Quantile Impulse Response Functions

Transcription

1 Quantile Impulse Response Functions Sulkhan Chavleishvili DG - Research, European Central Bank and Simone Manganelli DG - Research, European Central Bank Preliminary Draft Abstract This article develops a new approach for analyzing the impact and transmission of market shocks to the quantiles of financial return series. Our framework accommodates a bivariate system of dynamic conditional quantiles of random variables where one of the random variables evolves exogenously to the system. We provide the statistical framework to define structural quantile shocks and the associated quantile impulse response functions (QIRFs). We also derive the asymptotic distribution of the QIRFs and provide finite sample evidence with a series of Monte Carlo experiments. The empirical part of the paper investigates the effect and transmission of US equity market shocks to a set of financial institutions. Keywords: Correlation; Quantile regression; Tail shock. addresse: sulkhan.chavleishvili@ecb.europa.eu. addresse: simone.manganelli@ecb.int. 1

2 1 Introduction The tail dynamics of random variables can be conveniently modeled using quantile regression (QR), introduced by Koenker and Bassett (1978) and extensively reviewed in Koenker (2005). QR is a semi-parametric method that allows to capture tail specific dynamics which is fundamental for financial risk management. Many researchers have applied this methodology to analyze financial time series data. Engle and Manganelli (2004) and Kim, Manganelli and White (2010, 2015) have introduced a class of models for estimation and inference on the quantiles of financial returns, the conditionally autoregressive value-at-risk (CAViaR) and the vector autoregression for value-at-risk (VAR for VaR) models. However, when cast in a multivariate framework such as a VAR model, it remains unclear how to trace the transmission of shocks to one variable to the whole system. It is in fact unclear how to think about structural shocks in the first place in a multivariate quantile regression setup. This paper proposes a new approach to define quantile structural shocks and to derive quantile impulse-response functions (QIRFs) associated with these shocks. To make things concrete, we proceed by setting up a model of the interaction between the market and individual banks returns, where the tail dynamics of each bank return is exposed to spillovers from the lagged market return. Given the quantile setup, a tail market shock is conveniently defined as a stock market return being equal to its quantile associated with a sufficiently small probability. We next define the impulse response function as the difference between the expected tail dynamics with and without the market tail shock. The structural nature of the shock is derived from the assumption that market shocks can contemporaneously affect individual banks returns, but not vice versa. The first part of the paper introduces the system of the conditional quantiles and shows how the quantile shocks are related to the shocks of an analogous system modeled along a multivariate generalized autoregressive conditional heteroskedasticity (GARCH). In the second part we define the QIRFs and formalize the related asymptotic theory for inference on the parameters as well as on the confidence bands of the QIRFs. Furthermore, we propose the measure for asymmetry in responses between the left and right tails. The last part of the paper presents the empirical estimates of the model on a selected 2

3 sample of US banks. We first show how the standardized quantile shocks are characterized by properties similar to those of the standardized residuals of a GARCH model. We next use these standardized quantile residuals to obtain the structural shocks of the model and derive the associated quantile impulse response functions. We observe important differences in the magnitude, persistence and transmission dynamics across different financial institutions, as well as asymmetry between the left and right extreme tail responses. The paper is organized as following. Section 2 defines the bivariate conditional quantile system and links it to the GARCH framework. Section 3 defines the QIRFs. Section 4 provides the asymptotic properties. Section 5 discusses the finite sample properties. Section 6 presents the empirical estimates. Section 7 briefly concludes. Some of technical proofs and analytical results are provided in the Appendix. 2 The model In this article, we study the impact and transmission of tail shocks to financial institutions. We consider a set-up with two sequence of random variables {y mt } T t=1 and {y it } T t=1 representing returns of a market index and a financial institution and explore implications of a market tail shock. We first define the concept of tail shock and then develop a framework for quantile impulse response functions QIRFs. 2.1 Conditional quantile shocks Given the information set F t 1, generated by sequence of financial returns available at time t 1, a confidence level θ (0, 1) and a conditional quantile function q jt we introduce the following data generating processes for a market and individual financial return processes y mt = q mt + q mt ε mt, y it = q it + q it ε it, (1) with ε jt represents the reduced-form standardized quantile specific shock and satisfies the following conditional quantile restriction Pr(sgn(q jt )ε jt 0 F t 1 ) = θ, j (i, m), (2) 3

4 where sgn(q jt ) denotes the sign of q jt. It is easy to see that if the restriction (2) holds then Pr((y jt q jt )/q jt sgn(q jt )ε jt F t 1 ) = θ and q jt is a conditional quantile function of y jt. Therefore, the system of equations (1)-(2) specifies return processes in terms of the conditional quantile function q jt and the tail shocks ε jt, j (m, i). To have an intuition of a quantile specific data generating process, the following theorem establishes a relation with a standard location-scale model. Theorem 1 Let the daily return process be: y mt = σ mt z mt, y it = σ it z it, (3) and z it = ρ i z mt + 1 ρ 2 i w it, ρ i [ 1, 1], (4) where σ mt and σ it represent the time-varying volatilities and z mt and w it are i.i.d.(0,1) with continuous elliptical distribution function F ( ). Then, the tail specific shocks ε mt and ε it defined in (1) are related to z mt and w it as follows: ε mt = ρ 2 F 1 (θ) z mt, and ε it = ρ i 1 + ρ i ε mt + i F 1 (θ) w it. (5) It is important to note that the processes (3)-(4) reflect the standard identifying strategy (see, for example, Brownlees and Engle, 2015, among many others). We first introduce a concept of a structural shock in the context of tails. Definition 1 We refer to ε mt and w it as to the structural quantile shocks. Note that the correlation structure (5) maps into the following relationship ε it + 1 = ρ i (ε mt + 1) + σ i w it, (6) where the standard identifying assumption for the individual structural shock w it is that the structural market shock ε mt has a contemporaneous impact on individual reduced-form tail shock ε it and not vice versa. The above model falls into the class of the constant conditional correlation model by Bollerslev (1990). Similarly to Engle, Ito, and Lin (1990) and Lin (1997), the tail shock 4

5 propagates through the dynamic volatility structure and not the time-varying dynamic correlation structure. There are many ways to specify the time-varying correlation among standardized residuals (z it, z mt ). Here we focus for simplicity on the constant correlation model. 2.2 Conditional quantile functions In order to study a time-varying tail behavior, we need to introduce a specific data generating processes for the time-varying quantiles q mt and q it. We introduce a specification treating a market rate as an exogenous state process that drives dynamics of a financial institution s conditional quantile function. In particular, we assume daily quantile functions to be characterized by the following system q mt = ω m + β m q mt 1 + α m y mt 1, q it = ω i + β i q it 1 + α i y it 1 + α im y mt 1, (7) where the market quantile function q mt follows the univariate CAViaR model by Engle and Manganelli (2004) and the individual quantile functions q it is extended to take spillovers from the lagged market rate y mt 1. If α im = 0, the quantile q it reduces to the CAViaR specification. This formulation allows two specifications to be estimated independently from each other and therefore provides substantial computational simplicity. Furthermore, the following daily volatility components characterized by Taylor s (1986) linear GARCH (1,1) specification, can be viewed as the valid data-generating process for the model (7) σ mt = ω m + β m σ mt 1 + α m y mt 1, (8) σ it = ω i + β i σ it 1 + α i y it 1 + α im y mt 1, where the parameters of the conditional quantile processes in (7) are given as α j = α j F 1 (θ), ω j = ω j F 1 (θ) for j (m, i) and α im = α im F 1 (θ). 3 Impulse response analysis In this section, we present a framework for analyzing the propagation of a tail shock to the market rate y m,t through the system of conditional quantiles (7). For this purpose, we first 5

6 provide the definition of QIRFs and then derive its analytical expressions. The following definition summarizes our approach. Definition 2 (QIRF) The QIRF for y jt, j (m, i) in response to the market tail shock, i.e. ε mt = 0, is defined as h j = E t [q jt+h ε mt = 0] E t [q jt+h ], h = 1, 2,.... Definition 2 specifies a tail shock to a market rate as the level of return y mt that equals to its conditional quantile value q mt. Intuitively, under the market tail shock, this definition aims at understanding how differently does the tail of the financial return y jt, compared to its expected value over the horizon of interest h, behave. 1 Then, we make use of fact that for any real x, x = sgn(x)x, and since 1 and 5 % conditional quantiles of financial returns are always negative we can rewrite the system of quantiles in (1) as following q mt+1 = ω m + ε mt q mt, q it+1 = ω i + ε it q it + ˇε mt q mt, where the parameters ε mt = [β m α m 1+ε mt ], ε it = [β i α i 1+ε it ] and ˇε mt = α im 1+ε mt are functionals of the market and individual tail shocks. The system can be conveniently written in a matrix form as q mt+1 q it+1 = or somewhat more compactly as ω m ω i + ε mt 0 ˇε mt ε it q mt q it, q t+1 = ω + Γ t q t. (9) 1 For comparison, following the standard impulse response exposition (see, for instance, Section 11.4 in Hamilton (1994)), consider any process written as a vector autoregression (VAR), Y t = ΦY t 1 + V t, with eigenvalues of Φ lie inside the unit circle. Then as h, Y t+h = Φ h+1 Y t 1 + V t+n + + Φ h V t will have MA( ) representation. By setting {V t+2,..., V t+h }, Y t 1 and all elements of V t besides v jt to zero, the element y it+h / v jt δ will give the impulse response as a function of h. Furthermore, this approach can be formalized as I Y (h, δ, F t 1 ) E t 1 (Y t+h v jt = δ, F t 1 ) E t 1 (Y t+n F t 1 ). 6

7 For any h > 1, by applying the Law of Iterated Expectations we can write E t q t+h = ω + E t [E t+h 1 [Γ t+h 1 ] q t+h 1 ] = ω + Γ E t q t+h 1 = h 1 l=0 Γ l ω + Γ h 1 E t Γ t q t. where Γ = E t+h 1 Γ t+h 1 and the results follows due to the i.i.d. property of Γ t. Then, by Definition (2) and for moments µ m = E t [ 1 + ε mt ], µ i = E t [ 1 + ε it ] and µ i = E t ρ i + σ i w it, the QIRFs are given as h = Γ h 1 (E t [Γ t ε mt = 0] E t [Γ t ]) q t, where the impact of the shock at time t is given as E t [Γ t ε mt = 0] E t [Γ t ] = α m(µ m 1) 0 α im (µ m 1) α i (µ i µ i ) To finalize our definition and to guarantee that QIRFs are well defined, we impose the following stability assumption. Assumption 1 The matrix Γ has eigenvalues less than one in absolute value. Intuitively, QIRFs can be used for analyzing systemic stability and its resilience to the market shocks. In particular, the degree of resilience to the market shock can be well characterized by the number of consecutive periods required for getting back to the initial time t level. In particular, policymakers are often interested in the following quantity. ĥ s.t ĥj 0, j (m, i), (10) where ĥ will typically measure expected number of days it takes to reach back the time t level after the shock. The effect of the market tail shocks are not necessarily symmetric, implying the impact of the left tail shock is not the same as the right tail shock. Those institutions that tend to be overly sensitive to the left tail shock as oppose to the right tail shock, can at lesser extend build up a buffer to offset sharp market downturns. Therefore, at a certain confidence level θ standing for the left tail, the QIRFs are expected to satisfy h θ 0 and h 1 θ 0. Then, we construct the degree of asymmetry in responses using the following metric δ h h 1 θ + h θ, h = 1, 2,..., (11) 7

8 where the desired result is δ h > Asymptotic distribution of QIRFs In this section, we derive the asymptotic distribution of the QIRFs and the parameter vector γ = (γ m, γ i) = (ω m, β m, α m, ω i, β i, α i, α im ). Given the conditional quantile restriction (1), parameters of the model γ j solve the following asymmetric loss problem by Koenker and Basset (1978) ˆγ j = arg min T 1 γ j T [θ I (u jt < 0)]u jt, j (i, m), (12) t=1 where u jt = y jt q jt and using the estimated parameter vector ˆγ j the standardized quantile specific residuals can be recovered as ˆε jt = û jt /ˆq jt and used to estimate the moment values ˆµ i, ˆµ m and ˆ µ i. In order to establish consistency and asymptotic normality for the QR estimator ˆγ = (ˆγ m, ˆγ i) we first note that the system of conditional quantile functions (7) can be viewed as the special case of one considered in Kim, Manganelli and White (2015, KMW henceforth) and use the following Corollary to summarize its asymptotic properties. Corollary 1 (Corollary to KMW) Suppose that assumptions 1-6 of KMW hold, then the QR estimator ˆγ = (ˆγ m, ˆγ i) is consistent and asymptotically normal T ˆγ m γ m N (0, Ω), Ω = θ(1 θ) H 1 m J m Hm 1 0 ˆγ i γ i 0 H 1 i J i H 1 i where for a continuous density function f jt the positive definite matrices H j, J j, j (m, i) are given as J j = lim T T 1 T t=1 γ j ˆq jt γ j ˆq jt, H j = lim T T 1 T t=1 f jt(ˆq jt ) γj ˆq jt γ j ˆq jt. Since our model is nested in one considered by KMW, Corollary 1 directly flows from Theorem 2 by KWM and therefore will not be discussed in details. To infer on the estimated parameter vector ˆγ, we require a valid covariance matrix estimator ˆΩ. For some bandwidth parameter h T, satisfying h T 0 and h 2 T T, the 2 Since is straightforward to construct the right tail QIRF, we briefly outline it in Appendix B.1., 8

9 consistent estimates of the variance covariance matrix Ω can be obtained using Powell s (1986) approach as following with and ˆΩ = Ĥ jt = 1 2T h T Ĵ jt = 1 T T t=1 Ĥ 1 m ĴmĤ 1 m 0 0 Ĥ 1 i Ĵ i Ĥ 1 i, T I( y jt ˆq jt h T ) γj ˆq jt γ j ˆq jt, t=1 ( θ I(yjt ˆq yjt )) 2 γj ˆq jt γ j ˆq jt. Suppose that the results of the Corollary 1 hold and h, h > 0 is continuously differentiable function with values in two-dimensional Euclidean space. Then using the result from Serfling (1980, p. 122) we can reach the following conclusion T ( ˆ h h ) N (0, GΩG ), where G = γ h is nonzero matrix at γ and the result is then used to approximate the confidence bands for the QIRFs. 3 Furthermore, given the parameter vector γ = (γ 1 θ, γ θ ), where γ θ is evaluated at a confidence level θ, and the vector G = γ h, it follows that T (ˆδh δ h ) N (0, G Ω G ), where Ω is a diagonal matrix with matrices Ω 1 θ and Ω θ along the diagonal. 5 Monte Carlo Evidence This section explores the finite sample properties of the correlation parameter ρ i, vis-à-vis different dependence strength and data distribution. 3 We delegate related analytical expressions to Appendix B.2. 9

10 5.1 Setting We consider the data generating process (DGP) in the equation (8) by using i.i.d. normal and i.i.d. Student-t distributed data with 4 degrees of freedom and the sample size T (1500, 3000). In practice the conditional volatility is a persistent process therefore in order to mimic the true data the following set of parameters are considered DGP: ω m = ω i = 0.1, β m = β i = 0.9, α m = α i = α im = 0.05, ρ (0.3, 0.6, 0.9). Results are based on 1000 replications of the DGP. We report the root mean square error (RMSE) based on the difference between Monte Carlo estimates and the true values of the correlation coefficient. [Insert Table 1 around here] 5.2 Results Tables 1 reports results of Monte Carlo analysis. We report the OLS estimates ρ OLS, QR estimates ρ QR at quantile values θ (0.25, 0.5, 0.75) and the quasi maximum likelihood estimates (QMLE) ρ ccc using the constant conditional correlation (CCC)-GARCH specification in the spirit of Bollerslev (1990) evaluated at the normal likelihood function. As in the classical tradition of the IRF analysis, our suggested two-step estimators ρ OLS and ρ QR are confronted with the one-step likelihood based approach ρ ccc in order to understand how far we deviate from benchmark, because it by construction does not inherit the uncertainty from the first-step estimates. We use the Nelder-Mead (nm) algorithm in the spirit of Engle and Manganelli (2004) in order to deal with estimating the conditional quantile functions and the modified Newton-Raphson (nr) algorithm while dealing with the QML estimation. We note that all of the correlation estimators based our paper are displaying improved finite sample performance with the sample size T. Furthermore, our the OLS based estimator ρ OLS in most of cases either outperforms the QR based estimator or performs in a very similar manner and therefore, its performance is closest to the one-step QMLE ρ CCC estimator. 10

11 6 Impact of the US equity market tail shocks The recent financial crisis triggered numerous works in the field of the systemic risk (See Adrian and Brunnermeier (2016), Engle and Brownlees (2015), Acharya et al. (2010) among others). From the prospectives of the systemic risk, institutions that are overly sensitive to the market movements tend to be important for the system stability. In this empirical application we examine the sensitivity of the set of the US based financial institutions to the extreme equity market shock and study it s propagation dynamics. 6.1 Data description and computational issues The data is obtained from Datastream. We use the daily closing equity prices for the set of US based financial institutions and the market and transform it into the continuously compounded log returns covering the period from 3 January 2000 to 31 December 2014, constituting the sample of 3913 observations. We use last 500 observations for out-ofsample analysis. Table 2 reports the set of institutions as well as descriptive statistics for them. [Insert Table 2 around here] We use the Nelder-Mead (nm) simplex algorithm for solving computational problems in the spirit of Engle and Manganelli (2004). In particular, we generate vector of i.i.d.u( 1, 1) random variables and select first 1000 that minimizes the objective function. Then for each of those initial parameter values, we run the nm algorithm and select the optimal estimates. 4 Furthermore, we make sure that the data at hand closely matches the GARCH framework in (7) by recovering the serially uncorrelated series ŷ jt, j (i, m) by fitting autoregressive filter of certain order. In this application, we fit AR(6) for the S&P 500 series and use residual for modeling purposes. Furthermore, in order to construct the estimates for the variance-covariance matrix estimates, following Machado and Santos Silva (2013) we use the following bandwidth ĥt = ˆκ T [φ 1 (θ + c T ) φ 1 (θ + c T )] with ( ) 1.5(φ(Φ c T = T 1/3 (φ 1 ( ) 2/3 1 1/3 (θ))), 2(Φ 1 (θ)) All calculations are done using either Stata or Matlab. 11

12 where φ( ) and Φ( ) are standard normal cumulative distribution and probability density functions, and ˆκ T is the median absolute deviation of θth conditional quantile residual ε jt. 6.2 Specification and correlation tests If the model in (1) represents the true model, the conditional quantile restriction given by equation (2) holds. We use the in-sample and out-of-sample dynamic quantile (DQ) tests by Engle and Manganelli (2004) for testing whether the conditional quantile restriction holds or not. For this purpose we use the one step ahead forecast values of the conditional quantile functions ˆq jt and the fitted values of quantile exceedances 0) θ, j (i, m). ˆ Hitjt = I(y jt ˆq jt < Furthermore, we test the presence of serial correlation in level and squared tail specific shocks ˆε it and ˆε mt for the empirical justification of the model. In particular, we run the AR(5) regression for the standardized quantile and GARCH residuals and provide the test statistics for the joint significance of included lagged information. Moreover, we use them to estimate parameters of the equation (6) using OLS and test the hypothesis H 0 : ρ i = 0 by standard t-statistics. 5 the QR estimates for the range of quantile values. Furthermore, we report the CCC-GARCH estimates along with 6.3 Results Table 3 reports the parameters of the conditional quantile functions at θ = 1% and θ = 99% confidence levels for the market rate captured by the S&P 500 and the set of individual financial institutions such as J.P. Morgan, U.S. Bancorp and American Express. [Insert Table 3 around here] We note that the conditional quantile functions evaluated at 1 and 99% confidence levels, standing for extreme losses and gains, tends to be persistent processes with autoregressive coefficient values being in the vicinity of 0.9 uniformly across all series. The market news, captured by the absolute value of the market rate, has statistically significant at 10% 5 Since, the value of T we are working with is high, we can treat the generated regressors (ˆε it, ˆε mt ) as data, therefore it can be expected that the t-statistics is well defined within our framework. 12

13 negative impact on the left tails, implying both negative and positive news tend to amplify losses. This could be explained by well known empirical evidence that such news tend to amplify volatility as well. Whereas, the market impact on the individual institutions becomes insignificant for the right tail, implying that the general market conditions play less role in good times. Sensitivity of the conditional quantile functions to the own series tends to negative for the extreme left tails and is positive for the right tail. The centered unconditional exceedences tend to be in the vicinity of zero both in-and out-of-sample and the out-of-sample DQ test performs well only for the left tail. This indicates that the standard GARCH type tail specification may not be a good process for the right tail dynamics. [Insert Figure 1 and 2 around here] Figures 1 and 2 plot the time dynamics of the conditional tails. The severe spikes are observed around the period of the recent financial crisis. This evidence is commensurate with the amplified uncertainty that has been prevailing during periods following the default of Lehman Brothers. Furthermore, individual financial institutions display higher amplification compared to the market rate that stems from the fact that the market is better able to diversify existing financial risks. [Insert Tables 4, 5 and 6 around here] Tables 4, 5 and 6 report statistical properties of standardized conditional quantile and CCC-GARCH shocks, where for each of series we fit AR(5) model and test the included lagged information for statistical significance. Furthermore, we use the empirical quantile of sgn(q jt )ε jt in order to confirm the conditional quantile restriction (2). The results successfully rejects the significance of lagged information almost for all series and confirms the conditional quantile restriction for all series. Furthermore, as we see from the results the constant from the AR(5) regression is close to -1, again confirming our dependence specification in the equation (6). [Insert Tables 7, 8 and 9 around here] Tables 7, 8 and 9 report analogously the serial correlation properties of the squared quantile and GARCH specific residuals. We notice that conditional quantile shocks tend to 13

14 outperform the results by CCC-GARCH model, that fails to reject the conditional heteroskedasticity in the squared standardized residuals. [Insert Table 10 and Figure 3 around here] Table 10 reports the OLS estimates for the correlation parameter ˆρ i in equation (6) along with constant conditional correlation (CCC)-GARCH model using the system (8). We notice the significantly high loading on the structural market tail shock uniformly for each institution. Therefore indicating at presence of high degree co-movement between the reduced form individual tail shocks and the structural market shock. Furthermore, interestingly the estimates fall in the vicinity of the parameter estimates by CCC-GARCH, though slightly amplifying it in case of the θ = 1% and subduing in the case of θ = 99%. Figure 3 reports the QR estimates using the equation (6). Similarly to the OLS estimates we observe that the correlation coefficient for θ = 1% is typically higher than for θ = 99%, but still are close to each other. [Insert Figures 4, 5 and 6 around here] Figures 4 and 5 report the left and the right tail QIRFs for the set of financial institutions in response to the market quantile shock captured by the value of the S&P 500 return being equal to its quantile value. We accompany our plots by two standard deviation confidence bands. There are several comments in line. First, some set of financial institutions tend to react with a higher amplitude compared to the market rate. For example J.P. Morgan overreacts to the market shock as well as the impact tends to be more persistent compared to the market rate. Second, as we have seen, some institutions as have insignificant loading on the lagged absolute value of the market rate. This fact is well expressed in the QIRFs, because the impact is modest and most of the transmission period it coincides the zero line. Third, the impact of the extreme market decline is more pronounced and long-lasting compared to the impact of the extreme market gain. Last but not least, these feature is well captured in the asymmetric QIRFs displayed in the figure 6. As we see, the impact of the extreme losses is more pronounce, whereas the confidence interval still remains wide. 14

15 7 Conclusion This article proposed a unified framework for definition and measurement of the tail shocks within the context of financial returns. We introduced the novel conditional quantile function based data-generating process and developed the comprehensive statistical framework for systematic analysis of the impact and transmission of the severe market downturns. The empirical application demonstrates the relevance of the proposed approach for practical purposes. We delegate the various important extensions withing the context of the systemic risk to the future research. 15

16 Appendices A Proof of Theorem 1 From the GARCH data-generating process (3) and the correlation structure (4) we can write ε mt = ymt qmt q mt = σmtzmt σmtf 1 (θ) σ mtf 1 (θ) = F 1 (θ) z mt, ε it = F 1 (θ) z it = F 1 (θ) ( ρ i z mt + 1 ρ 2 i w it = ρ i 1 + ρ i ε mt + 1 ρ 2 i F 1 (θ) w it, ρ i [ 1, 1]. ) B Analytic results B.1 QIRF for the right tail Here we use the fact that the conditional quantile value q jt evaluated at the right tail is positive. Then the right tail QIRFs are given as h = Γ h 1 (E t [Γ t ε mt = 0] E t [Γ t ]) q t, E t [Γ t ε mt = 0] E t [Γ t ] = α m(1 µ m ) 0 α im (1 µ m ) α i ( µ i µ i ) q mt+1 = ω m + ε mt q mt, q it+1 = ω i + ε it q it + ˇε mt q mt, ε mt = β m + α m 1 + ε mt, ε it = β i + α i 1 + ε it, ˇε mt = α im 1 + ε mt, where values µ m, µ i and µ i remain unchanged. 16

17 B.2 Derivatives Here we provide analytic expressions for the components of γ h. parameter γ from the parameter vector γ we have ( ) Γ l = Γ Γ l 1 + Γ Γ l 1, l = 2,..., h. γ γ γ Note that for any Furthermore, for the left tail QIRFs we have [ γ Γ = ω j Γ, = γ (E t [Γ t ε mt = 0] E t [Γ t ]) = 0, 0, 0, β m Γ, α m Γ, ω i Γ, β i Γ, α i Γ, , µ m 0 0 0, 0, , 0 0, 0 µ i µ m , 0, 0, µ i µ i, ] α im Γ 0 0, µ m µ m 1 0 Similarly, given results in Appendix B.1, we can provide the analytic results for the QIRFs evaluate at the right tail as following 1 0 γ Γ = 0, 0 0, µ m 0 0 0, 0, , , 0 µ i µ m 0 1 µ m 0 γ (E t [Γ t ε mt = 0] E t [Γ t ]) = 0, 0, 0 0, 0, 0, ,. 0 µ i µ i 1 µ m 0 and Furthermore, for j (m, i) consider the following analytic derivatives q jt = ω j q jt = β j α j q jt = q it = α im 1 + β j ω j q jt 1, for t 2, 0, for t = 1, q jt 1 + β j β j q jt 1, for t 2, 0, for t = 1, y jt 1 + β j α j q jt 1, for t 2, 0, for t = 1, y mt 1 + β i α im q it 1, for t 2, 0, for t = 1, where we make use of fact that dynamic quantiles are initialized using the empirical quantile of first T In empirical application we set T 0 =

18 C Finite sample evidence Table 1: Monte Carlo simulation results. θ = 0.01, ω m = ω i = 0.1, β m = β i = 0.9, α m = α i = α im = z mt, w it i.i.d.n(0, 1). T=1500 T=3000 ρ ρ OLS ρ QR,0.25 ρ QR,0.5 ρ QR,0.75 ρ CCC ρ OLS ρ QR,0.25 ρ QR,0.5 ρ QR,0.75 ρ CCC z mt, w it i.i.d.t(4). T=1500 T=3000 ρ ρ OLS ρ QR,0.25 ρ QR,0.5 ρ QR,0.75 ρ CCC ρ OLS ρ QR,0.25 ρ QR,0.5 ρ QR,0.75 ρ CCC Note: The table reports the finite sample evidence for the correlation coefficient ρ using the OLS, QR and the QMLE CCC-GARCH estimators, where the QMLE is based on the normal log-likelihood function. 18

19 D Empirical results Table 2: Summary statistics of data. Mean Std. Dev. Max Min Skewness Kurtosis S&P JPM AXP USB Note: The table reports the summary statistics of the daily equity returns for the U.S. based financial institutions and the S&P 500 index. Statistics for S&P 500 is presented after applying AR(6) filter. The sample ranges from January 10, 2000, to January 29,

20 Table 3: Estimates for dynamic conditional quantile functions. θ = 0.01 θ = 0.99 S&P500 JPM AXP USB S&P500 JPM AXP USB ωm (0.0281) (0.0439) (0.0172) (0.0188) ( ) (0.0192) (0.0355) (0.0527) ωi ωm [0.0071] [0.0365] [0.1159] [0.0247] [ ] [0.0139] [0.2339] [0.3762] ωi βm (0.0195) (0.0296) (0.0144) (0.0159) ( ) (0.0077) (0.0253) (0.0339) βi βm [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] βi αm (0.0402) (0.0502) (0.0457) (0.0327) ( ) (0.0374) (0.0687) (0.0604) αi αm [0.0000] [0.0000] [0.0006] [0.0000] [ ] [0.0000] [0.0002] {0.0005} αi αim (0.2198) (0.0524) (0.0608) (0.0416) (0.0730) (0.0710) [0.0680] [0.0230] [0.0584] [ ] [0.6016] [0.8005] αim RQ Hits in-sample Hits out-of-sample DQ in-sample DQ out-of-sample Note: Estimates correspond to the system of conditional quantile functions in equation (7) at θ = 99% and θ = 1% levels. Standard deviations are reported in parentheses and p-values are in square brackets. RQ corresponds to the value of the objective function in equation (12) and DQ stands for the dynamic quantile test by Engle and Manganelli (2004). The hit values stand for unconditional means of centered exceedances. 20

21 Note: Conditional quantiles follow the specification in equation (7). The amplification corresponds to the recent financial crisis triggered by the Lehman Brothers default. Figure 1: 1% conditional quantile functions. 21

22 Note: Conditional quantiles follow the specification in equation (7). The amplification corresponds to the recent financial crisis triggered by the Lehman Brothers default. Figure 2: 99% conditional quantile functions. 22

23 Table 4: Time dependence of the standardized conditional quantile shocks at θ = Variable Constant t 1 t 2 t 3 t 4 t 5 Pr > F q ε,θ S&P500 (0.0376) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0843] [0.1698] [0.3137] [0.3948] [0.4974] JPM (0.0396) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0835] [0.9611] ] [0.2990] [0.8129] AXP (0.0408) (.01720) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0296] [0.1312] [0.3930] [0.1833] [0.5311] USB (0.0400) (0.0172) (0.0172) (0.0172) (0.0171) (0.0171) [0.0000] [0.4349] [0.5514] [0.2789] [0.9116] [0.6232] Note: The table reports the estimates of the AR(5) model for the standardized quantile shock ˆε jt at the θ = 1% confidence level. Along with the parameter values we report standard errors in brackets and the corresponding p-values in square brackets. Furthermore, p-values for the joint significance of the included lagged information and the empirical counterpart of the conditional quantile restriction (2) are presented. 23

24 Table 5: Time dependence of the standardized conditional quantile shocks at θ = Variable Constant t 1 t 2 t 3 t 4 t 5 Pr > F q ε,θ S&P500 (0.0377) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0525] [0.2352] [0.3500] [0.4653] [0.5496] JPM (0.0395) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0861] [0.6865] [0.7959] [0.2168] [0.5413] AXP ( (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0665] [0.1727] [0.463] [0.1818] [0.5461] USB (0.0398) (0.0172) (0.0172) (0.0172) (0.0172) (0.0171) [0.0000] [0.3906] [0.7338] [0.1717] [0.9009] [0.4389] Note: The table reports the estimates of the AR(5) model for the standardized quantile shock ˆε jt at the θ = 99% confidence level. Along with the parameter values we report standard errors in brackets and the corresponding p-values in square brackets. Furthermore, p-values for the joint significance of the included lagged information and the empirical counterpart of the conditional quantile restriction (2) are presented. 24

25 Table 6: Time dependence of the standardized CCC-GARCH shocks. Variable Constant t 1 t 2 t 3 t 4 t 5 Pr > F S&P500 (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.7972] [0.1876] [0.2117] [0.3547] [0.2843] [0.4958] JPM (0.0396) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.9571] [0.0547] [0.8552] [0.8490] [0.2449] [0.6015] AXP (.01720) (.01720) (0.0172) (0.0172) (0.0172) (0.0172) [0.6376] [0.0142] [0.1751] [0.3651] [0.2453] [0.4402] USB (.01720) (0.0172) (0.0172) (0.0172) (0.0171) (0.0171) [0.7396] [0.4488] [0.5938] [0.2596] [0.8828] [0.5645] Note: The table reports the estimates of the AR(5) model for the standardized shock ẑ jt using the normal CCC-GARCH model. Along with the parameter values we report standard errors in brackets and the corresponding p-values in square brackets. Furthermore, p-values for the joint significance of the included lagged information are presented. 25

26 Table 7: Time dependence of the squared standardized conditional quantile shocks at θ = Variable Constant t 1 t 2 t 3 t 4 t 5 Pr > F S&P500 (0.0466) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.8069] [0.2727] [0.0825] [0.3163] [0.5417] JPM (0.0482) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.1661] [0.1721] [0.3279] [0.4028] [0.7778] AXP (0.0505) (0.0172) (0.0171) (0.0171) (0.0171) (0.0172) [0.0000] [0.0004] [0.0022] [0.3202] [0.8668] [0.5706] USB (0.0490) (0.0172) (0.0172) (0.0171) (0.0171) (0.0171) [0.0000] [0.1796] [0.1509] [0.0646] [0.4401] [0.8639] Note: The table reports the estimates of the AR(5) model for the standardized quantile shock ˆε 2 jt at the θ = 1% confidence level. Along with the parameter values we report standard errors in brackets and the corresponding p-values in square brackets. Furthermore, p-values for the joint significance of the included lagged information are presented. 26

27 Table 8: Time dependence of the squared standardized conditional quantile shocks at θ = Variable Constant t 1 t 2 t 3 t 4 t 5 Pr > F S&P500 (0.0458) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0749] [0.0005] [0.8868] [0.0471] [0.7001] JPM (0.0458) (.01720) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.3047] [0.0162] [0.1112] [0.2939] [0.5971] AXP (0.0462) (0.0172) (0.0171) (0.0171) (0.0171) (0.0172) [0.0000] [0.4163] [0.6571] [0.4943] [0.0407] [0.2967] USB (0.0448) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.1375] [0.0751] [0.8578] [0.5197] [0.5064] Note: The table reports the estimates of the AR(5) model for the standardized quantile shock ˆε 2 jt at the θ = 99% confidence level. Along with the parameter values we report standard errors in brackets and the corresponding p-values in square brackets. Furthermore, p-values for the joint significance of the included lagged information are presented. 27

28 Table 9: Time dependence of the squared standardized CCC-GARCH shocks. Variable Constant t 1 t 2 t 3 t 4 t 5 Pr > F S&P500 (0.0493) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0420] [0.000] [0.1933] [0.0662] [0.0833] JPM (0.0504) (0.0172) (0.0172) (0.0172) (0.0172) (0.0172) [0.0000] [0.0009] [0.0358] [0.0900] [0.4160] [0.4989] AXP (0.0537) (.01710) (0.0172) (0.0172) (0.0172) (0.0171) [0.0000] [0.0000] [ [0.745] [0.8381] [0.0063] USB (0.0508) (0.0172) (0.0172) (0.0172) (0.0171) (0.0172) [0.0000] [0.0004] [0.0028] [0.5137] [0.7418] [0.7572] Note: The table reports the estimates of the AR(5) model for the squared standardized shock ẑjt 2 using the normal CCC-GARCH model. Along with the parameter values we report standard errors in brackets and the corresponding p-values in square brackets. Furthermore, p-values for the joint significance of the included lagged information are presented. 28

29 Table 10: OLS and QMLE estimates of the correlation coefficient. θ = 0.01 θ = 0.99 JPM AXP USB JPM AXP USB ρ OLS (0.0147) (0.0165) (0.0161) (0.0106) (0.0108) (0.0116) [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] σ OLS R RSS ρ CCC (0.0103) (0.0100) (0.0129) [0.0000] [0.0000] [0.0000] Log likelihood Note: The parameter estimates correspond the correlation among standardized quantile residuals of the market and individual financial institution using the specification in equation (6). Estimates are based on the ordinary least squares (OLS) and to constant conditional correlation (CCC) - GARCH model of Bollerslev (1990) based on the normal distribution function. Standard errors are reported in brackets and p-values in square brackets. RSS stands for the residual sum of squares from the OLS regression. 29

30 Note: The charts plot the quantile regression estimates for the correlation among standardized quantile residuals of the market and individual financial institution using the specification in equation (6). The left panel corresponds to the standardized quantile residuals at θ = 1%, whereas the right panels deals with θ = 99% confidence level. Plots are produced over the grid of quantile values. Figure 3: Quantile regression estimates for the correlation coefficient. 30

31 Note: The charts plot the quantile impulse-response functions to a market rate shock at θ = 1% confidence level, together with two standard deviation confidence bands. The responses are calculated using the system of conditional quantiles where the shock to a financial institution is contemporaneously affected by the structural market quantile shock. Figure 4: Quantile impulse-response functions to a shock to the market rate. 31

32 Note: The charts plot the quantile impulse-response functions to a market rate shock at θ = 99% confidence level, together with two standard deviation confidence bands. The responses are calculated using the system of conditional quantiles where the shock to a financial institution is contemporaneously affected by the structural market quantile shock. Figure 5: Quantile impulse-response functions to a shock to the market rate. 32

33 Note: The charts plot the asymmetric quantile impulse-response functions to a market rate shock at θ = 99% and θ = 1% confidence levels, together with two standard deviation confidence bands. The responses are calculated using the system of conditional quantiles where the shock to a financial institution is contemporaneously affected by the structural market quantile shock. Figure 6: Asymmetric quantile impulse-response functions to a shock to the market rate. 33

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