Armand Fouejieu

Alexandra Popescu

Patrick Villieu

Trade-offs between macroeconomic and financial stability objectives

Keywords: Monetary policy objectives, financial stability, trade-offs. JEL classification: E61, E44, E52, E58

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Introduction

Ten years after the 2008-09 global financial crisis (GFC), global economic growth has taken hold, supported by largely expansionary monetary policies in major advanced economies. Low interest rates have played a key role in supporting the economic recovery, and despite gradual normalization, the monetary policy stance remains accommodative especially given still low inflation. In parallel, vulnerabilities to the global financial system have increased: debt (both private and public) have reach historical highs, banks are highly leveraged, and investors' risk appetite is on the rise. Favorable financial conditions (low interest rates) have fueled financial risks. These developments not only remind us of the pre-2008 GFC period, but they also highlight the challenges faced by central banks. Loosening financial conditions and increasing risks may call for monetary policy tightening, while inflation far below targets would require opposite policy actions. Although the prudential toolkit aiming to ensure financial stability has further expended since 2008, especially with the broad-based adoption of macroprudential policies, the role of monetary policy remains subject to debate and disagreement among policymakers as well as researchers. This paper contributes to the debate by providing, based on a theoretical framework, new insights on policy trade-offs that central banks may face.

The existing literature discusses various channels through which monetary policy can affect financial sector stability. According to the risk-taking channel [START_REF] Borio | Capital regulation, risk-taking and monetary policy: A missing link in the transmission mechanism?[END_REF], maintaining low interest rates for a protracted period of time increases financial risks through higher incentives to search for yields [START_REF] Rajan | Has Financial Development Made the World Riskier? Working Paper 11728[END_REF]. For financial corporations, low interest rates can increase interest margins, boost firm's value and increase leverage which ultimately translates into higher risk exposure (Adrian and [START_REF] Villieu | Financial Intermediaries and Monetary Economics[END_REF]. A credible and predictable monetary policy framework can also encourage risk-taking behavior. Indeed, as sound policies reduce uncertainty, participants in the financial market may underestimate underlying risks. For the private non-financial sector low interest rates increase the value of collateral and therefore provide further room for borrowing.

In view of the above, should monetary policy respond to financial imbalances? The related discussion has long been summarized around the "lean versus clean" debate. The "clean" view supports a clear division of labor between monetary and prudential policies, with the former focusing on inflation (or macroeconomic) stability. It is argued that not only financial imbalances or bubbles are hard to detect, but also the policy interest rate is ineffective or too blunt an instrument to address specific financial risks. Moreover, the macroeconomic costs associated with policy tightening in response to potential financial stability risks may be higher than the costs of a crisis if the risk materializes. Therefore, monetary policy interventions are expected only to contain the consequences of crises when they occur. Proponents of the "lean" support a more active role of central banks in preventing "unsafe" developments in the financial sector. The policy interest rate can be used as a tool to curb bubbles' growth, thereby preventing financial crises and associated damaging effects for the rest of the economy. In addition, in period of crisis, monetary policy may have limited room for maneuver to cleanup, because of the zero lower bound on the interest rate. Hence, a more proactive intervention is warranted.

Before the 2008 GFC, a widespread consensus supported the "clean" approach, with most central banks adopting a "benign neglect" attitude toward financial imbalances, i.e. only reacting to these to the extent that they affect inflation. However, the 2008 experience reveals how costly and long-lasting the consequences of a crisis can be. Despite unprecedented policy measures, the global economy contracted sharply and only recovered very slowly over the past decade [START_REF] Woodford | Inflation Targeting and Financial Stability[END_REF]. The pre-crisis period also showed that central banks can face trade-offs between macroeconomic stability and financial stability. Despite remarkable achievements in terms of controlling inflation, monetary policy did not prevent financial risks from accumulating.

A decade after the GFC, the role of monetary policy in preventing financial imbalances remains subject to debate (IMF, 2015). However, the key question seems to be how best central banks can act, rather than whether they should act at all. [START_REF] Rudebusch | Monetary policy and asset price bubbles[END_REF] argues that, ideally, a moderate adjustment of the policy interest rate in response to financial imbalances could contain bubbles and reduce the risks of broader macroeconomic crises. Higher interest rates, implying higher borrowing costs can contribute to reduce leverage not only for households and the corporate sector, but also for banks [START_REF] Dellariccia | Real interest rates, leverage, and bank risk-taking[END_REF]. For [START_REF] Smets | Financial stability and monetary policy: How closely interlinked?[END_REF], central banks should keep an eye on financial stability, as this will allow them to lean against the wind when necessary while maintaining the price stability objective. However, as illustrated by the current global economic and financial conditions, such policies may give rise to trade-offs.

The Tinbergen principle commends using as many instruments as objectives to be achieved (and to assign the best instrument to the dedicated objective). A "leaning against the wind" policy can generate trade-offs because of the violation of this principle. De Grauwe and Gros (2009) show that a trade-off between inflation and financial stability can emerge when the economy faces a technological shock, or when investors' behavior is characterized by too optimistic beliefs in financial markets ("animal spirits"). It may be dangerous for central banks to pursue a financial stability objective because this may require a tighter monetary policy than what is needed to tame inflation (Dell'ariccia, 2010; Mishkin, 2011, among others). [START_REF] King | Twenty years of inflation targeting[END_REF] identifies three scenarios where a trade-off can emerge between macroeconomic stability and financial stability. First, (too optimistic) misperception on the part of households, businesses and financial institutions about future incomes, leading to unsustainable spending and increase in the level of debt. Second, the "cycle of confidence", reflecting the fact that a prolonged period of stability (both macroeconomic and financial stability) can encourage exuberant behavior and subsequently generate instability. Third, trade-offs can appear due to the risk-taking channel, as described earlier. [START_REF] King | Twenty years of inflation targeting[END_REF] further points out that adding financial shocks to the traditional Taylor curve1 moves the frontier upper and to the right, i.e.

adding financial stability to the traditional macroeconomic stabilization objectives increases the volatility of both inflation and output. Although [START_REF] Issing | Monetary and financial stability: Is there a trade-off?[END_REF] suggests that the trade-off between inflation stability and financial stability is more likely to occur in the short-term, a more recent analysis by [START_REF] Fahr | Macroprudential policy-closing the financial stability gap[END_REF] shows that even when considering financial cycles, monetary policy is always less effective than macro-prudential policies in addressing financial stability risks.

In most cases, the literature discussed above does not resort formally to a theoretical model when discussing trade-offs between monetary policy objectives in the "leaning against the wind" setup. Notable exceptions are cases where trade-offs are assessed in the presence of macroprudential policies, for example in several DSGE models [START_REF] Agenor | Capital Regulation, Monetary Policy and Financial Stability[END_REF][START_REF] Beau | Macro-Prudential Policy and the Conduct of Monetary Policy[END_REF] or [START_REF] Christensen | Bank leverage regulation and macroeconomic dynamics[END_REF]). 2 Shukayev and Ueberfeldt (2016) investigates trade-offs between policy objectives in a context where banks are exposed to runs on their short-term liabilities.

They find that relying on the short-term interest rate, monetary policy can mitigate this risk (and preserve investment) but at the cost of higher inflation and output volatility.

To the best of our knowledge, this paper is the first to explicitly rely on a theoretical model, with an endogenous financial bubble, to investigate the existence of trade-offs when the central bank applies a "leaning against the wind" type of policy. Our analytical framework starts with a hybrid reduced form three-equation New Keynesian model, which we supplement with a fourth equation reflecting an asset price bubble. We further assume that the policy interest rate can affect the probability that the bubble bursts (i.e. financial risk materializes), making the bubble endogenous to the macro-framework. Moreover, asset price deviations are intuitively assumed to affect aggregate demand. In this framework, we define financial instability as the volatility of asset prices' deviations from their fundamental values. Thus, the bubble process is assumed to capture risk accumulation in the financial sector. We explore changes in inflation, output and bubble volatilities in response to various types of shocks and different responses of the central bank.

Our model can be related to the work of [START_REF] Filardo | Monetary policy and asset prices[END_REF]Filardo ( , 2001Filardo ( and 2004). 3 All these papers treat the question of whether or not monetary policy should respond to asset prices via the interest rate, but the first major difference with our paper is that they do not address tradeoffs between macroeconomic and financial stability objectives. The emergence of such tradeoffs is our main research question. Additionally, these papers use a backward looking simple model with an augmented Taylor rule. We however opt for a hybrid framework with both backward and forward looking terms as this type of model allows us to not only better fit the data (Fuhrer andMoore, 1995, Estrella and[START_REF] Estrella | Dynamic Inconsistencies: Counterfactual Implications of a Class of Rational-Expectations Models[END_REF] but also to account for habits in consumption or the fact that not all firms adjust their prices in each period of time. Moreover, by considering a hybrid model we partly respond to the Lucas critique by taking into account rational expectations. Another difference between [START_REF] Filardo | Monetary policy and asset prices[END_REF] and our paper comes from the supposed impact of the bubble on the economy. While [START_REF] Filardo | Monetary policy and asset prices[END_REF] assumes that inflation in asset prices has a direct impact on general price inflation, we include the bubble in the IS equation, implying only an indirect impact on inflation. We justify this choice by the fact that asset prices influence aggregate demand through several well acknowledged transmission channels: the wealth effect, Tobin's Q or the financial accelerator mechanism. 4 In addition, while Filardo (2001) and Filardo (2004) clearly distinguish the fundamental and the speculative part of asset prices, we simply define the bubble as the difference between these two components.

This assumption, in line with Filardo (2004), implies that in our model we only care for the speculative part of asset prices, meaning the one which creates financial instability.

Our results suggest that, when the central bank directly responds to financial imbalances, a trade-off indeed emerges between its primary objective of price stability and the financial stability objective. This finding emphasizes the limits of the leaning against the wind strategy.

More precisely, employing such a strategy may generate a conflict between the objectives of the monetary authority. While financial stability remains a major concern, central banks should focus on their traditional inflation (and output gap) stability objective and rely on other (macroprudential) instruments to deal with financial imbalances.

The remainder of the paper is organized as follows. Section 2 presents the model with an emphasize on the bubble process. Section 3 discusses the results of the different scenarios considered. Robustness tests are conducted in section 4 and section 5 concludes.

The Model

This section first describes how we model the financial bubble, and then the rest of the macroeconomic framework, and especially how the bubble fits into it.

The Bubble

The buildup of financial imbalances follows a process a la [START_REF] Blanchard | Bubbles, Rational Expectations and Financial Markets[END_REF]. The rational asset price bubble takes the following linear form:

b t =        1+ ī q (b t-1 -q t ) + ε b t if the bubble persists, (q t ) ε b t otherwise, (1 -q t ) (1)
where b t denotes asset price deviations from its fundamental value at time t. The bubble is assumed to persist with probability q t and to burst with probability (1 -q t ). q is the threshold value of q t above which the bubble bursts. q t is the difference between q t and q. Finally, ī is the equilibrium interest rate and ε b t is an exogenous shock with zero mean. Equation (1) suggests the bubble is self-fulfilling and grows without any connection to fundamentals when 1+ ī q > 1. This characterizes a phenomenon where a bubble in a specific market can be driven by (too optimistic) expectations that these asset prices will keep growing over subsequent periods.

By construction, the bubble in equation ( 1) is strictly exogenous since it does not depend on any economic variables or policy changes. Although exogenous bubbles are commonly used in macroeconomic models [START_REF] Bernanke | Monetary policy and asset price volatility[END_REF][START_REF] Cecchetti | Asset Prices and Central Bank Policy[END_REF][START_REF] Badarau | Monetary policy and credit cycles: A DSGE analysis[END_REF] Popescu, 2014, among others), it is well recognized that asset prices are sensitive to changes in the economic environment. Especially, as discussed above, the experience of the 2008 GFC as well as the current global economic environment suggest that the monetary policy stance is likely to play a key role in (at least partly) driving market participants' behavior in the financial market. To incorporate this into our model, we realistically assume that the policy interest rate can affect the probability q t that the bubble bursts.

q t = -γ∆i t-1 (2)
where ∆i t-1 is the change in the short term interest rate from one period to another. Equation [START_REF] Agenor | Capital Regulation, Monetary Policy and Financial Stability[END_REF] suggests that an interest rate increase will tend to reduce the probability that the bubble persists, and by construction increases the probability that the bubble bursts. 5 Such a correction of the monetary policy stance may be warranted when the authorities are willing to stop the buildup of imbalances sooner than later, in order to reduce the subsequent costs in case the risk materializes. In other words, this can be seen as a preventive intervention through which the central bank may want the bubble to "burst" before risk accumulation becomes too excessive.

However, such policy affects not only the duration, but also the size of the bubble. Indeed, a higher interest rate will translate into an upward reaction of the bubble. 6 Equation (2) also assumes that bubbles are likely to be more affected by changes in the interest rate rather than levels. Consider for example a first scenario where the short-term interest rate increases from 1 to 2%, and a second scenario where the rate goes from 1.5 to 2%.

In both cases, the interest rate reaches the same level, however, in the first scenario, the rate increases by 1 percentage point, whereas the hike is of only 0.5 percentage point in the second one. It seems straightforward to assume that market reactions would be more important in the first scenario (this would be in line with current speculations regarding the pace of the Fed monetary policy normalization). 7,8 Substituting equation (2) into equation [START_REF] Villieu | Financial Intermediaries and Monetary Economics[END_REF]gives the following expression:

b t = 1 + ī q (b t-1 + γ∆i t-1 ) D + ε b t ( 3
)
where D is a dummy variable which takes the value of 1 as long as the bubble lasts, and 0 when it bursts.

For example, the housing market tends to tighten in a low interest rate environment, as it is currently the case in many advanced economies. A sharp correction of the policy interest rate not only increases the cost of housing, therefore affecting demand, it also increases the probability of default on mortgages (to the extent that these are flexible interest rates mortgages). This can contribute to significantly depress the housing market, but also the financial sector as whole depending on banks' exposure to mortgages. 6 Gali (2014) also argues that an increase in the policy rate in reaction to a growing bubble will entail a positive effect on the bubble's growth. 7 Note however that we also perform our simulations based on a version of the model that includes the level and not the change in the interest rate. This does not change our main findings. See the robustness section for more details.

8 A similar approach is used in [START_REF] Gruen | How Should Monetary Policy Respond to Asset-Price Bubbles?[END_REF]. Their macroeconomic model includes an asset price bubble and compares the optimal monetary policy response for two types of policymakers: a skeptic one which implements a (standard) inflation targeting-type of policy, and an activist one which responds to asset price bubbles. In their sensitivity analysis, it is assumed that the probability that the bubble burst is affected (with a lag) by the difference between the short-term interest rate and its optimal path, chosen by the skeptic policymaker.

The Macroeconomic Framework

The macroeconomic framework is modeled based on a three-equation New Keynesian model.

The log-linear inter-temporal transformations of these equations take the following forms:

π t = αE t (π t+1 ) + (1 -α)π t-1 + λy t + ε π t (4) y t = δE t (y t+1 ) + (1 -δ)y t-1 + σ i t -E t (π t+1 ) + ϕb t + ε y t (5) i t = β i i t-1 + (1 -β i )(β π π t + β y y t + β b b t ) (6)
where π t , y t and i t represent respectively the inflation rate, the output gap, and the short-term Coming back to our example on the housing market, overvaluations of housing increase the value of collateral, boost leverage and therefore consumption. This translates directly into higher output. 9 Filardo (2004), supporting the relationship between aggregate demand and the bubble, argues that the fundamental component of asset prices does not really matter for output or its components. Instead, the bubble (the non-fundamental component) can affect aggregate demand by distorting economic decisions: changes in consumption through a wealth effect, changes in investment via the cost of capital, changes in government spending through the tax channel.

Finally, equation ( 6) describes the central bank's reaction function. The monetary policy instrument is set in response to deviations of inflation from the target, the output gap, as well asset prices deviations from their fundamental value (the bubble). Moreover, a smoothing parameter is included in order to limit interest rate volatility. The central bank's reaction function portrays an augmented Taylor rule with a financial variable, framing a leaning against the wind policy. As discuss earlier, such policy may be warranted to reinforce financial stability.

Tightening the policy stance could contribute to contain increasing and unsustainable financial imbalances, which subsequently could generate a financial crisis with related significant macroeconomic effects.

Model solution and calibration

The four-equation model described above is solved relying on the Blanchard-Kahn method [START_REF] Blanchard | The Solution of Linear Difference Models under Rational Expectations[END_REF]. This is a commonly used approach in the literature. However, the specification of the bubble equation gives our model a less standard structure. In this respect, the model is solved for two states of nature: first, when the bubble persists (i.e. D = 1 in equation ( 3)) and second, when the bubble is simply a random shock (D is equal to 0). 10We switch between the two states of nature depending on the value of D. To decide whether the bubble continues to grow or bursts in the next period, we use a random variable drawn from the uniform distribution over the interval 0-1. We compare this random variable to the probability of the bubble to persists, q t : if its value is inferior to q t , D is equal to 1, otherwise D is 0.

The model is calibrated based on estimated parameters for the euro area, using monthly data over 1999-M09 to 2007-M12. While using standard definitions for variables in the model, a key issue is how to measure financial imbalances (the bubble). We use the MSCI (Morgan Stanley Capital International) index, which is an aggregate index measuring equity valuations.

The deviations from fundamental values are assessed based on the Arbitrage Pricing Model (APM), where actual equity prices are compared with their "fair values". Fair values of equity are estimated based on the presumption that equity price variations are affected by surprises due to shocks on various fundamental variables (See Appendix 1 for further details).

A second set of parameters from Smets (2000) and corresponding to annual data for the euro area, are used for simulating the model and check the robustness of our findings. However, two (non-standard parameters are introduced in our model specification compared to Smets (2000), namely γ and ϕ. As there are no previous estimates for these parameters, we set their values relying on some ad-hoc assumptions, 11 and we check the robustness of our main conclusions to changes in the values of those parameters . Table 1 in the appendix presents the baseline values of the parameters used to perform the simulations.

Results

The reduced-form model characterized by equations ( 3), ( 4), ( 5) and ( 6) is used to investigate the existence of trade-offs between macroeconomic (inflation and/or output) stability and financial stability when the central bank reacts to the financial bubble. We rely on a simple procedure which can be summarized as follows: the economy is hit by exogenous shocks (supply and/or bubble shocks corresponding to ε pi t and ε b t ) which are assumed to randomly occur each period of time. 12 Central banks respond to these shocks by setting the short-term interest rate more or less aggressively. More precisely, we consider different values for the parameters concerning inflation, output and the bubble (β π , β y and β b in equation 6). For each type of shock and each value of the βs, we generate series of variances, calculated over 1000 periods, for each argument of the central bank's reaction function (inflation, output gap and the bubble). We then graphically present the evolution of these variances, in the spirit of a Taylor-type curve, to assess the efficiency in achieving the monetary policy objectives.

Confronted with the same shocks, central banks may react differently, both in terms of measures undertaken and in terms of intensity (or aggressiveness) of the policy. For example, faced with an asset price bubble shock, a central bank may decide to react indirectly through a stronger response to output gap (since the bubble affects aggregate demand), thus increasing β y . Another central bank may rather react directly by strengthening its response to the bubble, increasing β b . Moreover, in both cases, the responses may be more or less aggressive (a sharp or a more progressive increase in β).

The analysis conducted here should not be view as an attempt to derive the optimal monetary policy stance. The purpose is much more modest and simply aims at investigating, through comparative statics, the challenges central banks can face when reacting to financial variables.

More precisely, we focus on trade-offs between policy objectives. In addition, the differences in parameters in the central bank's reaction function can be viewed as responses from different central banks to the same shocks, and not necessarily as changes in a single central banks response over time. Before addressing the issue of trade-offs, we investigate the response of the model to random bubble shocks, by looking at how the output gap, inflation and the interest rate react to deviations of the asset price from its fundamentals (Figure 1). Bubble shocks are drawn from the standard normal distribution and can be either positive or negative. In response to positive bubble shocks, the output gap increases as expected. Given the policy rule, the short-term interest rate rises in reaction to higher output gap, but also in response to the bubble. The higher level of the policy rate reduces inflation in the economy with respect to its steady state value. The opposite holds when shocks are negative (i.e. periods when the bubble is below 0).

We turn now to the investigation of potential trade-offs between central banks' objectives when responding to real or financial shocks. The main results are discussed considering successively the central banks responses to each type of shock: supply shocks, bubble shocks and a combination of the two.

Supply shocks

Faced with supply shocks, central banks may decide to be more or less aggressive in their reaction to the evolution of inflation. In our model, a central bank which is more aggressive to inflation has a larger β π . This parameter varies between 1.5 and 2.5 and the arrows in Figure 2 show how volatilities evolve when β π increases. A stronger response to inflation shocks results in better inflation stabilization, but at the cost of higher output and bubble volatility. The standard trade-off between inflation and output stabilization in a context of supply shocks emerges, but there also seems to be a trade-off between inflation and the asset price bubble.

This implies that the stronger the central bank's reaction to inflation shocks, the higher the asset price bubble volatility. This finding is in line with the argumentation in De Grauwe and Gros (2009) or Issing, 2003. In addition, there seems to be no trade-off between the output gap and the asset price bubble in case of cost push shocks as their volatilities both increase.

Asset price bubble shocks

Since the asset price bubble enters the central bank's reaction function, the short-term interest rate rises in response to positive bubble shocks and diminishes when negative bubble shocks occur. This type of reaction, intended to strengthen financial market stability, corresponds to the leaning against the wind strategy. According to Figure 3 this strategy is however unsuccessful as the volatilities of all interest variables -inflation, output and the bubble -increase.

Being more aggressive to the bubble (gradual increase in β b from 0 to 0.75) not only harms the primary objectives of the central bank, but does not even succeed in reducing the volatility of the bubble. Alternatively, the central bank may choose to react indirectly to bubble shocks through a stronger response to output gap (gradual increase in β y from 0.5 to 1.5), since it is assumed that the bubble positively affects aggregate demand. This approach seems to provide a better outcome than the former as it reduces not only the bubble, but also the output gap volatility.

However, inflation volatility increases, making this strategy questionable. As a result of the central bank's actions, a trade-off between inflation and financial stability emerges, but the bubble and the output gap seem to be better stabilized (Figure 4). 13 This results is in line with 13 Figure 4 also shows that there is a trade-off between inflation and the output gap. Such a trade-off is not to be expected in the context of demand shocks in standard models. Note however that our model is different from the standard New Keynesian one, in the sense that it includes a bubble process which is assumed to affect Granville and [START_REF] Mallick | Monetary and financial stability in the euro area: Pro-cyclicality versus trade-off[END_REF] who show through an empirical model that asset price shocks can generate trade-offs in the long run if asset prices are not stabilized.

To some extent, these results suggests that a standard Taylor rule provides a better outcome than a pure leaning against the wind strategy, even when facing financial shocks. Although not satisfactory, the standard rule seems to be less costly.

Output -Bubble

Note: Variances of inflation, output gap and the bubble following bubble shocks. The response to output gap varies between 0.5 and 1.5 and all other parameters remain the same. The arrows indicate an increase in βy.

Supply and asset price bubble shocks

We assume now that the economy is faced with supply and bubble shocks at the same time.

The results in this scenario are in line with the above discussion. A stronger response to output gap results in better bubble and output gap stabilization, while inflation variability increases as shown in Figure 5. A trade-off appears between inflation and the bubble stabilization, but also between price stability and output gap stability.

If monetary authorities are more adverse to the bubble (increase in β b ), once again all volatilities increase. The results in Figure 6 also show that when the central bank's reaction becomes more aggressive, all policy objectives are negatively affected (increase in output, inflation and bubble volatilities). This strategy proves to be counterproductive, leading not only to a deterioration in macroeconomic stability, but also to an increase in the bubble's volatility.

aggregate demand. When removing this assumption, we reach the common conclusion of no trade-off between inflation and output. This finding is in line with Gali (2014) who argues that increasing the interest rate in response to a growing bubble generates higher fluctuations in the latter, as the interest rate positively affects the bubble's growth. In a different framework with another definition for financial instability, Svensson (2013) also concludes to a counterproductive effect of tightening monetary policy to stabilize the financial system.14 In response to supply and bubble shocks, an increase in the reaction to inflation lowers its volatility, while there is an increase in output gap and bubble volatilities (Figure 7). A trade-off between inflation and bubble stability emerges, in addition to the standard inflation-output gap trade-off.

Globally, these results seem all against the leaning against the wind strategy. Regardless of the type of shock, a trade-off between inflation stability and financial stability is always present. 15 A reaction to the bubble (β b different than zero) seems to be the worst case scenario as all interest variables find their volatilities aggravated. The best outcome, although not ideal, is to increase the response of the policy rate to the output gap. This allows monetary authorities to attain both financial and output gap stability, but the price stability objective is left aside in this case. Several authors now (De Grauwe and Gros, 2009, Woodford, 2012) seem in favor of this strategy, where the central bank neglects its main objective in the short run, in order to avoid or to diminish the harmful effects of a financial bubble.

Robustness Checks

To check the robustness of our results, various analyses are conducted. First, the bubble is assumed to be affected by the levels of the short-term interest rate rather than the changes.

Second, we assess the extent to which our results are sensitive to changes in the values of the parameters ϕ and γ. Third, we examine different random selections for supply and financial shocks. And finally, an alternative random selection of the probability q t is considered.

Bubble affected by levels rather than changes in interest rate

In section 2.1, it is assumed that changes in the short-term interest rate matter more for the financial sector than levels. However, it can be argued that the levels of the policy rate also matter since they can have important effects on the private sector's risk-taking behavior. We assess the extent to which our main results remain relevant if the bubble is affected by the level of the short-term interest rate rather than its changes (i.e. in equation ( 3), ∆i t-1 is replaced by i t-1 ). Within such a framework, our findings are the same as in the benchmark model when supply shocks are considered. The main results are also validated when bubble shocks occur and the central bank reacts to the output gap. Our conclusions slightly change however when bubble shocks occur and the central bank increases its response to the bubble.

If in the main analysis, this strategy generated an increase in the variance of all variables, when the bubble is affected by the level of the short-term interest rate, the volatility of asset prices with respect to their fundamental value seems to be reduced. Nevertheless, we cannot conclude to a better outcome as a trade-off emerges between inflation and financial stability.

When both supply and bubble shocks hit the economy and the central bank leans against the wind in a more aggressive way (increase in β b ), once again the trade-off between inflation and financial objectives emerges. A second difference with respect to the main results appears when β y varies in response to bubble and supply shocks. Such a strategy succeeded in containing bubble volatility in the main analysis, whereas in this new framework the variance of the bubble increases. Although slightly different in some specific configurations, the results obtained when considering the levels of the policy rate point to the same conclusions. That is, they highlight the existence of trade-offs and the failure of the leaning strategy (Appendix 3, Figure 1).

Robustness to different values of the parameters ϕ and γ

The second robustness test concerns the model's parameters. First of all, our hybrid IS curve is different from the one usually used in the literature in that it includes the bubble. We therefore concentrate on the parameter of the bubble, ϕ in the IS curve capturing the sensitivity of demand to deviations of asset prices from fundamentals. We consider two alternative values in addition to the estimated one (see Appendix 1 for the estimation of the model's parameters).

This robustness test is done only with respect to the relation between the variances of inflation and the bubble as in our main analysis no trade-off occurs between the bubble and the output gap.

In a context of asset price bubble shocks and with the central bank responding either directly to the bubble or indirectly, through a response to the output gap, we investigate the existence of this trade-off for different values of ϕ (between 0.10 and 0.20). The results presented in Appendix 3, Figure 2 show that whatever the value of ϕ considered, the more aggressive the central bank is to the bubble, the more it destabilizes inflation and the bubble. In line with our main results, responding more aggressively to the bubble only worsens the volatility in inflation and asset prices. However, when the central bank reacts indirectly to bubble shocks, the importance of the bubble parameter in the IS curve has all its importance. If the bubble only plays a small role in influencing demand (small value for ϕ), responding more aggressively to output gap increases both the volatility of the bubble and of inflation. Yet, a higher impact of the bubble on demand (higher value for ϕ) generates a trade-off between objectives: the volatility of the bubble diminishes whereas the one of inflation goes up. This is an important results as it underlines that depending on the role played by the bubble in the economy, leaning against financial imbalances can indeed reduce them but at the cost of inflation.

The same investigation is conducted for different values of γ, the sensitivity of the bubble to changes in the interest rate (with γ taking values between 0.02 and 0.07). The economy is hit by asset price bubble shocks and the central bank responds more aggressively either to the bubble or to the output gap. The findings are in line with our previous conclusions. Responding to the bubble only worsens its volatility as well as the volatility of inflation, whatever the value of γ.

Considering responses to output gap, trade-offs appear between inflation and bubble stability as in the main analysis inflation volatility increases and bubble volatility decreases (Appendix 3, Figure 3). However, the gain from reducing inflation volatility is rather limited.

Robustness to an alternative selection of shocks

As discussed earlier, the shocks are drawn on a random basis from a standard normal distribution. We perform a robustness check by considering another random selection of shocks and we conduct the same analysis as above. Regarding the responses to supply shocks, the findings are the same as in the baseline model. We also reach the same conclusions when the central bank reacts to bubble shocks by increasing β y . These two strategies lead to trade-offs between inflation and the bubble, as well as between inflation and the output gap objectives.

When supply and bubble shocks hit the economy, we find the same results as in the baseline configuration. It is only when responding to shocks by increasing the response to the bubble that results seem to contradict our previous main analysis conclusion, but confirm the results obtained when using the level of the interest rate (Appendix 3, Figure 4). All in all, whatever the central bank's strategy or the type of shock to which the economy is confronted, we notice that a trade-off between inflation and asset prices appears.

Robustness to an alternative selection of q t

Our last robustness check concerns the values of q t which are selected randomly from a uniform distribution over the interval [0, 1]. An alternative series of q t is drawn from the same distribution and new results are computed based on these probabilities. Our findings remain the same in three scenarios: in case of supply shocks when the central bank react more aggressively to inflation, in case of bubble shocks when the central bank responds to the output gap, and in case of both supply and bubble shocks, when the monetary authority increases its reaction to inflation. The results in the other three scenarios differ from our baseline analysis in the evolution of the bubble's volatility. When bubble shocks or simultaneous supply and bubble shocks are considered, responding to the bubble leads to more asset prices instability. The evolution of the volatility in inflation and output gap does not change with respect to the baseline model. Finally, when both supply and financial shocks occur and the central bank reacts more aggressively to output gap, the results are similar to the ones obtained while using the level of the interest rate. 16 Overall, the alternative analyses conducted in this section highlight the robustness of our main result. Although some of the tests do not lead to exactly the same outcome with respect to the variance of the bubble, the main conclusions remain identical from a qualitative point of view. A leaning against the wind strategy, in which the central bank responds directly to financial instability, is subject to trade-offs between the primary inflation stabilization objective and financial stability.

Conclusions and Policy Implications

This paper explores the extent to which trade-offs can emerge between macroeconomic and financial stability when the central bank responds directly to financial imbalances. The macroeconomic framework employed is a New Keynesian model consisting of aggregate demand and aggregate supply equations and the central bank's reaction function. To reproduce the leaning against the wind strategy, the interest rate is modeled as an augmented Taylor-type rule in which the policy rate responds to a financial bubble. The novelty in our model is the introduction of a fourth equation meant to capture risk accumulation in the financial sector.

Moreover, this bubble is modeled endogenously as the monetary policy stance has an impact on its evolution.

Our analysis of trade-offs is performed by comparing the variances of policy objectives when the economy is confronted with various types of shocks and the central bank adopts different strategies to overcome them. The main conclusion shows that central banks practicing the leaning against the wind strategy will face trade-offs between inflation and financial stability. More precisely, when the central bank responds to financial imbalances, in the best-case scenario, such a policy can succeed in dampening financial risks, but at the cost of higher price instability. Our results also seem to highlight the worse-case scenario in which the policy is counterproductive with respect to all the monetary policy objectives (increase in macroeconomic and financial instability). 16 We do not present these results due to space constraints, but provide them upon request.

21

While the leaning against the wind strategy may be required, this paper argues that such a policy can lead to trade-offs between objectives or even destabilize the real economy and the financial sector. Although the risk-taking channel of monetary policy highlights the potential impact of interest rates on the evolution of financial bubbles, using this instrument to respond to these imbalances can be harmful for the monetary authorities. We therefore argue that the interest rate should be used for achieving traditional macroeconomic goals, and a second instrument should complement the policy rate to tackle financial risk accumulation. The progress made after the GFC in terms of prudential regulation, including the development of a macroprudential framework, is precisely meant to relieve monetary policy from this "two objectivesone instrument" burden. However, further analysis should be conducted to test whether or not monetary and prudential policies lead to unwanted trade-offs. This is left for future research. Note: Estimated using monthly data with GMM over the period 1999M09-2007M12. Robust standard errors are in parentheses.

All data are in percent. The output gap is measured as the percentage deviation of a linear trend.

bubble than the one to inflation or output gap. Therefore β b is first taken to be equal to 0.15 implying a rather soft reaction to the bubble. Furthermore, to imply more aggressive reactions of the central bank to different shocks, this parameter is allowed to go up to 0.75. Finally, β i is taken equal to 0.98. The next table summarizes all the parameters used in the model. β π = 1.5 0,0123 0,0065 0,0181 β π = 1.7 0,0131 0,0063 0,0200 β π = 1.9 0,0139 0,0061 0,0217 β π = 2.1 0,0147 0,0060 0,0233 β π = 2.3 0,0154 0,0060 0,0249 β π = 2.5 0,0161 0,0059 0,0263

Parameters α λ δ σ ϕ γ q ī β i β π β
Note: The loss functions are computed with variances of inflation, output gap and the bubble following supply shocks. The response to inflation varies between 1.5 and 2.5 and all other parameters remain the same.

Following supply shocks, a moderate central bank (with equal weights on inflation and output) obtains the best outcome when the parameter on inflation in the augmented Taylor rule has the smallest value considered (i.e. 1.5, Table 2). The same outcome is obtained for a central bank interested more in output stability. As the bubble does not explicitly represent a policy objective, the best response is thus a soft leaning against the wind (0.15) and moderate response to inflation. Of course, minimizing the loss when the central bank has a high preference for stabilizing inflation, implies a strong coefficient for β π .

When faced with asset price bubble shocks (Tables 3 and4), the moderate central bank minimizes social loss when it responds in a small extent to the bubble and to the output gap (0.5 and 0.15 respectively). Interestingly, responding aggressively to the bubble (high value for β y = 0.5 0,0053 0,0084 0,0022 β y = 0.7 0,0055 0,0089 0,0021 β y = 0.9 0,0058 0,0094 0,0021 β y = 1.1 0,0061 0,0100 0,0022 β y = 1.3 0,0064 0,0106 0,0022 β y = 1.5 0,0068 0,0113 0,0022

Note: The loss functions are computed using the variances of inflation, output gap and the bubble following bubble shocks. The response to output gap varies between 0.5 and 1.5 and all other parameters remain the same.

or 0.15 for the minimal loss).

Finally, when the economy faces both supply and asset prices shocks (Tables 5, 6 and7), the optimal strategy for the moderate central bank, as well as for the central bank concerned with activity stabilization, is to have a strong reaction to the output gap (β y equal to 1.3 and 1.5 respectively). When the central banks has a strong preference for price stability, the minimal loss is obtained when responding aggressively to inflation in the augmented Taylor rule. Note that responding to the bubble (increasing values of β b ) never leads to minimizing the loss function.

This analysis confirms that leaning against the wind is not the optimal strategy to stabilize the economy. The loss function is always minimized for small or null values of the bubble coefficient in the augmented Taylor rule. Note: The loss functions are computed using the variances of inflation, output gap and the bubble following bubble shocks. The response to output gap varies between 0.5 and 1.5 and all other parameters remain the same.

Figure (1f)

Robustness to an alternative selection of shocks

  nominal interest rate under the central bank's control. The εs are exogenous shocks normally distributed, and E t denotes the expectation operator. α, λ, δ, σ, ϕ, β i , β π , β y , and β b are the model's parameters. Equation (4) represents a hybrid New Keynesian Phillips curve where current inflation is a function of both past and expected inflation, and current output gap. The hybrid IS curve (equation (5)) describes the current output gap as a function of its lagged and expected values, and the real interest rate. This equation differs from the conventional specification of the IS curve because of the inclusion of a new term: the bubble. Equation (5) assumes a positive relationship between asset price deviations from fundamentals and the output gap.

Figure 1 :

 1 Figure 1: Model's response to bubble shocks deviation from the steady-state

Figure 2 :

 2 Figure 2: Variances following supply shocks when the inflation coefficient varies

Figure 3 :

 3 Figure 3: Variances following bubble shocks when the bubble coefficient varies

Figure 4 :

 4 Figure 4: Variances following bubble shocks when the output gap coefficient varies

Figure 5 :

 5 Figure 5: Variances following supply and bubble shocks when the output gap coefficient varies

Figure 6 :

 6 Figure 6: Variances following supply and bubble shocks when the bubble coefficient varies

Figure 7 :

 7 Figure 7: Variances following supply and bubble shocks when the inflation coefficient varies

15 Appendix 2 :

 152 Loss function analysisWe present hereafter the results of our baseline analysis in terms of loss functions. The following tables correspond to Figures 2 to 7 in the main text. They give the values of the loss function computed using the variances of inflation and of the output gap. Three versions of the loss functions are computed. The first one assumes equal weights on inflation and output gap volatilities (0.5 -0.5). The second one, closer to an inflation targeting central bank, supposes that the monetary authority values more inflation variance (0.9) , than output gap volatility (0.1). Last, the central bank is more interested in stabilizing output (with a weight of 0.9) than inflation (0.1). Minimal values appear in bold.

Table 3 :

 3 β b ) does not lead to minimizing losses in either scenario considered (the value of β b is either 0 Losses in case of asset price bubble shocks (1) loss functions are computed with the variances of inflation, output gap and the bubble following bubble shocks. The response to the bubble varies from 0 to 0.75 and all other parameters remain the same.

  loss functions are computed with the variances of inflation, output gap and the bubble following bubble shocks. The response to the bubble varies from 0 to 0.75 and all other parameters remain the same.

Figure ( 1b )

 1b Figure (1b) Variances following bubble shocks when the bubble coefficient varies

Figure ( 1c )

 1c Figure (1c) Variances following bubble shocks when the output gap coefficient varies

Figure ( 1d )

 1d Figure (1d) Variances following supply and bubble shocks when the output gap coefficient varies

Figure ( 1e )

 1e Figure (1e) Variances following supply and bubble shocks when the bubble coefficient varies

Figure ( 2 )

 2 Figure (2) Variances following bubble shocks (Response to the bubble [left] and to the output gap [right])

Figure ( 3 )

 3 Figure (3) Bubble shocks (Response to the bubble [left] and to the output gap [right])

Figure ( 4a ) 36 Figure ( 4b )

 4a364b Figure (4a) Variances following supply shocks when the inflation coefficient varies

Figure ( 4c )

 4c Figure (4c) Variances following bubble shocks when the output gap coefficient varies

Figure ( 4d )

 4d Figure (4d) Variances following supply and bubble shocks when the output gap coefficient varies

Figure ( 4e )

 4e Figure (4e) Variances following supply and bubble shocks when the bubble coefficient varies

Figure ( 4f )

 4f Figure (4f) Variances following supply and bubble shocks when the inflation coefficient varies

Table 1 :

 1 Estimated parameters of a forward-looking model for the euro area economy

	Output equation
	y t = 0.55	E t (y t+1 ) + 0.45 y t-1 -0.84	i t -E t (π t+1 ) + 0.15	b t + ε y t
	(0.06)	(0.21)	(0.08)
	Inflation equation
	π t = 0.49	E t (π t+1 ) + 0.51 π t-1 + 0.22 y t + ε π t
	(0.16)	(0.02)

Table 2 :

 2 Losses in case of supply shocks

	Parameter	Loss function	Loss function	Loss function
	value	(0.5 -0.5)	(0.9 -0.1)	(0.1 -0.9)

Table 4 :

 4 Losses in case of asset price bubble shocks[START_REF] Agenor | Capital Regulation, Monetary Policy and Financial Stability[END_REF]

	Parameter	Loss function	Loss function	Loss function
	value	(0.5 -0.5)	(0.9 -0.1)	(0.1 -0.9)

Table 5 :

 5 Losses in case of supply and asset price bubble shocks[START_REF] Villieu | Financial Intermediaries and Monetary Economics[END_REF]

	Parameter	Loss function	Loss function	Loss function
	value	(0.5 -0.5)	(0.9 -0.1)	(0.1 -0.9)
	β y = 0.5	0,1567	0,0863	0,2270
	β y = 0.7	0,1398	0,0960	0,1836
	β y = 0.9	0,1308	0,1075	0,1540
	β y = 1.1	0,1263	0,1198	0,1329
	β y = 1.3	0,1249	0,1324	0,1174
	β y = 1.5	0,1253	0,1450	0,1057

Table 6 :

 6 Losses in case of supply and asset price bubble shocks[START_REF] Agenor | Capital Regulation, Monetary Policy and Financial Stability[END_REF]

Table 7 :

 7 Losses in case of supply and asset price bubble shocks[START_REF] Badarau | Monetary policy and credit cycles: A DSGE analysis[END_REF]

	Parameter	Loss function	Loss function	Loss function
	value	(0.5 -0.5)	(0.9 -0.1)	(0.1 -0.9)
	β π = 1.5	0,1567	0,0863	0,2270
	β π = 1.7	0,1655	0,0819	0,2492
	β π = 1.9	0,1746	0,0790	0,2702
	β π = 2.1	0,1835	0,0770	0,2901
	β π = 2.3	0,1923	0,0757	0,3089
	β π = 2.5	0,2008	0,0748	0,3268

Note: The loss functions are computed with variances of inflation, output gap and the bubble following supply shocks. The response to inflation varies between 1.5 and 2.5 and all other parameters remain the same.

Note: Variances of inflation, output gap and the bubble following supply shocks. The response to inflation varies from 1.5 to 2.5 and all other parameters remain the same. The arrows indicate an increase in βπ.

0.32 0.34 0.36 y 2 Output -Bubble Note

  Variances following supply and bubble shocks when the inflation coefficient varies : Variances of inflation, output gap and the bubble following supply and bubble shocks. The response to inflation varies from 1.5 to 2.5 and all other parameters remain the same. The arrows indicate an increase in βπ.

	0.07		Inflation -Output			0.07		Inflation -Bubble
	0.065						0.065
	0.06							0.06
	2						2
	0.055						0.055							0.3
	0.05							0.05							0.28
	0.045						0.045							0.26
	0.04							0.04							0.24
	0.24	0.26	0.28	0.3	0.32	0.34	0.36	0.53	0.54	0.55	0.56	0.57	0.58	0.59	0.53	0.54	0.55	0.56	0.57	0.58	0.59
				2							2							2
				y							b							b

The Taylor curve depicts the standard central bank's objectives, namely inflation and output stabilization, and suggests that a trade-off can emerge between these objectives in the context of supply shocks.

For an empirical assessment of trade-offs, see[START_REF] End | Trading off monetary and financial stability: a balance of risk framework[END_REF].

[START_REF] Bordo | Boom-busts in asset prices, economic instability, and monetary policy[END_REF] and[START_REF] Kent | Asset-price bubbles and monetary policy[END_REF] are also close to our paper.[START_REF] Bordo | Boom-busts in asset prices, economic instability, and monetary policy[END_REF] use a more complex model, where private agents' credit is taken into account. They study the dilemma of whether the central bank should lean or clean by considering non-linear Taylor type rules that depend on both macroeconomic fundamentals and private sector expectations. Kent and Lowe (1997) use a model where, the probability of the asset price bubble to burst is influenced by the level of the interest rate. We argue that it is rather changes in interest rates that impact this probability and less the levels, but propose a robustness test on this issue. Moreover, in their model, the bubble only emerges once and no other bubbles occur after its collapse, whereas we consider the reaction of the economy to successive bubbles.

[START_REF] Mallick | Monetary and financial stability in the euro area: Pro-cyclicality versus trade-off[END_REF] also make the assumption in their New Keynesian model that asset prices influence demande. Empirical evidence on finacial stress conditions influencing output fluctuations can be found in[START_REF] Mallick | The real effects of financial stress in the Eurozone[END_REF].

A change in the monetary policy stance can affect the behavior in the financial market by diverting investments toward different (more profitable) segments of the market, or simply by restricting access to liquidity.

Although we do not provide micro-foundations for including the bubble in the demand equation, the very intuitive arguments we develop support such a framework. Note however that given the model's specification where the Phillips curve is a function of the aggregate demand, in the end the bubble also affects (indirectly) the aggregate supply.

When the dummy is 0, the Blanchard-Kahn conditions are verified and the model is stable. For D = 1, the model is unstable due to the bubbles equation specification. However, in the long run the economy converges in probability towards the steady state.

It is assumed that the effect of the interest rate on the bubble is of the same magnitude as the effect of the real interest rate on the output gap. It is also assumed that the asset price bubble affects the aggregate demand to a lesser extent than the real interest rate.

The shocks are drawn from the standard normal distribution.

Considering household indebtedness as an indicator of financial stability,[START_REF] Svensson | Leaning Against the Wind" Leads to Higher (Not Lower) Household Debt-to-GDP Ratio[END_REF] shows that a tighter monetary policy to control the level of debt leads to an increase in the debt to GDP ratio. Tightening monetary policy more than necessary for inflation stabilization, will raise the real household debt and dampen the nominal GDP, increasing the debt to GDP ratio.

A study based on loss functions driven from these results is presented in Appendix 2.

for insightful comments and suggestions on an earlier draft of this paper. We are indebted to Gregory Levieuge, Olivier Pierrard, and the participants at seminars at the University of Orleans and University of Luxembourg, as well as participants at the 16th International Network for Economic Research (INFER) annual conference, for fruitful discussions on several aspects of the paper. We are grateful to the INFER committee for the INFER Research Prize awarded to this paper.

Appendix 1: Parameter estimates and model calibration

This appendix first provides a very brief description of the estimation of misalignments in the equity market, and then the results of the macro-model estimated for the euro area.

As discussed in the main text, we use the APM (based on the Arbitrage Pricing Theory, [START_REF] Ross | The arbitrage theory of capital asset pricing[END_REF] approach to estimate excess returns in the equity market as a function of surprises due to shocks on various fundamentals. Especially, the estimated equation takes the form of :

where excess R is the excess return defined as the difference between the MSCI and the risk-free rate (3-month rate), and X is the vector of fundamentals. X includes: the change in the VIX index (confidence risk), the spread between the long-term and the short-term rate (time horizon risk), change in the industrial production index, inflation, change in the nominal exchange rate, commodity price index, and volatility of daily equity returns. The equation is estimated with simple (robust) OLS, using monthly data. Fair values of the equity indices are estimated as the sum of the risk-free rate and the estimated excess return. The misalignments (under-or overvaluations) are the differences between the observed MSCI index and its estimated fair value. Because of limited data availability, this is done only for 9 Euro area countries, 17 and the average is computed and used for the model calibration described below.

Regarding the parameters estimates, we follow Smets (2000) and use a GMM estimator with monthly data over 1999M09-2007M12. Results are reported hereafter.

We do not estimate the Taylor rule as the purpose of our study is precisely to analyze different reactions of the central bank. This translates in considering several values for each β parameter. Hence we depart from standard values -β π = 1.5 and β y = 0.5 -and gradually increase each of them to simulate more aggressive reactions to shocks. Moreover, given that central banks did not respond to bubbles during our sample period, no value can be found in the literature for the parameter β b . However, even if central banks decide to react to bubbles their primary objectives remain price and output stability, implying a smaller reaction to the 17 Austria, Belgium, France, Germany, Ireland, Italy, Netherlands, Spain, Sweden.


Mathias Pacaud

Katel Hervé-Aubert

Martin Soucé

Alaa Abdelrahman Makki

Franck Bonnier

Amir Fahmi

Alexey Feofanov

Igor Chourpa
email: igor.chourpa@univ-tours.fr.

One-step synthesis of gold nanoflowers of tunable size and absorption wavelength in the red & deep red range for SERS spectroscopy

Keywords: gold nanoflower AuNF, surface plasmon resonance (LSPR) band, surface-enhanced Raman scattering (SERS) spectroscopy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Gold nanoflowers (AuNF) have emerged during the past decade as novel plasmonic nanosystems absorbing light in the red -deep red range (600-900 nm, a part of the "tissue optical window" between 600 and 1200 nm) [START_REF] Norton | Vo-Dinh Theoretical Models and Algorithms in Optical Diffusion Tomography Biomedical Photonics Handbook[END_REF][START_REF] Weissleder | A clearer vision for in vivo imaging[END_REF][START_REF] Smith | Bioimaging: second window for in vivo imaging[END_REF]. The AuNF are interesting for many applications [START_REF] Sajanlal | Pradeep Anisotropic nanomaterials: structure, growth, assembly, and functions[END_REF][START_REF] Gong | Individual nanostructured materials: fabrication and surface-enhanced Raman scattering[END_REF], in particular as agents for photoinduced heating and substrates for surface-enhanced Raman scattering (SERS) spectroscopy [START_REF] Jamieson | Bioanalytical Measurements Enabled by Surface-Enhanced Raman Scattering (SERS) Probes[END_REF]. Due to its high sensitivity, molecular specificity and selectivity, SERS spectroscopy is a rapidly developing analytical and bio-analytical technique. SERS excitation in the 630-850 nm region is favourable for bio-analytical SERS applications, because this wavelength range allows a compromise between minimal light interaction with biological tissues [START_REF] Norton | Vo-Dinh Theoretical Models and Algorithms in Optical Diffusion Tomography Biomedical Photonics Handbook[END_REF][START_REF] Weissleder | A clearer vision for in vivo imaging[END_REF] and the good performance of laser sources and CCD detectors [START_REF] Smith | Bioimaging: second window for in vivo imaging[END_REF].

The SERS effect is observed for organic molecules situated in the vicinity of a nanostructured surface of a noble metal, often in the form of nanoparticles (NP), which displays collective oscillations of surface electrons called plasmons [START_REF] Jamieson | Bioanalytical Measurements Enabled by Surface-Enhanced Raman Scattering (SERS) Probes[END_REF]. The SERS effect implies that the excitation wavelength coincides with the localized surface plasmon resonance (LSPR) band of the noble metal NP. Since isotropic smooth AuNP have their LSPR band around 530 nm, their use for SERS with higher wavelength excitation sources is possible only in the case of NP aggregation. For aggregation-free LSPR absorption in the red-deep red range, anisotropic gold nanosystems like nanorods (AuNR), nanostars (AuNS) and nanoflowers (AuNF) are being developed [START_REF] Sajanlal | Pradeep Anisotropic nanomaterials: structure, growth, assembly, and functions[END_REF][START_REF] Gong | Individual nanostructured materials: fabrication and surface-enhanced Raman scattering[END_REF][START_REF] Fratoddi | How toxic are gold nanoparticles? The state-of-the-art[END_REF][START_REF] Rodriguez-Lorenzo | Surface Enhanced Raman Scattering Using Star-Shaped Gold Colloidal Nanoparticles[END_REF][START_REF] Yuan | Vo-Dinh Gold nanostars: surfactant-free synthesis, 3D modelling, and two-photon photoluminescence imaging[END_REF][START_REF] Li | Tan Gold-Coated Fe3O4 Nanoroses with Five Unique Functions for Cancer Cell Targeting[END_REF][START_REF] He | Optimizing gold nanostars as a colloid-based surface-enhanced Raman scattering (SERS) substrate[END_REF][START_REF] De Silva Indrasekara | Vo-Dinh Manipulation of the Geometry and Modulation of the Optical Response of Surfactant-Free Gold Nanostars: A Systematic Bottom-Up Synthesis ACS Omega[END_REF][START_REF] Jena | Surfactantless Room Temperature Synthesis of Single Crystalline Fluorescent Gold Nanoflowers with Pronounced SERS and Electrocatalytic Activity[END_REF]. It has to be noted that the morphology of some short-tipped AuNS is very similar to that of AuNF. Unfortunately, AuNR and some long-spiked AuNS may have an intrinsical biotoxicity [START_REF] Fratoddi | How toxic are gold nanoparticles? The state-of-the-art[END_REF], due to their morphology [START_REF] Rodriguez-Lorenzo | Surface Enhanced Raman Scattering Using Star-Shaped Gold Colloidal Nanoparticles[END_REF] and/or due to the presence of surfactants like cetrimonium bromide (CTAB) employed in their formulation. Indeed, surfactant-free AuNS synthesized by Yuan et al. [START_REF] Yuan | Vo-Dinh Gold nanostars: surfactant-free synthesis, 3D modelling, and two-photon photoluminescence imaging[END_REF] had reduced toxicity and permitted an easy surface functionalization. Thus, surfactant-free synthesis of AuNF has to be preferred for applications like biosensing.

Although a certain number of protocols of generating AuNF and AuNS has been reported by various research groups [START_REF] Sajanlal | Pradeep Anisotropic nanomaterials: structure, growth, assembly, and functions[END_REF][START_REF] Yuan | Vo-Dinh Gold nanostars: surfactant-free synthesis, 3D modelling, and two-photon photoluminescence imaging[END_REF][START_REF] Li | Tan Gold-Coated Fe3O4 Nanoroses with Five Unique Functions for Cancer Cell Targeting[END_REF][START_REF] He | Optimizing gold nanostars as a colloid-based surface-enhanced Raman scattering (SERS) substrate[END_REF][START_REF] De Silva Indrasekara | Vo-Dinh Manipulation of the Geometry and Modulation of the Optical Response of Surfactant-Free Gold Nanostars: A Systematic Bottom-Up Synthesis ACS Omega[END_REF][START_REF] Jena | Surfactantless Room Temperature Synthesis of Single Crystalline Fluorescent Gold Nanoflowers with Pronounced SERS and Electrocatalytic Activity[END_REF] they typically involve at least two synthesis steps, corresponding to a nucleation-growth mechanism [START_REF] Turkevich | A study of the nucleation and growth processes in the synthesis of colloidal gold Discuss[END_REF][START_REF] Frens | Controlled Nucleation for the Regulation of the Particle Size in Monodisperse Gold Suspensions[END_REF]. The first step consists in introducing some seeds (silver [START_REF] De Silva Indrasekara | Vo-Dinh Manipulation of the Geometry and Modulation of the Optical Response of Surfactant-Free Gold Nanostars: A Systematic Bottom-Up Synthesis ACS Omega[END_REF], gold [START_REF] Yuan | Vo-Dinh Gold nanostars: surfactant-free synthesis, 3D modelling, and two-photon photoluminescence imaging[END_REF] or magnetic NPs [START_REF] Li | Tan Gold-Coated Fe3O4 Nanoroses with Five Unique Functions for Cancer Cell Targeting[END_REF] of diameters ca. 5 nm), either synthesized independently or in situ. Nucleation of silver and gold seeds is usually reached with one of the stronger reducing agents such as NaBH4 [START_REF] Sajanlal | Pradeep Anisotropic nanomaterials: structure, growth, assembly, and functions[END_REF][START_REF] Gong | Individual nanostructured materials: fabrication and surface-enhanced Raman scattering[END_REF]. The second synthesis step is intended to favour slow anisotropic growth of AuNF or AuNS around the seeds, using a mild reducing agent like ascorbic acid or hydroxylamine.

Size, shape and surface properties are usually controlled by the amount and nature of reducing agents and their ratio to the Au precursor. Anisotropic Au nanosystems can be synthesized in the presence of organic solvents and/or of capping agents like silver or other metal ions, surfactant molecules or polymers [START_REF] Sajanlal | Pradeep Anisotropic nanomaterials: structure, growth, assembly, and functions[END_REF][START_REF] Gong | Individual nanostructured materials: fabrication and surface-enhanced Raman scattering[END_REF]. The latter are intended to direct the anisotropic growth along specific crystallographic directions or to stabilize the surface. As a result, prior to use of those AuNF or AuNS for SERS, further post-synthesis steps are usually required, in order to remove the solvents, the excess of seeds and reagents [START_REF] Sajanlal | Pradeep Anisotropic nanomaterials: structure, growth, assembly, and functions[END_REF][START_REF] Gong | Individual nanostructured materials: fabrication and surface-enhanced Raman scattering[END_REF]. Only few seed-less [START_REF] He | Optimizing gold nanostars as a colloid-based surface-enhanced Raman scattering (SERS) substrate[END_REF] or surfactant-free [START_REF] Yuan | Vo-Dinh Gold nanostars: surfactant-free synthesis, 3D modelling, and two-photon photoluminescence imaging[END_REF][START_REF] De Silva Indrasekara | Vo-Dinh Manipulation of the Geometry and Modulation of the Optical Response of Surfactant-Free Gold Nanostars: A Systematic Bottom-Up Synthesis ACS Omega[END_REF] approaches have been reported so far. For instance, He et al. [START_REF] He | Optimizing gold nanostars as a colloid-based surface-enhanced Raman scattering (SERS) substrate[END_REF] have proposed a one-pot seedless protocol for the synthesis of AuNS, using ascorbic acid in the presence of silver nitrate (AgNO3). The group of Vo-Dinh have published a systematic study of well-controlled geometry and optical response of gold nanostars using silver seeds but without surfactant [START_REF] De Silva Indrasekara | Vo-Dinh Manipulation of the Geometry and Modulation of the Optical Response of Surfactant-Free Gold Nanostars: A Systematic Bottom-Up Synthesis ACS Omega[END_REF].

To the best of our knowledge, the protocols of AuNF synthesis without seeds nor surfactant are even less common. For instance, Jena and Raj [START_REF] Jena | Surfactantless Room Temperature Synthesis of Single Crystalline Fluorescent Gold Nanoflowers with Pronounced SERS and Electrocatalytic Activity[END_REF] used 5-hydroxyindole-3-acetic acid to generate, at room temperature, flower-like gold nanoparticles of a large, ca. 150 nm size, surprisingly showing two absorption bands at 545 and 1250 nm. Thus, there is no 'gold standard method' and there is still a need of a more convenient, rapid and efficient protocol for seed-less and surfactant-free synthesis of AuNF of controlled size and tunable LSPR band position within the 600-800 nm range.

In response to this need, we propose here a novel one-step synthesis method based on a mild chemistry in aqueous medium at room temperature.

Materials and methods

Chemicals

Gold (III) chloride trihydrate (AuHCl4.3H2O) ( ≥ 99.9%), trisodium citrate dihydrate (HOC(COONa)(CH2COONa)2 • 2H2O) ( ≥ 99.0%), sodium hydroxide (NaOH) ( ≥ 97.0 %) and Crystal Violet were purchased from Sigma-Aldrich (Saint Quentin Fallavier, France). Hydroxylamine hydrochloride (NH2OH • HCl) (≥ 99.0%), and Polyvinylpyrrolidone (PVP, Mw 7,000 -11,000) were purchased from VWR (Fontenay-sous-Bois, France) and BASF (Levallois-Perret, France), respectively.

Nile blue A, pure certified, was purchased from Acros Organics, (New Jersey, USA). All the reagents were used without further purification. Ultrapure water (Milli-Q) was used for the preparation of all solutions.

AuNF synthesis

Gold nanoflowers synthesis was carried out as follows: to 20 mL of water under vigorous magnetic stirring, aqueous solutions of hydroxylamine (0.1 M, volumes 0.3, 0.4, 0.45, 0.54, 0.56 or 0.64 mL), trisodium citrate (0.1 M, 1.0 mL) and sodium hydroxide (0.01 M, volumes 1.0, 1.0, 0.9, 0.9, 0.8 or 0.8 mL) were added. Upon addition of a freshly prepared aqueous solution of HAuCl4 (0.01 M; 1.0 mL), the initial colourless solution immediately turned dark blue. The resultant colloid was stirred for ca. 30 seconds and transferred to a hermetically closed tube protected from light with aluminium foil. The final solution was kept at 20 ± 1 °C. In some cases, 0.25 mL of PVP aqueous solution (final concentration 13.5 µM) were added for better stability of the colloids.

Instrumental

UV-visible absorption spectra were acquired using a Genesys 10S UV-vis spectrophotometer (Thermo Scientific, France).

IR spectra were acquired with a Frontier spectrometer (Perkin Elmer, France) equipped with a Quest multiple reflection diamond attenuated total reflectance (ATR) accessory (Specac, UK) using dried 2 µL drops of NF suspension and citrate solution.

TEM experiments were carried out on a transmission electron microscope JEOL JEM-1011 (JEOL Inc., Peabody, MA, USA) operating at an acceleration voltage of 100 kV. AuNF in aqueous suspension were deposited on carbon-coated Cu grids. Excess water was removed by filter paper. The obtained samples were left to dry under ambient air. Image J software was used to measure the size distribution of AuNF, and for each particle, two perpendicular dimensions were measured. For each batch, the weighted average size was determined as the center of mass of the corresponding size distribution, calculated according to the equation:

= ∑ ∑
where Di is AuNF dimension range (0-10 nm presented as 5 nm, 10-20 nm presented as 15 nm, … 140-150 nm presented as 145 nm), where Ni is the number of AuNF in the dimension range Di.

The mean hydrodynamic diameter DH values (z-average) were measured using dynamic light scattering (DLS, Malvern HPPS, Malvern Instruments, UK), at 25°C and with a He-Ne laser (4 mW, 633 nm, scatter angle 173°). Each measurement was done in triplicate.

The SERS spectra were recorded with a confocal Raman microspectrometer LabRam (Horiba SA, France) equipped with an EM CCD air-cooled detector, a 690 nm laser source (RO691-PLR45, Ondax Inc, Monrovia, CA, USA) providing 26.9 mW on the sample. The spectra were collected via a 10× objective focused on the top of the sample solution (200 µL) placed in a well of a P96 multiwell plate.

The spectral data acquisition and treatment were done using a LabSpec software (Horiba SAS, Villeneuve d'Ascq, France).

Results and discussion

AuNF synthesis

Our protocol uses optimized quantities of well-known reagents that are typical for gold reduction (Scheme 1): hydroxylamine (NH2OH, Solution 1) as a reductant, trisodium citrate (Na3Citrate, Solution 2) as a stabiliser for particles, sodium hydroxide (NaOH, Solution 3) as a main pH regulator and tetrachloroauric(III) acid (HAuCl4, Solution 4) as a gold precursor (see Materials and methods for more details). The reagent concentrations and volumes are easy to remember, as described below. First of all, aqueous solutions of all the reagents are prepared, at concentrations of 0.1 M for the Solutions 1 -3 and 0.01 M for the Solution 4. The preparation is started by adding the Solutions 1 (0.3 -0.65 mL), 2 (1 mL) and 3 (0.8 -1 mL) to 20 mL of water under vigorous stirring at ambient temperature (21 ± 1 °C). The volumes of the Solutions 1 and 3 are varied in order to adjust the NH2OH/NaOH molar ratio (hereafter called R) to the values of 0.3, 0.4, 0.5, 0.6, 0.7, 0.8. Then, 1 mL of the Solution 4 is injected into this mixture: the colourless solution immediately turns dark blue (Scheme 1A) and, after ca. 30 seconds, the stirring can be stopped. No washing steps are needed, the AuNF colloids are useable for SERS spectroscopy as is, and are chemically and physically stable for several days, till about a week for certain batches. Nevertheless, they have to be protected against light by aluminium foil and stored hermetically closed to minimize their oxidation.

In the context of the nucleation-growth mechanism described above, our rapid AuNF synthesis seems to correspond to almost instantaneous nucleation of seeds (Step 1 in Scheme 1B) followed by appearance of anisotropic structures (Step 2) and their fast growth to give nanoflowers (Step 3) with characteristic petal-like structures (Scheme 1 C). This rapid growth in presence of citrate and chloride anions adsorbed on the gold surface may be the reason of its anisotropic nature in the absence of silver ions and surfactants.

Variations in citrate or gold concentration around the optimum value (10:1 molar ratio) do not allow controlling the LSPR position. This behaviour is different from that described by Ji et al [START_REF] Ji | Size Control of Gold Nanocrystals in Citrate Reduction: The Third Role of Citrate[END_REF] who had shown that the HAuCl4/Na3Ct ratio affected both the pH and the growth of isotropic gold nanocrystals at elevated temperatures. In our study, synthesis without any citrate leads to colloids of reddish colours, i.e. with the LSPR band below 600 nm. Therefore, the presence of citrate is mandatory for successful AuNF synthesis by our method.

AuNF characterisation

The ratio R is the most critical parameter that affects the LSPR band position. An increase in R from 0.3 to 0.8 leads to a red shift of the LSPR band from ca. 620 to ca. 800 nm (Fig. 1). Between 0.4 and 0.8, the LSPR shift is a nearly linear function of R (Fig. 1B) and it is possible to adjust the LSPR position to any of three wavelengths of the laser sources most widely used in bio-analytical spectroscopy: 633 nm (R=0.4), 690 nm (R=0.6) and 785 nm (R=0.8). The LSPR band position was reproducible with a ±5 nm variation. The maximum absorbance for these LSPR bands was between 1.4 and 1.7.

The pH plays a very important role in such a synthesis. During the reaction, it affects the reactivity of gold complexes [START_REF] Goia | Matijević Tailoring the particle size of monodispersed colloidal gold Colloids and Surfaces[END_REF] and therefore, the kinetics of nanocrystal nucleation and growth.

The high pH conditions are known to be favourable for gold reduction. Prior to the addition of gold precursor, the pH of the solutions ranges from 11.5 to 10.6 as a function of R (0.3 to 0.8), as expected from the respective concentrations of NH2OH.HCl and NaOH.

After the AuNF synthesis, the final pH decreases for all the colloids, down to about 10.9 for R = 0.3 and to 6.7 for R = 0.8. This is in agreement with the fact that both auric acid dissociation and its reduction generate H + species [START_REF] Soni | Mehrotra Kinetics and mechanism of oxidation of hydroxylamine by tetrachloroaurate(III) ion Transition Metal Chemistry[END_REF]. Counting only dissociation of auric acid, pH should still remain well above 10, permitting a rapid onset of reaction. The H + ions liberated due to the redox reaction further lower pH, reaching almost neutral for conditions of high R. At all the pH values described above the citrate is in anionic form (trivalent or partially divalent) to be adsorbed on the gold surface, thus playing a role both in the AuNF growth and colloidal stability. The presence of citrates in the AuNF batches can be confirmed by the characteristic bands of citrates the IR spectra (Figure SI2).

According to transmission electron microscopy (TEM) data (Fig. 2), the red shift of the LSPR band corresponds to bigger AuNF with an increased size and/or number of the 'petals' giving them a flower-like morphology. Coupling of plasmons of neighbouring petals is supposed to favour formation of 'hot spots' where the SERS effect is particularly strong. As seen from the TEM images and the histograms in Fig. 2, the size distribution was narrower in the populations of smaller AuNFs.

The centre of mass of the histograms corresponds to the size of 49, 56, 75 and 89 nm for the R values of 0.3, 0.5, 0.6 and 0.8, respectively. The hydrodynamic diameter DH (z-average data from DLS), which includes the hydration layers around AuNF, increases from ca. 84 nm to ca. 112 nm.

As noticed above, the colloids are stable for about a week. However, the AuNF batches of higher R (bigger size and lower pH) show more pronounced sedimentation with time and are more prone to aggregation with time.

It is possible to stabilize the colloids of AuNF for at least a month by post-synthesis addition of 13.5 µM of PVP. As shown in Fig S1 (SI), PVP reduces the absorbance at the longwavelength side of the LSPR band, presumably due to re-dispersion of some aggregates. As expected, the effect of PVP on the LSPR band shape is more significant for the bigger AuNFs at nearly neutral pH (absorption maximum around 690 nm and above, Fig. SI 1A) while it has almost no effect for the colloids of smaller AuNF at higher pH (absorption maximum at 620-660 nm, Fig. SI 1B). While useful to store big AuNF colloids for an extended time, the postsynthesis addition of PVP was optional, taking into account the ease and rapidity of producing a fresh batch whenever needed.

Analytical validation of the AuNFs

For the SERS validation experiments with a 690 nm laser excitation, the most appropriate AuNFs have the LSPR band maximum at ca. 690 nm (R = 0.6, pH 7.2). For this validation, we intentionally selected chromophores void of thiol groups and therefore having much lower affinity to the gold surface than rhodamine [START_REF] Jaworska | Baranska Rhodamine 6G conjugated to gold nanoparticles as labels for both SERS and fluorescence studies on live endothelial cells Mikrochim[END_REF] and thiolated dyes [START_REF] Gühlke | Kneipp Surface-Enhanced Raman and Surface-Enhanced Hyper-Raman Scattering of Thiol-Functionalized Carotene[END_REF] often used in the literature.

The sample preparation for SERS on our AuNF is very convenient: a 10-fold dilution of the analyte solution in the AuNF colloid placed into a well of a 96-multiwell plate. In this manner, we successfully obtained SERS spectra of two blue dyes, Nile Blue and Crystal Violet (absorption maxima in water at 635 and 590 nm, respectively) (Fig. 3A).

It has to be underlined that added PVP did not give its own signal and did not change the SERS signal of the NB dye (Fig. 3B). This is true under both neutral and basic conditions. As expected the pH increase from neutral to basic conditions affects the relative intensity of SERS bands of NB and contributes to a decrease of the fluorescence background. The only effect noticeable with PVP was seen as a need of about 5-10 min delay after the NB incubation to observe an optimal SERS/fluorescence ratio. This time would correspond to the diffusion of the dye trough the polymeric layer prior to reach the gold surface. This shows the interest of a substrate with a minimal presence of polymer or surfactant:

(i) to reduce its effect on the adsorption and thus the detection of analytes [START_REF] He | Optimizing gold nanostars as a colloid-based surface-enhanced Raman scattering (SERS) substrate[END_REF] and (ii) to facilitate further post-functionalization with some molecules of interest.

On the AuNFs coated with PVP, the SERS intensity was a linear function of the NB dye concentration from nanomolar to micromolar range (Fig. 4A). Using a rapid SERS spectra acquisition (16x1 sec), we observed the detection limits situated in the nanomolar range (2.5 nM for NB -Fig. 4B) while it was possible to go below using longer acquisition times. As shown in Fig. 4B, the signal stays the same (the fingerprint bands are recognizable), just noise goes up on dilution.

Conclusions

In conclusion, the novel protocol we propose herein presents several significant advantages compared to those previously published for synthesis of AuNF: in addition to be facile and rapid (one-step, seed-less), it avoids the presence of organic solvents or surfactants and of other metals than gold. Moreover, our method permits to tune the size of AuNFs between 50 and 90 nm and the corresponding LSPR band position between 620 and 800 nm. The most critical parameter of the synthesis has been determined to be the hydroxylamine/NaOH molar ratio, which regulates both the reductant content and the pH. Obviously, for a good reproducibility of the data, the temperature has to be controlled, since it plays a role in the reaction kinetics. As noted above, AuNF were not formed without addition of citrate.

Together with citrates, chloride ions binding to the surface may also be essential to the anisotropic structure of the surface. The surfactant-free colloids are stable in water for several days and addition of a low concentration of PVP improves their long-term stability without affecting their aggregation-free SERS activity for organic dyes with relatively low affinity for gold. We believe that such a user-friendly wet chemistry protocol for rapid synthesis of AuNF should have very promising perspectives for bio-analytical SERS spectroscopy as well as for other plasmonic methods in the red-deep red range.

Figure Captions Scheme 1 :

 1 Figure Captions

Figure 1 :

 1 Figure 1: (A) UV-vis spectra of AuNF synthesized using different R = NH2OH/NaOH molar ratio. (B) The LSPR band position vs R.

Figure 2 :

 2 Figure 2: Top: Selection of typical TEM images of AuNFs synthesized using R = 0.3, 0.5, 0.6 and 0.8. Bottom: size distribution histograms. The centre of mass of the respective histograms corresponds to the size of 49, 56, 75 and 89 nm.

Figure 3 :

 3 Figure 3: A. Chemical structures of two dyes, Crystal Violet (CV) and Nile Blue (NB) and their SERS spectra obtained on AuNFs with R = 0.6. B. Comparison of the SERS spectra of NB on AuNFs in absence and in presence of PVP at basic and neutral aqueous solutions.

Figure 4 :

 4 Figure 4: A. SERS intensity (area of Raman bands in the range between 250 and 1800 cm -1 ) vs NB concentration, from 2.5 to 100 nM (from 2.5 to 20 nM in the inset). The data presented are averages of three independent measurements. B. Typical SERS spectra of NB at micromolar and nanomolar concentrations. The intensities are normalized and offset for easier comparison.

Acknowledgements

The authors acknowledge funding from the following grants in the framework of the MINERVA project (ERA-NET RUS PLUS program): RFBR 16-54-76013 (Russia), 01DJ16019 (Germany) and MENESR N°321 (France). The authors are grateful to Dr. Stéphanie David for her help with experimental design, to Julien Burlaud-Gaillard (the IBiSA Electron Microscopy Facility, University of Tours) for TEM images and to Hiba Chamekh for technical assistance.

Conflicts of interests disclosure

The author declares no conflict of interest.