Explaining co movements in US and UK stock prices using

Explaining co-movements in US and UK stock prices using international information and time-varying correlations Nektarios Aslanidis* Denise R. Osborn† Marianne Sensier† *Department of Economics, Monash University, Australia †Centre for Growth and Business Cycles Research Economics, School of Social Sciences University of Manchester October 2007 Abstract This paper provides evidence on the role of macroeconomic and financial variables in a two-equation model for monthly US and UK stock price movements with timevarying conditional correlations. In addition to domestic effects, international variables play an important role, with the US Federal Funds rate having a negative influence on UK returns and UK bond yields negatively affecting the US market. Oil price inflation has negative effects on returns in both markets. Compared to a bivariate model without explanatory variables, the inclusion of macroeconomic and financial variables increases the persistence of time varying correlations in a DCCGARCH model. Further, a regime-switching STCC-GARCH specification finds that conditional correlations increase dramatically after 1999-2000 and also increase in periods of high volatility. JEL classifications: C32, C51, G15 Keywords: international stock returns, DCC-GARCH model, smooth transition conditional correlation GARCH model, model evaluation. This paper is preliminary. Please do not quote without permission from the authors. Comments on an earlier version of the paper from seminar participants at the University of Essex, University of Alicante and the Fondazione Eni Enrico Mattei (FEEM) are greatly appreciated. We would like to thank Annastiina Silvennoinen and Christos Savva for sharing their program codes with us. 1. Introduction There is a great deal of interest, and a correspondingly large literature, on the relationship between international financial markets. In particular, it is now well established that returns across important stock markets are not only strong, but also time-varying. Important contributions to understanding the nature of this phenomenon include Ang and Bekaert (2002), Cappiello, Engle and Sheppard (2006), Hamilton and Susmel (1994), King, Sentana and Wadhwani (1994), Longin and Solnik (1995, 2001), and Ramchand and Susmel (1998). However, in focusing on relationships across markets, this literature largely ignores the impact of other information, both economic and financial. Although King et al. (1994) is an exception, most studies that consider the role of such information do so in the context of a single market; see, for example, Cochrane (1991), Fama (1990), McMillan (2001), and Pesaran and Timmermann (1995, 2000). Our purpose in this paper is to draw these two strands of literature together by examining comovements in monthly US and UK stock market prices in the light of both economic conditions and time-varying correlations of shocks to these markets. Capturing time-varying correlations is an important aspect of recent research concerned with international financial markets. One approach is to assume stationary dynamics, leading to the dynamic conditional correlation (DCC) model of Engle (2002), with a similar methodology proposed by Tse and Tsui (2002). A different approach is to assume these correlations are regime-dependent, but constant within a regime. Although the regimes can be specified using a discrete-state Markovswitching framework (see, for example, Ang and Bekaert, 2002, and Ramchand and Susmel, 1998), a model where the regime is defined as a continuous function of an explanatory variable has the advantage of providing an explanation for regime 1 change, while also allowing for the possibility of intermediate positions between regimes. Applied in the context of correlations, this last possibility leads to the smooth transition conditional correlation (STCC) GARCH model recently proposed by Silvennoinen and Teräsvirta (2005). Indeed, Berben and Jansen (2005) introduce a special case of the STCC model using time as the transition variable, thereby allowing the correlations of shocks to international markets to smoothly increase (or decrease) over time. In order to allow the possibility that correlations may be driven by observable information, in addition to temporal change, Silvennoinen and Teräsvirta (2007) extend the model to admit a second transition variable, yielding the double smooth transition conditional correlation (DSTCC) model. Our study investigates the nature of co-movements between the US and UK stock markets using (D)STCC and DCC specifications within a GARCH framework. However, as noted above, relatively few studies of international co-movements of stock returns examine the impact of economic information. We address this deficiency by considering the role of key macroeconomic and financial variables in explaining monthly returns in these markets and allowing information from both countries to (potentially) influence returns in each market. Further, through the (D)STCC model we investigate the nature of potential time variation in the correlations of shocks to these two markets. The organisation of this paper is as follows. In Section 2 we describe the DCC and (D)STCC models and explain their desirable features for modelling stock market prices. This section also discusses the procedures we use for specifying, estimating and evaluating such models. In Section 3 we introduce the data set we analyse including the international and domestic variables. In Section 4 we report our results 2 with a range of tests for the estimated models. Conclusions in Section 5 complete the paper. 2. Time-Varying Correlation Models This section outlines the models that we employ, with the first subsection outlining the mean and volatility equations, with the following two subsections describing the DCC and STCC correlation models. Finally, we discuss issues of hypothesis testing and estimation in subsequent subsections. 2.1 Mean and Volatility Equations The two-dimensional vector of stock returns for the US and UK (yt) has mean equation y t = E[ y t / ℑt −1 ] + r t , t = 1, 2, …, T (1) where ℑt −1 is all information available at time t -1, together with values of exogenous variables for time t. Since we are interested in the role of macroeconomic information in the evolution of yt, ℑt −1 includes currently available observations on economic variables (such as output, inflation and interest rates), together with lagged stock returns. Allowing information from both the foreign and domestic economy to affect stock returns for each country enables us to capture correlations that are due to common economic information. The mean equations in (1) are assumed linear. The conditional covariances follow r t |ℑt −1 ~ N (0, H t) (2) 3 where N denotes the bivariate normal distribution. From (2), each univariate error process can be written 1 r i ,t = hii /,t2 ε i ,t , i = 1, 2 (3) where hii ,t = E (ri 2t / ℑt −1 ) and ε i, t is a sequence of independent random variables with , mean zero and variance one. As common in empirical analyses, each conditional variance is assumed to follow a univariate GARCH(1,1) process h ii ,t = α i 0 + α i1ri 2t −1 + β i1h ii ,t −1 , with non-negativity and stationarity restrictions imposed. (4) Rather than modelling the off-diagonal elements of Ht directly, the definition h12,t = ρ t (h11,t h 22,t )1/ 2 (5) allows the focus to be placed on the time-varying correlations ρt. The DCC and STCC models then differ in their definitions of ρt. The constant conditional correlation (CCC) model simply assumes that ρt is constant over time. 2.2 DCC Model Engle (2002) specifies the DCC model through the GARCH(1,1)-type process q i , j ,t = ρ 12 (1 − α − β ) + α ε 1,t −1ε 2,t −1 + β q i , j ,t −1 (6) where ρ 12 is the (assumed constant) unconditional correlation between ε 1,t and ε 2,t , α is the news coefficient and β is the decay coefficient. In order to constrain the conditional correlation ρt to lie between -1 and +1, q1, 2,t from (6) and the conditional correlation is obtained from ρt = qt (q11,t q 22,t )1 / 2 (7) 4 The model is mean-reverting provided α + β < 1 , and when the sum is equal to 1 the conditional correlation process in (6) is integrated. 2.3 STCC and DSTCC Models The STCC model considered by Berben and Jansen (2005) and Silvennoinen and Teräsvirta (2005)1 assumes the presence of two extreme states (or regimes) with statespecific constant correlations. These correlations are, however, allowed to change smoothly between the two regimes as a function of an observable transition variable. More specifically, the conditional correlation ρ t follows ρt = ρ1 (1 −G t (st ;γ , c )) + ρ2G t (st ;γ , c ) (8) in which the transition function G t (st ; γ , c ) is assumed continuous and bounded by zero and unity, γ and c are parameters, whereas st is the transition variable. Since (8) implies ρt = ρ1 when Gt = 0 and ρt = ρ2 when Gt = 1, extreme values of the transition function identify the distinct correlations that apply in these regimes. A weighted mixture of these two correlations applies when 0 0 1 + exp[−γ (st − c )] (9) where the parameter c is the threshold between the two regimes. The slope parameter γ > 0 determines the smoothness of the change in the value of the logistic function and thus the speed of the transition from one correlation state to the other. When γ → ∞ , G t (st ; γ , c ) becomes a step function ( G t (st ; γ , c ) = 0 if st ≤ c and Notice that Berben and Jansen (2005)’s model is bivariate and the transition variable is a time trend, while Silvennoinen and Teräsvirta (2005)’s framework is multivariate and their transition variable can be deterministic or stochastic. 1 5 G t (st ; γ , c ) = 1 if st > c ), and the transition between the two extreme correlation states becomes abrupt. In that case, the model approaches a threshold model in correlations. An important special case of the STCC model uses time as the transition, st = t / T , which gives rise to the time-varying conditional correlation (TVCC) model employed by Berben and Jansen (2005)2. This allows one (smooth) change between correlation regimes, and as γ → ∞ captures a structural break in the correlations. This transition variable may be particularly relevant in order to capture the effects of increasing integration of financial markets over the last twenty years. Silvennoinen and Teräsvirta (2007) generalise (7) to allow two transition variables, such that ρ t = (1 −G 2t )( ρ11 (1 −G1t ) + ρ 21G1t ) +G 2t ( ρ12 (1 −G1t ) + ρ 22G1t ) (10) where the transition functions G1t(s1t; γ1, c1) and G2t(s2t; γ2, c2) are logistic functions as defined in (9). The model in (10) then becomes a double smooth transition conditional correlation (STDCC) model. A particular case of interest, which is applied below, occurs where one of the transition variables is time; say s2t = t/T. At the beginning of the sample period when G2t = 0, the correlations vary smoothly between the extremes ρ11 (G1t = 0) and ρ21 (G1t = 0) depending on the values of s1t. As time passes, the correlations ρ11 and ρ21 change smoothly to ρ12 and ρ22, respectively. The use of four regimes, each with an associated correlation, allows the transition variable s1t to have distinctive effects in the earlier and later parts of the sample period, reflecting differing impacts as globalisation progresses. 2 The scaling implied by defining st = t/T aids interpretation; see Berben and Jansen (2005). 6 2.4 DCC Model Tests Bollerslev (1990) uses two tests of the assumption that the correlations ρt in (5) are constant over time. The first is the Ljung-Box statistic for testing autocorrelation up to m lags in the cross products of the standardised residuals, defined using (4), which under the null hypothesis is asymptotically distributed as χ 2 with m degrees of freedom (we use m = 18). The second test is a residual-based diagnostic computed as − − − an F test from a regression of the sample values of r 1t r 2t h121,t − 1 on h121,t , r12t −1h121,t , , − − r22,t −1h121,t and lags r 1,t −k r 2,t −k h121,t (in which we include k = 1, …, 12). The final test on the CCC model is Tse (2000) test. Tse (2000) derives a Lagrange Multiplier (LM) test where the alternative model assumes ARCH-type dynamics for the conditional correlations. In our bivariate application, this statistic is distributed as χ 2 with 1 degree of freedom. In applying these tests (and those for the STCC model below), we draw on the usual result that, with a linear mean equation in (1), the asymptotic covariance matrix is block diagonal for the mean and covariance parameters (see, for example, Tse, 2000). Consequently, and on the assumption that the mean equations are correctly specified, the effects of estimating these mean equations can be ignored in applying tests concerned with Ht. Clearly, rejection of the CCC specification points to the need to model the time variation in the correlations and may indicate a DCC model. To check the adequacy of the DCC model, we follow Engle (2002) in testing the standardised residuals for autocorrelation. The DCC standardised residuals are obtained from vt = H t−1 / 2 r t , which is implemented using the sample analogue of the triangular decomposition of Ht, namely 1 v1,t = r 1,t / h11/,2 t 7 v 2,t = r 2,t ρ 12,t 1 − r 1,t 2 1/ 2 2 (h22,t (1 − ρ12,t )) (h11,t (1 − ρ12,t ))1/ 2 (11) The Ljung-Box test is then applied to both the standardised errors and the squared standardised errors. Furthermore, we report results of F tests from regressions of vi ,t and vi2,t on 12 own lags plus an intercept. 2.5 STCC Model Tests The need for an STCC model is examined by performing the test of Silvennoinen and Teräsvirta (2005), who derive a Lagrange Multiplier test LMCCC for the constancy of the correlations against a particular transition variable. Since correlations arising from observed macroeconomic variables are captured in the mean equation (1), we perform this test for the possibility of a time transition in the correlations. Nevertheless, many studies, including Ang and Bekaert (2002), Longin and Solnik (1995, 2001), Hamilton and Susmel (1994) and Ramchand and Susmel (1998), find that comovements in returns are stronger in volatile times than in more tranquil times. To examine the potential importance of US stock market volatility3 for the correlations we also test constancy against a STCC-GARCH model in which the transition variable is the conditional variance of the US stock returns, obtained from the univariate GARCH(1,1) of (4). The resulting variable is labelled SPvolt4. Since evidence is uncovered for time-varying conditional correlations using both transition variables, we also employ the test developed more recently by Silvennoinen and Teräsvirta (2007) to examine directly the constancy of correlations against a STCC model with two transition variables, namely the DSTCC model. 3 4 We select the US market for this purpose, as it is the dominant world market. We also considered SPvolt-1 as a possible transition variable, but the contemporaneous conditional volatility provided stronger evidence. 8 2.6 Estimation Issues We estimate the CCC, DCC-GARCH and (D)STCC-GARCH models by quasimaximum likelihood (QML), with robust standard errors (Bollerslev and Wooldridge, 1992) used for the parameter estimates. Furthermore, all parameters of the models (that is, for the conditional means, the volatility and the correlation equations) are estimated jointly. Engle (2002) and Cappiello et al. (2006) use a two step approach in their estimation of DCC models. However, this does not allow for computation of QML standard errors that are robust to the violation of the assumption of normality in (1). Given that this is relevant for our stock return series, we estimate all parameters in the model jointly, so that QML standard errors can be obtained. Furthermore, through this joint estimation taking account of (changing) cross-market correlations, we aim for efficiency gains in the estimation of the impact of economic information on stock returns. In practice we estimate the CCC and DCC-GARCH models using RATS programs downloaded from the Estima web-site (GARCHMV.prg). The (D)STCCGARCH models are estimated using GAUSS5. An issue that deserves special attention is the selection of starting values for the QML estimation. Starting values for the DCC models are based on linear estimates for the mean equations with all parameters in the GARCH part of the equation set to 0.05. For the correlation parameters, the news parameter α is initialised at 0.05. While we experimented with different values for the decay parameter, the likelihood maximum was achieved with β initialised at 0.05. 5 The Gauss programs were adapted from code supplied to us by Christos Savva. 9 As far as the (single transition) STCC-GARCH models are concerned, we use starting values based on the results of initial OLS estimates of the mean equations (1) and initial univariate estimates of the volatility equation (4) to obtain estimates of r1, r2, h11,t and h22,t, selecting as initial values those that minimise a loss function defined as the square of the distance between the estimated cross products of the standardised residuals and the correlations, namely  r 1,t r 2,t   ˆ ˆ  min  − [ ρ 1(1 −G t ( s; γ , c) + ρ 2 G t ( s; γ , c)] . γ ,c , ρ 1, ρ 2 ( h ˆ h )1/ 2 ˆ  11,t 22,t    However, when this grid search procedure was extended to the DSTCC-case (involving four correlations and two parameters for each transition function), these initial values did not always lead to convergence of the estimation, with particular problems in estimating the transition function slope parameter associated with US volatility6. An alternative set of starting values was obtained by estimating the (D)STCC-GARCH models conditional on the results from linear estimation for the mean equations and the STCC model results from Ox7. In practice, we found that obtaining starting values in this way yielded the best results in terms of the highest log likelihood values and fewer problems in estimating the transition slope parameters. 2 When gamma gets large, increments do not change the shape of the transition function. The likelihood function becomes flat with respect to that parameter and numerical optimizers have difficulty in converging. In such cases, Teräsvirta (1994) suggests fixing gamma and estimating the remaining parameters conditionally. In order to estimate the DSTCC-GARCH models we found it necessary to fix the slope parameter for the time transition to 500. 7 These Ox programs were supplied by Annastiina Silvennoinen. These programs are written such as that the returns are the residuals from a filtered time series, so they do not allow for the mean equations to be estimated simultaneously. Also, they do not allow for the computation of QML standard errors. 6 10 3. Data Description We simultaneously model movements in the monthly index of US and UK stock prices using data over the sample 1980m1-2006m6. More precisely, the US stock price is the Standard and Poor’s composite index (SP) and the UK stock price is the Financial Times All Share Index (FT), with end-of-month values employed for each. Stock returns may be explained by many macroeconomic events. Based on the US and UK analyses of Pesaran and Timmermann (1995, 2000), we consider a benchmark set of explanatory variables for each stock market. These are the dividend yield for the corresponding market (SPDY, FTDY), industrial production (USIP, UKIP), retail sales volumes (USRS, UKRS), a short interest rate (the US Federal Funds Rate, USFF, and the UK 3-month Treasury Bill Rate, UKTB) a long bond rate (USLR and UKLR), nominal money stock (USM1 and UKM0) and a Consumer Price index (USCP and UKRP) and the oil price measured in US dollars (OIL). In addition, the exchange rate of US dollars to pounds sterling (ER) is considered as an explanatory variable for the UK. Full details of the series and data sources are given in Appendix 1. While we aimed to use the corresponding series for the US and UK, a precise matching was not always possible due to data availability. Most variables (including stock market prices) are used as growth rates, computed as 100 times the first difference of the logarithms. Exceptions are the interest rate series and the dividend yield, which first difference, and the consumer price indices which are transformed to annual inflation rates. To match the timing of monthly stock returns, we also use end-of-month values for the explanatory variables. Nevertheless, care must be taken in relation to the lag at which macroeconomic variables become available. While retail sales, consumer prices, money, oil prices and US industrial production relating to a specific 11 month are released during the immediately subsequent month, that for UK industrial production is not. Therefore, lags of one month are employed for most variables, but UKIP is lagged by two periods. Financial data on the exchange rate, short and long interest rates are available continuously, and hence current values for the end of month are used for these variables. Dividend yields are lagged by one month to avoid the simultaneity that would result if the current value was employed. Although we use the sample period January 1980 to June 2006, we initially investigated models using data from the mid-1970s, but found evidence of parameter change around the end of 1977, which may be associated with the economic (and, for the UK, political) uncertainty during the early and mid-1970s8. Therefore, in order to focus on the recent past, we exclude this unusual historical period. Our sample period includes the stock market crash in October 1987, which affects both UK and US stock prices and the corresponding dividend yield series. The effect of the Long Term Capital Management crisis in 1998 is marked for the US stock price index. To ensure these events do not unduly influence the estimated models, we replace these outliers by the average value of the series over the sample period, computed excluding the outlier observation. We also remove outliers associated with extreme events in the industrial production, retails sales and money series (see Appendix 1 for details). These were crisis years of accelerating inflation, rising unemployment, massive industrial unrest and the first oil price shock (Dow, 1998). In their Markov switching model for UK returns, Guidolin and Timmermann (2003) associate one regime with negative mean returns and a large variance primarily with this period. 8 12 4. Results 4.1 Mean Equations and CCC Model Tests We allow for the interdependence of the US and UK stock markets arising from available international information by allowing the macroeconomic variables for each country to enter the linear mean equations for both countries. However, to avoid overparameterisation, the number of explanatory variables in the resulting model is reduced by adopting a general to specific approach and eliminating the least significant variable one by one in order to achieve the minimum Akaike information criteria (AIC). This is undertaken in a single equation setting for each market, but recognising the possible presence of heteroscedasticity by using robust standard errors to judge the least significant variable. This leads to a set of variables that enter the mean equations for each country in our CCC, DCC, and (D)STCC-GARCH models9. To provide further information about the roles of these variables, Appendix 2 contains the results for models that do not include these exogenous explanatory variables in the mean equations. For comparison with later results that model heteroscedasticity and timevarying correlations, the OLS estimates of the resulting linear models are presented in Table 1, together with heteroscedasticity robust t-statistics. The linear model for the UK explains almost a quarter of the variation in stock market price movements, with the strongest significance from the exchange rate, where an increase in ER represents an appreciation of the pound which has a negative impact. Domestic factors play a substantial role, with changes in the long and short rates, FT dividend yield, lagged stock prices and industrial production all being individually significant at the 5 We also considered using the unanticipated changes in these variables as regressors in our models, where the unanticipated component for each series was extracted using an AR(12) model, and including the residuals in the linear model. Then we followed a general to specific approach based on the AIC to select the specific model. The selected specific model was very similar that obtained using the original series, and hence we proceed with the model based on observed values. 9 13 percent level and of the anticipated signs. However, the presence of lagged changes in the UK stock market prices are not in line with the weak form of the efficient market hypothesis (being positive and significant). Important international variables for the UK include changes in the S&P dividend yield, oil prices and the Federal Funds rates which all negatively affect the market. At a 10 percent level, US inflation enters with a positive coefficient. For the UK we find that the domestic variables of retail sales and M0 have no significant effect in this specification. Although the linear model for the US contains fewer variables, as for the UK we find strong negative influences from changes in the domestic interest rates and oil prices. The US retail sales exerts a positive influence on the US market and is the only other domestic variable to appear, but its significance is marginal at the 10% level. Interestingly, the most significant variable in this equation is the change in UK long-term interest rates, which has a strong negative impact on the US market. Nevertheless, despite the overall plausibility of the estimated linear model, the diagnostic tests results indicate some inadequacies, particularly in relation to the evidence of time varying conditional volatility (ARCH) in the residuals of the US model. There is also evidence of non-normality, especially for the UK, although this is not unexpected when modelling stock returns. However, Table 2 shows that taking account of volatility through a model consisting of these mean specifications combined with a GARCH(1,1) volatility specification for each market, but assuming constant conditional correlations, is not satisfactory, with the Ljung-Box and (particularly) the Bollerslev residual diagnostic tests rejecting the assumption of constant conditional correlations. Although the Tse (2000) test is less decisive, it also rejects this assumption at a marginal significance 14 level of 6 percent. Therefore, we next consider models that take account of timevariation in the correlations of shocks to these markets. 4.2 DCC Results Table 3 presents the results of a joint estimation of the mean equation, together with a DCC-GARCH model10. In this case, all diagnostics presented are satisfactory. The GARCH estimates show considerable persistence in the volatilities for both markets. The correlation equation, however, effectively represents an integrated process, with the sum of the coefficients of (6) being (marginally) in excess of 1 and hence violating the assumption of an underlying correlation of shocks that is constant over time, which points to the possibility of an STCC model being more appropriate. Interestingly, as shown in Appendix Table A.4, the problem of a failure of the stationarity assumption does not occur when the DCC model is fitted to a constant mean model without the inclusion of explanatory variables. Figure 1 illustrates the estimated time-varying correlations for the DCCGARCH model with exogenous variables (Figure A.1 in the appendix illustrates constant mean case), the implied correlations grow fairly dramatically from around 0.4 at the beginning of the sample to around 0.9 in 2002. This increase may reflect increasing globalisation and integration of stock markets. Although Cappiello et al. (2006) associate an increase in correlations of stock markets in the recent past with the introduction of the euro currency, there is no clear reason why this would affect the bivariate correlation between the US and UK. Further, Savva et al. (2005) find the correlation between these markets to be largely unaffected. At this stage we eliminate US retail sales from the S&P mean equation and the lag of S&P returns from the FT equation as these are insignificant. 10 15 4.3 STCC Results As indicated by the CCC versus STCC test of Silvennoinen and Teräsvirta (2005) shown in Table 2, there is strong evidence for non-constancy of the conditional correlations, especially when time is considered as the transition function. The dominance of time as the transition variable is reinforced by the results of Table A.2, which tests constant correlations against time-varying correlations in a constant-mean model. By using a constant mean specification, all co-movement is captured by the correlations of the disturbances, even when such co-movements are due to related responses common macroeconomic information. Although the test results in Table A.2 point to interest rate variables and also US consumer price inflation (which is, of course, correlated with interest rates) as possible transition variables in this case, nevertheless the p-values point to time being the single most appropriate transition variable. Table 4 presents the estimates of the STCC-GARCH model, described by (1), (4), (8) and (9), with explanatory variables in the mean equations. The parameters of all equations are estimated simultaneously by QML. In this table, c defines the middle of the transition period for the correlations, which is expressed as a fraction of the sample size and the corresponding estimated mid-point date of May 2005 also indicated. The results for the correlation equation of this model in panel (c) show that the time transition begins from a conditional correlation between the two markets of 0.52 and increases gradually to the substantially higher value of 0.90. Indeed, similar values are obtained in a constant-mean specification (see Appendix Table A.5), indicating that macroeconomic variables account for little of this correlation or its pattern over time. 16 These time patterns are also clearly seen in Figures 1 and A.1, where the correlations implied by the STCC time transitions are plotted with the conditional correlations estimated from the DCC model. It may be noted that with the inclusion of explanatory variables, the slope parameter of the time transition in correlations in Table 4 is 13.4, resulting in a relatively smooth change over time in the cross-market conditional correlations. This is also illustrated in Figure 2, where the correlation change is seen to take place over approximately five years. It should also be noted that modelling the change over time in the conditional correlations also affects the estimated effects of economic variables in the mean equations. More specifically, comparing the mean equation estimates in Tables 1 and 4. In particular, although the lagged value of FT remains significant in the UK equation in Table 1, the magnitude of this coefficient is substantially lower than for the OLS estimates of Table 1. Further, the US long interest rate, which (surprisingly) was not highly significant for the US equation in Table 1, is now highly significant in Table 4. 4.4 DSTCC Results Finally, we return to the possibility that cross-market correlations are higher in times of high volatility, for which the test results of Table 2 indicate some support. However, given the dominance of the time as a potential transition variable in that table, Table 5 estimates DSTCC model using both transitions. As noted above, in the light of its role as the leading world stock market, market volatility is represented by the US market11. As before, the parameters are estimated simultaneously by QML12. The series is obtained from the GARCH(1,1) volatility model for US stock returns, which is estimated as part of the system. 12 However, in order to achieve convergence in these models, the slope parameter had to be set to 500, both in Table 5 and for the corresponding constant-mean model in Appendix Table A.5. 11 17 Panel (c) of Table 5 indicates four regimes for the correlations. Firstly, in the sample period until around the end of 1998, the correlations of shocks to the markets varies between 0.43 and 0.67, depending on the value of SPvolt in relation to the threshold value c1 =12.713. Therefore, volatility does appear to drive the correlations over this period. Subsequently, from around 2004, these correlations rise to 0.89 (in the low volatility state) and 0.97 (high volatility state). Although the estimates for this later period have to be treated with considerable caution due to the relatively small number of observations available after the time transition takes effect, the results nevertheless suggest that volatility changes play little role for the correlations at the end of the sample. Figure 2 plots the estimated time-varying correlations obtained from the DSTCC model with explanatory variables in the mean (Figure A.2 in the appendix illustrates the constant mean case). In addition, these figures show the US volatility series and the corresponding estimated threshold value generating the volatility regimes. Both models show similar characteristics, with abrupt switches in correlations depending on volatility until the late 1990s, followed by an increase in correlations around the turn of the century and subsequently a lower effect of volatility on these. Although the coefficients of the estimated mean equations are very similar between Tables 4 and 5, the standard errors in the latter case are smaller in almost all cases, with this decline being particularly marked for the US equation. Finally, comparing the value of AIC across the models of Tables 3, 4 and 5, the DSTCC model is preferred. Not surprisingly, the models in these tables all deliver lower AIC values than the constant mean models reported in Appendix 2. 13 As explained in Section 2, the slope parameter here is fixed at 500 in order to obtain convergence, indicating an almost instantaneous switch between the “low volatility” and “high volatility” regimes. 18 5. Concluding remarks This paper provides evidence on the causes of co-movements in monthly US and UK stock prices. We examine the role of macroeconomic and financial variables, including their cross-country influences. The results show that such variables play an important role, with increases in the US Federal Funds rate having a negative influence on the UK market, while increases in the UK Government bond yield negatively affects the US market. In addition to these cross-country effects, domestic short and long interest rates also play a role in explaining stock market returns, while there is a negative effect in both markets from an increase in oil prices. The communality of these effects will result in positive correlations between movements in the stock markets in both countries. Nevertheless, the degree of explanation achieved for monthly changes in stock market prices is relatively modest, especially for the US. Therefore, in addition to modelling co-movements through the mean equations, we also explore the usefulness of a set of time-varying conditional correlation models. This type of approach is also supported by the tests that reject the null hypothesis that the correlations of shocks are constant over time. Of these approaches, the dynamic conditional model of Engle (2002) points to increasing correlations in the latter part of the sample, but the parameter estimates indicate the conditional correlations are not stationary. This situation is handled well by the smooth transition conditional correlation specification of Silvennoinen and Teräsvirta (2005, 2007) using time as a transition. The resulting STCC specification finds that correlations of shocks (unexplained by the macroeconomic and financial variables) increased dramatically from around 1999, a period which coincides with the advent of 19 the European Economic and Monetary Union (the Euro Area). Finally, the inclusion of a second transition indicates that correlations also increase in periods of high volatility, with some evidence that this volatility effect may be more muted in the recent past. The robustness of our results is verified using constant-mean models that do not admit explanatory variables in the mean equations. These yield similar results, confirming the high degree of co-movement between the US and UK equity markets in recent years14. Further, the estimated dates of change (threshold parameters) and the lengths of the transition periods (slope parameters) vary little across the single transition and double transition\ models, and whether time-varying or constant means are assumed. Since the increase in co-movement remains largely unexplained after exploring the implications of common responses to observed economic information through the mean equations, the increased correlations of shocks appears to be a manifestation of increased globalisation. 14 These results relate to Goetzmann el al (2005) who examine the correlation structure of world equity markets for a period of 150 years and find that during the late nineteenth century, the Great Depression and the late twentieth century the correlations between stock markets were relatively high. 20 References Ang A. and G. Bekaert (2002), International asset allocation with regime shifts, Journal of Financial Studies 15, 1137-1187. Berben R.P. and W.J. Jansen (2005), Comovement in international equity markets: A sectoral view, Journal of International Money and Finance 24, 832-857. Bollerslev T. (1990), Modeling the coherence in short-run nominal exchange-rates - a multivariate generalized ARCH model, Review of Economic and Statistics 72, 498505. Bollerslev T. and J.M. Wooldridge (1992), Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric Reviews 11, 143-172. Bredin D. and S. Hyde (2007), Regime change and the role of international markets on the stock returns of small open economies, European Financial Management, forthcoming. Cappiello L., R.F. Engle and K. Sheppard (2006), Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns, Journal of Financial Econometrics 4, 537-572. Cochrane J.H. (1991), Production-based asset pricing and the link between stock returns and economic fluctuations, Journal of Finance 46, 209-238. Dow C. (1998), Major recessions: Britain and the world, 1920-1995, Oxford: Oxford University Press. Ehrmann M. and M. Fratzscher (2005), Equal size, equal role? Interest rate interdependence between the Euro Area and the United States, The Economic Journal 115, 928-948. Engle R. (2002), Dynamic Conditional Correlation: A simple class of multivariate Generalized Autoregressive Conditional Heteroskedasticity Models, Journal of Business and Economic Statistics 20, 339-350. Fama E.F. (1990), Stock returns, expected returns and real activity, Journal of Finance 45, 1089-1108. Goetzmann W.N., L. Li and K.G. Rouwenhorst (2005), Long-term global market correlations, Journal of Business 78, 1-38. Guidolin M. and A. Timmermann (2003), Recursive modelling of nonlinear dynamics in UK stock returns, The Manchester School 71, 381-395. Hamilton J. and R. Susmel (1994), Autoregressive conditional heteroskedasticity and changes in regime, Journal of Econometrics 64, 307-333. King M., E. Sentana and S. Wadhwani (1994), Volatility and links between national stock markets, Econometrica 62, 901-933. Longin F. and B. Solnik (1995), Is the correlation in international equity returns constant: 1960-1990?, Journal of International Money and Finance 14, 3-26. Longin F. and B. Solnik (2001), Extreme correlation and international equity markets, Journal of Finance 56, 649-676. 21 McMillan D.G. (2001), Non-linear predictability of stock market returns: evidence from non-parametric and threshold models, International Review of Economics and Finance 10, 353-368. Michael P., A.R. Nobay and D.A. Peel (1997), Transactions costs and nonlinear adjustment in real exchange rates: an empirical investigation. Journal of Political Economy 104, 862-879. Öcal N. and D.R. Osborn (2000), Business cycles nonlinearities in UK consumption and production, Journal of Applied Econometrics 15, 27-43. Perez-Quiros G. and A. Timmermann (2000), Firm size and cyclical variations in stock returns, Journal of Finance 55, 1229-1262. Pesaran M.H., and A. Timmermann (1995), Predictability of stock returns: Robustness and economic significance, Journal of Finance 50, 1201-1228. Pesaran M.H. and A. Timmermann (2000), A recursive modelling approach to predicting UK stock returns, The Economic Journal 110, 159-191. Pelletier D. (2006), Regime switching for dynamic correlations, Journal of Econometrics 131, 445-473. Qi M. (1999), Nonlinear predictability of stock returns using financial and economic variables, Journal of Business and Economic Statistics 17, 419-429. Ramchand L. and R. Susmel (1998), Volatility and cross correlation across major stock markets, Journal of Empirical Finance 5, 397-416. Sarantis N. (2001), Nonlinearties, cyclical behaviour and predictability in stock markets: international evidence, International Journal of Forecasting 17, 459-482. Savva, C.S., D.R. Osborn and L. Gill (2005), Spillovers and correlations between US and major European markets: The role of the euro, Centre for Growth and Business Cycle Research, Discussion paper No. 064. Sensier M., D.R. Osborn and N. Öcal (2002), Asymmetric interest rate effects for the UK real economy, Oxford Bulletin of Economics and Statistics 64, 315-339. Silvennoinen A. and T. Teräsvirta (2005), Multivariate autoregressive conditional heteroskedasticity with smooth transitions in conditional correlations, SSE/EFI Working Paper Series in Economics and Finance No. 577. Silvennoinen A. and T. Teräsvirta (2007), Modelling multivariate conditional heteroskedasticity with the double smooth transition conditional correlation GARCH model, SSE/EFI Working Paper Series in Economics and Finance No. 652. Teräsvirta T. (1994), Specification, estimation and evaluation of smooth transition autoregressive models, Journal of American Statistical Association 89, 208-218. Teräsvirta T. (1998), Modelling economic relationships with smooth transition regressions, in A. Ullah and D.E.A. Giles (ed), Handbook of Applied Economic Statistics, New York, Dekker, 507-552. Tse Y.K. (2000), A test for constant correlations in a multivariate GARCH model, Journal of Econometrics 98, 107-127. Tse Y.K. and A.K.C. Tsui (2002), A multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with Time-Varying Correlations, Journal of Business and Economic Statistics 20, 351-362. 22 Table 1: OLS Estimated Linear Models for US and UK Stock Prices Variable Constant ∆USRSt-1 ∆USLRt ∆USFFt ∆USCPt-1 ∆OILt-1 ∆UKLRt ∆UKTBt ∆FTt-1 ∆SPt-1 ∆FTDYt-1 ∆SPDYt-1 ∆UKIPt-2 ∆ERt s AIC SIC R2 Diagnostic tests: Autocorrelation ARCH Normality RESET Heteroscedasticity ∆SPt 0.7213 [0.2299] 0.3275 [0.2121] -1.1260 [0.5731] -0.4121 [0.2140] -0.0621 [0.0252] -2.6305 [0.7597] ∆FTt -0.0244 [0.4599] 3.7569 5.5038 5.5747 0.1392 0.5495 0.0000 0.0655 0.1186 0.4240 -0.3547 [0.1910] 0.1555 [0.0968] -0.0697 [0.0275] -3.5021 [0.8712] -1.0649 [0.4688] 0.4523 [0.1790] -0.1404 [0.1167] 11.9207 [3.7722] -6.7356 [2.7762] 0.5662 [0.2294] -0.3148 [0.0695] 3.7772 5.5328 5.6748 0.2413 0.8237 0.1634 0.0016 0.8846 0.2678 Notes: Values in square brackets are heteroscedasticity-robust standard errors; results for the diagnostic tests are presented as p-values. Diagnostic tests for autocorrelation and ARCH are Lagrange multiplier tests using lags 1 to 12 inclusive. 23 Table 2: Tests of Constant Conditional Correlations Tests against DCC model Ljung Box test Bollerslev test Tse LM statistic Tests against STCC model t/T transition SPvolt transition t/T and SPvolt transitions 19.47 (0.0000) 6.7796 (0.009) 23.126 (0.0000) 32.29 (0.020) 2.923 (0.0003) 3.476 (0.062) Notes: The Ljung-Box statistic tests autocorrelation up to 18 lags in the cross products of the GARCH standardised residuals, distributed as χ2 with 18 degrees of freedom. Bollerslev’s (1990) residual based diagnostic is the F test from a regression of r i ,t r j ,t hi−1,t − 1 on hi−1, t , ,j ,j 2 −1 ri 2t −1hi−1, t , rj , t −1hi , j , t and r i , t −1r j ,t −1hi−1,t ,...,r i ,t −12 r j ,t −12 hi−1,t . ,j ,j , ,j Tse’s (2000) LM is the Lagrange Multiplier statistic for constant correlations, distributed as χ2 with 1 degree of freedom. Figures in the parentheses are p-values. Tests against a single transition STCC model are those of Silvennionen and Teräsvirta (2005), distributed as χ2 with 1 degree of freedom. The test against two transitions is that Silvennionen and Teräsvirta (2007), distributed as χ2 with 2 degree of freedom. 24 Table 3: Model Estimates using Dynamic Conditional Correlations ∆SPt a. Returns equations E ( y i , t / ℑt −1 ) = y i ,t − r i , t Constant 0.9044 [0.1737] ∆FTt-1 ∆FTDYt-1 ∆SPDYt-1 ∆UKTBt ∆UKLRt -2.1266 [0.6062] ∆USLRt -2.0233 [0.4866] ∆USFFt -0.4179 [0.2030] -0.0476 [0.0213] ∆OILt-1 ∆ERt ∆UKIPt-2 ∆USCPt-1 b. Volatility equations E (ri 2t / ℑt −1 ) = hi , t , Constant ri 2t −1 , 0.5098 [0.2576] 0.0689 [0.0197] 0.8881 [0.0267] ∆FTt 0.3117 [0.2882] 0.2391 [0.1040] 7.4812 [2.5643] -3.7320 [1.5437] -0.9615 [0.4063] -3.6858 [0.7437] -0.3525 [0.1483] -0.0601 [0.0211] -0.2784 [0.0457] 0.4770 [0.1476] 0.1264 [0.0710] 1.3797 [0.8219] 0.0754 [0.0317] 0.8251 [0.0604] hi ,t −1 c. Correlation equation E (ε i ,tε j , t/ ℑt −1 ) = q i , j , t ε i ,t −1ε j , t −1 qi , j , t −1 AIC SIC Diagnostics LB (v i ,t ,18) F (v i1,t ,12) 0.0447 [0.0113] 0.9559 [0.0164] 10.4172 10.7012 17.76 (0.471) 1.039 (0.412) 22.86 (0.196) 1.446 (0.144) 19.62 (0.354) 1.378 (0.175) 7.656 (0.983) 0.434 (0.949) LB(vi2,t ,18) F (vi2,t ,12) Notes: Values in square brackets are robust standard errors (Bollerslev-Wooldridge, 1992). The sample period is January 1980 to June 2006 (318 observations). LB(., 18) is the Ljung-Box statistic for testing autocorrelation up to 18 lags calculated for both the standardized residuals (νi,t) (see (11)) and the squared standardized residuals, both distributed as χ2 with 18 degrees of freedom under the null hypothesis (where 18 is approximately the square root of 318). F(νi,t, 12) is the F test from the regression of νi,t on 12 of own lags plus an intercept, and F (vi2,t ,12) is the F test from the regression of vi2,t on 12 of its lags plus an intercept (see Engle, 2002). Figures in parentheses are pvalues. 25 Table 4: Model Estimates with Time-Varying Conditional Correlations ∆SPt ∆FTt a. Returns equations E ( y i , t / ℑt −1 ) = y i ,t − r i , t Constant 0.8443 [0.1904] ∆FTt-1 ∆FTDYt-1 ∆SPDYt-1 ∆UKTBt ∆UKLRt -2.4112 [0.7343] ∆USLRt -1.7081 [0.5716] ∆USFFt -0.4584 [0.2255] -0.0507 [0.0243] ∆OILt-1 ∆ERt ∆UKIPt-2 ∆USCPt-1 b. Volatility equations E (ri 2t / ℑt −1 ) = hi , t , Constant ri 2t −1 , 0.4760 [0.2402] 0.0639 [0.0222] 0.8959 [0.0236] 0.3128 [0.3186] 0.2526 [0.1093] 8.0769 [2.6158] -3.5870 [1.4812] -1.0225 [0.4228] -3.6162 [0.8274] -0.3781 [0.1767] -0.0613 [0.0257] -0.2778 [0.0511] 0.4078 [0.1627] 0.1355 [0.0711] 1.4469 [0.8443] 0.0658 [0.0320] 0.8281 [0.0613] hi ,t −1 c. Correlation equation ρ t = ρ 1(1 −G t (t / T ; γ , c)) + ρ 2G t (t / T ; γ , c) 0.5175 [0.0527] ρ1 0.8997 [0.0340] ρ2 γ 13.431 [6.5918] c 0.7701 [0.0220] (Date: 2000:m5) AIC 10.393 SIC 10.434 Notes: Values in square brackets are robust standard errors (Bollerslev-Wooldridge, 1992). 26 Table 5: Bivariate DSTCC-GARCH Model Estimates ∆SPt ∆FTt a. Returns equations E ( y i , t / ℑt −1 ) = y i ,t − r i , t Constant 0.8431 [0.1267] ∆FTt-1 ∆FTDYt-1 ∆SPDYt-1 ∆UKTBt ∆UKLRt -3.0149 [0.2062] ∆USLRt -1.2945 [0.1070] ∆USFFt -0.4098 [0.0375] -0.0487 [0.0067] ∆OILt-1 ∆ERt ∆UKIPt-2 ∆USCPt-1 b. Volatility equations E (ri 2t / ℑt −1 ) = hi , t , Constant ri 2t −1 , 0.6789 [0.2609] 0.0893 [0.0218] 0.8580 [0.0045] 0.2916 [0.2952] 0.2483 [0.1016] 8.3292 [2.4399] -3.4694 [1.4516] -1.1260 [0.4217] -3.5894 [0.7225] -0.2749 [0.1410] -0.0611 [0.0182] -0.2715 [0.0483] 0.4075 [0.1559] 0.1461 [0.0700] 1.2283 [0.7005] 0.0782 [0.0306] 0.8372 [0.0457] hi ,t −1 c. Correlation equation ρ t = (1 −G 2t ( s 2t = t / T ))( ρ11 (1 −G1t ( s1t = SPvol t )) + ρ 21G1t ) +G 2t ( ρ12 (1 −G1t ) + ρ 22G1t ) ρ 11 ρ 21 ρ 12 ρ 22 γ1 γ2 c1 c2 AIC SIC 0.4279 [0.0681] 0.6708 [0.0595] 0.8929 [0.0462] 0.9701 [0.0522] 500.00 7.9439 [3.7477] 12.685 [0.8824] 0.8121 [0.0354] (Date: 2001:m6) 10.390 10.436 Notes: Values in square brackets are robust standard errors (Bollerslev-Wooldridge, 1992). 27 .9 DCCx ST CCx .8 .7 .6 .5 .4 .3 1980 1985 1990 1995 2000 2005 Figure 1: Monthly time-varying conditional correlations from the bivariate DCC-GARCH (DCCx) and fitted time transition from a STCC-GARCH model (STCCx), both in model with explanatory variables in mean equation. 1.0 DSTCCx 0.8 0.6 1980 40 SPhx 1985 C1x 1990 1995 2000 2005 30 20 10 1980 1985 1990 1995 2000 2005 Figure 2: Monthly time-varying conditional correlations from the bivariate DSTCC-GARCH model and S&P estimated conditional variance (SPhx), both in model with explanatory variables in mean equation. 28 Appendix 1: Data Table A.1: Variable Descriptions and Sources Variable Description Source Standard and Poors’ composite index (EP), Datastream NSA Standard and Poors’ 500 composite: Datastream dividend yield (EP), NSA Federal Funds Rate Market Rate, (EP), GFD NSA 10-year Bond Constant Maturity Yield, GFD (EP), NSA Industrial production index, SA FRED Total retail trade (Volume), SA OECD M1 Money Stock, SA Consumer Price Index for All Urban Consumers: All Items, NSA Financial Times all share index (EP), NSA F.T. all share index: dividend yield-(EP), NSA US $ TO £1 (WMR), exchange rate (EP), NSA West Texas. Intermediate Oil Price (EP), US$/Barrel, NSA Treasury bills: average discount rate, NSA Gross interest yield on 2.5% Consols, (EP) NSA M0 wide monetary base (EP): level £M, SA Retail sales volume index, SA Industrial production volume index, SA Retail price index, NSA FRED FRED Datastream Datastream Datastream GFD ONS Datastream ONS Datastream ONS Datastream Name SP SPDY USFF USLR USIP USRS USM1 USCP FT UKDY ER OIL UKTB UKLR UKM0 UKRS UKIP UKRP Code USS&PCOM S&PCOM(DY) _FFYD IGUSA10D INDPRO SLRTTO01 .IXOBSA M1SL CPIAUCNS UKFTALL. FTALLSH(DY) USDOLLR. __WTC_D AJNB UKCONSOL AVAE UKRETTOTG CKYW UKCONPRCF Notes: EP – end of period; SA – seasonally adjusted; NSA – not seasonally adjusted; ONS – Office for National Statistics; FRED – Federal Reserve Economic Data (http://research.stlouisfed.org/fred/); GFD – Global Financial Database (http). Table A.2: Outliers Removed UK Stock Market Prices Dividend Yields Industrial Production M0/1 Retail Sales 1981m9, 1987m10 1981m9, 1987m10, 1998m1 2002m6 1999m12, 2000m1 1979m6 US 1987m10, 1998m8 1987m10 N/A 2001m9 1987m1, 2001m10 29 Appendix 2: Results for Constant Mean Model Appendix Table A.3: Tests of Constant Conditional Correlations Test Statistic p-value Tests against DCC model Ljung Box test 29.56 0.042 Bollerslev test 2.007 0.017 8.866 0.003 Tse test Tests against STCC model ∆FTt-1 transition ∆SPt-1 transition ∆FTDYt-1 transition ∆SPDYt-1 transition ∆UKTBt transition ∆UKLRt transition ∆USLRt transition ∆USFFt transition ∆OILt-1 transition ∆ERt transition ∆UKIPt-2 transition ∆USCPt-1 transition ∆USRSt-1 transition t/T transition SPvolt transition SPvolt, t/T transitions Notes: See Table 2. 1.436 0.007 0.265 0.003 9.406 9.190 0.443 4.509 0.002 0.862 3.490 10.02 0.295 15.55 3.961 19.51 0.230 0.932 0.606 0.952 0.002 0.002 0.505 0.033 0.960 0.353 0.061 0.001 0.586 8.0100e-005 0.046 0.0002 30 Appendix Table A.4: Bivariate DCC-GARCH Estimates ∆SPt a. Returns equations E ( y i , t / ℑt −1 ) = y i ,t − r i , t Constant 1.013 [0.1841] 2 b. Volatility equations E (ri ,t / ℑt −1 ) =hi , t Constant ri 2t −1 , 0.7113 [0.2593] 0.0796 [0.0285] 0.8732 [0.0265] ∆FTt 0.9988 [0.1873] 1.1480 [0.5167] 0.0979 [0.0362] 0.8417 [0.0396] hi ,t −1 c. Correlation equation E (ε i ,tε j , t/ ℑt −1 ) = q i , j , t ε i ,t −1ε j , t −1 qi , j , t −1 AIC SIC Diagnostics LB (v i ,t ,18) F (v i ,t ,12) 0.0752 [0.0502] 0.8873 [0.1225] 10.7272 10.8455 16.36 (0.567) 0.699 (0.752) 16.96 (0.525) 1.258 (0.242) 9.916 (0.935) 0.688 (0.762) 14.56 (0.691) 0.897 (0.550) LB(vi2,t ,18) F (vi2,t ,12) Notes: See Table 3. Appendix Table A.5: Bivariate STCC-GARCH Estimates ∆SPt a. Returns equations E ( y i , t / ℑt −1 ) = y i ,t − r i , t Constant 0.9423 [0.2168] b. Volatility equations E (ri 2t / ℑt −1 ) =hi , t , Constant ri 2t −1 , 0.6487 [0.3512] 0.0679 [0.0230] 0.8887 [0.0235] ∆FTt 1.0714 [0.2205] 1.6799 [0.8645] 0.0813 [0.0367] 0.8236 [0.0554] hi ,t −1 c. Correlation equation ρ t = ρ 1(1 −G t (t / T ; γ , c)) + ρ 2G t (t / T ; γ , c) 0.5633 [0.0468] ρ1 0.8813 [0.0210] ρ2 γ 31.739 [22.759] c 0.7600 [0.0152] (Date: 2000:m2) AIC 10.697 SIC 10.716 Notes: Values in square brackets are robust standard errors (Bollerslev-Wooldridge, 1992). 31 Appendix Table A.6: Bivariate DSTCC-GARCH Estimates ∆SPt a. Returns equations E ( y i , t / ℑt −1 ) = y i ,t − r i , t Constant 0.9613 [0.2175] b. Volatility equations E (ri 2t / ℑt −1 ) =hi , t , Constant ri 2t −1 , 1.0873 [0.4054] 0.1030 [0.0297] 0.8263 [0.0038] ∆FTt 1.0670 [0.2166] 1.5062 [0.8089] 0.1012 [0.0415] 0.8185 [0.0497] hi ,t −1 c. Correlation equation ρ t = (1 −G 2t (s 2t = t / T ))( ρ11 (1 −G1t (s1t = SPvol t )) + ρ 21G1t ) +G 2t ( ρ12 (1 −G1t ) + ρ 22G1t ) ρ 11 ρ 21 ρ 12 ρ 22 γ1 γ2 c1 c2 AIC AIC 0.5316 [0.0553] 0.6861 [0.0536] 0.8400 [0.0408] 0.9104 [0.0217] 500.00 26.958 [8.6078] 18.049 [1.7406] 0.7537 [0.0155] (Date: 1999:m12) 10.7044 10.7281 Notes: Values in square brackets are robust standard errors (Bollerslev-Wooldridge, 1992). 32 .9 DCCm ST C Cm .8 .7 .6 .5 .4 .3 1980 1985 1990 1995 2000 2005 Figure A.1: Monthly time-varying conditional correlations from the bivariate DCC-GARCH (DCCm) and fitted time transition from a STCC-GARCH model (STCCm), both in model with constant mean. 1.0 0.9 0.8 0.7 0.6 1980 40 DSTCCm 1985 SPh C1 1990 1995 2000 2005 30 20 10 1980 1985 1990 1995 2000 2005 Figure A.2: Monthly time-varying conditional correlations from the bivariate DSTCC-GARCH model and S&P estimated conditional variance (SPh), both in model with constant mean. 33