Frédérique Bec

Patrick Kanda
email: patrick.kanda@u-cergy.fr

Is inflation driven by survey-based, VAR-based or myopic expectations? An empirical assessment from US real-time data *

Keywords: VAR-based expectations, Myopic expectations, Survey forecasts, New Keynesian Phillips Curve. JEL Classification: E27, E31

The relative importance of survey-based, VAR-based or myopic expectations is evaluated in accounting for US inflation dynamics in a New Keynesian Phillips Curve (NKPC) setting. Our contribution is three-fold. First, we estimate the NKPC with both final and real-time vintage data in order to control for large revisions in the real GDP data. Second, we distinguish between two different series for VAR-based inflation forecasts -derived by a recursive or rolling-window method -to account for changes in the conduct and transmission mechanisms of US monetary policy after World War II. Third, joint restrictions are tested in the NKPC to assess whether one of the expectational variables is able, on its own, to capture inflation dynamics. On a statistical basis, we find that there is no clear-cut winner between VAR-and surveybased inflation expectations. Most of our estimated NKPC variants conclude that survey inflation expectations tend to have the largest numerical weight. Nevertheless, the difference between VAR-and survey-based expectations' estimated coefficients is not statistically significant. Moreover, myopic expectations do not play any significant role in the majority of the estimated NKPC variants.

Introduction

Expectations play a fundamental role in the study of inflation dynamics in the New Keynesian framework (see for instance [START_REF] Roberts | New Keynesian economics and the Phillips curve[END_REF][START_REF] Gali | Inflation dynamics: a structural econometric analysis[END_REF]. According to this strand of research, inflation depends on a measure of economic activity (usually marginal costs or the output gap) as well as expected inflation. That said, the nature of expected inflation remains a contentious issue. Typically, New Keynesian models are based on the rational expectation hypothesis which posits that economic agents' expectations correspond to the model-implied forecasts. 1 .

Despite the popularity and theoretical appeal of the rational expectations hypothesis, some studies2 challenge its relevance because New Keynesian models based on this assumption cannot replicate key inflation dynamics such as persistence and the cost of disinflation. A number of papers, such as [START_REF] Roberts | Is inflation sticky?[END_REF] or [START_REF] Mavroeidis | Empirical evidence on inflation expectations in the New Keynesian Phillips Curve[END_REF], show that rational expectations-based New Keynesian monetary models require ad hoc extensions such as consumption habit formation, investment adjustment costs, variable capital utilization, autocorrelated shocks as well as wage and price-setting decisions indexation to past inflation to capture key inflation dynamics properties. Yet, there seems to be no microeconomic relevance to these extensions, as extensively documented in [START_REF] Milani | The modeling of expectations in empirical DSGE models: a survey[END_REF]. Similarly, the rational expectations hypothesis suggests that all economic agents share the same information set and have the same model-consistent expectations. However, this assumption is at odds with studies from e.g. [START_REF] Mankiw | Disagreement about inflation expectations[END_REF] or more recently [START_REF] Andrade | Inattentive professional forecasters[END_REF] and [START_REF] Coibion | The formation of expectations, inflation and the Phillips curve[END_REF], which all point out that economic agents form (HAVE ?) heterogeneous expectations.

In the following, rational expectations will in general refer to theoretical model-based expectations. Nevertheless, it is worth noticing that some authors, like for instance [START_REF] Fuhrer | The role of expectations in inflation dynamics[END_REF]; [START_REF] Fuhrer | The role of expectations and output in the inflation process: an empirical assessment[END_REF], use the same terminology to designate a PAS DE TIRET theoretical model-based -typically VAR-based -expectation. Indeed, as advocated by [START_REF] Fuhrer | The role of expectations in inflation dynamics[END_REF], " (...) vector autoregressive equations allow us to form rational expectations of inflation without imposing further structure on the model. " (p.146) Some studies show that once the rational expectations hypothesis is replaced by an alternative expectations scheme such as adaptive learning by economic agents, as in [START_REF] Milani | Expectations, learning and macroeconomic persistence[END_REF], or by inflation forecasts from surveys as in [START_REF] Fuhrer | Expectations as a source of macroeconomic persistence: evidence from survey expectations in a dynamic macro model[END_REF], ad hoc extensions become redundant and the resulting model closely emulates empirical inflation properties.

It is against this backdrop that a line of research seeks to evaluate the relative contribution of various expectations formation schemes, such as rational or VAR-based, survey-based and myopic schemes, in explaining US inflation dynamics, within the New Keynesian Philips Curve (NKPC hereafter) framework. As will be seen in the next section, the conclusions are at best mitigated, if not contradictory. Our paper contributes to this strand of research, but departs from existing work in three directions.

Firstly, to our knowledge, our paper is the first one to use real-time data in this empirical literature. Secondly, special care is taken for the computation of VAR-based proxy of rational inflation expectations so as to accommodate the major changes which occurred since World War II in the conduct and propagation mechanisms of the U.S. monetary policy. More specifically, our empirical analysis will be conducted using both rolling and recursive computations of these expectations. Thirdly, by contrast with previous empirical studies, we proceed with systematic joint hypothesis tests to assess whether a mix of the VAR-based, survey and/or myopic expectations is needed or if only one of them is enough to capture inflation dynamics.

Our main finding is that VAR-based and survey inflation expectations contribute significantly to inflation dynamics. Using final vintage data, their relative contributions to inflation dynamics depends on whether recursive or rolling-window VAR-based inflation forecasts are used. On the contrary, estimations using real-time data consistently show that survey inflation expectations have the largest weight. However, estimates of VAR-based and surveybased inflation expectations contributions are not significantly different from each other: Our tests generally cannot reject the null hypothesis that both expectation variables estimated weights are equal. Hence, contrary to the conflicting outcome of existing studies3 , we cannot conclude that any one of the two types of forward-looking inflation expectations is enough on its own to capture inflation dynamics in the NKPC framework. Finally, myopic (i.e. backward-looking) inflation expectations mostly do not play any significant role in explaining US inflation dynamics.

The remainder of the paper is structured as follows: Section 2 offers a short overview of the recent related literature. Section 3 discusses the data and methodology while Section 4 presents the results. Section 5 concludes.

Related literature

A few decades ago, studies seeking to compare the relevance of different types of inflation expectations schemes used to consider separately the various types of forward-looking expectations -one at a time -in the NKPC inflation equation and assessed the ability of the corresponding model to capture inflation dynamics (see for example [START_REF] Roberts | New Keynesian economics and the Phillips curve[END_REF].

The strand of research investigating the relative contributions of various expectations schemes to US inflations dynamics is relatively new. It aims at bringing together different practices of benchmarking various measures of inflation expectations in the NKPC literature. Basically, the following hybrid formulation of the NKPC equation featuring heterogeneous inflation expectations is considered:

π t = β e E t π t+1 + β s S t π t+1 + β m π t-1 + γmc t + u t (1)
where π t is the inflation rate, E t π t+1 is the model-consistent (i.e. rational or VAR-based) expectation of inflation in period t + 1, formed in period t, S t π t+1 is the one-period-ahead inflation survey forecast as reported in period t, mc t is a measure of marginal costs and u t is a disturbance term.

Given that forward-looking inflation expectations are unobservable, different studies use different proxies. Mainly, three ways are used to circumvent the non-observability of expectations: (1) substitute inflation expectations for realized inflation and use instruments -the so-called Generalized Instrumental Variables (GIV) approach; (2) use a Vector Auto-Regression (VAR hereafter) model to derive inflation expectations; (3) use direct measures of inflation expectations obtained from surveys [START_REF] Mavroeidis | Empirical evidence on inflation expectations in the New Keynesian Phillips Curve[END_REF]. These three methods are featured in studies that investigate the relative contributions of different expectations schemes (e.g. rational, survey and myopic) in explaining US inflation dynamics in the NKPC framework (see for example [START_REF] Nunes | Inflation dynamics: the role of expectations[END_REF][START_REF] Fuhrer | The role of expectations and output in the inflation process: an empirical assessment[END_REF][START_REF] Fuhrer | The role of expectations in inflation dynamics[END_REF].

In this literature, forward-looking inflation expectations are composed of both rational inflation expectations -which are not observable -and survey counterparts. To deal with unobservable expectations, [START_REF] Nunes | Inflation dynamics: the role of expectations[END_REF] uses the GIV approach. On the other hand, [START_REF] Fuhrer | The role of expectations and output in the inflation process: an empirical assessment[END_REF] and [START_REF] Fuhrer | The role of expectations in inflation dynamics[END_REF] derive rational inflation expectations from a reducedform VAR model. These studies methods, data and results are summarized in Table 1. Based on NKPC estimations featuring both rational and survey expectations, Nunes (2010) finds weak evidence in favor of survey expectations but strong evidence in favor of rational expectations in explaining inflation dynamics: while the maximum estimated weight found for the former ( βs ) is 0.22, it ranges from 0.56 to 0.96 for the latter ( βe ), depending on the model and on the proxy chosen for the marginal cost in the NKPC, Equation (1). However, this argument clashes with Fuhrer and Olivei (2010)'s finding that the role of survey expectations slightly dominates that of rational counterparts: Depending on the period considered, these authors find a contribution of survey expectations ranging from 0.2 to 0.4 while the one of rational expectations is always 0.2. In the same perspective, Fuhrer (2012) finds overwhelming evidence that survey expectations play a more important role than (VAR-based) rational counterparts in inflation dynamics explanation. The myopic expectations contribute significantly in half the cases. All in all, columns labelled βe and βs in Table 1 emphasize that the relative contributions of rational and survey-based inflation expectations in explaining inflation dynamics is still debated. Furthermore, the column labelled E t π t+1 denoting inflation rational expectations, reveals that there is no consensus in this literature regarding the ideal proxy for rational inflation expectations.

Data and Methodology

In the NKPC Equation ( 1), the model-consistent inflation expectations and the marginal cost variables are not directly observable. Hence, they need to be proxied. There are almost as many different flavors of NKPC estimated equations as empirical contributions to this literature, since the latter use different proxies for the marginal cost and/or model-consistent inflation expectations. In this section, we first describe the data used to build these proxies and then present the methodology retained for the NKPC estimation.

Data

Real-time output gap: In this study, the output gap is used as a proxy for marginal costs in the NKPC Eq.(1). Our benchmark analysis relies on final vintage (FV hereafter) data, that is, the most up-to-date data for the real GDP. However, real GDP data are submitted to large revision every quarters for years. Hence, using final vintage of observations to compute the output gap used in the VAR and the NKPC equation could be misleading. To compute the inflation expectations recursively from 1981Q3 on, one would rather use the observations that were available back then. Indeed, [START_REF] Koenig | The use and abuse of real-time data in economic forecasting[END_REF] and [START_REF] Clements | Real-time forecasting of inflation and output growth with autoregressive models in the presence of data revisions[END_REF] argue that the use of the latest available vintage data leads to an overestimation of independent variables' power to predict the dependent variable. As such, the authors advocate the use of real-time data. Furthermore, data revisions have been shown to have an impact on economic agents' expectation formation as well as the conduct of monetary policy and the response of policy to uncertainty (see [START_REF] Croushore | A real-time data set for macroeconomists[END_REF], for a review). Hence, the Philadelphia Fed's real-time database is used to extract real-time data for the real GDP. This database consists of quarterly snapshots or "vintages" of key macroeconomic variables. A vintage refers to data series on a variable as it appeared to an analyst at a specific point in time. For any given vintage date, the series are exactly those an analyst would have observed in published sources at that particular date (see Table A1 in the Appendix). To illustrate this, for each vintage at time t (in quarters), the series runs from 1947Q1 to time t -1. The combination of different vintages forms a real-time dataset. The first vintage date in the real-time database is 1965Q4. The Philadelphia Fed's real-time dataset comprises data as they appeared in the middle of each quarter. In fact, the timing of the real-time dataset was set so as to match the timing of the Survey of Professional Forecasters [START_REF] Croushore | A real-time data set for macroeconomists[END_REF]. Hence, at every date within the sample, we use the most up-to-date available estimate for the variable. For instance, in 1981Q4, the most up-to-date estimate is the first release of the 1981Q3 value of real GDP. This entails using the vector of diagonal elements of the real-time database as the real-time vintage (RTV hereafter) series for a variable.

For different vintages of output would correspond different vintages of the output gap. In other words, the final vintage output gap data does not correspond with the ones forecasters used in forming expectations in the past. Unfortunately, the potential output is not observable and no real-time measure of it is available to our knowledge. Hence, it needs to be estimated. There exists a number of estimation approaches for the potential output in the academic literature, but there is still no consensus on which approach yields the best estimate of it. Here, inspired by the paper of [START_REF] Guisinger | Comparing measures of potential output[END_REF], we have compared two methods for extracting the potential output: the quadratic trend and the Hodrick and Prescott filter. Then, using final vintage data, we have compared these series based on both methods to the Congressional Budget Office (CBO) official measure of the potential output which is not available in real-time vintage to our knowledge. As a result, it turned out that the HP filter was more correlated to the CBO measure than the nonlinear trend measure. Consequently, we have carried out all the subsequent estimations using the HP-filtered measure of potential output.

Model-consistent expectations: As a matter of fact, the computation of the empirical counterpart of the so-called rational expectations is far from reaching a consensus among macro-economists. As stressed in the Introduction, even the terminology regarding expectations computation is still unsettled: Fuhrer (2012) proxies the rational expectations by VAR-based forecasts while, for instance, the Federal Reserve Bank's model of the US economy clearly distinguishes rational expectations from VAR-based expectations (see [START_REF] Brayton | The role of expectations in the FRB/US macroeconomic model[END_REF].

Yet, Nason and Smith (2008b) argue that the many difficulties in estimating and testing the NKPC can be traced back to the fact that inflation expectations are unobservable. Morever, [START_REF] Fuhrer | The role of expectations and output in the inflation process: an empirical assessment[END_REF] point out that the model-consistent nature of rational expectations poses a difficulty when the model does not match key features of the economy.

Given the significant costs of accessing all the information as well as elaborating a model that mimics the economy's complex structure, agents may opt for limited information (a small set of key macroeconomic variables) and a forecasting model that closely represents the economic environment but does not capture the complexity of the economy. This approach motivates the use of VAR-based expectations [START_REF] Brayton | The role of expectations in the FRB/US macroeconomic model[END_REF][START_REF] Branch | The theory of rationally heterogeneous expectations: evidence from survey data on inflation expectations[END_REF]Fanelli, 2008b,a;[START_REF] Fanelli | Simulation-based tests of forward-looking models under VAR learning dynamics[END_REF][START_REF] Tulip | Fiscal Policy and the Inflation Target[END_REF]. Here, following the Fed's model of the US economy described in [START_REF] Brayton | The role of expectations in the FRB/US macroeconomic model[END_REF], a small unrestricted VAR model is used, consisting of an equation for each of the output gap, the inflation rate and the Federal Funds rate. Let X t = (ỹ t , π t , i t ), where ỹt is the output gap, π t is the inflation rate and i t is the Federal Funds rate (nominal interest rate). The following VAR system is considered:

X t = µ + j=1 P j X t-j + ξ t , ξ t ∼ W N (0 N ×1 , Σ ξ ) (2)
where P j (j = 1, . . . , ) are n × n matrices of parameters, is the lag length, and ξ t is a white-noise error with covariance matrix Σ ξ . All the data in X t come from the Federal Reserve Bank of St. Louis' FRED online database. The sample starts in 1954Q3 due to the effective Federal Funds rate (i t ) availability and ends in 2017Q4. Using quarterly US data, we compute the output gap as ỹ = 100 × (y t -ȳt ); where y t and ȳt are the logs of real GDP and potential GDP, respectively. The inflation rate is given by π t = 400 × (p t -p t-1 ), where p t is the log of the consumer price index (CPI). The output gap is stationary by construction and Augmented Dickey-Fuller tests applied to π t and i t reject the unit root null at the 5%-level in both cases4 . Hence, the vector X t is stationary.

First, the VAR model given in Equation ( 2) is estimated over the full sample, that is 1954Q3-2017Q4. In order to choose the number of lags, , the Lagrange Multiplier test for no serial correlation in residuals is sequentially implemented for the VAR( )'s residuals, ∀ = 1, 2, . . . 8): the smallest lag order for which the residuals are serially uncorrelated up to order 8 is selected. The chosen lag order, say ˆ , is kept as a key feature of the DGP. Next, the VAR( ˆ ) model is estimated over the period 1954Q3-1981Q2 (i.e. the quarter just before the survey inflation forecasts data is available).

The one-step-ahead VAR inflation expectations are computed recursively over the remaining part of the sample (1981Q3-2017Q4). By doing so, our measure of inflation expectations begins exactly at the same quarter as the survey forecasts. Given the possibility of major changes in the monetary policy over our sample, for instance moving from the Great Inflation period to the Great Moderation one, VAR-based inflation forecasts computed on a rolling window basis are also considered. Each window spans 134 quarters, starting from 1954Q3-1981Q2 and ending in 1991Q1-2017Q4.

Survey-based expectations:

The survey-based expectations data used in this paper come from the Federal Reserve Bank of Philadelphia Survey of Professional Forecasters (SPF) which is available online. This quarterly survey is conducted among private-sector economists who share the specificity that forecasting macroeconomic variables is a key part of their work. Its outcomes are released at the end of the middle month of a given quarter.

A key feature of the survey is that it is anonymous so as to ensure that forecasters do not feel compelled to adapt to the consensus forecast (see [START_REF] Croushore | Introducing: the survey of professional forecasters[END_REF]. We use the median CPI inflation survey forecasts.5 Following e.g. [START_REF] Fuhrer | The role of expectations in inflation dynamics[END_REF], the four-quarter-ahead inflation survey forecasts are used in the NKPC. As emphasized by this author, theoretical models of inflation do not explicitly consider relative price variation. However, in practice, forecasters usually take into consideration relative price variations (food, energy and import prices) when forecasting one-quarter-ahead inflation. Hence, using the four-quarter-ahead inflation expectations rather than one-period-ahead inflation forecasts addresses this issue to some extent. Indeed the four-quarter-ahead inflation forecasts series is smoother than its one-quarter-ahead analogue (see Figure A1d in the Appendix). Although the starting date of the SPF is 1968, the CPI forecasts are included in the survey only since 1981Q3. This explains why the estimation of the NKPC equation featuring survey inflation expectations cannot start before this date.

Estimation of the NKPC

To assess the relative importance of different schemes of inflation expectations in explaining inflation dynamics, we consider a specification of the NKPC featuring heterogeneous expectations, that is, a combination of survey, VAR-based as well as myopic (i.e. lagged) inflation expectations. The various specifications of the NKPC considered are nested in the following version of Equation ( 1):

π t = β var E t π t+1 + β s S t π t+4 + β m π t-1 + γ ỹt + u t (3)
where E t π t+1 is time t VAR-based forecast of inflation in period t + 1 and S t π t+4 is the t + 4 inflation survey forecast as reported in period t.

We restrict parameters pre-multiplying expectational variables to sum up to one, such that β var + β s + β m = 1 throughout the empirical analysis. This restriction is in the spirit of a strand of the literature where a fraction of firms sets prices by relying on either modelconsistent, survey or myopic inflation expectations [START_REF] Nunes | Inflation dynamics: the role of expectations[END_REF]. In addition, we consider two versions of the NKPC model: (1) the purely forward-looking model where β m is set to zero, and (2) the so-called hybrid one where β m can be different from zero, allowing for a fraction of firms to form myopic expectations.

Given that there are endogeneity issues as well as measurement errors due to the estimation of unobservable variables (VAR-based expectations and output gap) in the NKPC equation, it will be estimated using the method of Generalized Instrumental Variables, a special case6 of the Generalized Method of Moments (GMM). According to this method:

E [(π t -β var E t π t+1 -β s S t π t+4 -β m π t-1 -γ ỹt ) Z t ] = 0, (4)
which means that its residuals should have a zero mean and be orthogonal to the instruments contained in Z t . The instruments set used here is in the spirit of [START_REF] Gali | Robustness of the estimates of the hybrid New Keynesian Phillips Curve[END_REF] and Nunes (2010). It consists of four lags of inflation, two lags of each regressor as well as wage inflation and labor share7 . The estimation method is the continuously-updated GMM (here equivalent to GIV) with Newey-West weight matrix and Bartlett bandwidth selection. The estimation sample period is 1982Q3-2017Q4. [START_REF] Mavroeidis | Empirical evidence on inflation expectations in the New Keynesian Phillips Curve[END_REF] (and references therein) argue that weak identification is prevalent in the NKPC since it is difficult to forecast changes in inflation. Hence, lagged instruments would be close to irrelevant. As such, inference relying on the J-test for overidentifying restrictions may be misleading. Therefore, weak identification of the NKPC has to be handled with robust inference methods, which remain valid under weak identification, [START_REF] Dufour | Inflation dynamics and the New Keynesian Phillips Curve: an identification robust econometric analysis[END_REF]Nason and Smith, 2008a,b;[START_REF] Mavroeidis | Empirical evidence on inflation expectations in the New Keynesian Phillips Curve[END_REF]. One such method is the [START_REF] Anderson | Estimation of the parameters of a single equation in a complete system of stochastic equations[END_REF] test.

Following Nason and Smith (2008b), we illustrate how inference on a given parameter of the NKPC can be carried out. For this purpose, let us consider the parameter pre-multiplying the survey inflation forecast (β s ), and rewrite Equation (3) as:

π t -β s S t π t+4 = β var E t π t+1 + β m π t-1 + γ ỹt + ΓV t + u t (5)
where V t represent a list of v supplementary variables and γ is a (1 × v) vector of parameters.

In our case, V t contains the same variables as the instrument set. To compute the left-hand side variable of the equation, we should pick a value β s0 for β s . Testing the hypothesis that β s = β s0 involves performing the standard F -test of the hypothesis that the variables in V t are all insignificant, that is, Γ = 0. The reasoning is that if β s0 is the true value for β s , then (i) the main regressors in the NKPC will generate the dynamics of π t and (ii) the residuals will not exhibit any systematic pattern due to the inclusion of supplementary variables V t in the regression (Nason and Smith, 2008b). Using a range of values between 0 and 1 for β s0 , Equation ( 5) is estimated.8 For each round of estimation, both the F -statistic of the null that the parameters pre-multiplying the supplementary variables are zero, and its associated p-value are collected. All values of β s0 associated with p-values greater than 5% do not reject the null. Consequently, the boundaries of the interval over which these p-values are greater than 5% define the 95% confidence interval for the estimated value of β s in the above example. Of course, the same applies to all estimated coefficients of the NKPC equation.

Results

Following the lines described in subsection 3.1, a lag order of six is retained for the VAR model given in Equation ( 2). Figure A2 in the Appendix shows the VAR-based inflation expectations series. Figures A2a andA2b show the VAR-based inflation expectations obtained by the recursive forecasting approach using final and real-time vintage data, respectively. Similarly, Figures A2c andA2d plot VAR-based inflation expectations from the rollingwidow forecasting approach. Next, Figure A3 in the Appendix displays the HP filter-based real time vintage output gap series, along with the final vintage measure obtained using official data on potential output from the CBO. Given the choice between (i) final and real-time vintage data, (ii) recursive and rollingwindow estimation for the VAR-based forecasts, and (iii) hybrid and purely forward-looking (i.e. without myopic inflation expectations) model, eight variants of the NKPC are estimated. Table 2 presents the results. The first (respectively last) four columns correspond to final (resp. real-time) vintage data. Columns (a), (c), (e) and (g) report the estimated parameters for hybrid NKPC models while columns (b), (d), (f ) and (h) show purely forward-looking NKPC models' estimates.

For all models, the J-test for over-identifying restrictions, bottom line of Table 2, is associated with a large p-value. Hence, the restrictions are valid. However, in this framework of the NKPC estimated by GIV, inference may be misleading due to prevalence of weak identification. For this reason, we also report in Table 2 the 95 percent robust confidence intervals obtained from the [START_REF] Anderson | Estimation of the parameters of a single equation in a complete system of stochastic equations[END_REF] test. These intervals remain valid regardless of whether identification is weak or not. In general, point estimates fall within the boundaries of the confidence intervals. However, there is a lot of uncertainty as the intervals are very wide and always include zero (except for the survey inflation expectations' estimate). This corroborates findings in the literature that confidence sets for the NKPC model's parameters tend to be wide and contain zero, see for example [START_REF] Dufour | Inflation dynamics and the New Keynesian Phillips Curve: an identification robust econometric analysis[END_REF] or Nason and Smith (2008b) and the references therein.

A quick glance at the results reveals that in six out of the eight estimated models, the parameter pre-multiplying the survey inflation expectations variable ( βs ) has the largest weight (ranging between 0.58 and 0.7) and is statistically significantly different from zero, using the robust confidence intervals. The two exceptions are the NKPC models estimated using final vintage data and the rolling-window VAR-based inflation forecasts. In this instance, the estimated coefficient for survey inflation expectations is aound 0.45, while the coefficient for VAR-based inflation expectations ( βvar ) ranges between 0.54 and 0.6.

All in all, it can be seen that estimations of the NKPC using final vintage data yield different conclusions depending on the methods used to derive VAR-based inflation expectations. It is also worth noting that in the case of real-time vintage data, the estimated parameter for survey inflation expectations has the largest weight regardless of which type of VAR-based inflation forecasts variable used.

Except for the case corresponding to the NKPC model estimated with real-time data and recursive VAR-based inflation expectations, the coefficient pre-multiplying the myopic inflation expectations variable ( βm ) is not statistically different from zero. While the weight of the myopic inflation expectations parameter is very close to zero when the NKPC equation is estimated from final vintage data, it turns out to be roughly 0.2 in the models estimated with real-time data. Finally, the coefficient for the output gap (γ) is never significantly different from zero: This variable does not contribute significantly to the US inflation dynamics.

Table 3 reports Wald tests statistics (and their p-values) of hypotheses of interest on NKPC parameters estimates. Practically, there appears to be no statistically significant difference between the weights of survey and VAR-based inflation expectations, as confirmed by the third and sixth lines of this table. Except for one case (hybrid NKPC model estimated with real-time data and recursive VAR-based inflation forecasts), the hypothesis of equality between the VAR-based and survey inflation expectations cannot be rejected at the 5%-level. As such, these tests suggest that VAR-based and survey inflation expectations are equally relevant and important in explaining US inflation dynamics in the NKPC framework. Then, the hypothesis that only VAR-based inflation expectations matter is strongly rejected at the 5%-level across all models (first and fourth lines of Table 3). The same applies for survey inflation expectations (second and fifth lines).

As a robustness check of these findings, all variants of the NKPC model are estimated using four-quarter-ahead VAR-based inflation forecasts instead of one-quarter-ahead ones. In this fashion, VAR-based inflation expectations match the horizon of survey counterparts used in the study. Tables 4 and5 generally confirm our conclusions.

As shown in Table 4, survey inflation expectations generally tend to have larger weights than VAR-based expectations. Nonetheless, both expectation measures are statistically equivalent in most cases (see Table 5, third and sixth lines). The only difference with the main results is the case corresponding to the use of real-time data in the estimation: There, the hypothesis that only survey inflation expectations matter in explaining US inflation dynamics cannot be rejected.

Similar to the main findings, Table 4 shows that the overidentifying restrictions are valid based on the J test. Also, [START_REF] Anderson | Estimation of the parameters of a single equation in a complete system of stochastic equations[END_REF] 95% weak identification-robust confidence intervals are wide and they contain the point estimates in most cases. Again, they do not contain zero for the survey inflation expectations estimate in six out of eight models. On the other hand, they consistently contain zero for other expectational variables across all models.

Concluding remarks

Using both final and real-time vintage data, we assess the relative contributions of VARbased, survey-based and myopic inflation expectations in the NKPC. The computation of VAR-based inflation expectations data accommodates major changes in the conduct and propagation mechanisms of the U.S. monetary policy which occurred since World War II using recursive and rolling-widow forecasting methods. We find that VAR-and surveybased inflation expectations contribute significantly to inflation dynamics and statistically speaking, point estimates of these inflation expectations coefficients are not significantly different from each other: The null hypothesis H 0 : β var = β s is almost never rejected at the 5%-level. Contrarily to the conflicting outcomes in the literature [START_REF] Nunes | Inflation dynamics: the role of expectations[END_REF][START_REF] Fuhrer | The role of expectations in inflation dynamics[END_REF], we conclude that none of the two types of forward-looking inflation expectations is able, on its own, to capture inflation dynamics in the NKPC. Moreover, myopic inflation expectations play very little role, if at all, in explaining inflation dynamics. Even though our inflation data seems to be stationary for the samples under scrutiny in this paper, this variable is known for its strong persistence. For this reason, [START_REF] Cogley | Trend inflation, indexation, and inflation persistence in the New Keynesian Phillips Curve[END_REF] have suggested to introduce trend inflation among the explanatory variables in the NKPC equation (see also [START_REF] Stock | Why has US inflation become harder to forecast?[END_REF]; [START_REF] Nason | Sticky professional forecasts and the unobserved components model of US inflation[END_REF]; [START_REF] Cecchetti | Deflating Inflation Expectations: The Implications of Inflation's Simple Dynamics[END_REF]; [START_REF] Eusepi | The science of monetary policy: An imperfect knowledge perspective[END_REF]; [START_REF] Forbes | A trendy approach to UK inflation dynamics[END_REF]). Although very appealing, this extension has the drawback that this extra right hand side variable is not observable, and as such, it is not so convenient to implement. Nonetheless, the inclusion of such a trend inflation variable in the NKPC is on our research agenda.

  ,0.25] [-0.17,0.25] [-0.09,0.25] [-0.09,0.25] [-0.25,0.25] [-0.25,0errors are in ( ); Anderson and Rubin (1949)-based robust 95% confidence intervals are in [ ]; J gives the p-value for the GMM overidentifying restrictions test.

Figure A1 :

 A1 Figure A1: Plots of variables (1954Q3-2017Q4) (a) Output gap (ỹ t )

Table 1 :

 1 Relative weights of lagged inflation, rational and survey inflation expectations in a New Keynesian Phillips CurveNotes: Figures in bold denote significant coefficients at the 5%. π t : inflation rate; E t π t+1 : inflation rational expectations; β e and β s : coefficients pre-multiplying the rational and survey expectations variables, respectively; β m : lagged inflation parameter. β s = 1 -β e and β

	Paper	Country	Sample	π t	E t π t+1	Model	βe	βs	β s	βm	β m
							0.82	0.22	-	-	-
							(0.08) (0.09)
							0.76	-	0.24	-	-
						Detrended GDP	(0.09) 0.74	0.19	(0.09) -	0.10	-
							(0.10) (0.09)		(0.11)
							0.56	0.18	-	-	0.25
							(0.09) (0.09)			(0.09)
	Nunes (2010)	US	1968Q4 -2007Q4 GDP deflator GIV		0.96	0.07	-	-	-
							(0.05) (0.06)
							0.96	-	0.04	-	-
						Marginal cost	(0.05) 0.88	0.05	(0.05) -	0.09	-
							(0.09) (0.06)		(0.12)
							0.81	0.02	-	-	0.17
							(0.08) (0.05)			(0.11)
						Core CPI, output gap
						1983 -1992	0.20	0.40	-	0.30	-
	Fuhrer and Olivei (2010)	US	1983 -2009	Core CPI	VAR	1990 -1999	(n.a) (n.a) 0.20 0.30	-	(n.a) 0.40	-
							(n.a) (n.a)		(n.a)
						1999 -2008	0.20	0.20	-	0.20	-
							(n.a) (n.a)		(n.a)
						SPF TI / ML	0.00 (n.a) (0.25) 0.75	-	-	0.25 (0.10)
					VAR	CS TI / ML	0.00 (n.a) (0.17) 0.73	-	-	0.36 (0.11)
	Fuhrer (2012)	US	1990Q1 -2010Q3	CPI		SPF TI / ML (including 1980s)	0.03 (0.14) (0.30) 0.87	-	-	0.10 (0.06)
					GIV	TI / Optimal GMM TI / GMM	0.11 (0.21) (0.28) 0.57 0.77 0.22 (0.33) (0.56)	--	--	0.25 (0.09) -0.04 (0.13)

m = 1 -β e -β s . Standard errors are in ( ). "SPF": Survey of Professional Forecasters; "TI": trend inflation; "CS": Cogley-Sbodorne; "n.a": not available.

Table 2 :

 2 Estimates of the NKPC model

	Final vintage	Real-time vintage
	Recursive Etπ t+1	Rolling Etπ t+1	Recursive Etπ t+1	Rolling Etπ t+1

Table 3 :

 3 Wald tests for parameters restrictions in the NKPC model

		Final vintage	Real-time vintage
		Recursive Etπ t+1	Rolling Etπ t+1	Recursive Etπ t+1	Rolling Etπ t+1
	Hybrid model
	H 0 : βvar = 1, βs = 0, βm = 0	30.35	12.13	33.17	11.29
		(<0.01)	(<0.01)	(<0.01)	(<0.01)
	H 0 : βvar = 0, βs = 1, βm = 0	16.12	20.14	14.55	15.37
		(<0.01)	(<0.01)	(<0.01)	(<0.01)
	H 0 : βvar = βs, βm = 0	0.77	0.43	7.49	2.75
		(0.68)	(0.81)	(0.02)	(0.25)
	Purely forward-looking model
	H 0 : βvar = 1, βs = 0	29.76	14.16	29.28	13.39
		(<0.01)	(<0.01)	(<0.01)	(<0.01)
	H 0 : βvar = 0, βs = 1	15.53	20.05	6.76	6.08
		(<0.01)	(<0.01)	(0.01)	(0.01)
	H 0 : βvar = βs	0.57	0.13	1.97	0.36
		(0.45)	(0.72)	(0.16)	(0.55)
		Notes: p-values into ( ).

Table 4 :

 4 Estimates of the NKPC model with four-quarter VAR-based expectations Standard errors are in ( );[START_REF] Anderson | Estimation of the parameters of a single equation in a complete system of stochastic equations[END_REF]-based robust 95% confidence intervals are in [ ]; J gives the p-value for the GMM overidentifying restrictions test.

			Final vintage			Real-time vintage
		Recursive Etπ t+1	Rolling Etπ t+1	Recursive Etπ t+1	Rolling Etπ t+1
		(a)	(b)	(c)	(d)	(e)	(f )	(g)	(h)
	βvar	0.54	0.40	0.66	0.58	-0.02	0.25	0.04	0.30
		(0.18)	(0.11)	(0.17)	(0.12)	(0.19)	(0.13)	(0.27)	(0.18)
		[0.00, 0.54] [0.00, 0.54] [0.00, 0.80] [0.00, 0.80]	[0.00,0.45]	[0.00,0.45]	[0.00,0.79]	[0.00,0.79]
	βs	0.55	0.60	0.38	0.42	0.74	0.75	0.71	0.70
		(0.12)	(0.11)	(0.13)	(0.12)	(0.14)	(0.13)	(0.20)	(0.18)
		[0.09, 1.00] [0.09, 1.00] [0.03, 1.00] [0.03, 1.00]	[0.06,0.80]	[0.06,0.80]	[0.00,0.76]	[0.00,0.76]
	βm	-0.10	-	-0.04	-	0.28	-	0.25	-
		(0.10)		(0.09)		(0.09)		(0.11)
		[0.00, 0.49]		[0.00, 0.44]		[0.00,0.39]		[0.00,0.32]
	γ	0.12	0.12	0.02	0.03	0.03	-0.17	0.01	-0.11
		(0.08)	(0.08)	(0.08)	(0.08)	(0.16)	(0.15)	(0.14)	(0.14)
		[-0.16,0.25] [-0.16,0.25] [-0.17,0.25] [-0.17,0.25] [-0.25,0.25] [-0.25,0.25] [-0.25,0.25] [-0.25,0.25]
	J	71%	78%	76%	83%	76%	64%	77%	63%
	Notes:

Table 5 :

 5 Wald tests for parameter restrictions in the NKPC model with four-quarter VARbased expectations

		Final vintage	Real-time vintage
		Recursive Etπ t+1	Rolling Etπ t+1	Recursive Etπ t+1	Rolling Etπ t+1
	Hybrid model
	H 0 : βvar = 1, βs = 0, βm = 0	25.65	9.75	31.38	13.15
		(<0.01)	(0.01)	(<0.01)	(<0.01)
	H 0 : βvar = 0, βs = 1, βm = 0	13.81	22.92	16.00	14.12
		(<0.01)	(<0.01)	(<0.01)	(<0.01)
	H 0 : βvar = βs, βm = 0	1.37	0.92	10.28	5.11
		(0.50)	(0.63)	(0.01)	(0.08)
	Purely forward-looking model
	H 0 : βvar = 1, βs = 0	29.95	12.31	32.11	15.46
		(<0.01)	(<0.01)	(<0.01)	(<0.01)
	H 0 : βvar = 0, βs = 1	13.71	22.99	3.74	2.96
		(<0.01)	(<0.01)	(0.06)	(0.09)
	H 0 : βvar = βs	0.78	0.41	3.48	1.22
		(0.38)	(0.52)	(0.06)	(0.27)
		Notes: p-values into ( ).

See among others[START_REF] Gali | Inflation dynamics: a structural econometric analysis[END_REF],[START_REF] Sbordone | Prices and unit labor costs: a new test of price stickiness[END_REF],[START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the EURO area[END_REF],[START_REF] Christiano | Nominal rigidities and the dynamic effects of a shock to monetary policy[END_REF] or the survey by[START_REF] Milani | The modeling of expectations in empirical DSGE models: a survey[END_REF].

See for instance[START_REF] Fuhrer | Inflation persistence[END_REF],[START_REF] Roberts | Inflation expectations and the transmission of monetary policy[END_REF],[START_REF] Estrella | Dynamic inconsistencies: counterfactual implications of a class of rational-expectations models[END_REF],[START_REF] Milani | Adaptive learning and inflation persistence[END_REF] or[START_REF] Milani | Expectations, learning and macroeconomic persistence[END_REF] inter alia.

See next section.

These tests were conducted with an intercept only and with respectively 2 and

lags in first differences so as to eliminate residuals serial correlation up to order 8. The resulting tests statistics are respectively -3.61 and -2.87.5 These are median CPI values across forecasters, over time.

See for instanceMavroeidis et al. (2014), pages 133-134.

For real-time data estimations, the instrument set includes two lags of the inflation rate, survey inflation expectations, VAR-based inflation expectations and the real-time vintage output gap, in the spirit of[START_REF] Mavroeidis | Empirical evidence on inflation expectations in the New Keynesian Phillips Curve[END_REF]

 8 In practice, we have used a grid of 100 evenly spaced values between 0 and 1 for inflation expectation variables. For the output gap variable, the grid spans from -0.25 and 0.25 as in[START_REF] Mavroeidis | Empirical evidence on inflation expectations in the New Keynesian Phillips Curve[END_REF].

useful comments from two anonymous referees. We are also grateful to the participants to Cergy-Pontoise University -THEMA economics seminar for their stimulating comments. Of course, all remaining errors are ours. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01).

Appendix


Olivier Proux

Jean-Sébastien S Micha

Jean-René Regnard

Agnès Traverse

B Dieny

Franck Ernult

Pascale Bayle-Guillemaud

J-L Hazemann

L Giacomoni

J R Régnard

Structural investigations of Co/ZrO 2 discontinuous multilayers by x-ray absorption fine structure spectroscopy

Introduction

The spin dependent tunnel effect has attracted great interest in the last seven years. Besides its fundamental aspect, the tunnel magnetoresistance (MR) of magnetic tunnel junctions (MTJ) could be useful for applications, especially in magnetic random access memories [1][2][3][4]. As MTJ(two continuous ferromagnetic electrodes separated by an insulating layer),ferromagnetic nanoparticles embedded in an insulating matrix also exhibit tunnel MR but with a weaker amplitude [1,[5][6][7]. In order to obtain high MR sensitivity in metal/oxide systems, the ferromagnetic particles can be arranged to form discontinuous multilayers (DCM) [8,9]. Compared to MTJ, DCM are easier to prepare by physical vapour deposition such as sputtering. Besides the intrinsic physical properties of the magnetic metal and of the oxide, the local atomic order at the metal-oxide interface has been shown to have a strong influence on the spin dependent tunnel mechanisms, in particular on the polarization rate of the tunnelling electrons [10,11]. Dispersion of the metallic species into the insulating matrix may lead to a drastic change in the magnetic and electronic properties of the metal/oxide/metal device. Moreover, the relative orientation of the magnetic moments of neighbouring particles, which is modified by the applied magnetic field H, also determines the shape of the MR variation [12]. As a result, the variation of the magnetization and of the electrical resistance of the DCM with H is mostly governed by the morphology of the nanogranular metal-oxide layers. This paper presents the results of the characterization of the local structure around the Co atoms and of the morphology of the Co layers in Co/ZrO2 multilayers as a function of the nominal deposited metal thickness (λ) and deposition rate (v). We compare the results obtained by several techniques which are sensitive to different length scales, from 1 to a few tens of Å. The central technique of this study is x-ray absorption spectroscopy (XAS), which appears to be particularly suitable for such systems without long-range order and composed of nanometric particles [13]. X-ray absorption near edge structure (XANES) and extended x-ray absorption fine structure (EXAFS) investigations have been performed at the Co K-edge to probe the structural local environment of the Co atoms. A quantitative analysis of the EXAFS data allows the determination of the nature and the number of nearest atomic neighbours (NN) at distance R from the probed central Co atoms, and the socalled Debye-Waller (DW) factor, corresponding to the mean-square relative displacement of the R distribution. In this paper the sensitivity of the EXAFS signal to the morphology of this nanoscaled metallic layer is stressed. Moreover, the investigation of the pre-edge and XANES features provides information about the electronic structure of the Co atoms and the number that are located in metallic sites. Both structural and morphological results from XAS analyses are compared to those deduced from transmission electron microscopy (TEM) micrographs and transport and magnetic measurements.

Experimental details

DCM elaboration

Co/ZrO 2 multilayers were grown by alternative magnetron sputtering of two separate targets (Co and ZrO 2 ) on an Si wafer or a glass substrate maintained at room temperature (RT). The base pressure was 2×10-8 Torr and the argon pressure was 3×10 -3 Torr during deposition. The ZrO 2 layer nominal thickness was fixed at 40 Å whereas λ was varied from 2 to 62 Å. The bottom and top layers were ZrO 2 in order to (i) eliminate the possible interaction between the substrate and the Co aggregates; and (ii) protect the Co aggregates from oxidation by contact with air. The deposition of the [Co/ZrO 2 ] bilayer was repeated 40 times with either v = 0.5 or 0.9 Å s -1 . Specific ZrO 2 40 Å/Co λ/ZrO 2 40 Å trilayers were deposited with λ = 8 and 10 Å at v = 0.5 Å s-1 for TEM plane view observations.

Magnetic and transport measurements

The magnetic properties of the as-deposited Co/ZrO 2 multilayers (v = 0.9 Å s -1 ) were measured at RT using a vibrating sample magnetometer. The magnetic field (H), up to 5 T, was applied in the plane of the Co layers. Resistance measurements were performed at 100 and 300 K in the current-in-plane (CIP) geometry for the DCM deposited at v = 0.9 Å s -1 [14].

XAS measurements

The XAS measurements were performed at the Laboratoire pour l'Utilisation du Rayonnement Electromagnétique (LURE, Orsay, France) D42 station and at the European Synchrotron Radiation Facility (ESRF, Grenoble, France) BM32 beamline respectively in the conversion electron yield (CEY) and in the fluorescence yield (FY) mode. Some samples were studied in both the CEY and the FY modes, and no substantial differences were observed in their corresponding spectra. The CEY EXAFS measurements were carried out at RT or liquid nitrogen temperature [15], allowing in the latter case a good signal-to-noise ratio in an enlarged energy range due to the reduction of the dynamical part of the DW factor. Theoretical or semi-empirical models [16,17] can be used in order to estimate the probing depth of this detection mode. Taking the bulk density values of ZrO 2 and Co (5.8 and 8.9g cm -3 respectively), the probing depth is estimated to range between 600 and 700 Å in the λ-range covered. The probed thickness of the Co/ZrO 2 DCM then corresponds approximately to the 15 first top bilayers allowing the XAS signal relative to the Co atoms near the interface of the DCM and the SiO 2 substrate to be safely neglected. A (331) Si channel-cut monochromator was used in order to have an energy step of 0.5 eV in XANES part of the XAS spectra. The FY measurements were carried out at RT using a 30 element solid germanium Canberra detector. The monochromator was a (111) Si 2-crystal. The dynamical bending of the second crystal permits a horizontal focusing of the beam without losing flux. The size of the x-ray spot was reduced to 0.3×0.3 mm 2 in order to increase the signal-to-noise ratio for DCM presenting a low Co mean concentration (low value of λ).

EXAFS analysis

EXAFS oscillations χ(k) were extracted from the raw data with the Autobk program [30] and simulations were performed using the ab initio backscattering phases and amplitudes calculated by the FEFF 6.01 code [31]. All the analyses were performed with k-weighted χ(k) signals whose Fourier transform (FT) correspond to the mean pseudo radial distribution function around Co atoms. The nearest peak from the origin is related to the photoelectron backscattering by Co or O neighbouring atoms. The inverse FT of this peak was fitted with FEFFIT to the FEFF-calculated and filtered signal kχ(k). To validate the analysis, a metallic Co foil reference spectrum was investigated in the same conditions.

Macroscopic characterization results

Transmission electron microscopy observations

Figure 1 shows TEM electron energy filtered images of two as-deposited Co/ZrO 2 trilayers for two Co nominal thicknesses λ = 8 and 10 Å obtained at the Co L 2,3 edge. Micrographs were digitized in order to quantify the morphology of the discontinuous layer and to estimate the in-plane Co coverage on ZrO 2 . The in-plane views of the trilayers show that the Co layers (light contrast) form Co aggregates embedded in the ZrO 2 matrix (dark contrast). For λ = 8Å, Co aggregates with radius R A ≤ 25 Å are almost spherical. Larger Co aggregates are also formed, with a chain-like shape; it is likely that they are due to the coalescence of spherical and smaller Co aggregates. By counting the fraction of white pixels the oxide coverage ratio is estimated to be about 40%. For λ = 10 Å, there are no longer small spherical Co aggregates but instead large ones formed by touching nanoparticles. The Co layer has a 2D coverage ratio of 55% and is close to the percolation threshold.

Transport properties

For λ ≤ 10 Å, the CIP resistance is very large and decreases with the temperature (from about 10 7 Ω at 100 K to 2×10 6 Ω at 300 K). This non-metallic behaviour is due to the discontinuous nature of the Co layers: electrons cross the oxide barrier separating Co particles by the tunnel effect and thermally activated hopping [18,19]. For larger thicknesses (λ ≥ 14 Å), the CIP resistance falls to 10 2 Ω (100 K) and 2.10 2 Ω (300 K). It increases with temperature as in standard metals. Co layers are therefore continuous from the electrical point of view. The electric percolation threshold can be then located at λ ≈ 11 Å (for v = 0.9 Å s-1).

Magnetic properties

The magnetization loops are shown in figure 2. For λ = 6.3 Å, the absence of hysteresis indicates that Co aggregates are superparamagnetic at RT. For λ = 10.7Å, particles are clearly bigger due to the increase of initial susceptibility (dM/dH) H=0 . A slight hysteresis can be attributed to the non-spherical shape of the particles near the electric percolation threshold as suggested by the observation of a spin-glass-like behaviour at low temperature [14]. For λ = 15.8 Å, Co layers are continuous with a disordered soft ferromagnetic behaviour at RT. Other results concern the saturation magnetization (M S ). Even for the largest λ DCM, M S remains much smaller than the Co bulk saturation magnetization (M S bulk = 1445 emu cm -3 ). Two main effects explain this reduction. First, magnetic behaviour at the surface of the aggregates is different from that in the core. Canting and frustration of the surface spins may occur [20,21]. They lead to a lack of complete saturation even at high field [22] and a decrease of the aggregate magnetization despite an enhancement of the magnetic moment per atom (MMA) observed in free aggregates [23,24]. Secondly, due to the interactions with the atoms of the matrix, the electronic state of Co atoms not entirely surrounded by magnetic Co atoms may be modified as compared to the Co metallic ones [25]. It can be similar to that of Co ions. Co 2+ and Co 3+ are, for instance, paramagnetic and generally may be found in a low spin state [START_REF] Cox | Transition Metal Oxides. An Introduction to their Electronic Structure and Properties[END_REF]. They are far from being fully magnetized at RT even for high Hvalue. Exchange coupling between ferromagnetic Co atoms in the core and a disordered antiferromagnetic Co ions layer around them can also be present [START_REF] Meiklejohn | [END_REF]28]. The variation of MS as a function of λ probably comes from the variation of the proportion of magnetic atoms. Following this idea, the proportion of Co atoms participating in the magnetism of the Co aggregates can be roughly estimated by assuming that they are either magnetic (with the full MMA as in bulk Co) or non-magnetic (in the ionic form without any effective magnetization). According to this most basic model, by taking the ratio M S /M S bulk , the magnetic fraction is found to be7%, 15% and 24% for λ = 6.3, 10.7 and 15.8Å respectively.

XAS results

XANES

The two series of DCM differing by the value of v exhibit the same evolution as a function of λ of their XANES spectra. Since the series relative to v = 0.9 Å s -1 has been measured with a higher energy resolution, the corresponding spectra are reported in figure 3

EXAFS qualitative analysis

In order to optimize the magneto-electronic properties, the fraction of metallic Co atoms should be as high as possible. The DCM prepared at v = 0.5 Å s -1 have a higher Co metallic ratio than those prepared at v = 0.9 Å s -1 (not presented here).

EXAFS quantitative results

The values of the structural parameters (the first Co and O NN numbers, the Co-Co first distance and the corresponding DW factor) deduced from the fits of the kχ(k) signals are reported as a function of λ in figure 6. Two kinds of nearest atomic neighbourhood are used in the fits, depending on the value of λ. For λ < 15 Å, both O (characteristic of the first Co-O bond in the CoO reference) and Co (characteristic of Co-Co bond in metallic Co) first neighbours must be taken into account, whereas for larger λ only a single Co atoms shell is necessary. • Oxygen neighbourhood. For λ = 2 Å, a single O shell is necessary for the simulation without any additional Co shell. The number of O nearest neighbours is quite similar to that of Co in the CoO structure. However, the quality of the fit remains poor due to the high disorder around the Co central atoms even by introducing an asymmetric radial distribution (see the so-called cumulants method [35]). A more precise description of the oxygen neighbourhood around Co atoms would require EXAFS oscillations over a wider k-range. As a result, for higher λ-values, the number of nearest oxygen neighbours decreases with λ. This variation originates from the reduction of the fraction of probed central Co atoms with oxygen neighbours.

• Co-Co first distance. The Co-Co bond length is smaller than the Co-Co distance in Co metal (R = 2.505 Å): the bond length variation is negative for all the λ-values as reported in nanostructured aggregates due to the surface stress [33,34]. The contraction tends towards 0 when λ increases and the mean first Co-Co bond length has apparently reached the reference value for λ = 30 Å. This variation can be explained by two complementary reasons. The first reason is the average effect over all the probed Co atoms: the fraction of Co atoms in bulk-like sites increases. The second reason comes from the absence of a model for the distribution of the first Co-Co distance. As the structural disorder and the width of the distribution decrease, the Gaussian distribution more accurately models the real distribution (see below).

• Debye-Waller factor of the first Co-Co distance. The value of the DW factor (σ2), as λ increases, follows a general damping which is related to the increase of structural atomic ordering around an average Co central atom. The reduction of the mean local structural disorder around Co central atoms is attributed to a larger fraction of Co central atoms in well-ordered crystal sites. For the smallest values of λ, the structural disorder is so important that the harmonic distribution model of Co-Co first distance is not well adapted. The presence of the Co-O contribution in the same R-range cannot allow the extraction of the distance distribution or the application of the cumulants method. Finally, even for the thickest Co layers, one can note the remaining disorder (higher than that measured on a Co metallic foil).

• Co NN number. The number of Co nearest neighbours varies as a function of λ. This number tends towards 12 as λ increases, corresponding to the Co bulk value. It is a statistical average number calculated over all the central Co atoms. We shall discuss later how its variation with λ depends basically on the size of the particles and on the fraction of Co non-metallic sites.

Discussion

XAS analysis on Co/ZrO 2 at the Co K-edge gives a wealth of structural information about the local order around Co atoms. First, the presence of the oxygen atoms in the nearest vicinity of Co atoms is discussed. Secondly, an estimation of the number of this kind of Co atoms is proposed. Thirdly, at a greater length scale, a model of the variation of the number of Co nearest neighbours with respect to λ obtained by XAS is presented by taking into account the evolution of morphological parameters of the Co layers. Finally, information extracted from XAS measurements is inserted in the global structural and morphological diagram of Co/ZrO 2 multilayers as a function of λ.

Oxygen neighbours of Co atoms

For λ ≤ 5 Å (v = 0.5 Å s-1) and λ ≤ 10 Å (v = 0.9 Å s-1), XAS experiments indicate the presence of oxygen atoms in the Co nearest neighbourhood. According to XANES comparisons, most of the Co atoms surrounded by O atoms are likely to be in the 2+ valence state without excluding the 3+ valence state as in Co 3 O 4 oxide. Local order around these non-metallic atoms is limited to very short distances: only the first O shell is clearly defined and is similar to that of the first shell of the CoO oxide. However, with such poor local order, no longrange order structure as observed in CoO or Co 3 O 4 oxides can be evidenced in the DCM by macroscopic techniques. The Co-O bonds are likely to be due to Co atoms (i) in substitution of Zr atoms in the sputtered ZrO 2 matrix or/and (ii) at the interface between Co aggregates and the ZrO 2 oxide matrix. Previous studies on Fe/ZrO 2 DCM grown using the same apparatus in similar conditions corroborate this conclusion [32].

Basic estimation of the Co metallic ratio

As the magnetic fraction was deduced from the variation of MS with λ, so too the fraction of metallic Co atoms can be derived from the variation of the pre-edge features with λ. For small values of λ, the small pre-peak centred around 0 eV is at the same position as the pre-peak of the oxide references (figure 3). In contrast, for the higher nominal thickness, this edge feature seems to be intermediate between the Co metal and Co oxide features. Following the most simple assumption that Co atoms can be either in a metallic Co environment (metallic fraction F Co ) or surrounded by O atoms (1 -F Co ), F Co can be roughly estimated with the help of the corresponding normalised pre-edge spectrum (normalised pre-edge absorption signal), S DCM , expressed as a linear combination: (1) where S Co and S CoO are the normalised pre-edge spectra of the Co metallic foil and CoO references respectively. Pre-edge features are preferred to those of XANES since they are not sensitive to structural disorder but to chemical environment. Reference signals, S Co and S CoO , were measured within the same run as the XAS experiments on DCM. Fits of the experimental edge spectra according to this model are satisfactory. All the results are gathered in figure 7 for the two series of DCM. For a given value of λ, the DCM deposited at v = 0.5 Å s -1 have a higher Co metallic ratio than those grown at v = 0.9 Å s-1 indicating that low deposition rate favours the formation of Co particles. Previous Mössbauer spectroscopy and EXAFS experiments performed on Fe/ZrO 2 DCM with λ Fe from 6 to 15 Å [32] have strongly suggested the presence of dead layers composed of non-metallic Fe atoms. Concerning the Co/ZrO 2 DCM, an equivalent Co thickness (l Co-O ) corresponding to Co atoms surrounded by O atoms can be assumed, which is constant with respect to λ but depends on v. When λ is lower than l Co-O , all the Co atoms are assumed to be non-metallic with O neighbouring atoms (F Co = 0). When λ exceeds l Co-O , an equivalent l Co-O amount of Co is surrounded by O atoms whereas an equivalent (λl Co-O ) amount is in metallic Co sites. l Co-O can be seen as the total amount of non-metallic Co atoms roughly located on both sides of each nominally continuous Co layers. The F Co ratio can then be simply written as (2) Comparisons of the experimental data with this model are shown in figure 7 (solid curves). The good agreement suggests that a constant amount of Co is surrounded by O atoms in these Co/ZrO 2 DCM. These atoms can be considered as dead or lost for the MR properties since their magnetic and electronic properties differ from those of metallic Co atoms in a ferromagnetic and metallic crystal. The l Co-O thickness is found to be around 2.5 and 6 Å for v = 0.5 and 0.9 Å s-1 respectively. The greater value of l Co-O found for the higher deposition rate is likely to be related to the better activation of the atomic interdiffusion between the oxide and metallic layers and to the increase of structural disorder due to the higher number and energy of sputtered species impinging on the substrate. Since these mechanisms limit the aggregation process of the metallic element in ordered particles, l Co-O values can be regarded as a metallic 'aggregation threshold': for λ < l Co-O , no substantial amount of metallic Co can be found in the ZrO 2 matrix. Finally, the Co magnetic ratios obtained from the saturation magnetization values are smaller than the Co metallic ratios obtained from the XANES adjustments. Numerous Co atoms with reduced magnetic moments, near or at the surface of Co aggregates with chain-like shape or composed of touching nanocrystallites, actually contribute to the underestimation of the magnetic Co atoms.

Co metallic aggregates

The Co NN number is lower for the Co/ZrO 2 DCM than for the bulk metallic Co reference (figure 6(c)). Two main parameters can govern this reduction: (i) the fraction of under coordinated atoms at the surface of nanometric aggregates is not negligible, and (ii) Co atoms are roughly located in two different sites (pure Co aggregates and Co surrounded by O atoms). This repartition can be estimated from the F Co value (number of metallic Co with respect to the total number of Co atoms, see section 5.2). The measured Co NN number for the DCM must then be compared to the Co NN number of the reference weighted by F Co to account for this repartition. Numerous studies have established the correlation between the NN number reduction and the size of the aggregates [36][37][38]. However, these theoretical calculations are performed when atoms gather in a polyhedral aggregates. In this study, the Co aggregates evolve from a roughly spherical shape (below the electric percolation threshold) to larger chain-like particles (see figure 1). The Co layers in the DCM are discontinuous for low values of λ and continuous for higher ones. One can estimate the resulting NN number when the metallic Co atoms are gathered in spherical aggregates or in continuous layers. Similar calculations can be found in previous studies [39][40][41]. In the case of spherical particles, the mean NN number is expressed in good approximation as (3) where RA is the particle radius, FCo the fraction of non-metallic Co atoms deduced in section 5.2, NC = 12 (bulk Co NN number value) and d the first interatomic distance. The mean particle size can hence be estimated from the EXAFS NN number in the DCM. However, there is a large discrepancy between the size estimated by EXAFS and those measured by TEM. On one hand, the energy filtered TEM is sensitive to heavily concentrated Co regions irrespective of the valence state of Co atoms and structural local order around them. The size obtained is of the order of a few tens of Å. On the other hand, EXAFS is mainly sensitive to well-ordered atomic structure: probed Co atoms surrounded by large distance distributed Co atomic neighbours do not fully contribute to the corresponding EXAFS shell signal amplitude. Moreover, the generally accepted uncertainty on the measured NN number is about 10-20% and the nanoparticles are not strictly identical and spherical. The estimation of the mean nanoparticle size from the absolute value of the mean EXAFS NN number (even corrected by FCo) leads then to an approximate and very reduced size compared to that obtained by macroscopic techniques [41]. But the comparison of EXAFS NN numbers values can be safely exploited to obtain structural information. In our case, we put in the model morphological parameters. We assume that the Co layers are composed of spherical aggregates (radius RA) forming a 2D hexagonal lattice (distance Λ between the centres of two neighbouring aggregates). According to the Co deposited volume conservation, Previous grazing incidence small-angle x-ray scattering (GISAXS) experiments performed on Co/SiO 2 DCM [42,43] have suggested a proportional relationship between R A and Λ : Λ = αR A . By following these descriptions the mean NN number is then Two regions can be defined. For λ _ 16 Å, the experimental values are found to be in agreement with the simulated data. For λ > 20 Å, the previous model based on specific particle size, shape and organisation is no longer valid. EXAFS results hence agree with the morphological evolution of the Co layers from a discontinuous (Co aggregates) to a continuous layer.

Co/ZrO 2 DCM morphology diagram

The results concerning the morphology of the Co layers are summarised in figure 8. Three frontiers are reported.

(i) The Co aggregation threshold deduced from the XANES investigation and magnetic measurements, estimated to about 2.5 Å. This value is significant of the amount of both nonmetallic and non-magnetic Co atoms having neighbouring O atoms.

(ii) From TEM micrographs and transport measurements, the electrical percolation threshold has been estimated to about 10 Å.

(iii) A 3D-2D transition region has been defined by EXAFS between 16 and 20 Å corresponding to the transition froma discontinuous (surface to volume ratio characteristic of 3D aggregates i.e.with a rather spherical shape) to a continuous layer (surface to volume ratio characteristic of 2D layer i.e. continuous or composed of rather pancake and chainlike particles).

The percolation threshold is obviously not evidenced by the measurement of the EXAFS mean NN number. The positions of the three thresholds are highly dependent on the difference of the surface energy between the metal and the insulator matrix as well as on the substrate temperature during the deposition. For multilayers grown by sputtering at RT, it has been observed for small values of λ that Co does not wet SiO 2 [9,42] and Al 2 O 3 layers [44]: the electrical percolation threshold was found to be around 20 Å and between 15 and 20 Å respectively. These ranges are slightly higher than that measured in Co/ZrO 2 DCM. The reduced λ-range for preparing DCM by sputtering with Co and ZrO 2 limits then the optimisation of the MR properties as a function of λ. Grazing incidence small-angle scattering data are under analysis to determine the growth process in this metal-oxide system.

Conclusion

The structure of Co/ZrO 2 multilayers grown with two Co atoms deposition rates (v) has been studied. Macroscopic measurements have clearly established the discontinuity of the metallic layer for a Co nominal deposited layer thickness (λ) lower than 10 Å. A reduction of the saturation magnetization in DCM with respect to the bulk Co value is observed. Moreover, the XAS indicates that the mean number of Co nearest neighbours around Co atoms is lower than that in bulk Co due to the finite size of the layer. The evolution of this number with λ is governed by the variation of the fraction of Co atoms having an oxygen atom as a first neighbour and the size of the particles composing the layer. According to a basic model, the number of Co atoms with oxygen neighbours which are lost for the MR properties can be estimated to an equivalent deposited thickness of 2.5Å (v = 0.5Å s-1) and 6 Å (v = 0.9Å s-1) for any value of λ. The electronic state of the non-metallic Co atoms probed by XANES is rather similar to that of Co 2+ ions in CoO but the local order around them is limited to the first O shell. Three thresholds have been found: aggregation, percolation and 2D-3D layers thresholds. Their location has allowed the determination of the λ-range corresponding to the presence of Co particles with a negligible fraction of non-magnetic and non-metallic Co atoms. In addition to the structural disorder, this result represents an argument to explain the reduced efficiency of the spin-polarized transport in DCM compared to that predicted by theoretical calculations. The quantitative analysis of the XANES experiments is now in progress, in order to determine accurately the local structure and the oxidation state of Co in the various DCM. Grazing incidence xray diffraction and GISAXS data are under analysis in order to complete the structural and the morphological study of Co/ZrO 2 DCM.

Figure 1 .

 1 Figure 1. In-plane TEM electron energy filtered (Co edge) images of the as-deposited ZrO 2 40 Å/Co λ/ZrO 2 40 Å discontinuous trilayers (v = 0.5 Å s-1) for (a) λ = 8 Å and (b) λ = 10 Å. The ZrO 2 matrix and Co particles are respectively of dark and light contrast.

Figure 2 .

 2 Figure 2. RT magnetization curves of as-deposited Co/ZrO 2 DCM grown at a deposition rate of 0.9 Å s -1 .

Figure 3 .

 3 Figure 3. XANES spectra obtained at the Co K-edge of the as-deposited Co/ZrO 2 (CEY-300 K) DML grown at a deposition rate of 0.5 Å s-1, compared with those of metallic Co foil, CoO and Co 3 O 4 . Spectra are shifted for clarity.

  . The normalised Co K-edge absorption spectra of DCM are superimposed on those of the CoO (2+ valence state), Co 3 O 4 (both 2+ and 3+) and Co foil (metallic state) reference. The origin of the photoelectron energy is taken at the first inflection point of the Co foil spectrum. The spectra are normalised according to the height of the absorption jump at the edge given by the Autobk program. The Co K-edge energy position is shifted to higher energy by increasing the valence state of the probed Co atoms (see the edge positions of the CoO and Co 3 O 4 (arrow) spectra). The mean valence state of Co atoms in Co/ZrO 2 DCM is much closer to 2+ than 3+. Moreover, the peak located at the maximum of the absorption edge, the so-called white line (WL), is very marked for Co 3 O 4 . The fraction of Co atoms in DCM with the chemical neighbourhood of those in Co3O4 is then likely to be negligible as can be seen by the absence of a shoulder at the Co 3 O 4 WL location in DCM spectra. In the following, DCM spectra will hence only be compared to the CoO spectrum relative to a single 2+ valence state. The influence of the O and Co atomic shells around the central Co atoms on the XANES features has been previously studied in CoO ordered crystal [29]; the six nearest O atoms contribute to an enhancement of the height of theWL compared to that of metallic Co absorption spectrum; they also lead to the appearance of the pre-edge peak A, the bump B and the valley C; The further neighbours, 12 Co and 8 O, mostly sharpen the bump B and generate a large bump D. For the as-deposited Co/ZrO 2 DCM, features B and D are very weak and the WL height is reduced. For λ = 3 and 5 Å the mean first neighbourhood composed of O atoms around the central Co atom is quite similar in the DCM to the arrangement of the first O layer in the CoO oxide but with a structural disorder. With the increase of λ from 5 to 10 and 16 Å, several variations occur: (i) the C valley shifts to lower energy (dotted curve), (ii) the weak and broad peak D, and the WL heights decrease, and (iii) a weak peak located at around 50 eV, and characteristic of the Co metal, appears on the DCM spectrum for λ = 16 Å. The DCM spectra seem then to be intermediate between those obtained for λ = 3 Å and bulk Co. Co atoms can be approximately regarded as located in two different groups of sites. The first group corresponds to Co atoms with O neighbouring atoms without long-range order. The second group corresponds to Co atoms surrounded by Co atoms in a Co metal site. When λ increases, the fraction of the Co atoms present in such a Co metal site increases. Estimation of this metallic fraction will be presented in the discussion.

Figure 4 .

 4 Figure 4. χ(k) EXAFS spectra obtained (FY-300 K) at the Co K-edge of the as-deposited Co/ZrO 2 DML (v = 0.5 Å s-1) as a function of the Co layer nominal thickness and of the Co foil reference.

Figure 4

 4 shows the χ(k) EXAFS signal for the as-deposited Co/ZrO 2 DCM and for the Co foil reference. For small nominal thicknesses, the spectra display very weak oscillations due to a broad radial distribution of Co and O first neighbouring atoms around the Co central atoms and/or to the nanometric size of the probed aggregates. The λ = 2 Å EXAFS kχ(k) signal is mostly constituted of a single damped sinusoid similar to those observed in amorphous or ill-ordered materials. Due to the respective backscattering amplitude function of O atoms (the amplitude maximum occurs at low k-values) and Co atoms, kχ(k) oscillations correspond mostly to a highly disordered O shell. A large majority of the Co atoms then do not have a complete metallic environment. In the following discussion, these atoms will be referred as non-metallic atoms. They are located in an oxide-like site, even if this site does not obviously correspond in a strict manner to those existing in bulk Co oxides. For λ ≥ 10 Å, the kχ(k) spectra are quite similar to that of bulk Co. The amplitude of the oscillations clearly increases when λ increases. Most of the nearest neighbours around Co atoms are Co atoms. Corresponding DCM can be regarded as composed of Co atoms in a metallic environment. For the intermediate thickness λ = 6 Å, Co atoms can occupy both the metallic and 'oxide' sites.

Figure 5 .

 5 Figure 5. Moduli of the FT of the kχ(k) EXAFS spectra obtained (FY-300 K) at the Co K-edge from the asdeposited Co/ZrO 2 DCM (v = 0.5 Å s-1) as a function of the Co layer nominal thickness. The spectrum corresponding to the largest Co layer thickness is similar to that of the Co foil. Inset: back FT (R-window from 1.1 to 2.8 Å) spectra of Co 10 Å/ZrO 2 (open circles) and the result of the fit (solid curve).

Figure 6 .

 6 Figure 6. Results of the fits of the kχ(k) filtered signal obtained at the Co K-edge from the as-deposited Co/ZrO 2 DCM (v = 0.5 Å s-1). (a) Mean first Co-Co distance, (b) Debye-Waller factor of the first shell Co, (c) mean nearest neighbours number of cobalt (open circles) and oxygen atoms (full circles). The dotted lines correspond to reference values deduced from the fit of the Co metallic foil spectra in the same conditions of analysis. The full curve is the plot of the model function according to the equation (4). Error bars are the statistical errors resulting from the fits.

Figure 7 .

 7 Figure 7. Estimation of the Co metallic ratio (F Co ) from the XANES signals versus the nominal thickness λ Co of the DCM for two different deposition rates: v = 0.5 and 0.9 Å s -1 (symbols). Solid curves are Co metallic ratios calculated from equations (2) with l Co-O = 2.5 and 6 Å.

( 4 )

 4 Calculations have been performed from equations (4)with the following numerical values: d = 2.505 Å (Co-Co distance in bulk Co), lCo-O = 2.5 Å (as determined in section 5.2) and α = 3.2 fixed after [42]. The results obtained are shown as a solid curve in figure (6(c)) for comparison with the experimental data (open circles).

Figure 8 .

 8 Figure 8. Schematic representation of the structural evolution of the Co layers in the Co/ZrO 2 DCM(v = 0.5 Å s -1 ). The ZrO 2 matrix and Co particles are respectively in light and dark contrast.