Estmatng the Moments of Informaton Flow and Recoverng the Normalty of Asset Returns Ané and Geman (Journal of Fnance, 2000) Revsted Anthony Murphy, Nuffeld College, Oxford Marwan Izzeldn, Unversty of Lecester Paper presented at 2006 Econometrc Study Group Conference, Brstol
BACKGROUND A great deal of tme and effort has gone nto examnng the relatonshp between the volatlty of returns and measures of market actvty, such as volume and the number of transactons. Clark (1973) was the frst to propose usng a stochastc clock as a tme changer n order to recover the normalty of returns. Clark wrote the prce process for cotton futures as a subordnated process, where the economc nterpretaton of the subordnator was the cumulatve volume of traded contracts. - 2 -
Ané and Geman (AG, 2000), revst Clark's method of dealng wth the non-normalty of observed returns. AG consdered a general tme change process. Usng a novel non-parametrc procedure, AG clam to have: Recovered the moments of the tme change / nformaton flow process from return data; Fnd that the 2 nd... 6 th moments of the nformaton flow process match the 2 nd... 6 th moments of the observed number of transactons, but not those of volume, usng hgh frequency data for Csco and Intel. - 3 -
Geman (2005) summarzed the results as showng, "that n order to recover a quas perfect normalty of returns, the transactons clock s better represented by the number of trades than the volume". Wdely cted, albet rather controversal, results. On the one hand t s consstent wth the fndngs of Hasbrouk (1999) and others. On the other hand, L (2004) and Izzeldn (2005), for example, are unable to reproduce ths fndng. - 4 -
Our Contrbuton We nvestgate the procedure used by Ané and Geman (AG 2000) to recover the moments of the unobserved nformaton flow. We explan why ther procedure s lkely to be unrelable. Our Monte Carlo experments show that the thrd and hgher moments of the nformaton flow cannot be accurately recovered. We also show that, contrary to the clams n AG, returns condtoned on the recentered number of trades are not approxmately Gaussan. - 5 -
RECOVERING THE MOMENT OF THE STOCHASTIC TIME CHANGER AG consder a general return process: r(t) =x((t)) where: x(.) s a Brownan moton and (t) s some stochastc tme change or nformaton flow process, whch may nclude a jump component. Ths generalzes the subordnated process n Clark (1973). - 6 -
The Brownan moton assumpton s nnocuous, snce arbtrage-free return process can be expressed as a tme-changed Brownan moton process (Monroe, 1978). The dscrete tme verson of the AG return process s smpler to use. AG assume that, condtonal on the exogenous tme changer / nformaton flow t, returns rt are dstrbuted as a normal mxture: r N 2 t t ( µ rt, σ r t ) Let m 1,..., m 6 denote the frst sx (central) moments of t. - 7 -
Easy to show that the frst sx uncondtonal (central) moments of rt are: m = µ m r 1 r 1 m = σ m + µ m r 2 2 2 r 1 r 2 m = 3µσ m + µ m r 2 3 3 r r 2 r 3 m = µ m + 6σ µ m + 6σ µ mm + 3 σ ( m + ( m ) ) r 4 2 2 2 2 4 2 4 r 4 r r 3 r r 1 2 r 2 1 m = µ m + 10 σ µ m + (10 σ µ m + 15 µσ ) m + 30 µσ m r 5 2 3 2 3 4 4 5 r 5 r r 4 r r 1 r r 3 r r 2 m = µ m + 15 σ ( m + 3 mm + ( m ) ) + 15 µ σ r 6 6 3 4 6 r 6 r 3 2 1 1 r r + 45 µ σ ( m + 2 mm + m ( m ) ) 2 4 2 r r 4 3 1 2 1 2 ( m + mm ) 5 4 AG want to recover 2 1 6 µ r, σ r, m,..., m. Many dentfcaton ssues arse. - 8 -
Identfcaton Issues 2 Frstly, µ r and σ r are only dentfed up to scale, snce t s not observed. One soluton to ths scalng problem s to normalze 1 m, the mean of the unobserved nformaton flow process, to one. It s not clear how AG deal wth ths ssue. Secondly, wth hgh frequency data, returns are close to zero on average so, a pror, t s plausble to set µ r to zero. However, when µ r = 0, the three uneven moment condtons are dentcally zero so dentfcaton - 9 -
becomes even more problematc. Fnally, even when µ r s non zero and m 1 s set to one, 2 the seven remanng parameters ( µ r, σ r, m 2,, m 6 ) are not dentfed from sx moment condtons. They can only be recovered f addtonal restrctons are mposed or addtonal moment condtons are added. - 10 -
Addtonal Moments Approxmate MGF AG augment the sx moment condtons wth a number of approxmate moment generatng functon (MGF) condtons: 2 3 4 E [exp( r t )] exp( m 1 ) 1 α m α 2 m α β α + + 3 + m 4, 2 6 24 where = + and dfferent values of β are used. 1 2 2 α βµ r 2 β σ r Approach s smlar to the exact MGF approach set out n Quandt and Ramsey (1978) and Schmdt (1982). AG do not dscuss approxmaton error etc. - 11 -
The approxmate MGF condtons are not very nformatve about the hgher moments of t. The man reason s that the hgher order m 3 and m 4 terms n AG's approxmate MGF condtons are extremely small. As a result, they are poorly dentfed, f at all. Another reason s that very large sample szes are requred to estmate accurately E [exp( β r t )] usng ts sample analog 1 T t exp( β r t ). In addton, 1 T t exp( β r t ) and 1 T t exp( β jr t ) are hghly collnear when β and β j have the same sgn. - 12 -
0.025 Ln Emprcal MGF Ln AG Approx MGF 0.020 0.015 0.010 0.005 0.000-0.6-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fgure 1. The AG approxmate and emprcal MGFs for a NIG return process wth 10,000 observatons. The natural logs of the emprcal and approxmate MGFs are plotted aganst β, whch ranges from -0.7 to +0.7. In the DGP, 1 rt t N (0, 10 t ) and t s Inverse Gaussan wth parameters µ = 1 and λ = 2. AG's approxmate MGF s exp( ) 1, where 1 4 2 3 4 α m 1 + α 2 m 2 + α 6 m α 3 + 24 m 4 m..., m are the moments of t and α = 1 2 1 2 β 10 (snce µ r = 0 and σ = 1 2 r 10 n ths case). The emprcal MGF s just 1 T t exp( β rt ). - 13 -
For these reasons, the use of addtonal approxmate MGF condtons s of lttle help n dentfyng the hgher moments of t, especally wth hgh frequency data when µ r s bascally zero. Ths s not a small sample problem. - 14 -
Bvarate DGP Clark (1973), Tauchen and Ptts (1983), Harrs (1987) and Rchardson and Smth (1994), nter ala, consdred a bvarate DGP: 2 rt µ r t σ r t 0 t N, 2 a t µ a t 0 σ a t where at s some observed measure of "market actvty" (volume, log volume, the number of trades etc.). No compellng theoretcal reason why at should be normally dstrbuted and lnear n t. - 15 -
The unvarate and bvarate moments of rt and at can be 2 used to dentfy and estmate r, a, r µ µ σ and 2 σ a as well as the moments of the nformaton flow process m 2,.., m 6. AG s approxmate MGF moment condtons are not needed. The bvarate approach should, assumng a correctly specfed DGP, produce more precse estmates snce t explots more nformaton and uses exact moments. It also provdes more potental over-dentfyng restrctons. - 16 -
THE MONTE CARLO EXPERIMENTS We smulate returns from three normal mxture DGPs and see how well the AG procedure works. The three DGPs are: Normal lognormal (e.g. Clark, 1973); Normal nverse Gaussan (e.g. Barndorff-Nelsen, 1995) Normal gamma (e.g. Madan and Senata, 1990). All three DGP s have been wdely used n the emprcal fnance lterature. The DGP s are easy to smulate from. Just generate t and then generate rt gven t. - 17 -
In all three cases, the dstrbutons of t depend on two parameters. However snce the frst moment of t s normalzed to unty, the two parameters are not ndependent. Parameter Settngs We set the mean return equal to 0 or 0.1 and the varance of returns equal to 0.1. The zero mean settng s probably the approprate one to use when consderng hgh frequency data. The parameters of the nformaton flow dstrbutons were chosen so m1 = 1 (normlzaton) and m2 = ½. The other moments vary by dstrbuton. - 18 -
In the bvarate DGP, the mean and varance of at were based on usng the model to recover the moments of t usng 10 years hgh frequency return and volume data for Dell. Other Settngs Etc. GMM used to recover moments of t. In prncple, GMM should be more effcent than AG s method of moments lke procedure. In practce, the method of moments and GMM procedures produce very smlar results. We focused on the 2 nd, 3 rd and 4 th moments of t, as the - 19 -
hgher moments are more dffcult to estmate. Sample szes of 500, 1000, 2500, 5000 and 10000 obs were used. Sample szes of 5000 and 10000 obs are not uncommon n studes usng, respectvely, daly and hgh frequency data. Monte Carlo results are all based on 1000 replcatons. - 20 -
THE MONTE CARLO RESULTS Wll consder the results for the normal lognormal DGP (wth zero mean return) n Table 1. We use the 2 nd, 4 th and 6 th moment of the return process rt and two approxmate moments based on the MGF. As expected, the thrd and fourth order moments of the tme changer / nformaton flow t cannot be accurately recovered usng the AG procedure. Ths s true even n very large samples of 10,000 observatons. - 21 -
Table 1. Estmated Moments of the Tme Changer - Normal Lognormal DGP for Returns Zero mean for returns ( µ r = 0) and one over-dentfyng restrcton Sample Sze µ r Ave. GMM Parameter Estmates and Std. Errors Sze of GMM Test Statstc 2 σ r 1 m 2 m 3 m 4 m 10% Level 5% Level 1% Level T = 500 0.000 0.099 1.000 0.457 0.584-248.9 11% 5% 1% - (0.008) - (0.218) (0.847) (5330.2) T = 1000 0.000 0.099 1.000 0.471 0.671-135.2 9% 6% 1% - (0.005) - (0.160) (0.560) (3741.8) T = 2500 0.000 0.099 1.000 0.488 0.805-86.8 9% 5% 1% - (0.003) - (0.122) (0.602) (2565.0) T = 5000 0.000 0.099 1.000 0.497 0.839 15.7 9% 6% 1% - (0.002) - (0.094) (0.534) (1890.6) T = 10000 0.000 0.099 1.000 0.500 0.898-52.1 11% 5% 1% - (0.001) - (0.073) (0.495) (1387.8) True Parameters 0.000 0.100 1.000 0.500 0.875 3.890 - - - NOTE: The moments of the tme changer are estmated usng GMM and AG's unvarate procedure. The followng fve moment condtons are used - the 2nd, 4th and 6th moment of r and two approxmate MGF moments wthβ = -0.7 and 0.6. The results are based on 1000 converged replcatons. The table entres are the average parameter estmates and standard errors (n parentheses) n the 1000 Monte Carlo experments. Detals of the DGP are set out n Secton 3 of the paper. - 22 -
For example, when T = 1000, the average estmate of m 3 s 0.898. Ths s close to the true value of 0.875. However, the assocated average standard error of 0.495 s very large. As expected, the results for m 4 are consderably worse than those for m 3. Note that the GMM test statstc gves no ndcaton of ths poor performance. Robustness of our Fndngs? Obtaned smlar Monte Carlo results for normal gamma and normal nverse Gaussan DGPs n Tables 2 and 3. - 23 -
Also ran some Monte Carlo experments usng DGPs wth Non-zero means for returns; Returns expressed as percentages; Dfferent and/or addtonal moment condtons; Usng least squares rather than GMM to estmate the moments of the tme changer. In all cases, the AG procedure cannot recover the hgher moments of the unobserved nformaton flow t. AG s approxmate MGF moment condtons are just not nformatve about the moments of t. - 24 -
Bvarate DGP Results Some representatve GMM results for the bvarate DGP case are set out n Table 4. The thrd and fourth order moments of t can be recovered n ths bvarate setup. Of course, the bvarate setup nvolves addtonal assumptons and large samples are requred to estmate the hgher moments precsely. - 25 -
Table 4. Estmated Moments of the Tme Changer Bvarate Normal Lognormal DGP for Returns and "Actvty" Zero mean for returns and one over-dentfyng restrcton Sample Sze µ r Ave. GMM Parameter Estmates and Std. Errors Sze of GMM Test Statstc 2 σ µ r a 2 σ a 1 m 2 m 3 m 4 m 10% Level 5% Level 1% Level T = 500 0.000 0.096 2.980 1.818 1.000 0.439 0.745 2.508 22% 15% 7% - (0.007) (0.107) (0.955) - (0.106) (0.326) (1.806) T = 1000 0.000 0.098 2.988 1.697 1.000 0.462 0.758 2.963 20% 14% 7% - (0.005) (0.076) (0.714) - (0.082) (0.268) (1.741) T = 2500 0.000 0.099 2.995 1.593 1.000 0.482 0.844 3.565 17% 11% 6% - (0.003) (0.048) (0.487) - (0.058) (0.207) (1.565) T = 5000 0.000 0.099 2.998 1.524 1.000 0.491 0.858 3.755 16% 11% 5% - (0.002) (0.034) (0.357) - (0.043) (0.161) (1.377) T = 10000 0.000 0.099 2.998 1.525 1.000 0.495 0.863 3. 773 15% 10% 5% - (0.001) (0.024) (0.256) - (0.031) (0.120) (1.088) True Parameters 0.000 0.100 3.000 1.500 1.000 0.500 0.875 3.890 - - - NOTE: GMM results based on 1000 replcatons usng the followng seven moment condtons - the second and fourth moments of returns r, the frst four moments of "actvty" a and the covarance between r 2 and a. - 26 -
EVIDENCE USING CISCO AND INTEL RETURN DATA Tred (unsuccessfully) to replcate AG s emprcal results for Csco and Intel. Table 5 reports the estmated moments of the tme changer / nformaton flow usng AG's unvarate moments and two dfferent sets of bvarate moments usng the number of trades or volume as our actvty measure. The AG procedure produces large and mplausble estmates of m 3 and m 4, n lne wth our Monte Carlo results, whereas the bvarate results appear more plausble and sgnfcant. - 27 -
Table 5. Estmated Moments of the Tme Changer t for Csco and Intel Usng the Unvarate AG Procedure and Two Bvarate Procedures Moments Csco (10 Mnute) Returns Intel (15 Mnute) Returns m 2 3 m 4 m 2 m 3 m m 4 Unvarate moments 63.279 9992.91 1.15e+09 9.897 424.198 5.7e+08 (21.837) (3190.57) (8.8e+08) (1.994) (88.523) (1.8e+08) Bvarate moments wth volume 0.667 1.636 8.311 0.858 2.607 20.379 (0.094) (0.232) (2.038) (0.085) (0.554) (7.082) Bvarate moments wth 0.814 2.231 14.397 0.914 3.254 28.705 transactons (0.073) (0.304) (3.012) (0.094) (0.773) (12.906) Moments of re-centered volume 0.707 1.439 6.461 0.603 1.659 11.777 Moments of re-centered 0.540 0.851 3.114 0.520 1.208 6.194 transactons NOTE: GMM results wth standard errors n parentheses. The moment condtons used are the same as those n Tables 1 to 3. The bvarate moments are the same as n Table 4. The Csco and Intel results are based on 10 and 15 mnute return data respectvely. The Data are descrbed n Secton 5 of the text. The re-centered volume and transactons data are scaled so that they have unt means. - 28 -
The volume and trade results are reasonably smlar to each other, whch should come as no surprse snce the number of trades and volume are hghly correlated. The estmated moments of t do not match those of ether volume or the number of trades, contrary to AG's clam that the transactons clock should be based on the number of trades. Even though the AG procedure does not produce accurate estmates of the hgher moments of t, t s stll possble that returns condtoned by the re-centered number of transactons may be approxmately Gaussan. - 29 -
Table 6. The Moments of Csco and Intel Raw Returns and Returns Condtoned by the Re-centered Number of Trades and Volume Csco Systems (10 Mnutes) Intel (15 Mnutes) r r r r r r v t v t Mean 0.0009-0.0003-0.0008-0.003-0.008-0.007 Varance 0.243 0.239 0.277 0.169 0.156 0.154 Skewness 0.004 0.471-6.456-0.101-0.219-0.210 Excess Kurtoss 194.930 117.144 522.913 20.087 7.525 55.412 Jera-Barque 1.15e+07 5.60e+07 1.11e+08 109721.2 15449.1 839714.4 Test NOTE: r = returns, v = re-centered volume, t = re-centered no of transactons. The data are the same as n Table 5. - 30 -
150 100 CISCO Returns Normal 150 100 INTEL Returns Normal 50 50-0.03-0.02-0.01 0.00 0.01 0.02 0.03-0.03-0.02-0.01 0.00 0.01 0.02 0.03 100 CISCO r / sqrt(t) Normal 100 INTEL r / sqrt (t) Normal 50 50-0.02-0.01 0.00 0.01 0.02 0.03-0.03-0.02-0.01 0.00 0.01 0.02 0.03 100 CISCO r / sqrt(v) Normal 100 INTEL r / sqrt(v) Normal 50 50-0.02-0.01 0.00 0.01 0.02 0.03-0.02-0.01 0.00 0.01 0.02 0.03 Fgure 2. Estmated denstes of Csco Systems and Intel raw returns and returns condtoned by the re-centered numbers of trades and volume. The Csco 10 mnute returns are shown on the rght and Intel 15 mnute returns on the left. The denstes of the raw returns, returns condtoned by trades and returns condtoned by volume are dsplayed n top, mddle and bottom panels espectvely. Normal dstrbutons wth the same mean and varances are also shown. The data are descrbed n Secton 5 of the paper. - 31 -
However, Fgure 2 and the more formal results n Table 6 confrm that returns condtoned by re-centered volume or the number of transactons are not normally dstrbuted. (The recentred varables are scaled to have a mean of unty.) The condtoned returns are more Gaussan than the raw returns but the normal dstrbuton stll provdes a poor approxmaton. It would be extremely useful f all the mportant features of the latent nformaton flow were captured by a combnaton of returns and the number of trades or volume. Unfortunately, ths does not appear to be the case. - 32 -
SOME FURTHER EVIDENCE Also look at hgh frequency data for two other stocks - Dell and WorldCom stocks. Obtaned smlar results: The AG procedure agan produces large and mplausble estmates of moments of t. Bvarate procedures generate more plausble and sgnfcant estmates. Returns condtoned by re-centered volume or no. of trades are not normally dstrbuted. - 33 -
CONCLUSIONS The procedure used by Ané and Geman (2000) to recover the moments of the latent tme changer / nformaton flow from hgh frequency data s not relable. Monte Carlo results show that the thrd and hgher moments of the tme changer cannot be accurately recovered because the approxmate MGF condtons used are not nformatve. Bvarate procedures work farly well, assumng the mean and varance of market "actvty" s lnear n the tme changer. Returns condtoned on the re-centered number of trades or volume are not Gaussan. - 34 -