Regime changes in the relationship between stock returns and the macroeconomy

Transcription

1 Regime changes in the relationship between stock returns and the macroeconomy Don Bredin University College Dublin Gerard O Reilly Central Bank of Ireland Stuart Hyde University of Manchester Abstract The presence of nonlinear influences in the relationship between stock returns and the macroeconomy is examined for six countries. The markets chosen are Canada, France, Germany, Japan, U.K. and the U.S. Nonlinearity is accounted for via regime switching using a smooth transition regression (STR) model with the world market return as the transition variable. There is evidence of nonlinearity in all countries. Results show that covariance with the world market portfolio increases during crisis regimes, complementing the findings of Longin and Solnik (2001). Interest rate and inflation variables are strong determinants of stock returns while evidence of the role of industrial production growth, dividend yields and oil prices is identified in individual cases. Out-of-sample forecasting of the nonlinear models is superior to a random walk but struggles to out-perform that of the linear models. However, the smooth transition regression models predict direction more frequently than linear specifications. Keywords: Smooth transition, Regime switching JEL Classification: G14, G1 don.bredin@ucd.ie. stuart.hyde@mbs.ac.uk gerard.oreilly@centralbank.ie. Corresponding author: Stuart Hyde, Manchester Business School, University of Manchester, MBS Crawford House, Booth Street East, Manchester, UK M13 9PL. Tel: 44 (0) Fax: 44 (0) The views expressed here are our own and do not necessarily reflect the views of the ESCB or the staff of the Central Bank of Ireland.

2 1 Introduction Understanding the relationship between international stock markets and the relationship between stock markets and the macroeconomy is of importance and of relevance to market participants irrespective of their role. Consequently much previous research has been employed trying to analyze these relationships from different perspectives. Research on the the interaction between stock returns and financial and macroeconomic variables is well established. Numerous studies have investigated the relationship between stock returns, interest rates, inflation and real activity (see, inter alia, Fama (1981, 1990), James et al. (198), Mandelker and Tandon (198), Asprem (1989), Schwert (1990), Lee (1992) and Canova and De Nicoló (2000)) while others consider the relationship between stock returns and a wider spectrum of financial and macroeconomic variables (see, inter alia, Chen et al. (1986), Fama and French (1989) Ferson and Harvey (1991) and Cheung and Ng (1998)). However, these studies are all based on linear analysis of the relationships. Importantly they do not account for the time variation or state dependence in the relationship between these variables. Recent studies to take account of both possible regime switching and macroeconomic influences include Guidolin and Ono (2006) and Bredin and Hyde (2007). Guidolin and Ono (2006) focus on macro economic influences on US stock returns using a markov switching approach to account for non-linearity. Bredin and Hyde (2007) consider the influence of global and regional macro and financial variables with international regime switching influences on two small open markets. This paper investigates the relationship between stock returns and the macroeconomy accounting for nonlinearity for a range of countries. Although it was debated whether economic news had a significant impact on stock prices, e.g. Pearce and Roley (198) and Wasserfallen (1989). It is now widely understood that stock prices react in response to and that macroeconomic variables have explanatory power over prices and returns. McQueen and Roley (1993) show that the stock market response to macroeconomic news is dependent upon the state of the economy, while Errunza and Hogan (1998) and Flannery and Protopapakis (2002) highlight that macroeconomic factors influence both stock market volatility and returns. Significantly, it is the time varying nature of these relationships which is important. Moreover, recent work by Longin and Solnik (2001) and Ang, Chen and Xing (2006) highlight that correlations between stock returns vary over time. In particular, that correlations increase during down or bear markets compared with the correlation during up or bull markets. Thus any model incorporating stock returns should be able to capture this correlation behavior. Such variation is accounted in this study by considering state dependence through regime switching in the form of smooth transition models. This will capture the cyclical behavior of stock returns and indicate whether the relationship between stock returns and the candidate financial and macroeconomic factors is linear or nonlinear. A conventional set of financial and macroeconomic variables is adopted to investigate the impact of regime change (nonlinearity) on the relationship with stock returns for six G7 countries. 1 Initially we consider a specification where the world stock return dictates the regime change. Subsequently we decompose this effect to separately identify the roles of US returns and non-us returns. Financial factors are allowed to enter both contemporaneously and with a lag. The key results show that linear models fail to capture the true significance of macroeconomic influences on stock returns. In most cases, other than stock returns only changes 1 Results using Italian stock returns indicate that there is no evidence of non-linearity. Results are available from the authors. 1

3 in exchange rates have a significant impact. However, the nonlinear models highlight that inflation and interest rates are most important in describing the behavior of stock returns across all countries, while industrial production growth, dividend yields and changes in oil prices are also significant. Importantly, supporting the evidence of Longin and Solnik (2001) and Ang, Chen and Xing (2006), covariance with world stock returns increases in bear markets. Further, decomposing the world return into US and non-us influences reveals that covariance with non-us returns is witness to the greatest increase. The remainder of the paper is set out as follows. Section 2 discusses previous research which has investigated the relationship between stock returns and the macroeconomy and documents the importance of possible nonlinearities. Section 3 provides the details of the empirical methodology adopted in the study while section 4 provides a discussion of the data and presents the empirical results. Section reports a pseudo out-of-sample forecasting exercise while section 6 concludes. 2 Literature Review There is a rich history of literature which investigates the relationship between asset returns, real activity, inflation and money. Fama (1981) focusses upon the correlation between stock returns and expected and unexpected inflation in the U.S., showing that the observed negative relation is a proxy effect for more fundamental relationships between stock returns and real activity. Mandelker and Tandon (198) provides international evidence in support of this proxy effect using data from the U.S., U.K., France, Canada, Japan and Belgium. Geske and Roll (1983) offer a supplementary explanation suggesting that stock prices signal changes in expected inflation because money supply responds to changes in expected real activity. James et al. (198) uses a VARMA approach to model the relationships, finding strong links between stock returns, real activity and money. Investigating the stock return-inflation relation for the U.S., U.K., Canada and Germany, Kaul (1987) argues that the observed relation is a consequence of money demand and counter-cyclical monetary policy. Chen et al. (1986) identify the importance of the term spread, oil prices and industrial production growth in explaining stock return behavior. Evidence of the role of the dividend yield and the term spread in determining stock prices is provided by Fama and French (1989). Clare and Thomas (1994) investigate a wide range of factors in the U.K. reporting broadly consistent findings with those of Chen et al. (1986). Asprem (1989) documents a positive relationship between stock returns and real activity using data from European countries in addition to finding support for money supply, interest rate and exchange rate variables. The strength of the relation between stock returns and real activity or industrial production is further enhanced by the findings of Fama (1990) and Schwert (1990). Recently, Cheung and Ng (1998) provide evidence of long-run relationships between the stock market and the macroeconomy for five stock markets (the U.S., Canada, Germany, Italy and Japan). The long run relationships provide additional explanatory power for stock returns to that contained in dividend yields, default and term spreads and future GNP growth. Canova and De Nicoló (2000) examine the dynamic interrelationships between stock returns, real activity and inflation for the U.S., U.K., Japan and Germany using both open and closed economy VARs. Results show that innovations in stock returns are not significantly related to inflation or real activity. However, Lamont (2001) provides evidence that portfolios of asset returns can be used to forecast future movements in macroeconomic factors. 2

4 It is well established that returns across international stock markets are time varying. In order to take account of the time variation, GARCH type models became very popular, e.g. Flannery and Protopapakis (2002) and Connolly and Wang (2003) utilize such techniques to investigate the influence of domestic and foreign economic news on domestic stock market activity. However, recent empirical evidence has shown that correlations are higher during bear markets than bull markets, see Longin and Solnik (2001). Ang and Bekaert (2002) show that the asymmetric GARCH is unable to take account of this type of correlation pattern. Hence it is the regime or the nonlinearity that is important rather than the changing volatility. Recent evidence has shown that regime switching models that account for different phases in the business cycle are quite successful in this regard, see Schaller and van Norden (1997), Perez-Quiros and Timmermann (2000), Ang and Bekaert (2002), and Guidolin and Timmermann (2003, 200). These studies adopt the Markov switching approach of Hamilton (1989). In particular work by Guidolin and Ono (2006) investigate possible nonlinearity for US stock returns in relation to possible financial markets and macro-economy linkages. The authors find evidence of multiple regime and time varying covariances in relation to returns and their chosen set of macro and financial variables. However, a special class of regime switching model, where the state variable is observed, and also allows intermediate positions between the regimes (rather than abrupt changes as with Markov switching) is the smooth transition model, see Teräsvirta (1994), van Dijk et al. (2002) and Teräsvirta (2004). For example, if the stock market is characterized by a large number of individuals or firms, each of whom switch abruptly but at different times, this will produce a smooth transition. This is seen as an appropriate regime switching approach to examine the nature of the underlying regime and has proven to work well with specifications of two or more explanatory variables. A number of studies have applied these type of approaches to modeling stock returns, see Sarantis (2001), McMillan (2001), Bradley and Jansen (2004) and Bredin and Hyde (2007). Sarantis (2001) employs smooth transition autoregressive (STAR) models to investigate cyclical behavior of stock returns in the G7. The estimated models suggest that stock price behavior is characterized by asymmetric cycles with relatively slow rates of transition between regimes and outof-sample forecasts from the models outperform a random walk. McMillan (2001) finds evidence in the U.S. of a nonlinear relationship between stock market returns and macroeconomic and financial variables. Using a two regime STAR model, the results shows that while interest rates are important determinants in both regimes, the macroeconomic series (unemployment) only explains stock returns in one regime. Bredin and Hyde (2007) investigate the influence of global (US) and regional (UK and Gemany) macroeconomic and financial variables on equity returns in two small open markets (Ireland and Denmark). They identify that US stock returns, changes in US interest rates and changes in oil prices are important determinants of regime change. Bradley and Jansen (2004) model stock return and industrial production using various nonlinear models including STAR, reporting that out-of-sample forecasts from a linear model do as well or better than the forecasts from the STAR model. While both McMillan (2001) and Bredin and Hyde (2007) report that the in-sample performance and out-of-sample forecasts of the smooth transition regressions are superior to the linear specifications. 3

5 3 Methodology Initially the relationships between stock returns and the financial and macroeconomic factors are investigated using a simple linear regression. These specifications are then tested for smooth transition type nonlinearity using the approach of Luukkonen et al. (1988) and Teräsvirta (1994). y t = β 0 + β 1 z t + β 2 z t s t + β 3 z t s 2 t + β 4 z t s 3 t + η t (1) where y t is the dependent variable, country stock returns, z t represents the financial and macroeconomic factors and s t includes the candidate transition variables. Here the candidate variable is world stock returns. An LM-type test can be adopted to test the null hypothesis, H 0 : β 2 = β 3 = β 4 = 0 against a general alternative: LM = T (RSS l RSS) RSS l where RSS l is the sum of squared residuals from the linear equation and RSS is the sum of squared residuals from equation (1). The tests statistic is distributed as χ 2 with 3q degrees of freedom. Given the recent finding that correlations among stock markets are influence by the position on business cycle, to account for the existence of nonlinearity the stock return equations are modeled using a regime switching model. Specifically, a smooth transition regression (STR) model is adopted: y t = α 0z t + F (s t )α 1z t + u t (2) F (s t ) = {1 + exp[ γ(s t c)]} 1 where γ > 0 (3) where α 0 and α 1 are the coefficient vectors, z t is a (k + 1) 1 vector of explanatory variables, u t is the error term, which is iid(0, σ 2 ) and finally F (s t ) is the transition function defining the regime. The role of the explanatory variables in z t can differ between the two regimes through the coefficients α 1. Equation (3) shows the transition function, which is defined as a logistic function. Sarantis (2001), McMillan (2001) and Bredin and Hyde (2007) advocate the use of the logistic as opposed to the exponential function. The S-shaped logistic function is more intuitive to bull and bear stock market regimes or recessions versus expansions, as opposed to the U-shaped exponential function which cannot be used to identify expansion or contraction behavior. The transition function determines the regime and is itself governed by the transition variable, s t, and by the speed of transition, γ. As γ the transition becomes more and more abrupt, F (s t ) becomes a step function and the model approaches the standard threshold regression model. The transition function is bounded by zero and unity, which is determined by s t. The parameter c is the threshold variable and the transition function is dependent on the position of the transition variable relative to the threshold variable, with F (s t ) = 0. when s t = c. The modeling procedure to determine the appropriate transition variable is as follows. First, a two dimensional grid search of the residual sum of squares (RSS) over values of γ and c is 4

6 undertaken in order to identify the appropriate initial values for the transition function given the candidate transition variable s t is world stock returns, i.e. the values which minimize the RSS. 2 Next ordinary least squares estimates of the coefficient values are obtained as initial values of the model conditional on this choice of transition variable and the corresponding values of γ and c from the grid search. A more parsimonious model is obtained via a general to specific modeling procedure. The final model is estimated by non-linear least squares including estimation of the parameters γ and c. Estimation of the slope parameter, γ, can be problematic. Teräsvirta (1994) and others suggest that the transition function F (s t ) is standardized to make γ scale-free. This implies dividing the exponent in F (s t ) by the standard deviation of s t. Accurate estimation of the slope parameter relies on a large number of observations in the neighborhood of c. The validity of the estimated models is checked by applying a battery of diagnostic tests. Specifically the tests of Eitrheim and Teräsvirta (1996) are employed to check the validity of the STR models with respect to autocorrelation, additional nonlinearity and parameter constancy. Further, the residuals are also checked using the LM test for ARCH effects and the Lomnicki-Jarque-Bera test for normality. The models are additionally tested for any remaining smooth transition type nonlinearity using the Luukkonen et al. (1988) test. In-sample performance is also evaluated by standard statistical measures of model specification such as R 2, the residual standard deviation, RSS and the Akaike (AIC) and Schwarz (SBC) information criteria. The linear and STR models are also evaluated by examining their out-of-sample forecasting performance using one-step ahead forecasts. The linear forecasts are generated assuming that all the independent variables in the linear regression follow an AR(1) process. Nonlinear forecasts are obtained using a bootstrap with 00 replications. We compare these forecasts with those generated by a random walk process. In addition to conventional predictive error statistics such as the mean squared error (MSE), the median squared error (MedSE) and the mean absolute error (MAE), we also use a number of statistics to compare the forecasting performance of alternative models. We compute the three tests of Diebold and Mariano (199), the asymptotic test, the sign test and the Wilcoxon signed-rank test, and the tests of Pesaran and Timmermann (1992). 4 Data and Empirical Results The data employed in the study is monthly observations from 1979:04 to 2007:01 obtained from Datastream. For each of the six countries the series used are stock returns on the market index, R t, World market returns, Rt W, World market (excluding US) returns, Rt W xus, the dividend yield, DY t, changes in the short-term interest rate, SR t, the term structure, T ERM t, i.e. the difference between the short-term (3 month) interest rate and long term ( year) interest rates, inflation, Infl t, the change in the effective exchange rate, s t, industrial production growth, indp t, and changes in oil prices, oil t. 3 Tables 1 and 2 present the results of the linear model using all the explanatory variables, where the first table includes world returns, while the latter decomposes world returns into US and non-us 2 Each grid search involves γ = 1, 2,..., 20 and 40 values of c. 3 All series are tested for the presence of unit roots and differenced accordingly. Only stationary series are employed in the analysis.

7 returns. The most significant variable in describing stock returns in each of the countries is the contemporaneous return on the world market portfolio. This has a significant positive impact on stock returns in all countries. Although, lagged country stock returns do not add any incremental information, the sign of impact varies across countries. Exchange rate changes have significant impacts on most country stock returns while interest rate effects (either through changes in the short term interest rate or via the term structure) are surprisingly not important in most cases. Inflation, dividend yields and industrial production provide little or no explanatory power in explaining stock returns. In table 1 the impact of contemporaneous world returns is statistically significant with a large positive coefficient. When world returns are decomposed into the US and non-us world returns the dominance of the role played by the US is clear. The influence of US returns on each market clearly dominates the influence of world returns reported in table 1. The only exception is the case of Japan, which now returns a negative relationship relative to US returns, with a large positive influence with the remaining world returns. Despite the linear models explaining between 39% (Germany) and 69% (US) of the variation in stock returns for the models with world returns, and between 39% (US) and 63% (Canada) when we decompose returns, their performance with respect to the diagnostic tests is weak. 4 Although most of these specifications do not suffer from residual autocorrelation, there is evidence of significant ARCH effects and non normality. To analyze the models for nonlinearity, the linear model is adopted for testing linearity against smooth transition type nonlinearity with either the world returns or non-us returns as the candidate transition variable as described in section 3. As can be seen from the LST test statistic reported in tables 1 and 2, all the linear models exhibit significant nonlinearity. Table 3 reports the estimated single transition models for the case where world returns are included in isolation. With the exception of France, all the transition functions seem to have relatively steep slopes, implied by the high γ values, with few observations positioned on the slope. Thus, in contrast to the findings of Sarantis (2001) the estimated regime changes are abrupt. The threshold at which the switch in regimes takes place is typically for large negative returns. This suggests that market behavior for large negative returns is distinct to that for smaller falls and positive returns. Plots of the transition functions show that in most of the six markets switch in response to the October 1987 crash, August 1998 (the impact of the Asian crisis) and September 2001 (the impact of the 9/11 terrorist attack) in addition to the periods August and September 1990, September and November 2000 and February and March On the whole the coefficients are of similar magnitude to those reported for the linear models. World market returns have a significant positive impact on all six country returns. Importantly, the STR models clearly show (with the exception of Canada and Japan) that large negative returns on the world market portfolio induce larger negative country returns (i.e. an amplifying effect), while small negative or positive returns have a reduced impact (i.e. a dampening effect). That is the covariance between country and world returns is much greater during the crisis or bear regime. Changes in short-term (term structure) interest rates are important, especially in the U.K. and U.S., for describing stock returns. In contrast to the linear specifications, inflation now has an overall negative impact and is significant in the cases of Canada, Japan and the U.S. However, in the crisis regime, i.e. large negative returns, inflation typically has a large negative impact exacerbating the bear market. 6 From table 4, we can 4 The presence of US returns in the world measure accounts for why the US case has highest explanatory power when using world returns, but least when we adopt non-us returns. Not reported. Available from the authors on request. 6 The exception here is the US, which has a large positive effect during the bear market 6

8 see that with the exception of Japan, US returns dominate the influence of world returns for each market. Again the covariance between country and US returns is much greater during the crisis or bear regime. In all cases the US returns dominate in terms of the normal regime and the heightened effect during the bear regime. The exception is the case of Japan which is dominated by non-us world returns, an effect that is particularly sensitive to the crisis period. Both Canada and Japan appear to be heavily influenced by non-us world returns and the potential effect of bull-bear swings is likely to have a particularly large influence on both these markets. The diagnostics show that for all countries the AIC has been reduced and the coefficient of determination, R 2, has increased to between 46% (Germany) and 72% (US) for the models with world returns, and between 46% (US) and 67% (Canada) when we decompose returns. However there is evidence of problems of serial correlation and ARCH effects in most cases. The null hypothesis of parameter constancy also fails to be rejected for the majority of the models. Further, in most cases there is no evidence of remaining nonlinearity with respect to the transition variable, suggesting that the single transition models capture the nonlinearity and represent an improvement on the linear specifications. Out-of-sample forecasts In order to evaluate the economic implications of the estimated models, we perform a pseudo out-of-sample forecasting exercise. We undertake recursive estimation of our smooth transition regression models model over the period 1999: :12 to generate one period ahead forecasts. This means that the first estimation uses data for the interval 1979: :12 (i.e. 249 observations) and produces a forecast for 2000:01, the second for 1979: :01 (20 observations) for 2000:02, etc. This recursive updating of the parameter estimates captures the expanding learning of an investor who uses the model to characterize the dynamic properties of international equity markets. The results of the pseudo out-of-sample forecasting experiment are presented in table and shown in figures 1 and 2. It can clearly be seen that in virtually every case both the linear and the nonlinear models are superior to a simple random walk forecast. However the forecasts from the smooth transition regressions provide little improvement over those from the linear specifications. There are only small differences in the reported MSE, MedSE and MAE statistics across the models in each country. Typically the null hypothesis that the competing forecasts have the same accuracy cannot be rejected using the Diebold and Mariano (199) tests. However, significantly the Success ratio which captures the percentage of correctly predicted positive and negative returns (i.e. correctly predicted direction) is typically much higher for the STR models than the linear model. This shows that the nonlinear models forecast the direction of returns correctly more than the linear specification. 6 Conclusion This paper contributes to the rich literature on relationships between the stock market and the macroeconomy providing international evidence from 6 countries on the influence of macroeconomic 7

9 and financial variables on stock returns. Previous research has provided evidence of a role for such information in determining movements in the stock market. The novel aspect of this study is that regime switching is used to account for time variation in the relationships. The approach used is the smooth transition regression (STR) model. Similar to previous studies, the analysis highlights a range of important financial and macroeconomic factors for determining the behavior of stock returns. Key is that the results provide evidence that the real significance of the influence of factors such as interest rates, inflation, output and dividend yields are only revealed once regimes are explicitly modeled. Stock returns respond in a nonlinear fashion to financial and macroeconomic factors. The nonlinearity or cyclical behavior is captured by world market stock returns with large negative falls leading to a different regime to that of smaller negative or positive returns. The single transition models all isolate extreme falls in returns or crisis periods including key dates such as the market crash in October 1987, the consequence of the Asian financial crisis and the impact of the 9/11 terrorist attacks in September Significantly, the smooth transition models show that covariance between country stock returns and world stock returns increases during these periods supporting the findings of Longin and Solnik (2001) and Ang, Chen and Xing (2006). The in-sample performance of the smooth transition regressions is superior to the simple linear specifications. The out-of-sample performance is evaluated using one-period ahead forecasts and testing the competing forecasts from the different models with a range of statistics for predictive accuracy. There is little to choose between the different forecasts, particularly between the linear model and the STR model, and the null of the same underlying accuracy in the Diebold and Mariano (199) cannot be rejected. However, importantly for stock return prediction, the nonlinear models correctly forecast the direction of the stock return more often than the linear model. The findings highlight the importance of accounting for nonlinearity and the cyclical behavior of stock returns. The analysis reveals that the significance of the influence of interest rates, inflation, output and dividend yields on stock returns is not captured by the linear models, only once nonlinearity is accounted for. 8

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13 Table 1: Linear Model This table reports the OLS estimation of the linear model using World returns. S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki-Jarque-Bera test. LST is the Luukkonen et al (1988) nonlinearity test. * denotes % significance, ** significance at %, *** significance at 1%. Canada France Germany Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const ** R t * Rt W *** *** *** DY t SR t T ERM t ** Infl t s t *** *** *** indp t oil t S.E AIC R Diagnostics AU T O ARCH 0.24 < N ORM < < < LST < Parameter constancy All coeffs Intercept

14 Table 1: [cont.] Linear Model This table reports the OLS estimation of the linear model using World returns. S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki-Jarque-Bera test. LST is the Luukkonen et al (1988) nonlinearity test. * denotes % significance, ** significance at %, *** significance at 1%. Japan UK US Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const * ** R t ** * Rt W *** *** *** DY t ** ** SR t * T ERM t Infl t s t * *** indp t oil t ** S.E AIC R Diagnostics AU T O ARCH < N ORM < < LST < < Parameter constancy All coeffs < Intercept <

15 Table 2: Linear Model This table reports the OLS estimation of the linear model using US and non-us World returns. S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki- Jarque-Bera test. LST is the Luukkonen et al (1988) nonlinearity test. * denotes % significance, ** significance at %, *** significance at 1%. Canada France Germany Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const R t ** Rt US *** *** *** Rt W xus *** *** *** DY t SR t T ERM t *** Infl t s t ** *** ** indp t oil t S.E AIC R Diagnostics AU T O < ARCH < N ORM 0.20 < < LST < < Parameter constancy All coeffs Intercept

16 Table 2: [cont.] Linear Model This table reports the OLS estimation of the linear model using US and non-us World returns. S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki- Jarque-Bera test. LST is Luukkonen et al (1988) nonlinearity test. * denotes % significance, ** significance at %, *** significance at 1%. Japan UK US Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const ** ** R t Rt US ** *** Rt W xus *** *** *** DY t SR t T ERM t Infl t s t *** *** indp t oil t ** S.E AIC R Diagnostics AU T O ARCH N ORM < < LST < Parameter constancy All coeffs < Intercept

17 Table 3: Smooth Transition Regression (STR) Model This table reports the estimation of the smooth transition regression (STR) model using World returns, y t = α 0w t + F (s t )α 1w t + u t F (s t ) = {1 + exp[ γ(s t c)]} 1 where γ > 0 S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki-Jarque-Bera test. LST is the Luukkonen et al (1988) test for remaining STR nonlinearity. * denotes % significance, ** significance at %, *** significance at 1%. Canada France Germany Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const *** * R t *** Rt W *** *** DY t SR t *** *** T ERM t ** Infl t * s t *** *** *** indp t *** oil t F F R t *** F Rt W *** * F DY t F SR t *** *** F T ERM t ** * F Infl t * F s t *** F indp t *** F oil t γ c *** *** S.E AIC R Diagnostics AU T O ARCH < N ORM < LST Parameter constancy All coeffs Intercept

18 Table 3: [cont.] Smooth Transition Regression (STR) Model This table reports the estimation of the smooth transition regression (STR) model using World returns, y t = α 0w t + F (s t )α 1w t + u t F (s t ) = {1 + exp[ γ(s t c)]} 1 where γ > 0 S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki-Jarque-Bera test. LST is the Luukkonen et al (1988) test for remaining STR nonlinearity. * denotes % significance, ** significance at %, *** significance at 1%. Japan UK US Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const ** ** ** R t *** *** Rt W *** *** DY t *** ** SR t ** T ERM t *** ** Infl t *** *** s t *** ** *** indp t ** oil t *** F * *** F R t ** ** F Rt W *** *** *** F DY t ** F SR t *** *** F T ERM t *** ** F Infl t *** *** F s t *** ** F indp t * ** F oil t * γ c *** *** S.E AIC R Diagnostics AU T O ARCH N ORM < < LST Parameter constancy All coeffs < Intercept

19 Table 4: Smooth Transition Regression (STR) Model This table reports the estimation of the smooth transition regression (STR) model using US and non-us World returns, y t = α 0w t + F (s t )α 1w t + u t F (s t ) = {1 + exp[ γ(s t c)]} 1 where γ > 0 S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki-Jarque-Bera test. LST is the Luukkonen et al (1988) test for remaining STR nonlinearity. * denotes % significance, ** significance at %, *** significance at 1%. Canada France Germany Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const ** R t * * Rt US *** *** *** Rt W xus ** ** DY t SR t ** T ERM t *** Infl t ** s t *** ** indp t *** oil t *** F ** F R t * F Rt US *** * * F Rt W xus * F DY t F SR t F T ERM t F Infl t *** F s t *** F indp t *** F oil t γ * c *** *** S.E AIC R Diagnostics AU T O ARCH < N ORM < < LST Parameter constancy All coeffs Intercept

20 Table 4: [cont.] Smooth Transition Regression (STR) Model This table reports the estimation of the smooth transition regression (STR) model using US and non-us World returns, y t = α 0w t + F (s t )α 1w t + u t F (s t ) = {1 + exp[ γ(s t c)]} 1 where γ > 0 S.E. is the standard error of the regression and AIC is the Akaike Information Criterion. Diagnostic test results are p-values. Tests for autocorrelation (AUTO) and ARCH are LM tests up to lag 6. NORM is the Lomnicki-Jarque-Bera test. LST is the Luukkonen et al (1988) test for remaining STR nonlinearity. * denotes % significance, ** significance at %, *** significance at 1%. Japan UK US Variable Coeff. t-stat. Coeff. t-stat. Coeff. t-stat. const *** *** ** R t 1 R US R W xus t *** t *** DY t *** *** SR t *** T ERM t Infl t *** s t *** *** indp t ** oil t F *** *** ** F R t ** F Rt US *** F Rt W xus *** *** F DY t *** *** F SR t *** * F T ERM t F Infl t *** F s t ** F indp t * ** F oil t *** γ *** c *** ** S.E AIC R Diagnostics AU T O ARCH N ORM < LST Parameter constancy All coeffs < Intercept

21 Table : Out of Sample Forecasts Forecasts are one-step ahead forecasts. The models are initially estimated over the sample period 1979: :12 and a forecast produced for 2000:01. The parameters of the models are then re-estimated at each point in time over the period 2000:01 to 2006:12 and one-step ahead forecasts produced. This generates 8 one-period ahead forecasts for the period 2000:01 to 2007:01. RW the random walk model, Lin-W and STR-W refer to the linear model the smooth transition regression model with world stock returns respectively. Lin-WxUS and STR-WxUS refer respectively to the linear model and the smooth transition regression model with US and non-us returns. MSE, MedSE and MAE are the Mean-squared forecast errors, Median-squared forecast error and the Mean absolute forecast error respectively. DM, DM Sign and DM W ilcoxon are the p-value associated with various the Diebold-Mariano tests for competing forecasts. P T is the Pesaran-Timmermann statistic and Success is the associated Success ratio. RW Lin-W STR-W Lin-WxUS STR-WxUS Canada M SE M edse M AE DM DM Sign DM W ilcoxon < < < P T Success France M SE M edse M AE DM DM Sign DM W ilcoxon < < P T Success Germany M SE M edse M AE DM DM Sign DM W ilcoxon < < < < P T Success

22 Table : [cont.] Out of Sample Forecasts Forecasts are one-step ahead forecasts. The models are initially estimated over the sample period 1979: :12 and a forecast produced for 2000:01. The parameters of the models are then re-estimated at each point in time over the period 2000:01 to 2006:12 and one-step ahead forecasts produced. This generates 8 one-period ahead forecasts for the period 2000:01 to 2007:01. RW the random walk model, Lin-W and STR-W refer to the linear model the smooth transition regression model with world stock returns respectively. Lin-WxUS and STR-WxUS refer respectively to the linear model and the smooth transition regression model with US and non-us returns. MSE, MedSE and MAE are the Mean-squared forecast errors, Median-squared forecast error and the Mean absolute forecast error respectively. DM, DM Sign and DM W ilcoxon are the p-value associated with various the Diebold-Mariano tests for competing forecasts. P T is the Pesaran-Timmermann statistic and Success is the associated Success ratio. RW Lin-W STR-W Lin-WxUS STR-WxUS Japan M SE M edse M AE DM DM Sign DM W ilcoxon < < P T Success UK M SE M edse M AE DM DM Sign DM W ilcoxon < P T Success US M SE M edse M AE DM DM Sign DM W ilcoxon < P T Success