The Overlapping Data Problem

Quaniaive and Qualiaive Analysis in Social Sciences Volume 3, Issue 3, 009, 78-115 ISSN: 175-895 The Overlapping Daa Problem Ardian Harri a Mississipi Sae Universiy B. Wade Brorsen b Oklahoma Sae Universiy Absrac Overlapping daa are ofen used in finance and economics, bu applied work ofen uses inefficien esimaors. The aricle evaluaes possible reasons for using overlapping daa and provides a guide abou which esimaor o use in a given siuaion. JEL Classificaions: C13. Keywords: Auocorrelaion, Mone Carlo, Newey-Wes, overlapping daa. a Deparmen of Agriculural Economics, Mississippi Sae Universiy, PO Box 5187, Mississippi Sae, MS 3976, U.S.A.; el.: +1 66 35-044, email harri@agecon.mssae.edu b Corresponding auhor. Deparmen of Agriculural Economics, Oklahoma Sae Universiy, 308 Ag Hall, Sillwaer, OK 74075, U.S.A.; el.: +1 405 744-6836, email: wade.brorsen@oksae.edu

1 Inroducion Time series sudies esimaing muliple-period changes can use overlapping daa in order o achieve greaer efficiency. A common example is using annual reurns when monhly daa are available. A one-year change could be calculaed from January o December, anoher from February o January, and so on. In his example he January o December and February o January changes would overlap for eleven monhs. The overlapping of observaions creaes a moving average (MA) error erm and hus ordinary leas squares (OLS) parameer esimaes would be inefficien and hypohesis ess biased (Hansen and Hodrick, 1980). Pas lieraure has recognized he presence of he moving average error erm. Our aricle seeks o improve economeric pracice when dealing wih overlapping daa by synhesizing and adding o he lieraure on overlapping daa. We find limied saisical reasons for no using he disaggregae daa and ha he preferred esimaion mehod can vary depending on he specific problem. One way of dealing wih he overlapping observaions problem is o use a reduced sample in which none of he observaions overlap. For he example given above, he reduced sample will have only one observaion per year. Thus, for a 30-year period of monhly daa only 30 annual changes or observaions will be used insead of 49 (he maximum number of overlapping observaions ha can be creaed for his period) annual observaions. This procedure will eliminae he auocorrelaion problem bu i is obviously highly inefficien. A second way involves using average daa. For our example his means using he average of he 1 overlapping observaions ha can be creaed for each year. This procedure resuls in he same degree of daa reducion and apparenly uses all he informaion. In fac, no only is i inefficien, i also does no eliminae he moving average error erm (Gilber, 1986) and can inroduce auocorrelaion no presen in he original series (Working, 1960). A hird way is o use he overlapping daa and o accoun for he moving average error erm in hypohesis esing. Several heeroskedasiciy and auocovariance consisen (HAC) esimaors have been consruced ha can provide asympoically valid hypohesis ess when using daa wih overlapping observaions. These HAC esimaors include Hansen and Hodrick (HH), 1980, Newey-Wes (NW), 1987, Andrews and Monahan (AM), 1990, and Wes (1997). A fourh way is o ransform he long-horizon overlapping regression ino a non-overlapping regression of one-period reurns ono a se of ransformed 79

regressors Brien-Jones and Neuberger (B-JN), 004. A final way is o use OLS esimaion wih overlapping daa, which yields biased hypohesis ess. To illusrae he enormiy of he problem he number of empirical aricles involving he use of overlapping daa in regression analysis in hree journals during 1996 and 004 were couned. The journals were, The Journal of Finance, The American Economic Review, and The Journal of Fuures Markes. The mehods of esimaion are classified as OLS wih non-overlapping daa (OLSNO), OLS wih he Newey-Wes (1987) variance covariance esimaor, OLS wih any of he oher generalized mehod of momens (GMM) esimaors, and jus OLS. The porion of aricles using overlapping daa increased from 1996 o 004 (Table 1) so ha he majoriy of aricles in finance now use overlapping daa. Mos of he empirical aricles ha used overlapping daa sudied asse reurns or economic growh. A common feaure of hese aricles is ha reurns or growh are measured over a period longer han he observaion period. For example, daa are observed monhly and he esimaion is done annually. Auhors provide several possible reasons for using aggregaed daa. The mos common reason given is measuremen error in independen variables. For example, Jones and Kaul (1996, p. 469), sae ha hey selec use of quarerly daa on all variables as a compromise beween he measuremen errors in monhly daa.... Mos auhors provide no jusificaion for using overlapping daa, bu here mus be some advanage o using i or i would no be so widely used. Brien Jones and Neuberger (004) conend, he use of overlapping daa is based more on economic reasons raher han saisical ones. Here, we evaluae possible saisical reasons for using overlapping daa. Table 1: Number of aricles using overlapping daa, 1996-004. Number of aricles Journal Year OLSNO NW Oher a OLS Toal J. Finance Toal number of empirical aricles in he journal Percenage of aricles wih overlapping daa 1996 16 8 8-6 55 47.3 004 3 16 16 3 45 71 63.4 Amer. 1996 10 3-14 77 18. Econ. Rev. 004 19 4-0 109 18.3 J. Fu. Mks. 1996 1 3 5 19 43 44. 004 18 5 5 5 6 44 59.1 Noes: The sum of he columns 3 hrough 6 may be larger han he oal in column 7 since some aricles use more han one mehod of esimaion. a These include HH and AM esimaors. 80

Table 1 also shows each of he esimaion mehods frequency of use. The OLSNO and Newey-Wes esimaion mehods are used mos ofen. We defined OLSNO as esimaion using non-overlapping observaions. This means ha he daa exis o creae overlapping observaions bu he researchers chose o work wih non-overlapping observaions. I migh be more correc o say ha OLSNO is used simply because i is no a pracice o creae overlapping daa. The OLSNO mehod will yield unbiased and consisen parameer esimaes and valid hypohesis ess. Bu i will be inefficien since i hrows away informaion. We firs demonsrae ha he commonly used Newey-Wes and OLSNO mehods can be grossly inefficien ways of handling he overlapping daa problem. This is done by deermining and comparing he small-sample properies of Newey-Wes, OLSNO, maximum likelihood esimaion (MLE), and generalized leas squares (GLS) esimaes. Unresriced MLE is included as an alernaive o GLS o show wha happens when he MA coefficiens are esimaed. 1 Then, we consider possible saisical reasons for using overlapping daa such as nonnormaliy, missing daa, and errors in variables. Finally, we evaluae ways of handling overlapping daa when here are economic reasons for doing so. While Newey-Wes and OLSNO esimaion provide inefficien esimaes he GLS esimaion canno be applied in every siuaion involving overlapping daa. An example would be when lagged values of he dependen variable or some oher endogenous variable are used as an explanaory variable. In his case, as Hansen and Hodrick (1980) argue, he GLS esimaes will be inconsisen since an endogeneiy problem is creaed when he dependen and explanaory variables are ransformed. For he specific case of overlapping daa considered by Hansen and Hodrick (1980), we have lile o add o he previous lieraure (e.g. Mark, 1995) ha favors using he boosrap o correc he small sample bias in he Hansen and Hodrick approach. Wih a general mulivariae ime series model, ofen overlapping daa canno be used o recover esimaes of he disaggregae process ha generaed he daa. The percenage of cases where lagged values of he dependen variable are used as an explanaory variable is repored in Table 1. In The Journal of Finance less han 5 percen of aricles included a lagged dependen variable as an explanaory variable (half wih he Newey-Wes esimaor and half wih OLSNO). In he American Economic Review abou 7 percen (all wih he Newey-Wes esimaor) of he aricles included a 1 Wih normaliy, he GLS esimaor is he maximum likelihood esimaor. The rue MLE would have he parameers of he moving average process be known raher han esimaed. Such a resriced MLE should be considered wih large sample sizes since i uses less sorage han GLS. 81

lagged dependen variable. Thus, in mos cases where nonoverlapping daa are used, here are no lagged dependen variables and so more precise esimaion mehods are available. The res of he paper is srucured as follows. Secion considers he simples case where he daa represen aggregaes and he explanaory variables are sricly exogenous. Secion 3 discusses he OLSNO and Newey-Wes esimaion mehods and heir inefficiency. Secion 4 conducs a Mone Carlo sudy o deermine he size and power of he hypohesis ess when using overlapping daa and GLS, OLSNO, Newey-Wes, and unresriced MLE esimaion mehods. Secions 5 and 6 consider possible saisical and economic reasons, respecively, for using overlapping daa. Secion 7 discusses how o handle overlapping daa in several special cases ha do no fi any of he sandard procedures. Secion 8 concludes. The Sricly Exogenous Regressors Case There are many variaions on he overlapping daa problem. We firs consider he simples case where he daa represen aggregaes and he explanaory variables are sricly exogenous. This is he mos common case in he lieraure such as when annual daa are used for dependen and independen variables and monhly daa are available for boh. To consider he overlapping daa problem, sar wih he following regression equaion: y = β x + u, (1) where y is he dependen variable, x is he vecor of m sricly exogenous independen variables, and u is he error erm. Equaion (1) represens he basic daa ha are hen used o form he overlapping observaions. The error erms, u, in equaion (1) have he following properies: E[ u ] = 0, E[ u ] = σ, and cov[ u, u ] = 0 if s. u However, one migh wan o use aggregaed daa and insead of equaion (1) esimae he following equaion: 8 s Y = β X + e, () where he (1 1) scalar Y and ( m 1) vecor X represen an aggregaion of y and x, respecively. To esimae equaion () he overlapping observaions are creaed by summing he original observaions as follows: + k 1 + k 1 + k 1 (3) Y = y, X = x for i = 1,... m, i.e. X = ( X,..., X ), e = u, j i ij 1 m j j= j= j=

where k is he number of periods for which he changes are esimaed. If n is he original sample size, hen n k + 1 is he new sample size. These ransformaions of he dependen and independen variables induce an MA process in he error erms of equaion (). Because he original error erms were uncorrelaed wih zero mean, i follows ha: k 1 k 1 E [ e ] = E[ u + j ] = E[ u + j ] = 0. (4) j= 0 Also, since he successive values of u j are homoskedasic and uncorrelaed, he uncondiional variance of e is j= 0 var[ e ] = σ = E[ e ] = kσ. (5) e u Based on he fac ha wo differen error erms, e and e + s, have k - s common original error erms, u, for any k s > 0, he covariances beween he error erms are cov[ e, e s ] E[ e, e s ] = ( k s) σ u ( k ) > 0. (6) Dividing by kσ u gives he correlaions: + = + s k s corr [ e, e + s ] = ( k s) > 0. (7) k Collecing erms we have as an example he case of n = k + : k 1 k s 1 1...... 0 0 k k k k 1 k 1 k s 1 1...... 0 k k k k k 1 k 1 k s 1... 1...... k k k k........................ Ω=,........................ 1 k s k 1 k 1...... 1... k k k k 1 k s k 1 k 1 0...... 1 k k k k 1 k s k 1 0 0...... 1 k k k where, Ω is he correlaion marix. The correlaion marix, Ω, appears in Gilber s aricle and he presence of a moving average error erm is commonly recognized. Wih Ω derived analyically he GLS parameer esimaes and heir variance-covariance marix can be obained as follows: ˆ β = ( ' Ω ) ' Ω (8) 1 1 1 X X X Y (9) 83

and ˆ β = σ e X' Ω X (10) 1 1 var[ ] ( ), where X ' = ( X1,..., Xn k + 1) is an ( m n k + 1) marix and Y ' = ( Y1,..., Yn k + 1) is a (1 n k + 1) row vecor. (Equaion (3) defines Y and X, for = 1,..., n k + 1.) Under hese assumpions, he GLS esimaor of he aggregae model will be bes linear unbiased and asympoically efficien. If errors are normally disribued, hen GLS is efficien in small samples, sandard hypohesis es procedures would be valid in small samples, and he GLS esimaor would be he maximum likelihood esimaor. This case canno explain why auhors would choose o use overlapping daa. The disaggregae model does no lose observaions o aggregaion so i would sill be preferred in small samples. 3 Alernaive Esimaion Mehods The nex issue o be discussed is he OLSNO and Newey-Wes esimaion mehods and heir inefficiency. We consider only Newey-Wes raher han he alernaive GMM esimaors. As Davidson and MacKinnon (1993, p. 611) say he Newey-Wes esimaor is never grealy inferior o ha of he alernaives. Wih he Newey-Wes esimaion mehod, parameer esimaes are obained by using OLS. The OLS esimae ˆ β is unbiased and consisen bu inefficien. The OLS esimae of σ e is biased and inconsisen. The Newey and Wes (1987) auocorrelaion consisen covariance marix is compued using he OLS residuals. The OLSNO esimaion mehod esimaes parameers using OLS wih a reduced sample where he observaions do no overlap. The OLSNO esimaes of he variance are unbiased since wih no overlap here is no auocorrelaion. The OLSNO parameer esimaes are less efficien han he GLS esimaes because of he reduced number of observaions used in esimaion. While i is known ha GLS is he preferred esimaor, he loss from using one of he inferior esimaors in small samples is no known. We use a Mone Carlo sudy o provide informaion abou he small-sample differences among he esimaors. 84

4 Mone Carlo Sudy A Mone Carlo sudy was conduced o deermine he size and power of he hypohesis ess when using overlapping daa and GLS, OLSNO, Newey-Wes, and unresriced MLE, esimaion mehods. The Mone Carlo sudy also provides a measure of he efficiency los from using OLSNO, Newey-Wes, and when he MA coefficiens are esimaed. The mean and he variance of he parameer esimaes are calculaed o measure bias and efficiency. Mean-squared error (MSE) is also compued. To deermine he size of he hypohesis ess, he percenage of he rejecions of he rue null hypoheses are calculaed. To deermine he power of he hypohesis ess he percenages of he rejecions of false null hypoheses are calculaed. 4.1 Mone Carlo Procedure Daa are generaed using Mone Carlo mehods. A single independen variable x wih an i.i.d. uniform disribuion (0,1) and error erms u wih a sandard normal disribuion are generaed. We also considered a N(0,1) for x bu hese resuls are no included here since he conclusions did no change. The opions RANUNI and RANNOR in SAS sofware are used. The dependen variable y is calculaed based on he relaion represened in equaion (1). For simpliciy β is assumed equal o one. The daa se wih overlapping observaions of X and Y is creaed by summing he x s and y s as in equaion (3). The regression defined in equaion () was esimaed using he se of daa conaining X and Y. The number of replicaions is 000. For each of he 000 original samples, differen vecors x and u are used. This is based on Edgeron s (1996) findings ha using sochasic exogenous variables in Mone Carlo sudies improves considerably he precision of he esimaes of power and size. Six sample sizes T are used, respecively, 30, 100, 00, 500, 1000, and 000. Three levels of overlapping k-1 are used, respecively, 1, 11, and 9. The level 11 is chosen because i corresponds o using annual changes when monhly daa are available. When auocorrelaion in x is large and he error erm follows a firs-order auoregressive process, Greene (1997, p. 589) finds ha he inefficiency of OLS relaive o GLS increases when he x s are posiively auocorrelaed. Since many real-world daases have explanaory variables ha are posiively auocorrelaed, he inefficiency of OLS found here may be conservaive. 85

The OLSNO, he Newey-Wes, and GLS esimaes of β were obained for each of he 000 samples using PROC IML in SAS sofware. The unresriced MLE esimaes of β were obained using PROC ARIMA in SAS. The Ω marix o be used in GLS esimaion was derived in equaion (8). The Newey-Wes esimaion was validaed by comparing i wih he available programmed esimaor in SHAZAM sofware using he OLS... /AUTCOV opion. The power of he ess are calculaed for he null hypohesis β = 0. 4. Resuls for he Exogenous Regressor Case The means of he parameer esimaes and heir sandard deviaions as well as he MSE values for he hree overlapping levels 1, 11, and 9, for he OLSNO, Newey-Wes, and GLS are presened in Tables, 3, and 4. The rue sandard deviaions for he GLS esimaion are lower han hose for he OLSNO and Newey-Wes esimaion. This demonsraes ha he Newey-Wes and OLSNO parameer esimaes are less efficien han he GLS esimaes. The inefficiency is greaer as he degree of overlapping increases and as he sample size decreases. For a sample size of 100 and overlapping level 9, he sample variance of he GLS esimaes is 0.119 while he sample variance of he Newey-Wes and OLSNO esimaes is.544 and 7.969, respecively. Besides he more efficien parameer esimaes, he difference beween he esimaed and acual sandard deviaions of he parameer esimaes are almos negligible for he GLS esimaion regardless of sample size or overlapping level. The esimaed sandard deviaions for he OLSNO esimaion show no biases as expeced. The Newey-Wes esimaion ends o underesimae he acual sandard deviaions even for overlapping level 1. The degree of underesimaion increases wih he increase of overlapping level and as sample size decreases. Someimes he esimaed sandard deviaion is only one-fourh of he rue value. The Newey-Wes covariance esimaes have previously been found o be biased downward in small samples (e.g. Goezmann and Jorion, 1993; Nelson and Kim, 1993; Brien-Jones and Neuberger, 004). The parameric boosrap suggesed by Mark (1995) and used by Irwin, Zulauf and Jackson (1996) can lead o ess wih correc size, bu sill uses he inefficien OLS esimaor. The inferioriy of he Newey-Wes and OLSNO parameer esimaes compared o he GLS esimaes is also suppored by he MSE values compued for he hree mehods of esimaion. Thus, for he sample size 100 and he overlapping level 9, he MSE for he GLS, Newey-Wes, and OLSNO esimaion is respecively 0.1,.55, and 8.0. 86

Table : Parameer esimaes, sandard deviaions, and MSE for OLSNO, Newey- Wes, and GLS esimaion (overlapping 1). GLS Esimaion Newey-Wes Esimaion Non-overlapping Esimaion Sample Size Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes 30 0.981 0.639 a 0.663 b 0.440 0.971 100 1.005 00 0.993 500 1.001 1000 1.001 000 1.00 0.348 a 0.345 b 0.119 0.996 0.46 a 0.44 b 0.060 0.993 0.155 a 0.154 b 0.04 1.003 0.110 a 0.109 b 0.01 0.997 0.077 a 0.08 b 0.007 0.998 Sandard Deviaions MSE Parameer Esimaes 0.631 a 0.808 b 0.654 0.970 0.374 a 0.43 b 0.179 0.997 0.69 a 0.303 b 0.09 0.989 0.17 a 0.189 b 0.036 1.001 0.1 a 0.134 b 0.018 1.005 0.086 a 0.098 b 0.010 1.00 Sandard Deviaions MSE 0.893 a 0.930 b 0.865 0.490 a 0.497 b 0.47 0.346 a 0.345 b 0.119 0.19 a 0.18 b 0.048 0.155 a 0.156 b 0.04 0.110 a 0.116 b 0.014 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These are he esimaed sandard deviaions of he parameer esimaes. b These are he acual sandard deviaions of he parameer esimaes. The model esimaed is equaion (), where Y and X represen some aggregaion of he original disaggregaed variables. For simpliciy β is chosen equal o 1. The model is esimaed using Mone Carlo mehods involving 000 replicaions. The errors for he original process are generaed from a sandard normal disribuion and are homoskedasic and no auocorrelaed. As a resul of he aggregaion, e follows an MA process wih he degree of he process depending on he aggregaion level applied o x and y. Table 3: Parameer esimaes, sandard deviaions, and MSE for OLSNO, Newey- Wes, and GLS esimaion (overlapping 11). GLS Esimaion Newey-Wes Esimaion Non-overlapping Esimaion Sample Size Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE 30 1.001 0.647 a 0.647 b 0.418 1.03 0.665 a 1.878 b 3.57 1.0.940 a 4.601 b 1.16 100 0.998 0.348 a 0.359 b 0.19 1.003 0.651 a 1.047 b 1.096 1.008 1.56 a 1.308 b 1.711 00 0.994 0.45 a 0.36 b 0.056 0.989 0.57 a 0.698 b 0.487 0.993 0.871 a 0.895 b 0.80 500 1.005 0.155 a 0.155 b 0.04 1.005 0.363 a 0.455 b 0.07 1.06 0.540 a 0.54 b 0.94 1000 0.997 0.110 a 0.11 b 0.013 1.004 0.6 a 0.315 b 0.099 1.00 0.38 a 0.390 b 0.15 000 0.995 0.078 a 0.077 b 0.006 0.999 0.189 a 0.3 b 0.050 0.999 0.70 a 0.7 b 0.074 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These are he esimaed sandard deviaions of he parameer esimaes. b These are he acual sandard deviaions of he parameer esimaes. 87

Table 4: Parameer esimaes, sandard deviaions, and MSE for OLSNO, Newey- Wes, and GLS esimaion (overlapping 9). GLS Esimaion Newey-Wes Esimaion Non-overlapping Esimaion Sample Size Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE 30 0.996 0.648 a 0.668 b 0.446 0.996 0.539 a.04 b 4.858 -- c -- c -- c -- c 100 1.005 0.349 a 0.345 b 0.119 1.077 0.711 a 1.595 b.551 1.33.8 a.83 b 8.03 00 0.996 0.45 a 0.48 b 0.06 1.016 0.694 a 1.16 b 1.478 0.988 1.467 a 1.571 b.469 500 1.005 0.155 a 0.158 b 0.05 1.09 0.53 a 0.76 b 0.58 1.05 0.867 a 0.893 b 0.798 1000 1.004 0.110 a 0.110 b 0.01 1.011 0.394 a 0.496 b 0.46 1.010 0.605 a 0.611 b 0.374 000 1.00 0.077 a 0.078 b 0.006 1.00 0.90 a 0.343 b 0.118 1.004 0.47 a 0.45 b 0.181 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These are he esimaed sandard deviaions of he parameer esimaes. b These are he acual sandard deviaions of he parameer esimaes. c These values canno be esimaed because of he very small number of observaions. The means of he parameer esimaes and heir sandard deviaions as well as he MSE values for he hree overlapping levels 1, 11, and 9, for he unresriced MLE are presened in Table 5. The resuls are similar o he resuls presened for he GLS esimaion. However, in small samples he acual sandard deviaions of he MLE esimaes are larger han hose of he GLS esimaes. As he degree of overlapping increases, he sample size for which he sandard deviaions for boh mehods are similar, also increases (e.g. from 100 for overlapping 1 o 1000 for overlapping 9). 88

Table 5: Parameer esimaes, sandard deviaions, and MSE for he maximum likelihood esimaes assuming he MA coefficiens are unknown for hree levels of overlapping (1, 11, and 9). Overlapping 1 Overlapping 11 Overlapping 9 Sample Size Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE 30 0.975 0.6 a 0.64 b 0.391 1.019 0.541 a 0.833 b 0.694 - c - c - c - c 100 1.010 0.343 a 0.347 b 0.10 0.998 0.311 a 0.374 b 0.140 0.991 0.81 a 0.455 b 0.07 00 0.989 0.43 a 0.47 b 0.061 0.995 0.30 a 0.56 b 0.065 0.984 0.16 a 0.78 b 0.078 500 0.990 0.154 a 0.156 b 0.05 0.990 0.149 a 0.158 b 0.05 0.986 0.145 a 0.165 b 0.07 1000 0.991 0.11 a 0.109 b 0.013 0.991 0.107 a 0.11 b 0.013 0.990 0.105 a 0.11 b 0.013 000 0.995 0.078 a 0.077 b 0.006 0.995 0.076 a 0.078 b 0.006 0.995 0.075 a 0.080 b 0.006 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These are he esimaed sandard deviaions of he parameer esimaes. b These are he acual sandard deviaions of he parameer esimaes. c These values canno be esimaed because of he very small number of observaions. The Newey-Wes and OLSNO esimaion mehods also perform considerably poorer han he GLS esimaion in hypohesis esing. The hypohesis esing resuls are presened in Table 6. The Newey-Wes esimaor rejecs rue null hypoheses far oo ofen. In one exreme case, i rejeced a rue null hypohesis 50.0% of he ime insead of he expeced 5%. In spie of grealy underesimaing sandard deviaions, he Newey-Wes esimaor has considerably less power han GLS excep wih he smalles sample sizes considered. While he OLSNO esimaion has he correc size, he power of he hypohesis ess is much less han he power of he ess wih GLS. The resuls of he hypohesis ess for he unresriced MLE are presened in Table 7. While he power of he hypohesis ess is similar o he power for he GLS esimaion, he size is generally larger han he size for he GLS esimaion. Unresriced MLE ends o rejec rue null hypoheses more ofen han i should. However, his problem is reduced or eliminaed as larger samples are used, i.e. 500, 1000, 000 observaions. Table 7 also presens he number of replicaions as well as he number/percenage of replicaions ha converge. Fewer replicaions converge as he degree of overlap increases and as sample size decreases. Given he convergence problems, as shown in Table 7, i can be concluded 89

ha, when MLE is chosen as he mehod of esimaing equaion (), he MA coefficiens should be resriced raher han esimaed unless he sample size is quie large. Table 6: Power and Size Values of he Hypohesis Tess for OLSNO, Newey-Wes, and GLS Esimaion (Overlapping 1, 11, 9). Degree of Overlapping Sample Size GLS Esimaion Newey-Wes Esimaion Non-overlapping Esimaion Power Size Power Size Power Size 1 30 0.319 0.05 0.366 0.135 0.181 0.044 100 1 0.043 0.500 0.090 0.500 0.05 00 1 0.04 1 0.081 1 0.049 500 1 0.053 1 0.078 1 0.05 1000 1 0.049 1 0.075 1 0.056 000 1 0.058 1 0.089 1 0.07 11 30 0.315 0.044 0.500 0.49 0.045 0.044 100 1 0.056 0.434 0.54 0.111 0.046 00 1 0.039 0.486 0.169 0.194 0.045 500 1 0.048 0.500 0.14 0.455 0.050 1000 1 0.053 1 0.104 0.500 0.051 000 1 0.046 0.997 0.094 0.958 0.049 9 30 0.340 0.049 0.500 0.500 -- a -- a 100 1 0.044 0.500 0.417 0.070 0.056 00 1 0.055 0.449 0.91 0.070 0.046 500 1 0.061 0.500 0.176 0.03 0.044 1000 1 0.050 0.500 0.13 0.364 0.055 000 1 0.059 0.885 0.113 0.646 0.051 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These values canno be esimaed because of he very small number of observaions. 90

Table 7: Power and size values of he hypohesis ess for he maximum likelihood esimaes assuming he MA coefficiens are unknown for hree levels of overlap (1, 11, and 9). Degree of Overlap Sample Size Toal Number of Replicaions Replicaions ha Converge Power b Size b Number Percenage 1 30 1000 999 99.9 0.331 0.070 100 1000 1000 100 0.87 0.047 00 1000 1000 100 0.98 0.058 500 1000 1000 100 1.000 0.060 1000 1000 1000 100 1.000 0.06 000 1000 1000 100 1.000 0.051 11 30 1400 994 71.0 0.476 0.5 100 1000 995 99.5 0.884 0.109 00 1000 1000 100 0.980 0.085 500 1000 998 99.8 0.998 0.075 1000 1000 1000 100 1.000 0.069 000 1000 1000 100 1.000 0.056 9 30 -- a -- a -- a -- a -- a 100 1600 970 60.6 0.814 0.54 00 100 107 85.6 0.980 0.135 500 100 108 90. 1.000 0.081 1000 1100 1066 96.9 1.000 0.078 000 1000 93 93. 1.000 0.060 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These values canno be esimaed because of he very small number of observaions. b These are calculaed based on he number of replicaions ha converged. 5 Possible Saisical Reasons for Using Overlapping Daa If he explanaory variables were sricly exogenous, no observaions were missing, and he errors were disribued normally as assumed so far, here are no saisical reasons o use overlapping daa since he disaggregae model could be esimaed. We now consider possible saisical reasons for using overlapping daa. 91

5.1 Missing Observaions Missing observaions can be a reason o use overlapping daa. I is no unusual in sudies of economic growh o have key variables observed only every five or en years a he sar of he observaion period, bu every year in more recen years. Using overlapping daa allows using all of he daa. In his case, he disaggregae model canno be esimaed so OLSNO is wha has been used in he pas. When some observaions are missing, one can derive he correlaion marix in equaion (8) as if all observaions were available and hen delee he respecive rows and columns for he missing overlapping observaions and hus use GLS esimaion. The Newey-Wes esimaor assumes auocovariance saionariy and so available sofware packages ha include he Newey-Wes esimaor would no correcly handle missing observaions. I should, however, be possible o modify he Newey-Wes esimaor o handle missing observaions. From his discussion i can be argued ha he case of missing observaions is a saisical reason for using overlapping daa ha sands up o scruiny bu more efficien esimaors are available han he ofen used OLSNO. 5. Nonnormaliy The GLS esimaor does no assume normaliy, so esimaes wih GLS would remain bes linear unbiased and asympoically efficien even under nonnormaliy. The hypohesis ess derived, however, depend on normaliy. Hypohesis ess based on normaliy would sill be valid asympoically provided he assumpions of he cenral limi heorem hold. As he degree of overlapping increases, he residuals would approach normaliy, so nonnormaliy would be less of a concern. The Newey-Wes esimaor is also only asympoically valid. The GLS ransformaion of he residuals migh also speed he rae of convergence oward normaliy since i is averaging across more observaions han he OLS esimaor used wih Newey-Wes. We esimaed equaion () wih wo correlaed x s and wih he error erm u following a -disribuion wih four degrees of freedom. Resuls are repored in Table 8. The main difference wih he previous resuls is he increased sandard deviaions for all mehods of esimaion. Proporionally, he increase in sandard deviaions is slighly larger for Newey- Wes and OLSNO. Thus, he Mone Carlo resuls suppor our hypohesis ha he 9

advanages of GLS would be even greaer in he presence of nonnormaliy. This can also be seen from he hypohesis es resuls presened in Table 8. The power of he hree mehods of esimaion is reduced wih he bigges reducion occurring for he Newey-Wes and OLSNO. Finally, he increase of he sandard deviaions and he resuling reducion in he power of hypohesis ess, is larger when he correlaion beween he wo x s increases. This is rue for he hree mehods of esimaion. However, he GLS resuls are almos idenical o he resuls from he disaggregae model. This means ha lack of normaliy canno be a valid saisical reason for using overlapping daa when he disaggregae daa are available. 93

Table 8: Parameer esimaes, sandard deviaions, MSE, and power and size of hypohesis ess for OLSNO, Newey-Wes, and GLS esimaion wih wo Xs and nonnormal errors (overlapping 1, 11, and 9). Degree of Overlap Sample Size Parameer Esimaes GLS Esimaion Newey-Wes Esimaion Non-overlapping Esimaion Disaggregae Esimaion Sandard Deviaions 1 30 1.014 0.953 a 100 0.969 0.498 a 500 1.008 0.6 a 1000 1.004 0.159 a 11 30 1.019 0.943 a 100 0.994 0.507 a 500 1.008 0.6 a 1000 1.003 0.159 a 9 30 1.014 0.935 a 100 1.009 0.507 a 500 1.010 0.6 a 1000 1.000 0.160 a MSE Power Size Parameer Esimaes Sandard Deviaions MSE Power Size Parameer Esimaes Sandard Deviaions 1.003 b 1.007 0.08 0.046 0.997 0.898 a 1.67 b 1.606 0.88 0.15 1.049 1.334 a 0.510 b 0.61 0.494 0.053 0.969 0.56 a 0.61 b 0.386 0.460 0.095 0.999 0.700 a 0.3 b 0.050 0.988 0.051 1.005 0.49 a 0.73 b 0.074 0.956 0.08 0.996 0.317 a 0.155 b 0.04 1 0.04 1.001 0.177 a 0.19 b 0.037 0.999 0.070 1.00 0.5 a 0.943 b 0.890 0.0 0.049 0.977 0.830 a.585 b 6.684 0.579 0.541 -- c -- c MSE Power Size Parameer Esimaes Sandard Deviaions MSE Power Size 3.0 0.01 0.18 1.01 0.977 a 1.058 0.19 0.050 1.794 b 1.09 b 0.766 0.34 0.111 0.970 0.517 a 0.76 0.467 0.05 0.875 b 0.55 b 0.15 0.83 0.117 1.008 0.36 a 0.054 0.983 0.055 0.390 b 0.33 b 0.08 0.971 0.11 1.004 0.166 a 0.07 1 0.039 0.86 b 0.163 b -- c -- c -- c -- c 1.010 0.833 a 0.870 b 0.757 0.39 0.053 0.53 b 0.74 0.498 0.05 0.998 0.915 a 1.48 b.196 0.338 0.44 0.944.059 a.30 b 4.975 0.07 0.051 1.001 0.50 a 0.53 b 0.74 0.516 0.053 0.5 b 0.051 0.993 0.049 1.010 0.54 a 0.663 b 0.439 0.517 0.138 1.035 0.810 a 0.159 b 0.05 1 0.04 1.0 0.378 a 0.457 b 0.09 0.734 0.107 1.016 0.557 a 0.995 b 0.990 0.193 0.056 1.014 0.654 a.614 b 6.833 0.69 0.611 -- c -- c 0.543 b 0.94 0.513 0.046 0.995 0.911 a.38 b 5.40 0.505 0.455 0.98 4.919 a 0.5 b 0.051 0.989 0.050 0.958 0.759 a 1.041 b 1.085 0.335 0.177 0.950 1.350 a 0.16 b 0.06 1 0.058 1.008 0.570 a 0.739 b 0.547 0.464 0.143 1.03 0.898 a 0.687 0.36 0.056 1.007 0.33 a 0.054 0.988 0.05 0.88 b 0.33 b 0.33 0.43 0.057 1.00 0.164 a 0.07 1 0.040 0.568 b 0.166 b -- c -- c -- c -- c 0.995 0.680 a 0.76 b 0.57 0.319 0.051 81.94 0.063 0.059 1.00 0.466 a 0.37 0.599 0.041 9.05 b 0.486 b 1.90 0.103 0.05 1.009 0.8 a 0.05 0.988 0.046 1.385 b 0.9 b 0.818 0.00 0.056 1.001 0.164 a 0.08 1 0.061 0.904 b 0.168 b Noes: The sample sizes are he sizes for samples wih overlapping observaions. a These are he esimaed sandard deviaions of he parameer esimaes. b These are he acual sandard deviaions of he parameer esimaes. c These values canno be esimaed because of he very small number of observaions. 94

5.3 Errors in Variables The mos common reason auhors give for using overlapping daa is errors in he explanaory variables. Errors in he explanaory variables causes parameer esimaes o be biased oward zero, even asympoically. Using overlapping daa reduces his problem, bu he problem is only oally removed as he level of overlap, k, approaches infiniy. We added o he x in equaion (1) a measuremen error, ω, ha is disribued normally wih he same variance as he variance of x, ω ~ N(0, 1/1). We hen conduced he Mone Carlo sudy wih x no being auocorrelaed and also wih x being auocorrelaed wih an auoregressive coefficien of 0.8. In addiion o esimaing equaion () wih GLS, Newey- Wes, and OLSNO, we also esimaed equaion (1) using he disaggregae daa. The resuls are repored in Table 9. The esimaion was performed only for wo sample sizes, respecively 100 and 1000 observaions. In he case when x is no auocorrelaed, here is no gain in using overlapping observaions, in erms of reducing he bias due o measuremen error. This is rue for all mehods of esimaion. GLS would be he preferred esimaor since i is always superior o Newey-Wes and OLSNO in erms of MSE, especially as he overlapping level increases relaive o he sample size. In he case when x is auocorrelaed, for relaively low level of overlap in relaion o he sample size, Newey-Wes and OLSNO have he smaller MSE, as a resul of smaller bias. As he degree of overlap increases relaive o he same sample size, he GLS esimaor would be preferred compared o Newey-Wes and OLSNO esimaors based on smaller MSE as a resul of he smaller variance. Thus he rade-off for he researcher is beween less biased parameer esimaes wih Newey-Wes or OLSNO versus smaller sandard deviaions for he parameer esimaes wih GLS. However, he GLS ransformaion of he variables does no reduce furher he measuremen error producing esimaes ha are jus barely less biased han he disaggregae esimaes. On he oher hand, Newey-Wes and OLSNO sandard errors are sill biased wih he bias increasing as he overlapping level increases. So he preferred esimaion mehod in he presence of large errors in he variables would be OLS wih overlapping daa and wih sandard errors calculaed using Mone Carlo mehods. 95

Table 9: Parameer esimaes, sandard deviaions, and MSE, for GLS, Newey-Wes, OLSNO, and he disaggregae esimaion wih measuremen errors in X (overlapping 1, 11, and 9). Correlaion of X Sample Size Degree of Overlap Parameer Esimaes GLS Esimaion Newey-Wes Esimaion Non-overlapping Esimaion Disaggregae Esimaion Sandard Deviaions 0 100 1 0.494 0.5 a 11 0.509 0.5 a 9 0.495 0.53 a 1000 1 0.499 0.079 a 11 0.50 0.079 a 9 0.499 0.079 a 0.8 c 100 1 0.718 0.191 a 11 0.731 0.187 a 9 0.730 0.186 a 1000 1 0.735 0.058 a 11 0.733 0.058 a 9 0.736 0.058 a MSE Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions MSE Parameer Esimaes Sandard Deviaions 0.30 0.493 0.69 a 0.354 0.494 0.360 a 0.389 0.494 0.50 a 0.318 0.5 b 0.311 b 0.361 b 0.50 b 0.310 0.51 0.479 a 0.784 0.503 0.95 a 1.303 0.510 0.39 a 0.303 0.63 b 0.739 b 1.08 b 0.51 b 0.54 b 0.30 0.480 0.501 a 1.185 b 1.675 0.390 1.789 a.310 b 5.709 0.497 0. a 0.3 b 0.303 0.57 0.50 0.088 a 0.57 0.501 0.11 a 0.61 0.499 0.079 a 0.57 0.077 b 0.095 b 0.111 b 0.077 b 0.55 0.499 0.189 a 0.303 0.497 0.77 a 0.33 0.501 0.079 a 0.55 0.080 b 0.7 b 0.81 b 0.080 b 0.57 0.517 0.85 a 0.366 0.509 0.441 a 0.440 0.499 0.078 a 0.57 0.078 b 0.364 b 0.445 b 0.077 b 0.119 0.816 0.174 a 0.080 0.816 0.18 a 0.084 0.716 0.190 a 0.10 0.199 b 0.14 b 0.3 b 0.198 b 0.111 0.931 0.187 a 0.096 0.934 0.337 a 0.17 0.71 0.181 a 0.113 0.196 b 0.30 b 0.351 b 0.187 b 0.110 0.963 0.174 a 0.186 0.966 0.536 a 0.493 0.70 0.166 a 0.109 0.194 b 0.49 b 0.701 b 0.174 b 0.074 0.833 0.055 a 0.03 0.83 0.066 a 0.033 0.734 0.058 a 0.074 0.060 b 0.065 b 0.067 b 0.060 b 0.075 0.940 0.071 a 0.011 0.941 0.096 a 0.013 0.73 0.058 a 0.075 0.06 b 0.086 b 0.097 b 0.06 b 0.073 0.954 0.091 a 0.016 0.950 0.135 a 0.01 0.735 0.057 a 0.074 0.061 b 0.116 b 0.138 b 0.060 b Noe: The sample sizes are he sizes for samples wih overlapping observaions. a These are he esimaed sandard deviaions of he parameer esimaes. b These are he acual sandard deviaions of he parameer esimaes. c The x is generaed as follows: x = x 0 + ω, where x 0 ~ uniform (0, 1) and ω ~ N (0, 1/1). MSE 96

6 Possible Economic Reasons for Using Overlapping Daa The economic reasons for using overlapping daa ypically involve lagged variables. Even hough in mos cases where overlapping daa are used, here are no lagged variables, i is imporan o consider he case because i is he case ha has generaed he mos economeric research. The lagged variable may be sricly exogenous or be a lagged value(s) of he dependen variable. Wih lagged variables and overlapping daa, GLS is generally inconsisen and so he siuaion has generaed considerable research ineres. A second economic reason for using overlapping daa is he case of long-horizon regressions. Examples of long-horizon regressions include he case of expeced sock reurn (as dependen variable) and dividend yield (as explanaory variable) and also he case of he GDP growh and nominal money supply. Since he economic reason for using overlapping daa is increased predicion accuracy, we compare predicion errors when using overlapping daa o hose using disaggregae daa. 6.1 Lagged Dependen Variables The case of overlapping daa and a lagged dependen variable (or some oher variable ha is no sricly exogenous) was a primary moivaion for Hansen and Hodrick s (1980) esimaor. In he exbook case of auocorrelaion and a lagged dependen variable, ordinary leas squares esimaors are inconsisen. Engle (1969) shows ha when he firs lag of he aggregaed dependen variable is used as an explanaory variable, ha using OLS and aggregaed daa could lead o bias of eiher sign and almos any magniude. Generalized leas squares is also inconsisen. Consisen esimaes can be obained using he maximum likelihood mehods developed for ime-series models. Wih a lagged dependen variable in he righ-hand side (for simpliciy we are using a lag order of one), equaion (1) now becomes: y = α0 + α1y 1+ αx1 + u, u ~ N(0,1), (11) 97

where for simpliciyα 0 = 0 and α = 1. The value seleced for α 1 = 1 is 0.5. To ge he overlapping observaions, for k = 3 apply equaion (3) o (11) o obain he equivalen model of equaion () as Y = 0.5 Y + X + e, (1) 1 where Y = y1+ y 1+ y, X = x + x 1+ x, and e = u + u 1+ u. The model in equaion (1) also has he same variance-covariance marix, described by equaions (5) and (6), as our previous model in equaion (). If Y and X are observed in every ime period, he order of he lagged variables as well as he auoregressive (AR) and MA orders can be derived analyically. For a deailed discussion of he issues relaed o emporal aggregaion of ime series see Marcellino (1996, 1999). 3 If Y, or X, or none of hem are observed in every ime period, hen eiher lagged values for aggregae Y, (Y -1 and Y - ), or X, (X -1 and X - ), or boh Y and X are no observable. The model usually esimaed in his second siuaion is: 4 Y = 0.5 Y + β X + β X + υ, (13) k τ τ 1 τ 3 τ 1 τ where, τ represens every k h observaion. Assuming he daa are observed every ime period equaion (13) is equivalen o: 5 Y = 0.5 Y + X + 0.5X + 0.5 X + ε. (14) 3 3 1 The resuling error erm ε in equaion (14) is a MA process of order four of he error erm u in equaion (11): ε = u + 1.5u 1 + 1.75u + 0.75u 3 + 0. 5u 4. One poenial problem wih model (14) is he noise inroduced by aggregaion. The variables X 1 and X include x 1, x, x 3, and x 4, while only x 1, and x are relevan as shown by he model in equaion (1). This errors-in-variables problem biases parameer esimaes oward zero. The noise inroduced and he associaed bias would be greaer as he degree of overlap increases. The errors-in-variables problem is even bigger for model (13), where x 5 is also included in he model hrough X τ -1. An analyical soluion for he β s in equaion (13) canno be derived. This is because o be consisen wih our previous resul, X is sricly exogenous and no auocorrelaed. Based on he emporal aggregaion lieraure (Brewer, 1973, p. 141; Weiss, 1984, p. 7; and 3 See also Brewer (1973), Wei (1981), and Weiss (1984). 4 The model considered by Hansen and Hodrick (1980) is Y = β + βy + β Y. 1 3 4 5 Equaion (14) is obained by subsiuing for Y -1 and hen for Y - in equaion (1). 98

Marcellino, 1996, p. 3), no analyical soluion is possible unless x is generaed by some auocorrelaed process and he unobserved erms can be derived from he observed erms. Finally, anoher possible model using nonoverlapping observaions for Y and overlapping observaions for X is: Y = 0.5 Y + X + 0.5X + 0.5 X + η. (15) 3 τ τ 1 1 τ In general, in cases when nonoverlapping daa are used, as also is he case of Hansen and Hodrick s (1980) esimaor, Marcellino (1996, 1999) shows ha esimaes of he parameers of he disaggregaed process can no longer be recovered. Wih nonoverlapping daa, he ime-series process can be quie differen han he original process. We esimaed he models in equaions (1), (13), and (15) wih MLE employing PROC ARIMA in SAS sofware using a large Mone Carlo sample of 500,000 observaions. There is no need o esimae he model in equaion (14) since he model in equaion (1) can be esimaed when overlapping daa are available for boh Y and X. The resuls are repored in Table 10. The empirical esimaes of he AR and MA coefficiens and he coefficiens of he Xs for he models in equaion (1) fully suppor he analyic findings. The parameer esimaes for he exogenous variables in equaion (15) are similar o he analyical values. On he oher hand, he parameer esimaes for he exogenous variables in equaion (15) are very differen from he analyical values derived for eiher equaion (1) or (14) because of he differen lagged values of he exogenous variable included in he model. Boh models (13) and (15) resul in an ARMA(1,1) process wih he AR coefficien 0.118 for equaion (13) and 0.13 for equaion (15). The MA coefficien is he same for boh models, 0.163. As noed above, he AR and MA coefficiens for equaions (13) and (15) are differen from he respecive coefficiens of he disaggregae model. Wih overlapping daa and a lagged dependen value as an explanaory variable where he lag is less han he level of overlap, he only consisen esimaion mehod is maximum likelihood. Maximum likelihood provides consisen esimaes when he explanaory variables are predeermined wheher or no hey are sricly exogenous. When overlapping daa are used for boh he dependen and independen variables he parameers of he aggregae model are he same as hose of he disaggregae model. When nonoverlapping daa are used for he dependen or independen, or boh he dependen and independen variables he parameers of he aggregae model canno be used o recover hose of he disaggregae model. 99

Table 10: Parameer esimaes of differen models for he case of he lagged dependen variable. QASS, Vol. 3 (3), 009, 78-115 Equaion Number Mehod of Esimaion Daa Esimaed Model (1) MLE Overlapping Y = 0.0016 + 0.496Y 1+ 1.0065X + ε + ε 1 + 0. 99999ε (13) MLE Nonoverlapping Y τ = 0.019 + 0.118Yτ 3 + 1.413Xτ + 0.34Xτ 3 + ετ + 0. 163τ 3 (15) MLE Y Nonoverlapping X Overlapping Y τ = 0.019 + 0.13Y τ 1 + 1.00X 1 1+ 0.489 + 0.51X 3 + ετ + 0. 163ε τ 1 Noe: The models in Table 10 are esimaed using a large Mone Carlo sample of 500,000 observaions. The unresriced maximum likelihood esimaes are obained using PROC ARIMA in SAS. 100

6. The Case of Long-Horizon Regressions wih Overlapping Observaions The underlying daa-generaed processes commonly assumed are: y = + x + u (16) x α 1 β, μ ρ x 1 ν, = + + (17) E[( u, ν,) ( u, ν,)] = = [ σ σ ; σ σ ]. (18) 11 1 1 The effec of he assumpion in equaion (17) is ha he covariance beween he error erms in equaion (16) ha are one period apar conains addiional erms as compared o equaion (6): cov[ e, e 1] E[ ee 1] ( k 1) σ u β (1 ρ) + + σv. = = + + (19) The oher covariance erms are as in equaion (6). The long-horizon variables, Y and X -1 are creaed as in equaion (3). Then he esimaed models include: Y = + X + e (0) α k 1 β. As Valkanov (003, p. 05) argues Inuiively, he aggregaion of a series ino a longhorizon variable is hough o srenghen he signal, while eliminaing he noise. Several sudies have aemped o provide more efficien esimaors of he sandard errors compared o he ordinary leas squares esimaes. 6 Valkanov (003) suggess rescaling of he - saisic by he square roo of he sample size, Hjalmarsson (004) suggess rescaling he - saisic by he square roo of he level of aggregaion, k, and Hansen and Tuypens (004) sugges rescaling he -saisic by he square roo of ⅔ of he level of aggregaion, k. Two scenarios are considered. The firs one is he scenario where he reurns are unpredicable which implies ha β = 0. The second is when reurns are predicable, so ha β 0 (sudies usually assume β = 1). The firs scenario is exacly he case considered in Secion, so he conclusions from Secion apply here. For he case when reurns are predicable, β = 1, we perform Mone Carlo simulaions o compare he sandard errors of he GLS esimaor and he OLS sandard errors rescaled by he square roo of he sample size (Valkanov, 003), by he square roo of he level of 6 We also conduced Mone Carlo simulaions for he ransformaions proposed by Brien-Jones and Neuberger (004) o his model. Resuls are no repored since he β esimaes from he BJN ransformaions were inconsisen. 101

aggregaion (Hjalmarsson, 004) and by he square roo of ⅔ of he level of aggregaion (Hansen and Tuypens, 004). We use wo levels of aggregaion: k=1 wih sample sizes, 50, 100, 50, 500 and 1000, and k=75 wih sample sizes, 00, 50, 500, 750 and 1000. 5000 replicaions are done for each case. We conduc Mone Carlo simulaions for commonly used assumpions of ρ = 1 in equaion (19) and σ = σ 1 =± 0.9 in equaion 1 (18). We also conduc simulaions by changing ρ o 0.9 and σ 1 o 0.5 and 0.1. In addiion, we assume α = μ =0. A summary of he resuls from he above simulaion follows. In general, all esimaors and rescaling approaches produce very good power agains he alernaive hypohesis β = 0. However, his is no rue for he size of he ess. Size is he criical issue when using an approximaion like his because a es should be conservaive so ha if he es rejecs he null hypohesis, he researcher can be confiden ha he conclusion is correc. For he high absolue values of σ 1 considered above, respecively 0.9 and -0.9, he mos promising approach is he one suggesed by Hansen and Tuypens (004) where sandard errors are rescaled by he square roo of ⅔ of he level of aggregaion. However, his approach sill produces correc es sizes only when he raio level of aggregaion/sample size is close o 1/10. Tes sizes are greaer han he nominal size when he raio level of aggregaion/sample size is less han 1/10 and less han he nominal size when he raio level of aggregaion/sample size is greaer han 1/10. Our simulaions sugges ha beer es sizes are produced when he following adjusmen is applied o he Hansen and Tuypens approach. When he raio is less han 1/10, he adjusmen is (. k ) ; when he raio is greaer han 1/10 he adjusmen is 3 k k 09. + n 3 09 1. Tes sizes for he Hansen and Tuypens (HT) and our modified version of he HT approach are repored in Table 11. Table 11 also repors es sizes for he cases when σ 1 equals 0.5 and 0.1. In he case when σ 1 = 0.5, he modified Hjalmarsson (004) rescaling of he sandard errors by he square roo of he level of aggregaion produces beer es sizes for differen sample/aggregaion level combinaion. In his case, for raios less han 1/10, he adjusmen is ( 09. k 1), and for raios greaer han 1/10 he adjusmen is k k 09. + n. Finally, when σ 1 = 0.1, he GLS sandard errors produce good es sizes. This is no surprising since a small σ 1 brings us closer o he case of exogenous independen variables discussed in Secion. Therefore, no modified 10

rescaling is needed in his las case. However, we also repor in Table 11 he es sizes for he unmodified Hjalmarsson (004) rescaling as a comparison o he GLS es sizes. Table 11: Size of hypoheses ess (=1β ) for he Hansen and Tuypens (HT) and Hjalmarsson (H) rescaling and our modified HT and H approaches for long-horizon reurn regressions (overlapping 11 and 74). σ 1 = 0.9 Overlapping level 11 Sample Size 50 100 50 500 1000 Size of HT 0.06 0.0504 0.0314 0.030 0.080 Size of Modified HTa 0.0554 0.0504b 0.0464 0.0430 0.040 Overlapping level 74 Sample Size 00 50 500 750 1000 Size of HT 0.078 0.074 0.0570 0.0458 0.040 Size of Modified HTa 0.05 0.0538 0.054 0.0458b 0.0516 σ1 = 0.5 Overlapping level 11 Sample Size 50 100 50 500 1000 Size of H 0.0784 0.0608 0.0394 0.0308 0.0316 Size of Modified H c 0.0510 0.0608b 0.0574 0.0498 0.0488 Overlapping level 74 Sample Size 00 50 500 750 1000 Size of H 0.096 0.08 0.0668 0.0510 0.0504 Size of Modified Hc 0.0576 0.056 0.0550 0.0510b 0.0564 σ1 = 0.1 Overlapping level 11 Sample Size 50 100 50 500 1000 Size of H 0.1014 0.078 0.0558 0.0480 0.0446 Size of GLS 0.0506 0.0530 0.0510 0.0468 0.0446 Overlapping level 74 Sample Size 00 50 500 750 1000 Size of H 0.100 0.1096 0.0900 0.0734 0.0730 Size of GLS 0.0518 0.050 0.054 0.0504 0.054 Noes: The sample sizes are he sizes for samples wih overlapping observaions. a This is he modified Hansen and Tuypens rescaling. b No modificaion is performed when he raio equals 1/10. c This is he modified Hjalmarsson rescaling. 103

6.3 Predicion Accuracy In his secion we compare he predicion accuracy of he aggregae and disaggregae models. We also compare he predicion accuracy for he Hansen and Hodrick (HH) model given in foonoe 4. The disaggregae model wih he HH esimaor is similar o he aggregae model wih he only change being he dependen variable is disaggregae (oneperiod change). Numerous auhors have argued for using overlapping daa when predicing muliperiod changes. Bhansali (1999), however, reviews he heoreical lieraure and finds no heoreical advanage o using overlapping daa when he number of lags is known. Marcellino, Sock, and Wason (006) consider a number of ime series and find no empirical advanage o using overlapping daa even when using preesing o deermine he number of lags o include. Since he models here are known, heory predics no advanage o using overlapping daa. The Mone Carlo simulaion involves generaing 50,000 sample pairs. The firs sample is used o obain he parameer esimaes for he aggregae and disaggregae models for all cases. The second sample is used o obain prediced values by uilizing he parameer esimaes for he firs sample. Dynamic forecasing is used o obain prediced values of he lagged values in he disaggregae model. The means and sandard deviaions of he 50,000 roo-mean-squared forecas errors are repored in Table 1. Two levels of aggregaion, 1 wih five sample sizes, 50, 100, 50, 500, and 1000, and 75 wih sample sizes 100, 50, 500, 750, and 1000, are used in he simulaions. In he case of long horizon regressions he aggregae and disaggregae model are roughly equal in forecas accuracy as expeced. In he case of he HH model for low levels of aggregaion (level 1) he differences beween he aggregae and he disaggregae models are small. Wih he high level of aggregaion, he disaggregae model someimes ouperforms he aggregae one. If he economic goal is predicion, overlapping daa migh be preferred if hey make calculaions easier since here is lile difference in forecas accuracy. 104