Amélie Charles
email: acharles@audencia.com.

Olivier Darné
email: olivier.darne@univ-nantes.fr.

Zakaria Moussa
email: zakaria.moussa@univ-nantes.fr.

Nicholas Bloom

Rüdiger Bachmann

Kenneth French

Sydney Ludvington

Zakaria Olivier Darné

Moussa

Lemna

Charles Amélie

Darné Olivier

Kim Jae

Etienne Redor

Pham Thi Hong

Hanh N°

Pham Thi

Hong Hanh

Moussa Zakaria

The sensitivity of Fama-French factors to economic uncertainty *

Keywords: Fama-French factors, Economic uncertainty, Markov-switching model. JEL Classification: G10, G11, C32. The sensitivity of Fama-French factors Fama-French factors, Economic uncertainty, Markov-switching model. JEL Classification: G10, G11, C32

This paper analyzes the sensitivity of the three Fama-French factors in relation to the US economic uncertainty, by using three proxies of uncertainty measures in macroeconomics, financial markets or economic policy from January 1985 to December 2011. We examine the extent, speed and duration of response of the three (market, size and value) risk premia to movements in the US uncertainties under low and high volatility regimes through the Markov-regime switching VAR model. We find clearly two (high and low) volatility regimes, where each regime is highly persistent.

The high volatility regime is the prevailing regime between periods of 2000 to 2003, and 2008 to the end of 2012. We show a negative effect of changes in financial and economic policy uncertainties on value risk premia during the high volatility regime. This finding imply that investors move to growth stocks from value stocks in high volatility regime when volatility is expected to increase. The latter suggests that value firms can be more risky than growth firms during high volatility periods. We also propose an aggregate measure of economic uncertainty by using Principal Component Analysis based on the three uncertainty proxies. The results on value risk premia are confirmed. We find a negative relationship between the market risk premium and the change in the economic uncertainty index in high volatility regime. Finally, by adding a liquidity risk factor we find a positive effect of financial uncertainty on liquidity factor during the high volatility regime, suggesting that investors preferring liquidity stocks when market uncertainty increases.

Introduction

The Capital Asset Pricing Model (CAPM) of [START_REF] Sharpe | Capital asset prices: A theory of market equilibrium under conditions of risk[END_REF], [START_REF] Lintner | The valuation of risk assets and the selection of risky investments in stock portfolio and capital budgets[END_REF], and [START_REF] Mossin | Equilibrium in a capital asset market[END_REF] remains a benchmark asset pricing model for both financial economists and investment practitioners, which relates required or expected returns to systematic risk (or market beta) and not related to other variables. 1However, some firm-specific characteristics besides the market beta have been documented to have significant explanatory power for average returns, for example, firm size (e.g., [START_REF] Banz | The relationship between return and market value of common stocks[END_REF][START_REF] Reinganum | Misspecification of capital asset pricing: Empirical anomalies based on earnings' yields and market values[END_REF][START_REF] Reinganum | A direct test of Roll's conjecture on the firm size effect[END_REF], and book-to-market equity ratio (B/M) (e.g., De Bondt and Thaler, 1985; [START_REF] Fama | The cross-section of expected stock returns[END_REF]. Motivated by the growing empirical evidence on these CAPM anomalies, [START_REF] Fama | Common risk factors in the returns on stocks and bonds[END_REF] propose a three-factor model (FF) that adds two factors to the market risk premium, size and value (premia) factors.

It is considered that value (growth) stocks are riskier than growth (value) stocks in bad (good) times (e.g., [START_REF] Jagannathan | The conditional CAPM and the cross-section of expected returns[END_REF]Lettau and Ludvigson, 2001b;[START_REF] Petkova | Is value riskier than growth[END_REF][START_REF] Zhang | The value premium[END_REF][START_REF] Chen | The expected value premium[END_REF], suggesting that investors tend to switch from riskier assets to safer ones in bad times. Further, since small firms are relatively more sensitive to economic downturns than large firms (e.g., Gertler and Gilchrist, 1994;[START_REF] Fort | How firms respond to business cycles: The role of firm age and firm size[END_REF], investors tend to move out of small stocks in bad times.

It is also well-known that uncertainty about the future has real implications on economic agents' behavior (e.g., Dixit, 1989;Bernanke, 1983;[START_REF] Bloom | Uncertainty and investment dynamics[END_REF][START_REF] Bloom | The impact of uncertainty shocks[END_REF]. Growing literature provide evidence that economic uncertainty affect financial markets, especially firm fundamentals, such as cash flows, risk-adjusted discount factors, and investment opportunities (e.g., [START_REF] Bloom | The impact of uncertainty shocks[END_REF][START_REF] Bloom | Uncertainty and investment dynamics[END_REF], equity portfolios and individual stocks (e.g., [START_REF] Anderson | The impact of risk and uncertainty on expected returns[END_REF][START_REF] Bekaert | Risk, uncertainty, and asset prices[END_REF][START_REF] Bali | Risk, uncertainty, and expected returns[END_REF], and volatility (e.g., [START_REF] Veronesi | Stock market overreaction to bad news in good times: A rational expectations equilibrium model[END_REF][START_REF] Bansal | Risks for the long run: A potential resolution of asset pricing puzzles[END_REF][START_REF] Bloom | The impact of uncertainty shocks[END_REF]. Recently, [START_REF] Brogaard | The asset pricing implications of government economic policy uncertainty[END_REF] show that uncertainty related specifically to the economic policy of governments may impact financial markets. Economic uncertainty is difficult to quantify since it is intrinsically unobservable concept, and there are different sources of uncertainty, but it is possible to observe uncertainty indirectly using a number of proxy indicators [START_REF] Bloom | Fluctuations in uncertainty[END_REF][START_REF] Bloom | Held back by uncertainty[END_REF]. This paper analyzes the sensitivity of three Fama-French factors (1993) in relation to the US economic uncertainty, by using three proxies of uncertainty measures in macroeconomics, financial markets or economic policy. 2 We use the index of economic policy uncertainty proposed by Baker, Bloom and Davis (2013), the CBOE volatility index as proxy of financial market uncertainty, and the macro uncertainty factor developed by Jurado et al. (2013). Specifically, we investigate whether the uncertainty measures have a direct and systematic effect on equity returns by increasing or decreasing the returns of the systematically priced factors included in the [START_REF] Fama | Common risk factors in the returns on stocks and bonds[END_REF] model. In addition to specific measures of uncertainty, we use a statistical approach to develop an aggregate measure of economic uncertainty. To sufficiently capture the common variation among the correlated factors of economic policy, financial and macroeconomic uncertainty, we apply the principal component analysis that uses orthogonal transformation to convert a set of highly correlated indicators into a set of linearly uncorrelated variables called principal components. We then examine the sensitivity of three Fama-French factors to this economic uncertainty index.

In particular, we examine the extent, speed and duration of response of the three (market, size and value) risk premia to movements in the US uncertainties under low and high volatility regimes through the application of Markov regime switching (MS) analysis [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF]. The MS models have become popular in the financial literature because they can capture the instability of financial time series, such as sudden (transitory or short-lived) or persistent changes of behavior [START_REF] Ang | Regime changes and financial markets[END_REF]). 3 One of the major advantages of this approach is that it does not require prior specifications or dating of volatility regimes. Thus, the use of the MS model allows a more robust and informative analysis on the sensitivity of three Fama-French factors. To the knowledge of the authors, no study on the three Fama-French factors has yet utilized the MS approach. 4 In order to measure the extent of the response of three Fama-French factors to movements in the uncertainty in economic policy, financial markets and macroeconomics, a Markov Switching Vector Autoregressive Model (MS-VAR) is estimated. With this model, an impulse response analysis is then conducted afterwards to determine the speed and duration of the response. This approach allows us to analyze the effects of uncertainty in high and low volatility periods.

The remainder of this paper is organized as follows: Section 2 briefly describes the methodology of MS-VAR. The data are presented in Section 3. Section 4 discussed the empirical results. Finally, Section 5 concludes.

for which the probabilities of the different possible states of the world are known. Uncertainty refers to a future outcome that has unknown probabilities associated with the different possible states of the world. Note that some of what we call uncertainty may indeed be risk as defined by [START_REF] Knight | Risk, Uncertainty and Profit[END_REF]. Thus, we use different proxies for economic uncertainty, which can be different from Knightian uncertainty. 3 See Guidolin (2012) for a survey on applications of Markov regime switching models in empirical finance. 4 Durand et al. ( 2011) and Shamsuddin and Kim (2014) analyze the effect of the market uncertainty (using the VIX index) on the three Fama-French factors but using a standard VAR model without regime switching.

Data

We consider monthly data for three Fama-French factors from January 1985 to December 2011: market (MKT= Rm -R f )), size (Small Minus Big, SMB) and value (High Minus Low, HML) factors. The market return (Rm) is the value-weight return of all CRSP firms incorporated in the US and listed on the NYSE, AMEX, or NASDAQ, and the risk-free rate (R f ) is the one-month Treasury bill rate. The SMB factor is the difference between the returns on the portfolio of small size stocks and the returns on the portfolio of big size stocks. The HML factor is the difference between the returns on the portfolio of high book-to-market stocks and the returns on the portfolio of low book-to-market stocks. The data for Rm, R f , SMB and HLM come from Kenneth French's website (see Figure 11). 5 We use three proxies of US uncertainty measures in macroeconomics, financial markets or economic policy. The US macroeconomic uncertainty variable (UMACRO) is the macro uncertainty factor developed by Jurado et al. (2013), based on a large number of economic time series. For the uncertainty measure in US financial markets we employ the CBOE volatility index (VXO), also known as the "fear index" or the "fear gauge", based on trading of S&P 100 (OEX) options. 6 The VXO reflects market uncertainty associated with future stock price movements and might proxy risk aversion. The US economic policy uncertainty variable that we use is the index of economic policy uncertainty (EPU) proposed by Baker, Bloom and Davis (2013), built on three components: (i) the frequency of newspaper references to economic policy uncertainty, (ii) the number of federal tax code provisions set to expire, and (iii) the extent of forecaster disagreement over future inflation and government purchases. 7 The span of the data is 1985:1 to 2011:12 which enables us to look at the pre-and post-crisis periods to discover the impact volatility in different economic environments.

Figure 1 displays the three proxies of uncertainty. All uncertainty measures are higher during the 2001 economic recession and much higher during the 2008 global financial crisis. During the 1990 economic recession the economic policy uncertainty is higher, and the financial uncertainty is moderately high.

The economic policy and financial uncertainties also present a spike around the LTCM and Russian 5 The data are available on mba.tuck.dartmouth.edu/pages/faculty/ken.french/DataLibrary/. 6 As an alternative to the VXO index, we could have used the newer VIX index, which was introduced by the CBOE on September 22,2003. The VIX is obtained from the European style S&P500 index option prices and incorporates information from the volatility skew by using a broader range of strike prices than just at-the-money strike series as in the VXO. However, the daily data on VIX starts from January 2, 1990, which does not cover our full sample period, beginning in January 1986.

The pre-1986 VXO data are calculated by [START_REF] Bloom | The impact of uncertainty shocks[END_REF]. See [START_REF] Whaley | Understanding the VIX[END_REF] for a history of the VIX and a summary on its calculation. Durand et al. (2011) show that changes in the VIX drive variations in the expected returns of the factors included in the Fama-French three-factor model. We also find that uncertainty measures are negatively correlated with SMB, that might also be consistent with a flight-to-quality interpretation as increasing uncertainty may lead to investors being less willing to hold small stocks. [START_REF] Campbell | Intertemporal asset pricing without consumption data[END_REF][START_REF] Campbell | Understanding risk and return[END_REF] assume that investors want to hedge against the changes in the forecasts of future market volatilities. 9 The correlations between HML and uncertainty measures are positive and very low, as found in Durand et al. 9 Bali and Engle (2010) use implied volatility from the S&P100 index options (VXO) to test whether stocks that have higher correlation with the changes in future market volatility yield lower expected return in an ICAPM.

(2011). The results show that HML is negatively correlated to MKT (-0.27) and SMB (-0.33). Finally, the three measures of uncertainty are positively correlated, with the highest correlation between financial and economic policy uncertainties (0.39).

Y t =              a 1 + B 11 Y t-1 + . . . + B p1 Y t-p + A 1 u t i f s t = 1 . . . a m + B 1m Y t-1 + . . . + B pm Y t-p + A m u t i f s t = m (1)
The Y t is a vector of endogenous variables which depends upon an unobserved regime variable s t that controls the state of the economy. Each regime is characterized by an intercept a i , a K dimensional vectors of auto-regressive terms B 1i , . . . , B pi , a matrix A i , and fundamental disturbance u t , with u t ∼ N(0, I K ). Matrix A i is computed from the regime-dependent variance covariance matrix from the reduced form VAR, Σ i :

Σ i = E A i U t U ′ t A i ′ = A i IA i ′
In order to compute A i , which has K 2 elements and K being the number of variables, from Σ i having only

)
who found that stock market volatility follows a two-state Markov-switching process, with the market risk premium varying across the "low" and "high" volatility regimes, 11 we assume that the number of regimes m = 2, s t . 12 These two regimes conventionally corresponds to the low mean change in volatility or stable state and the high mean change in volatility or volatile state, respectively. s t = {1, 2} is assumed to follow the discrete time and discrete state stochastic process of a hidden Markov chain and controlled by transition probabilities p i, j = Pr(s t+1 = j|s t = i), and ∑ 2 j=1 p i j = 1∀i, j ∈ (1, 2). The stochastic process is defined by the transition matrix P as follows:

p =   p 11 p 12 p 21 p 22  
The model is estimated using the Expectation-Maximization (EM) algorithm as suggested by [START_REF] Krolzig | Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis[END_REF] (and in [START_REF] Hamilton | Time Series Analysis[END_REF] for the univariate case), which consists of two steps whereby 10 In this paper, we estimate three MSIAH(4)-VAR(p) models. In each model we include Fama-French factors and either ∆VIX or one of uncertainty measures, ∆EPU or ∆UMACRO. Of course it would be better to estimate only one model including all our variables, but we retain this solution in order to estimate a more parsimonious models as possible. Indeed, within VAR models, the number of variables cannot be enlarged very much, because of both estimation and identification problems. This problem is even more present within MS-VAR models as the number of parameters to estimate grows not only with the number of variables included in the model but also with regimes. 11 Schwert (1989), Schaller and van Norden (1997), [START_REF] Kim | Testing for mean reversion in heteroskedastic data based on Gibbs-sampling-augmented randomization[END_REF][START_REF] Kim | Does an intertemporal tradeoff between risk and return explain mean reversion in stock prices[END_REF][START_REF] Kim | Is there a positive relationship between stock market volatility and the equity premium?[END_REF], [START_REF] Hess | What drives Markov regime-switching behavior of stock markets? The Swiss case[END_REF] and [START_REF] Mayfield | Estimating the market risk premium[END_REF], among many others, have modeled monthly stock return volatility using a Markov-switching specification, with high volatility regimes typically corresponding to periods of recession and low volatility regimes typically corresponding to periods of expansion. 12 We have also estimated a three-state regime models but the third regime is merely capturing a few extreme outliers in the data, rather than persistent changes in volatility. This result is consistent with the finding of [START_REF] Hamilton | Autoregressive conditional heteroskedasticity and changes in regime[END_REF] and [START_REF] Abdymomunov | Time variation of CAPM betas across market volatility regimes[END_REF].

θ i k,h = ∂ E t Y t+h ∂U k,t |s t = .... = s t+h = i, f or h ≥ 0 (2)
where

θ i k,1 • • • θ i k,h are K dimensional response
vectors of the responses of the endogenous variables to a shock to the k th fundamental disturbance. To account for estimation uncertainty, we adopt the standard bootstrapping method to get the related confidence bands by retaining the mean along with the relevant percentiles of the numerical approximation of the distribution of the original estimates of the regime vectors. 144 Empirical results

Specific uncertainty measures

In this section, we examine the three MSIAH(4)-VARs, using one of the three proxies of US uncertainty in financial markets, macroeconomics or economic policy, respectively. The MSIAH(4)-VAR(1) model 15 is considered according to the specification tests (see Technical Appendix). In all MSIAH(4)-VAR(1) models, the linearity test suggests that the model is significantly non-linear and parameters switch substantially between regimes. Moreover, for the three MSIAH(4)-VAR(1) models, each regime is highly persistent according to the transition matrix (Table 6, Appendix A), with transition probabilities lying between 87% and 96% month-to-month probabilities of remaining in the low and high volatility regimes, respectively. Inferences regarding the turning points can be obtained from the smoothed probabilities of regimes (Figure 2). The timing of the change across regimes and the number of months for which factors were under the two regimes are very similar. These results suggest that there are clearly two different volatility regimes. Note that the first regime, corresponding to the high volatility regime, is the prevailing regime between periods of 2000 to 2003, and 2008 to the end of 2012. These periods correspond the bear market following the burst of the dot-com bubble and Fed's interventions, and the 2007-2008 financial crisis and the related recession 16 , respectively. The second regime, corresponding to the low volatility regime, coincides with the two bull market periods; the first was part of the dot-com bubble and the second corresponds to the mortgage market bubble.

We also find clear spikes, commune to the three MSIAH(4)-VAR(1) models, corresponding to stock market crash in October 1987 and the LTCM and Russian Debt crisis of 1998. A specific spike is found from MSIAH(4)-VAR(1) using EPU in 1992 associated with the presidential election (Figure 17b). These changes can be considered as extreme outliers, as suggested by [START_REF] Hamilton | Autoregressive conditional heteroskedasticity and changes in regime[END_REF] who find that "extremely large shocks, such as the October 1987 crash, arise from quite different causes and have different consequences for subsequent volatility than do small shocks."

We use MSIAH(4)-VAR(1) to examine the sensitivity of three risk premia (MKT, SMB and HML) to changes in macroeconomics (∆UMACRO), financial markets (∆VXO) or economic policy (∆EPU) uncertainties, under low and high volatility regimes. To analyze how changes in uncertainty impact the risk premia we examine the impulse response functions (IRFs) derived from the MS-VAR, according to the two volatility regimes. The IRFs are reported in Figure 3 along with 90% confidence band computed with standard bootstrap. Figures 3 to 5 display how the three risk premia respond to a one standard deviation innovation in ∆VXO, ∆EPU and ∆UMACRO, respectively. The impulse reaction period is chosen to be 6 months. Table 3 reports the effect size estimates from the impulse response functions with the cumulative responses. This indicates the response values of a (risk premia) variable to a standard deviation innovation (uncertainty) shock to other over the time horizon from 0 to 6 months.

The IRFs show a positive effect of a shock to ∆VXO on SMB only during the low volatility regime (3), implying that investors prefer larger stocks over smaller stocks in low volatility regime whereas they move to growth stocks from value stocks in high volatility regime when volatility is expected to increase.

We also find that a negative effect of ∆VXO on HML during the high volatility regime, suggesting that value firms can be more risky than growth firms during high volatility periods. This is consistent with the view that value is riskier than growth in bad times when the price of risk is high (e.g., Jagannathan the impact on HML is larger from a shock to ∆VXO (-16.2%) than from a shock to ∆EPU (-4.2%) (see Table 3). Further, the impact of EPU shock on HML is both larger and more persistent during periods when volatility is high as opposed to periods during which volatility is lower. A 1% shock to ∆EPU leads to 4.3% and 1.6% decline of HML during the high and low volatility regime, respectively. This is consistent with investors preferring growth stocks over value stocks when economic policy uncertainty increases.

The results presented in Figure 5 show that the risk premia respond to a shock to ∆UMACRO only on MKT during the high volatility regime. This shock has very higher impact (-48.8%) than that from a shock to ∆EPU (-6.4%), and this effect is more persistent. This finding indicates that a negative relationship between the market risk premium and the change in the macroeconomic uncertainty in high volatility regime. However, the change in the macroeconomic uncertainty seems to have no effect on HML and SMB factors.

Economic uncertainty index

Since the three measures of uncertainty are highly positively correlated with each other (Table 2)

and they tend to move together, suggesting there is a common uncertainty component to all the measures (Figure 1), we propose an aggregate measure of economic uncertainty (UFACTOR) by using Principal Component Analysis (PCA). 18 We use the PCA to extract the common component of the three uncertainty proxies that capture different dimensions of the economic uncertainty: economic policy, finance and macroeconomics. The first principal component from PCA sufficiently captures the common variation among the three uncertainty measures. who find that recessions appear in periods of significantly higher economic uncertainty. Table 4 displays the correlation between the economic uncertainty index and the three uncertainty measures, showing that they are highly correlated. We find similar correlations between the changes in the economic uncertainty index and the Fama-French factors, namely ∆UFACTOR is positively correlated to MKT and SMB, and negatively to HML.

From the MS-VAR model, both regimes are highly persistent according to the transition matrix (Table 6, Appendix A), with transition probabilities lying between 85% and 94% month-to-month probabilities of remaining in the low and high volatility regimes, respectively. The timing of the change across regimes and the number of months for which the economic uncertainty index is under the two regimes are similar to our findings from specific uncertainty measures. Figure 8 displays a negative effect of ∆UFACTOR on MKT during the high volatility regime, suggesting a negative relationship between the market risk premium and the change in the economic uncertainty index in high volatility regime. This shock to ∆UFACTOR has a higher effect on MKT than those from ∆EPU and ∆UMACRO, with a short horizon (3 months). The IRFs displays a negative effect of ∆UFACTOR on HML during both volatility regime, as for ∆EPU, with a longer horizon, namely 6 and 5 months, under the high and low regime, respectively, and a higher impact. This suggests that the impact of uncertainty shock on HML is larger during periods when volatility is high as opposed to periods during which volatility is lower. This is consistent with investors preferring growth stocks over value stocks when global uncertainty increases. 2011). The data for WML and LIQ comes from Kenneth French's website and from Lubos Pastor's website, respectively (Figure 12). 19 . We obtain the same results from the MSIAH(5)-VAR(1) model than from the MSIAH(4)-VAR(1) models for (i) the both highly persistent regimes with the same timing of the change across regimes and with slightly higher number of months for which the regime remains in the low or high volatility regimes (Figure 14); (ii) the effects of a shock to uncertainty on risk premia (MKT, HML and SMB) (Figure 15). These findings show the robustness of our results according to the number of variables in the MS-VAR model. ?? reports a short negative effect of ∆VXO on WML only during the high volatility regime, whereas we find a positive effect of ∆EPU (Figure ??) and ∆UMACRO (Figure Figure ??) on the momentum premium, especially a highly persistent effect from a shock of macroeconomic uncertainty. 20 These results suggest that investors appear to move to proven stocks (past winners) rather than "glamor" stocks when economic policy and macroeconomic uncertainties increase or when market uncertainty decreases during the high volatility regime.

More interesting, Figure

Further, Figure 15 displays a positive effect of ∆VXO on liquidity factor during the high volatility regime, with a short horizon (2 months), suggesting that investors preferring liquidity stocks when market uncertainty increases. This finding is consistent with the flight-to-liquidity phenomenon where 19 The data are available on http://faculty.chicagobooth.edu/lubos.pastor/research/. 20 Information uncertainty has been proposed as an explanation for the abnormal returns earned via momentum strategies.

Faced with greater uncertainty, investors are increasingly unable to accurately determine the true value of an asset and are more likely to misprice it. [START_REF] Zhang | Information uncertainty and stock returns[END_REF] finds that information uncertainty exacerbates momentum, using forecast dispersion as a measure of information uncertainty, whereas [START_REF] Verardo | Heterogeneous beliefs and momentum profits[END_REF] investors rebalance their portfolios toward more liquid assets. This is also consistent with Chung and Chuwonganant (2014) who found that market uncertainty (measured by the VIX) exerts a large marketwide impact on liquidity. We find the same result with a shock of ∆EPU (Figure 16). 21

PCA from 6 uncertainty measures

As robustness check of our results on the aggregate measure of economic uncertainty, we consider three others proxies of uncertainty measure to construct the economic uncertainty index in our sample. In this respect, we use a measure of equity market uncertainty with the equity market-related economic uncertainty (EMEU) proposed by Baker et al. (2013), a measure of financial uncertainty with the corporate bond spreads (SPREAD), defined as the monthly spread of the 30-year Baa-rated corporate bond yield index over the 30-year treasury bond yield, and a measure of (micro)economic (firm-specific) uncertainty with the forecast disagreement index (FDISP) based on the forecast dispersion in the general business situation question from the Business Outlook Survey proposed by Bachmann et al. (2013). 22 These uncertainty measures are significantly correlated with the three previous measures, except for FDISP (Table 5). The economic uncertainty index is obtained by extracting the common component of the six uncertainty proxies. Figure 13 presents the economic uncertainty index obtained from six uncertainty measures (UFACTOR6) and also that obtained from three measures (UFACTOR3). The evolution of both uncertainty indexes are very similar. Further, they are highly correlated (0.93, Table 4).

We find the same results from this economic uncertainty index than from the economic uncertainty 21 We obtained the same results for a shock of ∆UMACRO and ∆UFACTOR on the three Fama-French factors from the MSIAH(4)-VAR(1) and MSIAH(5)-VAR( 1) models (see Technical Appendix). However, the responses to ∆UMACRO and ∆UFACTOR on the liquidity factor are non significant.

Fama-French portfolios

We study the effect of uncertainty on the 6 Fama-French benchmark portfolios formed on Size and Book-to-Market: small value (SV), small neutral (SN), small growth (SG), big value (BV), big neutral (BN), and big growth (BG). These portfolios, which are constructed at the end of each June, are the intersections of 2 portfolios formed on size (market equity, ME) and 3 portfolios formed on the ratio of book equity to market equity (BE/ME). The size breakpoint for year t is the median NYSE market equity at the end of June of year t. BE/ME for June of year t is the book equity for the last fiscal year end in t -1 divided by ME for December of t -1. The BE/ME breakpoints are the 30th and 70th NYSE percentiles.

Figure 22 displays the IRFs for the 6 portfolios. The results show that a negative effect of ∆UFACTOR6 on 4 portfolios (SG, SN, SV and BV) during the high volatility regime, suggesting a negative relationship between these portfolios and the change in the economic uncertainty index in high volatility regime. This is consistent with the view that the small firms are more sensitive of uncertainty in bad times (high volatility). This shock to ∆UFACTOR6 has a higher effect on SG and SV portfolios than for SN and BV portfolios. The uncertainty shock have no effect on BG uncertainty, suggesting that big growth firms are not affected by economic uncertainty shock, even during high volatility regime. This is consistent with investors preferring growth stocks when economic uncertainty increases. Finally, We only find a negative effect of ∆UFACTOR6 on BN portfolio during the low volatility regime.

Conclusion

This paper analyzed the sensitivity of the three Fama-French factors in relation to the US economic uncertainty, by using three proxies of uncertainty measures in macroeconomics, financial markets or economic policy. We examined the extent, speed and duration of response of the three risk premia to movements in the US uncertainties under low and high volatility regimes through the MS-VAR model.

We found clearly two different volatility regimes, where each regime is highly persistent. The first regime, corresponding to the high volatility regime, is the prevailing regime between periods of 2000 to 2003, and 2008 to the end of 2012. These periods correspond to the bear market following the burst of the dot-com bubble and FedŠs interventions, and the 2007-08 financial crisis and the related recession, respectively. The low volatility regime coincides with the two bull market periods; the first was part of the dot-com bubble and the second corresponds to the mortgage market bubble.

Examining the sensitivity of three risk premia (market, size and value) to changes in macroeconomics, financial markets or economic policy uncertainties under low and high volatility regimes, we showed a negative effect of changes in financial and economic policy uncertainties on value risk premia during the high volatility regime. This finding implies that investors move to growth stocks from value stocks in high volatility regime when volatility is expected to increase. The latter suggests that value firms can be more risky than growth firms during high volatility periods. This is consistent with the view that value is riskier than growth in bad times when the price of risk is high (e.g., Jagannathan and Wang, 1996; Finally, we proposed an aggregate measure of economic uncertainty by using Principal Component Analysis based on the three uncertainty proxies. The results on value risk premia are confirmed. We also found a negative relationship between the market risk premium and the change in the economic uncertainty index in high volatility regime.

Overall, the results of our analysis point to the sensitivity of US market and value risk premia to economic uncertainty shock, especially during high volatility period, and warrants further research.

Figure 1 .

 1 Figure 1. Economic policy, Financial and Macroeconomic Uncertainties.

  the expectation step infers the hidden Markov chain conditioned on a given set of parameters, and the maximization step re-estimates the parameters based on the inferred unobserved Markov process. These steps are repeated until convergence.Our decision to use MS-VAR framework is also motivated by the possibility to derive regimedependent Impulse Response Functions (IRFs), which helps to determine the cyclical variation in the responses of factors to a particular shock. For the MS-VAR models,[START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF] have developed the regime-dependent IRFs which permit to simulate the responses of endogenous variables to exogenous shocks. Such response functions are conditional on the prevailing regime at the time of the shock and on the entire horizon of the response. The regime-dependent IRF13 is described by equation 2, which traces the expected path of the endogenous variables at time t + h following a one standard deviation shock to the kth initial disturbance at time t, conditional on regime i.

Figure 2 .

 2 Figure 2. Estimated smoothed probabilities for MSIAH(4)-VAR(1) models

Figure 4

 4 Figure4displays a negative effect of ∆EPU on MKT during the high volatility regime, suggesting a negative relationship between the market risk premium and the change in the economic policy uncertainty in high volatility regime. Contrarily to a shock to ∆VXO, the IRFs show a negative effect of a shock to ∆EPU on SMB only during the low volatility regime. This result indicates that investors move to large-cap firms from small-cap firms when the economic policy uncertainty increases in high volatility regime. Finally, the IRFs displays a negative effect of ∆EPU on HML during the high volatility regime, as for ∆VXO, with a longer horizon (5 months), and also during the low volatility regime. Note that17 Durand et al. (2011) also find that an effect of VIX is positive on HML but negligible on SMB whereas this effect is negligible on SMB and HML for Shamsuddin and Kim (2014). These authors use the VIX index (in first-difference for Durand et al., 2011; in level for Shamsuddin and Kim, 2014) for market uncertainty while we use the VXO index. Durand et al. (2011) use daily data from February 1, 1993 to July 30, 2007, and Shamsuddin and Kim (2014) employ weekly data from January 1990 to December 2011.

Figure 3 .0

 3 Figure 3. Response to VXO shock in MSIAH(4)-VAR(1)

Figure 4 .

 4 Figure 4. Response to EPU shock in MSIAH(4)-VAR(1)

Figure 6

 6 Figure 6 presents the economic uncertainty index obtained from the first principal component of the 18 Bali et al. (2014) also proposed a newly measure of macroeconomic risk associated to a quantitative indicator of economic uncertainty, using individual measures of macroeconomic risk obtained from estimating time-varying conditional volatility of the economic indicators based on a VAR-GARCH model. Their indicator is computed from January 1994 to March 2012. Haddow et al. (2013) also used PCA to construct an uncertainty index based on four indicators for the UK on the 1985-2013 period.

Figure 5 .

 5 Figure 5. Response to UMACRO shock in MSIAH(4)-VAR(1)

Figure 6 .

 6 Figure 6. Economic Uncertainty Index based on 3 uncertainty proxies.

Figure 7 .

 7 Figure 7. Estimated smoothed probabilities for MSIAH(4)-VAR(1): UFACTOR

Figure 8 .0

 8 Figure 8. Response to UFACTOR shock in MSIAH(4)-VAR(1)

4. 3

 3 Robustness check 4.3.1 Momentum and Liquidity Factors As robustness check of our results on IRFs from the MSIAH(4)-VAR(1) models with the different measures of uncertainty, we add two others risk factors with (i) the momentum factor (Winner Minus Loser, WML) introduced by Cahart (1997), and (ii) the aggregate liquidity factor (LIQ) proposed by Pastor and Stambaugh (2003). Cahart (1997) proposes a four-factor model by adding this risk factor into the Fama-French three-factor model. The phenomenon of price momentum is documented in several studies (see, e.g., Jegadeesh and Titman, 1993; Chan et al., 1996; Fama and French, 1996; Jegadeesh and Titman, 2001), and a number of studies show that liquidity-related risks are prices (see, e.g., Pastor and Stambaugh, 2003; Acharya and Pedersen, 2005; Korajczyk and Sadkha, 2008; Lee,

  shows that investor uncertainty, based on company's fundamentals, is associated with less momentum. Note that Durand et al. (2011) and Kim and Shamsuddin (2014) find that WML responds positively to a shock in VIX.

Figure 9 .

 9 Figure 9. Estimated smoothed probabilities for MSIAH(4)-VAR(1): UFACTOR6

Figure 10 .0

 10 Figure 10. Response to UFACTOR6 shock in MSIAH(4)-VAR(1)

  Lettau and Ludvigson, 2001b; Petkova and Zhang, 2005; Zhang, 2005; Chen et al., 2008).

Figure 11 .

 11 Figure 11. Market, size and value risk premia.

Figure 12 .

 12 Figure 12. Momentum and liquidity factors.

Figure 13 .

 13 Figure 13. Economic Uncertainty Index based on 6 uncertainty proxies.

Figure 14 .

 14 Figure 14. Estimated smoothed probabilities for MSIAH(5)-VAR(1) models with liquidity factor.

Figure 15 .

 15 Figure 15. Response to VXO shock in MSIAH(5)-VAR(1)

Figure 16 .Figure 17 .

 1617 Figure 16. Response to EPU shock in MSIAH(5)-VAR(1)

Figure 18 .

 18 Figure 18. Response to VXO shock

Figure 19 .

 19 Figure 19. Response to EPU shock

  of the risk premia (Rm-Rf), SMB, HML and the liquidity factor to a positive shock to EPU by one standard deviation. The impulse reaction period is chosen to be 6 months. Solid lines show impulse responses, while dashed lines represent confidence intervals using the 10th and 90th percentile values calculated on the basis of 1000 bootstrap replications.

Figure 20 . 6 -

 206 Figure 20. Response to UMACRO shock

Figure 21 .

 21 Figure 21. Response to Factor 3 shock

  Note: Responses of the risk premia (Rm-Rf), SMB, HML and the liquidity factor to a positive shock to EPU by one standard deviation. The impulse reaction period is chosen to be 6 months. Solid lines show impulse responses, while dashed lines represent confidence intervals using the 10th and 90th percentile values calculated on the basis of 1000 bootstrap replications. 9 Appendix D: MSIAH(6)-VAR model with Fama-French portfolios

Figure 22 .

 22 Figure 22. Response to UFACTOR6 shock in MSIAH(6)-VAR(1)

Figure 1 .

 1 Figure 1. Estimated smoothed probabilities for MSIAH(5)-VAR(1) models with liquidity factor.

Figure 2 .

 2 Figure 2. Response to VXO shock in MSIAH(5)-VAR(1)

Figure 3 .

 3 Figure 3. Response to EPU shock in MSIAH(5)-VAR(1)

Figure 4 .0

 4 Figure 4. Response to UMACRO shock

Figure 5 .

 5 Figure 5. Response to Factor 3 shock

Figure 6 .

 6 Figure 6. Estimated smoothed probabilities for MSIAH(5)-VAR(1) models with momentum factor.

Figure 7 .

 7 Figure 7. Response to VXO shock

Figure 8 .

 8 Figure 8. Response to EPU shock

  of the risk premia (Rm-Rf), SMB, HML and the momentum factor to a positive shock to EPU by one standard deviation. The impulse reaction period is chosen to be 6 months. Solid lines show impulse responses, while dashed lines represent confidence intervals using the 10th and 90th percentile values calculated on the basis of 1000 bootstrap replications.

Figure 9 . 6 -

 96 Figure 9. Response to UMACRO shock

Figure 10 .

 10 Figure 10. Response to Factor 3 shock

  Note: Responses of the risk premia (Rm-Rf), SMB, HML and the momentum factor to a positive shock to EPU by one standard deviation. The impulse reaction period is chosen to be 6 months. Solid lines show impulse responses, while dashed lines represent confidence intervals using the 10th and 90th percentile values calculated on the basis of 1000 bootstrap replications.

Figure 12 .

 12 Figure 12. Response to UFACTOR6 shock in MSIAH(6)-VAR(1)

  crisis of 1998. Higher uncertainty is also displayed during the October 1987 financial crisis for the VXO, and the July 2011 debt ceiling dispute.Table1provides summary statistics for the variables used in this study. The mean of the three factors is positive, except for SMB. The risk premium factor displays the higher mean and volatility, in terms of standard deviation. ∆VXO is the most volatile among the uncertainty measures. The three factors exhibit negative skewness, except HML, while the three uncertainty measures display positive skewness. Excess kurtosis is observed for all variables, showing that their empirical distributions are leptokurtic, i.e. with substantially fatter tails (than the normal distribution). The Jarque-Bera test statistic is significant at the 1% level of significance for all series, indicating that the variables are highly non-normal. We also conduct the LM test of Engle (1982) for ARCH conditional heteroscedasticity.

7

See

Baker et al. (2013)

for a detailed description of the EPU indexes.

The data are available on www.policyuncertainty.com/index.html.

Debt

8

This test statistic is significant for all uncertainty measures, indicating that they show strong conditional heteroscedasticity, whereas it is non significant for the risk premium and size factors.

Table 1 .

 1 Descriptive statistics

		Min	Max	Mean	S. dev.	Med.	Kurtosis Skewness	JB test *	Engle LM test * *
	MKT	-23.2400	12.4600	5.7019e-01	4.6160	1.1500	5.5445	-0.8832	129.1273 (1.0000e-03)	4.1559 (5.2720e-01)
	SMB	-22.0200	8.4600	-1.3344e-01 3.2564	-0.1600	11.0036	-1.3584	961.4616 (1.0000e-03)	5.5645 (3.5092e-01)
	HML	-9.8600	13.8700	3.9133e-01	3.0884	0.2500	5.6469	0.5737	112.0088 (1.0000e-03)

Table 2

 2

	displays the correlations for the whole sample. The results show a negative correlation
	between uncertainty measures and risk premium factors, suggesting that an increasing in economic
	uncertainty is associated with a falling market, especially from financial uncertainty (-0.56). This result
	is consistent with the findings of Merton (1980), Fleming et al. (1995), Ang et al. (2006), and Durand
	et al. (2011). Merton (1980) point out that the market risk premium should be positively related to
	the variance of the market portfolio and that greater levels of risk should induce a larger market risk
	premium. French et al. (1987) show that the expected market risk premium is positively related to
	expected volatility and negatively related to unexpected changes in volatility. Ang et al. (2006) report
	a negative relationship between returns and changes in expected volatility, using changes in the VXO.

Table 2 .

 2 Correlations.

		MKT SMB HML ∆VXO ∆EPU/100 ∆UMACRO/100
	MKT	1.00	0.19	-0.27	-0.56	-0.27	-0.18
	SMB		1.00	-0.33	-0.23	-0.14	-0.10
	HML			1.00	0.08	0.05	0.02
	∆VXO				1.00	0.39	0.25
	∆EPU/100					1.00	0.15
	∆UMACRO/100						1.00
	3 MS-VAR model
	Markov-Switching vector autogressive (MS-VAR) model developed by Krolzig (1997) provides a
	convenient framework to analysis multivariate changes in regimes. Applied to the Fama-French factors,
	the Markov-switching framework offers the possibility to model high and low volatility periods as
	switching regimes of the stochastic process that generates changes in the expectation of market volatility.
	The model is described by equation 1. In this general specification all parameters (mean, variance
	and autoregressive parameters) are allowed to switch between regimes according to hidden Markov
	chain. In the terminology of Krolzig (1997) this specification is a Markov switching intercept
	autoregressive heteroskedastic VAR model, MSIAH(m)-VAR(p) model, with m the number of variables
	and p the lag order.

Table 3 .

 3 Cumulative responses to shocks from period 0 to 6.This table reports cumulative responses to different uncertainty shocks to the three Fama-French factors.High and Low denote high and low volatility regimes, respectively.

	Response	MKT	SMB	HML
	Shock	High	Low	High	Low	High	Low
	∆VXO	-0.019 -0.033	0.040	0.048	-0.162 0.004
	∆EPU	-0.064 -0.005	0.023 -0.014 -0.043 -0.016
	∆UMACRO	-0.488 -0.034 -0.226 -0.018 -0.068 0.021
	∆UFACTOR3	-1.685 -0.117	0.364 -0.335 -1.312 -0.823

Table 4 .

 4 Correlations.

		∆VXO ∆EPU/100 ∆UMACRO/100	∆UFACTOR3	∆UFACTOR6
	∆VXO	1.00	0.39	0.25	0.82	0.82
	∆EPU/100		1.00	0.15	0.83	0.71
	∆UMACRO/100			1.00	0.38	0.36
	∆UFACTOR3				1.00	0.93
	∆UFACTOR6					1.00
		MKT	SMB	HML	∆UFACTOR	∆UFACTOR6
	MKT	1.00	0.19	-0.27	-0.50	-0.51
	SMB		1.00	-0.33	-0.23	-0.26
	HML			1.00	0.08	0.10
	∆UFACTOR3				1.00	0.93
	∆UFACTOR6					1.00

Table 5 .

 5 Correlations.

		∆VXO ∆EPU/100 ∆UMACRO/100 ∆SPREAD ∆EMEU ∆FDISP ∆UFACTOR6
	∆VXO	1.00	0.39	0.25	0.45	0.51	0.07	0.82
	∆EPU/100		1.00	0.15	0.07	0.54	0.02	0.71
	∆UMACRO/100			1.00	0.29	0.16	0.03	0.36
	∆SPREAD				1.00	0.12	0.10	0.49
	∆EMEU					1.00	0.02	0.77
	∆FDISP						1.00	0.25
	∆UFACTOR6							1.00

Table 6 .

 6 Transition matrix for MSIAH(4)-VAR models

			Regime 1	Regime 2
	VXO	Regime 1	0.8775	0.1225
		Regime 2	0.0351	0.9649
	UMACRO	Regime 1	0.8788	0.1212
		Regime 2	0.0282	0.9718
	EPU	Regime 1	0.8926	0.1074
		Regime 2	0.0378	0.9622
	UFACTOR	Regime 1	0,8532	0,1468
		Regime 2	0,0597	0,9403

7 Appendix B: MSIAH(5)-VAR models with liquidity factor

Table 1 .

 1 Linearity test: VAR model

	1 Appendix A: MSIAH(4)-VAR models
	1.1	MSIAH(4)-VAR(p) estimation results: VXO
		Lags	IC	Two regimes a	single regime
			AIC	20.1921*	21.1228
		Lag1	HQ	20.4823 *	21.2632
			SC	20.9189*	21.4745
			AIC	20.3035*	21.1397
		Lag2	HQ	20.7445*	21.3555
			SC	21.4079*	21.6802
			AIC	20.3558*	21.0405
		Lag3	HQ	20.9483*	21.3321
			SC	21.8396*	21.7707
		a All information criterion (values with an
		asterisk (*)) for all number of lags support the
		presence of regime shifts.

Table 2

 2

	. Lag length test: MSIAH(4)-VAR(p) model
		AIC a	HQ	SC
	Lag = 1 20.1660* 20.3813* 20.7053*
	Lag = 2	20.1940	20.4848	20.9224
	Lag = 3	20.1991	20.5659	21.1177
	a The lag length supported by the IC (values
	with an asterisk (*)) is one.

Table 3 .

 3 Transition matrix

		Regime 1 a Regime 2
	Regime 1	0.8775	0.1225
	Regime 2	0.0351	0.9649

Table 4 .

 4 Linearity test: VAR model

	Lags	IC	Two regimes a	single regime
		AIC	13.3697*	13.9413
	Lag1	HQ	13.6598*	14.0817
		SC	14.0965*	14.2930
		AIC	13.5314*	13.9615
	Lag2	HQ	13.9723*	14.1773
		SC	14.6358	14.5020*
		AIC	13.5449*	13.9674
	Lag3	HQ	14.1374*	14.2589
		SC	15.0287	14.6975*
	a All information criterion (values with an
	asterisk (*)) for all number of lags support
	the presence of regime shifts (except SC for
	Lag=2,3).

Table 5 .

 5 Lag length test: MSIAH(4)-VAR(p) model

		AIC a	HQ	SC
	Lag = 1 13.3697* 13.6598* 14.0965 *
	Lag = 2	13.5314	13.9723	14.6358
	Lag = 3	13.5449	14.1374	15.0287
	a The lag length supported by the IC (values
	with an asterisk (*)) is one.

Table 6 .

 6 Transition matrix

		Regime 1 a Regime 2
	Regime 1	0.8788	0.1212
	Regime 2	0.0282	0.9718

Table 7 .

 7 Linearity test: VAR model

	Lags	IC	Two regimes a	single regime
		AIC	23.6659*	24.3413
	Lag1	HQ	23.9561*	24.4817
		SC	24.3927*	24.6929
		AIC	23.9631*	24.3711
	Lag2	HQ	24.4041*	24.5869
		SC	25.0675	24.9115*
		AIC	23.9113*	24.3494
	Lag3	HQ	24.5038*	24.6410
		SC	25.3950	25.0795*
	a Information criterion (values with an aster-
	isk (*)) for all number of lags (except SC for
	Lag=2,3) support the presence of regime shifts.

Table 8 .

 8 Lag length test: MSIAH(4)-VAR(p) model

		AIC a	HQ	SC
	Lag = 1 23.6328* 23.8481* 24.1721*
	Lag = 2	23.8411	24.1320	24.5696
	Lag = 3	23.8345	24.2013	24.7531
	a The lag length supported by the IC (values
	with an asterisk (*)) is one.

Table 9 .

 9 Transition matrix

		Regime 1 a Regime 2
	Regime 1	0.8926	0.1074
	Regime 2	0.0378	0.9622
	a Note that p i, j = Pr(s t+1 = j|s t = i)

Table 10 .

 10 Linearity test: VAR model

	Lags	IC	Two regimes a	single regime
		AIC	16,32949*	17,1329
	Lag1	HQ	16,6196*	17,2733
		SC	17,0562*	17,4845
		AIC	16,5494*	17,1895
	Lag2	HQ	16,9903*	17,4053
		SC	17,6538*	17,73
		AIC	16,8641*	17,1853
	Lag3	HQ	17,4566*	17,4768
		SC	18,3479	17,9154*
	a Information criterion (values with an aster-
	isk (*)) for all number of lags (except SC for
	Lag=2,3) support the presence of regime shifts.

Table 11 .

 11 Lag length test: MSIAH(4)-VAR(p) model

		AIC a	HQ	SC
	Lag = 1 16,3294* 16,6196* 17,0562*
	Lag = 2	16,5494	16,9903	17,6538
	Lag = 3	16,8641	17,4566	18,3479
	a The lag length supported by the IC (values
	with an asterisk (*)) is one.

Table 12 .

 12 Transition matrix

		Regime 1 a Regime 2
	Regime 1	0,8532	0,1468
	Regime 2	0,0597	0,9403
	a Note that p i, j = Pr(s t+1 = j|s t = i)

Table 13 .

 13 Linearity test:VAR model

	Lags	IC	Two regimes a	single regime
		AIC	26.4611*	27.7957
	Lag1	HQ	26.8916*	28.0063
		SC	27.5395*	28.3232
		AIC	26.5940*	27.7418
	Lag2	HQ	27.2601*	28.0702
		SC	28.2624*	28.5643
		AIC	26.6210*	27.6222
	Lag3	HQ	27.5238*	28.0689
		SC	28.8820	28.7409*

a Information criterion (values with an asterisk (*)) for all number of lags (except SC for Lag=2,3) support the presence of regime shifts.

Table 14

 14

	. Lag length test: MSIAH(5)-VAR(1) model
		AIC a	HQ	SC
	Lag = 1 26.4611* 26.8916* 27.5395*
	Lag = 2	26.5940	27.2601	28.2624
	Lag = 3	26.6210	27.5238	28.8820

a The lag length supported by the IC (values with an asterisk (*)) is one.

Table 15 .

 15 Transition matrix

		Regime 1 a Regime 2
	Regime 1	0.9278	0.0722
	Regime 2	0.0406	0.9594

Table 16 .

 16 Linearity test:VAR model

	Lags	IC	Two regimes a	single regime
		AIC	30.0264*	31.0436
	Lag1	HQ	30.4569*	31.2542
		SC	31.1048*	31.5711
		AIC	30.2903*	31.0579
	Lag2	HQ	30.9565*	31.3862
		SC	31.9587*	31.8803
		AIC	30.3923*	31.0152
	Lag3	HQ	31.2951*	31.4619
		SC	32.6533	32.1339*

a Information criterion (values with an asterisk (*)) for all number of lags (except SC for Lag=2,3) support the presence of regime shifts.

Table 17 .

 17 Lag length test: MSIAH(5)-VAR(1) model

		AIC a	HQ	SC
	Lag = 1 30.0264* 30.4569* 31.1048*
	Lag = 2	30.2903	30.9565	31.9587
	Lag = 3	30.3923	31.2951	32.6533

a The lag length supported by the IC (values with an asterisk (*)) is one.

Table 18 .

 18 Transition matrix

		Regime 1 a Regime 2
	Regime 1	0.9130	0.0870
	Regime 2	0.0396	0.9604

a Note that p i, j = Pr(s t+1 = j|s t = i)

See Shih et al. (2014) for a survey on the evolution of CAPM during the last four decades.

[START_REF] Knight | Risk, Uncertainty and Profit[END_REF] established a distinction between risk and true uncertainty. Risk refers to the possibility of a future outcome

The LM test is applied on the residuals of the ARMA model, where the lag length is selected based on the Akaike information criterion.

The estimation method, identification and impulse response are detailed in[START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF].

In this analysis, we use 1000 bootstrap replicates.

The number of lags is set equal to one according to all information criteria displayed in the Technical Appendix.

This result confirms the findings of[START_REF] Schwert | Business cycles, financial crises, and stock volatility[END_REF],[START_REF] Hamilton | Stock market volatility and the business cycle[END_REF] and Charles and Darné (2014) that volatility of stock returns increases during (severe) recessions.

a Note that p i, j = Pr(s t+1 = j|s t = i)


Dominique Attali

Olivier Devillers

Marc Glisse

Sylvain Lazard

Recognizing shrinkable complexes is NP-complete ⋆

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable.

Introduction

Edge contraction is a useful operation for simplifying simplicial complexes. An edge contraction consists in merging two vertices, the result being a simplicial complex with one vertex less. By repeatedly applying edge contractions, one can thus reduce the size of a complex and significantly accelerate many computations. For instance, edge contractions are used in computer graphics to decimate triangulated surfaces for fast rendering [START_REF] Garland | Surface simplification using quadric error metrics[END_REF][START_REF] Hoppe | Progressive meshes[END_REF]. For such an application, it may be unimportant to modify topological details and ultimately reduce a surface to a single point since this corresponds to what the observer is expected to see if he is sufficiently far away from the scene [START_REF] Popović | Progressive simplicial complexes[END_REF]. However, for other applications, it may be desirable that every edge contraction preserves the topology. This is particularly true in the field of machine learning when simplicial complexes are used to approximate shapes that live in high-dimensional spaces [START_REF] Attali | Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes[END_REF][START_REF] Carlsson | Topological approximation by small simplicial complexes[END_REF][START_REF] Chazal | A sampling theory for compact sets in Euclidean space[END_REF][START_REF] De | A weak characterisation of the Delaunay triangulation[END_REF]. Such shapes cannot be visualized easily and their comprehension relies on our ability to extract reliable topological information from their approximating complexes [START_REF] Carlsson | On the local behavior of spaces of natural images[END_REF][START_REF] De | Topological estimation using witness complexes[END_REF][START_REF] Niyogi | Finding the Homology of Submanifolds with High Confidence from Random Samples[END_REF].

In this paper, we are interested in edge contractions that preserve the topology, actually the homotopy type, of simplicial complexes. It is known that contracting edges that satisfy the so-called link condition preserves the homotopy type of simplicial complexes [START_REF] Ehrenborg | The topology of the independence complex[END_REF] and, moreover, for triangulated surfaces and piecewise-linear manifolds, the link condition characterizes the edges whose contraction produces a complex that is homeomorphic to the original one (a constraint that is stronger than preserving the homotopy type) [START_REF] Dey | Topology preserving edge contraction[END_REF][START_REF] Nevo | Higher minors and Van Kampen's obstruction[END_REF]. An edge ab satisfies the link condition if the link of ab is equal to the intersection of the links Link(a) Link(b) Link(ab) a b of a and b, where the link of a face f is a simplicial complex defined as follows (see figure): consider the smallest simplicial complex that contains all the faces containing f , i.e. the star of f ; the link of f is the set of faces disjoint from f in that simplicial complex [START_REF] Dey | Topology preserving edge contraction[END_REF]. 3We only consider contractions of edges that satisfy the link condition, which implies that the homotopy type is preserved. We refer to such edge contractions as admissible; an admissible edge contraction is also called a shrink and the corresponding edge is said to be shrinkable. After some sequence of shrinks, the resulting complex (possibly a point) does not admit any more shrinkable edges and the complex is called (shrink) irreducible.

We are interested in long sequences of shrinks because they produce irreducible complexes of small size and it is natural to ask, in particular, whether a simplicial complex can be reduced to a point using admissible edge contractions. If this is the case, the simplicial complex is called shrinkable.

Barnette and Edelson [START_REF] Barnette | All orientable 2-manifolds have finitely many minimal triangulations[END_REF] proved that a topological disk is always shrinkable (by any sequence of shrinks). They use this property to prove that a compact 2-manifold (orientable or not) of fixed genus admits finitely many triangulations that are (shrink) irreducible [START_REF] Barnette | All orientable 2-manifolds have finitely many minimal triangulations[END_REF][START_REF]All 2-manifolds have finitely many minimal triangulations[END_REF]. For instance, the number of irreducible triangulations of the torus is 21 [START_REF] Lawrencenko | Irreducible triangulations of the torus[END_REF] and it is at most 396 784 for the double torus [START_REF] Sulanke | Irreducible triangulations of low genus surfaces[END_REF]. We address in this paper the problem of recognizing whether an arbitrary simplicial complex is shrinkable.

Tancer [START_REF] Tancer | Recognition of collapsible complexes is NP-complete[END_REF] recently addressed a similar problem where he considered admissible simplex collapses instead of admissible edge contractions. An admissible simplex collapse (called elementary collapse in [START_REF] Tancer | Recognition of collapsible complexes is NP-complete[END_REF]) is the operation of removing a simplex and one of its faces if this face belongs to no other simplex. 4 Such collapses preserve the homotopy type. Similarly to edge contractions, collapses are often used to simplify simplicial complexes, and a simplicial complex is said collapsible if it can be reduced to a single vertex by a sequence of admissible collapses. Tancer proved that it is NP-complete to decide whether a given (twodimensional) simplicial complex is collapsible [START_REF] Tancer | Recognition of collapsible complexes is NP-complete[END_REF]. The proof is by reduction from 3-SAT and gadgets are obtained by altering Bing's house [START_REF] Bing | Some Aspects of the Topology of 3-Manifolds Related to the Poincaré Conjecture[END_REF], a space that is contractible but whose triangulations are not collapsible.

Both questions of collapsibility and shrinkability are related to the question of contractibility: given a simplicial complex, is it contractible? This question is known to be undecidable for simplicial complexes of dimension 5. A proof given in Tancer's paper [START_REF] Tancer | Recognition of collapsible complexes is NP-complete[END_REF]Appendix] relies on a result of Novikov [24, page 169], which says that there is no algorithm to decide whether a given 5-dimensional triangulated manifold is the 5-sphere. We thus cannot expect shrinks and collapses, even combined, to detect all contractible complexes, but they still provide useful heuristics towards this goal (e.g. [START_REF]Collapsing Rips complexes[END_REF]) and can even be sufficient in specific situations [START_REF] Ehrenborg | The topology of the independence complex[END_REF]. Actually, it is always possible to reduce a contractible simplicial complex to a point if we allow another homotopy preserving operation: the anticollapse (the reverse operation of collapse) [START_REF] Cohen | A course in simple-homotopy theory[END_REF] but, of course, undecidability of contractibility implies that the length of the sequence is not bounded.

Contributions.

A shrinkable simplicial complex is clearly contractible and the converse is not true because of the above undecidability result. We first present a simple shrink-irreducible contractible simplicial complex with 7 vertices. This simple complex is interesting in its own right and it inspired the proof of our main result, which is that it is NP-complete to decide whether a given (threedimensional) simplicial complex is shrinkable. Our proof uses a reduction from 3-SAT similarly as in Tancer's NP-completeness proof of collapsibility [START_REF] Tancer | Recognition of collapsible complexes is NP-complete[END_REF] but, noticeably, our gadgets are much smaller than those used for collapsibility.

Our NP-completeness result on shrinkability together with Tancer's analog on collapsibility naturally raises the question of whether it is also NP-complete to decide if a given simplicial complex can be reduced to a single vertex by a sequence combining admissible edge contractions and admissible simplex collapses. In this direction, we present a contractible simplicial complex with 12 vertices that is irreducible for both shrinks and collapses.

Preliminaries

In this paper, simplicial complexes are abstract and their elements are (abstract) simplices, that is, finite non-empty collections of vertices. We can associate to every abstract simplicial complex a geometric realization that maps every abstract simplex to a geometric simplex of the same dimension. The union of the geometric simplices forms the underlying space of the complex.

As mentioned in the introduction, given a simplicial complex, we are interested in operations that preserve the homotopy type of the underlying space. One of these operations is the shrink, which is the contraction of an admissible edge, also called shrinkable edge. Below, we give a useful characterization of shrinkable edges in terms of blockers. Let K be a simplicial complex and recall that a face of a simplex is a non-empty subset of the simplex. The face is proper if it is distinct from the simplex. Definition 1. A blocker of K is a simplex that does not belong to K but whose proper faces all belong to K.

A blocker is also sometimes called a missing face [START_REF] Melikhov | Combinatorics of embeddings[END_REF], a minimal non-face [START_REF] Ehrenborg | The topology of the independence complex[END_REF], or a simplicial hole [START_REF] Goaoc | Transversal Helly numbers, pinning theorems and projections of simplicial complexes[END_REF].

Lemma 1 ([13]

). An edge ab of K is shrinkable if and only if ab is not contained in any blocker of K. Note that one of the direction is straightforward: if σ is a blocker containing ab, then σ \ {a, b} ∈ Link (a) ∩ Link (b) but σ \ {a, b} ∈ Link (ab).

As we contract shrinkable edges, blockers may appear or disappear and therefore edges may become non-shrinkable or shrinkable. For instance, consider the simplicial complex L = {a, b, c, d, ab, bc, cd, da} whose edges form a 4-circuit and the cone K on L with apex w, that is, the set of simplices of the form {w} ∪ σ where σ ∈ L. The complex K does not contain any blocker and therefore all edges are shrinkable. Note however that the contraction of edge ab creates a blocker which disappears as we contract wa. Hence, as we simplify the complex, an edge that used to be shrinkable (or not) may change its status several times later on during the course of the simplification. Interestingly, the only blockers we need to consider in the paper are triangles.

A simple non-shrinkable contractible simplicial complex

To construct a contractible simplicial complex that is shrink-irreducible, we start with the triangulation of the torus with 7 vertices described in Fig. 1-(a,b) (Császár polyhedron). Notice that the vertices and edges of this triangulation form a complete graph. Thus, every triple of vertices forms a cycle in this graph, which may or may not bound a face.

We now modify the complex as follows. The idea is to add two triangles so that every (arbitrary) cycle on the modified torus is contractible and to remove a triangle so as to open the cavity; see Fig. 1-(c). Namely, we add triangles 012 and 035 and remove triangle 145; see Fig. 1-(d). The resulting complex is contractible because it is collapsible; indeed all edges and vertices inside the "square" and on the boundary of the (expanding) hole can be collapsed until the hole fills the entire square, then it only remains triangles 012 and 035, which can also be trivially collapsed into a single vertex.

To see that the resulting complex is shrink irreducible, note that every edge is incident to at most 3 triangles; indeed, every edge is incident to 2 triangles in the initial triangulation of the torus, and we only added two triangles, which do not share edges. On the other hand, every edge belongs to exactly 5 cycles of length 3 since the graph is complete on 7 vertices. Hence, every edge belongs to at least 2 blockers, which implies that no edge is shrinkable, by Lemma 1.

NP-completeness of shrinkability

Theorem 1. Given an abstract simplicial complex of dimension 3 whose underlying space is contractible, it is NP-complete to decide whether the complex can be reduced to a point by a sequence of admissible edge contractions.

The proof is given in this section by reduction from 3-SAT. We show that any Boolean formula in 3-conjunctive normal form (3CNF) can be transformed, in polynomial time, to a contractible 3-dimensional simplicial complex, such that a satisfying assignment exists if and only if the complex is shrinkable.

Gadgets design

In the following, the gadgets are defined as abstract simplicial complexes but, for clarity, we describe geometric realizations of these gadgets in R 3 . Then the gadgets are assembled by identifying one triangle of one gadget with a triangle of another; this operation preserves the blockers and thus the unshrinkability of edges. A shrinkable edge remains shrinkable if it does not belong to the identified triangles or if it was shrinkable in both gadgets.

Forward gadget

Properties. The forward gadget has a special triangle with edges A, B, C such that A is the only shrinkable edge of the gadget and once A is contracted (thus identifying B and C) there is a sequence of shrinks that reduces the gadget to a single point.

Usage. By gluing the triangle ABC to a triangle of another construction, we enforce that A is contracted before B and C, thus preventing some sequences of shrinks.

Realization. Refer to Fig. 2. Start with four points a, b, x and y in convex position in R 3 and consider the tetrahedron abxy. Split this tetrahedron in four by adding a point o in its interior. The result is a simplicial complex with 5 vertices, 10 edges, 10 triangles and 4 tetrahedra. We then remove the 4 tetrahedra, by applying four triangle collapses. The first three collapses dig a gallery starting at triangle axy by successively removing the pair of simplices (axy, axyo), (oxy, oxyb), (oxb, oxba). The fourth collapse removes the pair (ayb, oayb). The obtained simplicial complex has 5 vertices, 10 edges, 6 triangles: 2 triangles of the initial tetrahedron (axb and xyb) and 4 triangles incident to o (oab, oax, oay and oyb). Notice that as we collapse these pairs of simplices (σ, Σ), the triangle σ becomes a blocker. Thus, the resulting simplicial complex has a unique blockerfree edge A = oa. Let B = ob and C = ab. If A is contracted, the resulting complex contains the triangles axb, xyb, oyb, thus any of the edges incident to b can be shrunk, which reduces the complex to a triangle, which is shrinkable.

Freezer gadget

Properties. The freezer gadget has a special triangle with edges A, B, C such that A and B are the only shrinkable edges of the gadget, and once A or B is contracted (identifying the other with C), there is a sequence of shrinks that reduce the gadget to a single point.

Usage. By gluing the triangle ABC to a triangle of another construction, we enforce that C is non-shrinkable (or frozen) until either A or B is contracted; such a contraction identifies C with the uncontracted remaining edge (B or A).

Realization. Refer to Fig. 3. We start with the same construction as for the forward gadget except that instead of collapsing the pair (oxb, oxba), we collapse the pair (xab, oxab). The list of blockers thus created is axy, oxy, xab, ayb, and the resulting complex contains only 1 triangle of the initial tetrahedron (xyb) and 5 triangles incident to o (oab, oax, oay, oyb and oxb). The result is a simplicial complex with exactly two blocker-free edges, A and B. Similarly as for the forward gadget, once edge A or B is contracted, the resulting complex is shrinkable.

Variable gadget

Properties. The variable gadget associated to a variable x has three special edges: X, X and L (lock). At the beginning X and X are shrinkable edges. When X or X has been contracted, the other one is not shrinkable before L and there is a sequence of shrinks that reduces the gadget to a single point.

Usage. Given a truth assignment, true (resp. false), for variable x, the edge X (resp. X) of the associated gadget is contracted before the other edge X (resp. X). Gluing the lock edge to some key edges (see the clause gadgets), we ensure that once an assignment is chosen for the variable, the other edge, X (resp. X), cannot be contracted unless all the keys needed to open the lock have been released (i.e. all the blockers passing through L have been removed).

Realization. Refer to Figure 4. We consider the four triangles of a squared-base pyramid. From a vertex of the base, X and X are the incident edges on the base and L is the third incident edge on the pyramid. We glue three freezer gadgets onto three triangles incident to the apex, as shown in Figure 4, to ensure that the 3 edges that are incident to the apex and distinct from L are contracted after L, and that the edges on the base remain shrinkable. Contracting any edge on the base transforms the base into a blocker and L remains the only shrinkable edge, ensuring that L will be shrunk before one of X or X.

Two-clause gadget

Properties. The two-clause gadget has three special edges: two literals V and W and a key K. We require that the key is not contracted before one of the two literals. Namely, at the beginning V and W are shrinkable edges and K is not shrinkable. K cannot be contracted before one of V or W and there are sequences of shrinks that contract any non-empty subset of {V, W } before K. Usage. Gluing the key edge to a lock edge of a variable gadget ensures that the lock will not be contracted before the key has been released (i.e. K has become shrinkable).

K V A K V = K V W W W K cannot be contracted before V K V W W K B K A V = W A K =V = W A B A ′ A ′ A ′
Realization. Consider a horizontal triangle and a vertical edge B that pierces it. Each of the triangle edges together with piercing edge define a tetrahedron, and we consider the simplicial complex defined by these three tetrahedra; see Fig. 5. The initial triangle we considered is not part of this complex and is thus a blocker. We place K on the blocker and take for V and W the edges incident to an endpoint of K not in the blocker. Finally, we glue a forward gadget to the face incident to V but not to K and another one for W , symmetrically. Let A (resp. A ′ ) be the third edge of the triangle defined by edges V (resp. W ) and K, and recall that B is the central edge. The only edges that are initially shrinkable are A, A ′ , B, V , and W . Contracting A identifies V and K, ensuring that K will not be contracted before V . Contracting A ′ is similar to contracting A (exchanging V and W ). Contracting B identifies V and W , and yields a configuration where A= A ′ and V = W are the only shrinkable edges; then contracting A identifies V , W , and K ensuring that K will not be contracted before V nor W . Thus, K cannot be contracted (strictly) before one of V or W . Finally, we can contract V then K, which yields a forward gadget whose only contractible edge is W . Hence possible ordering to shrink V , W , and K are V W K, W V K, V KW , or W KV .

Three-clause gadget

Properties. The three-clause gadget has four special edges: three literals U , V , and W and a key K. We enforce that the key is not contracted before one of the three literals. Namely, at the beginning U , V , and W are shrinkable and K is not. K cannot be contracted before one of U , V , or W and there is a sequence of shrinks that contracts any non-empty subset of {U, V, W } before K. Realization. Refer to Fig. 6. The realization is done by simple association of two two-clause gadgets, gluing the key of one clause on one literal of the other, as described in Fig. 6-left. We furthermore add the two triangles defined by KV and KW (note that the triangle KU already belongs to the gadget). These two extra triangles will be needed when gluing gadgets together. By construction, our two glued two-clause gadgets satisfies the properties we require for the three-clause gadgets. Adding the two triangles KV and KW does not invalidate these properties. Indeed, let A be the third edge of triangle KV ; the addition of A has created a blocker (in red in Fig. 6-right). Thus A cannot be contracted and it does not block the contraction of U , V , W , or K. Once V or K is contracted, A is identified with K or V and this extra triangle disappears. Thus, the gadget keeps its properties with these two additional triangles.

Wrap up

3-SAT and shrinkability. Given a 3CNF Boolean formula, we build a threeclause gadget per clause and a variable gadget per variable. The literal edge of each clause gadget is glued to the relevant edge of the variable gadget, that is, a literal x (resp. ¬x) is glued to the edge X (resp. X) of the variable gadget associated to x. The lock edge of each variable gadget is glued to the key edge of each clause it appears in. We assume that the obtained complex is connected, otherwise the 3-SAT problem can be decomposed into independent subproblems, which can be solved separately.

Notice that a pair of edges key/literal forms a triangle in the three-clause gadget and that the pair of edges lock/X (or lock/ X) also forms a triangle in the variable gadget. Thus, the third edges of these triangles are also glued. Actually, the effect of this construction is that the edges K and L of all gadgets are identified and become a single edge in the final complex. By construction, the complex is contractible since each gadget is contractible and we are gluing them by triangles that all have a common edge, K.

Our construction is 3 dimensional, thus it can be embedded in R 7 using general position for the vertices.

From a truth assignment to a sequence of shrinks. For every variable, if it is assigned true (resp. false), edge X (resp. X) is contracted in the associated gadget. These edges are identified to literal edges of the clause gadgets, so their contractions make edge K shrinkable from the point of view of all clause gadgets and K can thus be contracted. All edges corresponding to the other values of the variable gadgets become shrinkable and the complex can be contracted to a point. Fig. 7. Triangulation of a torus with 9 vertices. From left to right: the torus represented as a square with opposite edges identified and its embedding in R 3 as a polyhedron with 9 trapezoidal faces; a non-shrinkable triangulation; and its embedding.

From a sequence of shrinks to a truth assignment. For every variable gadget, if edge X (resp. X) is contracted before X (resp. X), we assign true (resp. false) to the variable associated to the gadget. All clauses are satisfied by this assignment since K cannot be contracted before all clause gadgets have one of their literal edge contracted.

A non-shrinkable Bing's house

In this section, we construct a contractible simplicial complex which is irreducible, both for shrinks and for collapses. 5 The idea is to triangulate carefully Bing's house, in such a way that no edge is shrinkable. Bing's house has two rooms, one above the other. The only access to the upper room is through an underground tunnel that passes through the lower room and the only access to the lower room is through a chimney that passes through the upper room; see Fig. 9-middle.

To triangulate the lower room (and the tunnel), we start with a triangulation of the torus with 9 vertices presented in Fig. 7. We now proceed to two successive alterations of the complex; see Fig. 8. First, we create a room inside the torus, by adding the two (pink hashed) triangles: 036 and 236 and removing triangle 013; the two added triangles delimit the room inside the torus and the removed triangle provides access to the room from outside. We then build a tunnel through the middle of the room by removing two triangles: 023 and 026 and by adding the (blue hashed) triangle 012.

To see that the resulting complex is shrink-irreducible, notice that the triangulation of the torus is shrink-irreducible to start with. During the modification, the only way an edge may become non-shrinkable is if there are more triangles incident to that edge that are added than the ones that are removed. The only edges that fulfill that condition are 36 and 12 and one can check that they are still covered by blockers at the end: 361 and 123 respectively. Similarly, one can check that the room has only three collapsible edges, namely 02, 03 and 13, all lying on the roof. Indeed, no edges are collapsible in the initial triangulation of the torus and an edge is collapsible in the final complex if and only if the number of added triangles incident to that edge is one less than the number of removed triangles. The only edges with this property are 02, 03 and 13.

To finish our construction of Bing's house, we consider a copy of the lower room, which we place above the original one; see Fig. 9. Renaming vertices x by x ′ in the copy, this boils down to the following identifications: vertices 0 with 0', 1 with 2', 2 with 1', 3 with 3', 4 with 5' and 5 with 4'. The result is a simplicial complex with 12 vertices which is still shrink-irreducible but in which no edge is collapsible anymore; see Fig. For the final version, we will make this material available on a web page. If you want to build the non-shrinkable Bing's house with two rooms, you can print the following pages (A4 format yields to a Bing's house of about 10 cm) on some reasonably rigid paper or cardboard. Fold it convex on the solid black lines, concave on the dashed black lines, and glue the parts with the same label together:

The match indicates the tunnel through the lower room to upper room entrance.

Fig. 1 .

 1 Fig. 1. (a) triangulation of the torus with 7 vertices, (d) a contractible non-shrinkable simplicial complex and (b,c) an embedding of their underlying spaces in R 3 . (e) highlights 8 blockers (015, 023, 123, 146, 246, 256, 345, 256) that suffice to cover all edges.

Fig. 2 .

 2 Fig.2. Left: The forward gadget with triangle ABC in blue. Its 1-skeleton is the complete graph with vertices o, a, b, x and y and its (dotted) blockers axy, oxy, oxb, ayb are the triangles that have been collapsed. Middle: Contracting edge A produces a complex with a unique blocker, axy. Right: Schematic representation of the gadget.

Fig. 3 .Fig. 4 .

 34 Fig. 3. The freezer gadget. Left: realization. Middle: contraction of edge A or B. Right: schematic representation.

Fig. 5 .

 5 Fig. 5. The two-clause gadget. Left: realization. Middle: various edge contractions. Right: schematic representation.

Fig. 6 .

 6 Fig. 6. Left: two glued two-clause gadgets. Right: The three-clause gadget.

Fig. 8 .

 8 Fig. 8. Triangulation of the lower room and underground tunnel using 18 triangles: 5 (blue) triangles are coplanar and form the roof, 7 triangles (5 pink and 2 hashed) bound the tunnel under the roof and 6 (pink) triangles lie on the outer walls of the room. The arrow indicates a passage through the tunnel from the underground entrance 678 to the roof exit 023. The red loops indicate the 3 triangles removed from the torus.

9

 9

  .

Fig. 9 .

 9 Fig. 9. Building the Bing's house. Left: triangulation of the lower room and schematic representation. Middle: the two rooms one above the other with the four arrows representing the way through the underground tunnel to the upper room and through the chimney to the lower room. Right. Triangulation of Bing's house.

In other words, in an abstract simplicial complex, the link of σ is the set of faces λ disjoint from σ such that σ ∪ λ is a face of the complex.

Strictly speaking, Tancer calls several of our admissible simplex collapses an elementary collapse. His elementary collapse is the removal of a nonempty non-maximal face σ and the removal of all the faces containing σ if σ is contained in a unique maximal face of the simplicial complex, where maximality is considered for the inclusion in an abstract simplicial complex[START_REF] Tancer | Recognition of collapsible complexes is NP-complete[END_REF].

Acknowledgements This work was initiated during the 12 th INRIA-McGill-Victoria Workshop on Computational Geometry at the Bellairs Research Institute. The authors wish to thank all the participants for creating a pleasant and stimulating atmosphere. The authors thank Uli Wagner for fruitful discussions and interesting references.

⋆ This work has been supported by ANR project TopData ANR-13-BS01-0008 and by the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions).