Tiempo de reacción

This chapter estimates the temporal interaction between the UK and US stock and bond markets following the QE operations. Specifically, a diagonal VECH (DVECH) model is used to examine the conditional covariance structure between the stock markets of the UK and US, and the stock and bond markets in the UK and US separately against the backdrop of the QE operations. In the first stage of the empirical analysis, the covariation between the UK and US equity markets is modelled to see whether this are affected by the QE operations in the UK and the US. It is not parsimonious to include the individual QE intensity and phase variables, as done in univariate analysis in the previous chapter, into the model in a multivariate setting, as this would further compound the curse of dimensionality, which afflicts multivariate GARCH models.

The solution adopted to enable an examination of the impact of QE and that of the financial crisis across the two markets, whilst still making the models relatively parsimonious is to include only the QE intensity variables (actual purchases), and combine these on a per country basis, into one single variable or index that captures daily QE intensity in that country throughout the entire

sample period rather than on an individual or periodic basis. Thus two new variables, QE_{𝑈𝐾,𝑡},

and QE_{𝑈𝑆,𝑡} are created as:

QE_{𝑢𝑠,𝑡} = [QE1_𝑖,𝑡;QE2_𝑖,𝑡;QE3_𝑖,𝑡;MEP𝑖,𝑡] 𝑖 ∈ 𝑈𝐾, 𝑈𝑆 (4.1) QE_{𝑢𝑘,𝑡} = [QE1_𝑖,𝑡;QE2_𝑖,𝑡;QE3_𝑖,𝑡 ] (4.2)

Thus including both of them in the multivariate specifications of the variances and covariance. In this case, the variance processes are similar but aggregated versions of the processes specified in the previous empirical chapter, that is

119

ℎ𝑖,𝑡 = 𝜔𝑖,𝑖+𝛼𝑖,1Lehman𝑖,𝑡+𝑏𝑖,𝑖𝜀𝑖,𝑡−12 + 𝑐𝑖,𝑖ℎ𝑖,𝑡−1+ 𝑒𝑖,𝑖,𝑈𝐾QE_{𝑈𝐾,𝑡} + 𝑒𝑖,𝑖,𝑈𝑆QE_{𝑈𝑆,𝑡−1} (4.3)

The variable 𝑒_{𝑖,𝑚,𝑈𝑆}QE_{𝑈𝑆,𝑡−1}measuring the impact of the US QE operations is lagged by

one day to address the issue of the overlap or non-synchronous trading times between the US and UK stock markets. While the companion inter-market (UK and US equity markets) covariance processes are specified as:

ℎ_{𝑖,𝑚,𝑡} = 𝜔_𝑖,𝑚+𝛼𝑖,1Lehman𝑖,𝑡+𝑏_𝑖,𝑚𝜀_{𝑖,𝑡−1}𝜀_{𝑚,𝑡−1}+ 𝑐_𝑖,𝑚ℎ_{𝑖,𝑚,𝑡−1}+ 𝑒𝑖,𝑚,𝑈𝐾QE_{𝑈𝐾,𝑡}+ 𝑒𝑖,𝑚,𝑈𝑆QE_{𝑈𝑆,𝑡−1}

(4.4)

Asymmetric effects in conditional covariance have been used by some studies in modelling the covariance between stock and bond returns. Kroner and Ng (1998) identify three possible forms of asymmetric behaviour-(i) the covariance matrix displays own variance asymmetry if hii,t (the

conditional variance of 𝜀𝑖,𝑡) is affected by the sign of the innovation in ri,t-1; (ii) cross variance

asymmetry if hii,t is affected by the sign of the innovation in rm,t-1;(iii) covariance asymmetry if ℎ𝑖,𝑚,𝑡 (the conditional covariance) is affected by the sign of the innovation in either ri,t-1 or rm,t-1.

For robustness and to accommodate possible asymmetric effects, in modelling the conditional covariance between the stock and bond markets i.e. the second stage in the empirical analyses in this chapter, two alternate DVECH models i.e. a symmetric and the asymmetric DVECH following the approach of Glosten et.al. (1993), allowing explicitly for asymmetric conditional covariance terms are estimated.

The choice of the DVECH for the analyses in this chapter is justified on the basis that unlike the BEKK model where it may be difficult to track responses in the conditional variances and

covariance to specific or individual parameters, due to the quadratic form of the BEKK specification, this is not the case with the DVECH specification. This permit with the DVECH, an investigation of the effect of specific variable(s) of interest on the variance-covariance matrix. The DVECH model used to model the intra-market (equity and bond markets) covariance is specified as:

ℎ_{𝑒,𝑏,𝑡}= 𝜔_𝑒,𝑏+ 𝛼_𝑖,1Lehman𝑖,𝑡+ 𝑏𝑒,𝑏𝜀𝑒,𝑡−1𝜀𝑏,𝑡−1+ 𝑐𝑒,𝑏ℎ𝑒,𝑏,𝑡−1+ 𝑑1𝑒,𝑏𝐼,𝜀𝑒,𝑡−1𝜀𝑒,𝑡−1𝐼,𝜀𝑏,𝑡−1𝜀𝑏𝑡−1+

+ 𝑒𝑒,𝑏,𝑈𝐾QE_{𝑈𝐾,𝑡} 𝑜𝑟 𝑒𝑒,𝑏,𝑈𝑆QE_{𝑈𝑆,𝑡} (4.5)

with ℎ𝑒,𝑏,𝑡 = covt {𝑟𝑒,𝑡−1,𝑟_{𝑏,𝑡−1}} i.e. covariance respectively of the equity and bond markets returns. The indicator variable 𝐼𝜀_{𝑘,𝑡−1} is equal to one if 𝜀_{𝑘,𝑡−1}< 0 (and zero otherwise), k =𝑒, 𝑏. It permits the effect of lagged return shocks in the both equity and bond markets to depend on their signs. In the above equation, 𝐼_{,𝜀𝑒,𝑡−1}𝜀_{𝑒,𝑡−1}𝐼_{,𝜀𝑏,𝑡−1}𝜀𝑏,𝑡−1 is nonzero for negative pairs of 𝜀𝑒,𝑡−1 and 𝜀𝑏,𝑡−1. This term assigns an asymmetric covariance effect to the covariance matrix. As the sample period begins prior to the financial crisis, an indicator variable 𝛼𝑖,1Lehman𝑖,𝑡 starting from the period of the crisis is included in the model. We start this period on September 15, 2008, the date that Lehman brothers collapsed or declare bankruptcy. In addition, one broad indicative measure of the intensity of the QE operations in either the UK or US is included in the relevant model(s).

The covariance equations are estimated by maximum likelihood assuming conditional normality about the error terms i.e. 𝜀𝑘,𝑡|Ω𝑡−1~𝑁(0, 𝐻𝑡), the log-likelihood function (for the sample 1……T) is given

by

ℓ(𝜃)= - 1₂ TN log 2𝜋 - 1₂∑𝑇 log det 𝐻_𝑡(𝜃)

121

where 𝜃 denotes the vector of unknown parameters, the N ×1 vector 𝜖𝑡 (𝜃) contains the error

elements 𝜀𝑘,𝑡(𝜃) , and 𝐻𝑡(𝜃) contains the covariance terms ℎ𝑒,𝑏,𝑡, as defined in equation (4.4).

The conditions under which the maximum likelihood is consistent and asymptotically normal are derived in Bollerslev and Wooldridge (1992), and they also derived the formulae for asymptotic standard errors that are robust to departures from the normality assumption. These robust standard errors are used to compute the t-statistics of the reported estimates which are obtained using the Berndt et.al (1974) algorithm.

The data employed for the analyses in this chapter consist of the daily closing observations, adjusted for dividends, of the FTSE100 and the S&P500 equity indices as proxy for the stock markets in the UK and US respectively, and the DataStream-constructed 10-year maturity UK and US government bond indices as proxy for longer-term UK and US Treasury bonds respectively. The DataStream 10-year Benchmark bond indices are composed of the most liquid government bonds and calculated using the European Federation of Financial Analysts (EFFAS) methodology. The benchmark indices are based on single bonds and the bond chosen for each series is the most representative bond available for the given maturity band at each point in time; consideration is also given to yield, liquidity, issue size and coupon.

In addition, to the bond indices, the 10 year yields i.e. the Treasury (US) and Gilt (UK) are used for robustness of the empirical analysis. The daily closing data or observations are collected from DataStream, and are used to calculate returns as the log daily change in index level for the stock and bond indices excluding days when either one or both of the markets were closed.