Estimating Earnings Trend Using Unobserved Components Framework

Transcription

1 Esimaing Earnings Trend Using Unobserved Componens Framework Arabinda Basisha and Alexander Kurov College of Business and Economics, Wes Virginia Universiy December 008 Absrac Regressions using valuaion raios for predicing long erm sock reurns ofen use a en-year moving average of earnings as a proxy for unobserved fuure earnings. This paper shows ha he earnings rend can be direcly esimaed using bivariae unobserved componens models. The resuls show ha he valuaion raios based on he esimaed rends improve he fi of sock reurn predicive regressions. However, he 90 percen confidence inervals around he esimaed rends are large and end o include he moving average rend. JEL Classificaion: C, C51, G1. Keywords: Valuaion Raios, Unobserved Componens Model Corresponding auhor. Dep. of Economics, PO Box 605, College of Business and Economics, Wes Virginia Universiy, Morganown, WV Tel: , arbasisha@mail.wvu.edu. Dep. of Finance, PO Box 605, College of Business and Economics, Wes Virginia Universiy, Morganown, WV Tel: , alkurov@mail.wvu.edu.

2 Inroducion Empirical sudies by Campbell and Shiller (1998, 001) show ha valuaion raios can be used for predicing long erm real sock reurns. In paricular, hey use he price-smoohed earnings raio compued by assuming a en-year moving average of real earnings as earnings rend, i.e., he componen ha capures long erm fuure earnings. 1 Compuing a muli-year moving average reduces he effec of cyclical flucuaions on corporae earnings. Averaging pas earnings in valuaion analysis is a common approach firs recommended by Graham and Dodd (1934), bu i may no be he bes way o model he earnings rend. This paper relaxes he assumpion of he moving average rend by leing he daa speak. We direcly esimae he earnings rend using unobserved componens models. Our models use boh he real earnings daa and long-erm real sock reurns daa. The unobserved componens framework also allows us o compare he performance of differen saisical assumpions by using he Schwarz Informaion Crierion. The heoreically and saisically appealing moivaion for esimaing he earnings rend is derived from he Beveridge-Nelson (1981) decomposiion of a ime series ino is rend and cycle componens. The rend can be inerpreed as a long erm condiional forecas of he ime series. Morley (00) and Morley, Nelson and Zivo (003) show ha he Beveridge-Nelson decomposiion can be usefully cas ino an unobserved componens framework which allows he rend and he cycle shocks o be correlaed. Furher sudies by Ord e al. (1998) and Anderson e al. (006) show ha he perfec correlaion beween he shocks, as in he Beveridge-Nelson decomposiion, can be modeled as a single source shock. Laubach (001) argues in a differen conex of esimaing NAIRUs ha bivariae modeling can also help o reduce he uncerainy around he esimaed unobserved componens. 1 Many oher empirical sudies also use he price-smoohed earnings raio for predicing sock reurns. These sudies are no menioned due o space consideraions. 1

3 This paper uses he above developmens in he unobserved componens modeling o esimae he earnings rend under hree differen assumpions abou correlaion of shocks. The resuls indicae a fairly volaile earnings rend in all hree cases. The fi of he long-run sock reurns predicive regression is higher in all he cases relaive o he moving average rend. However, he 90 percen confidence inervals of he esimaed rends end o include he moving average rend. The unobserved componens model and he esimaes The following unobserved componens model is used for esimaing he earnings rend. The firs measuremen equaion uses log real earnings, e, o be decomposed ino wo unobserved componens: is permanen (or sochasic rend) par p and he cyclical par c. e = p + c (1) The permanen par is assumed o follow a random walk wih a consan drif μ in equaion (). The cyclical par c is assumed o follow an auoregressive process in equaion (3). Following Morley (001) and Morley e al. (003), we use an auoregressive process of order wo. p = + p 1 μ + ε () φ ( L) = ω (3) c The second measuremen equaion, equaion (4), is he fuure sock reurns equaion. The sock reurn over he nex en years, 10 r +1, depends on a consan, he valuaion raio shown by log real sock price s ) minus he permanen par of log real earnings p ) and a serially correlaed ( unobserved componen o accoun for oher omied facors. The componen is assumed ( f +1 f + 1

4 o ake a moving average form based on comparison of SICs (no repored) in equaion (5). We use en lags in he moving average process. r 10 = + β ( s p ) + f α (4) f θ (5) = ( L) v We use hree specificaions of he correlaions beween he hree shocksε, ω and ν. In he firs specificaion, hey are assumed o be uncorrelaed as in Clark (1987). In he second and hird specificaions, following Anderson e al. (006) which uses he Beveridge-Nelson resul of perfecly negaive correlaion of he shocks, ε and ω are assumed o be perfecly correlaed and modeled as a single source shock using ω = γε. The shock ε is assumed o be uncorrelaed o ν + 1 in he second specificaion and we allow for ha correlaion o be esimaed in he hird specificaion. We use annual average daa ranging from 1871 o 007 obained from Rober Shiller s web sie. The en-year sock reurn is compued from 1871 o 1997 where he 1997 sock reurn denoes he sock reurn from 1997 o 007. Therefore, based on he daa availabiliy, we are able o esimae he earnings rend from 1871 o We esimae he parameers of he model using maximum likelihood and hen use he Kalman filer o obain he esimaes of he rend. The sandard errors of he esimaed rends are compued using Hamilon s (1986) procedure and are based on 1000 Mone Carlo replicaions. The resuls are repored in Table 1 and he esimaed rends are shown in Figure 1. Panel A of Table 1 shows ha he valuaion raio based on he en-year moving average of earnings predics fuure long erm sock reurns wih a fi of 19 percen. The upper-lef panel of Figure 1 shows he moving average rend. Panel B of Table 1 shows he parameer esimaes from he 3

5 bivariae unobserved componens model wih uncorrelaed shocks. The esimae of μ shows ha real earning grew a an average annual rae of 1.4 percen. The sandard deviaions of all shocks are moderae and fairly precisely esimaed. The esimaed earnings rend is shown in he upperrigh of Figure 1 along wih is 90 percen confidence inerval and he moving average rend. One can observe ha alhough he esimaed earnings rend appears o be fairly differen han he moving average rend, he 90 percen confidence inerval does include he moving average rend almos all of he ime. The regression of he long-erm fuure sock reurns on he esimaed earnings rend has a fi of abou 9 percen, which is 10 percen higher han he moving average rend is used. Panel C of Table 1 repors he parameer esimaes using a single source of shock beween he rend and cycle of real earnings bu uncorrelaed wih shocks o fuure real sock reurns ν + 1. The esimaes are largely similar o hose in Panel B, alhough he poin esimae of he sandard deviaion of he shock o he earnings rend is higher, implying a more volaile rend. The lower-lef panel of Figure 1 shows he esimaed rend, which appears o be more volaile han he rend in he uncorrelaed case. The fi of he sock reurns regression on he valuaion raio based on he esimaed rend is sill abou 9 percen. Panel D of Table 1 repors he parameer esimaes using a single source of shock beween he rend and cycle of real earnings and correlaed wih shocks o fuure real sock reurns ν + 1. The esimaes are largely similar o hose in Panel C. The poin esimae of he sandard deviaion of he shock o earnings rend is higher han ha repored in Panel B, implying a more volaile rend. The correlaion beween he rend shock and he fuure sock reurns shock is esimaed o be -0.3, low bu precisely esimaed. The lower-righ panel of Figure 1 shows he esimaed rend which appears o be more volaile han he rend in he 4

6 uncorrelaed case. The fi of he sock reurns regression on he valuaion raio based on he esimaed rend is marginally lower a abou 8 percen bu sill higher he 19 percen fi repored in Panel A. This model also shows he lowes SIC of he hree unobserved componens models, implying ha i provides he bes descripion for he daa. The s p ) ( erm in equaion (4) can be viewed (up o a consan) as a deviaion of he aggregae sock prices from he earnings rend, or as a proxy for marke mispricing. Brown and Cliff (005) find ha sock marke valuaion errors are posiively correlaed wih invesor senimen. We examined he relaion beween our mispricing proxy and he invesor senimen index from Baker and Wurgler (006). The senimen index is esimaed as he firs principal componen of he closed-end fund discoun, equiy share in new securiy issues, and lagged NYSE urnover. The coefficien of he senimen index in a linear regression using he period was posiive and significan a he 10 percen level. Conclusion The main conribuion of his paper is o show ha he earnings rend can be esimaed by using a bivariae unobserved componens framework under differen assumpions abou he correlaions of he shocks. All hree esimaed rends show a beer fi of he regression equaion using he price-earnings raio o predic long erm sock reurns han he radiional specificaion using he moving average earnings rend. The drawback of he esimaed rends is ha heir 90 percen confidence inerval almos always includes he moving average rend. However, his drawback will become less of an issue wih availabiliy of more daa, making he unobserved componens models an asympoically aracive choice. These resuls are no abulaed o save space, bu are available upon reques. 5

7 References Anderson, H. M., Low, C. N. and R. Snyder (006); Single source of error sae space approach o he Beveridge Nelson decomposiion, Economics Leers, 91, Baker, M. and J. Wurgler (006); Invesor Senimen and he Cross-Secion of Sock Reurns, Journal of Finance, 61, Beveridge, S. and C. Nelson (1981); A New Approach o he Decomposiion of Economic Time Series ino Permanen and Transiory Componens wih Paricular Aenion o he Measuremen of he Business Cycle, Journal of Moneary Economics, 7, Brown, G. W. and M. T. Cliff (005); Invesor Senimen and Asse Valuaion, Journal of Business, 78, Campbell, J. Y. and R. J. Shiller (1998); Valuaion Raios and he Long-Run Sock Marke Oulook, Journal of Porfolio Managemen, 4, Campbell, J. Y. and R. J. Shiller (001); Valuaion Raios and he Long-Run Sock Marke Oulook: An Updae, NBER Working Paper Series, no. 81. Clark, P. K. (1987); The Cyclical Componen of US Economic Aciviy, Quarerly Journal of Economics, 10, Graham, B. and D. L. Dodd (1934); Securiy Analysis, Firs Ediion. New York: McGraw Hill. Hamilon, J. (1986), A Sandard Error for he Esimaed Sae Vecor of a Sae Space Model, Journal of Economerics, 33, Laubach, T. (001); Measuring he NAIRU: Evidence from Seven Economies, The Review of Economics and Saisics, 83, Morley, J. (00); A sae-space approach o calculaing he Beveridge-Nelson decomposiion, Economics Leers, 75, Morley, J., Nelson, C. and E. Zivo (003); Why Are Beveridge-Nelson and Unobserved- Componen Decomposiions of GDP So Differen?, Review of Economics and Saisics, 85, Ord, J. K., Koehler, A. B. and R. Snyder (1997); Esimaion and Predicion for a class of dynamic nonlinear saisical models, Journal of American Saisical Associaion, 9,

8 Table 1: Parameer Esimaes from he Unobserved Componens Models Panel A: The 10 Year Moving Average Regression α β R (0.35) (0.13) Panel B: The UC Model wih Uncorrelaed Shocks μ ε ν ω R SIC (0.01) (0.0) (0.01) 0.13 (0.01) Panel C: The UC Model wih Single Shock for Earnings and Uncorrelaed wih Sock Reurns μ ε ν R SIC (0.01) (0.0) (0.01) Panel D: The UC Model wih Single Shock for Earnings and Correlaed wih Sock Reurns μ ε ν ρ εν R SIC (0.01) (0.0) (0.01) -0.7 (0.09) Noe: The numbers in he parenheses are sandard errors. The R s are compued by regressing he 10 year fuure sock reurns on a consan and he difference of log real sock price and esimaed rend from he given model. The SIC s represen Schwarz Informaion Crierion and are compued for he bivariae UC models only. 7

9 Figure 1: Esimaes of Earnings Trends from Unobserved Componens Models Log Real Earnings 10 Moving Average 10 Year Moving Average Earnings Trend, Uncorrelaed Lower Bound Upper Bound Year Moving Average Earnings Trend, Single Source Uncorrelaed Lower Bound Upper Bound 10 Year Moving Average Earnings Trend, Single Source Correlaed Lower Bound Upper Bound Noe: The upper-lef panel shows he logarihm of real earnings daa and is 10-year moving average. The upper-righ panel shows he esimaed log real earnings rend and is 90 percen confidence inerval when he shocks are uncorrelaed wih each oher. The lower-lef panel shows he esimaed log real earnings rend and is 90 percen confidence inerval when he permanen shocks o real earnings are perfecly correlaed o is ransiory shocks bu uncorrelaed o fuure sock reurns shock. The lower-righ panel shows he esimaed log real earnings rend and is 90 percen confidence inerval when he permanen shocks o real earnings are perfecly correlaed o is ransiory shocks and correlaed o fuure sock reurns shock. The sandard errors of he rends are based on 1000 Mone Carlo replicaions. 8

10 No for Publicaion Appendix: Noes for he Referees 1. The univariae Beveridge-Nelson decomposiion of log real earnings yielded an R of abou 9 percen in he sock reurns regression, lower han he valuaion raio using he en-year moving average.. The correlaion beween he rend and he cycle shocks o real earnings was also esimaed following Morley, Nelson and Zivo (003). The esimaes showed a perfecly negaive correlaion as in he Beveridge-Nelson decomposiion. This creaed a problem for geing he sandard error of he esimaed rend following Hamilon (1986), as some parameers were on he border of he parameer space. So we decided o use he model wih a single source of shocks following Anderson e al. (006) ha does he Beveridge-Nelson decomposiion under he assumpion of perfec correlaion. 3. Hamilon s (1986) procedure for compuing he confidence inerval for unobserved componens accouns for boh parameric uncerainy and filering uncerainy. 4. If we limi our sample o he period, implying he las observaion for he enyear fuure sock reurns covers he period 1987 o 1996, he fi of he regression increases o 0.31, similar o ha of Campbell and Shiller (1998). The fi of he regression based on he esimaed rend from he correlaed model also increases o The fi of he regression based on he esimaed rend from he uncorrelaed model increases o The following graph compares he esimaed price-earnings raio from correlaed rends model wih he price-earnings raio based on he en-year moving average of earnings. 9

11 Price-Earnings Raio (10 year moving average) Price-Earnings Raio (Correlaed rend) 10