Federal Reserve Bank of New York Staff Reports

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1 Federal Reserve Bank of New York Saff Repors Time-Varying Consumpion Correlaion and he Dynamics of he Equiy Premium: Evidence from he G-7 Counries Asani Sarkar Lingjia Zhang Saff Repor no. 8 April 2004 This paper presens preliminary findings and is being disribued o economiss and oher ineresed readers solely o simulae discussion and elici commens. The views expressed in he paper are hose of he auhors and are no necessarily reflecive of views a he Federal Reserve Bank of New York or he Federal Reserve Sysem. Any errors or omissions are he responsibiliy of he auhors.

2 Time-Varying Consumpion Correlaion and he Dynamics of he Equiy Premium: Evidence from he G-7 Counries Asani Sarkar and Lingjia Zhang Federal Reserve Bank of New York Saff Repors, no. 8 April 2004 JEL classificaion: G2, G5 Absrac We examine he implicaions of ime variaion in he correlaion beween he equiy premium and nondurable consumpion growh for equiy reurn dynamics in G-7 counries. Using a VAR-GARCH (,) model, we find ha he correlaion increases wih recession indicaors such as above-average unemploymen growh and wih proxies for sock marke wealh. The combined effec is ha he correlaion increases during a recession. We find ha he effec of a counercyclical correlaion is ha he equiy premium, Sharpe raio, and risk aversion are also generally counercyclical. These findings survive several robusness checks such as allowing he mean reurn o depend on is condiional variance and conrolling for lower consumpion volailiy during he pos-990 period. The evidence is sronger for counries ha have larger sock marke capializaion relaive o GDP. Our resuls show he imporance of combining financial and macroeconomic indicaors for explaining ime variaion in he consumpion correlaion and he equiy premium. Sarkar: Research and Marke Analysis Group, Federal Reserve Bank of New York ( asani.sarkar@ny.frb.org). Zhang: Porfolio Managemen Division, Bank of New York ( zhanglingjia@yahoo.com). The views expressed in he paper are hose of he auhors and do no necessarily reflec he posiion of he Federal Reserve Bank of New York or he Federal Reserve Sysem.

3 I is a challenge for economics o explain he ime-series variaion of he mean and volailiy of he equiy premium EP. The mean EP is predicable a long horizons (Fama and French, 988) and increases during recessions, peaking near business cycle roughs (Fama and French, 989; Ferson and Harvey, 99). Explanaions for he cyclical properies of EP include ime-varying risk aversion (Consaninides, 990; Campbell and Cochrane, 999; Barberis e al. 200), consumpion commimens or ransacions coss ha mus be paid o change consumpion (Chey and Szeidel, 2003), and ime-varying correlaion beween durable consumpion growh and reurns (Yogo, 2003). I is harder o explain evidence of counercyclical variaions in he Sharpe raio SR (Brand and Kang, 200), where SR is he condiional mean reurn per uni of is condiional sandard deviaion. Leau and Ludvigson (2003) show ha leading asse-pricing models generae swings in he SR much smaller han wha is visible in he daa. In his paper, using daa for he G7 counries, we examine wheher ime-varying correlaion beween he EP and nondurable consumpion growh (he consumpion correlaion) can accoun for cyclical properies of he EP and SR. While counercyclical variaion in he correlaion can poenially lead o similar variaion in he EP and SR, he empirical evidence remains unclear. Duffee (2004) shows ha he correlaion is high when sock marke wealh is high relaive o consumpion (he composiion effec ) and infers ha he correlaion is procyclical. However, Sanos and Veronesi (2003) show ha, if invesors have dividend and labor income, hen expeced reurns are negaively relaed o he raio of labor income o consumpion. Furher, large and predicable income shocks imply counercyclical correlaion when consumers infrequenly updae or monior spending decisions, as discussed in secion. Finally, he sample correlaion is higher during recessions. For example, in U.S. quarerly daa, he sample Harvey (989) shows ha he condiional covariance is high in January, bu no enough o explain he January

4 2 correlaion is 0.46 (0.24) afer quarers wih above (below) average unemploymen growh. We esimae a VAR-GARCH model ha allows us o simulaneously characerize he correlaion and obain implicaions for he mean and variance of consumpion growh and reurns. The mean equaions follow a Vecor Auo Regression (VAR) process, augmened by exogenous predicion variables such as he erm spread. We characerize he condiional correlaion using a GARCH(,) model. The correlaion varies boh wih changes in he unemploymen rae UG and he composiion effec C (proxied by he raio of sock marke wealh o consumpion MEC, used by Duffee (2004), or he VAR reurn residual RR). Specifically, he correlaion depends on wheher, in he previous period, UG and C were above or below heir sample averages. The means, variances and he correlaion are esimaed joinly, as a sysem. We find ha he consan correlaion model is rejeced for all counries excep France and Ialy. 2 During recessions, C alone causes lower correlaion, consisen wih Duffee (2004), while UG alone causes higher correlaion, consisen wih Sanos and Veronesi (2003). However, plos of he esimaes and regression analysis show ha he combined effec of UG and C is for he correlaion o increase during recessions. The ime variaion in correlaion is large. For example, in U.S. quarerly daa, he correlaion is 0.60 (saisically zero) afer quarers wih above (below) average UG and RR. Nex, we plo he condiional equiy premium implied by ime-varying correlaion, and show ha i increases during recessions. Variaions in he Sharpe raio SR are also counercyclical and large in magniude Finally, he counercyclical correlaion ends o make risk-aversion premium in sock reurns. In cross-secional evidence, Leau and Ludvigson (200b) find ha value socks earn higher average reurns han growh socks because of higher consumpion correlaion during bad imes. 2 Previously, he hypohesis of consan condiional covariance was rejeced in U.S. daa by Schwer and Seguin (989) and Harvey (989), and for U.S., UK, and Japanese daa by Cumby (990).

5 3 (implied by he condiional Euler equaion) procyclical, while he counercyclical SR has he opposie effec. Overall, he risk-aversion generally increases during recessions. Comparing cross-counry, we find ha he evidence favoring counercyclical correlaion, EP, SR and risk-aversion is sronger for counries wih relaively large shares of sock marke capializaion o GDP, such as he U.S. and Canada. In conras, France and Ialy have relaively small sock markes and small variaion in he correlaion, EP and SR. Our resuls survive several robusness checks. We use differen recession proxies such as real GDP growh GG or flucuaions in he aggregae consumpion-wealh raio CAY (Leau and Ludvigson, 200a). The correlaion is greaer when GG is below average or for high values of CAY. Since expeced reurns increase wih CAY, his resul again implies counercyclical correlaion. We also implemen a GARCH-M model where he mean reurn depends on he condiional reurn variance, and we conrol for he decline in consumpion volailiy in he pos- 990 period. In all cases, our basic resuls coninue o hold. Of relaed papers, Duffee (2004) shows ha he correlaion is posiively relaed o MEC and infers ha i is procyclical. In conras, Sanos and Veronesi (2003) show ha he raio of labor income o consumpion LIC is he main (negaive) deerminan of sock reurns. 3 We resolve his confusion by showing ha he correlaion increases wih MEC and unemploymen growh UG bu he laer effec dominaes, so ha he correlaion is counercyclical. Similar o LIC, UG is a direcly observable macro variable raher han an esimaed or a price-scaled variable. Yogo (2003) finds ha durable consumpion growh is high relaive o nondurable consumpion growh during recessions and ha sock reurns decrease (increase) wih durable (nondurable) consumpion growh, implying counercyclical variaion in he equiy premium. In conras, we 3 In unrepored resuls, he auhors sae ha, in a previous version of heir paper, hey regress he condiional

6 4 obain counercyclical variaion in he equiy premium hrough ime varying correlaion of reurns wih nondurable consumpion growh. Unlike Yogo (2003), we relae he correlaion o macro indicaors such as unemploymen, GDP and CAY. Differen from he above papers, we sudy implicaions of ime-varying correlaion for he Sharpe raio and risk-aversion, and exend our analysis o he G7 counries. However, unlike us, Duffee (2004) ess differen models of ime-varying consumpion risk, and Sanos and Veronesi (2003) and Yogo (2003) sudy he cross-secional variaion in he equiy premium. Finally, we are unable o explain he equiy premium puzzle (Mehra and Presco, 985). Our esimaes of he level of covariance imply unreasonably large risk aversion for all counries. Leau and Ludvigson (2003) find ha CAY predics he volailiy of reurns. We find ha CAY is informaive of he correlaion even afer condiioning on unemploymen: boh UG and CAY significanly deermine he correlaion and, furher, ime-variaion in he correlaion is amplified when CAY is included in he mean reurn equaion. More generally, since we combine financial wealh and labor income shocks o explain he risk-reurn radeoff, our approach is complemenary o Leau and Ludvigson s (2003) use of CAY for he same purpose. The remainder of his paper is as follows. In secion, we discuss why he correlaion may vary over ime. Secions 2 and 3 describe he VAR-GARCH framework and he daa, respecively. Secion 4 presens descripive saisics of consumpion growh and sock reurns. Resuls for he U.S. are in secions 5 and 6. In secion 7, we presen resuls for he oher G7 counries. In secion 8, we plo esimaes from our model and illusrae he dynamic properies of he equiy premium, Sharpe raio and risk-aversion. Secion 9 concludes. covariance obained from a GARCH(,) model on LIC and find a significan coefficien.

7 5. Time-varying Consumpion Correlaion: Discussion Consumpion-based asse pricing models predic ha expeced asse reurns should be proporional o he covariance beween reurns and some funcion of real consumpion, such as expeced consumpion growh (Hansen and Singleon, 983). If he covariance is posiive, hen asses have high reurns when consumpion is high (i.e. marginal uiliy is low), and require a higher risk premium. However, he risk-aversion implied by he model is oo high in all G7 counries (Table ), varying beween 50 and 62. An imporan reason for he model s failure is he low (and even negaive) value of he uncondiional correlaion (Cochrane and Hansen, 992). Recen lieraure has focused on he condiional Euler equaion. Given power uiliy and lognormally disribued consumpion growh, he condiional equiy premium is approximaely: E ( EP + ) γσ ( CGRO+ ) σ ( EP + ) ρ ( CGRO+, EP + ) () where EP is he equiy premium, γ is he risk-aversion, ρ is he correlaion, CGRO is he log consumpion growh and E is an expecaion condiional on informaion a ime. Thus, predicable variaion in EP may arise from variaions in γ, ρ or he condiional variances of EP and CGRO. Our approach is o incorporae ime-variaion in ρ and he variances. The condiional and uncondiional correlaion may differ, excep in he special case of i.i.d. disribuion. Furher, he sample correlaion varies subsanially over differen calendar periods and macro condiions for he G7 counries (Table 2). Alhough ime variaion in correlaion is presen in he daa, i may be difficul o inerpre (Cochrane, 200). Neverheless, in secion A, we argue ha he observed behavior of consumers and invesors is consisen wih higher correlaion during bad economic imes. In secion B, we discuss models implying, insead, ha he correlaion may be procyclical. We rea he overall cyclical propery of he correlaion as an empirical issue.

8 6 A. Models ha may imply counercyclical correlaion Below, we discuss some recen models of consumer and invesor behavior. Alhough some models do no address he correlaion direcly, heir resuls have implicaions for he correlaion. Labor income shocks. In Sanos and Veronesi (2003), invesors have boh dividend and labor income. When he share of labor income in consumpion decreases, he covariance beween consumpion growh and dividend growh is higher, which ranslaes o a higher covariance beween reurns and consumpion growh. Inaenive consumer. Gabaix and Laibson (200) argue ha slow updaing of consumpion (e.g. due o decision or aenion allocaion coss) leads o a downward bias in he measured covariance beween consumpion growh and reurns. They show ha he covariance increases wih he frequency of updaing. In addiion, if households adjus consumpion quicker afer large sock reurn shocks, hen he covariance is increasing in he size of reurn shocks. Reis (2003) models consumers coss of informaion acquisiion and finds ha consumers adjus quicker in response o large and predicable shocks. If unemploymen is predicable, hen an implicaion is higher consumpion correlaion during imes of high unemploymen. Absen-minded consumer. Ameriks e al. (2004) find ha many consumers are highly uncerain of heir spending behavior. Their model of he absen-minded consumer implies ha, in a cyclical downurn, hose wih fewer resources and more ime (such as he unemployed) may decide o monior heir spending more closely. This may exacerbae he decline in consumpion, and cause he correlaion o be higher during economic recessions. Less consumpion smoohing during recessions. Borrowing consrains imply ha consumpion smoohing is weaker during recessions han expansions. Zeldes (989) finds ha an inabiliy o borrow agains fuure labor income affecs mos consumpion. Oher mehods of consumpion

9 7 smoohing may also be less effecive during recessions. For example, Sephens (200) repors ha, following he head s unemploymen, he reducion in family income is less han he reducion in he head s earnings, possibly due o increased spousal earnings and ransfers from relaives. Such forms of consumpion smoohing are less likely during recessions when spouses would find i harder o increase heir labor supply. B. Models ha may imply procyclical correlaion In his secion, we describe he composiion effec and ime-varying sock marke paricipaion, and heir implicaions for he correlaion. Composiion effec. When he share of sock marke wealh (SMW) in oal wealh is high, consumpion will be low relaive o SMW and more sensiive o changes in SMW. Thus, Duffee (2004) argues, he correlaion is higher when he share of SMW in oal wealh is above average. Time-varying sock marke paricipaion. Shareholders consumpion is more closely correlaed wih sock reurns han non-shareholder consumpion (Mankiw and Zeldes, 99). Furher, here is large urnover in he se of sock marke paricipans and, among paricipans, large changes in he porfolio shares of equiy over ime (Vissing-Jorgensen, 2002). Thus, if high sock reurns arac increased paricipaion, he correlaion may increase wih reurns. 4 On he oher hand, Polkovnichenko (200) shows ha, wih fixed paricipaion coss, he consumpion correlaion of shareholders endowed wih labor income is lower compared o a model where hey have no labor income. An implicaion is ha he correlaion increases afer a negaive labor income shock. In he following secion, we discuss he saisical model for esimaing condiional momens of consumpion growh and sock reurns. 4 Hong e al. (2004) illusrae ha muliple paricipaion equilibria can occur if equiy invesmen is influenced by social ineracion (i.e. an invesor s cos of paricipaion is decreasing in he number of his peers who also paricipae). Anunovich and Sarkar (2003) provide empirical evidence for paricipaion exernaliy.

10 8 2. Empirical Mehodology In he previous secion, we argued ha ime-varying correlaion is heoreically plausible. Sample evidence, for he G7 counries, also shows ime variaion in he mean and volailiy of reurns and consumpion growh, as well as in he consumpion correlaion (Table 2). A naural way o model hese feaures of he daa is o combine a VAR model for means wih a GARCH model for second momens. We firs describe he VAR-GARCH process and hen elaborae, based on our discussion in secion, on economic facors driving he correlaion. Le R be a vecor of consumpion growh CGRO and he equiy premium EP. We assume ha he mulivariae process governing R is: R = m + e, m E( Ω ) = R (2) e Ω ~ N(0, H ) where Ω is he informaion se a ime -. e is a vecor of innovaions, assumed condiionally normal wih a condiional covariance marix H. Elemens of H are h ij, i j, (off-diagonal erms, or condiional covariance) and h ii (diagonal erms, or condiional variances). We model he means of CGRO and EP as a VAR, augmened wih one-period lagged values of exogenous predicion variables: 2 L P i = αi0 + α jτ R j τ + j= τ = j= R β z + e ij j. i (3) where i= (CGRO), or 2 (EP). L is he order of he VAR, chosen according o various informaion crieria. z j., j=,..,p is he j-h exogenous predicion variable. The VAR model assumes ha consumers can predic consumpion growh and he equiy

11 9 premium. There is weak evidence ha consumpion growh is predicable. Campbell and Mankiw (989) find ha, for he G7 counries, consumpion growh is prediced by is lags and by he erm srucure. Moivaed by habi persisence models, Deaon (987) esimaes a regression of consumpion growh on is lags. 5 Kandel and Sambaugh (990) regress consumpion growh on he dividend yield, he defaul and erm spreads. Evidence in favor of reurn predicabiliy is sronger. For U.S. quarerly reurns, Fama and French (989) documen ha he Baa-Aa corporae bond yield, he S&P 500 dividend yield and he erm spread predic reurns. Whielaw (994) uses similar variables o predic monhly and quarerly U.S. reurns. We use a GARCH(,) model for he condiional variances and covariance of he innovaions. The condiional variance is assumed o be linear in is own lag and he squared pas innovaion. We do no include z j,- because high-dimensional GARCH models are difficul o esimae. Iniially, he condiional correlaion r ij is assumed o be consan (Bollerslev, 990): h i=, 2 (4) ii = ai + bi ei ei + ci hii h = r h h i j (5) ij ij ii jj where bi represens he ARCH effec while ci represens he GARCH effec. Nex, we es wheher r ij varies over ime by inroducing dummy variables in (5): ij S h = ( r D ) h h i j (6) k k= k, ii jj Secion indicaes ha he correlaion increases wih large labor income shocks ha occur during periods of high unemploymen. Such periods may also winess reducions in aggregae oupu measures, such as GDP or indusrial producion. Thus, we hypohesize ha he 5 Campbell and Mankiw (989) omi he firs lag of consumpion growh, due o he ime averaging of consumpion (Breedon e al., 989). Deaon (987), however, includes he firs lag of consumpion growh.

12 0 correlaion depends on one of he following macro facors M: growh in real indusrial producion (IG), he unemploymen rae (UG) or real GDP (GG). Le E(M) be he sample mean of M. Then, for ime, we define dummy variables as follows: D, = if M > E( M ), D2, = if M E( M ), (7) (8) and hey are 0 oherwise. The correlaion also varies wih he composiion effec C, where C={MEC, RR}. MEC is he raio of sock marke wealh o consumpion and RR is he reurn residual e 2, from he VAR. Since consumpion has low variaion, mos of he variaion in MEC is likely o come from sock reurns, so RR may be viewed as anoher proxy for he composiion effec. Alernaively, if he sock marke anicipaes changes in he real economy, RR may pick up macro informaion no conained in M. Le E(C) be he sample mean of C. We define: D3, = if C > E( C), D4, = if C E( C), (9) (0) and hey are 0 oherwise. Finally, Gabaix and Laibson (200) imply ha he correlaion is higher for larger reurn shocks. Suppose a large reurn shock means ha RR is greaer in absolue value han is mean E(RR) plus sandard deviaion σ(rr). Then, we redefine D 3 and D 4 as: D 3, = if RR > E( RR) + σ ( RR), D 4, = if RR E( RR) + σ ( RR), () (2) and hey are 0 oherwise. We combine informaion on macro and reurn condiions o define: D 5, = if D, = and D 3, =, and D 5, =0 oherwise. D 6, = if D, = and D 4, =, and D 6, =0 oherwise. D 7, = if D 2. = and D 3, =, and D 7, =0 oherwise. (3)

13 D 8, = if D 2, = and D 4, =, and D 8, =0 oherwise. As a robusness check, we also esimae a GARCH-in-means or GARCH-M(,) model by including he condiional reurn variance h 22 in he mean reurn equaion: 6 2 L P R2 = α 20 + α j R j τ + β2 jz j. + γh22, + e2 j= τ = j= τ (4) In all cases, we esimae he VAR-GARCH sysem joinly. The condiional log-likelihood funcion for he GARCH(,) process can be expressed as: T / Lik( x) = [ln(2π ) + ln H + e ( H ) e ] (5) 2 = where x is he vecor of all he parameers o be esimaed and T is he sample size. Lik(x) is maximized by he BFGS Quasi-Newon mehod wih a mixed quadraic and cubic line search procedure. We iniialize he condiional variances o heir uncondiional values, and use he Simplex mehod for a few ieraions o sraighen ou he iniial condiions. To es for coefficien resricions, we use a likelihood raio es LR: 2 LR = 2 * ( ULOGL RLOGL) ~ χ ( k ) (6) where ULOGL (RLOGL) is he value of he unresriced (resriced) likelihood funcion and k is he number of resricions. 3. Daa A complee descripion of sources and sample daes is in he daa appendix. The sample sars in he 960s for he U.S., UK and Canada, in 970 for ohers, and ends 2003 Q or Q2. We use monhly daa for he U.S., and quarerly daa for he non-u.s. G7 counries. 6 We hank an anonymous referee for suggesing his o us.

14 2 Consumpion is he sum of seasonally adjused expendiures on nondurables and services if disaggregaed daa is available, and aggregae expendiures oherwise. We obain per capia consumpion expendiures afer dividing by he populaion. 7 Consumpion growh is he log difference in curren and lagged consumpion, 8 while he equiy premium is he oal reurn (capial gains plus dividend yield) minus he local 3-monh Treasury bill rae. We mosly use he MSCI sock index excep for he U.S., UK and Canada, where we use a local sock index o mach he longer hisory of consumpion daa in hose counries. All nominal daa are convered o real erms using he local Consumer Price Index. We use lagged values of DEF, he defaul spread, DIVY, he dividend yield, TERM, he 0- year noe yield minus he 3-monh bill rae 9 and PE, he price-earnings raio, o predic he equiy premium and consumpion growh. For non-u.s. G7 counries, oher han Canada, we define DEF as he corporae bond yield minus he long-erm Treasury yield since separae daa for high-risk and low-risk corporae bonds is unavailable. 0 For Canada, we use he corporae bond yield minus he 3-monh prime corporae paper rae. For U.S. daa, DEF is he Baa-Aa corporae bond yield, and DIVY is he S&P 500 dividend yield. 4. Descripive Saisics for Consumpion Growh and Sock Reurns Campbell (2002) repors a small or negaive correlaion in he quarerly daa for mos counries. He noes ha variaion in he raio of sock marke capializaion o GDP may imply 7 For Germany, we use Wes Germany s populaion before 99 Q and use Unified Germany s populaion afer ha. This is consisen wih sandard pracice of he main daa agencies. 8 Since consumpion daa is ime-averaged, a iming convenion is needed. Our definiion assumes ha consumpion is measured a he end of a period. Alernaively, consumpion growh may be defined relaive o nex period s consumpion, which resuls in higher conemporaneous correlaion wih reurns, especially a quarerly horizons (Campbell, 2002). 9 We use consan mauriy Treasury raes o define TERM for U.S. daa 0 Kandel and Sambaugh (990) use he difference beween Aaa and he Treasury bill raes. We use long-erm raher han shor-erm Treasury raes o mach he corporae bond mauriies more closely.

15 3 similar variaion in he sock marke s share of oal wealh, and in he sock marke claim o oal consumpion. In secion A, we discuss he sample correlaion of consumpion growh and sock reurns for he G7 counries, and relae i o he size of he sock marke. The correlaion may be high condiionally, even when is uncondiional value is low. Thus, in secion B, we compare he correlaion for differen calendar periods and macro condiions. A. Sample disribuion and correlaion of consumpion growh and sock reurns Table shows he disribuion and correlaion (CORR) of consumpion growh CGRO and he equiy premium EP. The U.S. has he highes correlaion (0.24 for monhly and 0.34 for quarerly daa). Canada has double-digi correlaion, bu he oher counries have small or negaive correlaion. The correlaion is weakly associaed wih Mcap and Mgdp. Thus, while he U.S. and Canada have relaively large sock markes and moderae correlaion, Japan and he U.K. have large sock markes and negaive correlaion. The mean annualized CGRO is around 2% for mos counries bu, compared o he U.S., he volailiy of CGRO is higher for he G7 counries. 2 The mean equiy premium is high in all counries relaive o he consumpion volailiy, as indicaed by he high risk-aversion implied by he Euler equaion. B. Correlaion by decade and macroeconomic condiions Earlier, we saw ha he uncondiional consumpion correlaion is low or negaive for many counries. Now, we examine wheher he correlaion is high in paricular calendar periods and Daa issues may influence he correlaion. For example, he consumpion daa for some counries do no separae ou nondurables and services expendiures. Also, he summaion bias in consumpion daa reduces he variance of consumpion changes whereas asse reurns daa do no have his bias. Variaion in he size of his bias across counries may affec he comparabiliy of he correlaions. 2 In paricular, he sandard deviaion of CGRO exceeds 3% for Germany. The high volailiy of consumpion growh in Germany is consisen wih he high variabiliy of is oher real macro variables. For example, he

16 4 macro economic condiions. Specifically, we compue he correlaion for ime condiional on he realizaions of D and D 2 a ime -, where D and D 2 are defined in (7) and (8). Table 2 presens CORR and COV, he correlaion and covariance beween CGRO and EP. There is subsanial calendar ime variaion in CORR, primarily reflecing similar variaion in COV. For example, CORR and COV are relaively high in he 960s for he UK, in he 970s for France and he U.S., and in he 980s for Canada and Germany. Variaions in he volailiy of CGRO and EP also lead o variaion in CORR. In paricular, CORR is relaively high for he U.S. and UK in he pos-990 period, a ime of low consumpion volailiy in hese counries. However, for oher G7 counries, even wih low consumpion volailiy in he pos-990 period, CORR is low or negaive because COV is low or negaive. Turning o macro condiions, for all counries excep France and Ialy, he correlaion is higher in bad imes. For example, in U.S. quarerly daa, CORR is 0.24 (0.46) following quarers when unemploymen growh is high (low). Similarly, he correlaion is relaively high following periods wih low growh in GDP or indusrial producion. Mean reurns are generally low when he economy is in a bad way. However, anicipaing fuure resuls, in he U.S. and Canada he mean reurn is no lower during higher unemploymen periods. In summary, he correlaion varies over ime and macro condiions, and is ypically higher when economic condiions are poor. The volailiy of reurns and consumpion growh also varies over ime. Nex, we esimae condiional momens of consumpion and reurns, and examine how he correlaion varies wih macro and sock marke condiions. 5. VAR-GARCH(,) Resuls for U.S. daa In his secion, we esimae he VAR-GARCH(,) model for U.S. daa. Consumers may variance of real indusrial producion for Germany is muliples of he U.S. variance.

17 5 form beer forecass of mean reurns and consumpion growh by condiioning on variables such as he dividend yield. Table 3 shows ha CGRO and EP are mos correlaed wih lagged values of TERM, while EP also shows moderae correlaion wih lagged DIVY and lagged DEF. To describe ime variaion in he mean CGRO and EP, we discuss he VAR resuls in secion A. In secion B, we discuss resuls from he GARCH(,) model, firs when he correlaion is consan, and hen when i varies wih reurn shocks and macro facors UG, IG and GG. A. VAR Resuls for U.S. daa Table 4 presens resuls from esimaing he VAR model for U.S. daa. Of significan forecasing variables, TERM is posiively relaed o CGRO and EP. Furher, EP is posiively associaed wih DIVY and he price-earnings raio. Turning o he auoregressive par of he model, CGRO is negaively auocorrelaed a lag and posiively auocorrelaed a lag 3. 3 The F-Tess show ha we can rejec he hypohesis ha lagged consumpion growh does no explain variaions in CGRO, consisen wih Flavin (98) and Reis (2003). Lagged consumpion and lagged reurns are no significan in predicing EP in U.S. quarerly daa. Finally, in he monhly daa, lagged reurns are significan in explaining variaions in CGRO and EP. In unrepored resuls, we find ha he mean innovaions and auocorrelaions are nearly zero. A chi-square es canno rejec he null hypohesis of zero auocorrelaion of innovaions for all lags up o eigh. B. VAR-GARCH(,) resuls for U.S. monhly and quarerly daa Table 5 presens esimaes of he condiional variances and correlaion from he VAR- GARCH(,). Panel A shows resuls when he condiional correlaion (CCORR) is resriced o be consan. The ARCH coefficiens for CGRO and EP are significan in boh he monhly and 3 Time-averaging of consumpion daa biases he auocorrelaion upward for U.S. daa, and downward for counries

18 6 quarerly daa. The GARCH coefficiens are significan in U.S. quarerly bu no monhly daa. CCORR is 0.26 for U.S. monhly daa and 0.34 for U.S. quarerly daa and boh esimaes are saisically significan. These esimaes are similar o he sample values (Table ). Nex, we es for ime-variaion in he condiional correlaion. In Panel B of Table 4, for U.S. monhly daa, CCORR varies wih UG, IG and reurn residuals RR. Column UG shows ha he correlaion is 0.33 following monhs when UG is above average (D =) and 0.20 oherwise (D 2 =). Column RR shows ha CCORR is 0.29 afer monhs when RR is above average or posiive (D 3 =) and 0.23 oherwise (D 4 =). In boh cases, he LR es rejecs he consan correlaion model a he 5% level or less. In column UG+RR, we combine informaion on unemploymen and reurns. Condiioning on D =, CORR is 0.35 when RR is posiive (D 5 =) and 0.32 when RR is negaive (D 6 =). Condiioning on D 2 =, CORR is 0.26 when RR is posiive (D 7 =) and 0.5 when RR is negaive (D 8 =). However, wih an LR es (no shown), we canno rejec he hypoheses ha D 5 = D 6 or D 7 = D 8. Finally, we find ha he correlaion is similar wheher IG is above or below average, and he consan correlaion model canno be rejeced. In panel C of Table 4, we repea he ess for U.S. quarerly daa. CCORR is wice as high in high raher han low unemploymen saes (0.45 versus 0.22), and higher afer posiive reurn shocks (0.47 versus 0.25). Furher, CCORR is 0.60 when D 5 =, 0.30 when D 6 =, 0.36 when D 7 = and saisically zero when D 8 =. An LR es (no shown) confirms ha we can rejec equaliy of D 5 and D 6, or D 7 and D 8. Thus, given he unemploymen growh, he correlaion is higher wih higher reurn shocks; and, given he reurn shock, he correlaion is higher wih higher unemploymen. Turning o GG, we canno rejec he consan correlaion model when condiioning on GG alone, bu we can rejec i when we combine reurn and GDP informaion where he daa includes durable consumpion.

19 7 (column GG+RR). Given below average GG, he correlaion is 0.57 afer quarers wih posiive RR and saisically zero oherwise, and hese esimaes are significanly differen. In summary, we find ha he correlaion varies subsanially over ime. The correlaion is higher in bad imes (i.e. when unemploymen growh is high or GDP growh is low). The resuls are consisen wih infrequen updaing and monioring of spending decisions by consumers, resuling in higher correlaion during recessions (Ameriks e al, 2004 and Reis, 2003). The resuls also imply a negaive relaion beween expeced reurns and labor income shocks, consisen wih Sanos and Veronesi (2003). Finally, he correlaion is higher wih posiive reurn shocks, consisen wih he composiion effec or increased sock marke paricipaion. In he nex secion, we perform several robusness checks, including using differen proxies for he composiion effec (such as MEC) and macro condiions (such as CAY). 6. Robusness Checks for U.S. Daa Previously, we found ha he correlaion is higher for above average unemploymen growh and reurn shocks. In his secion, we check he sensiiviy of our resuls o alernaive proxies for macro condiions and sock marke wealh (secion A). We also esimae a GARCH-M model, where he condiional mean reurn depends on he reurn variance (secion B). Finally, we conrol for he decline in consumpion volailiy in he pos-990 period (secion C). A. Alernaive specificaions for he correlaion Duffee (2004) finds ha he correlaion increases wih MEC, he raio of sock marke wealh o consumpion. We replace RR wih MEC in he correlaion equaion. The MEC series is only available unil 2002 Q2, a year earlier han he RR series. The resuls are in Panel D of Table 5, column MEC. Consisen wih Duffee (2004), CCORR is 0.49 when MEC is above

20 8 average (D 3 =) and 0.29 when i is below average (D 4 =). Column UG+MEC shows ha CCORR is 0.47 when D 5 =, 0.32 when D 6 =, 0.43 when D 7 = and saisically zero when D 8 =. Thus, given he sae of unemploymen, he correlaion is higher when MEC is above average raher han below average; and, given MEC, he correlaion is higher wih higher unemploymen. These resuls are qualiaively similar o our previous resuls using RR and UG. Gabaix and Laibson (200) predic ha he correlaion should be higher when he reurn shock is large. We replace RR wih ARR, he absolue value of he reurn residual, and redefine dummies D 3 and D 4 as specified in () and (2). The resuls are in Panel D of Table 5, column ARR. We find ha CCORR is 0.46 for large shocks (D 3 =) and 0.3 for small shocks (D 4 =). Furher, condiioning on high UG, CCORR is 0.55 wih large shocks and 0.35 wih small reurn shocks. Condiioning on low UG, CCORR is 0.39 wih large reurn shocks and saisically zero when reurn shocks are small. These resuls are consisen wih Gabaix and Laibson (200). Exending he basic UG model, we allow he mean reurn and consumpion growh, he condiional reurn variance and correlaion o depend on a one-quarer lag of CAY. 4 In he VAR, lagged CAY is significanly associaed wih CGRO and EP, negaively wih he former and posiively wih he laer, consisen wih Leau and Ludvigson (200a). The GARCH resuls are in Panel D of Table 5, column UG+CAY. We find ha CAY is negaively relaed o he reurn variance, consisen wih Leau and Ludvigson (2003). UG and CAY are significan, and he correlaion increases wih CAY. 5 Since high CAY predics high reurns, he resul implies ha he correlaion is counercyclical. Hence, CAY is informaive of he correlaion, even afer condiioning on unemploymen. 4 CAY is no significan when i is included in he consumpion volailiy equaion. 5 In conras o our resul, Duffee (2004) finds he correlaion is negaively relaed o CAY. This may be because, in Duffee s analysis, he correlaion depends on MEC and CAY, which have a high negaive correlaion of 0.6, making heir relaive significance hard o disinguish. In conras, we use UG and CAY, which have a low

21 9 We ry oher specificaions of he condiioning variables in he covariance equaion. For example, we allow CCORR o vary wih he level of unemploymen in addiion o is change; and wih unemploymen and GDP growh combined. We also ry a coninuous specificaion raher han one-zero dummy variables. For example, le U + =max(0,ug) and U - =min(0,ug). Then, we allow CCORR o vary wih an inercep, U + and U -. In all cases, he earlier resuls remain valid. In summary, we show ha he correlaion is higher wih proxies for recessionary periods (UG or CAY) and higher wih proxies for sock marke wealh (MEC or RR). The correlaion also increases wih he size of reurn shocks, consisen wih Gabaix and Laibson (200). B. GARCH-M resuls While we expec reurns and volailiy o be posiively relaed, he evidence so far is mixed. Some (e.g. French, Schwer and Sambaugh, 987), find a posiive correlaion bu ohers (e.g. Campbell, 987; Glosen e. al., 993), find a negaive or no relaion beween reurns and volailiy. The VAR-GARCH-M(,) resuls, repored in Table 6, show ha he mean reurn is negaively and significanly relaed o he reurn variance, consisen wih recen evidence in Brand and Kang (200), Leau and Ludvigson (2003), and Whielaw (2000). This is rue for he consan correlaion model (no repored) and he ime-varying correlaion model. Unlike in he base model, we include CAY in he mean equaions. While he qualiaive resuls are unchanged, we find ha he ime-variaion in he condiional correlaion is enhanced. For example, in column UG+RR, he correlaion ranges from 0.69 in he high UG, high RR sae o saisically zero in he low UG, low RR sae. This compares o a correlaion range of 0.60 o (saisically) zero in he same saes when using model UG+RR in Table 5 Panel C. correlaion of 0.4.

22 20 C. Reducion in consumpion volailiy in he pos-990 period Sock and Wason (2002) show ha macro volailiy, including he consumpion growh variance, is lower in he 990s. Leau, Ludvigson and Wacher (2004) show ha his shif is associaed wih higher sock prices. We include a dummy variable for he pos-990 period and find ha he condiional variance of consumpion growh is lower and he condiional correlaion is higher in he 990s. However, our resuls do no change qualiaively. 7. VAR-GARCH(,) Resuls for non-u.s. G7 Counries For U.S. daa, we have shown ha he condiional correlaion CCORR increases wih proxies for recessions and sock marke wealh. Do he resuls also hold for non-u.s. G7 counries? Cross-counry variaions in he duraion of unemploymen insurance (Tasiramos, 2003) and he size of he equiy marke (Table ) may lead o cross-counry differences in how CCORR reacs o unemploymen and sock marke wealh. We presen descripive saisics and VAR resuls followed by he GARCH esimaes. There are wo specificaions for he correlaion. Firs, CCORR varies wih unemploymen growh UG or wih GDP growh GG; second, CCORR varies wih UG and reurn shocks RR, or wih GG and RR. When CCORR varies wih RR alone, he esimaes are no significan, so we do no repor hese resuls. We do no condiion on MEC and CAY for lack of daa. Nor do we condiion on he size of reurn shocks, parly for breviy and parly because he U.S. resuls are similar wheher using he size or he sign of reurn shocks. Table 7, Panel A shows he disribuion of predicion variables for non-u.s. G7 counries. The mean values are broadly comparable across counries, and wih hose of he U.S. The mean defaul spread is around % for Canada and he U.K. and 0.5% or below for oher counries (compared o 0.7% for he U.S.). The mean dividend yield is beween 3% and 4.5% for mos

23 2 counries (compared o 3.34% for he U.S.) excep Japan. The PE raio is around 20 for mos counries (compared o 8.68 for he U.S.), excep Ialy and Japan. The erm spread is beween.24% and 2% for mos counries (compared o.39% for he U.S.), excep Ialy and he U.K. Panel B shows he correlaion beween CGRO, EP and he predicion variables. Lagged predicion variables have low correlaion wih consumpion growh and equiy premium wih few excepions, such as DIVY and TERM in Canada and DIVY and PE in he UK. A. VAR Resuls for non-u.s. G7 daa Table 8 shows resuls from esimaing he VAR model for non-u.s. G7 counries. The F- ess rejec he hypoheses ha lagged consumpion and lagged reurns do no explain variaions in CGRO in all counries excep France, for lagged CGRO, and UK, for lagged reurns). Consisen wih U.S. daa, consumpion growh is generally negaively auocorrelaed a shor lags and posiively auocorrelaed a longer lags. However, he auoregressive model does poorly in predicing reurns. Of he predicion variables, DIVY is posiively correlaed wih reurns in Canada and he U.K. PE is posiively correlaed wih reurns in he U.K. and negaively correlaed wih CGRO in Japan and France. TERM is posiively correlaed wih reurns in Canada and France, and wih CGRO in Germany. Overall, here is moderae-o-srong evidence of predicabiliy for CGRO bu lile evidence of predicabiliy for reurns. B. VAR-GARCH(,) resuls for non-u.s. G7 daa In Table 9, we esimae he condiional correlaion CCORR for non-u.s. G7 counries. For he consan correlaion model (panel A), he Arch and Garch effecs for EP are posiive and significan in all counries, excep Japan, Ialy and France. The Garch effec for CGRO is significan for 4 counries. CCORR is saisically zero for all counries.

24 22 In panel B of Table 9, CCORR varies wih UG and RR. Ialy is excluded as i has no unemploymen daa before 980. Oher counries also have missing daa in he early years. The resuls show ha, for Canada and Germany, CCORR is posiive when UG is above average (D =) and saisically zero oherwise (D 2 =). For Japan, UG does no predic CCORR by iself, bu does so in combinaion wih reurns. Thus, in column UG+RR for Japan, CCORR is 0.33 afer quarers wih high UG and negaive RR (D 6 =) and saisically zero in oher saes. For Germany, oo, CCORR is posiive when D 6 =, and saisically zero oherwise. In panel C of Table 9, CCORR varies wih GG and RR. For Canada and he UK, CCORR is posiive following quarers wih below average GG (D 2 =) and saisically zero oherwise. For Japan, GG does no predic CCORR by iself, bu does so in combinaion wih reurns. Thus, in column GG+RR for Japan, CCORR is 0.43 afer quarers wih low GG and negaive RR (D 8 =) and saisically zero in oher saes. For UK, oo, CCORR is posiive when D 8 =, and saisically zero oherwise. In conras, for Canada, CCORR is posiive during periods of low GDP growh and posiive reurn shocks (D 7 =). Summarizing, he consan correlaion model is rejeced for 4 of 6 non-u.s. G7 counries: Canada, he UK, Germany and Japan. Similar o U.S. daa, he condiional correlaion varies subsanially over ime. Whereas he uncondiional correlaion is mosly zero or negaive, he condiional correlaion is posiive and moderae during economic conracions (i.e. when unemploymen growh is high or GDP growh is low). However, unlike U.S. daa, reurn shocks alone do no affec he correlaion. Furher, condiional on low GDP growh, he correlaion is higher afer negaive reurn shocks in he U.K., Japan and Germany. These resul may indicae ha he composiion effec is weak hese counries; alernaively, ouside he U.S, he reurn shock may be a poor proxy for he raio of sock marke wealh o consumpion.

25 23 8. Cyclical Properies of he Correlaion, Equiy Premium, Sharpe Raio and Risk Aversion Wha are he implicaions of ime-varying correlaion for he equiy premium EP, Sharpe raio SR, and he risk-aversion RA? EP and SR are proporional o he correlaion, so we expec ha counries wih counercyclical correlaion may also have counercyclical EP and SR. Implicaions for RA are ambiguous, since i varies inversely wih he correlaion and direcly wih SR. In his secion, we plo esimaes of he covariance, correlaion, EP and SR from our model using G7 quarerly daa 6 and examine heir behavior over business cycles daed by he NBER (for he U.S.) and OECD (for non-u.s. counries). We omi cycles denoed minor by he OECD, since he heory discussed in secion mosly applies o large income shocks. We also plo he risk-aversion implied by he condiional Euler equaion: γ = σ ( CGRO + E ( EP + ) ) σ ( EP ) ρ ( CGRO + +, EP + ) (7) For U.S. daa, we plo esimaes from 3 models ha differ by he variables deermining he correlaion: RR, reurn shocks; UG+RR, unemploymen growh UG and RR; and UG+ CAY. For he non-u.s. counries, we only use model UG+RR. For France and Ialy, we use he consan correlaion model since i canno be saisically rejeced. We firs discuss how momens of he condiional EP and SR rack sample momens, he range of SR and he mean of γ. Then, we discuss he plos. Secion A repors resuls for U.S. daa and secion B for he oher counries. A. U.S. quarerly daa Table 0 shows annualized momens of he condiional and sample EP and SR for he G7 counries. For U.S. quarerly daa (USQ), he mean condiional EP maches is sample counerpar more closely when he correlaion varies han when i is consan. For example, in

26 24 he UG+RR model, he mean EP is 4.55% compared o 4.08% in he sample. When he correlaion is consan, he mean EP is 6.4%. Thus, ime-varying correlaion is imporan in maching he mean equiy premium in he daa. However, he sandard deviaion (SD) of he condiional EP underesimaes he sample SD by a facor of abou 2.5. We find large swings in he condiional SR, ranging from around 2.70 o 3.28, compared o sample range (derived from daily daa) of 2.45 o The average implied RA remains oo high, even allowing for imevarying correlaion. This is consisen wih Campbell (2002), who found ha he implied RA is oo high for mos counries, varying beween 9 and 50, even when he correlaion is assumed. We nex urn o he business cycle properies of he covariance and correlaion. Figure plos hese variables, averaged over each cycle of conracion (i.e. pos peak o rough quarers) and expansion (i.e. pos rough o peak quarers), wih shaded areas indicaing NBER-daed conracions or recessions. For model SR, he correlaion mosly decreases in recessions, consisen wih Duffee (2004). In some recessions, he covariance increases bu i is offse by increases in he volailiy, so ha he correlaion decreases. In conras, for model UG+RR, he covariance and correlaion always increases during recessions, consisen wih Sanos and Veronesi (2003). Table confirms his resul by regressing he correlaion on a recession dummy, and obaining a posiive and significan coefficien. The correlaion is also counercyclical for model UG+CAY. The EP is counercyclical for all models, increasing jus before recessions and reaching a maximum a or near he rough, consisen wih Whielaw (994). For model SR, he EP increases even in recessions where he covariance decreases as risk-aversion increases sufficienly o offse he decline in he covariance (see Figure 2). Figure 2 plos he SR and RA, he laer averaged over each expansion and conracion. Like 6 Resuls for U.S. monhly daa are similar o quarerly daa, and are no repored (hey are available upon reques).

27 25 he equiy premium, he SR increases during recessions and peaks jus before a rough. Flucuaions in he SR are greaes for model UG+RR, and leas for model SR. The RA generally increases during recessions, suggesing ha he higher correlaion during recessions is more han offse by he increase in he SR. Excepions are he shor recession of 980 Q2-Q3 and he mos recession. Campbell and Cochrane (999) s model also did wors for he mos recen recession. B. Non-U.S. G7 quarerly daa Table 0 shows ha, similar o USQ daa, he model racks he mean EP while underesimaing volailiy. For example, in UK daa, he mean EP and SD for model UG+RR are 8.8% and 7.64%, respecively, compared o 8.5% and 20.49% in he daa. Figure 3 clearly shows he counercyclical propery of he correlaion for all counries, and his resul is confirmed by regression analysis in Table. Figures 4 and 5 show ha he EP and SR generally increase during recessions for Canada, UK and Japan, while showing no discernible paern for he ohers. Figure 6 plos he implied risk-aversion averaged over expansions and conracions and shows a clear paern of higher risk-aversion during recessions for Canada, UK and Japan. Summarizing, he correlaion is counercyclical for all 4 counries where he consan correlaion model is rejeced. In Canada, UK and Japan---counries wih large sock markes relaive o GDP---he EP, SR and RA are all generally counercyclical. For he oher counries, noably France and Ialy, here is lile evidence of any cyclical paern in hese variables. 9. Conclusion We examine he implicaions of ime-variaion in he correlaion beween equiy premium and nondurable consumpion growh for equiy reurn dynamics for he G7 counries. Using a

28 26 VAR-GARCH(,) model, we find ha he correlaion is higher wih recession proxies, such as above average unemploymen growh. This resul is consisen wih asse pricing models wih labor income (Sanos and Veronesi, 2003) and models where consumers infrequenly updae and monior heir spending decisions. The correlaion also increases wih proxies for sock marke wealh, consisen wih he composiion effec (Duffee, 2004). Graphs and regression analysis show ha he correlaion is counercyclical when hese wo effecs are combined. Plos of he equiy premium and Sharpe raio also show a counercyclical paern. During recessions, he increase in he Sharpe raio offses he increase in he correlaion, so ha he risk-aversion generally increases. We ry differen recession proxies, allow he mean reurn o depend on is condiional variance, and conrol for lower consumpion volailiy during he pos-990 period, o find our resuls qualiaively unchanged in all cases. The evidence is sronger (weaker) for counries wih larger (smaller) sock marke capializaion relaive o GDP. The low uncondiional correlaion in he daa has promped some auhors o find alernaives o aggregae consumpion risk for explaining risk and reurns. Our resuls, however, show ha ime-varying consumpion correlaion reains an imporan role for undersanding he cyclical properies of a wide range of variables of ineres: he equiy premium, Sharpe raio and risk aversion. Echoing Jagannahan and Wang (996) and ohers, we show he imporan role of labor income for undersanding he risk-reurn radeoff. In paricular, we sress ha combining financial and macro indicaors is a fruiful approach. However, our approach canno explain he level of he covariance beween consumpion and reurns and, hence, he equiy premium puzzle. Incorporaing realisic feaures of consumer behavior, like habi-formaion and adjusmen coss, may be necessary o explain boh he level and variaion in he covariance and he equiy premium.

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