Risk-free rate effects on conditional variances and conditional correlations of stock returns

Journal of Empirical Finance 25 (2014) 95–111

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Journal of Empirical Finance journal homepage: www.elsevier.com/locate/jempfin

Risk-free rate effects on conditional variances and conditional correlations of stock returns Alessandro Palandri Warwick Business School, University of Warwick, Coventry CV4 7AL, United Kingdom

a r t i c l e

i n f o

Article history: Received 17 July 2012 Received in revised form 27 August 2013 Accepted 4 December 2013 Available online 12 December 2013 JEL classification: G10 G19 C50

a b s t r a c t This paper investigates whether the risk-free rate may explain the movements observed in the conditional second moments of asset returns. Original results are derived, within the C-CAPM framework, that attest the existence of a channel connecting these seemingly unrelated quantities. The empirical results, involving 165 time series of stock returns quoted at the NYSE, show that the risk-free rate does contain information that is relevant in predicting the 165 conditional variances and 13,530 conditional correlations. These findings are particularly pronounced at lower frequencies where the persistence of the conditional second moments is significantly weaker. © 2013 Elsevier B.V. All rights reserved.

Keywords: Conditional variance Conditional correlations Interest rate Capital asset pricing model Sequential conditional correlations

1. Introduction Models of the conditional second moments of stock returns successfully fit and predict the underlying unobservable variables by making use of the information contained in the corresponding time series. While these models accurately describe and replicate the phenomena, they do not provide further understanding of the latent mechanism and the possible determinants behind it. Specifically, whether time varying volatilities and correlations are, to some extent, driven by observable economic variables. Various studies have investigated the link between the conditional variance of financial returns and exogenous explanatory variables. Factors that may potentially influence stock volatility have been considered by Schwert (1989). In particular, with respect to the linkages between financial and macro volatility “the puzzle highlighted by the results [..] is that stock volatility is not more closely related to other measures of economic volatility.” Nonetheless, economic recessions are found to be the primary factor that drives fluctuations in volatility, a finding that is consistent with a later study by Hamilton and Lin (1996). Economic recessions aside, King et al. (1994) find that “…only a small proportion of the covariances between national stock markets and their time-variation can be accounted for by observable economic variables. Changes in correlations between markets are driven primarily by movements in unobservable variables.” Engle and Rangel (2008), separately model fast- and slow-moving components of equity volatilities and find that the latter “…is greater when the macroeconomic factors of GDP, inflation and short-term interest rates are more volatile or when inflation is high and output growth is low.” Other studies have considered the effects of news announcements on volatility. Cutler et al. (1990) find that contemporaneous news events explain only a fraction of volatility ex post. With respect to scheduled macroeconomic news announcements on E-mail address: [email protected]. 0927-5398/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jempfin.2013.12.002

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interest rate and foreign futures markets, Ederington and Lee (1993) observe that “…volatility remains substantially higher than normal for roughly fifteen minutes and slightly elevated for several hours.” Similarly, in Balduzzi et al. (2001) it is found that “public news can explain a substantial fraction of price volatility in the aftermath of announcements…” and that “…volatility increase[s] immediately after the announcements and persist[s] for up to 60 min after the announcements.” Fleming and Remolona (1999) distinguish between a first and a second stage after the release of a major macroeconomic announcement and find that, in the latter, the announcement induces price volatility persistence. Using intra-daily data, Andersen and Bollerslev (1998) “… conclude that these effects (release of public information and, in particular, certain macroeconomic announcements) are secondary when explaining overall volatility.” This paper investigates whether the risk-free rate may explain the movements observed in the conditional variances and correlations of stock returns. A possible explanation for the connection between these seemingly unrelated quantities is put forward within the Consumption Capital Asset Pricing Model (C-CAPM) framework. The results are derived under the unique assumption that the product of the relative risk aversion coefficient and the marginal utility is a monotonic function of consumption. A logical connection is shown to exist between the conditional expectation and the conditional variance of the stochastic discount factor, henceforth SDF. Through the well established inverse relationship between the risk-free rate and the expected value of the discount factor, it is then possible to characterize the SDF's conditional variance as related to the risk-free rate. An analysis of the degree of association between asset returns conditional second moments and the conditional variance of the discount factor completes the description of the channel linking risk-free rate variations to movements of variances and correlations. Under the stated assumption, no restriction is imposed on the sign of the response of the conditional moments to changes in the risk-free rate. However, it is possible to identify the (exogenous) conditions for which these quantities will move in the same or in the opposite direction. The approach followed for the parametrization of the conditional variance and covariance matrix as a function of the risk-free rate is that of the Sequential Conditional Correlations (SCC) methodology introduced by Palandri (2009) which ensures positive definiteness of the conditional correlation matrix under the necessary and sufficient condition that each correlation and partial correlation is bounded between plus and minus one.1 The associated bivariate Autoregressive Conditional Correlation (ACC) model, based on the Fisher transformation of the correlations, satisfies the required bounds by construction and can therefore effortlessly accommodate the inclusion of exogenous variables. The empirical study is based on 165 time series of stock returns quoted at the NYSE, involving 165 conditional variances and 13,530 conditional correlations. A previous study by Glosten et al. (1993) found that short term interest rates positively predict stock market volatility. However, using a large number of stocks observed for more than 40 years, this paper finds that the relationship between risk-free rate and assets' volatility is nonlinear. Specifically, at low frequencies more than three quarters of the stocks considered exhibit volatilities that are concave functions (here approximated by negative quadratic functions) of the risk-free rate. Similarly, the sign of the risk-free rate effects on the conditional correlations is found to differ for different pairs of assets. Furthermore, the results highlight the relevance of the informational content of the risk-free rate at low frequencies. Key to the success of the purely dynamic autoregressive models is the persistence in the conditional volatilities and correlations. However, over longer horizons such persistence gradually vanishes until the variances and correlations of consecutive observations are essentially uncorrelated as implied by stationarity. Accordingly, the findings in this paper feature the evanescence of the purely dynamic GARCH and GARCH-like specifications accompanied by the simultaneous rise of the risk-free rate to relevant predictor. The paper is organized as follows. Section 2 presents the theoretical analysis leading to the identification of the relationship between the risk-free rate and the conditional second moments of assets' returns. The data set is described in Section 3 followed by the description of the econometric model specifying the conditional second moments as functions of exogenous variables in Section 4. Details on the estimation are contained in Section 5 and the empirical results are discussed in Section 6. Section 7 concludes. Appendix A and B contain proofs and derivations of results needed in the Section 2. A preliminary analysis of the relation between the risk-free rate and the conditional second moments of the SDF in the case of Epstein and Zin (1989) and Weil (1989) recursive preferences may be found in Appendix C. Appendix D lays out the conditions for uniform ergodicity of the Autoregressive Conditional Correlations model. 2. Information content of the risk-free rate This section tries to identify the theoretical connections between the risk-free rate and the conditional variances and correlations of risky assets within the C-CAPM framework. The argument made is quite simple: assets loading on the SDF will exhibit conditional variances and correlations that respond to changes in the conditional variance of the SDF itself. Under the fairly general assumptions (i) and (ii) below, conditional mean and variance of the SDF respond monotonically to changes in the consumption process. Hence, movements in the risk-free rate, which is the conditional mean of the SDF, proxy movements in the conditional variance of the SDF which is a component of the conditional second moments of the risky returns. It must be 1 The specification of a Multivariate GARCH model suited for the inclusion of exogenous variables brings about a series of complications due to the necessity of preserving positive definiteness of the conditional variance–covariance matrix. As a result, in most specifications, the vector of exogenous determinants enters the dynamic equations through some quadratic form. Clearly, this amounts to the imposition of arbitrary restrictions on the functional form and the set of parameters.

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emphasized that no causality relation between the risk-free rate and the quantities of interest is being suggested. The only causality is between the SDF and all the other quantities. The risk-free rate is the observable conditional mean of the SDF and as such is potentially informative about its conditional variance and therefore the moments of the assets loading on the SDF. 2.1. Stochastic discount factor and risk-free rate Consider a representative agent with preferences described by a differentiable utility function u with u′ N 0 and u″ b 0. The asset pricing restrictions for the return Ri,t + 1 satisfy: h i Et mtþ1 Ri;tþ1 ¼ 1 ð1Þ ′

where Et is the time t conditional expectation, mtþ1 ¼ βuuðcðc Þ Þ is the SDF, ct + 1 next period's consumption and β the inter-temporal discount factor.2 The risk-free return is Rf,t + 1 with certainty and may therefore be pulled out of the conditional expectation in Eq. (1) from which it follows that the product of the expected value of the SDF and Rf,t + 1 equals one: changes in Et ½mtþ1  imply opposite movements of the risk-free rate. Furthermore, since the first is not an observable quantity, its levels and variations may be inferred from observations of the latter. Eq. (1) does not seem to suggest any other immediate relationship between Rf,t + 1 and higher order conditional moments of the random variable mt + 1. To unravel possible links between first and second order moments of the SDF and therefore the risk-free rate, a simple stylized economy is analyzed. The underlying assumptions are: ′

tþ1 t

(i) The log-consumption process at time t + 1 has a stochastic growth rate with time t conditional mean μt, time t conditional variance σ2t and independent and identically distributed standardized innovations zt + 1 with support in ℝ and density p(z):   ctþ1 ¼ ct  exp μ t þ σ t ztþ1 (ii) The product of the relative risk aversion AðcÞ and the marginal utility u′ is a monotonic function of consumption: i dh ′ AðcÞ  u ðcÞ ¼ vðcÞ dc with either: (iia) þ

vðcÞ≤0; ∀c ∈ ℝ →−

u‴ ðcÞ  c u″ ðcÞ

≥1; ∀c ∈ ℝ

þ

(iib) þ

vðcÞ≥ 0; ∀c ∈ ℝ →−

þ u‴ ðcÞ  c ≤1; ∀c ∈ ℝ u″ ðcÞ

Assumption (i) is a general statement about the dynamic properties of consumption growth rate. Assumption (ii) allows to establish precise relationships between the conditional first two moments of the SDF. For three times differentiable utility functions, the monotonicity assumption (ii) may be interpreted in terms of restrictions on the curvature of the utility function or relative prudence. Specifically, (iia) admits relative risk aversion that is decreasing in consumption, constant and increasing at a rate bounded above by the decrease in the marginal utility. Relative risk aversion increasing at a rate bounded below by the decrease in the marginal utility is consistent with Assumption (iib). Notice that while Assumption (ii) allows to determine the sign of Eqs. (3)–(5), its violation does not necessarily imply that Eqs. (4) and (5) are identically zero and therefore that the variance of the SDF does not depend on the first two moments of the consumption process. For Constant Absolute Risk Aversion (CARA) utility functions the SDF is not scale-invariant with respect to consumption. Nevertheless, for c ≥ α−1, where α is the coefficient of absolute risk aversion, the derivative of AðcÞu′ ðcÞ with respect to consumption is negative, satisfying condition (iia). For CRRA utility functions, the product of the relative risk aversion AðcÞ and the marginal utility u′ is equal to γ ⋅ c−γ and its derivative with respect to consumption is − γ2 ⋅ c−γ − 1, satisfying monotonicity condition (iia). The categorization of the monotonicity behavior applies also to preferences with habit persistence. In particular, in Campbell and Cochrane (1999) the time (t + 1) utility is given by:  1−γ   c −xtþ1 −1 u ctþ1 ; xtþ1 ¼ tþ1 1−γ

2 The representative agent's utility function is introduced to easily generate the sought after relation between the first two conditional moments of the SDF. Even though the chosen class of utility functions may give rise to the risk-free rate and equity puzzles it serves the scope of the paper which is not to correctly price securities for reasonable values of the agent's parameters.

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where xt + 1 is the level of habit. Since in the agent's optimization problem the level of habit xt + 1, whether internal or external, is not a choice variable u′(c) = (c − x)−γ and AðcÞ ¼ γcðc−xÞ−1. Hence, the derivative of AðcÞu′ ðcÞ with respect to consumption is v(c) = − γ(c − x)−γ − 2(γc + x) ≤ 0 which satisfies monotonicity condition (iia). Epstein and Zin (1989) and Weil (1989) recursive preferences allow for the separation between the intertemporal elasticity of substitution and risk aversion. Log-normality of mt + 1, as in Bansal and Yaron (2004), implies that the first two conditional moments of the SDF itself are linked to the risk-free-rate. Monotonicity conditions at (ii) when the log-normality assumption is relaxed are discussed in Appendix C. A comparative statics approach may be used to conduct a qualitative analysis of the effects of variations in the first two conditional moments of consumption's growth rate on the first two conditional moments of the SDF. Let vt and s2t denote the conditional expectation and variance of mt + 1, respectively. Differentiating these moments with respect to consumption's conditional mean μt and standard deviation σt yields: "    # −A ctþ1 u′ ctþ1 ∂νt ¼ β  Et ∂μ t u′ ðct Þ

ð2Þ

" #     −A ctþ1 u′ ctþ1 ∂ν t ¼ β  ℂOVt ; ztþ1 ∂σ t u′ ðct Þ

ð3Þ

"      # 2 −A ctþ1 u′ ctþ1 u′ ctþ1 ∂st 2 ¼ 2β  ℂOVt ; ∂μ t u′ ðct Þ u′ ðct Þ

ð4Þ

"      # −A ctþ1 u′ ctþ1 ztþ1 u′ ctþ1 ∂s2t 2 ¼ 2β  ℂOVt ; ∂σ t u′ ðct Þ u′ ðct Þ

ð5Þ

where ℂOVt is the time t conditional covariance. Signing derivative (2) is straightforward as the random variable inside the expectation only takes negative values: ∂νt/∂μt b 0. Derivatives (3) and (4) may be easily signed recalling that monotonic non-decreasing functions of a random variable exhibit positive covariances as proved in Appendix A. In particular, by assumption (i) consumption is an increasing function of the innovation zt + 1. Together with assumption (ii) this implies that the first term inside the covariances of Eqs. (3) and (4) are either increasing (iia) or decreasing (iib) functions of zt + 1. It follows immediately that either ∂νt/∂σt N 0 (iia) or ∂νt/∂σt b 0 (iib). The second term in Eq. (4) is decreasing in zt + 1 and hence either ∂s2t /∂μt b 0 (iia) or ∂s2t /∂μt N 0 (iib). Similarly, it may be shown that derivative (5) is positive under both assumptions (iia) and (iib). Derivation of this result may be found in Appendix B. From the signs of derivatives (2)–(5) it is possible to determine the directions of the adjustments of the first two conditional moments of the SDF resulting from variations in the conditional mean and variance of consumption's growth rate. Under assumption (i) and (iia), a higher (lower) μt and a lower (higher) σ2t imply a lower (higher) Et ½mtþ1  and Vt ½mtþ1 . Since the risk-free rate moves in the opposite direction of the conditional expectation of mt + 1, it follows that an increase (decrease) in the level of the risk-free rate may be indicative3 of a decrease (increase) in the conditional variance of the SDF. Under assumptions (i) and (iib), instead, higher (lower) μt and σ2t imply lower (higher) Et ½mtþ1  and higher (lower) Vt ½mtþ1 .4 In turn, this implies that an increase (decrease) in the risk-free rate may be indicative of an increase (decrease) in the conditional variance of mt + 1. A summary of the derivatives' sign may be found in Table 7 2.2. Risky assets and stochastic discount factor The next period return on any risky asset i can be decomposed into a part correlated with the SDF and a conditionally orthogonal idiosyncratic component: Ri;tþ1 ¼ ai;t þ bi;t  mtþ1 þ σ i;t  i;tþ1 h i     with Et i;tþ1 ¼ 0, Et 2i;tþ1 ¼ 1 and Et mtþ1  i;tþ1 ¼ 0 by construction. Since nonlinear relations between the risky returns and the SDF may only be approximated by such one-factor model, as a result, the conditional variance σ2i,t of the residuals will in general be a function of the conditional moments of mt + 1. The conditional variance of Ri,t + 1 is equal to: h i 2 2 2 Vt Ri;tþ1 ¼ bi;t  st þ σ i;t

ð6Þ

3 The use of the expression “may be indicative” is due to the fact that simultaneous variations in the conditional mean and variance of consumption's growth rate may offset their effects on the first two moments of the SDF. As a result, the nature of the relation between the conditional mean and the conditional variance of the SDF may change in time. Under these circumstances, variations in the risk-free rate will not be able to predict with certainty the direction of the changes in the variance of the SDF. Nevertheless, Rf,t + 1 may still be used as an imperfect predictor. 4 In general, there are no implications in the direction that variances and correlations should respond in the same way for all traded assets.

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Table 1 Symbols of included stocks. AA BA CHG CTB DUK F GIS HNZ IRF MHP NOC PG RTN TR WAG

ABI BAX CL CVS EAS FE GLW HP JCP MMM OGE PGN SCG TXN WEC

ABT BC CLF CVX ED FL GM HPQ JNJ MO OKE PNW SCX TXT WGL

ADM BCO CMS CW EDE FMC GMT HRS K MOT OLN PPG SJI TY WHR

AEP BGG CNP DD EGN FOE GNI HSC KO MRK PBI PPL SLE UEP WR

AME BMY COP DE EIX FPL GR HSY KR MSB PCG PSD SO UGI WWY

AP BWS CPB DIS EK GAM GT IBM LG MUR PE PVH SP UIS WYE

APD CAT CQB DOV EMR GAP GXP IDA LMT NC PEG R STR USG XEL

AVA CBE CR DOW EQT GCO GY ILA LTR NEM PEO ROH SUN UTX XOM

AVT CCK CSK DPL ETN GD HAL IP MDU NFG PEP ROK TKR UVV XRX

AYE CEG CSL DTE ETR GE HMX IR MEE NL PFE RSH TPL VAR Y

Table 2 Models of the conditional variance. In columns V1–V3 the corresponding models are information criteria compared and the relative percentages reported. Since all three specifications have the same number of parameters, Akaike (AIC) and Schwarz Information Criteria (SIC) provide identical rankings. Columns V1–V4 report SIC-comparison percentages and AIC-comparison percentages, in brackets, when the set of competing models is extended to the nesting specification V4. Column for constant variance model V0 not reported because dominated by other specifications. Freq.

V1

1

100.00%

V2 0.00%

V3 0.00%

5

100.00%

0.00%

0.00%

20

94.54%

3.64%

1.82%

40

77.58%

16.36%

6.06%

80

44.85%

36.97%

18.18%

120

27.88%

46.67%

25.45%

V1

V2

V3

V4

54.55% [18.79%] 26.67% [3.03%] 9.70% [2.42%] 2.43% [0.00%] 1.21% [0.00%] 0.61% [0.00%]

0.00% [0.00%] 0.00% [0.00%] 0.00% [0.00%] 4.24% [2.42%] 13.94% [12.73%] 20.00% [19.39%]

0.00% [0.00%] 0.00% [0.00%] 0.00% [0.00%] 0.00% [0.00%] 1.21% [0.60%] 2.42% [0.61%]

45.45% [81.21%] 73.33% [96.97%] 90.30% [97.58%] 93.33% [97.58%] 83.64% [86.67%] 76.97% [80.00%]

Thus, in general, changes in the first two conditional moments of the SDF may be seen as affecting the conditional variance of a risky asset through three main channels: the factor loading bi,t, the conditional variance s2t of mt + 1 and the conditional variance σ2i,t of the projection residuals. Eq. (6) outlines the transmission mechanism between the moments of mt + 1 and Ri,t + 1 while suggesting that the direction and magnitude of such co-movements need not to be the same across assets. There is no transmission mechanism when Eq. (6) does not depend on the moments of the SDF.5 The conditional correlation ρi,j,t between any two returns i and j is given by:  1=2  1=2 2 2  ρi; j;t ¼ ρi;m;t ρ j;m;t þ 1−ρi;m;t 1−ρ j;m;t ρi; j;t where ρi,j,t is the conditional correlation between the components of the two returns that are orthogonal to the SDF. In general, variations in the first two conditional moments of the SDF translate into adjustments of the conditional correlations of the assets and mt + 1 and therefore of weights being shifted between the standardized SDF and the projection residual. This in turn leads to a realignment of the conditional correlation between assets i and j. At the same time, the risk-free rate will also respond to the movements in vt and s2t and react accordingly. For the classic one-factor model with constant factor loading and σ2i,t that does not depend on the conditional moments of mt + 1 it is possible to establish very precisely that an increase in the conditional variance of the SDF produces an increase in the 5

Specifically, this is the case either when the conditional variance of the idiosyncratic term adjusts to offset the effects of mt + 1:    2 2 1=2 Ri;tþ1 ¼ ai;t þ bi;t  mtþ1 þ Vt Ri;tþ1 −bi;t st i;tþ1 when the factor loading adjusts:   

 2 1=2 mtþ1 −ν t þ σ i;t  i;tþ1 Ri;tþ1 ¼ ai;t  Vt Ri;tþ1 −σ i;t st or when a combination of both effects takes place.

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Table 3 Interest rate effects in V2. Summary statistics of the functional form linking interest rates and conditional variances in V2. Legend: positive quadratic curve (∪), negative quadratic curve (∩), increasing convex curve (⌟), increasing concave curve(⌜), decreasing convex curve (⌞) and decreasing concave curve (⌝). Last column reports the number of times V2 is the Schwarz Information Criteria preferred model in the comparison V1–V4. Freq.

∪

∩

⌟

⌜

⌞

⌝

Total

1 5 20 40 80 120

0.00% 0.00% 0.00% 28.57% 8.70% 12.12%

0.00% 0.00% 0.00% 71.43% 69.57% 78.79%

0.00% 0.00% 0.00% 0.00% 0.00% 3.03%

0.00% 0.00% 0.00% 0.00% 21.73% 6.06%

0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

0 0 0 7 23 33

Table 4 Interest Rate Effects in V4. Summary statistics of the functional form linking interest rates and conditional variances in V4. Legend: positive quadratic curve (∪), negative quadratic curve (∩), increasing convex curve (⌟), increasing concave curve (⌜), decreasing convex curve (⌞) and decreasing concave curve (⌝). Last column reports the number of times V4 is the Schwarz Information Criteria preferred model in the comparison V1–V4. Freq.

∪

∩

⌟

⌜

⌞

⌝

Total

1 5 20 40 80 120

21.33% 21.49% 6.04% 5.19% 5.80% 8.66%

48.00% 57.02% 79.87% 85.07% 88.41% 80.32%

16.00% 9.92% 4.03% 2.60% 2.17% 1.57%

14.67% 11.57% 6.71% 5.19% 0.72% 4.72%

0.00% 0.00% 2.01% 1.95% 1.45% 3.94%

0.00% 0.00% 1.34% 0.00% 1.45% 0.79%

75 121 149 154 138 127

conditional variance of all risky assets. In good times, the risk-free rate is high (pro-cyclical), the variance of the SDF is low (iia) and so is the variance of stock returns. Counter-cyclical variance of stock returns is documented in Schwert (1989), Brandt and Kang (2004) and Mele (2007). Tables 3 and 4 show a clear lack of support in the data for this specification. For individual stock, the counter-cyclicality of variance is rejected: at low frequencies more than 85% of the stocks considered exhibit variances that are negative quadratic functions of the risk-free rate. Furthermore, different assets respond differently to risk-free rate variations. Nonetheless, the risk-free rate is found to be a good proxy for the variance of the unobservable SDF and a relevant predictor of future variance. The derivative of the conditional correlation between assets i and j with respect to the conditional variance of mt + 1 is proportional to ρi,j,t. Since assets that pay a positive risk-premium have negative factor loadings (bi b 0, bj b 0) they are positively correlated (ρi,j,t N 0) and therefore an increase (decrease) in the conditional variance of the SDF results in an increase (decrease) in the correlation between all assets that pay a positive premium. This effect is due to the fact that an increase (decrease) in s2t is the same as an increase (decrease) in the magnitude of the loadings on the standardized factor. The immediate consequence is that these assets will exhibit conditional correlations that move in the same direction of s2t and therefore in the opposite direction of the risk-free rate under (i) + (iia) and in the same direction under (i) + (iib).

3. Data Ideally, the time dimension should be long enough to make the potential findings robust to spurious local relations and the cross section large enough to ensure that the results do not pertain to few assets alone. In practice, there is a trade-off between the two dimensions as increasingly fewer assets have been quoted for increasingly longer periods of time. Focusing the analysis on the 165 companies listed in the New York Stock Exchange directory (www.nyse.com) which have been quoted continuously6 from January 1962 through December 2006 seems a reasonable compromise between the two dimensions.7 The list of ticker symbols may be found in Table 1. Source of daily closing prices and factors to adjust for splits is the CRSP database. The returns characteristics have been studied for six different frequency levels beginning with the daily frequency which comprises 11,327 time series observations per asset. Lower frequency returns have been constructed by aggregating the daily returns over 5, 20, 40, 80 and 120 days corresponding roughly to 1 week, 1, 2, 4 and 6 months respectively. For each frequency, all

6 The number of continuously quoted companies over the time period is 167. However, two have been excluded from this study due to corrupted price listings in the CRSP database at the time of the download. Corrupted data files regard Exelon Corporation (EXC) and Ameren Corporation (AEE) which did not cover the whole sample period. Moreover, EXC contained fragments of price listings of Peco Energy Co. (PE) while AEE contained fragments of Union Electric Co. (UEP). 7 While focusing on assets that have been quoted for long periods of time may distort analyses of their returns due to survivorship bias, it is not evident how to relate, if at all, such bias to returns' conditional second moments. In any case, as with all analyses based on a subset of traded stocks over a specific time interval, broader conclusions about asset returns in general should be drawn cum grano salis.

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Table 5 Models of the Conditional Correlations. PANEL A reports the number of parameters and log-likelihood values of the corresponding specifications C1–C3 for the whole conditional correlation matrix when all conditional variances are filtered by specification V1. Results when all conditional variances are filtered by specification V2 and V3 are reported in Panel B and Panel C, respectively. s indicates the SIC preferred specification and a the AIC preferred. Column for constant correlation model C0 not reported because dominated by other specifications. C1

C2

C3

Freq.

Par.

Log-Lik

Par.

Log-Lik

Par.

Log-Lik

Panel A: V1 1 5 20 40 80 120

14,002 14,540 16,368 16,784 17,230 17,490

−709,720a,s −565,270s −343,848 −172,894 36,522 216,039

14,046 15,622 21,724 25,096 27,430 28,210

−713,428 −561,507 −286,726 −59,987 210,482s 415,177a,s

14,020 15,660 22,408 25,802 27,830 28,626

−714,046 −560,781a −282,330a,s −54,449a,s 260,590a 393,193

Panel B: V2 1 5 20 40 80 120

13,940 14,736 16,430 16,964 17,438 17,698

−687,104 −551,778 −332,470 −163,768 36,417 207,453

14,312 16,248 22,232 25,300 27,208 28,116

−684,692a −542,060 −270,711 −46,213s 218,968a,s 423,935a,s

14,264 16,194 22,562 25,936 27,802 28,264

−684,899s −539,873a,s −267,486a,s −43,864a 208,927 387,125

Panel C: V3 1 5 20 40 80 120

13,954 14,702 16,456 17,026 17,528 17,740

−688,028 −552,499 −332,324 −159,991 52,147 229,517

14,286 16,332 22,246 25,188 27,422 28,170

−685,520a,s −541,600 −271,390 −46,057 225,525a,s 430,025a,s

14,248 16,212 22,594 25,730 27,876 28,696

−686,030 −540,186a,s −268,624a,s −41,133a,s 224,679 415,328

possible aggregation schemes have been considered, resulting in 5 sub-samples of 2264 observations, 20 sub-samples of 565 observations, 40 sub-samples of 282 observations, 80 sub-samples of 140 observations and 120 sub-samples of 93 observations. The index a = 1, …, A is used to refer to a particular sub-sample with A being the aggregation level: 1, 5, 20, 40, 80 or 120 days. In Table 6 Models of the Conditional Correlations. For each asset the best SIC variance specification is selected among V0–V4. For each element of the conditional correlation matrix the best SIC specification is selected among C0–C4. The top panel reports the number of parameters, log-likelihood values, SIC-comparison percentages and AIC-comparison percentages, in brackets. The bottom panel columns reports the SIC- and AIC-comparison percentages relative to the time-varying elements only of the correlation matrix. Freq.

Par.

Log-Lik

C0

C1

C2

C3

C4

1

14,377

−706,583

5

17,070

−550,275

20

27,181

−244,842

40

32,419

−1,873

80

36,261

260,981

120

37,048

424,251

96.87% [85.28%] 87.25% [60.85%] 54.71% [36.96%] 42.14% [30.10%] 34.62% [24.72%] 33.07% [24.38%]

1.68% [4.17%] 3.07% [9.04%] 7.58% [12.23%] 9.56% [13.27%] 10.67% [14.38%] 11.11% [14.36%]

0.95% [6.80%] 5.26% [16.68%] 17.74% [20.71%] 20.01% [20.09%] 20.90% [20.07%] 21.38% [19.23%]

0.49% [3.39%] 4.20% [10.89%] 16.53% [17.44%] 20.33% [16.76%] 21.40% [15.67%] 21.12% [15.15%]

0.01% [0.36%] 0.22% [2.54%] 3.44% [12.66%] 7.96% [19.78%] 12.41% [25.16%] 13.32% [26.88%]

1

–

5

–

20

–

40

–

80

–

120

–

53.79% [28.31%] 24.06% [23.09%] 16.74% [19.40%] 16.53% [18.98%] 16.32% [19.10%] 16.60% [18.99%]

30.33% [46.19%] 41.28% [42.60%] 39.16% [32.85%] 34.58% [28.74%] 31.97% [26.66%] 31.95% [25.42%]

15.64% [23.04%] 32.93% [27.82%] 36.50% [27.66%] 35.14% [23.98%] 32.73% [20.82%] 31.56% [20.04%]

0.24% [2.46%] 1.74% [6.49%] 7.60% [20.09%] 13.75% [28.30%] 18.98% [33.42%] 19.89% [35.55%]

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Table 7 Signs of the derivatives. (i) dν t dν t ds2t

dμ t .

(i) + (ii)

dσ t

dμ . t 2

dst

(i) + (iia)

(i) + (iib)

N0 b0

b0 N0

b0

N0

dσ t

each sub-sample a the non-overlapping aggregate time interval has been normalized to unity so that a time index may be defined as t = 1, …, T where T is the floor of (11,327 − A + 1)/A. As for the corresponding time intervals, returns in each of the sub-samples do not overlap. The cross-sectional dimension is spanned using the index i = 1, …, 165 and the returns are therefore denoted by ri,a,t. The 1-Year Treasury Constant Maturity Rate is used as a proxy for the risk-free rate. Observations are at the daily frequency over the same 45-year period. Instead of aggregating to generate low frequency returns, only the most recent observation within each time interval is selected. In particular, the time t, a-th aggregate return of asset i over A days ri,a,t ≡ (ri,a + A(t − 1) + ⋯ + ri,a + At − 1) is associated to the risk-free rate rf,a + At − 1. Thus, for the given aggregate risky return, lags of the constant maturity rate have been defined to be rf,a,t − 1 ≡ rf,a + A(t − 1) − 1 at one lag, rf,a,t − 2 ≡ rf,a + A(t − 2) − 1 at two lags, etc. In this way, rather than the cumulated interest rate, only the most recent and available level of rf will be used as a covariate, consistently with the argument that it is the most informative about the current conditional moments of the SDF. 3.1. Polynomial regressions If unaccounted, risk-free rate effects on the conditional mean of ri,a,t would be carried over to the conditional second moments and erroneously detected in conditional variances and conditional correlations. Thus, polynomial filtrations are carried out to remove potential risk-free rate effects on the assets' conditional means. This is the sole purpose of the polynomial regressions and no attempt is being made at predicting future stock returns. Specifically, for each aggregation level, a base of orthogonal polynomials xp,a,t is constructed from the constant maturity rate: x0;a;t ¼ 1  1 ^ x x1;a;t ¼ r f ;a;t −λ 1;0 0;a;t  2 ^ x −λ ^ x ¼ r −λ f ;a;t

2;a;t

2;0 0;a;t

2;1 x1;a;t

⋮

−1  P PX ^ x λ xP;a;t ¼ r f ;a;t − P;p p;a;t p¼0 p−1 ^ P;p are ordinary least By definition, xp,a,t is the projection residual of (rf,a,t)p on {xl,a,t}l = 0 , x0,a,t − 1 is equal to unity and the λ squares coefficients. Risky returns are then projected on such space by means of ordinary least squares:

r i;a;t ¼

P X

ci;p;a xp;a;t−1 þ i;a;t

ð7Þ

p¼0

The optimal degree P of the polynomial is determined on the basis of the Schwarz Information Criterion (SIC) for each of the 165 assets. Section 4 describes the econometric models for the conditional variances and correlations used as filters of the returns' conditional variance–covariance matrix. 4. Multivariate modeling Let M denote the cross-sectional dimension of 165 time series of stock returns and a,t be the M-dimensional vector of residuals i,a,t of the polynomial filtrations at Eq. (7): a,t′ ≡ (1,a,t,2,a,t, …,M,a,t). Furthermore, let Ha,t be the (M × M) conditional variance–covariance matrix of the time t M-dimensional vector of residuals a,t of a-aggregated asset returns. Bollerslev (1990) and Engle (2002) have exploited the possibility of separating the conditional variances from the conditional correlations: Ha;t ¼ Da;t Ra;t Da;t where Da,t is the (M × M) diagonal matrix of time-varying standard deviations and Ra,t is the (M × M) matrix of conditional correlations. The Sequential Conditional Correlations methodology allows to further decompose the conditional correlation matrix

A. Palandri / Journal of Empirical Finance 25 (2014) 95–111

103

into its constituting components: the K-matrices. In particular, it is shown how any correlation matrix may be expressed as the product of a sequence of K-matrices and that the product of any sequence of K-matrices (with |ρ| b 1) is a correlation matrix8: Ha;t ¼ Da;t

M−1

M

∏ ∏ K i; j;a;t

i¼1 j¼iþ1

!

M−1

!′

M

∏ ∏ K i; j;a;t

i¼1 j¼iþ1

Da;t

The Ki,j,a,t matrices are lower triangular with generic element [row, col] given by: 8 ρ > < i; j;a;t 1=2 2 Ki; j;a;t ½row; col ¼ 1−ρi; j;a;t > : I½row; col

if row ¼ j and col ¼ i if row ¼ j and col ¼ j otherwise

where I is the identity matrix and the element ρi,j,a,t is the time t correlation (i = 1) or partial correlation (i N 1) between the time t returns of assets i and j. Thus, SCC eliminates the dimensionality problem inherent to MGARCH models by making it possible to separately model and estimate the elements ρi,j,a,t without violating positive definiteness and without imposing parametric restrictions. This is achieved thanks to the resulting block-triangular structure of the moment conditions for which one- and multi-step estimations coincide both numerically and asymptotically.9 4.1. Models of the conditional variances In order to model possible relations between the risk-free rate and the conditional variances of risky assets, the following univariate models have been evaluated and compared: V0 :

hi;a;t ¼ ωi;a

V1 :

hi;a;t ¼ ωi;a þ βi;a hi;a;t−1 þ α i;a i;a;t−1

2

V2 : ln hi;a;t ¼ ωi;a þ γ i;a;1 x1;a;t−1 þ γ i;a;2 x2;a;t−1 V3 : ln hi;a;t ¼ ωi;a þ βi;a ln hi;a;t−1 þ γ i;a;1 x1;a;t−1 8 > < hi;a;t ¼ ν i;a;t  si;a;t ln ν i;a;t ¼ ωi;a þ δi;a ln νi;a;t−1 þ γi;a;1 x1;a;t−1 þ γ i;a;2 x2;a;t−1 V4 :   > 2 : s i;a;t ¼ 1−β i;a −α i;a þ β i;a si;a;t−1 þ α i;a i;a;t−1 =ν i;a;t−1 V0 describes constant conditional variances while specification V1 is the GARCH(1,1) of Bollerslev (1986).10 Here, the conditional 2 variance hi,a,t of asset i, in the a-th aggregation scheme and at time t is a function of the squared residual i,a,t − 1 of the same asset, same aggregation and at time t − 1. V2 and V3 are functionally similar to the GARCH specification but with a crucial difference: the lagged realizations are replaced by functions of the risk-free rate. In V2, the relationship between the risk-free rate and asset i log-variance is approximated by a second degree polynomial. x1,a,t − 1 refers to the unaggregated and demeaned risk-free rate observable just prior to time t in the a-th aggregation. Similarly, x2,a,t − 1 refers to the component of the squared risk-free rate that is orthogonal to a constant and x1,a,t − 1. In V3, instead, the asset's conditional log-variance is a linear function of the demeaned risk-free rate, augmented by a GARCH-type memory parameter. The magnitude of β will reflect the degree of persistence in the volatility as opposed to a specification like V2 in which any persistence in the variance may arise solely from that of its covariates. Furthermore, the memory parameter in V3 allows shocks to the risk-free rate to affect the conditional variances over time as opposed to V2 where such effects are exhausted in a single period. Evaluating specifications V2 and V3, which do not contain GARCH effects, against V1, which does, will provide insight on the relative strength of the risk-free rate and GARCH effects at different frequencies. Finally, V4 includes both GARCH and risk-free rate effects in a specification that nests V0–V3. The rationale for the multiplicative form of V4 is to allow risk-free rate and GARCH components of volatility to have different dynamics and therefore levels of persistence. Comparing specifications V1 and V4 makes it possible to assess whether there is any information content in the risk-free rate beyond that contained in the squared returns. 8 The K-matrix decomposition is unique for given R, however a permutation of the order of the series implies a permutation of the elements of R, leading to a different sequence of K-matrices. Following Palandri (2009), in this study the 165 assets have been ordered in decreasing order of total square correlation: Euclidean norms of the columns of the unconditional correlation matrix. 9 SCC methodology does not deal with the inferential curse of dimensionality which, for large cross-sectional dimensions, occurs also for constant correlations specifications as discussed in Palandri (2009). 10 The GARCH literature has produced a vast number of alternative functional forms as well as specifications that may accommodate threshold and asymmetric effects. However, it is beyond the scope of the paper to select the GARCH specification that best fits the individual series. The specification of Bollerslev has been adopted because it is relatively easy to estimate and it has not been shown to be consistently outperformed by other GARCH specifications. Furthermore, while threshold and asymmetric effects are relevant in equity markets at high frequencies (daily and lower), to keep the analysis simple they have not been included in this study. Notice that in any case these effects are not the main volatility component and given their relevance at the high frequencies they would not change the empirical results which already favor the GARCH specification rather than those based on the risk-free rate.

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4.2. Models of the conditional correlations Within the SCC methodology, the conditional correlation matrix is decomposed into its constituting elements: conditional correlations and partial correlations. It is convenient, in this setting, to parametrize the Fisher transformation χi,j,a,t of the conditional correlation ρi,j,a,t between assets i and j:

χ i; j;a;t ¼

1 þ ρi; j;a;t 1 ln 1−ρi; j;a;t 2

!

Mapping the interval (− 1, 1) into (−∞, + ∞), this transformation allows χi,j,a,t to take any value on the real line. Thus, there will be no need to bound the process by imposing constraints on the model's parameters. These quantities are modeled by the bivariate specifications C0–C4 mimicking V0–V4 of the conditional variances: C0 : χ i; j;a;t C1 : χ i; j;a;t C2 : χ i; j;a;t C3 : χ i; j;a;t C4 : χ i; j;a;t

¼ ωi; j;a ¼ ωi; j;a þ θi; j;a ψi; j;a;t−1 ¼ ωi; j;a þ γ i; j;a;1 x1;a;t−1 þ γ i; j;a;2 x2;a;t−1 ¼ ωi; j;a þ δi; j;a χ i; j;a;t−1 þ γ i; j;a;1 x1;a;t−1 ¼ ωi; j;a þ δi; j;a χ i; j;a;t−1 þ θi; j;a ψi; j;a;t−1 þ γi; j;a;1 x1;a;t−1 þ γ i; j;a;2 x2;a;t−1

In C0, the conditional correlation is constant. C1 is the Autoregressive Conditional Correlations (ACC) model11 where the ψi,j,a, t − 1 is the Fisher transformation of the time (t − 1) realized correlation ϕi,j,a,t − 1: ! 1 þ ϕi; j;a;t 1 ψi; j;a;t ¼ ln 1−ϕi; j;a;t 2 Q i; j;a;t ½1; 2 ϕi; j;a;t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q i; j;a;t ½1; 1  Q i; j;a;t ½2; 2  Q i; j;a;t ¼ α i; j;a Q i; j;a;t−1 þ 1−α i; j;a i; j;a;t ′i; j;a;t

where i,j,a,t is the bidimensional vector containing i,a,t and j,a,t, Qi,j,a,t is the (2 × 2) covariance matrix, ϕi,j,a,t is the corresponding correlation and ψi,j,a,t its Fisher transformation.12 C2 and C3 are the transposition to the modeling of the conditional correlations of V2 and V3 respectively. Evaluating both specifications, for which correlations do not depend on past realizations, against C1 will allow to discriminate between the relative strength of the risk-free rate and GARCH-type effects. Similarly to V2 and V3, shocks to the risk-free rate have an instantaneous effect on the conditional correlations in C2 while the same effect spreads over time in C3. Specification C4 includes both autoregressive and risk-free rate effects in a parametrization that nests C0–C3. Since for correlations there is no immediate multiplicative decomposition, the various components of C4 enter through a more parsimonious additive functional form.

5. Estimation The parameters of the models for the conditional variances and conditional correlations are estimated by GMM based on the scores of Gaussian likelihood functions. In particular, there are M scores pertaining to the variance models and M(M − 1)/2 scores pertaining to the correlation models. Standard GMM estimates are obtained by setting all moment conditions to zero and simultaneously solving for the whole set of parameters. However, given the block-triangular structure of the moment conditions, this is the same as solving for one set of moments at the time. Nevertheless, the theoretical convenience of standard GMM asymptotics does not solve the practical issues related to the computation of the parameters' variance–covariance matrix. In particular, the estimated variance of the moments as the outer-product average of the moment conditions is not full rank for large cross sections M (N T1/2). This feature is inherent to all MGARCH models and for large M shows even in constant correlations models. 11 The ACC model has been proposed, together with the SCC methodology, in Palandri (2009) and exhibited substantial improvements with respect to the standard Dynamic Conditional Correlations. The ACC(0,1) specification in C1 is the correlations' counterpart of the GARCH(1,1) for the conditional variances. The similarities between these two functional forms are sketched in Appendix D.1. 12 Uniform ergodicity conditions are derived in Appendix D.2.

A. Palandri / Journal of Empirical Finance 25 (2014) 95–111

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5.1. Score of the variances Let i be the (A ⋅ T × 1) vector of residuals from the polynomial regression of asset i: ′i = (i,1′,i,2′, …,i,A′) with i,a′ = (i,a,1,i,a,2, …,i,a,T). Then, the concentrated Gaussian log-likelihood Lc is: ′

−1

Lc ¼ −lnjΩi j−i Ωi i where Ωi is the conditional variance–covariance matrix: 0

Ωi;1;2

Ωi;1;1 B Ω′ B Ωi ¼ @ i;1;2 ⋮

…

′

1

… Ωi;2;A C C ⋱ ⋮ A … Ωi;A;A

Ωi;2;2 ′

Ωi;1;A

Ωi;1;A

Ωi;2;A

The Ωi,a,a are diagonal matrices of time-varying variances hi,a,t while the Ωi,a,l matrices, with a ≠ l, are bidiagonal and contain the time-varying covariances between residuals overlapping across aggregation schemes. These covariances arise as a byproduct of the different aggregation schemes and, with respect to this study, do not contain any useful information. Hence, the matrix Ωi is replaced with the misspecified block diagonal matrix Ω∗i which disregards the artificially induced covariances. Hence, the log-likelihood becomes:    ′   −1 i Lc ¼ −lnΩi −i Ωi A h  i X  ′ −1 ¼− lnΩi;a;a  þ i;a Ωi;a;a i;a a¼1 A X T h X

¼−

2

−1

lnhi;a;t þ i;a;t hi;a;t

i

a¼1 t¼1

5.2. Score of the correlations −1/2 ′Ω−1/2 ′Ω−1/2 ⋅ i,a,t is Let η′i ≡ (i,1′Ω−1/2 i,1,1 , i,2 i,2,2 , …, i,A i,A,A ) be the (A ⋅ T × 1) vector of standardized residuals. Specifically, ηi,a,t = hi,a,t homoscedastic with unit variance for any aggregation level a. Letting ηi,j′ = (η′,η i ′), j the joint concentrated Gaussian log-likelihood is:

    ′ −1 Lc ¼ −lnQ i; j −ηi; j Q i; j ηi; j where Qi,j is the conditional variance–covariance matrix: Q i; j ¼

Q i; j;1;1

Q i; j;1;2

Q i; j;1;2

Q i; j;2;2

′

!

All elements on the main diagonals of Qi,j,1,1 and Qi,j,2,2 are equal to unity. The off-diagonal elements contain time-varying correlations between different aggregation schemes that are of no particular interest as they are artificially induced by the aggregation schemes themselves. The main diagonal of Qi,j,1,2 contains the contemporaneous conditional correlations on which this part of the analysis focuses while the off-diagonal elements of Qi,j,1,2 contain lead and lag correlations induced by the data aggregation. Thus, Qi,j is replaced with the misspecified matrix Q∗i,j containing only the quantities of interest while ignoring all the artificial features:



Q i; j ¼





I

Q i; j;1;2 I

Q i; j;1;2

!

where Q∗i,j,1,2 is the diagonal matrix of the conditional correlations ρi,j,a,t and its inverse is equal to: h



Q i; j

i−1

1;1

¼

1;2

Q i; j

Q i; j

Q i; j

Q i; j

1;2

2;2

!

0

 −1 2 I−Q i; j;1;2 B  −1 ¼@ 2 −Q i; j;1;2 I−Q i; j;1;2

 −1 1 2 −Q i; j;1;2 I−Q i; j;1;2 C  −1 A 2 I−Q i; j;1;2

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A. Palandri / Journal of Empirical Finance 25 (2014) 95–111

The concentrated log-likelihood is then equal to:   h i   ′  −1 ηi; j Lc ¼ −lnQ i; j −ηi; j Q i; j       ′ 1;1 ′ 1;2 ′ 2;2 ¼ −lnI−Q i; j;1;2  Q i; j;1;2 −ηi Q i; j ηi −2ηi Q i; j η j −η j Q i; j η j  2 A X T X η j;a;t −ρi; j;a;t ηi;a;t  A T  2 2 2 ¼ −ln ∏ ∏ 1−ρi; j;a;t − 1−ρi; j;a;t þ ηi;a;t a¼1t¼1

a¼1 t¼1

2  2 3 A X T  η −ρ η X i; j;a;t j;a;t i;a;t 2 6 7 ¼− 4ln 1−ρi; j;a;t þ 5 1−ρ2i; j;a;t a¼1 t¼1

The sum of the η2i,a,t terms, on the third line, is dropped from the concentrated log-likelihood as it does not depend on ρi,j,a,t. Since a certain number of partial correlations may not be time-varying, it would be more appropriate to treat them as constant rather than impose a dynamic specification to all the elements of Rt. In order to do this, each model is compared to a constant correlation model in terms of the Schwarz Information Criteria (SIC) and the best specification is selected. This results in parsimonious specifications that still may explain very well the variations in the conditional correlations.

6. Results 6.1. Conditional variances Models V0–V3 have been fitted to the projection residuals of the 165 asset returns on the risk-free rate polynomial. For each asset and a given aggregation level, the models were ranked according to the SIC and the percentage of times that each model ranked first have been reported in Table 2. Rankings with respect to the Akaike Information Criteria (AIC) are also reported. Nevertheless, throughout the paper the analysis of the results is based on the SIC as it is a more conservative criterion than the AIC. Results for the constant variance model V0 have not been reported because dominated by those of the alternative specifications V1–V4. At the daily and weekly level, the GARCH effects prevail and for none of the assets included in the sample the conditional variance can be better explained in terms of risk-free rate movements alone. Nevertheless, at the 1 and 2-month frequencies, while still overshadowed by GARCH effects, the risk-free rate effects gradually begin appearing. For the lowest frequencies considered, four and six months, the aggregation reduced the idiosyncratic noise and the associated GARCH effects. At the same time it allowed the relationship between the assets' volatilities and the risk-free rate to emerge in all its strength. What the numbers further suggest is that on average both GARCH and risk-free rate effects are present but while the first dominate at higher frequencies the second are stronger at lower frequencies mainly due to the idiosyncratic noise reduction resulting from the time aggregation of the returns. The first three columns of Table 2 draw a rough outline of which effect is the main component of the time varying variances at a given frequency. Introducing model V4 (which nests V1–V3) in the comparison, allows to determine whether the variance component explained by the risk-free rate is only a low-frequency phenomena or whether it is also relevant at higher frequencies. For daily and weekly data, a model that includes both effects is preferred to the pure GARCH V1 for nearly half and three quarters of the assets respectively. At the intermediate frequencies of 1 and 2 months, V4 is the preferred model for more than 90% of the assets. When the frequencies are lowered to 4 and 6 months and the GARCH effects mitigate, for 15% to 20% of the assets a purely risk-free rate based conditional variance model is preferred. Hence, both GARCH and risk-free rate induced conditional variance components appear to be present in the 165 time series of asset returns considered. The sign of the risk-free rate effects on the conditional variances are summarized in Tables 3 and 4 for models V2 and V4 respectively. For the observed range, the functional form linking the conditional variance and the risk-free rate is approximated by a positive quadratic curve (∪), a negative quadratic curve (∩), an increasing convex curve (⌟), an increasing concave curve (⌜), a decreasing convex curve (⌞) and a decreasing concave curve (⌝). The negative quadratic curve (∩) is the functional form that overall best approximates the response of the conditional variances to the risk-free rates. In particular, it is the shape taken by the polynomial in V2 for no less than two thirds of the assets. The polynomial in V4 reproduces such shape for approximately half of the assets at high frequencies and no less than three quarters of the assets at lower frequencies. Hence, for most of the returns considered, the conditional variances are lower, ceteris paribus, either when the risk-free rate is low or when it is high. It should be noted that the location of the vertices of the quadratic approximations differ across assets. Therefore, an increase in the level of the risk-free rate may result in a higher conditional variance for some assets while lower for others.13 13 The robustness of these findings has been confirmed by running kernel regressions of (i,a,t)2 on the risk-free rate at the low frequency of 120 days. Despite the inability of conditioning, and therefore controlling, for the previous period variance hi,a,t − 1 and realizations i,a,t − 1, the ∩ relation, as a whole or in parts, arises for roughly 60% of the 165 assets considered.

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6.2. Conditional correlations Table 5 reports the estimation results for the conditional correlation matrices of the 165 assets when the variances are modeled by V1, V2 and V3. For any given variance specification, the data is standardized accordingly and the correlation models C0–C3 estimated. Results for the constant correlation model C0 have not been reported because dominated by those of the alternative specifications C1–C4. The preferred SIC and AIC specifications are then reported for each frequency considered. At the daily frequency, the overall preferred specification of the conditional correlations is quite sensitive to the choice of conditional variances' specification. However, at all other frequencies the correlations' results are quite robust to the choice of the model for the conditional variances. In particular, at medium frequencies (5–40 days) C3 is the preferred specification and that at low frequencies (80–120 days) conditional correlations are best described by C2. Thus, the second degree polynomial of C2 is generally a better description of the time-varying correlations at low frequencies while the memory parameter of C3 helps to better fit the data at medium frequencies (5–40 days). Regardless of the variances' specification, at low frequencies the risk-free rate is better at describing the evolution of the correlations than the purely dynamic GARCH-like model C1. Therefore, there is a clear indication that the risk-free rate contains relevant information which is not contained in the realized correlations and which is informative of the evolution of the time-varying correlations, especially at low frequencies. Table 6 reports the results of the model obtained selecting the best variance specification for each asset among V0–V4 and the best specification for each pair of conditional correlations and partial correlations among C0–C4, in terms of the SIC. The so obtained model is preferred to the constant correlation model C0 at any of the considered frequencies. It should be noted that the overall number of parameters of the preferred model increases as the frequency of the data decreases.14 Given that for each of the estimated models C0–C4 the number of parameters is fixed, it follows that at lower frequencies there are less correlations and partial correlations that are best described by the constant correlation specification C0. Hence, there are dynamic and risk-free rate effects in the correlations that may be detected at low but not at high frequencies. For the specification determined by the SIC, the distributions of the selected conditional variance and correlation models are reported in Tables 2 and 6 respectively. Column C0 confirms that the number of partial correlations that are best modeled as constant decreases with the frequency. In particular, it is somewhat striking that while at the daily frequency more than 95% of the partial correlations are constant,15 at the 120-day frequency only 33% of them are constant.16 The relative distribution of the selected models that allow for time-varying correlations is reported in the lower panel of Table 6. At the daily frequency, more than half of the elements of the conditional correlation matrix are best described by a purely dynamic model and less than 1% of such elements exhibit both dynamic and risk-free rate effects as seen from column C4. At the weekly frequency, the dynamic specification C1 is preferred in roughly 25% of the cases while the most general specification C4 is preferred in no more than 2% of the cases. As the frequencies are lowered further, the number of elements best described by C1 stabilizes around 17%. Conversely, C4 is increasingly selected as the best model at lower frequencies. At 80 and 120 days more elements of the conditional correlation matrix exhibit both dynamic and risk-free rate effects than just dynamic effects. The purely exogenous models C2 and C3, in which the risk-free rate is the sole determinant of the level of the correlations, are each selected as the best specifications in roughly 30% of the cases. On the other hand, C2 exhibits a peak of preferences in the 5–20 day range, while C1 exhibits a significant low of 15.64% at the daily frequency. Relative to the time varying components, specifications C2–C4 which include risk-free rate effects are preferred to C1 in terms of the SIC in more than 83% of the cases at frequencies of 20–120 days. At the higher frequencies of 1 and 5 days, C2–C4 are selected in 46.21% and 75.94% of the cases, respectively. 7. Conclusions This paper extrapolates a theoretical connection between the risk-free rate and the first two conditional moments of the SDF and, in turn, with the conditional second moments of stock returns. The ACC specification is extended to include exogenous covariates in a way that is consistent with the natural bounds of the correlations. Within the SCC methodology it easily parametrizes positive definite correlation matrices. The empirical analysis, conducted by modeling the 165 variances and 13530 correlations in the data set, confirms that the risk-free rate has explanatory power with respect to the conditional second moments. Specifically, it is found that at low frequencies GARCH and GARCH-like effects in the conditional second moments tend to vanish (consequence of stationarity). At the same time the risk-free rate is found to contain relevant

14 The number of parameters is found to be increasing as the frequency of the data decreases also for the various combinations of variances and correlations specifications of Table 5. 15 This does not mean that 95% of the elements of the conditional correlation matrix do not vary with time. For example, if all correlations on the first row of the correlation matrix are time varying, then all elements of the matrix will be time varying even if all partial correlations are constant. This would correspond to a 98.78% of constant correlations and partial correlations. Furthermore, the term “constant” should be taken cum grano salis: correlations and partial correlations whose dynamics are either too noisy or don't behave like C1–C4. Thus, given the alternatives, modeling by a constant may be the best statistical description. 16 These results highlight the inappropriateness of traditional MGARCH models that constrain elements of the conditional correlation matrix to share common dynamics. In particular, the same functional form would apply to the time varying and the constant correlations. For the time varying components, this results in biased parameters' estimates and biased dynamics and forecasts. For the constant correlations, it results in time varying fitted values and forecasts which are solely driven by noise.

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predictive information. In particular, risk-free rate effects are found to be present at all frequencies and to dominate the standard purely dynamic effects at the low-end of the considered spectrum. Differently from previous studies, the extensive analysis in this paper indicates that interest rate changes have different effects for different assets. Nonetheless, a substantial commonality, best approximated by a negative quadratic curve, has been detected in the response of the conditional variances to the risk-free rate. The inclusion of additional macroeconomic explanatory variables and further evaluations of the explanatory power of the risk-free rate are left as an area for future research. Appendix A. Covariance

Theorem A.1. Let X be a continuous random variable and f, g be two non-constant, monotonic non-decreasing functions of X. Then:

ℂOV½ f ðX Þ; g ðX Þ N0 ∘

∘

∘

Proof of Theorem A.1. Define f ¼ f −Ef and g ¼ g−Eg. Since f, g are increasing in X, ∃ ! x1 and x2 such that f ðx1 Þ ¼ 0 and ∘

∘

g ðx2 Þ ¼ 0. Without loss of generality, assume x1 ≤ x2 and let A = (−∞, x1], B = (x1,x2] and C = (x2, + ∞). Then, f ðxÞ≤0 ∀x ∘

∘

∘

∈ A, f ðxÞN0 ∀x ∈ B∪C , g ðxÞ ≤0 ∀x ∈ A∪B and g ðxÞN0 ∀x ∈ C . Rewriting the covariance between f, g as the expectation over disjoint sets:

∘ 

∘ 

∘  ∘ ∘ ∘ ℂOV½ f ðX Þ; g ðX Þ ¼ EA f g þ EB f g þ EC f g

∘  ∘ h∘i ∘ h∘i ∘ NEA f g þ f ðx2 ÞEB g þ f ðx2 ÞEC g

∘  ∘ n h ∘ io ∘ NEA f g þ f ðx2 Þ 0−EA g N0

∘  ∘ ∘ ∘ The last step is obtained by noticing that in the set hA:ithe random variable f g is non-negative and therefore EA f g N0 and the ∘ ∘ random variable g is non-positive and therefore −EA g N0. Appendix B. Monotonicity results Derivative (5) may be signed by letting z⁎ be the value of the innovation zt + 1 such that: h  i     ′       ′  A c z u c z z ¼ Et A ctþ1 u ctþ1 ztþ1

ð8Þ

The right hand side of Eq. (8) is either negative (iia) or positive (iib): notice the opposite sign with respect to the derivative in Eq. (3). Since the coefficient of relative risk aversion and the marginal utility are positive, Eq. (8) implies that z∗ b 0 (iia) or z∗ N 0 (iib). Let's consider the case in which (iia) holds and rewrite the expected value of the deviations from the mean as the sum of three integrals over disjoint subsets: Z

z −∞

Z

0 z

Z



h

h  i ′ ′ Au z−Et Au z pðzÞdzþ

 i ′ ′ Au z−Et Au z pðzÞdzþ

þ∞ h 0

 i ′ ′ Au z−Et Au z pðzÞdz ¼ 0

The derivative of Au′ z with respect to z is equal to:  ′ dAu′ z d Au ′  z þ Au ¼ dz dz

ð9Þ ð10Þ ð11Þ

ð12Þ

A. Palandri / Journal of Empirical Finance 25 (2014) 95–111

109

The product Au′ is positive and by assumptions (i) and (iia) is decreasing    in z. Therefore for any z ∈ (−∞, 0) the derivative in Eq. (12) is positive. Hence, ∀ z ∈ (−∞, z∗) the quantity Au′ z−Et Au′ z is negative and so is integral (9). For z ∈ (z ∗,0) the    sign of the derivative in Eq. (12) is still positive, making the integrand Au′ z−Et Au′ z as well as integral (10) positive.   Finally, the integral in Eq. (11) is positive as both terms of the integrand Au′ z and Et Au′ z are positive. Collecting positive terms: Z

þ∞ h z

Z  i ′ ′ Au z−Et Au z pðzÞdz ¼ −

h

z −∞

 i ′ ′ Au z−Et Au z pðzÞdz

Since u′ is decreasing in z, if multiplied by the integrands of Eq. (13) it yields: Z

þ∞ z



Z

Z h  i ′ ′ ′ u Au z−Et Au zÞ pðzÞdz b−

z −∞

ð13Þ

h  i ′ ′ ′ u Au z−Et Au z pðzÞdz

h

ℝ

 i ′ ′ ′ u Au z−Et Au z pðzÞdz b 0⇒ h i ′ ′ ℂOVt u ; Au z b 0⇒ ∂s2t N0 ∂σ t

It may be shown, mutatis mutandis, that the sign of derivative (5) is positive even under assumption (iib). Appendix C. Recursive utility (θ − 1) With Epstein and Zin (1989) and Weil (1989) recursive preferences the stochastic discount factor is given by mt + 1 = δθG−θ/ψ t + 1Ra,t + 1 where Gt + 1 is the gross growth rate of consumption, Ra,t + 1 is the gross return on an asset paying aggregate consumption, δ ∈ (0,1) is the time discount factor, θ = (1 − γ)(1 − 1/ψ)−1 with γ N 0 the risk-aversion coefficient and ψ N 0 the intertemporal elasticity of substitution. The return on the wealth portfolio may be defined as Ra,t + 1 = Wt + 1/(Wt − Ct) where wealth Wt is defined as the present discounted value of the consumption stream or recursively as W t ¼ C t þ Et ½mtþ1 W tþ1 . Absorbing the terms of the stochastic discount factor that do not depend on Ct + 1 in the constant κ yields: γ−1

−γψ−1

ψ−1 mtþ1 ¼ κ  C ψ−1 tþ1 W tþ1

Monotonicity conditions (ii) transpose to monotonicity conditions on −∂m C tþ1 : ∂C tþ1

tþ1

h i ∂mtþ1 −1 C ¼ B  C tþ1 W tþ1 −A  mtþ1 ∂C tþ1 tþ1   with A ≡ γ−1 and B ≡ ðγ þ AÞ ¼ γψ−1 ψ−1 ψ−1 . Letting v(ηt + 1) be its derivative w.r.t. Ct + 1 and disregarding positive proportionality terms: −

  2 2 v ηtþ1 ∝−Bð1 þ BÞηtþ1 þ Bð1 þ 2AÞηtþ1 −A where ηt + 1 ≡ Ct + 1/Wt + 1. In the case of γ, ψ N 1 as in Bansal and Yaron (2004) the quadratic function v(η) has positive discriminant and thus has two real roots:

η1;2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 2AÞB  B2 þ 4AB2 −4A2 B ¼ 2ð1 þ BÞB

such that η1 N 0 and η2 b 1. Therefore, ∀ η ∈ (0,η1) ∪ (η2,1) v(η) b 0 satisfies monotonicity assumption (iia) while ∀ η ∈ (η1,η2) v(η) N 0 satisfies monotonicity assumption (iib). It should be noticed that recursive utilities do not necessarily imply sign changes for v(η). In fact, for reasonable values of the risk-aversion γ and intertemporal elasticity of substitution ψ the root η1 can be larger than 25%. Hence, for consumption that never exceeds one fourth of the present value of the consumption stream the stochastic discount factor will satisfy monotonicity condition (iia). Similarly, different sets of reasonable values of γ and ψ yield η1 ≈ 0. Thus, for consumption that is not negligible relative to the present value of the consumption stream, the stochastic discount factor will satisfy monotonicity condition (iib).

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Appendix D. Properties of ACC Appendix D.1. Similarities between ACC and GARCH Consider the ACC(1,1) specification applied to the modeling of volatilities rather than correlations. The time t conditional variance ht is then given by: ht ¼ ω þ δht−1 þ θψt−1

ð14Þ

In this setting, the Fisher transformation of ACC may be simply replaced with the identity function: ψt − 1 = ϕt − 1. Furthermore, ϕt − 1 may be set equal to qt − 1, where the latter is the realization of the variance estimated by the exponential smoothing: 2

ψt−1 ¼ αψt−2 þ ð1−α Þt−1

ð15Þ

Substituting Eq. (15) in Eq. (14) yields: 2

ht ¼ ð1−α Þω þ ðα þ δÞht−1 −αδht−2 þ ð1−α Þθt−1 Hence, the ACC(1,1) model is the correlations' analog of the GARCH(1,2). In turn, the ACC(0,1) specification at C1, with δ = 0, is the analog of the GARCH(1,1). Appendix D.2. Ergodicity of ACC Consider the exponential smoothing with parameter α ∈ (0,1) associated to the ACC(1,1) specification: qt ¼ α  qt−1 þ ð1−α Þ  U t  where qt ≡ vech(Qt), U ′t ¼ u2i;t ui;t u j;t Then:

u2j;t



    with uj,t = ρtui,t + (1 − ρ2t )1/2j,t, Et−1 u2i;t ¼ 1, Et−1 2j;t ¼ 1 and Et−1 ui;t  j;t ¼ 0.

0

1 1 @ qt ¼ α  qt−1 þ ð1−α Þ  ρt A þ ð1−α Þ  Z t 1 where Z t ≡ U t −Et−1 ðZ t Þ. Hence, qt is a function of qt − 1 and χt ≡ IF(ρt) where IF(•) denotes the inverse Fisher transformation. In the ACC(1,1), χt is a function of χt − 1 and ψt − 1 = F(ϕt − 1) = F(C(qt − 1)) where f(•) is the Fisher transformation and C(•) is the function that computes the correlation ϕt − 1 from the vector of variances and covariances qt − 1. Jointly, these equalities allow for the following nonlinear Markovian representation: 

qt χt



 ¼f

qt−1 χ t−1



 þ

Zt 0



The skeleton f(•) of the Markov chain has a unique fixed point. Proof. Let q⁎, ρ⁎, ϕ⁎, ψ⁎ and χ⁎ denote the fixed points of qt, ρt, ϕt, ψt and χt respectively. From the exponential smoothing equation, it follows that q∗1 = 1, q∗2 = ρ∗ and q∗3 = 1. Then: ϕ∗ = ρ∗, ψ∗ = F(ϕ∗) = F(ρ∗) = χ∗. Substituted in the ACC(1,1) equation it yields: 





−1

χ ¼ ω þ δχ þ θχ ¼ ωð1−δ−θÞ □

The mapping of the skeleton has the whole Euclidean space as domain of attraction. Proof. Let the index n denote the mapping of the skeleton of the exponential smoothing equation q2,n = αq2,n − 1 + (1 − α)ρn and take the Fisher transformation: h i ψn ¼ F αq2;n−1 þ ð1−α Þρn ¼ F ½αIF ðψn−1 Þ þ ð1−α ÞIF ðχ n Þ

The Fisher transformation is an odd function, convex (concave) for positive (negative) arguments: ψn ≤ ð N ÞαF ½IF ðψn−1 Þ þ ð1−α ÞF ½IF ðχ n Þ≤ ð N Þαψn−1 þ ð1−α Þχ n

ð16Þ

A. Palandri / Journal of Empirical Finance 25 (2014) 95–111

111

Therefore, when ψt is positive (negative) the skeleton is bounded above (below) by ψn = αψn − 1 + (1 − α)χn. Thus, the nonlinear mapping may be replaced by the linear mapping (16) that bounds it. It follows that: χ n ¼ ð1−α Þω þ ½α þ δ þ ð1−α Þθχ n−1 −αδχ n−2 This is the skeleton  of an AR(2) for which the domain of attraction is the whole Euclidean space under the conditions αδ b 1, □ δ + θ b 1, and δ þ 1−α 1þα θ N−1. The ACC(1,1) process is uniformly ergodic. Proof. Being bounded by an AR(2), the ACC(1,1) process automatically satisfies the absorbing requirements of assumptions (A1)–(A5) of Theorem 3.3.2 of Chan and Tong (2001) for uniform ergodicity. □ Conditions for uniform ergodicity may be easily generalized to the ACC(p,q) specification even though they will not be available in analytical form. In particular, the mapping of the skeleton of an ACC(p,q) is bounded by the skeleton of an AR(m) with m = max(p + 1, q) and therefore, the ACC(p,q) is uniformly ergodic when its parameters satisfy the analogous conditions of the associated AR(m). References Andersen, T.G., Bollerslev, T., 1998. Deutsche Mark–Dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies. J. Financ. 53, 219–265. Balduzzi, P., Elton, E., Green, T., 2001. Economic news and bond prices: evidence from US Treasury market. J. Financ. Quant. Anal. 36, 523–543. Bansal, R., Yaron, A., 2004. Risks for the long run: a potential resolution of asset pricing puzzles. J. Financ. 59, 1481–1509. Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. J. Econ. 31, 307–327. Bollerslev, T., 1990. Modeling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Rev. Econ. Stat. 72, 498–595. Brandt, M.W., Kang, Q., 2004. On the relationship between the conditional mean and volatility of stock returns: a latent VAR approach. J. Financ. Econ. 72, 217–257. Campbell, J.Y., Cochrane, J.H., 1999. By force of habit: a consumption-based explanation of aggregate stock market behavior. J. Polit. Econ. 107, 205–251. Chan, K.S., Tong, H., 2001. Chaos: A Statistical Perspective. Series in StatisticsSpringer. Cutler, D., Poterba, J., Summers, L.H., 1990. Speculative dynamics and the role of feedback traders. Am. Econ. Rev. 80, 63–68. Ederington, L.H., Lee, J.H., 1993. How markets process information: news releases and volatility. J. Financ. 48 (4), 1161–1191. Engle, R.F., 2002. Dynamic conditional correlation —A simple class of multivariate GARCH models. J. Bus. Econ. Stat. 17 (5). Engle, R.F., Rangel, J.G., 2008. The spline-GARCH model for low-frequency volatility and its global macroeconomic causes. Rev. Financ. Stud. 21, 1187–1222. Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57, 937–969. Fleming, M., Remolona, E., 1999. Price formation and liquidity in the US treasury market: the response to public information. J. Financ. 54, 1901–1915. Glosten, L.R., Jagannathan, R., Runkle, D.E., 1993. On the relation between the expected value and the volatility of the nominal excess returns on stocks. J. Financ. 47, 1779–1801. Hamilton, J., Lin, G., 1996. Stock market volatility and the business cycle. J. Appl. Econ. 5, 573–593. King, M., Sentana, E., Wadhwani, S., 1994. Volatility and links between national stock markets. Econometrica 62, 901–933. Mele, A., 2007. Asymmetric stock market volatility and the cyclical behavior of expected returns. J. Financ. Econ. 86, 446–478. Palandri, A., 2009. Sequential conditional correlations: inference and evaluation. J. Econ. 153, 122–132. Schwert, G., 1989. Why does stock market volatility change over time? J. Financ. 44, 1115–1153. Weil, P., 1989. The equity premium puzzle and the risk-free rate puzzle. J. Monet. Econ. 24, 401–421.