WORKING PAPER CMVM MODELING AND FORECASTING LIQUIDITY USING PRINCIPAL COMPONENT ANALYSIS AN ILLIQUIDITY COMPOSITE INDICATOR PROPOSAL

Transcription

1 WORKING PAPER CMVM C O M I S S Ã O DO M E R C A D O DE V A L O R E S M O B I L I Á R I O S * N º 0 3 / MODELING AND FORECASTING LIQUIDITY USING PRINCIPAL COMPONENT ANALYSIS AND DYNAMIC FACTOR MODELS AN ILLIQUIDITY COMPOSITE INDICATOR PROPOSAL

2 WORKING PAPER CMVM Modeling and Forecasting Liquidity using Principal Component Analysis and Dynamic Factor Models An Illiquidity Composite Indicator Proposal Paulo Pereira da Silva * CMVM-Portuguese Securities Commission Rua Laura Alves nº 4 Apartado LISBOA paulosilva@cmvm.pt * The views stated herein are those of the authors and not those of the Portuguese Securities Commission.

3 W O R K I N G P A P E R N º 3 / ABSTRACT I survey and describe the main liquidity proxies used in the literature, highlighting some of their merits. Some theoretical background and motivation for the usage of PCA and DFM in the design of a liquidity composite indicator is provided. I apply the PCA/ DFM to a set of nine liquidity proxies over a group of four western European equity markets. The emphasis is placed in extracting a latent variable a liquidity component that captures the co-movement of the proxies. Besides the signal extraction, stress testing for equity market liquidity is illustrated. Finally, I also present some applications regarding the suitability of DFM to model and forecast future liquidity. 03

4 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y INTRODUCTION The microeconomic concept of liquidity is multidimensional and comprehends several dimensions, of which five are well documented in the literature: (i) tightness, (ii) immediacy, (iii) depth, (iv) breadth and (v) resilience (see Sarr and Lybek, 2002 for a quick survey). Tightness refers to reduced transaction costs. Immediacy reflects the velocity by which orders are transmitted to the market and settled. Depth concerns the presence of abundant orders both above and below the price at which the security is trading. Breadth refers to the existence of numerous and large in volume orders with minimal impact on prices. Finally, resilience is associated to the market ability to correct order imbalances, which tend to move the price away from the intrinsic value of the security. In short, market participants perceive a security as liquid if they can quickly sell large amounts of the security without affecting its price. Liquidity is not directly observable. Since there are several dimensions of liquidity, there are also numerous different empirical measures. Several proxies are often used but none captures all the dimensions of the concept. In this regard, Goyenko et al. (2009) perform a horserace of both monthly and annual liquidity measures to evaluate their merits. Sarr and Lybek (2002), Lesmond et al. (1999), Hasbrouck (2004, 2009) and Lesmond (2005) also compare several liquidity proxies based on monthly and daily data. In this paper, I propose the use of a composite indicator of liquidity based on a well -known static method, Principal Component Analysis (PCA, henceforth), and dynamic factor models. One of the main characteristics of these methods is their ability to capture the main features of the data. Regarding PCA techniques, I will use them to extract a few key, uncorrelated liquidity latent variables which are called the principal components from a larger set of correlated liquidity proxies. The suitability of these techniques will depend on the correlation of the proxies: the higher the correlation between the original set of variables, the better this technique will perform. In effect, a highly correlated set of variables means that it will require only a few principal components to characterize the latent(s) variable (s). PCA takes historical data on movements in the proxies and attempts to define a set of orthogonal components that explain the movements. The PCA methodology is derived from an eigenvalue analysis of a large covariance matrix of several commonly used variables that proxy liquidity. The basic idea is that the main 04

5 W O R K I N G P A P E R N º 3 / factors represent the common trend of liquidity over the analyzed time span. PCA permits to reduce the number of liquidity proxies to a manageable dimension and to detect its sources of variability. Notice that proxies with higher correlations are considered more capable of capturing liquidity, provided that they convey the same information (Naes et al., 2011). A good liquidity proxy should capture time-series variation in liquidity. So, PCA and dynamic factor models can be used to assess the liquidity of stock markets and to capture the co-movement of different correlated proxies. In addition, dynamic factor models also capture persistence in liquidity and allow making one-step-ahead predictions for the liquidity proxies and latent liquidity. In the second section, I survey and describe the liquidity proxies used in the study highlighting some of their merits. I use the majority of the proxies proposed by Sarr and Lybek (2002), Zhang (2010) and by Goyenko et al. (2005). In the third section some theoretical background and motivation for the usage of PCA and dynamic factor models in the design of the liquidity composite indicator is provided. In the fourth section, I apply the PCA/ dynamic factor models to a set of nine liquidity proxies over a group of four western European equity markets. The emphasis is placed in extracting a latent variable a liquidity component from the proxies described in section 2. Besides the signal extraction, stress testing for equity market liquidity is performed. In this section I also present some applications regarding the suitability of dynamic factor models to model and forecast future liquidity. 2. Liquidity proxies Nine liquidity proxies are used in this paper. Three of them are closely related to transaction costs (bid-ask spread, effective bid-ask spread and Roll s modified measure), four are associated to market impact (Amihud illiquidity indicator, HHL, Zeros and Market-Efficiency coefficient) and the final two are related to breadth and depth (value turnover and turnover ratio). i) Value turnover: indicator of realized liquidity that is computed as the daily sum of the value of all the transactions. Benston and Hagerman (1974) and Stoll (1978) argue that value turnover, volatility and price influence liquidity. 05

6 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... ii) Turnover ratio: defined as the ratio between the value turnover and the market capitalization of a listed company: iii) Bid-ask spread: measured as the absolute difference between bid and ask prices or as a percentage spread. The latter is more convenient in comparisons of different securities provided that higher prices tend to exhibit higher absolute spreads. The bid-ask spread is a measure of implicit transaction costs: high transaction costs reduce the demand for trades and, thus, the number of potentially active participants in a market. Concurrently, the reduction of the number of participants in the market due to high transaction costs influences market breadth and resilience. According to Glosten and Milgrom (1985), bid-ask spreads may also reflect the degree of information asymmetry. The absolute bid-ask spread is expressed as: where and are the ask and bid prices, respectively. The percentage spread is defined as: iv) Effective bid-ask spread is also used to capture transaction costs. where is the trading price of the security and the prevailing mid-quote when the trade occurs. v) Roll (1984) proposes an estimator of the effective spread based on the serial covariance of the changes in prices. Suppose that the unobservable fundamental value of a stock is a random walk with the following stochastic behavior: where is a white noise. The last observed trade price on day t is given by where is the effective spread and is a categorical variable that equals 1 if the last trade was buyer initiated and -1 otherwise. Roll (1984) assumes equal probabilities for each of the possible values. In addition, he considers that is serially uncorrelated and independent of such that: 06

7 W O R K I N G P A P E R N º 3 / The serial covariance might be written as: The effective bid-ask spread (Roll s estimator) can be expressed as: When the sample covariance is positive, the formula is undefined. Thus, a modified version of the Roll s estimator presented by Goyenko et al. (2009) is defined as vi) Amihud (2000) 1 suggests the following ratio as an indicator of market impact: where is the stock return at t and is the value turnover in Euro at t. One drawback of this indicator is its non-definition for zero volume days. Nonetheless, it is useful to capture the price impact of trades and is widely used as a liquidity proxy. Hasbrouck (2009) argues that among the daily proxies, the Amihud illiquidity measure is most strongly correlated with the transactions and quotes based price impact coefficient. On the other hand, the liquidity effect of asymmetric information is most likely captured in the price impact of a trade (Glosten and Harris, 1988). Acharya and Pederson (2005), Watanabe and Watanabe (2008), Spigiel and Wang (2005), Avramov et al. (2006) and Kamara et al. (2008) use the Amihud proxy to assess commonality in liquidity among stocks. vii) Zeros. Lesmond et al. (1999) compute the proportion of days with zero returns as a proxy for illiquidity. They present two reasons to support this indicator: (i) securities with lower liquidity are more likely to have zero volume days and thus more likely to have zero return days; (ii) stocks with higher transaction costs have less private information acquisition (since it is more difficult to overcome higher transaction costs) and thus, even on positive volume days, they are more likely to have no-information-revelation, zero return days. 1- Acharya and Pedersen (2004) also adopted this indicator. 07

8 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... viii) Hui-Heubel Liquidity ratio (HHL) attempts to capture the price impact, breadth and resilience dimensions of liquidity. It relates the volumes of trades and their impact on prices, and is computed as an average of 5-day periods in a sample, in order to smooth volatility. ithe Hui-Heubel Liquidity ratio uses the turnover ratio in the denominator, scaling price movements by the speed of rotation of the equity in the markets. The higher the liquidity of an asset, the lower will HHL be. ix) The Market-Efficiency Coefficient (MEC) was proposed by Hasbrouck and Schwartz (1988) to distinguish short-term from long-term price changes. Indeed, price movements are more continuous in liquid markets, even if new information influences equilibrium prices and consequently, for a given permanent price change, the transitory changes to that price should be minimal in resilient markets. where is the variance of returns over the longer period, is the variance of the return of the shorter period and T is the number of shorter periods embedded in the longer period. MEC should be close to one in more resilient markets (even though, slight lower than one), in the sense that overreaction and underreaction to new information should be minimal. Prices of assets with high market resilience may exhibit lower volatility (less transitory changes) between periods in which the equilibrium price is changing. Excessive short term volatility/overshooting leads to significantly lower than one MEC figures. 2. EXTRACTING LATENT VARIABLES A. The Principal Component Approach Principal component analysis (PCA) is a method for detecting patterns in data and to emphasize similarities and differences in variables. PCA reduces the dimension of the data, that is, attempts to reduce the number of variables to analyse without 08

9 W O R K I N G P A P E R N º 3 / much loss of information. Put differently, it aims to explain the variability of a set of variables through a new smaller set of new non-correlated/orthogonal latent variables. Thus, one may describe the variation in a set of correlated variables using a smaller new set of uncorrelated factors. Each component is computed in order to consider for the maximum possible variation of the initial dataset. The first component will be the most relevant. It denotes the linear combination of the original variables that yields the higher sample variance (eigenvalues) among all the possible linear combinations. If the variables display a high correlation altogether, the first component usually denotes a common trend. Fewer components ease the analyst task of providing an intuitive meaning to the set of components. The interpretation of the components is usually guided by the level of correlation of each variable with a particular component. Principal component analysis consists in the spectral decomposition of a covariance matrix or of a correlation matrix. Performing PCA is equivalent to determining the eigenvalues and eigenvectors of the covariance (correlation) matrix. If PCA is calculated using the correlation matrix then the outcome will only be affected by the correlations of variables, but if the input to PCA is the covariance matrix the results will depend not only on the correlations of the variables but also on their standard errors. Indeed, the representation of the principal components based on the covariance (correlation) provides a linear representation for the (standardized) liquidity proxies. It can be shown that there is no general association between the spectral decomposition of a covariance matrix and the spectral decomposition of the corresponding correlation matrix. Accordingly, there is no general association between the principal components of a covariance matrix and those of its correlation matrix. Though, if the variables have similar standard errors, both methods should yield similar results. PCA is sensitive to the units of measurement, which determine variances and covariances. In our case, it is preferable to work with correlation matrix because correlation is not affected by the scale of the variables. One should also note that the PCA is one of the simplest of many dimension reduction methodologies that transform a set of correlated variables into a set of uncorrelated variables. The main difference between the PCA and other factor analysis methods derives from the fact that the former seeks to identify a small number of factors to explain the total variation of the dataset while the latter place 09

10 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... the emphasis on using a small number of hypothetical random variables to explain the correlations or covariances in a multivariate dataset. PCA can be applied to any set of stationary time series, regardless of the level of correlation of the set of variables. The assumption that the variables are normally distributed is not required, only that they have finite variances and covariances. Standard variances and covariances are not robust and are sensitive to outliers. In order to make PCA insensitive to outliers, robust versions of variances and covariances are necessary. PCA can be implemented in three steps: 1. Calculate the covariance or correlation matrix of the original dataset. 2. Derive the eigenvalues and the eigenvectors of that matrix. Next, rank/order the eigenvalues by their value. The first principal component is associated to the higher eigenvalue; the second principal component is associated to the second higher eigenvalue, and so on. The first component explains the most variation of the dataset. In very highly correlated datasets, this component captures an almost parallel shift in all variables, and more generally it is labelled the common trend component (in our case it captures the most often experienced type of common movement in all the liquidity proxies). The second eigenvector belongs to the second largest eigenvalue, and therefore the second component explains the second most variation in the dataset. 3 - Let X be the time series dataset, V the covariance matrix (correlation matrix) and P the principal components. There is a representation of the data such that: where W is a p-by-p matrix whose columns are the eigenvectors of X T X (factor scores matrix). PCA allows transforming the original data into a system of orthogonal factors. Consider the following PCA representation of k liquidity proxies: 10

11 W O R K I N G P A P E R N º 3 / where denotes the liquidity proxy ( and is the factor loading of liquidity proxy. Thus, if the j principal components moves by, the liquidity proxy will move by, ceteris paribus. Factor Score i is provided by the following expression: is the i largest eigenvalue and corresponds to the eigenvector associated to eigenvalue i. The correlation/covariance matrix of the principal components is diagonal, given that the factors are uncorrelated. As for the variance of the principal components it will be equal to the corresponding eigenvalue. The total variation of the original time series is provided by the sum of the eigenvalues of the covariance (correlation) matrix. Consequently, one can assess the contribution of each factor by dividing its eigenvalue by the sum of the eigenvalues: The capacity of PCA to reduce dimensions, combined with the use of orthogonal variables for risk factors, makes this technique an extremely attractive option for Monte Carlo simulation and scenario analysis. B. Dynamic Factor Models Dynamic Factor Models (DFM) are flexible models for multivariate time series. DFM aim to combine the cross-section analysis through Principal Components Analysis and the time series dimension of data through linear regression modelling (Federici and Mazzitelli, 2010). These models allow for serial and mutual correlation of the idiosyncratic errors. One advantage of factor models lies in the fact that they may use information from many variables without running into scarce degrees of freedom, which is a problem frequently faced in regression analyses. Because of their ability to simultaneously and consistently model data sets in which the number of series exceeds the number of time series observations, these types of models have received considerable attention in the past decade. Breitung and Eickmeier (2005) point two other reasons to use factor models: the idiosyncratic movements which possibly include measurement errors and local shocks can be eliminated with this technique and one does not need to rely on overly tight assumptions as is sometimes the case in structural models. 11

12 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Coppi and Zannela (1978) introduced Dynamic Factor Models. They seek to decompose the covariance matrix (V) of a set of time series variables into three distinct covariance matrices: where represents the variability of the data structure without taking the time dimension into account (it equals the covariance matrix of the average of the units with respect to time); reflects the variability, due to the time dimension, of the average of the units, regardless of the dynamics of the single units; measures the variability due to the difference between the dynamics of the overall average of the units, that is the average dynamics, and the dynamics of the single units. The observed endogenous variables are linear functions of exogenous covariates and unobserved factors, which have a vector autoregressive structure, and thus are persistent over time. In this framework, the unobserved factors can also be a function of exogenous covariates. The error terms in the equations for the dependent variables may be autocorrelated. Stock and Watson (2010) divide the time-domain estimation of DFM into three generations. The first generation is based in low-dimensional (small N) parametric models estimated in the time domain using Gaussian maximum likelihood estimation and the Kalman filter. This methodology provides optimal estimates of the factors (and optimal forecasts) under the model assumptions and parameters. Notwithstanding, this estimation method requires nonlinear optimization, which may be a serious drawback due to convergence issues. The second generation of estimators involves nonparametric estimation with a large set of variables using crosssectional averaging methods, in particular principal components and related methods. The principal components estimator of the space spanned by the factors is consistent. Moreover, if N is sufficiently large, then the factors are estimated precisely enough to be used as data in later regressions (Stock and Watson, 2010). The third generation uses consistent nonparametric estimates of the factors to estimate the parameters of the state space model used in the first generation solving the dimensionality problem of first-generation models. As Principal Components, latent factors estimated this way is sometimes referred to as extracting or estimating an indicator. The principle of a dynamic factor model is that a few latent dynamic factors lead the comovements of a high-dimensional set of time-series variables, which is also influenced by a vector of mean-zero idiosyncratic disturbances. The error term arises from measurement errors and 12

13 W O R K I N G P A P E R N º 3 / from special aspects of the individual series. The latent factors follow a time series process, which is commonly taken to be a vector autoregression (VAR) (Stock and Watson, 2010). For each of the analyzed countries, I estimate the following dynamic factor model: where denotes a latent variable that represents a common movement in liquidity and follows an AR(1) process. The model is estimated in its state space representation using stationary Kalman Filter. represents persistence in liquidity. Thus, if liquidity is persistent it may also be foreseeable, which is an additional advantage over the standard PCA method. represents the expected value for the proxy i during normal periods and is the sensibility of proxy i to movements in the latent variable. Notice that I use seasonally adjusted variables in the estimation, and consequently there is no need to model seasonality. 4. APPLICATION TO FOUR WESTERN EUROPEAN COUNTRIES A. Principal Components Analysis In this section, I apply the PCA methodology to measure the liquidity of four western European equity markets. As said before, liquidity is a latent variable, that is, it is not directly observed. Nine well documented proxies of liquidity are used to capture the movements of liquidity: bid-ask spread, effective bid-ask spread, Roll s modified measure, Amihud illiquidity indicator, HHL, Zeros, MEC, turnover and turnover ratio. The analysis is based on monthly data, given that some of the proxies are only available at a monthly basis (e.g. zeros, MEC, Roll s modified measure). In order to calculate the values of the proxies, I collect daily data from Bloomberg, namely last trade prices, bid and ask prices, market capitalization and turnover. The data collected covers the period that ranges between 2000 and 2012, and 2043 securities traded in France, Italy, Spain and Portugal. All the securities (active or inactive) 13

14 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... present in MiFid database at are included in the main sample. In order to obtain country aggregate values for the liquidity proxies, the following procedure is conducted: i) Firstly, logs are introduced to smooth the path of some of the proxies (Amihud indicator, HHL, turnover, bid-ask spread and effective bid-ask spread); ii) next, the (average) monthly proxies for each of the individual securities is calculated; iii) at last, the weighted averages of the monthly liquidity proxies are computed for each market, using the securities market capitalization as weights (a 20% cap is introduced to reduce the dependency of the aggregate liquidity proxies to a few set of securities). The first step in PCA consists in the computation and analysis of the correlation matrix. As expected, the reported results show some similarities in the data: the correlation matrix shows a high linear statistical association between the (ln) Amihud Indicator and the (ln) HHL; and between the Turnover Ratio and the Turnover; MEC exhibits a low correlation with the other variables (Table 1). In a first stage, the PCA method is applied to the nine variables (Table 2). The KMO measure and Bartlett's Test of Sphericity suggest that the application of the PCA method provides good results for Portugal and Italy (KMO higher than 0.7) and acceptable for Spain and France (KMO higher than 0.5). In the case of Portugal, the first component accounts for 44.9% of the variance of the data. The (ln) Amihud Indicator, (ln) HHL, and (ln) Turnover are the variables with a higher percentage of the variance explained by the extracted component. For Italy, the first component explains 37.7% of the total variance, whereas for France and Spain that percentage drops to 32.6% and 29.7%, respectively. In general, the (ln) Amihud Indicator, the (ln) HHL and effective bid-ask spread are the proxies with higher contribution to the first principal component. In a second stage, I reduce the number of variables to six. Roll s modified measure, Zeros and Market-Efficiency coefficient seem to have little impact in the co-movement of the proxies according to the factor scores. They are very indirect measures of liquidity that account for little correlation with the first principal component, and for that reason they are dropped in further analyses. Notice that I am not taking seasonality into consideration. Two different approaches 14

15 W O R K I N G P A P E R N º 3 / are used to address seasonality. In the first, raw data is used to run the PCA and then seasonality is extracted from the principal component using the additive method of seasonality decomposition. In the second, seasonality is removed from the raw data using again the additive seasonality decomposition method. The different approaches yield similar results. Although both approaches are performed simultaneously, I put more emphasis in the second one in the subsequent analysis. After dropping Roll s modified measure, Zeros and Market-Efficiency coefficient, the performance of PCA increases dramatically for some of the analyzed equity markets (Table 3). In Portugal and Italy the first principal component now represents 65.7% and 46.5% of the variance of the sub dataset. In Spain and France, the total variance explained increases to 40.2% and 46.5% of the variance of the proxies, respectively. Communality analysis shows that the Amihud indicator, HHL, bid-ask spread or effective bid-ask spread are the proxies that capture a higher percentage of the variation of the first principal component. Figure 1 displays the first component evolution between January 2000 and December 2012 and allows identifying the pattern of the latent variable. For instance, all the analyzed countries exhibit a decline of liquidity after 2008, due to the international financial crisis. PCA indicates that at the end of 2012 the liquidity level was still above the level displayed in 2008 in three countries (the exception is France). In 2010, Spain had already recovered the liquidity level displayed before the crisis. Notwithstanding, the recovery process was reverted in the end of 2011 with the European sovereign debt crisis. One of the advantages of using the PCA approach is its flexibility to model the correlations between variables. The factor loadings obtained from PCA allow designing a stress test approach, where the impact of aggregate liquidity shocks over the proxies is simulated. This stress test exercise is presented in Figure 2. For example, regarding Spain the effective bid-ask spread rises from 0.362% in normal times, to 0.841% in stress periods. In France and Italy, the HHL and Amihud indicators more than double in highly stress periods, meaning that negative shocks in aggregate liquidity affects price impact measures in a large extent in these countries. Instead of performing PCA in the liquidity proxies, one might consider as an alternative their changes over time. Table 4 shows the correlation matrix of the first differences of the liquidity proxies. Changes in the Amihud indicator and HHL are highly correlated in the four countries. The same occurs with the (ln) turnover and 15

16 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... the turnover ratio (except for Italy). Table 5 presents the total variance explained by the first and second principal component, communalities, factor loadings and factor scores for each proxy. With the exception of Italy, the first two principal components account for more than 60% of the variance of the six liquidity proxies. These principal components represent different dimensions of liquidity. One way of assessing the economic interpretation of the principal components is through the analysis of the factor loadings, which in some sense represent the correlation between the factors and the liquidity proxies. For instance, in the case of Spain the first principal component is associated with breath (provided its correlation with the turnover ratio and log turnover), whereas the second principal component denotes price impact and transaction costs. In the case of France, the first principal component represents price impact, whilst the second denotes breadth. At last, in Italy and Portugal the first principal component is highly correlated with the price impact measures. Using differences instead of levels in PCA also permits simulating the impact of shocks in the latent variables over the liquidity proxies. Figure 3 shows 95% confidence intervals for the liquidity proxies (corrected for seasonality). In the case of Portugal, stress testing indicates a decline of the turnover ratio of 0.37% in the event of a standard deviation shock in the liquidity latent variables. In Spain, the turnover ratio is not particularly affected by changes in the latent variable, and in France and Italy that impact is also of minor importance. However, a standard deviation liquidity shock in the latent variables may have serious repercussions in price impact measures: the HHL indicator more than triples in Spain and the effective bid-ask spread increases by 30 basis points in France. B. Dynamic Factor Models DFM permit modelling the dynamics of our variables of interest and unobserved components in a VAR framework. I estimate a DFM for the liquidity proxies of each country assuming that the latent variable displays first order autocorrelation and that the behaviour of the proxies is explained by this latent variable and a disturbance term. Moreover, I am assuming that the dynamics of the proxies are solely described by the dynamics of the unobserved liquidity. This analysis focuses on the bid-ask spread, effective bid-ask spread, Roll s modified measure, Amihud illiquidity indicator, HHL and turnover ratio. Value turnover is excluded due to convergence issues and possible non-stationary of the series. 16

17 W O R K I N G P A P E R N º 3 / The estimated models reveal that the unobserved component is very persistent over time. The value of is statistically different from zero and ranges between 0.86 and Moreover, coefficients associated to the variables ln(1+ Amihud indicator), ln(1+ HHL), ln(1+ bid-ask spread) and ln(1+ effective bid-ask spread) are statistically significant at the 5% significance level for all countries. The liquidity component does not seem to explain turnover ratio in France, Spain and Italy. The forecasting ability of the DFM is also tested. In order to do so, the sample is divided in two subsamples: from January 2000 to December 2010 and from January 2011 to December I re-estimate the model for the first period and use the second for out-of-sample forecast. Two different measures of forecasting accuracy are computed, namely MAPE and RMSE. I also compare the accuracy of DFM with the use of the historical mean, in terms of forecasting. To do so, RMSE-R squares is computed and the Diebold and Mariano test is run. DFM presents a remarkable forecasting accuracy in the cases of Portugal and Spain, where the Diebold and Mariano t-stat is statistically significant at the 90% level in all the liquidity proxies, with the exception of turnover ratio in Spain. In the cases of France and Italy, DFM only appears to provide higher predictive accuracy than the historical average in the forecast of the bid-ask spread and the Amihud illiquidity indicator, respectively. RMSE based R-squared also suggests that the forecasting ability of DFM is higher amid price impact measures (Amihud indicator and HHL) than transaction costs measures. 5. FINAL REMARKS PCA methodology is widely used to capture unobserved variables through the analysis of other observable proxies. In this paper, I apply PCA to capture the evolution of liquidity, which is not directly observed. In doing so, I use a set of nine liquidity proxies. Concurrently I show how to simulate the impact of liquidity shocks in proxies of liquidity such as bid-ask spread, turnover ratio and Amihud Indicator. In that sense, PCA can be useful for measuring aggregate liquidity risk and for stress test reporting. In terms of the results, the unobserved liquidity variable evidences a significant downturn after the Lehman Brothers bankruptcy in the four analysed markets. Even though this event affected the liquidity of all markets, it was particularly severe in Spain and France, but with transitory effects. Both markets recovered to their long term liquidity level before mid Italy also exhibits a decline of the liquidity component after the Lehman Brothers bankruptcy, but contrary to Spain and France that liquidity shock assumes a more permanent effect. 17

18 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Furthermore, I present an extension of PCA methodology, Dynamic Factor Models. DFM reveal that the unobserved component is very persistent over time, and thus it is predictable. DFM presents a remarkable forecasting accuracy, particularly in Portugal and Spain. Even in the cases of France and Italy, DFM appears to provide higher predictive accuracy than the historical average in forecasting the bid-ask spread and the Amihud illiquidity indicator. This analysis also shows communalities across the liquidity components of the four markets. In other words, the liquidity of different European markets tends to co-move. Comparing the two approaches, PCA has the advantage of being easier to implement and is more flexible, whereas DFM computation is slow and sometimes convergence is not achieved. On the other hand, DFM permits modelling the time series structure of the proxies and to compute forecasts of the liquidity component and of the proxies. 18

19 W O R K I N G P A P E R N º 3 / REFERENCES Amihud, Y. (2002). Illiquidity and stock returns: Cross-section and time-series effects. Journal of Financial Markets 5, Acharya, V. and L. H. Pedersen (2005). Asset pricing with liquidity risk. Journal of Financial Economics 77, Avramov, D., T. Chordia and A. Goyal (2005). Liquidity and autocorrelations in individual stock returns. Working Paper. Benston, G. and R. Hagerman (1974). Determinant of bid-asked spreads in the over-the-counter market. Journal of Financial Economics 1(4), Breitung, J. and S. Eickmeier (2009). Testing for structural breaks in dynamic factor models. Deutsche Bundesbank Economic Studies Discussion Paper No. Coppi, R. and F. Zanella (1978). L analisi fattoriale di una serie temporale múltipla relativa allo stesso insieme di unità statistiche. Società Italiana di Statistica, XXIX riunione. Federici, A. and A. Mazzitelli (2010). Dynamic factor analysis with Stata. 2nd Italian Stata Users Group meeting. Glosten, L. and L. Harris (1988). Estimating the components of the bid/ask spread. Journal of Financial Economics, 21, Glosten, L. and P.R. Milgrom (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics, Glosten, L. (1987). Components of the bid ask spread and the statistical properties of transaction prices. Journal of Finance 42, Goyenko, R.Y., C.W. Holden and C.A. Trzcinka (2009). Do liquidity measures measure liquidity?. Journal of Financial Economics 92, Hasbrouck, J. (2004). Liquidity in the futures pits: inferring market dynamics from incomplete data. Journal of Financial and Quantitative Analysis 39, Hasbrouck, J. (2009). Trading costs and returns for US equities: estimating effective costs from daily data. Journal of Finance. Hasbrouck, J. and R. A. Schwartz (1988). An assessment of stock exchange and over-the-counter markets. Journal of Portfolio Management 14,

20 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... REFERENCES Kamara, A., X. Lou and R. Sadka (2008). The divergence of liquidity commonality in the cross-section of stocks. Journal of Financial Economics 89, Kyle, A. (1985). Continuous auctions and insider trading. Econometrica 53 (6), Lesmond, D. A., J. P. Ogden. and C. Trzcinka (1999). A new estimate of transaction costs. Review of Financial Studies 12 (5). Lesmond, D. (2005). Liquidity of emerging markets. Journal of Financial Economics 77, Naes, R, J. A. Skjeltorp, B. A. Ødegaard (2011). Stock market liquidity and the business cycle. The Journal of Finance 66, Roll, R. (1984). A simple implicit measure of the effective bid ask spread in an efficient market. Journal of Finance 39, Sarr, A. and T. Lybek (2002). Measuring liquidity in financial markets. IMF Working Paper No. 02/232. Spiegel, M. I. and X. Wang (2005). Cross-sectional Variation in Stock Returns: Liquidity and Idiosyncratic Risk. Yale ICF Working Paper No Stock, J. H. and M. W. Watson (2010). Dynamic factor models. Oxford Handbook of Economic Forecasting. Stoll, H. (1978). The pricing of security dealers services: An empirical study of NASDAQ Stocks. Journal of Finance 33, Watanabe, A. and M. Watanabe (2008). Time-varying liquidity risk and the cross section of stock returns. Review of Financial Studies, 21, Zhang, H. (2010). Measuring liquidity in emerging markets. Working Paper. 20

21 W O R K I N G P A P E R N º 3 / T A B L E S Table 1 Correlation Matrix Panel A Portugal Panel B Spain Panel C Italy Panel D France 21

22 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Table 2 Total Variance Explained by the First Principal Component and Communalities Using the 9 Liquidity Proxies Panel A - Portugal Panel C - Italy Panel B - Spain Panel D - France 22

23 W O R K I N G P A P E R N º 3 / Table 3 Total Variance Explained by the First Principal Component and Communalities Using 6 Liquidity Proxies Panel A - Raw Series Portugal Panel B - Seasonally Adjusted Series Panel A - Raw Series Spain Panel B - Seasonally Adjusted Series Panel A - Raw Series Italy Panel B - Seasonally Adjusted Series 23

24 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Panel A - Raw Series France Panel B - Seasonally Adjusted Series Figure 1 First Principal Component Evolution Between 2000 and 2012 Panel A - Portugal 24

25 W O R K I N G P A P E R N º 3 / Panel B - Spain Panel C - Italy 25

26 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Panel D - France Figure 2 Liquidity Shocks Impact of a Liquidity Aggregate Factor Shock on the Liquidity Proxies Panel A - Portugal Panel B - Spain 26

27 W O R K I N G P A P E R N º 3 / Panel C - Italy Panel D - France Table 4 Correlation Matrix of Differentiated Variables Panel A - Portugal 27

28 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Panel B - Spain Panel C - Italy Panel D - France Table 5 Total Variance Explained by the First and Second Principal Component, Factor Scores, Factor Loadings and Communalities Using 6 First-Differenced Liquidity Proxies Panel A - Portugal 28

29 W O R K I N G P A P E R N º 3 / Panel B - Spain Panel C - Italy Panel D - France 29

30 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Figure 3 Liquidity Shocks Impact of a Shock in the Liquidity Aggregate Factor on the Liquidity Proxies, Corrected for Seasonality Panel A - Portugal Panel B - Spain 30

31 W O R K I N G P A P E R N º 3 / Panel C - Italy Panel D - France 31

32 M O D E L I N G A N D F O R E C A S T I N G L I Q U I D I T Y... Table 6 Dynamic Factor Model Estimation Figure 4 Latent Liquidity Derived From a Dynamic Factor Model 32

33 W O R K I N G P A P E R N º 3 / Table 7 Out-of-Sample Forecasting Accuracy: DFM versus Sample Average Table 8 Out-of-Sample (One-Step-Ahead) Forecasting Accuracy: DFM versus Historical Average 33

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