Macroeconomic Stress , Equity Market Liquidity Spirals and Markov Regime Switching

This paper makes an attempt to identify the periods of high illiquidity spiral and loss spiral fitting into Markov switching regimes model with Constant Transition Probability and Time-Varying Transition Probability models in US equity market. We identified two different states of the illiquidity spiral and loss spiral in the data associated with the said variables under the CPT and TVTP. However the time-varying transition probabilities for illiquidity spiral and loss spiral have changed significantly during the period under analysis and the explanatory variables are very informative in dating the evolution of the state of the illiquidity spiral and loss spiral over a period of 27 years starting with 1983. Hence TVTP model is preferred over the CTP model in identifying the illiquidity spiral and loss spiral regime switching. In particular, the probability of remaining in the high illiquidity spiral and high loss spiral regimes increases with a decrease in S&P 500 return.


Introduction
Liquidity plays an important role in the well-functioning of the economic system.It is more closely intertwined with the financial markets of the economy (Fuerst, 1992;Brunnermeier, 2008;Naes, Skjeltorp, & Odegaard, 2011).Market liquidity is a function of the information flow (Glosten & Milgrom, 1985;Klibanoff, Lamont, & Wizman, 1998), trading rules (Amihud & Mendelson, 1988), and sentiments of market participants (Baker & Stein, 2004;Chen, Hong, & Stein, 2002).Sudden dry out of the liquidity from the system disrupts business and economic activities in the economy.Brunnermeier and Pedersen's (2009) observe that during 2008 financial crisis liquidity suddenly dried up due to evolvement of liquidity spiral.Some literature suggests that market liquidity dryness occurs due to various triggers such as asset price bubbles (Brunnermeir, 2010), credit bubbles (Kiyotaki & Moore, 1997) and liquidity spirals (Brunnermeier & Pedersen, 2009).Further, Brunnermeier and Oehmke (2012) study points out a severe mismatch between funding structure and potential investment venues dried out liquidity from the market during 2008 financial crisis.Jain, Mishra and McInish (2013) also empirically examine and affirm the existence of liquidity spirals during the financial crisis periods in US market.
Whether the existence of such liquidity spiral phenomenon was only limited to 2008 financial crisis, remains an open question for empirical investigation?This phenomenon might be associated with the other periods of financial crisis as well.Against this backdrop, this study uses the spiral measures proposed in Jain, Mishra and McInish (2013) and examine its applicability to identify illiquidity spiral and loss spiral dynamics across different crisis periods from 1983 to 2010 for S&P 500 constituent stocks using Markov-Switching Regime Models (MSRM).
This study contributes to the existing stock of finance literature in three ways.Firstly, we model the dynamic behavior of spiral measure namely illiquidity spiral and loss spiral and their state dependencies perhaps for first time to the best of our knowledge.Secondly, we also examine whether liquidity spiral measure actually captures the variation in the liquidity states together.Thirdly, the correspondence of illiquidity spiral with the loss spiral during the market stress period extends the scope to empirically examine the dynamic co-movement behavior of these two components of liquidity spiral phenomenon.Nonetheless empirical findings of this study extend support partly towards the proposed theoretical liquidity spiral phenomenon by Brunnermeier and Pedersen's (2009).
The rest of the paper is structured as follows.Section 2 provides review of literature on liquidity and liquidity spirals.Section 3 presents materials and methods.Section 4 outlines the econometric methodology that models the periods of illiquidity and loss across the study period.Section 5 delineates the preliminary and empirical findings of the study.Conclusion and limitations are discussed in section 6.

Liquidity and Liquidity Measure
There is no such unique definition of 'liquidity'.Being multidimensional in nature, it is neither observed nor measured directly (Amihud, 2002).Measurement without definition is, however, difficult if not impossible.Researchers have used different proxies to measure different dimensions of asset liquidity.According to Larry Harris, liquidity has four major dimensions namely, immediacy, width, depth, and resiliency.Immediacy shows how quickly a given size of asset can be arranged, width or market breadth refers to cost involved in trading asset, depth refers to size of asset at a given trading cost and last dimension resiliency indicates how quickly prices revert back to fundamental level.In empirical research, measures the liquidity under a few broad categories (i) volume based measures, which are captured by transaction cost (Stoll, 1978) and market frictions (Stoll, 2000), (ii) price based measure that reflects the resiliency of assets, which is commonly captured by price volatility and market efficiency coefficients (Hasbrouck & Schwartz 1988), and (iii) market impact which indicates the differential impact of liquidity on price (Cvitanić & Malamud, 2011;Ren & Zhong, 2012).
Thus, the finance literature identifies a wide array of proxies for the liquidity measurement, some of them are bid-ask spread, effective spread, trade volume, Amihud illiquidity measure, Roll's estimate, Gibbs sample estimates, Lesmond, Ogden, and Trzcinka (LOT) estimate (1999), and Stambaugh Gamma price impact estimator.However, the usage of such liquidity proxy normally differs based on frequency of data, and richness of data.Despite its importance, problems in measuring and monitoring liquidity risk persist.There are no consensus of using single efficient liquidity measure which captures all the dimensions (Goyenko, Holden, & Trzcinka, 2009;Corwin & Schultz, 2012).Brunnermeier and Pedersen's (2009) theoretical study triggered another debate on the characteristics of liquidity during the severe crisis periods.It is observed that liquidity dynamics and price movement behave in different way and document a reinforcing relationship between illiquidity and price movement.They named such liquidity dynamics as "liquidity spirals".In a study of 2007-2009crisis period, Hameed, Kang, and Viswanathan (2010) also find the dynamic relationship between sudden liquidity-dry up and the severity of crisis.The liquidity and crisis association becomes more prominent when liquidity is tied up with the funding availability.Rösch and Kaserer (2013) also document the spiral effect between the financial sector's funding liquidity and an asset's market liquidity.This effect is more prominent during the marker downturn periods.As under the uncertain and panic situations the asset funding becomes difficult and in result an increase in liquidity commonality which then leads to market-wide liquidity dry-ups.In order to define the liquidity spiral phenomenon, Jain, Mishra and McInish (2013) proposed measure to capture liquidity spiral phenomenon.The proposed measure includes two proxies called illiquidity spiral and loss spiral.The illiquidity spiral quantifies the intensity of illiquidity whereas loss spiral measure scales the severity of loss due to decline in the stock prices.

Data Sources
The study is based on the secondary data which is obtained majorly from the Center for Research in Security Prices (CRSP) database, provided by The University of Chicago where sample stocks related data are restricted to S&P 500 composite index and the daily stock data from January 1983 to December 2010.The each stock specific data obtained from CRSP are daily stock prices, daily high and low price, bid-ask prices, trading volume data, market capitalization, standard industrial classification, ticker symbol and permanent company code.

Liquidity Spiral Measure
Liquidity spiral is a new phenomenon documented around 2008 financial crisis and not much studied in depth in the finance literature.However, Jain, Mishra, and McInish (2013) study that examines the existence of liquidity spirals in equity market as predicted by Brunnermeier and Pedersen's (2009).Our work follows the liquidity spiral construction methodology of Jain, Mishra, and McInish (2013) and in brief such construction methodology is presented below:

Illiquidity Spiral
The construction of the illiquidity spiral measure is based on two conditions i.e. assigning direction to 'day wise state of the liquidity' and aggregation of the state of the liquidity for two weeks for each stock.In assigning values for the state of the liquidity, following conditions are resorted to: (i) if today's stock spread is simultaneously greater than the previous day's stock spread and benchmark spread , the liquidity is deteriorating for the stock which is captured by assigning '+1' value, (ii) if today's stock spread is simultaneously lesser than the previous day's stock spread and benchmark spread, the liquidity is improving for the stock which is captured by assigning '-1' value and (iii) violation of any of the aforesaid condition, a value '0' is assigned, which indicates the unchanged state of the stock's liquidity.This expression captures the depth of the illiquidity spiral in terms its duration for each individual stock.For example, a value +10 for illiquidity spiral (S spiral10 ) on a given day shows a high level of illiquidity for a stock.

Loss Spiral
The loss spiral measure assumes that higher is the value of loss spiral, higher is deterioration in the price level.The loss spiral measure is based on two conditions i.e. assigning direction to 'day wise state of the stock price' and aggregation of the state of the price changes for two weeks for each stock.In assigning values for the state of the loss, following conditions are resorted to: (i) if today's stock price is simultaneously lesser than the previous day's stock price and benchmark price, this indicate deterioration in the stock price which is captured by assigning '+1' value (ii) if today's stock price is simultaneously greater than the previous day's stock price and benchmark stock price.The price of a stock is improving which is captured by assigning '-1' value and (iii) violation of any of the aforesaid conditions, a value '0' is assigned, which indicates the unchanged state of the stock's price.This measure captures the behavior of the price series over the previous 10 consecutive days.For example a value -10 for loss spiral (P spiral10 ) indicate improvement in stock prices which is a case of a booming market.

Term Spread
It is measured as difference between 10 years Govt.bond rate and 91 days Treasury Bill Rate and market return is measured as S&P500 index return.Various studies (see Hameed, Kang, & Viswanathan, 2010) has shown that market wide illiquidity get reflected as increase in Term spread and decrease in market return.

Market Return
It is computed S&P 500 composite index return.

TED Spread
It is computed as difference between LIBOR and 91 days Treasury Bill Rate.This variable captures short term market liquidity (Note 1).
We use two different variants of the MSRM i.e.Constant Transition Probability Model (CTP) and Time-Varying Transition Probability Model (TVTP).We compare the estimated results under CTP and TVTP model.The CTP estimation include an intercept and the three lags of the dependent variable and a random variable with two states, where regression coefficients and the variance of the error terms are all assumed to be state-dependent as per the Markov switching model.Thus, there are only two possible states, only three explanatory variable and that the error process is normally distributed and homoscedastic in each state.The Markov switching model may be written as: where latent state variable ( ) The state variable is assumed to follow a first-order Markov chain (eq.2) where the transition probabilities for the two states are assumed to be constant.Denoting by p ij the probability of switching from state i to state j, the matrix of transition probabilities can be written as: The model is estimated using maximum likelihood, where errors are assumed to be normally distributed in each state.The log likelihood function for the purpose is denoted by φ(x, μ, σ 2 ) the normal density function with expectation μ and standard deviation σ: It is very often argued that constant transition probabilities are too restrictive to explain the behavior of financial or economic variables under the examination as they are not allowed to affect transitional probabilities.As explained by Filardo (1994) and Diebold et al. (1999), the Markov switching model with time-varying transition probability (TVTP) has the advantage over the fixed transition probabilities (CTP) in terms of flexibility.It can recognize systematic changes in the transition probabilities before and after turnings points, capture more complex temporal persistence and allow expected duration to vary across time.In this context, economic fundamentals and policy shocks can influence the regime transition probabilities.
To estimate Markov switching model with TVTP, we have followed Diebold et al. (1999).In the process we endogenized probabilities of changes of regime by incorporating economic variables as their determinants.Then, equation (3) becomes: The transition probabilities are modelled as a logistic functional form such as (6): To estimate this regime switching model, we must specify the complete data likelihood function.Following Diebold et al. (1999), let y t be the sample path of a time series conditional upon as follows: Where Thus the conditional density function of y t is specified as: f ( ) where i=0, 1.
Following Diebold et al. (1999) 1) determines the behavior of the switching from one state to other under CTP.
Equation (3), we specify that the switching of regimes follows a first-order Markov chain, where probabilities are noted by p 11 and p 22 , where p 11 is the probability of remaining in state 1 at t, given that the economy is in regime 1 at t-1, and p 22 is the probability of staying in regime 2 at t, given that the economy is in state 2 at t-1; 1p 11 and 1-p 22 are the transition probabilities for switching from one regime to the other under CTP.The maximum log likelihood function is specified in equation ( 4), which is deployed in estimating the MSRM with CTP.In equation ( 5) we specify the MSRM with TVTP.Equation ( 6) is the mathematical representation of the transitional probabilities under MSRM with TVTP.Equation ( 7) and ( 8) represent the complete data likelihood function and conditional density function under MSRM with TVTP respectively.

Preliminary Findings
The summary statistics for major variables used in the study including two spiral measures for full panel period consisting 476 common stocks that are included in the S&P 500 index for the study period is presented in Table 1.Total number of filtered stock sample contains 1,704,907 observations for stock prices.The minimum value of stock price is $0.35, average value is $44.81 while maximum value is $996.74.
We computed relative quoted spread (spread), mid-quote, illiquidity spiral and loss spiral as defined in previous sections.In data sample, the mean value of ask and bid price for stocks is $44.91 and $44.7 respectively.Ask and bid price series also show a high kurtosis which indicate variability in the sample.The average high and low trade price is $44.81 and $44.01 respectively.The trade volume shows a high variability in sample.The average value of trade volume is 3,554,600 and minimum value is 100.The volume series shows a very high kurtosis 2839.54 and skewness 40.67.Our sample contains constituent of S&P 500 index where there is remarkable variation in stock's market capitalization.Data descriptive also points this variation where minimum market capitalization of a firm is $16.2 millions and maximum is $6.14billions.The computed spread value varies from minimum of 0.002 % points to maximum of 196.04%.The spread variable is highly asymmetric in nature with kurtosis value 2476.53.Our main interest variables-loss and illiquidity spirals, value moves in the range of -10 to +10.The average value of illiquidity and loss spiral is -1.08 and -0.58 respectively.However, the skewness and kurtosis value are also not very high for our main interest variables.tatistically significant for both the models at least at the 5% level of significance.The estimated parameters and the LR statistics for illiquidity spiral and loss spiral are observed to be 9884.92and 11068.92respectively, which suggest the rejection of the null hypotheses of no regime switching against alternative of regime switching for both the variables.Thus, the estimated results support the assumption that the two different states occurred in the data α st for state 1 and state 2 are statistically different in both the models.In particular, the estimated results suggest that an average increase in illiquidity spiral and decrease in loss spiral of 0.273 and -2.074 unit in bear market regime (intense illiquidity and loss state) and decrease illiquidity spiral (increase in liquidity) of -2.188 unit and increase loss spiral of 0.798 unit in bull market regime(intense liquidity and profit spiral state) respectively.Further the relatively large posterior standard deviation, which is inferred from the variance of the parameter of the state of the illiquidity and loss spiral both reflect that there are a few observations in that state.
While examining the transition probability matrix (TMP) and the expected durations of the both the states it is affirmed that there is considerable state dependence in the transition probabilities with a relatively higher probability of remaining in the origin regime for illiquidity spiral and alternate regime for loss spiral switching regressions.The closer examination of the constant transition probabilities affirm on one hand that the probability of staying in liquidity spiral state (p11) at time (t), given that the market is in the same state at time (t-1) is 0.9935.On the other hand, the probability of staying in illiquidity spiral states (p22) in time (t), given that the market is in the same state at time (t-1) is found to be 0.9996.Similarly examining the probabilities of remaining in the loss spiral state and non-loss spiral state are affirmed to be very high with a tune of 0.9909 and 0.9912 respectively.The transition probabilities results are observed to be very large and statistically significant at 1% in both the states for illiquidity spiral (p11 = 0.9935 and p22 = 0.9964) which suggest that both states (illiquid and liquid spiral) are highly persistent.It is also further evident for the loss spiral that the state transition probabilities are found to be very large and statistically significant 1%, (p11 = 0.9909 and p22 = 0.9912) which suggests that both the states (loss spiral and non-loss spiral) are highly persistent (Table 2).These high probabilities either in liquidity or illiquidity spiral state and either in loss spiral or non-loss spiral state correspond that it is likely to be in such regimes.Thus the analysis here suggests that the periods can be easily identified under two states both for illiquidity spiral and loss spiral in the US equity market under the study period.2).However, it is evident that the illiquidity spiral duration is relatively observed to be longer than that of the liquidity spiral duration and the loss spiral duration is shorter than the non-loss spiral duration in the US equity market during our sample study period.Thus it is affirmed that the change from illiquidity and non-loss spiral state to liquidity and loss spiral state is more likely than change from liquidity and loss spiral state to illiquidity and non-loss spiral state in the US equity market.
It can be inferred from the above analysis that relatively illiquidity spiral state is longer than that of the loss spiral state, which corresponds that there might have some stretch of periods either in the state 1 or state 2 where illiquidity spiral and loss spiral don't move together in the US equity market.This non occurrence of simultaneous illiquidity spiral and loss spiral in certain cases of the both the states would be partially examined graphically in filtered and smoothened regime probabilities obtained from both the illiquidity and loss spiral Markov Regime Switching CTP models.

Illiquid Spiral and Loss Spiral under TVTP
We estimate the Markov Regime Switching TVTP both for illiquidity spiral and loss spiral, where we allowed a set of broad financial market variables to explain the evolution of such probabilities.The initial set of proxies used as explanatory variables Markov Regime switching TVTP frameworks selected are broad equity market return (S&P 500 index return), Term spread and TED Spread.With this information we established different models to select the one that presents the smooth transition probabilities consistent with the state of upturn and down turn of the US equity market history since 1983.Table 3 presents the results of the final selected model, which includes only S&P 500 return as explanatory variable in the TVTP model both for Illiquidity spiral and loss spiral.As expected, the external shocks have significantly affected the evolution of both illiquidity spiral and loss spiral in the US equity market.The model was selected based on the gradients and on a likelihood test that compares the Hamilton model, with CTP, with the model of TVTP.
The estimated results of the TVTP model indicate that the US equity market experiences two different states both in the context of illiquidity spiral and loss spiral.The magnitudes of the illiquidity spiral and loss spiral states significantly differ from the liquidity spiral state and non loss state.The illiquidity spiral and loss spiral states are identified with a positive mean value of 0.268 and 1.129 and the liquidity spiral and non loss spiral states are identified with a negative mean value of -2.196 and -1.812 respectively.However on the one hand TVTP and CTP models remained equally efficient in estimating the coefficients for illiquidity spiral under two different regimes and on the other hand TVTP model better discriminates compare to the CTP in segmenting the loss spiral under two different regimes.
Further the coefficients of the S&P 500 return in the TVTP model both for illiquidity spiral and loss spiral differ from zero with opposite (statistically significant) signs under the two different states.As to the transition matrix parameters, we find that increases in the illiquidity are associated with higher probabilities of being in the illiquidity spiral regime, lowering the transition probability out of regime 1 and increasing the transition probability from regime 2 into regime 1.Similarly, the transition matrix parameters, we see that increases in the loss are associated with higher probabilities of being in the loss spiral regime, lowering the transition probability out of regime 1 and increasing the transition probability from regime 2 into regime 1 (Table 3).
While examining the transition probability matrix and the expected durations of the TVTP model it is affirmed that there is considerable state dependence in the transition probabilities with a relatively higher probability of remaining in the origin regime both for illiquidity and liquidity spiral state and loss and non loss spiral states.The corresponding expected duration of illiquidity and liquidity regimes are approximately 218.58 and 149.75 periods and expected duration of loss and non loss regimes are approximately 109.78 and 78.22 periods respectively (Table 3).
Finally based on the discriminating power of the model in segmenting the states and likelihood test carried out in the line with Diebold et al. (1999), and the result supported the MSRM TVTP model over the CTP model both for illiquidity spiral and loss spiral identification.Thus the illiquid and liquid spiral states and loss and non-loss spiral states are identified in the line of the filtered transition and smoothen transition results obtained from the MSRM under TVTP.

Regime Identification for Illiquidity and Loss Spiral
The identification of the regimes under TVTP both for illiquidity spiral and loss spiral are based on standard deviations, which is obtained from the variance and covariance matrix.The dating of the two regimes both for Illiquidity spiral are schematically presented in Figure 2 and Figure 3 respectively, which plot simultaneously the filtered transition and smoothing (posterior) probabilities of state = 2 are summarized in Table 3.We combine both the filtered and smoothing transition probabilities to determine the illiquidity and liquidity spiral states and loss and non loss spiral states in the US equity market.In addition to that we have taken 0.5 as the cut off value for State =1 or 2. That is, the periods with the filtered and smoothing probabilities of State = 2 greater (less) than 0.5 are more likely to be in the state of high illiquidity spiral and high loss spiral periods respectively.According to this approach, since 1983 the US equity market has been experiencing 15 number of illiquidity spiral periods and 11 number of loss spiral periods.However in general, we observed a few number of occasions, where illiquidity spiral and loss spirals are observed to be persistent in the US equity market during the sample study period.A detailed analysis is resorted hereunder to relate with the events of the switching of illiquidity and loss spiral regimes in US equity market corresponding to our study period.
The high illiquidity and the loss spiral spikes during 1987 corresponds to the periods around the Black Monday of October 1987 when stock market around the world crashed, when a few short lived jumps that forced the US equity market to be in high volatile regime.The illiquidity and loss spiral regimes of 1991 can be attributed to the periods in which US engaged in Gulf war especially with the Iraq that culminated in high volatility regimes in the US equity market.Higher jumps in illiquidity spiral and loss spirals during 1997 corresponds to the Asian Financial crisis that led to massive deterioration of wealth in the East Asian economies real estate space, along with the onset of capital controls in Malaysia and blemish advice and numerous conditionality of the IMF to such economies ultimately buckled the confidence of the investors across the world.As a result of such crisis equity market across the world including US went on rampage in the fronts of equity prices and market volatility.The dot com bubble burst could have been responsible for shifting the regimes of illiquidity and loss spiral state around the turn up of the 21st century.
that means s t takes value 1, if state 1 governs at time 't' and s t takes value 2 if state 2 governs at time 't'.
t s , the MSRM TVTP parameters includes the mean and variances of each state

Table 1 .
Descriptive statistics on full sample

Table 2 .
Maximum likelihood parameter estimates and standard errors of the first order two state Markov regime switching CPT model for illiquidity and loss spiral to illiquid spiral state (P 12 ) is almost 0.0065 and the probability of changing from illiquidity spiral state to liquidity spiral state (P 21 ) is close to 0.0045.The corresponding expected durations to be in illiquidity spiral and liquidity spiral regimes are approximately 222.692 and 153.873 periods respectively.While the probability of switching from the loss spiral state to non-loss state (P 12 ) is almost 0.0092 and the probability of changing from non-loss spiral state to loss spiral state (P 21 ) is close to 0.0088.The corresponding expected duration to be in loss spiral state regime and non-loss spiral regimes are approximately 109.3 and 113.2 respectively (Table Further an attempt has been made hereunder to conduct the horizontal switching and the duration analysis for the illiquidity spiral and loss spiral in US equity market.It is observed that the probability of switching from a liquid spiral state

Table 3 .
Maximum likelihood parameter estimates and standard errors of the first order two states Markov regime switching TVTP model for illiquidity spiral and loss spiral Further the 9/11 attack of World Trade Center coupled with the recession