Journal of Financial. Economics, Vol. 43, No. 1, pp. 29− 77. Bekaert, G. & Wu, G.
... Engle, R. F. & Mustafa, C. (1992): “Implied ARCH Models from Options Prices.
MEDDELANDEN FRÅN SVENSKA HANDELSHÖGSKOLAN SWEDISH SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION WORKING PAPERS
441 Jan Antell
THE INFLUENCE OF WORLD FACTORS ON FINNISH STOCK MARKET VOLATILITY
DECEMBER 2000
Key words: Asymmetric volatility, world influence, asset pricing JEL Classification: G 12, G 15
© Swedish School of Economics and Business Administration, Jan Antell
Jan Antell Department of Finance and Statistics Swedish School of Economics and Business Administration P.O.Box 479 00101 Helsinki, Finland
Distributor: Library Swedish School of Economics and Business Administration P.O.Box 479 00101 Helsinki Finland Phone: +358-9-431 33 376, +358-9-431 33 265 Fax: +358-9-431 33 425 E-mail:
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SHS intressebyrå IB (Oy Casa Security Ab), Helsingfors 2000
ISBN 951-555-667-8 ISSN 0357-4598
The Influence of World Factors on Finnish Stock Market Volatility Jan Antell Swedish School of Economics and Business Administration P.O. Box 479 (Arkadiagatan 22) FIN− 00101 HELSINKI FINLAND
[email protected] Abstract This paper investigates to what extent the volatility of Finnish stock portfolios is transmitted through the “world volatility”. We operationalize the volatility processes of Finnish leverage, industry, and size portfolio returns by asymmetric GARCH specifications according to Glosten et al. (1993). We use daily return data for January, 2, 1987 to December 30, 1998. We find that the world shock significantly enters the domestic models, and that the impact has increased over time. This applies also for the variance ratios, and the correlations to the world. The larger the firm, the larger is the world impact. The conditional variance is higher during recessions. The asymmetry parameter is surprisingly non-significant, and the leverage hypothesis cannot be verified. The return generating process of the domestic portfolio returns does usually not include the world information set, thus indicating that the returns are generated by a segmented conditional asset pricing model. Key words: asymmetric volatility, world influence, asset pricing, GARCH. JEL Classification: G12, G15.
Acknowledgments We thank Gregory Koutmos, Eva Liljeblom, Anders Löflund, and participants at the 6th Annual GSFFA Research Workshop 2000 in Helsinki for helpful comments. Special thanks go to Mika Vaihekoski for providing the portfolio data. The usual disclaimer applies.
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1 Introduction According to portfolio theory the riskiness of an investment determines its expected return. Asset pricing models postulates that higher systematic risk is associated with a higher expected return. Another measure of the riskiness of common stocks, the volatility of the stocks’returns, is also an important factor in determining the cost of capital for the investment project underlying the stock or portfolio of stocks in question. Other things being equal, higher volatility increases the required rate of return and hence the cost of capital. However, what properties does the volatility process have, and which elements do influence it?
During the 1980s and 1990s a wave of deregulation of financial markets took place. The liberalization removed obstacles to international trade, investments, and financial decisions. This development should have lead each country to being more subjected to international shocks than before, and hence a deeper degree of market comovement or even integration. Previous research on the predictability of equity returns has found an increasing importance of “world factors” on stock returns (Bekaert and Harvey, 1995). However, as economies and financial markets have become increasingly integrated the influence of these world factors on total volatility should have increased as well. This should be the case for whole countries as well as for individual assets within a country. The research in this area is important since the overall volatility has increased over the last decades.
Another interesting feature of stock returns is the impact of negative shocks on stock volatility. Many studies suggest that a negative shock will generate more volatility than a positive shock of equal magnitude. The most popular explanation for this asymmetry of volatility is the leverage hypothesis: a negative shock increases the leverage of the firm, which increases its riskiness, i.e., its volatility. This explanation was first proposed by Black (1976), and Christie (1982). However, it seems that the leverage effect is too small to fully account for the asymmetry (see e.g. Schwert, 1989). Thus, another explanation for the asymmetric nature of volatility response to return shocks is timevarying risk-premia, or the volatility feedback model (see Pindyck, 1984; French et al., 1987; Campbell and Hentschel, 1992). The causality according to the volatility feedback model is opposite to that of the leverage hypothesis: the former claims that return shocks are driven by changes in volatility, whereas the leverage theory contends that return shocks lead to changes in volatility. In a conditional CAPM framework with a
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GARCH-in-mean parameterization Bekaert and Wu (2000) compare the leverage hypothesis with the volatility feedback model. They find that the main mechanism behind volatility asymmetry is covariance asymmetry, and that only a small part of this effect can be attributed to a pure leverage effect. They conclude that “the leverage effect may be a misnomer”. What still increases the difficulties is the fact that there are conflicting empirical findings regarding the asymmetries. While French et al. (1987), and Campbell and Hentschel (1992) find the relation between volatility and expected return to be positive, Nelson (1991), and Glosten et al. (1993) find it negative. For the index returns of nine countries, Koutmos (1998) finds that the volatilities are negatively asymmetric. On the firm level, Duffee (1995) finds that there is a contemporaneous positive relation between stock returns and volatility which, however, is reversed at the aggregate level. Whatever the source, volatility asymmetry is a stylized fact and failure to account for it is bound to lead to misspecification.
The volatility of financial markets has been subjected to a vast amount of research. For good surveys of the theoretical volatility models and the empirical research, see Bollerslev et al. (1992, 1994). Significant ARCH effects for individual stock returns are found among others by Engle and Mustafa (1992) and for indices by Akgiray (1989). In a non-ARCH setting Schwert (1989) analyzes US volatility over a long time horizon. Volatility is higher during recessions, which could be caused by increased “operating leverage”. However, while the level of leverage is a significant factor for explaining volatility, it does not explain changes in it. Stock volatility is affected by financial leverage. Arshanapalli et al. (1997) find that US, European, and Pacific Rim series show time-varying volatility. Their main conclusion is that the volatilities are driven by a common volatility process.
In Koutmos and Booth (1995) asymmetric volatility and international volatility spillovers are combined. They use a multivariate EGARCH model to analyze price and especially volatility spillovers between the New York, Tokyo and London stock markets. The volatility spillovers are two-way and asymmetric in the sense that a negative shock in one market increases foreign volatility more than a positive innovation does. Koutmos (1996) utilizes a multivariate VAR–EGARCH specification for the UK, France, Germany, and Italy, and finds significant second moment interactions. Moreover, with one exception, the volatility transmission mechanism is asymmetric, i.e., negative innovations in one country increases volatility in another country considerably more
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than positive innovations. Booth et al. (1997) also use EGARCH to analyze four Scandinavian stock markets. Each market’s returns and volatilities are dependent on their own past values. The volatilities are asymmetric in that negative shocks give rise to more volatility than positive shocks do. The impact (the time it is in effect) of a shock is largest for Finland and lowest for Denmark
Based on the hypothesis of increased world market influence on stock characteristics the purpose of this paper is to empirically investigate to what extent the volatility of Finnish stock portfolios is transmitted through the “world volatility”. It can be argued that different industries and firms of different leverages and of different sizes exhibit differing degrees to which they have become subjected to international shocks. Especially large, internationally acting and export oriented firms should be influenced by international market conditions. Further, the impact of this world volatility can be expected to have changed (increased) over time. We operationalize the volatility processes of Finnish leverage, industry, and size portfolios by asymmetric GARCH specifications according to Glosten et al. (1993). If there exists asymmetric volatility, using leverage portfolios in the tests should give a hint whether it is caused by the leverage hypothesis. The impact of world factors as well as the asymmetries are interesting research topics in the light of the total liberalization of the Finnish financial markets in the beginning of 1993. Further, the Finnish economy faced a sharp upturn in the late 1980’s and an exceptionally severe economic recession during the first half of the 1990’s and a subsequent recovery, which means we are able to estimate volatility in vastly differing economic conditions.
To sum up, our research agenda consists of the following three parts. First, we investigate the return and volatility connection between the Finnish stock markets and the world markets. Second, we test for the volatility asymmetries often encountered in financial data, and especially wheteher the leverage hypothesis is the explanation for it. Finally, based on previous empirical evidence, we examine the effect of the economic recession in the beginning of the 1990’s on stock market volatility. We also study the return and volatility effects of the liberalization of the Finnish financial markets in 1993.
The paper is organized as follows. Section 2 outlines the conditional asymmetric GARCH model and how the domestic variance can be decomposed into a “world part” and an idiosyncratic “domestic part”. Section 3 describes the leverage, industry, and
4
size portfolios, and the information variables. Section 4 gives the empirical results, first for the world model and then for the Finnish market and the portfolios. Section 5 concludes the paper.
2 Conditional Variance Decomposition The general setting of this paper is that of Bekaert and Harvey (1997), who analyze emerging markets on index level, based on the methodology of Glosten et al. (1993), and Bollerslev (1986). Operating with logarithmic returns, let ri,t+1 be the daily return on asset or portfolio i, i = 1, … , N, over a single period from time t to t+1 in excess of the riskfree rate. Let the return process be defined by a function ri,t+1 = f(It; b) + εi,t+1, where It is a vector of conditioning information variables, and b a vector of coefficients. Let the return be represented in the following way: rw,t+1 = E(rw,t+1|It) + εw,t+1,
(1a)
ri,t+1 = E(ri,t+1|It) + εi,t+1,
(1b)
where rw, t+1 is the world return. Further, εi,t+1 = νi,t εw,t+1 + ei,t+1.
(2)
E(ri,t+1|It) and E(rw,t+1|It) are the conditional expectations for ri,t+1 and rw,t+1 given information publicly available at time t. εi,t+1 is a random shock not accounted for by the conditional expectation. This shock is decomposed into the effect attributable to world shocks, νi,t εw,t+1, and a purely idiosyncratic part, ei,t+1. νi,t is a weighting parameter reflecting the relative importance of the world shock at each point in time. It can be a constant or modeled as a time-varying parameter.
We model the conditional expectation of the square of the world residual as 2 E ( εw2 ,t + 1 | I t ) = σw2 ,t + 1 = c w + α w σw2 ,t + βw εw2 ,t + γ w d w ,t + 1εw ,t ,
(3a)
and the pure asset specific shock, i.e., the idiosyncratic variance not accounted for by world factors as 2 E (ei2,t + 1 | I t ) = (σip,t + 1 ) 2 = c i + α i ( σip,t ) 2 + βi ei2,t + γ i ,1d i 1,t + 1e i ,t + γ i ,2 d i 2,t + 1 + γ i ,3 d i 3,t + 1 , (3b)
where It is a vector containing publicly available information. di1,t+1 (dw,t+1) is an indicator variable taking value one when the idiosyncratic (world) shock is negative, and zero otherwise. Hence γ i,1 (γ w) is the asymmetry parameter testing whether negative shocks have a larger impact on volatility than positive shocks. While the contribution of a positive innovation to the conditional variance is αi, the impact of a negative shock is
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αi+γ i,1. A natural metric for asymmetry is therefore (αi+γ i,1)/αi. di2,t+1 is another indicator variable. It takes value one from january 1993 onwards. It reflects the fact that the Finnish markets were totally opened up for foreign investors in January 1993. Since for example Schwert (1989) finds that volatility is higher in recessions, we also include a dummy variable, di3,t+1, reflecting the severe Finnish economic recession in the beginning of the 1990’s. It takes value one from the 2nd quarter 1990 to the first quarter 1993. σw,t and σi,pt are the conditional volatilities of the world return and the asset specific returns, respectively. Equations (3a) and (3b) are GARCH(1,1)–GJR representations. This parameterization was first proposed by Glosten et al. (1993).1 Simulation studies have shown these kinds of modified models to perform better than many other asymmetric models (Engle and Ng, 1993).
There is assumed to be no conditional contemporaneous correlation between the idiosyncratic shocks of two portfolios i and j, i.e., E(ei,t+1ej,t+1| It) = 0, ∀ i ≠ j. Further, the corresponding correlation between the unique shock of portfolio i and the world market shock is also assumed to be zero, i.e., E(ei,t+1εw,t+1| It) = 0, ∀ i. These assumptions along with equations (1)− (2) imply E( ε i2,t + 1 | I t ) = σi2,t + 1 = νi2,t σw2 ,t + 1 + ( σip,t + 1 ) 2 ,
(4)
E(εi,t+1εw, t+1|It) = νi ,t σw2 ,t + 1 = σiw ,t + 1 .
(5)
Within the framework outlined above, the ratio of portfolio specific variance accounted for by world factors can on the basis of equation (4) be represented as a variance ratio, which combined to equation (5) yields the following expression: VR i ,t + 1 =
νi2,t σw2 ,t + 1 σi2,t + 1
=
νi ,t σiw ,t + 1 σi2,t + 1
∈ [0, 1].
(6)
The variance ratio can hence be decomposed into three components, reflecting market integration, the ratio of standard deviations, and the correlation between the return of portfolio i and the world return, respectively: VRi,t+1 = νi,t×
σw ,t + 1 σi ,t + 1
×
σiw ,t + 1 σi ,t + 1σw ,t + 1
.
(7)
The correlation coefficient can also be denoted as:
1
Note, however, that Glosten et al. (1993) defines the dummy for positive shocks, not negative as in our paper.
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ρiw ,t + 1 =
νi ,t σw2 ,t + 1 σi ,t + 1σw ,t + 1
= νi ,t
σw ,t + 1 σi ,t + 1
.
(8)
Now, in order to conduct the tests we need to model the conditionally expected returns, E(rw,t+1|It) and E(ri,t+1|It). Traditionally asset pricing literature has utilized linear projections of the information variables.2 The conditional expectation of the world return and the return on asset i are modeled as a linear projection as follows: E(rw,t+1|It) = δw′ Xt,
(9a)
E ( r i ,t + 1 | I t ) = δi ,1 ′ X t + δi ,2 ′ X tl ,
(9b)
where Xt is the conditioning information set for the world return and X tl the corresponding information set holding only local variables. If the parameters on Xt in the local model are zero, assuming a conditional framework, the asset is probably not very much integrated to the world. The information sets are chosen by the econometrician to approximate the true information set. δ w, δ i,1, and δ i,2 are vectors of projection coefficients. Finally, the integration parameter is modeled as follows: νi, t = qi, 0 + [1 + exp(–(t–α)/β)]–1.
(10)
The second expression on the right hand side is the S-shaped cumulative distribution function of the logistic distribution. t is the time trend, and α and β scale parameters. α gives the inflection point of the curve, while β gives the curvature. The estimation of the GARCH(1,1)–GJR models involves maximization of the sample log likelihood function. Theoretically the most appealing would be to maximize the joint likelihood of all the data. Due to the size of the system this is in practice not possible. Another alternative, restricting the correlation between portfolios i and j to zero (but not between portfolio i and the world), is to use bivariate specifications. In this paper, however, we will use a two-stage procedure, first estimating the world market return model, then the asset specific models, conditioning on the world model estimates. As established in Bekaert and Harvey (1997), this approach yields consistent but not necessarily efficient estimates.
2
If the joint distribution of the asset return(s) and the information variables belong to the family of spherically invariant distributions, the asset returns can be modeled as a linear function of the information variables (Harvey, 1989).
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Using the maximum likelihood methodology makes it easy to accommodate different distributional assumptions. The most common distributional assumption is the normal distribution, εt ~ N(0, σ2t ), where εt and σ2t are general notations for a residual and the conditional variance, respectively. While the normality assumption in GARCH models generates some degree of unconditional excess kurtosis, it might not be adequate to fully account for the fat-tailedness often found in financial data. E.g. Bollerslev (1987) uses the t distribution, εt ~ t(η), where η denotes the degrees of freedom parameter. Another alternative is the generalized error distribution (also called the power exponential distribution), used for example by Nelson (1991), εt ~ ged(φ), where φ is a scale parameter. The tail-thickness parameter φ=2 corresponds to a normal distribution, and φ=2 to the double exponential (Laplace) distribution. For φ2 results in a distribution with thinner tails than the normal. When φ=1, εt has a double exponential distribution. Since the generalized error distribution allows both for the normal distribution and thicker-tailed distributions, we will use the ged(φ) in the estimation. This also reduces the number of estimations to be done. The estimation is conducted by using standard maximum likelihood procedures. Initial parameter estimates are obtained by a simplex algorithm, while the Berndt–Hall–Hall–Hausman (1974) algorithm is used for the actual estimation. Defining εt by equations (1a) or (2), and σ2t by (3a) or (3b), the log of the likelihood function for the generalized error distribution is given by φ
ε ln Lt = ln(φ/ λ) − 0.5 t − (1 + φ− 1 ) ln(2) − ln Γ( φ− 1 ) − 0.5 ln(σ2t ) , σt λ
(13a)
where Γ(·) is the gamma function, and 1
2( − 2 / φ) Γ(1/ φ) 2 λ= . Γ(3 / φ)
(13b)
The scale parameter φ is to be estimated in the maximization procedure. Since the ged(φ) for φ=2 boils down to the normal distribution, the φ=2 restriction can be tested by the likelihood ratio test.
In summary, the three items in the research agenda outlined in Section 1 can thus be tested by the methodology presented in this section. First, the link between the domestic markets and the world markets is modeled with the integration parameter in
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equations (2) and (10), and the volatility links through the variance ratios in equation (6). The correlation to the world is defined in equation (8). Second, the volatility asymmetry is modeled in equations (3a) and (3b). Finally, the question of the impact of the economic recession, and the financial market liberalization in 1993 on volatility is operationalized in equation (3b).
3 Data The data used in this study consist of ten Finnish leverage, seven industry, and ten size stock portfolios. The following industry branches are used: banking and finance, forestry, trade and transport, metal and electronics, food, housing and construction, and multi-business. Of the leverage (size) portfolios portfolio 1 has the smallest leverage (size) and portfolio 10 the largest. The time period considered is January 2, 1987 to December 30, 1998, giving a total of T = 3009 daily observations. Up to 1990 the Finnish market return is approximated by the value-weighted WI-Index constructed at the Swedish School of Economics and Business Administration. From 1991 onwards the index is HEX-Yield. The world return is the logarithmic change of the Morgan Stanley Capital International world index.3 All returns are considered from the point of view of a Finnish investor, and are calculated in excess of the one day return of the Finnish one-month Helibor interest rate.
To obtain descriptive statistics for our data, we use the method resting on Generalized Method of Moments (GMM) (cf. Hansen (1982), and Hansen and Singleton (1982)) outlined in Richardson and Smith (1993) and utilized in Harvey (1995), and Bekaert and Harvey (1997). Generally, let xit be a variable. To obtain the mean, variance, skewness, and excess kurtosis, we estimate the following exactly identified GMM system: u1it = xit –µi,
(14a)
u2it = (xit –µi)2–σi2,
(14b)
u3it = (xit –µi)3/σi3–skewi,
(14c)
u4it = (xit –µi)4/σi4–3–xkurti,
(14d)
where µ, σ, skew, and xkurt depict the mean, standard deviation, skewness, and excess kurtosis, respectively.
3
Since the MSCI Gross Dividends Index is calculated only on a monthly basis, we use the standard price index.
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Let ut = [u1it, u2it, u3it, u4it] be a 4×1 vector of disturbances. The GMM methodology rests on the moment condition E[ut+1|It]=0, which in practice usually means E[ht+1(θ)] = E[ut+1⊗ It]=0, where θ is the q×1 parameter vector and It a vector of conditioning information. The optimal GMM parameter vector is obtained by minimizing the norm J(θ) = gt(θ)′ Wgt(θ), where gt(θ) is the sample counterpart to ht(θ) and W the weighting matrix. It can be shown that the optimal weighting matrix is the inverse of the variancecovariance matrix of ht(θ). Since Ferson and Foerster (1994) showed that the iterative GMM procedure is superior to the two-stage procedure, the weighting matrix is obtained by the iterative procedure. The variance-covariance matrix is corrected for autocorrelated and heteroskedastic disturbances according to Newey and West (1987a). If there are equally many moment conditions (r) and parameters to estimate (q), the system is exactly identified and J(θ) is zero. However, if there are more moment conditions than parameters, Q = TJ(θ)∼ χ2(r–q) is a test of the over-identified restrictions and is thus a test of the restrictions imposed by economic theory. Setting the skewness and excess kurtosis parameters equal to zero gives an overidentified test for normality. Another alternative is to use the Wald test for GMM, outlined in Newey and West (1987b), testing whether these two parameters are jointly zero. Both tests are χ2(2). Using equations (14a–d), Table 1 provides some descriptive statistics for the daily asset returns for the world return, the Finnish market return, the ten leverage portfolios, the seven industry portfolios, and ten size portfolios. The mean world daily (yearly) excess return is 0.0069 (1.7) per cent and the Finnish stock market excess return is 0.0296 (7.5) per cent. The daily (yearly) standard deviations are 0.0103 (16.4) per cent and 0.0126 (20.0) per cent. Of the industry portfolios only two of seven exhibit a positive mean excess return. The year-to-year mean return for banking and finance is –13.8 per cent, and –16.2 per cent for housing and construction. These two industries were severely hit by the deep Finnish economic recession in the first half of the 1990’s. The mean return for the leverage and size portfolios is negative in six and four cases of ten, respectively. While the negative occasions are randomly distributed for the size portfolios, there is a clear pattern for the leverage portfolios. With minor exceptions, the mean excess return becomes smaller the higher the leverage is, being positive for the four first cases and negative for the following six. Generally, the excess returns are fairly low. This is explained by the negative returns and high interest rates during the economic recession.
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The GMM system in (14) also gives the coefficients of skewness and excess kurtosis. The skewness is negative for most of the portfolios. However, it is usually not significant. The excess kurtosis is highly positive, and usually also significant. Table 1 provides two normality tests. The heteroskedasticity consistent GMM Wald test rejects normality of the Finnish market at the five per cent level, but not of the world excess return. According to the Wald test, the normality null hypothesis is rejected especially for the industry portfolios. The GMM overidentifying normality test, Q, rejects the normality null in all cases.4
Further, Table 1 gives the first and second order autocorrelations for the returns. The first order autocorrelation is positive and significant at the five per cent level both for the world return and the Finnish market return. The second order autocorrelation is negative but insignificant for both. For the portfolios both the first and second order autocorrelations are more positive than negative. Finally, BP(·) and BP2(·) are the Box and Pierce (1970) Portmanteau tests for the returns and squared returns, respectively.5 They test for the joint significance of the autocorrelation lag up to the ith order. For the returns the statistics are rejected at the one per cent level for 6 of 7 industry portfolios, and in nine of ten cases for both the size and leverage portfolios. Thus, there seems to be some temporal dependencies in the first moments of the return series. When applied to squared returns, there are evident higher order dependencies. These are much stronger than for the returns, implicating that higher moment nonlinear temporal dependencies are more pronounced, and that it is easier to predict variances than returns. Since the t+1 expected return on the asset portfolios as well as on the world return are conditional upon some information set, information variables are needed. The information set consists of readily observable variables and are similar to those traditionally used in conditional asset pricing research, and aimed at capturing changes in expectations about the business cycle. The conditioning variables for the world return, Xt, consist of the constant term, the change in the default spread between Moody’s Baa and
4
We also conducted Jarque–Bera tests for normality (not reported). They were about ten times higher than the Q statistics. 5
The BP statistics are defined as BP(s) = T
∑
s i =1
number of observations. It is distributed as χ2(s).
ρˆ2i , where s is the number of lags, and T the
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Aaa rated corporate bonds, dDSt, the change in the one-month London Interbank Offered Rate, dL1t, and the change in the term structure spread, measured as the change in the yield difference between US 30-year bonds and three-month T-bills, dTSt. Corresponding variables are needed for the domestic portfolio returns as well. This information set consists of the world set, Xt, and a purely domestic set, X tl . The latter includes the domestic market return (not excess), Rmt, the change in the short-term term spread, dSTSt, measured as the change in the difference between the 12-month and the one-month interbank rates, the change in the one-month helibor interbank rate, dH1t, the log change in the FIM/USD exchange rate, dFIMUSDt, and a dummy variable for the Finnish capital market liberalization, d93t. It takes value one from the beginning of January 1993 onwards.
The procedure for obtaining descriptive statistics in Table 2 is the same as for the portfolio returns. Normality, measured with the heteroskedasticity consistent Wald for GMM test, suggests that many of the information variables are non-normal. The BP(5) and to some extent the sample autocorrelation coefficients suggest there is some autocorrelation. More importantly, the Augmented Dickey–Fuller tests including constant and trend, reject the non-stationarity null at high levels of significance. The correlation coefficients between the variables are low, the highest being –0.1678. There should be no problems with multicollinearity.
4 Empirical Results 4.1 Specification tests In order to assess the goodness of the estimated models we conduct a number of diagnostic tests. Let zˆi ,t = eˆi ,t / σ ˆi ,t , i = 1, … , N, w, be the general notation for a standardized estimated residual. It should be mean zero with unit variance. For the symmetric generalized error distribution considered itsexpected skewness is zero. The expression for the kurtosis of the generalized error distribution is a complicated function ˆ. Given that the model is correctly specified, the of the estimated scale parameter, φ
following expectations should apply: E [zˆi,t ] = 0 ,
(15a)
E [zˆi,t2 - 1] = 0 ,
(15b)
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E [zˆi,t , zˆi,t - j ] = 0 ,
j = 1, … , k,
(15c)
E [zˆ3i,t ] = 0 ,
(15d)
E [zˆi,t4 - ku i ] = 0 ,
(15e)
E [(zˆi,t2 - 1)(zˆi,t2 - j - 1)] = 0 , j = 1, … , k.
(15f)
System (15) is a modification of the moments tests in Nelson (1991), and used in Bekaert and Harvey (1997). The specification can conveniently be tested by the GMM. The correct specification of the conditional mean is implicit in the autocovariance equation (15c) (mean test with k degrees of freedom), and the conditional variance in (15f) (variance test with k degrees of freedom). The four first moments are depicted in equations (15a,b,d,e) (moment test with 4 degrees of freedom). Finally, all six equations can be tested (joint test with 2k+4 degrees of freedom). The lag length is chosen to be k = 5.
The second set of tests is aimed to capture any remaining asymmetry in the conditional volatility. The tests are based on Engle and Ng (1993). Let Si,t = 1 if the standardized residual at t–1 is negative, and zero otherwise. The tests are defined as follows: zˆi2,t = a + bSi,t + ut,
(16a)
zˆi2,t = a + bSi,t εi,t–1+ ui,t,
(16b)
zˆi2,t = a + b(1–Si,t)εi,t–1+ ui,t,
(16c)
zˆi2,t = a + b1Si,t + b2Si,t εi,t–1+ b3(1–Si,t)εi,t–1+ ui,t,
(16d)
The tests are called the sign bias test, the negative size bias test, the positive size bias test, and the joint test. The test statistic in (16a,b,c) is the t statistic on the b coefficient, and in (16d) the χ2(3) Wald statistic for H0: b1 = b2 = b3 = 0. These tests are extended in Koutmos and Knif (2000) to test for the correct specification of covariances. Let zˆi ,w ,t = (eˆi ,t / σ ˆi ,t )(εˆw ,t / σ ˆw ,t ) be the product of standardized residuals. The tests are
then: zˆi ,w ,t = a + b1Si,t + b2Sw,t + ui,t,
(17a)
zˆi ,w ,t = a + b1Si,t εi,t–1 + b2Sw,t εw,t–1+ ui,t,
(17b)
zˆi ,w ,t = a + b1(1–Si,t)εi,t–1 + b2(1–Sw,t)εw,t–1+ ui,t.
(17c)
We test H0: b1 = b2 = 0 by the χ2(2) Wald statistic. The joint statistic is χ2(6).
13
4.2 The world model Table 3 summarizes the results for the world model. Of the GMM statistics defined by (15) the moment and joint tests seem to cause some model misspecification. The sign, negative size, and positive size bias diagnostics indicate of no misspecification.6 The coefficient of skewness is positive but fairly small. On the other hand, the kurtosis is very high and positive. Of the Box–Pierce Portmanteau statistics only one is significant, BP(10), leaving some unmodeled autocorrelation in the mean equation. The statistics for the squared normalized residuals are not significant. The conditional volatility models are able to capture the time-varying properties of the volatility. Panel B of Table 3 shows some of the estimation results. The χ2(3) distributed Wald test for the three world information variables is 59.8, and thus highly significant. The asymmetry parameter is 0.1227 and highly significant, which implies a metric for asymmetry [(αw+γ w)/αw] as high as 3.64. The impact on volatility of a negative past innovation is thus 264 per cent higher than that of a positive innovation. The world volatility thus exhibits strong asymmetry. The degrees of freedom parameter, φ, is estimated to 1.22, implying a theoretical kurtosis of 4.67, which is much lower than the empirical kurtosis of 29.45. Needless to mention, all parameters in the variance equation are highly significant.
4.3 Analyzing Finnish Stock Market Volatility The first column of Table 4 shows the results for the Finnish market return. The Wald test for the four local information variables in Panel A is highly significant. This is not the case for the three world information variables, as tested by Wald 2. With respect to the return generating process the Finnish market is thus locally oriented. The parameter for the 1993 liberalization dummy, δ i,93, indicates that the daily return had a vertical shift of 0.084 percentage points in the beginning of 1993. However, since this coincides with the recovery of the Finnish economy from the severe crisis faced in the beginning of the 1990’s, the liberalization might not be the only explanation. Wald 3, testing for the constant and the logistic function in equation (10), is very significant. This means that the world contemporaneous shock is an important element in describing Finnish stock returns. Further, the integration parameter has increased over time.
6
We estimated models also for the normal distribution and the t-distribution, including/excluding asymmetric variance (not reported). The specification tests shown were usually much better for the asymmetric models than for the non-asymmetric. The joint bias test was significant for all non-asymmetric models.
14
All parameters in the variance equation are highly significant. This is the case also for the two additional variables in the variance equation, tested by Wald 4. The parameters for the 1993 dummy as well as for the recession dummy are positive and significant, the former being higher. Thus, the liberalization dummy enters positively significantly both the mean equation and the variance equation. The asymmetry parameter, γ i,1, is 0.1339 and significant at the five per cent level. The Finnish stock market return thus exhibits asymmetric variance, negative shocks increasing variance more than positive. The metric for asymmetry is 1.49.
Panel B of Table 4 exhibits the specification diagnostics. The GMM means, and through this test also the joint tests seem to cause some model specification problems, while the variance and moments tests are well-behaving. Neither the sign bias, negative size bias, positive size bias, nor the joint bias test indicate of any unmodeled variance asymmetries. At the covariance level the negative size bias test is significant, indicating that some of the asymmetries might be due to covariance rather than variance asymmetries. The skewness is close to zero, while the kurtosis is 5.49, somewhat higher than the theoretical kurtosis of 3.86 indicated by the scale parameter of 1.46. The worst problems arise with the Box–Pierce statistics. They are significant both for levels and squares of the normalized residuals.
Figures 1a–d show the idiosyncratic conditional volatility, the variance ratio of equation (6), the correlation to the world of equation (8), and the integration parameter defined by equation (10). There is a clear positive shift in volatility both during the recession and after the financial market liberalization in 1993. The constant term for the integration parameter in equation (10), qi,0, is 0.0041 but nonsignificant. The scale parameters, α, and β in (10) get the values 2215 and 557, respectively. α=2215 depicts the inflection point, which is in October 1995. This can be seen in Figure 1d. The variance ratio and correlation coefficient in figures 1b and 1c follow the shape of the integration parameter. The mean values of both the variance ratio, correlation coefficient, and the integration parameter are much higher post-1993 than pre-1993.
15
4.4 Finnish Portfolio Models Columns 1–8 of Table 4 present the selected model estimates, specification diagnostics, and the mean of the variance ratios, the correlations to the world, and the integration parameters both pre-1993 and post-1993 for the Finnish market portfolio and industry portfolios. The local information set, tested by Wald 1, is highly significant for all portfolios. The world set (Wald 2) is significant only for metal and electronics and multi-business. This is probably due to the fact that Nokia Corporation in the first half of the period belonged to the latter industry, and to the former in more recent years. The expected return process is thus based to a higher degree on local factors than on global. However, as shown by Wald 3, contemporaneous world shocks are very strongly incorporated in the portfolio returns. The α terms in the cumulative distribution function of the logistic distribution (equation 10), denoting the inflection points, are usually between 2000 (December 1994) and 3000 (December 1998). For low values of β, the slope of the S-shaped curve is high. The 1993 liberalization dummy enters positively the mean equation in five cases of which one (multi-business) is significant at the five per cent level.
While the asymmetry parameter for the Finnish market return is positively significant, it is significant at the one per cent level only for multi-business, +0.1011. It is positive for all industries. This means that the metric for asymmetry is larger than 1. The mean value of the asymmetry parameter (the metric for asymmetry) is 0.044 (1.322) with a standard error of 0.011 (0.224). The two additional exogenous variables in the variance equation are usually highly significant (Wald 4). The parameter on the 1993 dummy, γ i,2, is significant in five cases, of which one is negative, and one non-significantly negative. The parameter on the recession dummy (γ i,3) is positive and significant at the one per cent level for six of the seven portfolios, and non-significantly negative for one portfolio. There is thus a positive shift in stock market volatility during an economic recession, as found for example by Schwert (1989).
Panel B of Table 4 presents specification diagnostics for the seven industry portfolios. There are some specification problems for some of the portfolios, especially regarding the joint test of equation system (15). Especially the variance tests are rejected too often. The sign bias, negative size bias, positive size bias, and the joint tests according to Engle and Ng (1993) show that the asymmetric part of the volatility process is well-
16
defined. However, the covariance tests according to Koutmos and Knif (2000) indicate of some misspecification problems especially for the negative size bias test. The skewness of the standardized residuals is negative for all portfolios. There still exists some kurtosis in the residuals, albeit less than in the raw returns of Table 1. There are also some problems regarding first moment autocorrelation, shown by the BP(10) statistic.
Finally, Panel C of Table 4 shows the mean of the variance ratios, the correlation coefficients, and the parameter of integration to the world. The lowest VRi is 0.0181 for the food industry, and the highest 0.0643 for metal and electronics. The food industry has the lowest ρiw, 0.1051, while metal and electronics has the highest, 0.2062. The highest individual correlations range from approximately 0.60 to 0.80 (not reported), usually quite in the end of the period. The mean of the integration parameters, νi, range from 0.1198 for trade and transport to 0.3344 for metal and electronics. What is interesting to note is that all average values are higher after the 1993 Finnish financial market liberalization than before it. This indicates that world shocks have become more pronounced during the latter half of the research period.
One characteristic of the Finnish economy during the sample period was the severe economic recession in the beginning of the 1990’s. The banking and finance sector was particularly severly hit by it. Therefore Figures 2a–d show the idiosyncratic conditional volatility, the variance ratio, the correlation to the world, and the integration parameter for this sector. The conditional volatility is clearly higher during the recession and the banking sector crisis 1992–1994. Also, as seen in Figure 1a and γ i,2×10000 = 0.2949, the level of the volatility process is higher post-1993 than pre-1993. The unconditional mean of the banking and finance sector volatility (not reported) is the highest among the seven industries. Further, the second highest value is observed for housing and construction, another industry severly hit by the recession. To show the general development of the impact of world shocks, Figures 3a and 3b exhibit the integration parameters, νi, for the seven industry portfolios. The results for the leverage portfolios can be seen in Table 5. The local information set is significant at the 0.1 per cent level for all portfolios, and the global set at least at the five per cent level for only three of the ten portfolios. The world shock, measured by Wald 3, is highly significant. The impact of the 1993 dummy on expected return is
17
generally positive, but significant only for two portfolios. Contrary to the industry portfolios its impact on volatility is inconclusive. The recession dummy in the variance equation is positive and significant for seven of the ten portfolios. If there are volatility asymmetries and the leverage hypothesis is the explanation, the asymmetry parameter, γ i,1, for the leverage portfolios should be increasingly positive for portfolios based on more levered firms. However, while the parameters generally are positive, only one of them is significant. There do not seem to be any patterns when observing more levered firms.
Panel B of Table 5 shows the specification diagnostics. While autocorrelation caused some model misspecification for the industry portfolios, it is not a problem for the leverage portfolios, especially if measured by the Box–Pierce statistics. The GMM based diagnostics show some misspecification. However, the Engle and Ng (1993) asymmetry bias tests are well-behaving. And although some of the covariance bias tests are significant, they show no systematic covariance misspecification. The mean of the variance ratios, shown in Panel C of Table 5, range from 0.0143 for leverage portfolio 2 to 0.0518 for portfolio 7. The mean of the correlation to the world ranges from 0.0646 for portfolio 3 to 0.1914 for portfolio 7. The corresponding values for the integration to the world are 0.0741 to 0.2828 for the same portfolios. The maximum values for VRi, ρiw, and νi (not reported) are about 0.67 (usually less than 0.50), 0.82, and 1.1, respectively.
Finally, Table 6 presents the results for the ten size portfolios. The results are similar to the two previous portfolio sets: the local information set is always significant, while it is significant only in three occasions for the world set. The 1993 liberalization dummy in the mean equation is positively significant for three of the ten size portfolios. The impact in terms of the variance equation is more inconclusive. The recession dummy is positively significant for six portfolios of ten. Although none of the asymmetry parameters is significant, there seems to be a weak tendency of them to increase with size. The GMM specification diagnostics in Panel B of Table 6 are similar to the statistics for the industry portfolios and the leverage portfolios. Only few of the bias and covariance bias statistics are significant. Some residual autocorrelation is still left unmodeled. Some of the model misspecification features might be explained by the fact that the world index is constructed on the basis of calendar time, not trading time. This induces some non-contemporaneousity in the data. The most interesting feature
18
of the size portfolios can be seen in Panel C of Table 6. Except for size portfolios 5 and 6 the mean of the variance ratios, the correlations to the world, and the integration parameters increase monotonically with size. Thus, the larger the size, the more pronounced are world factors. This is logical since large firms usually are more internationally oriented than smaller ones. The ci, αi, and βi parameters in the variance equation (3b) (not reported) are as expected always very significant for all types of portfolios. However, while the Finnish market model includes a positive and significant asymmetry parameter, it is usually not significant for the individual portfolios. Further, the asymmetry parameter does not show any regular patterns even for the leverage portfolios.7 We cannot thus find evidence for the leverage hypothesis.
5 Summary and conclusion In this paper we have analyzed Finnish stock market volatility and the impact of world factors on volatility. The model is an asymmetric GARCH(1,1) model according to Glosten et al. (1993), and Bekaert and Harvey (1997). We use the daily returns for Finnish leverage, industry, and size portfolios. The sample period is January 2, 1987 to December 30, 1998.
The asymmetric part of Finnish volatility is surprisingly non-significant. Since this is the case even for the leverage portfolios, our data are not able to verify the leverage hypothesis. Our setting allows the contemporaneous world shock to enter the domestic models. This is modeled as the S-shaped cumulative distribution function of the logistic distribution. Although the world information set usually is non-significant, the contemporaneous world shock is significant, thus indicating that the Finnish stock markets to some degree are linked to the world. The tie has become stronger in the post-1993 period. 1993 was the year when the Finnish financial markets were totally opened up for foreign investors. Also the volatility process faced an upward shift in 1993. This year was thus an important year, firstly, due to the liberalization of the financial markets, and, secondly, due to the recovery of the Finnish economy. The Finnish economic recession had an even more pronounced effect on the volatility
7
We conducted some estimations using also the normal distribution. In these estimations the asymmetry parameter was positive and significant much more often than for the generalized error distribution. In all other respects the results were comparable.
19
process than the liberalization in 1993. The recession dummy is mostly positive, and quite often also significant. This is in line with for example Schwert (1989), who found that stock market volatility is higher during recessions than during other times.
Since global information usually is not that important, the expected returns seem to follow a segmented or at least partially segmented conditional asset pricing model. On the
other
hand,
the
domestic
shock
is
significantly
decomposed
into
a
contemporaneous world shock and a purely idiosyncratic part. Its impact grows significantly with time. The impact of contemporaneous world shocks is always larger post-1993 than pre-1993. Besides the integration parameters, also the volatility ratios, and correlations to the world increase when moving from the first subperiod to the second subperiod.
20
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Table 1. Descriptive statistics for Finnish stock market excess returns. Daily data January 2, 1987–December 30, 1998 Mean (%) World return Finnish market return
0.0069 0.0296
Std. dev.
0.0103 0.0126
Skewness
Excess kurtosis
Wald
Q
ρ$ 1
ρ$ 2
0.682 –0.492
28.008** 10.831**
4.23 8.51*
178.13** 45.66**
0.0367* –0.0189 0.1546* –0.0177
12.8 91.4
–0.620 –0.971 –0.481 0.531 –0.345 –0.184 –0.189 –1.342** –1.027 –0.904
62.869** 12.545* 13.627** 26.55** 4.62** 7.39** 4.483** 19.712 33.198 20.983**
77.65** 6.71* 33.52** 14.50** 27.28** 31.26** 29.15** 8.03* 0.74 39.54**
165.56** 19.64** 284.54** 190.09** 53.43** 68.07** 55.02** 82.76** 178.22** 245.28**
–0.2345*
0.0534* –0.1208* 0.0293 0.0010 0.0671* 0.0103 0.0783* 0.0059 0.0924* 0.0273 0.0327 –0.0718* 0.1010* –0.0280
24.1 72.1 18.0 20.5 27.4 44.2 53.1 51.2
–0.803 0.105 –0.469 0.107 –0.551 –0.798 –0.751
17.755** 13.009** 7.244** 5.623** 7.260** 9.975** 11.243**
28.91** 20.88** 28.40** 15.07** 17.55** 34.82** 13.88**
173.78** 110.27** 70.55** 26.64** 41.42** 138.45** 46.30**
0.1343* –0.0574* 0.1090* –0.0212 0.0128 0.0269 0.0881* –0.0475* –0.1070* 0.0043 0.0860* 0.0025 0.1539* 0.0471*
15.5 40.4 10.4 42.3 40.9 37.1 86.9
Panel A. Leverage portfolios. Leverage 1 (lowest) Leverage 2 Leverage 3 Leverage 4 Leverage 5 Leverage 6 Leverage 7 Leverage 8 Leverage 9 Leverage 10 (highest)
0.0192 0.0133 0.0126 0.0327 –0.0040 –0.0087 –0.0078 –0.0198 –0.0458 –0.0565
0.0207 0.0135 0.0159 0.0207 0.0158 0.0130 0.0158 0.0155 0.0200 0.0238
0.0283 –0.0697*
0.1168* 0.0078 0.0476*
244.9
Panel B. Industry portfolios. Banking and finance Forestry Trade and transport Metal and electronics Food Housing and constr. Multi-business
–0.0547 –0.0013 –0.0129 0.0415 –0.0033 –0.0642 0.0173
0.0210 0.0144 0.0127 0.0154 0.0162 0.0179 0.0154
Table 1 (cont.) Mean (%)
Std. dev.
Skewness
Excess kurtosis
Wald
Q
ρ$ 1
1.37 4.14 3.15 17.32** 1.74 15.08** 32.54** 26.28** 14.39** 14.62**
446.05** 91.07** 138.94** 60.24** 219.20** 90.60** 82.20** 67.78** 33.59** 6.00**
–0.0664*
ρ$ 2
Panel C. Size portfolios. Size 1 (smallest) Size 2 Size 3 Size 4 Size 5 Size 6 Size 7 Size 8 Size 9 Size 10 (largest)
0.0127 –0.0434 0.0065 –0.0529 0.0014 0.0132 –0.0230 0.0037 –0.0291 0.0280
0.0207 0.0155 0.0140 0.0140 0.0148 0.0143 0.0141 0.0139 0.0132 0.0157
–4.598 0.113 –1.089 –0.969* –3.146 –0.475 –0.553* –0.590 –0.586 –0.652
103.671 17.287* 22.268 11.448** 59.652 12.788** 7.154** 7.23** 8.020** 14.087**
0.0707* 0.0164 0.0199 0.0154 0.0768* –0.0170 0.0600* –0.0110 0.0196 –0.0211 0.0644* 0.0858* 0.0549* 0.0974* 0.0237 0.1585* –0.0388* 0.0999* –0.0455*
39.5
41.5 28.6 22.4 18.5 48.1 44.3 93.4 42.9
* Statistically significant at the 5 per cent level. ** Statistically significant at the 1 per cent level. Notes: The mean, standard deviation, skewness, and excess kurtosis are obtained from a univariate GMM system as in equ Q is the overidentifying GMM test for normality, distributed asχ2(2). ρ$ is the sample autocorrelation coefficient.| ρˆ |> 1/ 2T
zero. BP(s) and BP2(s) are the Box− Pierce tests for autocorrelation up to orders for the variable itself and its square, respective
Table 2. Descriptive statistics for information variables. Daily data January 2, 1987–December 30, 1998. Mean (×100)
Wald
$τ ADF, τ (lags)
13.318** 6.177** 72.933**
8.40* 23.59** 88.40**
–19.38** (9) –26.38** (4)
–0.1442 0.0350 –16.56** (10) 0.118
–0.507 8.941 4.298
10.952** 426.06** 81.505
7.61* 37.05** 2.00
–16.95** (8)
0.1538
–19.50** (10)
–0.0637
–18.77** (7)
–0.0934
dTS
dL1
Rm
dSTS
dFIMUSD
d93
1 –0.0118 –0.0107 –0.0205 0.0018 –0.0307
1 –0.0264 0.0111 0.0149 0.0261
1 –0.1024 0.1314 0.0568
1 –0.1678 –0.0001
1 –0.0055
Std. dev.
Skewness
Excess kurtosis
ρ
World information, Xt Change in US default spread d( DS) Change in US term spread (dTS) Change in one-month Libor rate dL1) (
–0.0150 –0.0366 –0.0705
0.0215 0.0646 0.0812
0.0638 –0.0013 0.0018
0.0126 0.2949 0.0077
0.049 0.536* 0.970
Local information, X tl Finnish equity market return R ( m) Change in short term spread d ( STS) Change in exchange rate d( FIMUSD) Correlation coefficients Change in US default spread d( DS) Change in US term spread (dTS) Change in one-month Libor rate dL1) ( Finnish equity market return R ( m) Change in short-term spread d ( STS) Change in exchange rate d( FIMUSD) Liberalization dummy (d93)
dDS 1 0.0185 0.0002 –0.0497 –0.0104 0.0485 0.0125
1
* Statistically significant at the 5 per cent level. ** Statistically significant at the 1 per cent level. Notes: The mean, standard deviation, skewness, and kurtosis are obtained from a univariate GMM system as in equation
2 normality, distributed as χ (2). ADF is the Augmented Dickey–Fuller test for unit roots.ρ$ is the sample autocorrelation c significantly different from zero. BP(s) and BP2(s) are the Box− Pierce tests for autocorrelation up to orders for the variable itself
26
Table 3. Characteristics of the world model. Daily data January 2, 1987–December 30, 1998. Panel A. Specification diagnostics for the standardized residuals. Mean test 8.800
Variance test 2.775
Moment test 9.920*
Joint test 31.332**
Sign bias test 0.254
Negative size –0.282
Positive size –0.233
Joint bias test 2.727
Skewness
Kurtosis
ρ$ 1
ρ$ 2
0.0437*
0.0128
1.477 BP(5) 9.31
29.150 BP2(5) 0.39
BP(10) 18.68*
BP2(10) 0.77
Panel B. Estimation results. Wald test for world information, χ2(3) p-value
ˆw Asymmetry parameter, γ Standard error Metric for asymmetry ˆ Scale parameter, φ w Standard error
59.811** [ 1/ 2T is regarded as significantly different from zero. BP(s) and BP2(s) are the Box− Pierce tests for autocorrelation up to order s for the standardized residual and its square, respectively. p-values are in brackets, robust standard errors in parentheses.
27
Table 4. Characteristics of the the Finnish market return and the industry portfolios. Daily data January 2, 1987–December 30, 1998. Fi. market
Banking and finance
Forestry
Trade and Metal and Food transport electronics
Housing and constr.
Multibusiness
Panel A. Model estimates. Wald 1
246.21** [
