Claudiu Tiberiu Albulescu

Christian Aubin

Daniel Goyeau

Stock prices, inflation and inflation uncertainty in the U.S.: Testing the long-run relationship considering Dow Jones sector average indexes

Keywords: stock prices, inflation uncertainty, cointegration with structural breaks, U.S. JEL codes: C22, E31, G15

We test for the long-run relationship between stock prices, inflation and its uncertainty for different U.S. sector stock indexes, over the recent time-span, namely 2006M1-2015M5. For this purpose we use a cointegration analysis with one structural break to capture the crisis effect, and two alternative measures of inflation uncertainty relying on a time-varying unobserved component model. In line with recent empirical studies we discover that in the long-run, the inflation and its uncertainty negatively impact the stock prices, opposed to the well-known Fisher effect. However, in the short-run the results are mixed, providing evidence for complex interdependences between stock prices, inflation and its uncertainty.

Our results are robust regarding the use of a bounded or unbounded inflation trend for measuring the uncertainty, and a slight difference is noticed between different sector indexes.

Introduction

Studies regarding the impact of inflation on asset prices have a long tradition in the financial field. The starting point is considered the "The Theory of Interest" by [START_REF] Fisher | The Theory of Interest[END_REF], who advanced the idea that expected nominal return on an asset should equal the expected real return and the expected inflation (the so-called "Fisher effect"). However, the empirical literature hardly succeeded to demonstrate this effect, even if this was the common view before the 1970s. As such, noteworthy works demonstrated on contrary, that the inflation negatively impact the stock returns [START_REF] Bodie | Common stocks as a hedge against inflation[END_REF][START_REF] Jaffe | The 'Fisher Effect' for risky assets: an empirical investigation[END_REF][START_REF] Nelson | Inflation and rates of returns on common stocks[END_REF][START_REF] Fama | Asset returns and inflation[END_REF], while subsequent papers find explanation for a potential negative relationship.

First, [START_REF] Modigliani | Inflation, rational valuation and the market[END_REF] suggest that stock market investors fail to understand the inflation's effects on the nominal cash flow, and during the periods of high inflation, the valuation errors induce an undervaluation in stocks, and vice-versa. Second, [START_REF] Feldstein | Inflation and stock market[END_REF] shows that the inflation generates artificial capital gains, which are subject to taxation. Therefore, if the inflation increases the firms face higher tax liabilities. In this context, rational investors reduce common stock valuation to take into account the effect of inflation, which negatively affect the stock prices (tax-effects hypothesis). Third, [START_REF] Fama | Stock returns, real activity, inflation, and money[END_REF] shows that the inverse relationship between real stock returns and expected inflation is generated by a positive relationship between equity market and the output growth, combined with a negative relationship between expected inflation and real economic activity (the proxy hypothesis). Fourth, [START_REF] Geske | The fiscal and monetary linkage between stock returns and inflation[END_REF] and [START_REF] Kaul | Stock returns and inflation: the role of the monetary sector[END_REF] explain the negative relationship using the argument of a counter-cyclical monetary policy. They suggest that positive shocks to real output generate monetary tightening which reduces the inflation, while triggering an increase in stock prices. Fifth, [START_REF] Hess | Stock returns and inflation with supply and demand disturbances[END_REF] explain the negative relationship as a combination of demand and supply disturbances. Sixth, [START_REF] Boudoukh | Stock returns and inflation: a long-horizon perspective[END_REF] followed by others, provide a more complex explanation, arguing that stock returns are inversely related to realized and expected inflation in the short-run, but may be positively related to inflation in the long-run. However, in the same spirit, [START_REF] Sharpe | Reexamining stock valuation and inflation: the implications of analysts' earnings forecasts[END_REF] draws a new perspective on the relationship between stock prices and inflation, underlining the existence of a potential negative relationship in the long-run also. He shows that a rise in expected inflation is accompanied by either a decline in expected long-run real earnings, or by a rise in the long-run real return, or both. the link between stock prices and inflation. Starting with [START_REF] Levi | Fisher, Phillips, Friedman and the Measured Impact of Inflation on Interest[END_REF] and [START_REF] Kolluri | Anticipated price changes, inflation uncertainty, and capital stock returns[END_REF], economist describe the Friedman effect 1 and its role in a Fisherian-type relationship.

Starting from the Friedman's second reasoning which shows that the inflation uncertainty negatively impact the real output, [START_REF] Hu | The variability of inflation and real stock returns[END_REF] and [START_REF] Park | Real activity, inflation, stock returns, and monetary policy[END_REF] demonstrate that expectations of an output decline depress current stock prices, effect reinforced by greater economic uncertainty during high inflationary periods.

However, the existing empirical literature addressing the relationship between inflation and stock prices makes abstraction of inflation uncertainty, with few exceptions (for a discussion see [START_REF] Azar | The Spurious Relation between Inflation Uncertainty and Stock Returns: Evidence from the U[END_REF]. Moreover, the mixed results and the lack of distinction between the short-and long-run horizons in estimating the nexus between stock prices, inflation and its uncertainty require supplementary investigations. Therefore, the present paper's contributions to the empirical literature are three fold.

First, we focus on the characteristics of the inflation and its uncertainty, which prove to be non-stationary processes, and on the endogeneity which exists between stock prices, inflation and its uncertainty. Given these evidence, we use a cointegration approach, focusing on the recent period and employing U.S. statistics over the period 2006M1 to 2015M5 (Dow Jones index). In order to underline the effect of the recent financial crisis we resort to the Gregory-Hansen cointegration test with one structural break (Gregory and Hansen, 1996a).

The cointegration analysis allows to investigate the above-mentioned relationship at different time-horizons, and to explore the hypotheses advanced by [START_REF] Sharpe | Reexamining stock valuation and inflation: the implications of analysts' earnings forecasts[END_REF]. We focus on stock price level and not on returns as [START_REF] Cochran | Inflation's negative effects on real stock prices: new evidence and a test for a proxy effect hypothesis[END_REF] and [START_REF] Alexakis | Inflation volatility and stock prices: Evidence from ARCH effects[END_REF] did. Second, different from previous studies which associate the inflation uncertainty with the inflation variability and use different generalized autoregressive conditional heteroskedasticity (GARCH)-type models, we consider that expected inflation is generally unobservable (see [START_REF] Cukierman | A theory of ambiguity, credibility, and inflation under discretion and asymmetric information[END_REF][START_REF] Kolluri | Stock returns and expected inflation: evidence from an asymmetric test specification[END_REF]. Further, the use of GARCH-type models to measure inflation uncertainty requires stationary inflation rates, which is not the case. Therefore, we use two alternatives measures of inflation uncertainty based on the unobserved component model. On the one hand, the [START_REF] Stock | Why has U.S. inflation become harder to forecast?[END_REF] 1 The link between inflation and its uncertainty on the one hand, and between the inflation uncertainty and output on the other hand, became famous with the Friedman's Nobel lecture [START_REF] Friedman | Nobel lecture: inflation and unemployment[END_REF]. The first hypothesis of Friedman, showing the role of inflation uncertainty in explaining the level of inflation, was formalized by [START_REF] Ball | Why does high inflation raise inflation uncertainty[END_REF] (we call this Friedman -Ball hypothesis). Afterwards, several competing hypothesis where advanced and become famous, showing the positive impact of inflation on its uncertainty [START_REF] Cukierman | A theory of ambiguity, credibility, and inflation under discretion and asymmetric information[END_REF], or on contrary, a negative relationship where the inflation leads uncertainty [START_REF] Pourgerami | The effects of inflation on the predictability of price changes in Latin America: some estimates and policy implications[END_REF], or where the inflation is leaded by its uncertainty [START_REF] Holland | Inflation and uncertainty: tests for temporal ordering[END_REF]. approach, called the unobserved component model with stochastic volatility, where the inflation trend is not unbounded. On the other hand, we employ the recently proposed timevarying unobserved component model which allows for the trend inflation to be bounded in specific limits [START_REF] Chan | A New Model of Trend Inflation[END_REF]. The second approach is recommended for measuring uncertainty during inflation-targeting regimes. Although the Federal Reserve is not formally inflation targeting, it is believed to be an implicit inflation targeter [START_REF] Thornton | How Did We Get to Inflation Targeting and Where Do We Need to Go to Now? A Perspective from the U.S. Experience[END_REF]. [START_REF] Bullard | Inflation targeting in the USA[END_REF] argue that starting with January 2012, the Federal Open Market Committee (FOMC) explicitly named a numerical inflation target for the U.S. of 2%, as measured by the personal consumption expenditures price index. In this context it is believed that trend inflation and the long-term inflation expectations are not allowed to evolve in an unbounded fashion.

Third, existing studies, however, do not focus on stock markets sector indexes.

Nevertheless, the impact of inflation and its uncertainty on stock prices might differ for various categories of prices (i.e. free-market established prices versus regulated prices, raw material versus final consumption goods, etc.). Therefore, we use in our analysis the Dow Jones Composite Average index and three sector indexes namely, industrial, transportations and utilities.

The rest of the paper is structure as follows. Section 2 presents a brief literature review of papers addressing the link between stock prices and inflation, and between stock prices, inflation and its uncertainty. Section 3 describes the research methodology. Section 4 presents the data and the empirical results, while the last section concludes.

Empirical literature review

The empirical evidence suggests the presence of complexities, regarding the stock pricesinflation nexus, and it is heavily oriented toward the U.S. case. While oldest empirical studies examined this relationship at relatively short horizons, subsequent works focused on longer time-horizons [START_REF] Boudoukh | Stock returns and inflation: a long-horizon perspective[END_REF][START_REF] Solnik | A multi-country test of the Fisher model for stock returns[END_REF][START_REF] Schotman | Horizon sensitivity of the inflation hedge of stocks[END_REF], and find evidence for the Fisher effect. On contrary, other studies like [START_REF] Engsted | The relation between asset returns and inflation at short and long horizons[END_REF] discover that the Fisher effect diminishes with the horizon's increases.

Recent empirical works focus on nonlinearities which may exist between stock prices, inflation and its uncertainty [START_REF] Boyd | The impact of inflation on financial sector performance[END_REF][START_REF] Kim | The stock return-inflation puzzle and the asymmetric causality in stock returns, inflation and real activity[END_REF][START_REF] Liu | Regime switching in the dynamic relationship between stock returns and inflation[END_REF][START_REF] Maghyereh | The long-run relationship between stock returns and inflation in developing countries: further evidence from a nonparametric cointegration test[END_REF][START_REF] Karagianni | Analysing the Dynamics between U.S. Inflation and Dow Jones Index Using Non-Linear Methods[END_REF]. While for example [START_REF] Liu | Regime switching in the dynamic relationship between stock returns and inflation[END_REF] employ a regimeswitching model, [START_REF] Maghyereh | The long-run relationship between stock returns and inflation in developing countries: further evidence from a nonparametric cointegration test[END_REF] resorts to a nonparametric cointegration test. Further, for structural breaks and nonlinear causality, documenting negative nonlinear linkages between the inherent dynamics of inflation and stock returns. Further, several studies combines the time-frequency domain resorting to wavelets. [START_REF] Kim | The relationship between stock returns and inflation: new evidence from wavelet analysis[END_REF] use a six level decomposition of the U.S. inflation and stock returns and show that there is a positive relationship at the shortest and longest time-scales, while at intermediate scales stock return and inflation are negatively correlated. More recently, [START_REF] Tiwari | Stock returns and inflation in Pakistan[END_REF] use a continuous wavelet transform and document a positive relations for higher time scales, which however lacks in robustness when a different measure of inflation is used.

Few studies investigate the link between stock prices, inflation and its uncertainty (see [START_REF] Alexakis | Inflation volatility and stock prices: Evidence from ARCH effects[END_REF][START_REF] Azar | The Spurious Relation between Inflation Uncertainty and Stock Returns: Evidence from the U[END_REF]. Along these, [START_REF] Alexakis | Inflation volatility and stock prices: Evidence from ARCH effects[END_REF] examines the link between inflation uncertainty and stock prices for a group of developed and emerging economies over the period 1980M1 to 1993M12, and report a negative relationship. Recently, [START_REF] Azar | The Spurious Relation between Inflation Uncertainty and Stock Returns: Evidence from the U[END_REF] shows that inflation uncertainty dominates the inflation in explaining stock prices in the case of the U.S. over the time-span 1950M1-2011M3. However, both variables become redundant when other fundamental variables are included in the regression.

In line with these papers we investigate the nexus between stock prices, inflation and its uncertainty for the U.S. However, different from previous works, we focus on the long-run relationship and perform a sector-level analysis. Moreover, we use two newly proposed measures of inflation uncertainty, based on time-varying unobserved component models.

Methodology

Measuring the inflation uncertainty

While many researches associate the inflation uncertainty with the inflation volatility and use different GRACH-type approaches for computing the volatility, other studies relate the inflation uncertainty to the inflation gap. In the second case the unobserved component (UC) model is employed. [START_REF] Stock | Why has U.S. inflation become harder to forecast?[END_REF] approach [START_REF] Stock | Why has U.S. inflation become harder to forecast?[END_REF] propose a generalized form of the UC model in which the variances of the permanent and transitory disturbances evolve randomly over time. The new model, called the unobserved component model with stochastic volatility (UC-SV), is the following:

3.2.1.

𝜋 𝑡 = 𝜏 𝑡 + 𝜂 𝑡 , where 𝜂 𝑡 = 𝜎 𝜂,𝑡 𝜁 𝜂,𝑡 (1)
𝜏 𝑡 = 𝜏 𝑡 + 𝜀 𝑡 , where 𝜀 𝑡 = 𝜎 𝜀,𝑡 𝜁 𝜀,𝑡

(2)

𝑙𝑛𝜎 𝜂,𝑡 2 = 𝑙𝑛𝜎 𝜂,𝑡-1 2 + 𝜐 𝜂,𝑡 (3)
𝑙𝑛𝜎 𝜀,𝑡 2 = 𝑙𝑛𝜎 𝜀,𝑡-1 2 + 𝜐 𝜀,𝑡 (4)
where: 𝜋 𝑡 is the level of inflation, 𝜏 𝑡 is the inflation stochastic trend, and the 𝜂 𝑡 represents the inflation gap, with the property that lim 𝑗→∞ 𝐸 𝑡 [𝜂 𝑡+𝑗 ] = 0.

In this framework, a constant parameter model supposes 𝜏 𝑡 = 𝜏 and 𝜂 𝑡 = 𝜌 1 𝜂 𝑡-1 -⋯ -𝜌 𝑝 𝜂 𝑡-𝑝 + 𝜀 𝑡 , with 𝜀 𝑡 ~𝑁(0, 𝜎 𝜂 2 ). A particular, simply case of this model, is a local trend model with stochastic volatility in the inflation equation, which imposes that any unit root behavior manifests in the inflation trend component:

𝜏 𝑡 = 𝜏 𝑡-1 + 𝜀 𝑡 𝜏 𝜂 𝑡 = 𝜀 𝑡 𝑒𝑥𝑝 ( ℎ 𝑡 2 ) (5) ℎ 𝑡 = ℎ 𝑡-1 + 𝜀 𝑡 ℎ
where: 𝜀 𝑡 𝜏 ~𝑁(0, 𝜎 𝜏 2 ), 𝜀 𝑡 ~𝑁(0,1) and 𝜀 𝑡 ℎ ~𝑁(0, 𝜎 ℎ 2 ), errors which are independent of one another at all leads and lags.

Defined in this manner, the trend inflation is a driftless random walk, which evolves in an unbounded fashion. [START_REF] Stock | Why has U.S. inflation become harder to forecast?[END_REF] generalize the model allowing for the inflation trend to vary over time. Therefore, the stochastic volatility in the innovation to the inflation trend is:

𝜀 𝑡 𝜏 ~𝑁(0, exp(𝑔 𝑡 )), (6)
where: 𝑔 𝑡 = 𝑔 𝑡-1 + 𝜀 𝑡 𝑔 and 𝜀 𝑡 𝑔 ~𝑁(0, 𝜎 𝑔 2 ).

Chan et al.'s (2013) approach

The unbounded model of [START_REF] Stock | Why has U.S. inflation become harder to forecast?[END_REF] might be inconsistent if the monetary authority decides to intervene if inflation moves outside of a desirable range, that is, if an inflation targeting regime is in place. To overcome this limit, [START_REF] Chan | A New Model of Trend Inflation[END_REF] build upon [START_REF] Cogley | Inflation-gap persistence in the U[END_REF], who propose a different specification for the time-varying inflation process, which allows for stochastic volatility in the state equation and not in the inflation trend.

The Cogley et al.'s (2010) inflation specification is:

𝜋 𝑡 = ∅ 0𝑡-1 + ∅ 1𝑡-1 𝜋 𝑡-1 + 𝜀 𝑡 𝑒𝑥𝑝 ( ℎ 𝑡 2 ), (7)
where: ℎ 𝑡 = ℎ 𝑡-1 + 𝜀 𝑡 ℎ and ∅ 𝑡 = ∅ 𝑡-1 + 𝜀 𝑡 ∅ represents the vector of coefficients.

The authors restrict the autoregressive parameter to lies inside the unit circle, and the inflation gap becomes:

𝜂 𝑡 = 𝜏 𝑡+1 - ∅ 0𝑡 1-∅ 1𝑡 (8)
where:

𝜂 𝑡+1 = ∅ 1𝑡 (𝜋 𝑡 - ∅ 0𝑡 1-∅ 1𝑡 ) + 𝜀 𝑡+1 𝑒𝑥𝑝 ( ℎ 𝑡+1 2
). [START_REF] Chan | A New Model of Trend Inflation[END_REF] advance in their turn an unobserved components model with an autoregression in the transitory component, considering the first order autoregression case:

(𝜋 𝑡 -𝜏 𝑡 ) = 𝜌 𝑡 (𝜋 𝑡-1 -𝜏 𝑡-1 ) + 𝜀 𝑡 𝑒𝑥𝑝 ( ℎ 𝑡 2 ) (9)
where: 𝜏 𝑡 = 𝜏 𝑡-1 + 𝜀 𝑡 𝜏 , ℎ 𝑡 = ℎ 𝑡-1 + 𝜀 𝑡 ℎ and 𝜌 𝑡 = 𝜌 𝑡-1 + 𝜀 𝑡 𝜌 , while 𝜀 𝑡 ~𝑁(0,1) and

𝜀 𝑡 ℎ ~𝑁(0, 𝜎 ℎ 2 ).
For bounding the inflation trend, the authors propose an innovation in the state equation:

𝜀 𝑡 𝑇 ~𝑇𝑁(𝑎 -𝜏 𝑡-1 , 𝑏 -𝜏 𝑡-1 ; 0, 𝜎 𝜏 2 ) ( 10
)
where: 𝑇𝑁(𝑎, 𝑏; 𝜇, 𝜎 2 ) is the Gaussian distribution with the mean 𝜇 and the variance 𝜎 2 truncated to the interval (𝑎, 𝑏) and

𝐸 𝑡 [𝜏 𝑡+1 ] = 𝜏 𝑡 + 𝜎 𝜏 [ ∅( 𝑎-𝜏 𝑡 𝜎 𝑡 )-∅( 𝑏-𝜏 𝑡 𝜎 𝜏 ) Φ( 𝑏-𝜏 𝑡 𝜎 𝜏 )-Φ( 𝑎-𝜏 𝑡 𝜎 𝜏 )
] is the conditional expectation with 𝑎 ≤ 𝜏 𝑡 ≤ 𝑏. ], with 𝑎 𝜌 ≤ 𝜌 𝑡 ≤ 𝑏 𝜌 , and the bounds are generated in order to satisfy the inflation gap long-run zero convergence condition (𝜌 𝑡 is limited to lies inside the unit circle or within constant limits).

Gregory-Hansen cointegration test

Our general equation is:

𝐿𝑁𝐷𝐽 = 𝑐 + 𝛼 1 𝐼 + 𝛼 2 𝑈 + 𝜀 𝑡 (12)
where: 𝐿𝑁𝐷𝐽 is the natural log of the DJ composite and sector average indexes (DJCA, DJIA, DJTA, DJUA); 𝑐 is the intercept; 𝐼 is the CPI inflation; 𝑈 represents the inflation uncertainty (U1, U2); 𝜀 𝑡 are the error terms.

In order to capture the long-run relationship between variables, usually 𝑐 and 𝛼 are considered time-invariant. However, Gregory and Hansen (2006a) consider that, if the cointegration holds over some periods of time, it may shift toward a new long-run relationship. They treat the timing and shifts as unknown and allows changes in the intercept and slope, defining the following dummy variable:

𝜑 𝑡𝜏 = { 0, 𝑖𝑓 𝑡 ≤ [𝑛𝜏] 1, 𝑖𝑓 𝑡 > [𝑛𝜏] (13)
The cointegration test proposed by Gregory and Hansen (2006a)

The test statistics computation in Gregory and Hansen (1996a) is based on the Ordinary Least Square (OLS) approach and each model yields the residuals 𝜀t 𝜏 , where the subscript τ

shows that the residual sequence depends on the choice of the change point τ. The first-order serial correlation coefficient is:

𝜌 ̂𝜏 = ∑ 𝜀̂𝑡 𝜏 𝑛-1 𝑡=1 𝜀̂𝑡 +1𝜏 ∑ 𝜀̂𝑡 𝜏 2 𝑛-1 𝑡=1 ⁄ (18)
The second-stage residuals are defined as 𝜐 ̂𝑡𝜏 = 𝜀̂𝑡 𝜏 -𝜌 ̂𝜏𝜀̂𝑡 -1𝜏 , while the estimate of long-run variance of 𝜐 ̂𝑡𝜏 is 𝜎 ̂𝜏2 = 𝛾 ̂𝜏(0) + 2𝜆 ̂𝜏. The estimate of the bias-corrected first-order serial correlation coefficient is: Gregory and Hansen (1996a) propose three test statistics, namely two Phillips statistics and one ADF statistic. The Phillips statistics are the following:

𝜌 ̂𝑡 * = ∑ (𝜀̂𝑡 𝜏 𝑛-1 𝑡=1 𝜀̂𝑡 +1𝜏 -𝜆 ̂𝜏) ∑ 𝜀̂𝑡 𝜏 2 𝑛-1 𝑡=1 ⁄ (19)
𝑍 𝛼 (𝜏) = 𝑛(𝜌 ̂𝑡 * -1) (20)
𝑍 𝑡 (𝜏) = (𝜌 ̂𝑡 * -1)/𝑠̂𝜏 (21)

where:

𝑠τ 2 = 𝜎 ̂𝜏2 ∑ 𝜀̂𝑡 𝜏 2 𝑛-1 1 ⁄ .
The ADF statistics is calculated by regressing Δ𝜀t 𝜏 upon 𝜀t -1𝜏 and Δ𝜀t -1𝜏 , …, Δ𝜀t -𝐾𝜏 for suitably chosen lag truncation K. As such, the ADF statistics is the t-statistics for the regressor 𝜀t -1𝜏 , denoted by:

𝐴𝐷𝐹(𝜏) = 𝑡𝑠𝑡𝑎𝑡(𝜀t -1𝜏 ) (22

Unit root tests results

For checking the presence of unit roots in our series, we resort to the classic Augmented Dicky-Fuller test (ADF) and Phillips-Perron test (PP). The ADF test has a low power against the null and usually if the test indicates that the variables are stationary in first difference, there is no point to use alternative tests that have more power against the null. However, because the Gregory-Hansen's approach is an extension of unit root tests with structural breaks, we also use the Zivot and Andrews's test with one structural break (ZA) for assessing the presence of unit roots.

Cointegration test

Stock and Watson's (2007) uncertainty results

The results of the Gregory-Hansen cointegration tests for the [START_REF] Stock | Why has U.S. inflation become harder to forecast?[END_REF] inflation uncertainty (U1) are presented in Table 2. Several findings can be advanced. First, for all the indexes except for transportation (DJTA), two out of four models (level shift with trend and regime shift with trend changes) confirm the cointegration relationship. Second, there is a large agreement between the ADF and Phillips statistics. Third, it appears that the cointegration between stock prices, inflation and inflation uncertainty is stronger for the DJIA and DJCA as compared to DJUA, which can be explained by the fact that energy prices are not completed "deregulated". Finally, in the case of DJTA, the cointegration appears for the level shift model, both in the case of Phillips and ADF tests, but only at 10% and respectively 5% level of significance. However, there is no consensus between the two Phillips tests, which put in question the long-run relationship between our variables in the case of the DJTA index. The last observation shows that the transportation sector is not very affected by the level of inflation and its uncertainty probably because the fuel prices are not highly correlated with overall consumption prices in the U.S.

Chan et al.'s (2013) uncertainty results

In order to check for the robustness of these findings and to compare the results in the case of alternative uncertainty definition, we proceed to test the cointegration relationship using the U2 uncertainty measure proposed by [START_REF] Chan | A New Model of Trend Inflation[END_REF]. Table 3 shows similar results for the DJCA, DJIA and DJUA, where the cointegration is noticed for the level shift with trend model, and for the regime shift with trend model. However, different form the U1 results, for the U2 in the case of DJTA, the cointegration appears for the regime shift model, but only at 10% level of significance. All in all we conclude that our results are robust.

Cointegration equations

Tables 2 and3 show that two out of four models reject the null hypothesis of no cointegration for DJCA, DJIA and DJUA. Therefore, in order to select the best model we proceed to the estimation of cointegration equations for these two models, using the Engle-Granger method and the OLS technique. For DJTA index the cointegration equation is computed for that model for which both the ADF and Phillips statistics confirm the existence of the long-run relationship. The results for U1 are presented in Table 4 bellow.

In the case of the three indexes where the cointegration is documented by the GH-LST and GH-RST models, we notice that the regime shift model with trend (GH-RST) better explains the influence of inflation and its uncertainty on stock prices. In all the cases the trend and the dummy variable are significant, meaning that this relationship is nonlinear over time.

Nevertheless, several differences appear when we compare the DJCA, DJIA and DJUA indexes. On the one hand, for the DJIA both the inflation and its uncertainty negatively impact the stock prices. On the other hand, in the case of DJCA only the inflation has a negative impact, while in the case of DJUA the uncertainty is the one influencing the stock prices. In all the situations the impact is negative, showing that investors are sensitive to macroeconomic environment and to the uncertainty regarding the macroeconomic context (i.e. the general level of prices), pointing against the Fisher effect. This evidence also point in the favor of the Friedman -Ball and Cukierman -Meltzer hypotheses, which sustain that the level of inflation and its uncertainty move in the same direction. Finally, in the case of DJTA a different situation appears. For the GH-LS model the inflation has a positive impact on the stock prices, while the uncertainty has a negative influence. We continue our analysis with the cointegration equations for U2. Table 5 shows similar results in the case of DJCA, DJIA and DJUA indexes. Moreover, in this case the GH-RST model documents the negative influence of both the inflation and the inflation uncertainty, for all three indexes. Consequently, when the inflation uncertainty is computed based on a bounded inflation trend, the role of macroeconomic context in explaining the stock prices is stronger. The results confirm than the [START_REF] Fama | Stock returns, real activity, inflation, and money[END_REF] argument who sustain that stock prices are correlated with business cycles, being negatively impacted by the level of inflation. 5 the GH-RS model employed in the case of DJTA index shows that both inflation and its uncertainty negatively impact the stock prices. Once again the findings of previous papers are validated, and the fact that the inflation and its uncertainty move together, provide indirect evidence in the favor of Friedman -Ball and Cukierman -Meltzer hypotheses.

Different from Table 4, in Table

After estimating and interpreting the cointegration relationships, we proceed to the estimation of the short-run relationship between stock prices, inflation and inflation uncertainty in the U.S., using an Error Correction Model (ECM).

Error correction model

To estimate the short-run dynamic equation we obtain the residuals from the representatives models (the GD-RST model for DJCA, DJIA, DJUA and LS/RS model for DJTA). After obtaining residual series (called ECM), we can estimate the error correction models for the relationship between stock prices, inflation and inflation uncertainty. Table 6 presents the results for U1. We notice that for DJCA and DJIA indexes the inflation and its uncertainty have no influence on the stock prices in the short-run. However, a positive short-run influence is documented for the DJTA and DJUA indexes. In the case of the DJUA index the short-run result contrasts the long-run evidence. While in the long-run the inflation and its uncertainty negatively impact the stock prices, in the short-run the opposite phenomenon is recorded. The result can be explained by the fact that energy prices are connected to the general level of prices in the short-run. An increased demand of consumption goods on the market leads to higher prices and an increased production, which trigger the energy prices, and vice-versa.

Further, the speculative traders contribute to this mechanism, being guided by the present economic context. On contrary, the institutional long-run investors, which are sensitive to the macroeconomic fundamentals, perceive the level of inflation and its increased uncertainty as a threat for the economy. This negative sentiment determines a decreased interest for the stock market if the inflation increases. Finally, the role of business cycles cannot be neglected in explaining these results.

We proceed further to test for the stability of the analyzed relationship. In this respect we use a CUSUM test for the coefficient stability (Figure 1) and a CUSUM SQUARES test for the residuals stability (see Appendix). The results show that all coefficients are stable in the case of ECM models, but instability in the residuals is recorded around the start-up of the global crisis. The last part of our analysis is dedicated to the robustness check for the results of ECM models, using this time U2. The coefficient of ECM t-1 is negative and significant which prove the existence of the long-run relationship between our variables. The results are similar for the DJCA and DJIA indexes, and expected for the DJTA index, where a different model applies compared to U1 (GH-RS instead of GH-LS). However, in the case of DJUA the findings are different from those reported in Table 6. The new results show a negative shortrun influence of the inflation uncertainty, in agreement with the long-run situation. In this case also the CUSUM test documents the stability of coefficients (Figure 2) but a relative instability of the residuals in crisis times (Appendix). On the whole, we document that the short-run relationship between stock prices, inflation and inflation uncertainty is less consistent compared to the long-run relationship.

Further, in the case of the short-run equation the empirical evidence is mixed and the results are influenced by the way inflation uncertainty is defined. However, in the long-run there is a clear and robust relationship between our variables and in general the inflation and its uncertainty negatively impact the stock prices, results in agreement with the findings reported by [START_REF] Cochran | Inflation's negative effects on real stock prices: new evidence and a test for a proxy effect hypothesis[END_REF], [START_REF] Alexakis | Inflation volatility and stock prices: Evidence from ARCH effects[END_REF] and [START_REF] Karagianni | Analysing the Dynamics between U.S. Inflation and Dow Jones Index Using Non-Linear Methods[END_REF], and opposed to those advanced by [START_REF] Boudoukh | Stock returns and inflation: a long-horizon perspective[END_REF] early on.

Conclusions

In this paper, we use the 2013) approach, with a bounded inflation trend, the second approach being recommended in the case of inflation targeting regimes.

Our findings document the existence of the long-run relationship between variables. In particular, the cointegration relationship is shown by two Gregory-Hansen tests for DJCA, DJIA and DJUA (level shift with trend and regime shift with trend), and by one test for DJTA. The cointegration equations provide evidence in the favor of a negative impact of the inflation and its uncertainty on the stock prices, except for the DJTA index where the results are mixed. These results are opposed to the Fisher's effect but in agreement with recent findings advanced by the empirical literature and with the arguments advanced by [START_REF] Fama | Stock returns, real activity, inflation, and money[END_REF], underlining the role of business cycles and the correlation existing between the real economy and equity markets.

However, the short-run equations show less consistent results, where the influence of the inflation and its uncertainty either do not exists, or it is influenced by the way uncertainty is defined. In general the inflation and its uncertainty have a similar impact on stock prices, meaning that they move together, indirectly supporting the Friedman -Ball and Cukierman -Meltzer hypotheses.

The present paper presents however some limits because we test the so called long-run relationship over a relatively reduced time-span (10 years). Therefore, a development of this work will be to address a longer time-horizon and more sector indexes to underline the differences which might exists in the perception of inflation and its uncertainty for different industries and economic sectors.

  from FED St. Louis (FRED database), having a monthly frequency and covering the period 2006M1 to 2015M5. The DJ average indexes data are available in the FRED database starting with 2005M12 only. This constraint forced us to limit the analysis to the indicated time-span, which however covers the pre-and post-crisis period. The CPI inflation is considered, and four DJ average indexes expressed in natural log (Industrial, Utility, and Transportation). The DJCA index is a combination of all other three major DJ sector average indexes. Each DJ sector average index is made up of a select group of prominent stocks. DJIA includes 30 companies acting in the manufacture of industrial and consumer goods but also financial services, entertainment and information technology. DJIA is one of the most representative indexes of the U.S. stock market and by extension, of the U.S. economy. DJTA is the oldest American index which includes different transportation companies. Given the fact that transportations represent the link between the manufacturers of goods and the customers, it needs to confirm the DJIA trend. Finally, the DJUA index refers to electricity and natural gas companies. In the U.S. the utilities' prices have been significantly "deregulated" starting with the 1990s. Compared to the other sector indexes, the utility index composition has remained largely unaltered over time.

  i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively; (ii) standard errors in brackets; (iii) Dum means that the dummy variable is unity after this date and zero otherwise; (iv) the break is considered based on the Z t test.

Figure 1 .

 1 Figure 1. CUSUM test -Stock and Watson's (2007) inflation uncertainty

Figure 2 .

 2 Figure 2. CUSUM test -Chan et al.'s (2013) inflation uncertainty

  Gregory-Hansen cointegration test to estimate the long-run relationship between stock prices, inflation level and inflation uncertainty in the U.S. The stock prices are estimated based on the DJ composite average index and there different sector indexes (industrial, transportation and utilities), during the period from 2006M1 to 2015M5. The inflation uncertainty is associated with the inflation gap and is computed based on the Stock and Watson's (2007) model with time-varying unbounded inflation trend, and based on the Chan et al.' (

  𝐿𝑁𝐷𝐽 = 𝑐 1 + 𝑐 2 𝜑 𝑡𝜏 + 𝛽 1 𝑡 + +𝛽 1 𝑡𝜑 𝑡𝜏 + 𝛼 1 𝐼 + 𝛼 2 𝐼𝜑 𝑡𝜏 + 𝛼 3 𝑈 + 𝛼 4 𝑈𝜑 𝑡𝜏 + 𝜀 𝑡

		accommodates a
	single endogenous break. The authors propose three models with assumptions about
	structural breaks in the intercept and slope. Afterwards, in Gregory and Hansen (2006b) a
	forth model is advanced, allowing for breaks in the trend also. Based on these four models,
	our equations become:
	Model 1 -Gregory and Hansen's (1996a) test: Level shift (GH-LS)
	𝐿𝑁𝐷𝐽 = 𝑐 1 + 𝑐 2 𝜑 𝑡𝜏 + 𝛼 1 𝐼 + 𝛽𝛼 2 𝑈 + 𝜀 𝑡	(14)
	Model 2 -Gregory and Hansen's (1996a) test: Level shift with trend (GH-LST)
	𝐿𝑁𝐷𝐽 = 𝑐 1 + 𝑐 2 𝜑 𝑡𝜏 + 𝛽𝑡 + 𝛼 1 𝐼 + 𝛼 2 𝑈 + 𝜀 𝑡	(15)

Model 3 -

Gregory and Hansen's (1996a)

test: Regime shift (GH-RS)

𝐿𝑁𝐷𝐽 = 𝑐 1 + 𝑐 2 𝜑 𝑡𝜏 + 𝛼 1 𝐼 + 𝛼 2 𝐼𝜑 𝑡𝜏 + 𝛼 3 𝑈 + 𝛼 4 𝑈𝜑 𝑡𝜏 + 𝜀 𝑡

(16)

Model 4 -

Gregory and Hansen's (1996b)

test: Regime shift with trend change (GH-RST)

Table 1

 1 shows that the ADF and PP tests cannot reject the null of unit root presence in level, for any of the selected variables. On contrary the stationarity is documented in the first difference for all variables, although to a smaller extent for theChan et al.'s (2013) inflation uncertainty variable (U2). The ZA test presents similar findings except for the stock prices for which the stationarity is documented in level also. All in all we conclude that our variables are non-stationary in level but stationary in first difference and we can use the cointegration analysis.

Table 1 .

 1 Unit root tests

	Variables	ADF test		PP test
		Level	First difference	Level	First difference
	DJCA	-0.20	-9.92***	-0.21	-10.7***
	DJIA	0.05	-8.49***	0.12	-8.38***
	DJTA	-0.28	-8.82***	-0.64	-8.90***
	DJUA	-0.78	-8.73***	-1.03	-8.70***
	I	-1.79	-6.22***	-2.50	-6.10***
	U1	3.32	-4.33***	1.41	-4.50***
	U2	-0.91	-1.84	-1.74	-2.84*
	Variables	ZA test
		Level	Break	First difference	Break
	DJCA	-9.64***	2007M9	-10.3***	2007M10
	DJIA	-8.96***	2007M9	-10.3***	2007M10
	DJTA	-4.79*	2008M9	-9.88***	2009M4
	DJUA	-6.29***	2008M8	-9.20***	2009M4
	I	-4.30	2008M10	-6.95***	2009M8
	U1	-1.01	2007M10	-6.91***	2013M11
	U2	-3.48	2010M7	-6.01***	2009M4
	Notes: (i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively and rejecting H0 of unit
	root presence; (ii) the natural log of stock indexes is considered.

Table 2 .

 2 Gregory-Hansen cointegration tests -Stock and Watson's (2007) inflation uncertainty

	Models	ADF	Break	Z t	Break	Z α	Break
	DJCA
	GH-LS	-3.20	2008M8	-3.68	2008M8	-16.7	2008M8
	GH-LST	-9.09***	2007M5	-7.29***	2010M5	-70.0***	2010M5
	GH-RS	-3.11	2010M1	-3.57	2010M1	-17.2	2010M1
	GH-RST	-9.06***	2012M3	-7.32***	2007M5	-71.5***	2007M5
	DJIA
	GH-LS	-3.19	2008M10	-3.68	2008M8	-16.6	2008M8
	GH-LST	-8.72***	2007M5	-5.81***	2010M12	-50.8*	2010M12
	GH-RS	-3.07	2010M1	-3.52	2010M1	-17.7	2010M1
	GH-RST	-9.18***	2008M9	-7.43***	2007M5	-73.5**	2007M5
	DJTA
	GH-LS	-4.85*	2013M3	-5.25**	2013M3	-34.0	2013M3
	GH-LST	-4.84	2008M4	-5.16*	2013M3	-34.5	2013M3
	GH-RS	-4.86	2013M3	-5.27*	2013M3	-34.1	2013M3
	GH-RST	-4.70	2011M5	-5.00	2008M10	-34.7	2008M10
	DJUA
	GH-LS	-3.64	2012M9	-3.91	2012M8	-19.4	2012M8
	GH-LST	-6.05***	2008M11	-5.86***	2008M10	-53.5*	2008M10
	GH-RS	-3.84	2008M4	-4.03	2008M5	-22.6	2008M5
	GH-RST	-6.91***	2008M9	-6.94***	2008M10	-67.3*	2008M10
	Notes: (i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively and rejecting H0 of no
	cointegration; (ii) the natural log of stock indexes is considered.

Table 3 .

 3 Gregory-Hansen cointegration tests -Chan et al.'s (2013) inflation uncertainty

	Models	ADF	Break	Z t	Break	Z α	Break
	DJCA
	GH-LS	-4.10	2007M12	-4.03	2008M1	-23.7	2008M1
	GH-LST	-8.98***	2007M6	-7.23***	2007M11	-70.0***	2007M11
	GH-RS	-3.75	2007M12	-3.77	2007M7	-21.6	2007M7
	GH-RST	-11.2***	2007M5	-8.13***	2007M7	-82.2***	2007M7
	DJIA
	GH-LS	-4.05	2007M12	-3.97	2008M1	-23.4	2008M1
	GH-LST	-8.65***	2007M6	-5.93***	2007M9	-52.0*	2007M9
	GH-RS	-3.75	2007M12	-3.82	2007M7	-21.4	2007M7
	GH-RST	-11.8***	2007M7	-10.7***	2007M5	-114.***	2007M5
	DJTA
	GH-LS	-4.36	2013M12	-4.49	2013M8	-30.6	2013M8
	GH-LST	-4.29	2008M10	-4.53	2008M11	-33.0	2008M11
	GH-RS	-5.42*	2013M4	-5.48*	2013M5	-40.8	2013M5
	GH-RST	-5.58	2011M5	-5.64	2011M6	-44.0	2011M6
	DJUA
	GH-LS	-2.80	2013M6	-3.07	2013M11	-14.0	2013M11
	GH-LST	-6.05***	2008M11	-5.85***	2008M8	-53.5*	2008M8
	GH-RS	-3.29	2013M5	-3.47	2013M5	-17.2	2013M5
	GH-RST	-6.61***	2009M1	-6.34**	2009M4	-55.5	2009M4
	Notes: (i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively and rejecting H0 of no
	cointegration; (ii) the natural log of stock indexes is considered.

Table 4 .

 4 Cointegration equations between stock prices, inflation and Stock and Watson's

	(2007) inflation uncertainty
	DJCA	GH-LS (Dummy 2008M8)	GH-LST (Dummy 2010M5)	GH-RS (Dummy 2010M1)	GH-RST (Dummy 2007M5)
	Intercept		8.210*** (0.004)		8.250*** (0.045)
	Dum × Intercept		0.000	(0.001)		-0.041	(0.046)
	Trend		0.008*** (0.000)		0.007*** (0.000)
	Dum × Trend					0.000** (0.000)
	I		-0.000	(0.000)		-0.005** (0.002)
	Dum × I					0.005*** (0.002)
	U1		-0.000	(0.001)		-0.009	(0.023)
	Dum × U1					0.009	(0.023)

Table 5 .

 5 Cointegration equations between stock prices, inflation andChan et al.'s (2013)

	inflation uncertainty
	DJCA	GH-LS (Dummy 2008M1)	GH-LST (Dummy 2007M11)	GH-RS (Dummy 2007M7)	GH-RST (Dummy 2007M7)
	Intercept		8.211*** (0.002)		8.286*** (0.011)
	Dum × Intercept		0.000	(0.001)		-0.077*** (0.012)
	Trend		0.008*** (0.000)		0.007*** (0.000)
	Dum × Trend					0.000** (0.000)
	I		-0.000	(0.000)		-0.013*** (0.002)
	Dum × I					0.013*** (0.002)
	U2		-0.000	(0.000)		-0.080*** (0.014)
	Dum × U2					0.080*** (0.014)
	DJIA	GH-LS (Dummy 2008M1)	GH-LST (Dummy 2007M9)	GH-RS (Dummy 2007M7)	GH-RST (Dummy 2007M5)
	Intercept		9.286*** (0.003)		9.483*** (0.045)
	Dum × Intercept		0.003	(0.002)		-0.204*** (0.045)
	Trend		0.008*** (0.000)		0.007*** (0.000)
	Dum × Trend					0.001*** (0.000)
	I		-0.001** (0.000)		-0.010*** (0.002)
	Dum × I					0.010*** (0.002)
	U2		-0.002*	(0.001)		-0.075*** (0.023)
	Dum × U2					0.075*** (0.023)
	DJTA	GH-LS

(Dummy 2013M8) GH-LST (Dummy 2008M11) GH-RS (Dummy 2013M5) GH-RST (Dummy 2011M6)

  Dum means that the dummy variable is unity after this date and zero otherwise; (iv) the break is considered based on the Z t test.

	Intercept				8.621*** (0.028)
	Dum × Intercept				-0.104	(0.157)
	Trend
	Dum × Trend
	I				-0.017** (0.009)
	Dum × I				0.126** (0.057)
	U2				-0.179*** (0.013)
	Dum × U2				0.906*** (0.206)
	DJUA	GH-LS (Dummy 2013M11)	GH-LST (Dummy 2008M8)	GH-RS (Dummy 2013M5)	GH-RST (Dummy 2009M4)
	Intercept		6.014*** (0.026)		6.165*** (0.025)
	Dum × Intercept		-0.429*** (0.021)		-0.629*** (0.052)
	Trend		0.006*** (0.000)		0.011*** (0.000)
	Dum × Trend					-0.003*** (0.000)
	I		0.006	(0.005)		-0.024*** (0.006)
	Dum × I					0.031*** (0.010)
	U2		0.005	(0.010)		-0.253*** (0.016)
	Dum × U2					0.268*** (0.021)
	Notes: (i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively; (ii) standard errors in
	brackets; (iii)

Table 6 .

 6 ECM -Stock and Watson's (2007) inflation uncertainty

		DJCA: GH-RST	DJIA: GH-RST	DJTA: GH-LS	DJUA: GH-RST
		(Dummy 2007M5)	(Dummy 2007M5)	(Dummy 2013M3)	(Dummy 2008M10)
	Intercept	0.008*** (0.000)	0.009*** (0.000)	0.008	(0.005)	0.004	(0.003)
	ECM t-1	-0.859*** (0.079)	-1.008*** (0.078)	-0.041	(0.049)	-0.200*** (0.068)
	ΔI	0.001	(0.000)	-0.000	(0.000)	0.018*	(0.010)	0.018*** (0.006)
	ΔU1	0.003	(0.005)	0.002	(0.005)	0.085	(0.075)	0.083*	(0.048)
	R 2	0.524		0.622		0.058		0.125
	Notes: (i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively; (ii) standard errors in
	brackets; (iii) ECM t-1 is the lagged error correction term.

Table 7 .

 7 ECM -Chan et al.'s (2013) inflation uncertainty

		DJCA: GH-RST	DJIA: GH-RST	DJTA: GH-RS	DJUA: GH-RST
		(Dummy 2007M7)	(Dummy 2007M5)	(Dummy 2013M5)	(Dummy 2009M4)
	Intercept	0.008*** (0.000)	0.009*** (0.000)	0.007	(0.004)	0.003	(0.003)
	ECM t-1	-1.091*** (0.073)	-1.003*** (0.078)	-0.168*** (0.055)	-0.345*** (0.078)
	ΔI	0.001*	(0.000)	-0.001	(0.000)	0.000	(0.010)	0.005	(0.006)
	ΔU2	0.001	(0.002)	-0.001	(0.003)	-0.152*** (0.041)	-0.081*** (0.028)
	R 2	0.678		0.621		0.206		0.678
	Notes: (i) *, ** and *** implies significance at 10%, 5% and 1% levels respectively; (ii) standard errors in
	brackets; (iii) ECM t-1 is the lagged error correction term.

Appendix -CUSUM SQAURES test


K Komatsu

D L'hôte

S Nakamae

F Ladieu

V Mosser

A Kerlain

M Konczykowski

E Dubois

V Dupuis

R Perzynski

Magnetic noise of a frozen ferrofluid

The magnetic noise of a frozen ferrofluid made of maghemite nanoparticles dispersed in glycerin is measured using a two-dimensional electron gas based quantum well Hall sensor ͑QWHS͒ with the spinning current technique. The frozen ferrofluid shows a superspin glass transition at 67 K. Below this glass transition temperature, the relation between the imaginary part of the ac susceptibility of a bulk ferrofluid sample and the magnetic noise measured with the QWHS gives an indication of a violation of the fluctuation dissipation theorem. © 2010 American Institute of Physics. ͓doi:10.1063/1.3366613͔

I. INTRODUCTION

Understanding the collective behavior of nanomagnet assemblies is an important problem for statistical physics as well as for applications such as magnetic storage, spintronics, magnetic sensors, and biomedicine. Magnetic noise measurement is a remarkably useful way to understand the microscopic dynamics of interacting nanomagnets because it directly reflects the fluctuations of the magnetic moments. At equilibrium, the magnetic fluctuations are related to the response to a small field through the fluctuation-dissipation theorem ͑FDT͒: Љ ϰ Sf / T, where Љ is the dissipative part of the magnetic susceptibility, S is the magnetic noise power spectrum, f is the measurement frequency, and T is the temperature. A frozen "ferrofluid" consists of monodomain nanomagnets suspended in a solid matrix ͑in our case, frozen glycerin͒. The individual nanomagnets have a large magnetic moment of the order of ϳ10 4 B , and thus it is called "su- perspin." When the nanomagnets are sufficiently concentrated at low enough temperature, long-range dipolar interactions among them produce spin-glass behavior ͑aging, memory, etc.͒. 1 The system is thus called a "superspin glass."

In the 1990s, theoretical investigations of the spin and structural glasses relaxation through dynamical studies started and allowed to make some predictions on the nature of the violations of FDT, which could be expected in such out-of-equilibrium systems. 2 In particular, the concept of "effective temperature" ͑T eff ͒, which allows to extend the FDT to the off-equilibrium state was proposed. It is defined by Љ ϰ Sf / T eff , with the same proportionality factor as that at equilibrium.

These FDT violations have also been experimentally investigated for a few structural glasses 3 and one spin glass system 4 at temperatures lower than the glass transition temperature where the system is off equilibrium. The system then shows aging, which is an evolution toward its equilibrium state with time. Although spin glass is one of the most studied glassy systems, the FDT violation in spin glasses has been measured for only one system due to the difficulty of detecting extremely small noises ϳ10 -7 G. 4 To overcome this problem, we have measured the magnetic noise of a frozen ferrofluid in the superspin glass state with a high resolution two-dimensional electron gas ͑2DEG͒ based quantum well Hall sensor ͑QWHS͒. The left panel in Fig. 1 shows the optical micrograph of the QWHS. Note two advantages of using a superspin glass and QWHS. First the large susceptibility ͑real part of ac susceptibility Ј and imaginary part of ac susceptibility Љ͒ of superspin glasses 1 ensures a large noise power. Second the distance from the 2DEG layer to the sample is ϳ1 m, thus the measured magnetic noise is much larger than that of the bulk sample measurements using superconducting quantum interference device ͑SQUID͒ sensor because the magnetic field of the nanomagnets decreases with the distance r as 1 / r 3 . 5

II. EXPERIMENT

We use a QWHS based on pseudomorphic AlGaAs/ InGaAs/GaAs heterostructures. [6][START_REF] Mosser | Proc. SPIE[END_REF][8] The nominal Hall cross a͒ Author to whom correspondence should be addressed. Electronic mail: katsuyoshi.komatsu@cea.fr. area of ϳ2 ϫ 2 m 2 ͑estimated effective area ϳ1.6 ϫ 1.6 m 2 ͒ is located ϳ1 m ͑according to the maker's design specification; estimated real distance is ϳ650 nm͒ beneath the probe surface. We also use the spinning current technique, which simultaneously suppresses the low frequency noise and the Hall voltage offset. This is achieved by switching the current injection and the voltage detection directions at a spinning frequency f spin . In our experiment, f spin = 1 kHz. The effectiveness of this method has been recently reported by Kerlain and Mosser. 8 Using this technique, we achieve a sensitivity of ϳ2 mG/ ͱ Hz at 0.01 Hz and at temperatures ranging from 20 to 85 K.

The ferrofluid used in this experiment is made of maghemite ␥-Fe 2 O 3 nanoparticles dispersed in glycerin with a volume fraction of 15%, prepared following the method of Ref. 9. The nanoparticles average diameter and magnetic moment are 8.6 nm and 10 4 B , respectively. A small 200 pl drop of ferrofluid was deposited directly on the probe surface. At low temperatures below 190 K, the fluid ͑glycerin͒ is frozen and the only remaining magnetic degree of freedom is that of the particle magnetic moments. These moments ͑superspins͒ interact through magnetic dipolar interactions leading to a superspin-glass transition at the temperature, T g Ϸ 67 K. 10 The detailed characteristics of the sample can be found in Ref. 10. The dc and ac susceptibilities of the bulk ferrofluid ͑approximately 1.5 l͒ were measured with a commercial SQUID magnetometer.

III. RESULTS AND DISCUSSION

Prior to measuring the magnetic fluctuation signal of the frozen ferrofluid, we characterized the Hall voltage noise spectra of a pristine Hall probe at various temperatures. Subsequently, the small drop of the ferrofluid was deposited on the probe and cooled down to temperatures ranging from 20 to 85 K. The right panel of Fig. 1 shows the optical micrograph of the ferrofluid dropped on the surface of the QWHS. The ferrofluid is liquid at room temperature but is frozen by cooling below 190 K.

The left panel of Fig. 2 shows the power spectrum density of the frozen ferrofluid ͑denoted by open symbols͒ obtained by subtracting the Hall probe noise ͑cross symbol͒ from the total power spectrum density ͑filled symbol͒ as a function of frequency f, at 45 K in zero applied field. The right panel of Fig. 2 shows the power spectrum density of the ferrofluid at various temperatures between 20 and 85 K. It exhibits an almost 1 / f dependence in the frequency range from 0.08 to 8 Hz. Here the power spectrum density was averaged over about 1 h after waiting for the temperature stabilization cooling from 85 K ͑about 10 min͒. The left panel of Fig. 3 shows the temperature dependence of the magnetic zero-field-cooled and field-cooled susceptibilities for dc measurement and Ј and Љ for ac measurement with a SQUID magnetometer measured at temperatures from 5 to 200 K. The measurement field was 1 Oe, which is sufficiently small for the linear response, for both dc and ac measurements. If the power spectrum density S, obeys FDT, then Љ ϰ Sf / T should hold. The right panel of Fig. 3 shows Љ of the bulk sample plotted as a function of Sf / T for various temperatures and frequencies. The solid line indicates the linear relation corresponding to Љ ϰ Sf / T, which is true only in the high temperature region above T g , where the system is in the equilibrium ͑super͒paramagnetic state. Interestingly, the plots deviate from this equilibrium line at temperatures below T g where the system is in the out-of-equilibrium ͑su-per͒spin glass state. This deviation, therefore, can be considered as an indication of the FDT violation below T g . A beyond-doubt confirmation of the existence of the FDT violation should be obtained through further investigations using various samples, other QWHS's detectors and improved experimental procedures ͑e.g., waiting time dependence͒.

IV. SUMMARY

In summary, using a 2DEG QWHS with spinning current, we succeeded in measuring the magnetic noise of a frozen ferrofluid that shows a superspin glass transition at T g = 67 K. Below T g , the relation between the imaginary part of the bulk ferrofluid sample ac susceptibility and the magnetic noise measured with the QWHS indicates a possible violation of FDT. Further studies, by this same method, of the nonequilibrium dynamics of the ferrofluid, such as aging effect are under development.

  ͑Presented 21 January 2010; received 30 October 2009; accepted 15 December 2009; published online 6 May 2010͒

FIG. 1 .

 1 FIG. 1. ͑Color online͒ Left panel: Optical micrograph of the 2DEG Hall microsensor. Right panel: Optical micrograph of the same microsensor with the ferrofluid drop on its surface.

FIG. 2 .

 2 FIG.2. ͑Color online͒ Left panel: Power spectrum density of the frozen ferrofluid ͑open symbols͒ obtained by subtracting the Hall probe noise ͑cross symbol͒ from the power spectrum density which contains both of them ͑filled symbol͒, as a function of frequency f, at 45 K in zero applied field. Right panel: The power spectrum density of the ferrofluid at various temperatures from 20 to 85 K.

0021-8979/2010/107͑9͒/09E140/3/$30.00 © 2010 American Institute of Physics 107, 09E140-1 Downloaded 10 May 2010 to 132.166.18.152. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

J. Appl. Phys. 107, 09E140 ͑2010͒ Downloaded 10 May 2010 to 132.166.18.152. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

ACKNOWLEDGMENTS

This work was supported by the RTRA-Triangle de la Physique ͑MicroHall͒.


Jean-Baptiste Barreau

Théophane Nicolas

Guillaume Bruniaux

Emilien Petit

Quentin Petit

Yann Bernard

Ronan Gaugne

Valérie Gouranton

J-B Barreau

Photogrammetry Based Study of Ceramics Fragments

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

INTRODUCTION

Today, photogrammetry and 3D modelling are an integral part of the methods used in archeology and heritage management. They provide answers to scientific needs in the fields of conservation, preservation, restoration and mediation of architectural, archaeological and cultural heritage [START_REF] Barreau | The West Digital Conservatory of Archaeological Heritage project[END_REF] [6] [START_REF] Héno | Archéologie et photogrammétrie -Les nouveaux potentiels[END_REF] [START_REF] Varea | Photogrammétrie et muséologie -Les sèvres du musée de Rouen[END_REF]. Photogrammetry on ceramic fragments was one of the first applications contemporary of the development of this technique applied in the archaeological community [START_REF] Kalantari | Implementation of a Low-Cost Photogrammetric Methodology for 3D Modelling of Ceramic Fragments[END_REF]. More recently and due to its democratization, it was applied more generally to artifacts [START_REF] Nicolae | Photogrammetry applied to problematic artefacts[END_REF]. Finally joined today by the rise of 3D printing [START_REF] Rutland | Non-contact 3D laser scanning as a tool to aid identification and interpretation of archaeological artefacts; the case of a Middle Bronze Age Hittite dice[END_REF] [10], it can restore fragmented artifacts [START_REF] Arbace | Innovative uses of 3D digital technologies to assist the restoration of a fragmented terracotta statue[END_REF] [START_REF] Powell | An ancient statue, re-created[END_REF]. These examples target one or several particular objects and use different types of equipment that can be expensive. These aspects can put off uninitiated archaeologists. So it would be appropriate to see if these techniques could be generalized to a whole class of geometrically simple and common artifacts, such as ceramics.

From these observations, associated to ceramics specialists with fragments of broken ceramics, we aimed at arranging different tools and methods, including photogrammetry, to explore opportunities for a cheap and attainable reconstruction methodology and its possible applications. Our first objective was to establish a protocol for scanning fragments with photogrammetry, and for reconstruction of original ceramics. We used the digital reconstitutions of the ceramics we got following our process to calculate some metrics and to design and 3D print a display for the remaining fragments of one pottery.

Archaeological context

The archaeological material studied in this work consists in domestic ceramics from an Iron Age site in Rezé (Loire-Atlantique -France) and from a Bronze Age site in Lannion Penn An Alé (fig 1).

Rezé ceramics

The study addresses seven vases resulting from a set of archaeological materials excavated in the "Jaguère" site at Reze (Loire-Atlantique, France). Dating from the second Iron Age, these ceramic artifacts were initially studied by Charlotte Choisy-Guillou (Université de Bretagne Sud). The seven vases are entitled Gr1C, Gr5E, MGr2E, MGr11D, Gr1E, Gr7D, and Gr8B. For all vases, we disposed of one single fragment, except for the Gr8B for which we had three fragments.

Lannion Penn An Alé ceramic

The study addresses one container from a set of archaeological materials revealed during a preventive excavation operation on the "Penn an Alé" activities zone in Lannion (Côte d'Armor, France) performed by Inrap (the French National Institute for Preventive Archaeological Research). This operation led on nearly 3.5 hectares, under the guidance of S. Blanchet, allowed to uncover an archaeological site from the Bronze Age; attesting a long human occupation between 1800 and 1200 BCE. It gave an abundant ceramic set which is a real baseline for the region (fig 1).

PROCESS OVERVIEW

The global workflow designed for the study of ceramics presented in this paper consists in five main steps, presented in Figure 2.:

1. First, in the step "Camera capture", ceramic fragments are scanned with photos.

This activity is detailed in sections 3.1 and 3.2.

2. In the step "Picture processing", the pictures are processed with a photogrammetry software to generate a point cloud. This step is detailed in section 3.3.

3. In the step "Mesh generation", a highly detailed mesh is generated for each fragment, scaled and textured, as detailed in section 3.4.

4. In the step "Shape reconstruction", full shapes of the ceramics are extrapolated from the meshes of the fragments. The extrapolated shapes are the used to get a full volume on which measurements can be taken. This step is detailed in section 4.

5. The last step "Mesh processing and 3D printing" aimed at building a fragment support for the ceramics of "Penn an Alé", using a 3D printer, as presented in section 4.3.

3D DIGITIZATION

We present now the protocol we followed to perform the 3D digitization of ceramic fragments by photogrammetry, the different steps to obtain satisfactory results in 3D reconstruction, the technique used to evaluate the volume of ceramics, and the problems we encountered during the work.

Equipment

As the study is based on a reconstitution of archaeological ceramics with photogrammetry, it was essential to use a high quality digital single-lens reflex camera with a good lens and good image resolution. Indeed, this allows to obtain very good quality photos and better results during the search for common points with the photogrammetry software. We used a Nikon D60 Digital SLR to take photos of the various ceramic fragments [START_REF] Labracherie | Test du reflex Nikon D60, Focus Numérique[END_REF]. Its technical features include a 10megapixel CCD sensor, an 18-55mm optical AF-S with a 40mm focal distance and a Nikon autofocus module Multi-Cam 530 (3 AF points).

The environment used to take photos was composed of two light tables, a white background with its support and a halogen lamp (fig 3). This equipment allowed getting a light from below and above to minimize the shadow games. Homogeneity of brightness was difficult to obtain with only one single halogen lamp. This was a major problem because we needed a uniform and constant light regardless of the orientation of the shooting. Indeed, if the light intensity varies too much, then common points determined on a photo by photogrammetry software will not be the same from one photo to another. The goal is here to shoot the object rotating on itself in order to get every viewing angle [START_REF] Kraus | Manuel de photogrammétrie -principes et procédés fondamentaux[END_REF]. We took a first series of photos, describing a semicircle, with a distance camera-object as regular as possible. Then, we rotated the object through 180° in order to complete the series and to get a first coverage of the object. Other series of photographs were taken from different angles to cover a maximum area. Following this procedure, five series of captures were performed: one on the top, one at 45° over the top, one horizontal, one at 45° over the bottom, and the last, at bottom (Fig. 4).

Picture processing

For 3D restitution, we used the photogrammetry software Agisoft PhotoScan version 0.8.3 beta 64bits. This software has been chosen for its ease of use and its ability to generate highly detailed 3D models. Photos of the object were then imported into PhotoScan, in order to rebuild 3D information, in the form of point clouds. Before starting the search for common points, a mask constraint (clipping of the object) must be included on each photo to focus the calculation only on the object and not all the environment. The search for common points and pictures alignment is then performed with a "strong" accuracy. Figure 5 illustrates picture alignment in Photoscan interface.

Figure 5. Picture alignment in PhotoScan interface

Correct scaling of the object is usually performed directly in the software by placing two markers whose distance is known, on a photo of the object. The distance between the markers is set in Photoscan as a reference distance that enables the software to generate a 3D model with a correct scale. This functionality requires a sufficiently uniform light, which was not the case in our installation. According to this limitation, we chose to handle the scaling of the model in another further step. The maximum number of common points was set to 60000 instead of 40000 (default) to get a better rendering. The generated point cloud, resulting from the determination of connection points and photos alignment, allowed to verify the coherency of the digital output with respect to the object (fig 6). The goal of this step is to generate a textured mesh for each fragment. The generation of geometries and textures was processed in PhotoScan with "arbitrary" option for the object type, "high" for the quality and "sharp" for geometric precision. The number of expected surfaces was set to 250000. The color of the mesh was computed from the color information contained in the points cloud. Photoscan generated the texture from several pieces of images extracted from the pictures, and applied it on the surface of the mesh (fig 7). The options for this step were a "generic" mapping mode and "average" fusion mode. The resulting mesh was then exported in ".obj" format and imported in Meshlab to perform scaling. A measure of distance was computed with the tool "rule" between two points whose real distance is known. The ratio between the real distance and the measured distance on the mesh provided the proportionality factor between these two distances. Thanks to the "Transform: scale" MeshLab function, distances on the X, Y and Z axis were multiplied by this factor achieving the correct scaling of the mesh.

RECONSTRUCTIONS AND APPLICATIONS

Technical environment

3D reconstruction from scanned fragments was performed via Blender software. The scale and volume calculation were processed with MeshLab software.

Rezé ceramics

In the case of the Rezé ceramics, the available fragments cover only a partial part of each object. Therefore, the reconstruction was theoretical and performed by extrapolating the different shapes from the fragments in Blender. Most of the fragments contain a part of the lip, the neck, the edge, the body or the bottom. Therefore, if the fragment has a piece of ceramic bottom, it can be aligned with the horizontal plane XY for an easier handling. The lip of the ceramic fragment has a curvature. Using this curvature, a circle (circular mesh) was created and the perimeter of this circle adjusted to coincide perfectly with the contour of the lip. This provided the theoretical circumference of the upper ceramic part. The same operation was repeated several times by shifting the other circles on the Z-axis which were then adjusted on the other parts of the ceramic (kept and visible on the fragment). Therefore, the assembly of these circles on the fragment provided the "skeleton" of the ceramic. Once this frame done, it was imported in MeshLab again to calculate the volume of the ceramic. The "convex hull" function, in the "remeshing, simplification and reconstruction" tab allowed to create a mesh from the different circles of the frame (fig 8). This mesh, constituting a completely closed shape, was used to calculate a volume with the "compute geometric measures" function. The table 1 gathers the characteristics of the reconstitutions of the different Rezé ceramics.

Lannion Penn An Alé ceramic

This Bronze Age ceramic is "almost" full, so we tried to assemble the different fragments in Blender. Fragments of ceramic having a part of the lip were first assembled and wedged on a circle perfectly matching its shape. The other pieces were then imported and positioned one by one. One difficulty, however, was met with several small fragments. For them, only the inner or outer part of the fragment was reconstituted. The reasons for this failure are many and difficult to determine. Indeed, this can be induced by low quality of light, insufficient image quality or a too thin thickness of the ceramic fragments that corrupted the detection of common points on the edges. To solve this problem, the inner and outer parts were rescanned separately and then scaled in MeshLab. The two parts were then positioned parallel to the horizontal plane XY, superimposed on each other by playing with a rotation on the Z axis, and then merged by sliding the inner part relative to the Z axis.

The reconstruction approach was different than in Rezé context. After deleting the handles, we used the function "Surface reconstruction: Poisson" in Meshlab allowing to extrapolate the initial ceramic shape from the assembly of fragments (fig 9). This shape was used to compute the volume of the ceramic whose result is 2742,9 cm3. A last step of the work consisted in using the generated shape to build a support for the fragments, in order to get a 3D tangible representation of the pottery. We used a 3D printer MarkerBot Replicator 2x, which is an ABS printer, to produce the support display. The mesh required some corrections made by a graphic designer under 3DSmax (fig 10), because the Poisson reconstruction introduced some approximation at the frontiers between the real fragments and the computed shape.

DISCUSSION

The digitization process has highlighted difficulties often masked by the apparent simplicity of photogrammetry. However, we believe we reached our goal regarding the very low cost of the digitization of fragments, and the resulting data and information.. The different approaches about reconstruction discussed here provided practical useful data for the analysis work performed by archaeologists. The volumes estimate, the 3D visualization of the shape, and the tangible 3D presentation represented concrete help in the study of the ceramics.

The 3D printer used for this work is a relatively inexpensive one, recently bought by the computer science laboratory involved in the project. The generation of a support display for the Lannion ceramic was a proof of concept for presenting fragments physically held around the content. However, problems have arisen. Because the Bronze Age ceramic is too large for this printer, especially because of the size of the handles, we had to print the support in two parts. In addition, the Poisson reconstruction of the internal volume of the ceramic generated approximations, which implied that the fragments did not exactly fit the display. Modifications on the 3d model of the display were required to match the shape of the fragments.

The result of this experimental work convinced us that the production of such display could well be generalized within the museum community due to its low cost. A first reflection on its use has led us to believe that it would be wise to add on top of the display (not covered by a fragment) signs indicating metadata in the print-ready 3D model (fig 11).

CONCLUSION AND FUTURE WORKS

We proposed a simple protocol to digitize fragments of ceramics. This protocol is accurate enough to provide a digital reconstitution of ceramics with heterogeneous morphologies and compute basic metrics. Furthermore, from the digital reconstitution of the Lannion Penn An Alé ceramics, we designed and 3D printed a display for the remaining fragments. This kind of display could be used for exhibition of artefacts in museums.

The next step of this work is to validate these methods on more ceramics. To do this, we must solve the problems of accuracy in digitization and printing of large volumes. We also wish to extend these methods to other kinds of less symmetrical archaeological fragmented artifacts. Finally, it would be interesting to integrate museologists to our future works on 3D printed displays.

Figure 1 .

 1 Figure 1. Location of sites and "Penn an Alé" site (© Inrap)

Figure 2 .

 2 Figure 2. Process diagram

Figure 3 .

 3 Figure 3. Shooting equipment and ceramic fragments

Figure 4 .

 4 Figure 4. Different camera positions for the series top, top 45°, and horizontal.

Figure 6 .

 6 Figure 6. Piece of the Bronze Age ceramic with connection points (blue) determined by PhotoScan / Resulting point cloud

Figure 7 .

 7 Figure 7. Raw mesh (top left) / smoothed mesh (top middle) / mesh with colors (top right) / textured meshs (bottom)

Figure 8 .

 8 Figure 8. Fragments of eating ceramics reconstituted by photogrammetry / Frame of circles aligned along the Z axis and dimensioned according to the curvature of the fragments / "convex hull" function on the Gr8B ceramic

Figure 9 .

 9 Figure 9. Separated parts / reconstruction step / reconstruction with the function "Surface reconstruction: Poisson"

Figure 10 .

 10 Figure 10. 3D models of the display without and with fragments / empty display / display with fragments

Figure 11 .

 11 Figure 11. Separated part of the display / metadata inscription on the 3d model

Table 1 .

 1 Summary of the characteristics of each reconstituted ceramic fragments

	Gr1C	36	24101	249 999	130234	40m	536,4
	Gr5E	78	55470	246768	128077	1h14m	1722,8
	MGr2E	89	31204	249999	139674	52m	88,9
	MGr11D	75	18354	250000	129272	43m	138,8
	Gr1E	44	29396	17911	11036	39m	338,7
	Gr7D	62	45250	154286	77517	41m	211,1
	Gr8B	111	59623	250000	149449	1h15m	1938,7