcasting would be done in real-time, we run estimations for the first, second, and third months of the quarter. The information taken into consideration differs for each month 𝑚 𝑖 to reflect what would have been available to the forecaster in real-time. We generally consider that the nowcast takes place around the 25 th day of month 𝑚 𝑖 + 1: 6 last quarter growth is then available only from the 3 rd month onwards -in line with an average publication lag of 2 months for GDP statistics; monthly variables are available up to the month 𝑚 𝑖 , 7 and weekly variables up to the 3 rd week of month 𝑚 𝑖 + 1 given their timeliness. Data availability is recapitulated in Table 1. 8 To further mimic a real-time set-up, the ranking of regressors and the calibration of the optimal number of regressors are re-estimated at each date, following the procedure shown 6 For example, it means that the nowcasting for "1 st month" is performed around day 25 of the 2 nd month. The numbering relates to the availability of the monthly factor -rather than to the moment of the nowcast: "1 st month" means that the monthly factor for 1 st month is available, not that the nowcast takes place in the 1 st month. 7 The "vertical realignment" procedure of [START_REF] Altissimo | New Eurocoin: Tracking Economic Growth in Real Time[END_REF] is particularly convenient at this stage since it transposes past publication lags. Indeed, if publication lags are constant over time, the value at any month 𝑚 0 in the past of the factor estimated contemporaneously at 𝑚 = 𝑚 0 + 𝜏 is the exact same value as what would have been estimated with information available at 𝑚 0 -not accounting for the revisions between 𝑚 0 and 𝑚. 8 However, due to the data unavailability of the vintages for many series, the analysis can only be conducted in pseudo real-time: it is not based on vintages that would have reflected the exact information set available to forecasters at the time of forecasting but rather mimics the availability of series due to publication delays. Another difference between our approach and a "pure" real-time approach is that the seasonal adjustment is performed once over the entire sample -and not once at each point with the data that would have been available at the moment of the nowcasting. This first reflects the fact that most series in the dataset are retrieved already seasonally adjusted from the data sources (and for some, e.g. PMIs, without having the possibility to retrieve the unadjusted series). In addition, empirical checks show that the difference between the two approaches (once for the entire sample or progressively with data available up to this point) yields very similar profiles for the resulting series.

Only the Covid-19 period is found to induce distortions for some weekly data whose behaviour has been hugely unusual in 2020 (e.g. US initial jobless claims which reached a peak at 6.6 million in the last week of March 2020, 1,000 more than the maximum level over 1990-2019). The Covid-19 has therefore been excluded from the sample on which parameters for seasonal adjustment are calibrated. 2) mimicking the exact information that would have been available to the real-time forecaster at date 𝑡. This means that the correlation of potential regressors with the target variable over the out-of-sample period is not considered: while closer to a real-time setup, this approach however contrasts with most of the literature in which the ranking appears to be performed only once and over the full sample (e.g. [START_REF] Bulligan | Forecasting economic activity with targeted predictors[END_REF][START_REF] Sousa | Forecasting euro area inflation using targeted predictors: is money coming back?[END_REF], thereby giving further advantage to pre-selection. In addition, to be fully consistent with a real-time exercise, the optimal number of regressors (𝑘 * ) is also re-calibrated at each date, also only on the in-sample period. To do so, the in-sample period is split between "train" (2001Q1-2005Q1) and "test" (2005Q2 to 𝑡) samples. Then pseudo-out-of-sample nowcasts 9 are made over the "test" sample: at each quarter 𝑞, the model is estimated from 2001Q1 to 𝑞 -1 and produce a one-period ahead forecasts for 𝑞. This gives a forecast error at 𝑞; repeating this procedure over the whole "test" sample gives a pseudo-out-of-sample RMSE.

This procedure (step 2 in Figure 2) is repeated across all possible values 𝑘 of potential regressors. In the end, the optimal number of regressors 𝑘 * chosen as the one minimizing the RMSE over the "test" sample. 10 The FA-MIDAS model with 𝑘 * regressors is then used to produce an out-of-sample one-period-ahead of world GDP growth at 𝑡 + 1 (step 3 in Figure 2). This way, the optimal number of regressors varies at each date 𝑡, as well as the ranking of regressors. 9 "Pseudo out-of-sample" since the ranking of regressors is made over the entire in-sample period, thereby also including the "test" period. 10 For model 1 with both weekly and monthly factors, this procedure is applied across all possible couples (𝑘 𝑚 , 𝑘 𝑤 ) indicating the number of respectively monthly and weekly series included in the factors.

Train Test (n quarters)

Out-of-sample 2001 Q1 2005 Q1 t

…

Step 1

Step 2

Step 3

Step n

In-sample 2 RMSE for one-period ahead nowcasts over the "test" sample in order to calibrate the optimal number of regressors (k*)

One-period ahead nowcast (out-of-sample) based on the optimal number of regressors (k*) calibrated in-sample at step 2 3 1

Rank the potential regressors over the full in-sample period based on a pre-selection rule (t-stat, SIS, LARS)

Section 4: Results

Pre-selection techniques

Results for the different pre-selection techniques are presented in Table 2. 11 Pre-selection in a FA-MIDAS yields accuracy gains up to 13% in terms of out-of-sample performances, on average over the three months, against the alternative of no pre-selection. However, gains are only timid for the first month: in the case of the LARS, 6% accuracy gains at month 1 compared with around 18% at months 2 and 3. It is notable that such gains arise even with a strict realtime set-up. When the pre-selection is not run in real-time (i.e. when the ranking of regressors is instead performed only once and over the entire sample, as in some of the literature), accuracy gains can reach up to 66% as shown in Table A1.1 in Annex 1.

A second finding in Table 2 is that the LARS procedure [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF] The factors obtained for the monthly and weekly datasets at 𝑡 = 2021Q1 and for the month 𝑚 3 are displayed in Figure 3. 12 Both appear to track adequately world GDP and to be leading indicators of turning points in the global economy with a 1-3 months lead.

11 Results are reported for the model 1. While not reported, the results for other models are similar.

12 Based on the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] information criterion and on its modified version in [START_REF] Alessi | Improved penalization for determining the number of factors inapproximate factor models[END_REF], the optimal number of factors to include in the regression is determined to be one. World GDP q-o-q growth (LHS)

Weekly factor (RHS) Weekly factor, 13-week moving average (RHS)

Note: factors are computed using the optimal number of regressors for a LARS technique, for month 3, and at 2021Q1 (meaning with ranking of regressors over 2001Q1-2020Q4 and calibration of optimal number of variables for one-period out-of-sample forecasts over 2005Q2-2020Q4, cf. section 3.2). the optimal number of regressors can be different for monthly (LHS) and weekly (RHS) frequencies.

In the lower part of Table 2, we also compare performances of the FA-MIDAS with the LASSO-MIDAS, another dimension reduction technique suited for a mixed-frequency setting.

LASSO is a sparse approach based on a penalized regression. Contrary to the FA-MIDAS, it performs both the selection of regressors and the estimation of the mixed-frequency equation in a single step. The LASSO-MIDAS is a recent addition to the literature [START_REF] Marsilli | Variable Selection in Predictive MIDAS Models[END_REF][START_REF] Uematsu | High-dimensional macroeconomic forecasting and variable selection via penalized regression[END_REF][START_REF] Mogliani | Bayesian MIDAS penalized regressions: estimation, selection, and prediction[END_REF] and the model used in this paper is the sparse-group LASSO of [START_REF] Babii | Machine Learning Time Series Regressions with an Application to Nowcasting[END_REF]. The interest of the "group-LASSO" over the unstructured LASSO is that for a given regressor, either all high-frequency lags enter the regression, or all are set to zero. Said otherwise, it does not select only some of the lags of a high-frequency regressor while setting other lags of the same regressor at zero, as would do an unstructured LASSO-MIDAS. This approach is therefore more consistent with our FA-MIDAS, and also more interpretable for a practitioner. Two other advantages of the group-LASSO are its computational efficiency and -as shown by [START_REF] Babii | Machine Learning Time Series Regressions with an Application to Nowcasting[END_REF] -the fact that it outperforms the unstructured LASSO.

The comparison shows that the FA-MIDAS outperforms the LASSO-MIDAS. This might seem in line with [START_REF] Giannone | Economic Predictions with Big Data: The Illusion of Sparsity[END_REF] showing that "sparse" methods -such as the LASSOperform better than "dense" approaches -such as factor models -only if the data generating process is of very low dimension. In the case of this paper, where the prediction of a global variable relies on various national indicators, such sparsity might be unlikely.13

Model comparison

Model performances are reported in Table 3. Both in-and out-of-sample RMSE suggest that weekly data improve nowcasting accuracy. Models with weekly data (1 and 2) generally outperform the model with only monthly data (3), and largely outperform the AR model.

Among models with weekly data, it appears that model 1 which includes a separated weekly factor yields better performances -with the exception of month 1 in the out-of-sample exercise -than the model incorporating weekly series with monthly data in a unique factor (model 2). We assess the significance of differences in predictive accuracy using the [START_REF] Diebold | Comparing predictive accuracy[END_REF] test with the [START_REF] Harvey | Testing the equality of prediction mean squared errors[END_REF] corrected variance for small-sample bias. Results are reported in Table 4. Model 1 is found to have significantly better predictive accuracy than model 3 and the AR model at month 2, while also outperforming the AR model at month 1.

No significant differences however arise at month 3. It confirms that relying on weekly data can drive significant improvements in accuracy against models that rely solely on monthly or quarterly data, though these gains are broad-based across all months of the quarter. Since nested models can weaken the inference in Diebold-Mariano tests [START_REF] Clark | Tests of equal forecast accuracy and encompassing for nested models[END_REF], we complement this analysis with a Model Confidence Set (MCS) test of Hansen et al.

(2011)14 in Table A1.2 in Annex 1: for months 2 and 3 of the quarter, the model 1 is always elected as the best model and the only remaining in the ℳ 50 * .

Table 4. [START_REF] Diebold | Comparing predictive accuracy[END_REF]

"Crisis" vs. "normal" periods

A recent strand of literature has documented state-dependent performances of forecasting models [START_REF] Chauvet | Forecasting output[END_REF]Siliverstovs and Wochner 2020;[START_REF] Siliverstovs | Assessing nowcast accuracy of US GDP growth in real time: The role of booms and busts[END_REF][START_REF] Siliverstovs | New York FED Staff Nowcasts and Reality: What Can We Learn about the Future, the Present, and the Past?[END_REF] which can differ depending on the state of the business cycle such as recessionary vs. expansionary period. The possibility of asymmetries in forecasting performances is even greater with highfrequency data given the general trade-off between timeliness and accuracy [START_REF] Ahnert | Trade-off between timeliness and accuracy[END_REF]: such data provide a timely signal which can enhance nowcasting performance if economic conditions suddenly deteriorate but during "normal" periods, their contribution might be only of second order.

Against this background, we distinguish our sample between "normal" and "crisis" episodes, and then compute the in-and out-of-sample RMSE for both sub-samples. 15 Results are reported in Table 5 where it appears that models 1 and 2 with weekly data heavily outperform others during "crisis" episodes. During those periods, the upside of providing a very timely signal likely exceeds the downside of noise in the weekly data. However, during "normal"

periods, those models have performances closer to those of models relying on monthly data (model 3) or on AR terms. In particular, the relative performances of FA-MIDAS models over the AR benchmark tend to be higher during "crisis" periods. For instance, model 1 entails on average a 42% improvement in predictive accuracy compared with the AR during "crisis"

episodes, but 33% during "normal" episodes.

Crisis episodes

In-sample 15 Regressions are still estimated over full sample -or for out-of-sample over the sample preceding target quarter. appear to outperform other models during the Great Financial Crisis and that gains can be reached with model 1 also at month 1 of the quarter. This calls for caution when using preselection since using a "pseudo real-time" set-up can lead to different results than a "realtime" framework.

𝐶𝑆𝑆𝐹𝐸𝐷 [0,𝑇] = ∑[𝑒̂𝑖 ,𝑡 2 -𝑒̂𝑗 ,𝑡 2 𝑇 𝑡=0 ],

Annual GDP forecasts

Back to the issue highlighted in the introduction, the main concern for "projection-takers" macroeconomists during the Covid-19 crisis has been the rapid obsolescence of "benchmark"

world GDP forecasts. Building on our results that weekly data significantly improve forecast performance during these crisis episodes, we construct a nowcasting model for the annual growth rate of world GDP based on a FA-MIDAS specification. This model predicts the annual growth rate of world GDP 𝑦 𝑡 𝑎𝑛𝑛𝑢𝑎𝑙 using the carry-over 𝑐 𝑡 𝑞 provided by the known quarterly growth rates -and whose coefficient is fixed to 1 in the regression, the monthly factor (𝑓 𝑡 𝑚 )

and the weekly factor (𝑓 𝑡 𝑤 ). 16 Formally: in Annex 1 where we compare the model described above with three alternatives: a model without the weekly factor; a model with only the carry-over; and an AR model. Our baseline model appears to outperform others for all weeks of the year.

𝑦 𝑡 𝑎𝑛𝑛𝑢𝑎𝑙 =
We finally compare in Figure 5 the pseudo real-time forecasts of our model with the projections released by the OECD and the IMF. We run a "pseudo real-time" comparison in which FA-MIDAS is re-estimated at the end of each week with the information available at this stage. 17 It shows that the nowcasting has the advantage of greater timeliness and decent accuracy compared with IMF's or OECD's forecasts as the former seems to have a 1 to 3 months lead on the latter. This suggests that a nowcasting based on high-frequency data can provide an alternative "benchmark" projection during crisis episodes. Fortunately, this is the moment when it is most needed (due the obsolescence of usual "benchmark" projections) and this coincides with the moment when high-frequency data most greatly enhance predictive performances.

Conclusion

Our paper provides evidence that including high-frequency (weekly) variables can significantly improve the in-and out-of-sample performances in a FA-MIDAS specification, but only during specific "crisis" episodes. Further analysis reveals that performances of models with weekly data are broadly similar to those of an AR model during "normal" periods when activity remains broadly stable. Large differences in predictive accuracy between models with weekly data and AR models happen only during "crisis" episodes (2008-2009 and 2020-2021, i.e. 12 quarterly observations on a total estimation sample of 81 observations). This finding points to strong asymmetries in nowcasting performances following the literature on state-dependent relative performances of sophisticated models vs.

the AR benchmark (e.g. [START_REF] Siliverstovs | Assessing nowcast accuracy of US GDP growth in real time: The role of booms and busts[END_REF][START_REF] Siliverstovs | New York FED Staff Nowcasts and Reality: What Can We Learn about the Future, the Present, and the Past?[END_REF]. In addition, when comparing models with weekly data to models relying on monthly data, performances are alike during "normal"

periods but also during the Great Financial Crisis (2008)(2009). The only differences arise at the moment of the Covid-19 crisis. In other terms, weekly data greatly contributes to nowcasting period only during the Covid-19 episode when its timeliness could make up for the long sample period. Instead we rely on a ranking of regressors performed once -over the full sample -and a fixed number of regressors (see footnote above). Therefore, this is rather a "pseudo out-of-sample" exercise in contrast with the set-up for quarterly nowcasts. -The first step is to identify the predictor 𝑥 𝑗 most correlated with the target variable 𝑦, then move its coefficient 𝛽 𝑗 from 0 to the direction given by its least-squares coefficient.

Annex 1: Additional tables

This causes the correlation of this regressor 𝑥 𝑗 with the evolving residual (𝑦 -𝛽 𝑗 • 𝑥 𝑗 )

to decrease in absolute value. As soon as another regressor 𝑥 𝑘 has as much correlation with the evolving residual, the process is paused and 𝑥 𝑘 joins the active set.

-Then coefficients 𝛽 𝑗 and 𝛽 𝑘 are moved equiangularly in the direction of their leastsquares coefficients until another predictor 𝑥 𝑙 has as much correlation with the updated evolving residual (𝑦 -𝛽 𝑗 • 𝑥 𝑗 -𝛽 𝑘 • 𝑥 𝑘 ).

-This procedure is continued until all predictors entered the active set, formally noted 𝔸 𝑘 . The size of the latter increases by one at each step. After 𝑘 steps, 𝑘 variables are in 𝔸 𝑘 or in other words, the coefficients corresponding to the remaining 𝑁 -𝑘 predictors are set to zero in the regression.

More formally, at step 𝑘:

-Let's note 𝜇̂𝔸 𝑘 the current estimate of 𝑦 with the regressors corresponding to the active set 𝔸 𝑘 .

-The vector of current correlations is 𝑐̂= 𝑋′(𝑦 -𝜇̂𝔸 𝑘 ). By definition, the active set 𝔸 𝑘 is the set of indices corresponding to predictors with the highest absolute correlation noted 𝐶 ̂= 𝑚𝑎𝑥 𝑗 {|𝑐̂𝑗|}, i.e. 𝔸 𝑘 = {𝑗 ∶ |𝑐̂𝑗| = 𝐶 ̂}.

-The active set 𝔸 𝑘 defines the matrix of predictors corresponding to this set 𝑋 𝔸 𝑘 = (𝑠 𝑗 𝑥 𝑗 ) 𝑗𝜖𝔸 𝑘 with 𝑠 𝑗 = 𝑠𝑖𝑔𝑛(𝑐̂𝑗).

-𝑢 𝑘 = 𝑋 𝔸 𝑘 𝑤 𝑘 is the equiangular vector: the name "least angle" arises from a geometrical interpretation of this process as 𝑢 𝑘 makes the smallest and equal angle with each of the predictors in 𝔸 𝑘 .

o 𝑤 𝑘 is defined as 𝑤 𝑘 = 𝐴 𝔸 𝑘 (𝑋 𝔸 𝑘 ′𝑋 𝔸 𝑘 ) -1 1 𝔸 𝑘 with 𝐴 𝔸 𝑘 = (1 𝔸 𝑘 ′ (𝑋 𝔸 𝑘 ′𝑋 𝔸 𝑘 ) -1 1 𝔸 𝑘 ) -1/2 with 1 𝔸 𝑘 a vector of ones equalling the size of 𝔸 𝑘 .

o We also define the inner product 𝑎 𝔸 𝑘 = 𝑋′𝑢 𝑘 .

The next step is to move the estimation 𝜇̂𝔸 𝑘 in the direction set by the equiangular vector 𝑢 𝑘 until another regressor from 𝔸 𝑘 𝑐 (i.e. the complement of 𝔸 𝑘 or in other words the regressors that are not yet included in the active set) joins the active set. Formally noted, estimation will reach 𝜇̂𝔸 𝑘 + = 𝜇̂𝔸 𝑘 + 𝛾 ̂• 𝑢 𝑘 with 𝛾 ̂ minimizing the following criterion:

Figure 1 .

 1 Figure 1. Approach to nowcasting annual / quarterly world GDP growth

  Comparing model 1 vs. model 2 allows us to test whether a three-frequency model performs better than a two-frequency model where weekly indicators are averaged at a monthly frequency. -Model 3 includes the monthly factor (𝑓 𝑡 𝑚 ). Comparing model 1 (model 2) vs. model 3 allows us to test whether weekly data can improve nowcasting performances. Model comparisons are based on the root mean squared errors (RMSE). Out-of-sample errors are computed over 2005Q2-2021Q1 to capture both the Great Financial Crisis and the Great Lockdown. Nowcasts are one-period ahead forecasts: the initial estimation sample is 2001Q1

Figure 2 .

 2 Figure 2. Real-time set-up for the pre-selection

Figure 3 .

 3 Figure 3. Factors and world GDP growth Monthly factor

Figure 4 .Figure 4

 44 Figure 4. CSSFED (out-of-sample) of model 1 compared to other models Panel A. Month 1 2005Q1-2019Q4 2019Q4 -2021Q1

Figure 5 .

 5 Figure 5. Pseudo real-time nowcast and forecasts for the annual world GDP growth in 2020

Figure A1. 1 .

 1 Figure A1.1. CSSFED (pseudo out-of-sample) of model 1 compared to other models Panel A. Month 1 2005Q1-2019Q4 2019Q4 -2021Q1

Table 1 .

 1 Data availability for each "month" of the nowcasting At date 𝑡, the ranking of regressors is performed over the in-sample period only (step 1 in Figure

		1 st month	2 nd month	3 rd month
	Quarterly variable	2 quarters lag	2 quarters lag	1 quarter lag
	Monthly variables	Up to 1 st month	Up to 2 nd month	Up to 3 rd month
	Weekly variables	Up to 3 rd week of 2 nd month	Up to 3 rd week of 3 rd month	Up to final week of 3 rd month

Table 2 .

 2 Out-of-sample RMSE across pre-selection techniques and months of the quarter

		1 st month	2 nd month	3 rd month	Average
	No pre-selection	0.973	1.350	1.635	1.319
	LARS	0.913	1.107	1.365	1.147
	FA-MIDAS
	SIS	1.108	1.279	1.440	1.275
	t-stat	1.264	1.269	1.386	1.307
	LASSO-MIDAS	1.585	1.532	1.556	1.558
	Grey cells indicate best performance for a given month (or the average of the three months)

exhibits better performances than other techniques. In the rest of the paper, FA-MIDAS models are based on this pre-selection technique with, as described in section 3.2, the optimal number of regressors 𝑘 𝑚 𝑖 ,𝑡,𝑗 * specific to each month 𝑚 𝑖 of the quarter (𝑖 = {1,2,3}), each date 𝑡 (between 2005Q2 and 2021Q1), and each model 𝑗 (𝑗 = {1,2,3}). In addition, for model 1 with both weekly and monthly factors, the optimal numbers of respectively monthly and weekly series is a couple (𝑘 𝑚 𝑖 ,𝑡,𝑗 𝑚, * , 𝑘 𝑚 𝑖 ,𝑡,𝑗 𝑤, * ) with 𝑘 𝑚 𝑖 ,𝑡,𝑗 𝑚, * possibly different from 𝑘 𝑚 𝑖 ,𝑡,𝑗 𝑤, * .

Table 3 .

 3 Performances (RMSE) relative to the AR model

			FA-MIDAS
					AR model
		Model 1	Model 2	Model 3
	In-sample
	1 st month	0.291	0.341	0.350	1.000
	2 nd month	0.215	0.299	0.352	1.000
	3 rd month	0.195	0.334	0.260	1.000
	Out-of-sample
	1 st month	0.521	0.467	0.483	1.000
	2 nd month	0.631	0.677	0.771	1.000
	3 rd month	0.601	0.631	0.633	1.000

Grey cells indicate best performance for a given month. Model 1 includes both a weekly and a monthly factor, model 2 includes a unique factor with both monthly and weekly data (averaged over the month), model 3 is based only on monthly data.

Table 5 .

 5 Performances (RMSE) relative to the AR model -by sub-periods

		FA-MIDAS
			AR model
	Model 1	Model 2	Model 3

  FollowingWelsh and Goyal (2008) or more recently[START_REF] Siliverstovs | Assessing nowcast accuracy of US GDP growth in real time: The role of booms and busts[END_REF][START_REF] Siliverstovs | New York FED Staff Nowcasts and Reality: What Can We Learn about the Future, the Present, and the Past?[END_REF], we complement this assessment with Cumulative Sum of Squared Forecast Errors (CSSFED). Unlike the analysis based on RMSE -which averages over several quarters, the CSSFED allows to follow the evolution of the relative forecasting performances of two models 𝑖 and 𝑗. A decreasing CSSFED indicates that model 𝑖 outperforms model 𝑗. When CSSFED is broadly stable, predictive performances of model 𝑖 and 𝑗 can be considered as similar. Formally if 𝑒̂𝑖 ,𝑡 is model 𝑖's forecast error, the CSSFED is defined as follows:

	1 st month	0.257	0.317	0.324	1.000
	2 nd month	0.146	0.279	0.325	1.000
	3 rd month	0.089	0.255	0.199	1.000
	Out-of-sample
	1 st month	0.518	0.461	0.477	1.000
	2 nd month	0.633	0.680	0.777	1.000
	3 rd month	0.599	0.630	0.631	1.000
	Non-crisis episodes
	In-sample
	1 st month	0.595	0.535	0.586	1.000
	2 nd month	0.649	0.512	0.576	1.000
	3 rd month	0.760	0.978	0.763	1.000
	Out-of-sample
	1 st month	0.595	0.635	0.648	1.000
	2 nd month	0.563	0.566	0.541	1.000
	3 rd month	0.845	0.805	0.867	1.000
	Grey cells indicate best performance for a given month. Model 1 includes both a weekly and a monthly factor,
	model 2 includes a unique factor with both monthly and weekly data (averaged over the month), model 3 is
	based only on monthly data.

Table A1 .

 A1 1. Out-of-sample RMSE across pre-selection techniques and months of the quarter (alternative with pre-selection not done in a real-time manner)

		1 st month	2 nd month	3 rd month	Average
	No pre-selection	0.973	1.350	1.635	1.319
	t-stat	0.834	0.432	0.856	0.644
	FA-MIDAS
	SIS	0.547	0.473	0.543	0.508
	LARS	0.493	0.382	0.971	0.677
	LASSO-MIDAS	1.585	1.532	1.556	1.558
	Grey cells indicate best performance for a given month (or for the average of the 3 months)

Table A1 .

 A1 2. Model Confidence Set[START_REF] Hansen | The model confidence set[END_REF] test results

			FA-MIDAS
					AR model
		Model 1	Model 2	Model 3
	1 st month	0.493	1	0.480	0.109
	2 nd month	1	0.317	0.329	0.199
	3 rd month	1	0.398	0.364	0.491
	Results report p-values of the statistic 𝑇 𝑚𝑎𝑥 of Hansen et al. (2011) based on a squared errors loss function.
	Model 1 includes both a weekly and a monthly factor, model 2 includes a monthly factor incorporating
	monthly and weekly data (averaged over the month), model 3 is based only on monthly data.

Table A2 .

 A2 1. Description of data

	Annex 3:
		Number of series	Seasonally adjusted 1	Sources
	Quarterly dependent variable
	GDP growth	-	Yes	ECB macroeconomic projections
	Monthly regressors (total = 718)
	EPU	26	No	Bloom et al. (2016) and others
	Housing (e.g. new buildings)	22	Yes	National statistics, Eurostat
	Car registrations	35	Yes	National statistics, ACEA 2
	Retail sales	33	Yes	National statistics, OECD
	Number of employees	23	Yes	National statistics
	Unemployment rates	30	Yes	National statistics, Eurostat
	Industrial production	34	Yes	National statistics, Eurostat, OECD
	Consumers Prices Inflation	37	Yes	National statistics, Thomson Reuters, OECD
	Producers Prices Inflation	35	Yes	National statistics, Eurostat, IMF, OECD
	Household confidence	33	Yes	National statistics, OECD, EU Commission
	World trade	4	Yes	Centraal Plan Bureau
	REER	37	No	BIS
	M2 money aggregate	36	Yes	National statistics
	PMI composite headline	20	Yes	IHS Markit, ISM, NBS
	PMI manuf. headline	40	Yes	IHS Markit, ISM, NBS
	PMI manuf. new export orders	35	Yes	IHS Markit, SIM, NBS
	PMI manuf. new orders	39	Yes	IHS Markit, ISM, NBS
	PMI manuf. output	39	Yes	IHS Markit, NBS
	PMI manuf. delivery time	40	Yes	IHS Markit, ISM, NBS
	PMI services headline	18	Yes	IHS Markit, NBS
	PMI services new export orders	16	Yes	IHS Markit, NBS
	PMI services new orders	6	Yes	IHS Markit, NBS
	PMI services output prices	17	Yes	IHS Markit
	PMI "whole" headline	13	Yes	IHS Markit
	PMI "whole" new export orders	11	Yes	IHS Markit
	PMI "whole" new orders	13	Yes	IHS Markit

Least-Angle Regression (LARS) algorithm of Efron et al. (2004)

  Suppose the objective is to regress a target variable 𝑦 on a dataset 𝑋 containing a large number 𝑁 of variables. The LARS algorithm starts with no variables selected.

The authors show that only PMIs and industrial production data -included in our monthly dataset -display a higher degree of correlation with world GDP growth than the indicators for the US economy.

We impose a maximum lag of 12 months so that the number of observations deleted at the beginning of the sample -due to the vertical realignment -is at most 12 (i.e. 4 in quarterly terms).

Specifically, the authors also test for the EM-algorithm of[START_REF] Stock | Macroeconomic Forecasting Using Diffusion Indexes[END_REF] and the Kalman smoother estimates of[START_REF] Doz | A quasi maximum likelihood approach for large approximate dynamic factor models[END_REF].

This transformation is not applied to series that are already stationary, notably interest rates (3-month interbank and 10-year sovereign), spreads and some US indicators (e.g. VIX, US National Financial Conditions Index of the Chicago Fed, US business condition index of[START_REF] Aruoba | Real-Time Measurement of Business Conditions[END_REF].

In this case, it should be noted that the ranking of potential regressors is made over the joint sample of monthly series and weekly series averaged over the month: in other words, this is a common ranking of both weekly and monthly variables.

Also, very few variables are kept with the LASSO-MIDAS (2 monthly and 2 weekly for the last iteration with estimation up to 2020Q4), which might explain the limited performances of such a model.

The MCS test estimates a set of superior models from an initial set of models where the "superiority" is defined by a user-specified loss function. It consists in a stepwise procedure in which the null hypothesis H0 of "all selected models perform identically" is repeatedly tested with respect to the expected loss function. Upon rejection, model is removed from the set and the procedure is the repeated. In addition to the issue of possibly nested models, one other advantage of MCS over pairwise tests -such as the Diebold-Mariano test -is that it allows for an asymptotic control of the family-wise error rate which might be problematic when conducting sequences of pairwise tests.

For the annual exercise, contrary to the quarterly exercise, the optimal number of variables is not selected for each week of the year. The monthly (weekly) factor is the principal component from the 60 (20) first series that LARS select.

We however do not apply the full procedure described in section 3.2 at the annual frequency, since the very limited number of annual data points (22) would prevent any distinction of a train and test sample for the in-

authors, and not those of the Banque de France, AMSE or European Central Bank. This work was supported by the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -A*MIDEX.

data that support the findings of this study are available from Datastream and ECB. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors with the permission of Datastream and ECB. The views expressed in this paper are those of the

publication lags in standard monthly indicators but does not yield significant benefits during other periods, even the Great Financial Crisis. Finally, the paper shows that higher accuracy can be reached when weekly data is introduced in a specific weekly factor rather than when it is averaged over the month and incorporated along monthly variables.

The paper also provides evidence that a pre-selection of variables enhances markedly the forecasting performances of a FA-MIDAS. Among the different techniques tested in this paper, the LARS [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF] gives more substantial gains than alternatives approaches based on univariate relations (SIS and t-stat).

It should be kept in mind that the above results are obtained in a purely data-driven exercise with minimal intervention from the forecaster: large datasets are assembled by the forecaster (around 1,000 variables in total) but the pre-selection of variables and the calibration of the models -notably of the choice of the optimal number of regressors to include -are performed automatically. In addition to being purely data-driven, the set-up is strictly out-of-sample with pre-selection of variables and calibration of the model done only in-sample. Higher and more significant accuracy gains can be reached when relaxing these constraints, for example by running the pre-selection over the full sample: this calls for caution as results of preselection under a "pseudo real-time" set-up can largely differ those in a "real-time" set-up.

We also build a real-time nowcasting model for the annual growth of world GDP which has the double advantage of: (i) timeliness as it provides a forecast every week while institutional forecasts are updated maximum four times a year; (ii) accuracy as it produces forecasts close to IMF's or OECD's projections but with a 1 to 3 month lead. This can provide an alternative "benchmark" for world GDP growth projection to economists. Interestingly during "crisis" episodes, not only such an alternative "benchmark" is needed -since the usual "benchmarks" provided by institutions are rapidly outdated given large and sudden swings in activity -but it is also when high-frequency data would provide the greater gains in predictive accuracy.

An avenue for future research could be the inclusion of further high-frequency data such as card transactions [START_REF] Cerdeiro | World Seaborne Trade in Real Time: A Proof of Concept for Building AIS-based Nowcasts from Scratch[END_REF], satellite data (Bricongne et al., 2021a), marine traffic [START_REF] Carvalho | Tracking the COVID-19 Crisis with High-Resolution Transaction Data[END_REF], or web-scrapped data [START_REF] Bricongne | Web Scraping Housing Prices in Realtime: the Covid-19 Crisis in the UK[END_REF]. Their limited timespan -e.g. Google mobility data only starts in January 2020 -or their high cost (e.g. using marine traffic requires several thousand euros) is however a strong limitation. Finally, our approach can be extended to nowcast GDP for a specific country or group of countries (e.g. euro area). where the 𝑚𝑖𝑛 + indicates that the minimum is taken over only positive components of each choice of 𝑗.

The interpretation is that 𝜇̂𝔸 𝑘 is moved in the direction of 𝑢 𝑘 so that 𝜇 𝔸 𝑘 (𝛾) = 𝜇̂𝔸 𝑘 + 𝛾 • 𝑢 𝑘 . Then Equating these two last equations provides the criterion for 𝛾 ̂ set in equation (A5.1). 𝛾 ̂ can be interpreted as the smallest positive value of 𝛾 so that a new index 𝑙 joins the active set (i.e.

whose absolute correlation with the residual will equal the absolute correlation of the variables in the active set). 𝑙 is the minimizing index for (A5.1) and the new active set becomes 𝔸 𝑘+1 = 𝔸 𝑘 ∪ {𝑙} and the new maximum correlation becomes 𝐶 ̂-𝛾 ̂• 𝐴 𝔸 𝑘

A main difference with the traditional forward stepwise algorithm is that 𝛾 ̂ in the updating rule is endogenously chosen in the LARS algorithm, so that it proceeds equiangularly between the variables in the active set ("least angle direction") until the next variable to be added in the active set is found. By contrast, in the forward stepwise algorithm, the updating rule is 𝜇̂𝑘 +1 = 𝜇̂𝑘 + 𝛾 ̂𝑠𝑖𝑔𝑛(𝑐̂𝑗)𝑥 𝑗 with 𝛾 ̂= |𝑐̂𝑗|. In other terms, it updates the least squares fit to include active regressors. LARS only enters predictors only by "as much" as needed before another regressor is achieve as much correlation with the residual. This way, regressors mildly correlated with other variables already in the active set have a chance to enter -correcting the excessive parsimony of forward stepwise selection [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF][START_REF] Hastie | Les données « haute fréquence » sont surtout utiles à la prévision économique en période de crise brutale[END_REF][START_REF] Bulligan | Forecasting economic activity with targeted predictors[END_REF].

Finally, [START_REF] Efron | Least angle regression[END_REF] show that the LARS algorithm encompasses other shrinkage methods, including LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] and the Elastic Net [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]. The LASSO modification of the LARS algorithm is that if a non-zero coefficient for a regressor in the active set hits zero, then this regressor is dropped from the active set and the current joint least squares direction is recomputed. [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF] also show that, in order to reformulate the Elastic Net as a LASSO problem, it is sufficient to apply a variable transformation, which can therefore be obtained through the LARS algorithm.


Nolwen Huet
email: nolwen.huet@math.univ-toulouse.fr.2000

Isoperimetry for spherically symmetric log-concave probability measures

Keywords: Mathematics Subject Classification: 26D10, 60E15, 28A75

We prove an isoperimetric inequality for probability measures µ on R n with density proportional to exp(-φ(λ|x|)), where |x| is the euclidean norm on R n and φ is a non-decreasing convex function. It applies in particular when φ(x) = x α with α ≥ 1. Under mild assumptions on φ, the inequality is dimension-free if λ is chosen such that the covariance of µ is the identity.

Introduction

In his paper [START_REF] Sergey | Spectral gap and concentration for some spherically symmetric probability measures[END_REF], Bobkov studies the spectral gap for spherically symmetric probability measures µ on R n with density dµ(x) dx = ρ(|x|),

where ρ is log-concave. His main result can be stated as follows.

Theorem 1 (Bobkov [START_REF] Sergey | Spectral gap and concentration for some spherically symmetric probability measures[END_REF]). The best constant P µ in the Poincaré inequality

Var µ (f ) ≤ P µ |∇f | 2 dµ, ∀f smooth satisfies E µ (|X| 2 ) n ≤ P µ ≤ 12 E µ (|X| 2 ) n .
In particular, if µ is isotropic, we get

1 ≤ P µ ≤ 12,
which means a spectral gap not depending on n.

Here "µ is isotropic" means that the covariance of µ is the identity. However, we already know from the spherically invariance of µ that the covariance is proportional to the identity. So in our case, the isotropy of µ reduces merely to E µ (|X| 2 ) = n.

If we assume furthermore that µ itself is log-concave (see [START_REF] Borell | Convex measures on locally convex spaces[END_REF] for precisions about log-concave measures), that is to say that ρ is non-decreasing, then one can show an isoperimetric inequality for µ, thanks to a result of Ledoux [START_REF] Ledoux | Spectral gap, logarithmic Sobolev constant, and geometric bounds[END_REF] (generalized in [START_REF] Milman | On the role of convexity in isoperimetry, spectral-gap and concentration[END_REF] by E. Milman) bounding the Cheeger constant from below by the spectral gap.

Theorem 2. There exists a universal constant c > 0 such that, for any n ∈ N, all log-concave measures µ on R n spherically symmetric and isotropic satisfy the following isoperimetric inequality:

Is µ (a) ≥ c a ∧ (1 -a). (1)
Here Is µ denote the isoperimetric function of µ and a ∧ b = min(a, b). We need some notation to define Is µ properly. Let A be a Borel set in R n . We define its ε-neighborhood by A ε = {x ∈ X; d(x, A) ≤ ε}.

The boundary measure of A is

µ + (∂A) = lim inf ε→0 + µ(A ε ) -µ(A) ε .
Now the isoperimetric function of µ is the largest function Is µ on [0, 1] such that for all Borel sets A, µ + (∂A) ≥ Is µ µ(A) .

The result of Bobkov answer the KLS-conjecture ( [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF]) in the particular case of spherically symmetric measures. This conjecture asserts that (1) is true for all log-concave and isotropic measures µ, with a universal constant c. Our aim in this note is to sharpen Theorem 2 when ρ is "better" than log-concave. For instance, the Gaussian measure γ n corresponding to ρ(t) = (2π) -n 2 exp -t 2 2 , is known to satisfy the log-Sobolev inequality and the following isoperimetric inequality:

Is γn (a) ≥ c a ∧ (1 -a) log 1 a ∧ (1 -a)
with constants not depending on n either. We can ask what happens for regimes between exponential and Gaussian or even beyond the Gaussian case. This idea has already be developed in [START_REF] La | Between Sobolev and Poincaré[END_REF][START_REF] Barthe | Levels of concentration between exponential and Gaussian[END_REF][START_REF] Barthe | Sobolev inequalities for probability measures on the real line[END_REF][START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF][START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF] for product measures.

Let φ : R + → R + be a convex non-decreasing function of class C 2 such that φ(0) = 0. Then we consider the probability measure on R n µ n,φ (dx) = e -φ(|x|) dx Z n,φ

and its associated radial measure on [0, +∞)

ν n,φ (dr) = |S n-1 | r n-1 e -φ(r) dr Z n,φ .

In the particular case φ(x) = φ α (x) = x α with α ≥ 1, we note µ n,α = µ n,φα and ν n,α = ν n,φα . We denote by σ n-1 the uniform probability measure on the unit sphere S n-1 of R n . If X is a random variable of law µ n,φ , then |X| has the distribution ν n,φ . Conversely, if r and θ are independent random variables whose distributions are respectively ν n,φ and σ n-1 , then X = rθ has the distribution µ n,φ . In view of this representation, we will derive inequalities for µ n,φ from inequalities for ν n,φ and σ n-1 .

In the subgaussian case, following the results for µ ⊗n 1,φ from [START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF], we expect the isoperimetric function of µ n,α to be equal to a constant depending on n, times a symmetric function defined for a ∈ [0, 1 2 ] by

L α (a) = a log 1 a 1-1 α ,
and more generally for µ n,φ ,

L φ (a) = a log 1 a φ -1 log 1 a .
Otherwise, since we are aiming at results which do not depend on n and because of the Central-Limit Theorem ( [START_REF] Klartag | A central limit theorem for convex sets[END_REF]), we cannot expect better isoperimetric profile than the one of the Gaussian measure, proportional to

L 2 (a) = a log 1 a .
The point is to know the exact dependence in n of the constant in front of the term in a, and in particular to know whether we recover universal constants in the isotropic case. The main theorems of this paper are stated next. Theorem 3. There exists a universal constant C > 0 such that, for every α ≥ 1, for every n ∈ N * , and every a ∈ [0, 1], it holds

Is µn,α (a) ≥ Cn 1 2 -1 α a ∧ (1 -a) log 1 a ∧ (1 -a) 1-1 α∧2 .
It can be seen as a corollary of the following more general theorem. Theorem 4. Let φ : R + → R + be a convex non-decreasing function of class C 2 such that φ(0) = 0. If moreover we assume that i) √ φ is concave, then for every n ∈ N * , and every a ∈ [0, 1], it holds

Is µ n,φ (a) ≥ C √ n φ -1 (n) φ -1 (1) a ∧ (1 -a) log 1 a∧(1-a) φ -1 log 1 a∧(1-a)
, ii) x → φ(x)/x is increasing, then for every n ∈ N * , and every a ∈ [0, 1], it holds

Is µ n,φ (a) ≥ C √ n φ -1 (n) a ∧ (1 -a) log 1 a ∧ (1 -a) ,
where C > 0 is a universal constant.

Further hypotheses ensure the optimality of these bounds among products of functions of n and functions of a, and lead to dimension-free inequalities when normalizing measures to obtain isotropic ones. See Theorem 19 for more precise statement.

Note that a straightforward application of Bobkov's inequality for log-concave measures (Theorem 7) leads to the good profile but with the wrong dimension dependent constant in front of the isoperimetric inequality. For instance, Lemma 4 of [START_REF] Barthe | Levels of concentration between exponential and Gaussian[END_REF] and the computation of exponential moments imply the Theorem 3 with n -1 α instead of n 1 2 -1 α . We introduce in Section 2 the different hypotheses made on φ. Then we establish in Section 3 the isoperimetric inequality for the radial measure. The proof relies on an inequality for log-concave measures due to Bobkov and some estimates of probabilities of balls. Section 4 is devoted to the argument of tensorization which yields the isoperimetric inequality from the ones for the radial measure and the uniform probability measure on the sphere. A cut-off argument is needed to get rid of the case of large radius. This tensorization relies on a functional version of the inequality whose proof is postponed to Section 5. We combine the previous results in Section 6 to prove Theorem 4. Eventually, we discuss the isotropic case and the optimality of the inequalities in Section 7.

Hypotheses on φ

We make different assumptions on φ, corresponding to the different cases in Theorems 4, 6, and 19.

Hypotheses (H0) φ : R + → R + is a non-decreasing convex function of class C 2 such that φ(0) = 0.

Hypotheses (H1) φ satisfies (H0) and x → φ(x)/x is non-increasing.

Hypotheses (H1') φ satisfies (H0) and √ φ is concave.

Hypotheses (H2) φ satisfies (H0) and x → φ(x)/x is non-decreasing.

Hypotheses (H2') φ satisfies (H2) and there exists α ≥ 2 such that x → φ(x)/x α is non-increasing.

The next lemma sums up some properties of φ under our assumptions.

Lemma 5.

• Under (H0), it holds:

i. For all t ≥ 1 and x ≥ 0, φ(tx) ≥ tφ(x).

ii. For all t ≥ 1 and y ≥ 0, φ -1 (ty) ≤ tφ -1 (y).

iii. For all x ≥ 0, xφ ′ (x) ≥ φ(x).

• Under (H1), it holds:

i. For all t ≥ 1 and x ≥ 0, tφ(x) ≤ φ(tx) ≤ t 2 φ(x).

ii. For all t ≥ 1 and y ≥ 0, √ tφ -1 (y) ≤ φ -1 (ty) ≤ tφ -1 (y).

iii. For all x ≥ 0,

φ(x) ≤ xφ ′ (x) ≤ 2φ(x).
iv. For all t ≥ 1 and x ≥ 0, φ ′ (tx) ≤ 2tφ ′ (x).

• Under (H2), it holds:

i. For all t ≥ 1 and x ≥ 0, φ(tx) ≥ t 2 φ(x).

ii. For all t ≥ 1 and y ≥ 0,

φ -1 (ty) ≤ √ tφ -1 (y).
iii. For all x ≥ 0, xφ ′ (x) ≥ 2φ(x).

3 Isoperimetry for the radial measure ν n,φ

In order to deal with µ n,φ , a first step is to establish a similar isoperimetric inequality for its radial marginal.

Theorem 6. There exists a universal constant C > 0 such that for every n ∈ N * , every a ∈ [0, 1 2 ], and every function φ, i) if φ satisfies (H1) then

Is ν n,φ (a) ≥ C √ n φ -1 (n) φ -1 (1) a log 1 a φ -1 log 1 a .
ii) if φ satisfies (H2) then

Is ν n,φ (a) ≥ C √ n φ -1 (n) a log 1 a .
As ν n,φ is a log-concave measure, we can apply the isoperimetric inequality shown by Bobkov in [START_REF] Sergey | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF].

Theorem 7 (Bobkov [START_REF] Sergey | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF]). If µ is a log-concave measure on R n , then for all Borel sets A, for all r > 0, and for all x 0 ∈ R n ,

2rµ + (∂A) ≥ µ(A) log 1 µ(A) + µ(A ∁ ) log 1 µ(A ∁ ) + log µ{|x -x 0 | ≤ r}, (2)
where A ∁ denotes the complement of A.

One chooses r as small as possible but with µ{|x-x 0 | ≤ r} large enough, such that the sum of the two last terms is non-negative. This requires explicit estimates of probabilities of balls. In our case, we will use two different estimates valid for two ranges of r, leading to inequalities for two ranges of a.

The first lemma is due to Klartag [START_REF] Klartag | A central limit theorem for convex sets[END_REF]. The balls are centered at the maximum of density in order to capture a large fraction of the mass.

Lemma 8 (Klartag [START_REF] Klartag | A central limit theorem for convex sets[END_REF]). Let ν(dr) = r n-1 ρ(r) dr be a probability measure on R + with ρ a log-concave function of class C 2 . Let r 0 be the point where the density reaches its maximum. Then,

∀δ ∈ [0, 1], ν{|r -r 0 | ≥ δr 0 } ≤ C 1 e -c 1 nδ 2
where C 1 > 1 and 0 < c 1 < 1 are universal constants.

Bobkov's inequality combined with the latter lemma leads to the following proposition.

Proposition 9. There exist two universal constants c > 0 and C > 0 such that for all functions φ satisfying (H0) and all n large enough to ensure e -cn < 1 2 , it holds

∀a ∈ e -cn , 1 2 , Is ν n,φ (a) ≥ C √ n φ -1 (n) a log 1 a .
Proof. Let C 1 and c 1 be the constants given by Lemma 8. Let K > 0 and set

δ = K log 1 a c 1 n . Choose a ∈ exp -c 1 n K , 1 2 and K > log C 1 log 2 . It follows that δ ≤ 1 and 1 -C 1 a K > 0. Then Lemma 8 implies (1 -a) log 1 1 -a + log ν n,φ {|r -r 0 | ≤ δr 0 } ≥ (1 -a) log 1 1 -a + log(1 -C 1 a K ). ( 3
)
The right-hand term of (3) cancels at 0 and is concave in a on [0, 1 2 ] if K ≥ 1. Take K large enough such that it is also non-negative at 1 2 . Thus, by concavity, it is non-negative on [0, 1 2 ]. So Bobkov's formula (2) yields

Is ν n,φ (a) ≥ 1 2 c 1 n Kr 2 0 a log 1 a .
It remains to estimate the point r 0 where the density of ν n,φ reaches its maximum. The differentiation of the density leads to

r 0 φ ′ (r 0 ) = n -1. By Lemma 5, φ(r 0 ) ≤ n -1. Thus r 0 ≤ φ -1 (n).
Let us remark that under (H1), for all a

≤ 1 2 , log 1 a ≥ log 2 φ -1 (1) log 1 a φ -1 log 1 a .
So the latter proposition implies a stronger inequality that the one required under (H1), but only for large enough sets.

To cope with smaller sets, we need another estimate for balls with greater radius.

Lemma 10. Let φ be a function satisfying (H0) and n ∈ N * . Then for all r ≥ φ -1 (2n),

ν n,φ {(r, +∞)} ≤ F n,φ (r) = er φ -1 (n) n e -φ(r) ≤ 1.
Note that this tail bound gives estimates of probability of balls centered at 0 for ν n,φ , but also for µ n,φ since

ν n,φ {(r, +∞)} = µ n,φ {|x| ≥ r}.
This lemma can thereby be used to derive isoperimetric inequalities from Bobkov's formula for both measures.

Proof. The main tool is integration by part.

+∞ r t n-1 e -φ(t) dt = +∞ r t n-1 φ ′ (t) φ ′ (t)e -φ(t) dt = r n-1 φ ′ (r) e -φ(r) + +∞ r n -1 tφ ′ (t) - φ ′′ (t) φ ′ (t) 2 t n-1 e -φ(t) dt ≤ r n-1 φ ′ (r) e -φ(r) + +∞ r n -1 tφ ′ (t) t n-1 e -φ(t) dt. If t ≥ r ≥ φ -1 (2n) ≥ φ -1 2(n -1) , then tφ ′ (t) ≥ 2(n -1)
. So the last integral in the above inequality is less than r) .

1 2 +∞ r t n-1 e -φ(t) dt. Moreover rφ ′ (r) ≥ 2n. Hence +∞ r t n-1 e -φ(t) dt ≤ 2 r n-1 φ ′ (r) e -φ(r) ≤ r n n e -φ(
It remains to deal with the normalization constant which makes ν n,φ a probability measure:

+∞ 0 nt n-1 e -φ(t) dt ≥ φ -1 (n) 0 nt n-1 e -φ(t) dt ≥ e -n φ -1 (n) 0 nt n-1 dt = φ -1 (n) e n .
Putting all together, we get the desired bound on the tail of ν n,φ :

ν n,φ {(r, +∞)} = +∞ r t n-1 e -φ(t) dt +∞ 0 t n-1 e -φ(t) dt ≤ er φ -1 (n) n e -φ(r) .
Then one can show that the bound is non-increasing for r ≥ φ -1 (n) and is equal to 1 for r = φ -1 (n).

Then, we show an isoperimetric inequality simultaneously for µ n,φ and ν n,φ in the range of small sets.

Proposition 11. For every c > 0, there exists C > 0 such that for all functions φ satisfying the hypotheses of (H0),

∀a ∈ 0, e -cn ∧ 1 2 , Is µ (a) ≥ C a log 1 a φ -1 log 1 a ,
where µ stands for µ n,φ or ν n,φ .

Note that this is worth showing the result for every c > 0. Indeed, to prove Theorem 6, we combine this result with Proposition 9 where this constant is already fixed but unknown.

Proof. As before, we start from (2) and set r(a

) = φ -1 K log 1 a , where K is a constant large enough to ensure Kc ≥ 2, (4)
K -1 ≥ 1 c , (5)
eKc exp (-(K -1)c) ≤ 1 2 . ( 6
)
By Lemma 5, r ≤ Kφ -1 log 1 a , as K > 1. So the result is deduced from Bobkov's inequality (2) provided that

(1 -a) log 1 1 -a + log µ n,α {|x| ≤ r} ≥ 0. (7)
Now, by concavity,

∀x ∈ 0, 1 2 , (1 -x) log 1 1 -x ≥ log 2 x, and log(1 -x) ≥ -2 log 2 x.
So, for all a ∈ [0, 1 2 ],

(

1 -a) log 1 1 -a + log µ n,α {|x| ≤ r} ≥ log 2 a -2F n,φ (r) ≥ 0, as soon as r ≥ φ -1 (2n) and F n,φ (r) ≤ a 2 .
Assume that a ≤ exp(-cn) ∧ 1 2 . Then r ≥ φ -1 (Kcn) ≥ φ -1 (2n) by ( 4). Let us define the function G by

G(a) = F n,φ r(a) a .
Then [START_REF] Sergey | Extremal properties of half-spaces for log-concave distributions[END_REF] holds as soon as G(a) ≤ 1 2 . To handle this, it is easier to look on G as a function of r. We know that a = exp -φ(r) K . So

G(a) = er φ -1 (n) n exp -φ(r) 1 - 1 K .
This function is non-increasing in r when

rφ ′ (r) ≥ n 1 -1 K . This is the case if r ≥ φ -1 Kn K-1 . Moreover φ -1 (Kcn) ≥ φ -1 Kn K-1 by (5). Thus, when a ≤ exp(-cn), G(a) ≤ G exp(-cn) ≤ eKc exp -(K -1)c n ≤ 1 2 n ≤ 1 2 .
Under (H1), this result is again stronger than the one required since then

1 ≥ √ n φ -1 (n) φ -1 (1).
We could also derive the required inequality under (H2), but with √ n/φ -1 (cn) instead of √ n/φ -1 (n). So we prefer to prove it directly, following the above proof.

Proposition 12. For every c > 0, there exists C > 0 such that for all functions φ satisfying (H2),

∀a ∈ 0, e -cn ∧ 1 2 , Is µ (a) ≥ C √ n φ -1 (n) a log 1 a ,
where µ stands for µ n,φ or ν n,φ .

Proof. We set

r(a) = K φ -1 (n) 2 n log 1 a , ( 8
)
where K is a constant large enough to verify

Kc ≥ 2, K -1 ≥ 1 2c , e √ Kc exp (-(K -1)c) ≤ 1 2 . Assume that a ≤ exp(-cn) ∧ 1 2 , then r ≥ √ Kc φ -1 (n) ≥ Kc 2 φ -1 (2n) ≥ φ -1 (2n).
So we can use the estimate from Lemma 10. Consider as before

G(a) = F n,φ r(a) a .
Then, as explained in the proof of Proposition 11, Bobkov's formula (2) yields the required isoperimetric inequality as soon as

G(a) ≤ 1 2 .
From ( 8), we deduce

a = exp - nr 2 K φ -1 (n) 2 .
So if we express G as a function of r,

G(a) = er φ -1 (n) n exp -φ(r) + nr 2 K φ -1 (n) 2 .
The derivative ∂ r G

1 n is of the same sign as

1 + 2r 2 K φ -1 (n) 2 - rφ ′ (r) n .
Under hypothesis (H2), rφ ′ (r) ≥ 2φ(r) ≥ 2n (r/φ -1 (n)) 2 as soon as r ≥ φ -1 (n). Thus, when r ≥ √ Kc φ -1 (n),

1 + 2r 2 K φ -1 (n) 2 - rφ ′ (r) n ≤ 1 + 2r 2 φ -1 (n) 2 1 K -1 ≤ 1 + 2Kc 1 K -1 ≤ 0, since 1 K -1 < 0. So G is non-increasing in r when r ≥ √ Kc φ -1 (n)
, and for all a ≤ exp(-cn), it holds

G(a) ≤ e √ Kc n exp cn -φ √ Kc φ -1 (n) ≤ e √ Kc exp (-(K -1)c) n ≤ 1 2 .
We have again used Hypothesis (H2) which ensures φ √ Kc φ -1 (n) ≥ Kcn.

Combining Proposition 9 for big sets, and Proposition 11 or Proposition 12 for small sets yields Theorem 6.

Tensorization and cut-off argument

We derive the isoperimetric inequality for µ n,α by tensorization from the ones for the radial measure and the uniform probability measure on the sphere, following the idea of the proof by Bobkov of Theorem 1. For that purpose, we need a functional version of our isoperimetric inequality. In [START_REF] Sergey | Isoperimetric problem for uniform enlargement[END_REF] and [START_REF] Barthe | Log-concave and spherical models in isoperimetry[END_REF], the authors give conditions so that isoperimetric inequalities translate into functional inequalities. Actually this works in our setting as explained in Section 5.

Let κ > 0. Let J : [0, 1] → R + be a continuous convex function symmetric with respect to 1/2, with J(0)=J(1)=0, and such that the following property holds : for any measure µ on R d and constant C ≥ 0, if

Is µ ≥ CJ, then for all smooth functions f : R d → [0, 1], κJ f dµ ≤ J(f ) dµ + 1 C |∇f | dµ.
Remark. Ideally, one would expect κ = 1. For instance the latter inequality implies the former one and is tight for constant functions only in the case κ = 1. However this does not matter here as we tensorize only once.

For such profiles J, we can show the following proposition.

Proposition 13. Let µ be a measure on R n with radial measure ν. Assume that there exists positive constants C ν and C σ n-1 such that

Is ν ≥ C ν J and Is σ n-1 ≥ C σ n-1 J.
There exist κ 1 , κ 2 > 0 depending only on κ such that, for every n ∈ N * , for every r 2 > r 1 > 0 and a such that

r 2 -r 1 ≥ 1 C ν J( 1 2 ) , (9)
κ 1 ν{[r 1 , +∞)} ≤ a ≤ 1 2 , ( 10
)
it holds Is µ (a) ≥ κ 2 min C ν , C σ n-1 r 2 J(a).
Proof. Let f : R n → [0, 1] be a smooth function. We recall some facts on radial and spherical differentiation. If we define g on R + × S n-1 by g(r, θ) = f (rθ), then the partial derivatives of g can be computed as follows:

∂ r g = ∇f, θ , ∇ θ g = r Π θ ⊥ (∇f ),
where Π θ ⊥ is the orthogonal projection on θ ⊥ . Hence,

∇f = ∂ r g θ + 1 r ∇ θ g, |∇f | 2 = |∂ r g| 2 + 1 r 2 |∇ θ g| 2 .
First, we apply the functional inequality for σ n-1 to the function F defined on S n-1 by

F (θ) = f (rθ) dν(r). As F dσ n-1 = f dµ, this yields κJ f dµ ≤ J(F ) dσ n-1 + 1 C σ n-1 |∇ S n-1 F | dσ n-1 .
On one hand,

∇ S n-1 F (θ) = r Π θ ⊥ (∇f )(rθ) dν(r).
On the other hand, we can use the inequality for ν to bound J(F ). Indeed, for all

θ ∈ S n-1 , κJ F (θ) ≤ J f (rθ) dν(r) + 1 C ν |∂ r f (rθ)| dν(r).
Putting all together,

κ 2 J f dµ ≤ J(f ) dµ + 1 C ν |∂ r f | dµ + κ C σ n-1 |x| |Π θ ⊥ (∇f )| dµ(x). ( 11
)
We would like to get |x| out of the last integral. As it is not bounded, we use a cut-off argument similar to the one in Sodin's article [START_REF] Sodin | An isoperimetric inequality on the ℓ p balls[END_REF], while simpler in our case. Heuristically, we use the fact that on "a set of large measure", |x| is almost constant, close to its expectation for instance. Let us introduce a cut-off function h(rθ) = h 1 (r) with

h 1 =      1 on [0, r 1 ) r 2 -r r 2 -r 1 on [r 1 , r 2 ] 0 on (r 2 , +∞)
with 0 < r 1 < r 2 to be chosen later (typically of the same order as E µ |X|). It holds

∇(f h) = h∇f + f ∇h, thus |∂ r (f h)| ≤ |∂ r f | + ||f || ∞ |∂ r h|, |Π θ ⊥ ∇(f h) | ≤ h |Π θ ⊥ (∇f )|. As h = 0 if |x| > r 2 , |x| Π θ ⊥ ∇(f h) dµ(x) ≤ r 2 |Π θ ⊥ (∇f )| dµ(x).
Besides, we can bound the derivative of h so that

|∂ r h| dµ ≤ ν [r 1 , r 2 ] r 2 -r 1 .
Finally, Inequality [START_REF] Borell | Convex measures on locally convex spaces[END_REF] applied to f h yields

κ 2 J f h dµ -J(f h) dµ - ||f || ∞ ν [r 1 , r 2 ] C ν (r 2 -r 1 ) ≤ max 1 C ν , κr 2 C σ n-1 |∂ r f | + |Π θ ⊥ (∇f )| dµ ≤ √ 2 max 1 C ν , κr 2 C σ n-1 |∇f | dµ. (12)
Hence we have almost the functional inequality for f and µ with an additional term that we expect to be negligible. It is easier to look at functions approximating characteristic functions to go back from f h to f in the left hand term. Let A ⊂ R n be a closed set of measure a ≤ 1 2 . Let K > 0 and t ∈ (0, 1) constants to be chosen later. Assume the following constraints on r 1 , r 2 , and a:

C ν (r 2 -r 1 ) ≥ K, ν{[r 1 , +∞)} ≤ ta.

Then it holds

µ{1 A h = 1} ≥ µ A \ {h < 1} ≥ (1 -t)a, µ{1 A h > 0} ≤ a ≤ 1 2 .
As J is non-decreasing on (0, 1 2 ), concave, and J(0) = 0,

J 1 A h dµ ≥ J (1 -t)a ≥ (1 -t)J(a).
Besides J cancels at 0 and 1, and reaches its maximum at 1 2 , so

J(1 A h) dµ ≤ J( 1 2 ) µ{0 < 1 A h < 1} ≤ J( 1 2 ) µ{1 A h > 0} -µ{1 A h = 1} ≤ J( 1
2 ) ta. As for the third term of [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF], it is bounded by

ν [r 1 , r 2 ] C ν (r 2 -r 1 ) ≤ ta K .
For ε > 0, we approximate 1 A by a smooth function f ε : R n → [0, 1] with f ε = 1 on A and f ε = 0 outside A ε . Then we apply [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] to f ε and let ε to 0, taking advantage of the continuity of J:

√ 2 max 1 C ν , κr 2 C σ n-1 µ + (∂A) ≥ κ 2 (1 -t)J(a) -J( 1 2 ) + 1 K ta.
Now by concavity, J(a) ≥ 2J( 12 )a on 0,

1 2 . Hence √ 2 max 1 C ν , κr 2 C σ n-1 µ + (∂A) ≥ κ 2 (1 -t) - J( 1 2 ) + 1 K 2J( 1 2 ) t J(a) = κ 2 -t κ 2 + 1 2 + 1 2KJ( 1 2 )
J(a).

Taking for instance K = J( 12 ) -1 and t = κ 2 /(2(κ 2 + 1)) yields a non-trivial result.

Note that looking at closed sets was not a real restriction. Indeed, if lim inf ε→0 + µ(A ε )µ(A) > 0 then µ + (∂A) = +∞.

Getting functional inequalities

To apply Proposition 13 to our case, we need to know how to pass from an isoperimetric inequality to a functional inequality. Actually we can approximate L φ by an other profile satisfying the hypotheses made in Section 4, assuming furthermore that √ φ is concave. This new profile appears to be the isoperimetric function Is µ 1,φ of µ 1,φ , denoted by I φ henceforth.

Lemma 14. There exist universal constants d 1 > 0 and d 2 > 0 such that for all φ satisfying (H1'),

d 1 I φ ≤ L φ ≤ d 2 I φ .
The second inequality is a consequence of Proposition 2.3 from [START_REF] Milman | An isoperimetric inequality for uniformly logconcave measures and uniformly convex bodies[END_REF] by Milman and Sodin, up to the uniform estimation of the normalizing constant of µ 1φ . However we give a self-contained proof of Lemma 14 at the end of this section for completeness. In the next lemma, I φ is shown to satisfy the required properties.

Lemma 15. Let φ satisfying (H1').

i) The function I φ is continuous and concave on [0, 1], symmetric with respect to 1/2, and I φ (0) = I φ (1) = 0.

ii) Let µ be a measure on R d and C ≥ 0. If

Is µ ≥ CI φ ,
then for all smooth functions f : R d → [0, 1],

κI φ f dµ ≤ I φ (f ) dµ + 1 C |∇f | dµ,
where κ > 0 is a universal constant.

Proof. Let us first remark that µ 1,φ is an even log-concave probability measure on the real line. Hence half-lines solve the isoperimetric problem and we can express explicitly I φ (see e.g. [START_REF] Sergey | Extremal properties of half-spaces for log-concave distributions[END_REF]). Let f φ : x → e -φ(|x|)

Z φ be the density of µ 1,φ , F φ (x) = µ 1,φ (-∞, x) its cumulative distribution function, and G φ (x) = µ 1,φ (x, +∞) . Then

I φ = f φ • F -1 φ = f φ • G -1
φ and the properties stated in i) are clearly satisfied. Besides the transfer principle emphasized by Barthe in [START_REF] Barthe | Levels of concentration between exponential and Gaussian[END_REF] holds : if Is µ ≥ cI φ then µ satisfies essentially the same functional inequalities as µ 1,φ . As a consequence, it remains to establish that for all smooth functions f : R → [0, 1],

κI φ f dµ 1,φ ≤ I φ (f ) dµ 1,φ + |f ′ | dµ 1,φ .
Now, applying the 2-dimensional isoperimetric inequality to the set (x, y) ∈ R 2 ; y ≤ F -1 φ (f (x)) , one can show (see e.g. [START_REF] Barthe | Log-concave and spherical models in isoperimetry[END_REF]) that

Is µ 1,φ ⊗2 f dµ 1,φ ≤ I φ (f ) dµ 1,φ + |f ′ | dµ 1,φ .
So, ii) is shown if there exists a universal κ > 0 such that

Is µ 1,φ ⊗2 ≥ κI φ .
Actually, a stronger dimension-free inequality holds and is stated in the next lemma.

Lemma 16. There exists κ > 0 such that for all φ satisfying (H1') and all n,

Is µ 1,φ ⊗n ≥ κI φ .
Barthe, Roberto, and Cattiaux prove it in [START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF] without verifying the universality of κ. However one can check that their constant can be uniformly controlled for every φ satisfying (H1'), by using the same estimates for G φ and Z φ as in the proof of Proposition Lemma 14. Indeed, a Beckner inequality is shown to hold with a constant uniform in φ, thanks to their explicit bound. It tensorizes and implies a super-Poincaré with a constant uniform in n and φ, which translates into the isoperimetric inequality of Lemma 16.

One can also check the simple criterion given by E. Milman in [START_REF] Milman | On the role of convexity in functional and isoperimetric inequalities[END_REF] for a tensorization result. As the function defined by

t → L φ (t) L 2 (t) = log 1 t φ -1 log 1 t
is non-decreasing under (H1) -all the more under (H1') -then by Lemma 14 there exists a universal constant D > 0 such that

∀0 < t ≤ s ≤ 1 2 , I φ (t) L 2 (t) ≤ D I φ (s) L 2 (s) .
This also implies Lemma 16. So, up to the proof of Lemma 14, we are done.

Proof of Lemma 14. We can restrict ourselves to the case φ(1) = 1. Indeed if we set φ λ (x) = φ(λx), one can show L φ λ = λL φ and I φ λ = λI φ . This hypothesis ensures that 1 ≤ φ ′ (1) ≤ 2 and also that

t 2 ≤ φ(t) ≤ t on [0, 1] and t ≤ φ(t) ≤ t 2 on [1, +∞).
Let r ≥ 0. By integration by part,

+∞ r e -φ = e -φ(r) φ ′ (r) - +∞ r φ ′′ (φ ′ ) 2 e -φ .
By the properties of φ and especially as (

√ φ) ′′ ≤ 0, 0 ≤ +∞ r φ ′′ (φ ′ ) 2 e -φ ≤ +∞ r e -φ 2φ = e -φ(r) 2φ(r)φ ′ (r) - +∞ r (φ ′ ) 2 + φφ ′′ 2(φφ ′ ) 2 e -φ ≤ e -φ(r) 2φ(r)φ ′ (r)
.

Hence e -φ(r) φ ′ (r) 1 - 1 2φ(r) ≤ +∞ r e -φ ≤ e -φ(r) φ ′ (r) . In particular, if r ≥ 1, e -φ(r) 2φ ′ (r) ≤ +∞ r e -φ ≤ e -φ(r) φ ′ (r) . (13)
Now let us estimate the normalizing constant for µ 1,φ , denoted by Z φ .

Z φ = 2 +∞ 0 e -φ = 2 1 0 e -φ + +∞ 1 e -φ ≤ 2(1 + e -1 )
.

Moreover 1 0 e -φ ≥ 1 0 e -x dx = 1 -e -1 , so Z φ ≥ 2(1 -e -1 ) > 1.
By symmetry, we consider the case a ∈ 0, 1 2 . We set a = G φ (r) =

+∞ r e -φ Z φ . It follows that r ≥ 0. Then, to prove the lemma, we need only to compare

e -φ(r) Z φ with L φ G φ (r) = G φ (r) log 1 G φ (r) φ -1 log 1 G φ (r)
.

Recall that φ(t) ≤ tφ ′ (t) ≤ 2φ(t) under (H1) so that

1 2 φ ′ • φ -1 (x) ≤ x φ -1 (x) ≤ φ ′ • φ -1 (x)
and

G φ (r) 2 φ ′ • φ -1 log 1 G φ (r) ≤ L φ G φ (r) ≤ G φ (r) φ ′ • φ -1 log 1 G φ (r)
.

Assume first that r ≥ 1 so that ( 13) holds. On one hand,

L φ G φ (r) ≥ G φ (r) 2 φ ′ • φ -1 log 1 G φ (r) ≥ e -φ(r) 4Z φ φ ′ (r) φ ′ • φ -1 log Z φ φ ′ (r)e φ(r) ≥ e -φ(r) 4Z φ since Z φ φ ′ (r) ≥ 1 and φ ′ • φ -1 is non-decreasing.
On the other hand, r) .

L φ G φ (r) ≤ G φ (r) φ ′ • φ -1 log 1 G φ (r) ≤ e -φ(r) Z φ φ ′ (r) φ ′ • φ -1 log 2Z φ φ ′ (r)e φ(
One can show that 2Z φ φ ′ (r)e φ(r) ≤ 2Z φ rφ ′ (r)e φ(r) ≤ 4Z φ φ(r)e φ(r) ≤ φ(4Z φ r)e φ(r)

≤ e φ(4Z φ r) e φ(r) ≤ e 2φ(4Z φ r) ≤ e φ(8Z φ r) .

Thus

G φ (r) log 1 G φ (r) φ -1 log 1 G φ (r) ≤ e -φ(r) Z φ φ ′ (r) φ ′ (8Z φ r) ≤ 32(1 + e -1 ) e -φ(r) Z φ . Now assume that r ≤ 1. Let us remark that L φ : a → a log 1 a φ -1 (log 1 a )
is non-decreasing on [0, 1 2 ]. Indeed, L φ ′ is of the same sign as

(x -1)φ -1 (x)φ ′ • φ -1 (x) + x, with x = log 1 a . If x > 1, (x -1)φ -1 (x)φ ′ • φ -1 (x) + x ≥ (x -1)x + x ≥ 0. Else x ∈ [log 2, 1] and (x -1)φ -1 (x)φ ′ • φ -1 (x) + x ≥ (x -1)2x + x = x(2x -1) ≥ 0. So L φ G φ (r) ≥ L φ G φ (1) ≥ G φ (1) 2 φ ′ • φ -1 log 1 G φ (1) ≥ e -φ(1) 4Z φ φ ′ (1) φ ′ • φ -1 log Z φ φ ′ (1)e φ(1) ≥ e -1 4Z φ ≥ e -1 4
e -φ(r) Z φ .

Similarly for the lower bound,

L φ G φ (r) ≤ L φ G φ (0) ≤ G φ (0) φ ′ • φ -1 log 1 G φ (0) ≤ 1 2 φ ′ • φ -1 (log 2) ≤ 1 ≤ 2(1 + e -1 ) e -1
e -φ(r) Z φ .

6 Isoperimetry for µ n,φ

Now we can apply Proposition 13 to µ n,φ with J = I φ when φ satisfies (H1') or J = Is γ the Gaussian isoperimetric function when φ satisfies (H2). Indeed by Theorem 6 and Lemma 14,

Is ν n,φ ≥ C ν n,φ J with C ν n,φ = Cφ -1 (1) √ n φ -1 (n) under (H1) and C ν n,φ = C √ n φ -1 (n) under (H2)
, where C > 0 is a universal constant. As for the sphere, it is known that σ n-1 satisfies Gaussian isoperimetry with a constant of order √ n, e.g. by a curvature-dimension criterion (cf [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses[END_REF]). That means that for every a ≤ 1 2 and every φ satisfying (H1),

Is σ n-1 (a) ≥ C √ n Is γ (a) ≥ CK √ n a log 1 a ≥ CK log 2 √ n φ -1 (1) log 1 a φ -1 log 1 a
where K > 0 is a universal constant.

Proposition 17. For every c > 0, there exists C > 0 such that if e -cn < 1 2 , then for every function φ,

i) if φ satisfies (H1') then ∀a ∈ e -cn , 1 2 , Is µ n,φ (a) ≥ C √ n φ -1 (n) φ -1 (1) a log 1 a φ -1 log 1 a . ii) if φ satisfies (H2) then ∀a ∈ e -cn , 1 2 , Is µ n,φ (a) ≥ C √ n φ -1 (n) a log 1 a .
Proof. We only prove i). We can restrict ourselves to the case φ(1) = 1. Let κ be the constant coming from Lemma 15, then let κ 1 and κ 2 be the corresponding constants given by Proposition 13. Set c 1 large enough to ensure

c 1 ≥ 2, max(κ 1 , 1)ec 1 e -c 1 ≤ e -c .
If we take r 1 = φ -1 (c 1 n), then we know by Lemma 10 that

κ 1 ν n,φ {(r 1 , +∞)} ≤ κ 1 ec 1 e -c 1 n ≤ e -cn .
Here we use that φ -1 (c 1 n) ≤ c 1 φ -1 (n). So for all φ, for all n, and all a ∈ e -cn , 1 2 , Condition (10) holds, i.e.

κ 1 ν n,φ {[r 1 , +∞)} ≤ a ≤ 1 2 . Now there exists a universal C > 0 such that C ν n,φ ≥ C √ n φ -1 (n)
by Theorem 6 and Lemma 15 as explained at the beginning of the section (recall that here φ -1 (1) = 1). So, if we set

r 2 = (1 + 1 CI φ ( 1 2 ) )φ -1 (c 1 n), then Condition (9) is also satisfied, i.e. r 2 -r 1 ≥ 1 C ν n,φ I φ ( 1 2 )
.

Thus Proposition 13 yields

Is µ n,φ (a) ≥ κ 2 min C ν n,φ , C σ n-1 r 2 I φ (a).
Besides, there exists a universal d > 0 such that I φ ≥ d L φ according to Lemma 14. In particular,

I φ 1 2 ≥ d L φ 1 2 = d log 2 2φ -1 (log 2) ≥ d √ log 2 2 .
We can deduce an upper bound for r 2 . Finally, we have established

Is µ n,φ (a) ≥ κ 2 Cd min 1, c 1 1 + 2 Cd √ log 2 -1 √ n φ -1 (n) L φ (a).
Therefore we have proved Theorem 4 at least for a large enough. We complete the proof with Proposition 11 or Proposition 12 for smaller sets.

Optimality and the isotropic case

One can ask whether the isoperimetric inequalities obtained are optimal at least up to universal constants, and whether we recover dimension-free results in the case of isotropic measures.

We consider only bounds for the isoperimetric profile constructed as product of a function of n times a function of a. When φ satisfies (H1'), inequalities of Theorem 4 are optimal in a for n = 1, according to Lemma 14. In the supergaussian case, the central limit theorem for convex bodies of Klartag (see [START_REF] Klartag | A central limit theorem for convex sets[END_REF]), in the simpler case of spherically symmetric distributions, ensures that we cannot find a profile bounding from below Is µ n,φ for all n, better than the Gaussian one (times a constant depending possibly on n). Else we should have concentration properties stronger than Gaussian. However by Klartag's theorem, there exists a sequence of positive number ε n → 0 such that for every Borel set A ⊂ R and every r > 0,

1 -µ n,φ A × R n-1 r ≥ 1 -γ(A r ) -ε n
, where γ denotes the standard normal distribution. Thus we cannot have a rate of concentration valid for all n better than the Gaussian one.

So optimal inequalities should be of the type

∀a ∈ 0, 1 2 , Is µ n,φ (a) ≥ C µ n,φ (n) φ -1 (1) a log 1 a φ -1 log 1 a under (H1'), ≥ C µ n,φ (n) a log 1 a under (H2).
This implies

∀a ∈ 0, 1 2 , Is µ n,φ (a) ≥ c C µ n,φ (n) a,
where c > 0 is universal. Now Poincaré inequalities are equivalent up to universal constants to Cheeger inequalities (see [START_REF] Milman | On the role of convexity in isoperimetry, spectral-gap and concentration[END_REF]), so the optimal constant in n should be

C µ n,φ (n) = C n E µ n,φ (|X| 2 )
, in view of Theorem 1, with C > 0 a universal constant. Thus, the two questions raised at the beginning of the section appear to be connected to the same property, namely E µ n,φ (|X| 2 ) ≃ (φ -1 (n))

2 . Undoubtedly, this must be quite standard, nevertheless we state and prove the next lemma for completeness. Lemma 18. i) Let φ be a function satisfying (H0). Define r n (φ) the point where the density of the radial measure ν n,φ reaches its maximum, and E µ n,φ |X| 2 the second moment of µ nφ . For every M > 1, there exists n 0 ∈ N not depending on φ such that, for all n ≥ n 0 ,

1 M E µ n,φ |X| 2 ≤ r n (φ) ≤ M E µ n,φ |X| 2 .
ii) Besides, if there exists α ≥ 1 such that x → φ(x)/x α is non-increasing, then

φ -1 (n) ≥ r n (φ) ≥ e -1 e φ -1 (n).
Proof. To prove the first point, we can assume that µ n,φ is isotropic, that is to say that E µ n,φ (|X| 2 ) = n. Let X be a random variable with distribution µ n,φ . In the following, we denote by P, E, and Var the corresponding probability, esperance, and variance. Let δ ∈ (0, 1). In view of Lemma 8, there exist universal constants c > 0 and C > 0 such that P r n (φ) -|X| ≥ δr n (φ) ≤ Ce -cnδ 2 .

On the other hand, Bobkov proved in [START_REF] Sergey | Spectral gap and concentration for some spherically symmetric probability measures[END_REF] the following upper bound for the variance of |X| to establish Theorem 1:

Var|X| ≤ (E|X|) 2 n , which can also be reformulate

nE|X| 2 ≤ (n + 1) (E|X|) 2 . Then E E|X| 2 -|X| 2 = 2 E|X| 2 E|X| 2 -E|X| ≤ 2 1 + 1 n -1 E|X| 2 E|X| ≤ E|X| 2 n .
So, by Chebychev's inequality it holds for all t > 0:

P E|X| 2 -|X| ≥ t E|X| 2 ≤ 1 nt 2 .
Fix δ ∈ (0, 1), and choose n large enough to ensure Ce -cnδ 2 + 1/nδ 2 < 1. Then there exist x > 0 such that |r n (φ) -x| ≤ δr n (φ) and | E|X| 2 -x| ≤ δ E|X| 2 . It follows

1 -δ 1 + δ √ n ≤ r n (φ) ≤ 1 + δ 1 -δ √ n.
Now, r n (φ) satisfies r n (φ)φ ′ r n (φ) = n -1. Therefore, as already mentioned, (H0) ensures that r n (φ) ≤ φ -1 (n). Assume moreover the existence of α ≥ 1 such that x → φ(x)/x α is non-increasing. Then

r n (φ) ≥ φ -1 n -1 α ≥ φ -1 n 2α ≥ 1 2α 1 2α
φ -1 (n) ≥ e -1 e φ -1 (n).

Eventually, we can state the following theorem.

Theorem 19.

• If φ satisfies (H1 ′ ) or if φ satisfies (H2 ′ ), then the inequality proved in Theorem 4 is optimal.

• For any n ∈ N, let us choose λ > 0 such that µ n,φ λ is isotropic, when replacing φ by φ λ : x → φ(λx). Then it holds a dimension-free isoperimetric inequality. More precisely, there exist a universal C > 0 and a universal n 0 ∈ N such that i) if φ satisfies (H1 ′ ) then ∀a ∈ 0,

1 2 , 1 ) a log 1 aφ -1 log 1 a; 2 ,

 21112 ∀n ≥ n 0 , Is µ n,φ λ (a) ≥ C φ -1 (ii) if φ satisfies (H2 ′ ), then ∀a ∈ 0, 1 ∀n ≥ n 0 , Is µ n,φ λ (a) ≥ C a log 1 a .

Acknowledgments. I would like to thank Franck Barthe for his support and fruitful discussions, and also Emanuel Milman for helpful suggestions.