are short term stock asset returns predictable? an extended empirical analysis
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Are Short Term Stock Asset Returns Predictable?An Extended Empirical Analysis
Thomas Mazzoni
Diskussionsbeitrag Nr. 449Mrz 2010
Diskussionsbeitrge der Fakultt fr Wirtschaftswissenschaftder FernUniversitt in Hagen
Herausgegeben vom Dekan der Fakultt
Alle Rechte liegen bei den Autoren
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Are Short Term Stock Asset Returns Predictable?
An Extended Empirical Analysis
Thomas Mazzoni
March 16, 2010
Abstract
In this paper, the question is addressed, whether or not short termstock asset returns are predictable from the knowledge of the past re-turn series. Several methods like Hodrick-Prescott-filtering, Kalman-filtering, GARCH-M, EWMA and non-parametric regression are bench-marked against naive mean predictions. The extended empirical analysisincludes data of several hundred stocks and indexes over the last ten years.The root mean square error of the concerning prediction methods is cal-culated for several rolling windows of different length, in order to assesswhether or not local model fit is favorable.
Keywords: Hodrick-Prescot-Filter; Kalman-Filter; GARCH-Model;Non-Parametric Regression; Maximum-Likelihood Estimation; Nadaraya-
Watson-Estimator.
1. Introduction
The question whether or not stock asset returns are predictable is a mostactive and controversially discussed issue since the influential works of Famaand French (1988a,b), Campbell and Shiller (1988) and Poterba and Summers(1988). Subsequent research has focused on middle- and long-term predictabil-ity. Additional variables, beyond the stock asset price and its dividend, wereused to enhance predictive power. An incomplete list includes changes in inter-est rates and yield spreads between short- and long-term interest rates (Keimand Stambaugh, 1986; Campbell, 1987, 1991), book-to-market ratios (Kothariand Shanken, 1997; Pontiff and Schall, 1998), price-earnings ratios (Lamont,1998), consumption-wealth ratios (Lettau and Ludvigson, 2001) and stock mar-ket volatility (Goyal and Santa-Clara, 2003). A comparative survey of aggre-gated stock prediction with financial ratios is provided in Lewellen (2004). Anexcellent review is given in Rey (2003).
Unfortunately, short-term returns are exceedingly difficult to predict for dif-ferent reasons. First, the signal-to-noise ratio is extremely unfavorable for short-term returns or log-returns, respectively. Second, additional variables, like those
Thomas Mazzoni, Department for Applied Statistics, University of Hagen, Germany,
Tel.: 0049 2331 9872106, Email: Thomas.Mazzoni@FernUni-Hagen.de
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mentioned in the previous paragraph, usually contribute to the trend of a stockasset over an extended period. Thus, in case of daily returns, their influence
is not detectable and one relies solely on statistical instruments and the infor-mation contained in the observation history. Nonetheless, predicting short-termreturns of stock assets is of great importance for example in risk-managementand related fields.
This paper contributes to the existing literature in that an extended em-pirical analysis of short-term (daily) log-returns of stock assets and indexes isconducted to answer the question, whether or not they are predictable to a cer-tain degree. For this purpose, more than 360 stocks are analyzed over the last10 years. Several statistical prediction methods are applied and benchmarkedagainst each other by assessing their root mean square errors.
The remainder of the paper is organized as follows: Section 2 introduces
the statistical instruments used in this survey. Section 3 provides a detailedanalysis of eight influential stock indexes. Based on the results, an optimizedstudy design is established for the analysis of the 355 stock assets contained inthe indexes. In section 4 the procedure is exercised on thirty DAX stocks. Theresults are summarized in table form. Additional results from the remainingstock assets are provided in appendix B. Section 5 summarizes the findings anddiscusses the implications.
2. Prediction Methods
In this section the prediction methods used in this survey are introduced. Prior
to this it is necessary to distinguish between local and global methods. In whatfollows, local methods are understood as prediction methods, which incorporateonly a limited amount of past information. A typical information horizon reaches40 or 100 days into the past. Global methods are prediction methods that rely onthe complete information available; in this case log-return data from 01-January-2000 to 31-January-2010. On the one hand, local methods often experience largevariability in their parameter estimates because of the small sample size. Onthe other hand, structural shifts due to environmental influences can cause themodel parameters themselves to vary over time. Local methods can betteradjust to such structural changes than global ones.
2.1. Mean Prediction
Log-returns, when analyzed with time series methods, generally turn out to bewhite noise processes. Therefore, the mean is the optimal linear predictor. It isalthough the most simple predictor to compute and hence serves as benchmarkin this survey, both locally and globally.
Let xt = log St be the logarithmic stock asset price at time t. Let further = 1B be the difference operator and B the usual backward shift operator,then the log-return is xt = xt xt1 and its linear local predictor is
xt+1 =1
K
K1
k=0 xtk =xt xtK
K
. (1)
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K defines the size of the window for the rolling mean calculation. The globalpredictor is calculated analogously, but the telescoping series yields xT x1.
Remark: Notice that in case of global prediction future information is in-volved in the prediction at time t < T. This is not a problem if the log-returnprocess is ergodic. In this case the distribution of the predictor is identical tothe distribution of another predictor, using the same amount of information butshifted back into the past by T t days. Thus, equation (1) applies for theglobal predictor as well with K = T 1.
2.2. Hodrick-Prescott-Filter
The Hodrick-Prescott-Filter (Hodrick and Prescott, 1997) basically divides atime series into its trend and cyclical component. Let
xt = gt + ct, (2)
again with xt = log St, gt the growth component, ct the cyclical (noise) com-ponent and t = 1, . . . , K . Then the HP-Filter is determined by solving theoptimization problem
min{gt}Kt=1
Kt=1
(xt gt)2 +
K1t=2
(2gt+1)2
. (3)
Because of the logarithmic transformation the trend of xt can be expected tobe linear. If changes in the trend are only due to random fluctuations, one
can formulate a random walk model for the growth rate gt = gt1 + 2gt.It can be shown (for example Reeves et al., 2000) that for 2gt N(0,
2g),
ct N(0, 2c ),
2gt and ct independent and = 2c/
2g , minimizing (3) yields
the Maximum-Likelihood estimator for g = (g1, . . . , gK)
g() = (I+ FF)1x, (4a)
with
F =
1 2 1 0 0
0 1 2 1 0...
. . .. . .
. . ....
0 0 1 2 1
. (4b)
In most cases the variances 2c and 2g , and hence , will not be known (they
may even not be constant in time, see section 2.4). Remarkably, Schlicht (2005)was able to derive the log-likelihood function for
l(;x) = log detI+ FF
Klog
xx xg()
+ (K 2)log . (5)
Equation (5) has to be maximized numerically with Newton-Raphson or a similaralgorithm (for an excellent treatment on this subject see Dennis and Schnabel,1983). The first and second derivative with respect to is derived analyticallyin appendix A.1.
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0 20 40 60 80 100
8.80
8.85
8.90
8.95
9.00
9.05
9.10
t
xt,
gt
10.26
0 5 10 15 20
0.5
0.0
0.5
Lag
PACF
Figure 1: Original and HP-Filtered Logarithmic DJI (left) Partial Sample
Autocorrelation Function for Log-Returns (right)
In figure 1 (left) a portion of the Dow Jones Industrial Average index isshown in logarithmic scale (gray) and HP-Filtered (black). The estimated vari-ance ratio is = 10.263. Obviously, the filtered log-return series has partialautocorrelations at lag one and two (figure 1 right), whereas the original log-returns are white noise. The dashed lines indicate the 95% confidence bandunder the white noise hypothesis. The correlation structure ofgt is used to fitan AR(p) model, where the model order is selected with BIC (Schwarz, 1978).In this case a AR(2)-model is identified with 1 = 1.522 and 2 = 0.629.
The general prediction formula under Hodrick-Prescott-filtering is thus
xt+1 = + pk=1
(kgtk+1 ), (6a)
with
=gt gtK
K. (6b)
The HP-prediction is clearly a local method because it involves a (K+1)(K+1)matrix inversion and the predictor cannot be updated recursively.
2.3. Kalman-Filter
The Kalman-Filter (Kalman, 1960) is a sophisticated estimator for a partially
or completely unobservable system state, conditioned on an information set pro-vided by a corresponding measurement process. If both, system and measure-ment process, are linear with Gaussian noise, the Kalman-Filter is the optimalestimator. For nonlinear processes, it is still the best linear estimator.
Because the Kalman-Filter is used as local predictor, the variance of the log-return process is assumed locally constant1. Thus, a basic structural state-space
1Prediction methods allowing for time varying variance, even locally, are introduced insections 2.4 and 2.5.
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model called local linear trend model can be formulated
ytt = 1 10 1yt1t1+ tt (7a)xt =
1 0
ytt
. (7b)
The noise vector is assumed identically and independently N(0,Q)-distributed.The variance of t can be estimated by the sample variance of xt, Var[t] =E[(xt )
2] = 2. To determine the variance of t is a little more involved,because the mean process is unobserved. Because t represents the variabilityof the estimated mean, its variance can be estimated by cross validation of .Let k be the estimated mean, omitting the k-th log-return. Then the cross
validated mean estimator is
=1
K
Kk=1
k =1
K(K 1)
K1k=0
j=k
xtj . (8)
It is easy to see that is unbiased, because E[] = E[], and its sample varianceis Var[] = 2/K(K1), which is an estimator for the variance oft. Because ytand t are assumed mutually independent, the off-diagonal entries ofQ vanishand one obtains
Q =
2 0
0 2
K(K1)
. (9)
The rekursive Kalman-scheme provides an optimal estimator for the unob-served system state. It is convenient to write (7a) and (7b) in vector/matrix-form as yt = Ayt1 + t and xt = Hyt, respectively. Let further yt be thesystem state expectation at time t and Pt the state error covariance. Then theoptimal filter is provided by the Kalman-recursions
yt+1|t = Ayt|t
Pt+1|t = APt|tA +Q
(10a)
Kt = Pt|t1H(HPt|t1H
)1
yt|t = yt|t1 +Kt(xt Hyt|t1)Pt|t = (IKtH)Pt|t1.
(10b)
Finally, the Kalman-Filter has to be initialized with prior state expectation anderror covariance. It is assumed that 0 = and y0 = x1 and therefore,Var[0] = Var[y0] =
2/K. Then the initial prior moments are
y1|0 =
x1
and P1|0 =
2
K
K + 2 1
1 KK1
. (11)
Figure 2 illustrates the filtered mean process and log-predictions for the
Deutscher Aktien Index (DAX) from November-2009 until January-2010.
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0 10 20 30 40 50 60
0.4
0.2
0.0
0.2
0.4
0.6
t
t
t
in
0 10 20 30 40 50 60
8.60
8.65
8.70
t
yt
1
t
Figure 2: Filtered Mean Process and 95% HPD-Band for DAX (left) Predicted
Logarithmic DAX and 95% HPD-Band (right)
The Kalman-predicted log-return takes the particularly simple form
xt+1 = yt+1|t xt. (12)The Kalman-Filter is clearly a local instrument because the variance is assumedconstant. In the next two subsections methods are introduced that do not sufferfrom this restriction.
2.4. GARCH-in-Mean
In this subsection the GARCH-in-Mean-model is introduced. ARCH- and
GARCH-models (Engle, 1982; Bollerslev, 1986) are extraordinary successful ineconometrics and finance because they account for heteroscedasticity and otherstylized facts of financial time series (e.g. volatility clustering). This is accom-plished by conditioning the variance on former variances and squared errors.The ARCH-in-Mean model (Engle et al., 1987) involves an additional point ofgreat importance. It relates the expected return to the level of present volatil-ity. This is significant from an economic point of view, because in arbitragepricing theory it is assumed that an agent has to be compensated for the riskhe is taking in form of a risk premium. Thus, GARCH-in-Mean models are thestarting point for GARCH-based option valuation methods (e.g. Duan, 1995;Duan et al., 1999, 2004; Heston and Nandi, 1997, 2000; Mazzoni, 2008). Recent
research challenges the calibration of option valuation models under the physicalprobability measure (Christoffersen and Jacobs, 2004). Therefore, the results ofthis survey may provide some insights on this topic.
For the empirical analysis a standard GARCH(1,1)-model has been chosenand the risk premium is modeled linearly regarding volatility. The model equa-tions are
xt = + t + t (13a)
2t = + 2t1 +
2t1, (13b)
with t = tzt and zt N(0, 1). The parameters have to be estimated with
Maximum-Likelihood. This can cause problems because on the one hand, ML
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is only asymptotically optimal. In small samples ML-estimates can be seriouslybiased and the distribution of the estimates can suffer from large deviations
from normality. On the other hand, in small samples the likelihood-surfacemight be nearly flat, which causes problems in numerical maximization routines.Therefore the model parameters are estimated using the full time series, whichmakes the prediction a global one. Under GARCH-in-Mean, the predictionformula is xt+1 = + + 2t + 2t . (14)2.5. EWMA
Exponentially Weighted Moving Average (EWMA, McNeil et al., 2005, sect. 4.4)is an easy to calculate alternative to ARCH/GARCH specification. It is alsoable to track changing volatility, but its theoretical rational is not as strong asfor the heteroscedastic models of section 2.4. The variance equation (13b) isreplaced by a simpler version
2t = 2t1 + (1 )(xt1)
2, (15)
where is a persistence parameter, yet to be determined. If it is known, andthe initial value 2tK is approximated, for example by the sample variance, thewhole series 2tK, . . . ,
2t+1 can be calculated. Thus, the parameters in (13a)
can be estimated directly with Generalized Least-Squares (GLS). Let = (, )
then = (XWX)1XWx, (16a)
with
X =
1 t... ...1 tK
and W =
2t 0
. . .
0 2tK
. (16b)The volatility persistence can either be obtained by external estimation
or be fixed at = 0.9, which is a practitioners setting. Figure 3 shows log-returns of the FTSE 100 index of the last ten years. A GARCH-in-Mean modelwas fitted and the local volatility t was calculated. The upper limit of the95% confidence area in figure 3 is calculated from these GARCH-volatilities.The lower limit is calculated from an EWMA-model with volatility persistence
= 0.9. Both limits are virtually symmetric, but the GARCH-estimation ismuch more demanding. Furthermore, the EWMA-method can be used as localpredictor and the corresponding prediction formula is
xt+1 = + 2t + (1 )(xt)2. (17)
2.6. Non-Parametric Regression
The general idea of the non-parametric regression method is to establish a non-
linear function for the conditional expectation m(x) = E[xt|xt1 = x] and
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2000 2002 2004 2006 2008 2010
10
5
0
5
10
Time
LogReturnsin
Figure 3: FTSE 100 Log-Returns and Confidence Band GARCH-in-Mean (upper) and
EWMA (lower)
estimate this function from the data under the model
xt = m(xt1) + t, (18)
(cf. Franke et al., 2001, sect. 13.1). The non-linear function m(x) can be ap-proximated by Taylor-series expansion in the neighborhood of an arbitrary ob-
servation xt
m(xt)
pj=0
m(j)(x)
j!(xt x)
j (19)
and hence, one can formulate a Least-Squares problem of the kind
K1k=0
2tk =K1k=0
xtk
pj=0
j(xtk1 x)j2
, (20)
with the coefficients j = m(j)(x)/j!. Notice that (20) is only a local approxi-
mation. Thus, the observations have to be weighted according to their distance
to x. This is usually done by using kernel-functions.Kernel-functions are mainly used in non-parametric density estimation2.
Certain characteristics are required by a kernel function, which are already sat-isfied, if a valid probability density function is used as kernel function. In thisapplication, the standard normal density function is used as kernel function,because it is sufficiently efficient compared to the optimal kernel (Silverman,1986, tab. 3.1 on p. 43) and it has smooth derivatives of arbitrary order
Kh(x) =1
hx
h
. (21)
2For an excellent treatment on this subject see Silverman (1986).
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The parameter h is the bandwidth or smoothing parameter of the kernel func-tion; its determination has a strong impact on the results of the nonparametric
regression. Now the coefficient vector
(x) can be estimated by solving theoptimization problem
= arg min
Kk=1
xtk+1
pj=0
j(xtk x)j2
Kh(xtk x). (22)
This optimization problem can be solved by GLS, if the bandwidth of the kernelfunction is determined prior to the estimation procedure. This is usually done bycross validation (e.g. Bhar and Hamori, 2005, sect. 4.5). But if the distributionof the log-returns is assumed locally normal, the optimal bandwidth can becalculated analytically (Silverman, 1986, sect. 3.4.2)
hopt =54
3K. (23)
Estimating and inserting the resulting optimal bandwidth in (22), the opti-mization problem has the solution
= (XWX)1XWx, (24a)
with
X = 1 xt1 x . . . (xt1 x)
p
......
. . ....
1 xtK x . . . (xtK x)p , x =
xt...
xtK+1and W =
Kh(xt1 x) 0. . .0 Kh(xtK x)
.(24b)
The desired non-parametric estimator for the conditional expectation is givenby
m(x) = 0(x). (25)
Notice: for p = 0 the optimization problem, and hence the non-parametricestimator (24a) and (24b), reduces to the ordinary Nadaraya-Watson-estimator(Nadaraya, 1964; Watson, 1964).
In figure 4 non-parametric regression estimates of orders p = 0 (solid black),p = 1 (dashed) and p = 2 (gray) are given. Observe that higher order polynomialfits tend to diverge for peripheral values. This is the case for p = 2 in figure 4left. Divergent behavior of the approximation can be counteracted to a certaindegree by increasing the bandwidth of the kernel function, see figure 4 right.Obviously, the non-parametric regression yields a roughly parabolic structure,which could explain the absence of linear correlation.
Non-parametric regression is clearly a local method and the log-return pre-dictor is
xt+1 = m(xt). (26)9
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0.04 0.02 0.02 0.04xt1
0.02
0.01
0.01
0.02
0.03
Extxt1
hhopt
0.04 0.02 0.02 0.04xt1
0.01
0.02
0.03
Extxt1
h1.5hopt
Figure 4: Non-Parametric Regression of Hang Seng Index
3. Empirical Analysis of Index Returns
In this section, the log-returns of eight representative stock indexes are ana-lyzed. The stock assets contained in each index are analyzed in detail in thenext section. The indexes are Deutscher Aktienindex (DAX), Dow Jones EuroStoxx 50 (STOXX50E), Dow Jones Industrial Average Index (DJI), FinancialTimes Stock Exchange 100 Index (FTSE), Hang Seng Index (HSI), NasdaqCanada (CND), Nasdaq Technology Index (NDXT) and TecDAX (TECDAX).The analyzed log-return data ranges from 01-January-2000 to 31-January-2010,if available3.
Table 1 gives the root mean square error of the applied prediction methodin %. The table is organized as follows:
Index indicates the analyzed stock index. All abbreviations refer to Wol-frams financial and economic database.
K gives the window size, used in local prediction methods.
Mean indicates the overall mean value (global) and the ordinary rollingmean (local), respectively.
The abbreviation HP indicates prediction with the Hodrick-Prescott-Filter. HP() means that the filter was applied with the fixed smoothingparameter . This is computationally more convenient because numericalmaximization routines are no longer required.
KF indicates prediction with the Kalman-Filter.
EW() indicates prediction with the EWMA method. Volatility persis-tence is given in %.
NR(p) means non-parametric regression. The parameter p indicates theorder of polynomial approximation. For p = 0 the method results in theordinary Nadaraya-Watson-estimator.
GARCH indicates the prediction with a globally fitted GARCH-in-Meanmodel.
3The TecDAX index gained approval to the Prime Standard of Deutsche Brse AG at24-March-2003. It replaced the Nemax50 index.
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LocalRMSE
GlobalRMSE
Index
K
Mean
HP
HP(6)
HP(10)
HP(20)
KF
EW(80)
EW(90)
EW(95)NR(0)
NR(1)
NR(2)
Mean
GARCH
DAX
40
1.6884
1.810
2
1.8168
1.8026
1.7858
1.6901
1.7540
1.7629
1.7613
1.7335
1.8769
7.7156
1.
6670
1.6695
STOXX50E
40
1.5758
1.698
4
1.7109
1.6975
1.6790
1.5785
1.6319
1.6407
1.6378
1.6089
1.7167
23.3640
1.
5560
1.5584
DJI
40
1.3305
1.427
9
1.4379
1.4240
1.4070
1.3321
1.3809
1.3821
1.3818
1.3605
1.6544
11.7722
1.
3115
1.3129
FTSE
40
1.3548
1.463
2
1.4704
1.4572
1.4396
1.3570
1.4128
1.4156
1.4132
1.3948
1.5477
5.0607
1.
3365
1.3375
HSI
40
1.7137
1.847
5
1.8392
1.8279
1.8118
1.7149
1.8263
1.8290
1.8164
1.7815
2.5769
16.9932
1.
6928
1.6938
CND
40
2.2914
2.442
3
2.4331
2.4190
2.4030
2.2899
2.4074
2.3952
2.3795
2.3462
3.1807
52.0094
2.2709
2.
2705
NDXT
40
2.1137
2.280
6
2.2971
2.2716
2.2399
2.1172
2.2145
2.2180
2.2206
2.1524
2.4040
6.0588
2.
0938
2.0981
TECDAX
40
1.7523
1.863
1.8652
1.8591
1.8508
1.7527
1.8195
1.814
1.8133
1.8183
2.0185
6.8387
1.
7327
1.7337
DAX
100
1.6733
1.816
3
1.8228
1.8070
1.7872
1.6743
1.7128
1.7167
1.7208
1.7118
1.8093
4.4208
1.
6668
1.6691
STOXX50E
100
1.4818
1.616
2
1.6266
1.6101
1.5918
1.4832
1.5228
1.5269
1.5328
1.5146
2.0347
24.1967
1.
4765
1.4783
DJI
100
1.3056
1.418
9
1.4320
1.4157
1.3970
1.3068
1.3354
1.3400
1.3444
1.3224
1.5263
9.7191
1.
3003
1.3020
FTSE
100
1.3392
1.464
3
1.4729
1.4583
1.4396
1.3407
1.3767
1.3769
1.3801
1.3700
1.6050
6.8907
1.
3342
1.3354
HSI
100
1.6836
1.828
4
1.8288
1.8140
1.7961
1.6841
1.7555
1.7437
1.7387
1.7572
4.6404
53.0186
1.
6763
1.6768
CND
100
2.2862
2.451
7
2.4498
2.4333
2.4139
2.2853
2.3258
2.3248
2.3293
2.3222
3.2307
37.2829
2.2795
2.
2791
NDXT
100
2.1649
2.360
2
2.3799
2.3503
2.3141
2.1658
2.2219
2.2268
2.2335
2.1876
2.3724
3.2032
2.
1584
2.1629
TECDAX
100
1.7402
1.865
9
1.8706
1.8646
1.8554
1.7407
1.7844
1.7841
1.7851
1.7808
2.0986
12.6642
1.
7346
1.7359
DAX
250
1.6909
1.851
7
1.8530
1.8357
1.8138
1.6909
1.7173
1.7176
1.7163
1.7214
2.1478
7.8891
1.
6874
1.6896
STOXX50E
250
1.4021
1.538
8
1.5427
1.5257
1.5062
1.4019
1.4362
1.4328
1.4328
1.4202
1.6333
1.6621
1.
3986
1.4005
DJI
250
1.3168
1.437
3
1.4486
1.4321
1.4125
1.3169
1.3422
1.3411
1.3416
1.3356
1.5753
1.8065
1.
3137
1.3154
FTSE
250
1.3561
1.494
1
1.4996
1.4831
1.4624
1.3564
1.3874
1.3825
1.3826
1.3725
1.4975
2.2697
1.
3534
1.3545
HSI
250
1.6747
1.831
2
1.8328
1.8156
1.7955
1.6745
1.7212
1.7069
1.6980
1.7241
6.3393
86.4545
1.
6708
1.6715
CND
250
2.3212
2.491
0
2.4947
2.4773
2.4557
2.3201
2.3572
2.3631
2.3625
2.3729
3.1625
12.6406
2.3131
2.
3121
NDXT
250
2.3515
2.568
4
2.5897
2.5562
2.5164
2.3512
2.3882
2.3910
2.3966
2.3733
2.5462
6.3705
2.
3453
2.3507
TECDAX
250
1.7688
1.903
4
1.9104
1.9019
1.8904
1.7682
1.7931
1.7945
1.7954
1.8203
2.6563
79.7466
1.
7632
1.7649
Table1:RootMeanSquareErrorin%ofIndexLog-ReturnPrediction
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The minimum root mean square error in % over all prediction methods is indi-cated in boldface.
In particular, each prediction is generated by considering the last K valuesxt, . . . , xtK and then calculating the one step log-return prediction xt+1. Thisis also done for the global predictors, even if their parameter estimates do notdepend on the rolling window position or size.
Obviously, the global mean provides the best predictor for log-returns inalmost all cases. Its RMSE differs only slightly from the RMSE of the GARCH-in-Mean prediction. Because all root mean square errors are in %, the differenceis roughly O(105). Hence, the first surprising result is that using local methodsdoes not seem to enhance prediction quality. Among all local predictors, rollingmean and Kalman-Filter provide the most precise forecasts. Observe furtherthat the RMSE for the Hodrick-Prescott-predictor nearly coincides with the
HP-Filter prediction with fixed smoothing parameter = 10. Thus, in furthercomputations HP(10) is used to speed up calculations. All EWMA predictionsprovide very similar results. Therefore, only EW(90) will be used in furtheranalysis, because it provides a good average. The non-parametric regressionmethod clearly indicates that higher order Taylor-series expansion is ineffectivein this situation. Only NR(0), which coincides with the Nadaraya-Watson-estimator, generates reasonable RMSEs. Thus, higher order NR-predictors areabandoned in further analysis.
From this first survey it can be concluded that the benefit of local predictionis questionable. Furthermore, it seems that the global mean value is the simplestand most efficient predictor.
4. Empirical Analysis of Stock Asset Returns
In this section the stock assets contained in the Deutscher Aktien Index (DAX)are analyzed in detail. A similar analysis was conducted for the stocks containedin the other indexes of section 3. The results are summarized in appendix B, andwill be discussed comprehensively in section 5. The results of local and globallog-return predictions of DAX stocks are summarized in table 2. The organiza-tion of the data is analogous to table 1, with two exceptions. First, smoothing-or persistence-parameters are fixed as indicated in the previous section andtherefore no longer listed, and second, the abbreviation NW now indicates the
Nadaraya-Watson-prediction method.Again, the mean prediction is the dominant method. But despite the results
for indexes, a smaller RMSE can be achieved by local prediction in roughlyone half of the cases. For some stock assets, the most efficient prediction isaccomplished by the Kalman-Filter, which virtually tracks the changes in thelog-return expectation. Furthermore, in those cases, where the GARCH-in-Mean method appears superior, GARCH-prediction RMSEs are barely smallerthan the corresponding global mean RMSEs.
Some preliminary conclusions can be drawn from this observation. First,obviously market portfolios, or indexes as their proxies, are unaffected by lo-cal trends in the expected log-returns, whereas stock assets may be. Second,
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LocalRMSEK
=40
LocalRM
SEK=100
Lo
calRMSEK=250
GlobalRMSE
Stock
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
GARCH
ADS.D
E
3.7735
3.9
933
3.7
764
3.8
518
3.8
086
3.7
738
4.0
1383.
7759
3.8
049
3.8
035
3.9
075
4.16
02
3.9
083
3.9
361
3.9
366
3.8
997
3.9
466
ALV.D
E
2.6
845
2.8
746
2.6
880
2.8
269
2.7
880
2.
6455
2.8
5662.
6467
2.7
142
2.7
380
2.6
912
2.92
19
2.6
910
2.7
368
2.7
880
2.6
854
2.6
886
BAS.D
E
2.3
546
2.5
275
2.3
599
2.5
527
2.3
725
2.3
423
2.5
4282.
3451
2.4
623
2.
3173
2.3
310
2.54
55
2.3
316
2.3
853
2.3
585
2.3
254
2.3
320
BAYN.D
E
2.6
902
2.8
828
2.6
960
2.8
976
2.8
113
2.6
873
2.9
0892.
6890
2.8
537
2.6
674
2.6
155
2.85
89
2.6
160
2.6
632
2.6
253
2.
6091
2.6
330
BEI.DE
3.0
727
3.2
356
3.0
739
3.1
150
3.1
101
3.
0703
3.2
4503.
0705
3.0
781
3.0
969
3.1
591
3.34
57
3.1
594
3.1
645
3.1
656
3.1
542
4.0
011
BMW.D
E
2.1
269
2.2
777
2.1
305
2.1
934
2.2
330
2.0
551
2.2
2852.
0569
2.0
867
2.1
205
2.0
585
2.23
82
2.0
589
2.0
767
2.1
322
2.
0540
2.0
543
CBK.D
E
3.2
402
3.4
465
3.2
438
3.3
841
3.3
113
3.1
955
3.4
321
3.
1951
3.2
719
3.2
455
3.2
358
3.48
49
3.2
334
3.2
887
3.2
906
3.2
246
3.4
090
DAI.DE
2.3
475
2.4
922
2.3
479
2.4
751
2.4
163
2.
3008
2.4
7512.
3031
2.3
669
2.3
478
2.3
334
2.51
48
2.3
330
2.3
796
2.3
734
2.3
276
2.3
304
DB1.D
E
5.6
528
5.9
800
5.6
569
5.7
368
5.6
800
2.8
094
2.9
8372.
8054
2.9
097
2.8
400
2.8
157
3.03
42
2.8
136
2.8
679
2.8
356
2.
8038
2.9
608
DBK.D
E
2.7
007
2.8
958
2.7
037
2.8
189
2.8
080
2.6
744
2.8
868
2.
6741
2.7
476
2.7
655
2.7
293
2.95
74
2.7
284
2.7
789
2.8
702
2.7
208
2.7
224
DPW.D
E
2.4
675
2.6
191
2.4
694
2.7
354
2.5
444
2.2
589
2.4
3462.
2614
2.3
734
2.3
590
2.1
819
2.36
12
2.1
819
2.2
352
2.3
186
2.
1769
2.1
775
DTE.D
E
2.5
324
2.7
233
2.5
364
2.6
314
2.5
900
2.4
383
2.6
5192.
4405
2.4
951
2.4
957
2.3
659
2.59
17
2.3
669
2.4
001
2.4
000
2.
3621
2.3
626
EOAN.D
E
2.5
767
2.7
669
2.5
837
2.8
690
2.7
113
2.5
640
2.7
8242.
5670
2.8
354
2.7
138
2.5
441
2.78
07
2.5
451
2.6
046
2.6
600
2.5387
2.6
286
FME.D
E
3.0
104
3.1
915
3.0
138
3.0
669
3.0
326
2.
9965
3.1
8332.
9979
3.0
188
3.0
063
3.0
767
3.27
54
3.0
775
3.0
839
3.0
925
3.0
699
3.0
724
FRE3.D
E
3.2
477
3.4
341
3.2
499
3.3
138
3.2
847
3.
2377
3.4
3113.
2387
3.2
665
3.2
639
3.3
315
3.53
92
3.3
315
3.3
416
3.3
525
3.3
234
3.4
660
HEN3.D
E
3.1
154
3.2
858
3.1
157
3.1
674
3.1
991
3.
1105
3.2
9273.
1111
3.1
280
3.1
259
3.2
174
3.41
50
3.2
178
3.2
228
3.2
289
3.2
105
3.7
094
IFX.D
E
4.0
161
4.1
975
4.0
122
4.1
376
4.1
019
4.0
061
4.2
248
4.
0014
4.0
544
4.0
851
4.0
268
4.26
26
4.0
254
4.0
697
4.0
752
4.0
181
4.1
817
LHA.D
E
2.0
820
2.2
221
2.0
841
2.1
818
2.1
396
1.9
863
2.1
4741.
9883
2.0
128
1.9
899
1.9
051
2.05
95
1.9
046
1.9
291
1.9
238
1.8
997
1.
8996
LIN.D
E
1.9
550
2.0
973
1.9
581
2.0
051
1.9
757
1.8
774
2.0
3861.
8789
1.9
189
1.8
965
1.8
186
1.98
08
1.8
187
1.8
540
1.8
419
1.8
137
1.
8133
MAN.D
E
2.9
191
3.1
218
2.9
236
3.1
359
2.9
635
2.8
709
3.1
0782.
8734
3.1
494
2.9
340
2.5
208
2.73
56
2.5
200
2.5
395
2.5
768
2.5149
2.5
151
MEO.D
E
2.1
569
2.3
137
2.1
583
2.3
072
2.2
152
2.0
645
2.2
3772.
0667
2.1
576
2.1
448
2.0
407
2.21
28
2.0
407
2.1
249
2.1
529
2.
0360
2.0
382
MRK.D
E
1.9
590
2.0
925
1.9
605
2.0
312
1.9
911
1.8
947
2.0
5741.
8965
1.9
219
1.9
216
1.8
262
1.99
53
1.8
267
1.8
397
1.8
698
1.8
227
1.
8207
MUV2.D
E
2.0
257
2.1
634
2.0
279
2.1
330
2.0
633
1.8
117
1.9
6021.
8131
1.8
852
1.8
601
1.7
226
1.87
78
1.7
233
1.7
739
1.8
029
1.7
187
1.7186
RWE.D
E
1.8
070
1.9
346
1.8
092
2.0
040
1.9
080
1.6
653
1.7
9901.
6660
1.7
606
1.7
906
1.6
582
1.80
38
1.6
588
1.7
068
1.7
697
1.
6560
1.6
582
SAP.D
E
1.9
209
2.0
391
1.9
226
1.9
641
1.9
643
1.8
371
1.9
7321.
8378
1.8
788
1.8
753
1.7
193
1.85
99
1.7
199
1.7
559
1.7
475
1.7151
1.7
650
SDF.D
E
2.4
995
2.6
578
2.5
016
2.5
890
2.5
795
2.4735
2.6
5782.
4737
2.5
177
2.5
365
2.5
189
2.72
31
2.5
192
2.5
509
2.5
795
2.5
137
2.5
144
SIE.D
E
2.1
927
2.3
413
2.1
940
2.2
982
2.2
636
2.
1478
2.3
2582.
1492
2.2
123
2.2
068
2.1
757
2.36
46
2.1
756
2.2
095
2.2
424
2.1
686
2.2
493
SZG.D
E
2.9
912
3.2
117
2.9
955
3.1
138
3.0
947
2.
9904
3.2
4422.
9921
3.0
832
3.0
422
3.0
753
3.35
61
3.0
760
3.1
272
3.1
570
3.0
738
3.0
760
TKA.D
E
3.0
377
3.2
372
3.0
425
3.3
715
3.0
946
2.
9912
3.2
1862.
9934
3.4
251
3.0
302
3.0
366
3.28
70
3.0
369
3.5
320
3.0
912
3.0
291
3.0
547
VOW3.D
E
2.5
967
2.7
513
2.5
948
2.7
760
2.7
069
2.5
704
2.7
460
2.
5680
2.6
758
2.6
936
2.5
947
2.78
43
2.5
936
2.6
878
2.7
431
2.5
874
2.5
838
Table2:Roo
tMeanSquareErrorin%ofLog-ReturnPredictionforDeutscherAktienIndex(DAX)Stocks
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there seems to be very little exploitable systematic information to infer from theknowledge of previous log-returns to future log-returns. The GARCH-in-Mean
prediction, which associates information about the current volatility with thedesired risk-premium, generates barely more efficient forecasts than the corre-sponding mean prediction, and only for a few stock assets. Only for BASF theincorporation of nonlinear information results in more precise forecasts. Thus,for most stocks the log-return process seems to be a noise process.
5. Summary and Conclusions
Daily log-returns of 355 stock assets and 8 indexes were analyzed over more thanten years. Several non trivial methods were used to calculate a one-step-aheadprediction. The efficiency of the predictions was assessed by the root mean
square error criterium.In roughly half the cases, the global mean prediction generated the smallest
RMSE. The GARCH-in-Mean prediction achieved RMSEs of similar magnitudesin some cases. Especially indexes seem to be dominated by global parameters,whereas stock assets occasionally are better predicted by local methods. Amongall local prediction methods the rolling mean is superior in almost all cases. In asmall number of occasions the Kalman-Filter method generates the best predic-tions. The Nadaraya-Watson-estimator only prevails on very rare exceptions.Other prediction methods seem to be inferior.
What conclusions can be drawn from this analysis? First, short-term stockasset log-returns are modeled adequately by noise processes. The same is true
of indexes. Nonlinear dependencies, if they exist, are obviously of minor impor-tance. As a consequence, knowledge of the past is not instrumental in predictingthe next value of the log-return process. Second, the parameters of those noisemodels have not necessarily to be invariant in time. It is common knowledgethat volatilities are not constant. One possible way to account for this fact math-ematically are GARCH-models. But the results of the current analysis suggestthat expectation values might be subject to local trends as well. This can beinferred from the fact that local prediction methods are successful in roughlyhalf the cases. Third, one would have expected the GARCH-in-Mean predictionto be superior, because it involves well established economic assumptions aboutrisk premia. This is not the case, which is problematic for the arbitrage pricing
theory. GARCH-in-Mean models are used for option valuation (Duan, 1995;Heston and Nandi, 2000; Mazzoni, 2008) in that they are estimated under aphysical probability measure P and subsequently translated into an equivalentmodel with risk-neutral measure Q, under which the option is valuated. Thismight explain why recent approaches to calibrate the model directly under therisk-neutral measure Q are partially more successful (Christoffersen and Jacobs,2004).
The current investigation clearly indicates that daily log-return processes arenot predictable, at least not beyond their mean value. Therefore, they are notsuited for profit-oriented investment strategies. This clearly classifies financialmarkets as risk transfer markets, at least in short term.
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A. Mathematical Appendix
A.1. Log-Likelihood Derivatives
Let M() = (I + FF)1. For notational convenience, functional argumentsare omitted in what follows. Then the log-likelihood function (5) can be written
l = log detM1 Klogx(IM)x
+ (K 2)log . (27)
The differential of the determinant of an arbitrary quadratic and invertiblematrix A is d detA = detA tr
A1dA
(cf. Magnus and Neudecker, 2007,
sect. 8.3). In particular, d log detA = trA1dA
holds. Further, the dif-
ferential of the inverse is dA1 = A1(dA)A1 (Magnus and Neudecker,2007, sect. 8.4). Observe that M can be expanded into a Taylor-series
M = IF
F+ (F
F)
2
. . . and hence, M and F
F commute. Therefore, thederivative ofM with respect to is M2FF. Now the derivative of (27) canbe calculated and one obtains
dl
d= tr
MFF
K
xM2FFx
x(IM)x+
K 2
. (28)
Because dtrA = tr dA (Magnus and Neudecker, 2007, p. 148), the calculationof the second derivative is straight forward
d2l
d2= tr
(MFF)2
+ K
xM2FFx
x(IM)x
2+ 2K
xM3(FF)2x
x(IM)x
K 2
2. (29)
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B.
Tables
B.1.
RMSEofDowJone
sEuroStoxx50Index(S
TOXX50E)Stocks
LocalRMSEK
=40
LocalRM
SEK=100
Lo
calRMSEK=250
GlobalRMSE
Stock
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
GARCH
ABI.BR
5.7713
6.1
225
5.7
724
5.9
143
5.9
220
5.8
001
6.1
7915.
7991
5.9
630
6.0
225
5.9
648
6.36
20
5.9
641
7.1
517
6.2
110
5.9
518
6.7
439
ACA.P
A
2.6
277
2.8
167
2.6
312
2.7
293
2.7
077
2.6
271
2.8
5342.
6295
2.6
778
2.6
947
2.5
515
2.78
11
2.5
516
2.5
737
2.6
068
2.5449
2.5
452
AGN.A
S
3.4
509
3.6
843
3.4
548
3.5
978
3.5
550
3.
2917
3.5
5273.
2957
3.3
650
3.4
038
3.3
260
3.59
35
3.3
265
3.3
755
3.4
430
3.3
172
3.4
623
AI.PA
3.
3125
3.5
281
3.3
219
3.8
965
3.7
945
3.3
147
3.5
5383.
3187
4.1
444
3.7
617
3.4
294
3.68
48
3.4
309
4.2
989
3.8
959
3.4
223
5.3
999
ALO.P
A
2.
9533
3.1
702
2.9
578
3.0
579
3.0
253
2.9
651
3.2
1522.
9676
3.0
135
3.0
453
2.9
926
3.27
24
2.9
934
3.0
325
3.0
398
2.9
880
2.9
877
ALV.D
E
2.6
845
2.8
746
2.6
880
2.8
269
2.7
880
2.
6455
2.8
5662.
6467
2.7
142
2.7
380
2.6
912
2.92
19
2.6
910
2.7
368
2.7
880
2.6
854
2.6
886
BAS.D
E
2.3
546
2.5
275
2.3
599
2.5
527
2.3
725
2.3
423
2.5
4282.
3451
2.4
623
2.
3173
2.3
310
2.54
55
2.3
316
2.3
853
2.3
585
2.3
254
2.3
320
BAYN.D
E
2.6
902
2.8
828
2.6
960
2.8
976
2.8
113
2.6
873
2.9
0892.
6890
2.8
537
2.6
674
2.6
155
2.85
89
2.6
160
2.6
632
2.6
253
2.
6091
2.6
330
BBVA.M
C
1.9
364
2.0
548
1.9
361
2.0
110
1.9
784
1.
8775
2.0
1441.
8788
1.9
213
1.8
951
1.9
080
2.04
95
1.9
077
1.9
338
1.9
073
1.9
031
1.9
069
BN.P
A
2.3
207
2.4
707
2.3
220
2.4
041
2.3
537
2.
3036
2.4
7562.
3056
2.3
455
2.3
328
2.3
825
2.56
68
2.3
835
2.4
112
2.4
016
2.3
791
2.3
964
BNP.P
A
2.4
473
2.6
121
2.4
512
2.5
483
2.5
281
2.4165
2.5
9672.
4166
2.4
499
2.4
685
2.4
718
2.67
47
2.4
722
2.4
927
2.4
958
2.4
658
2.4
832
CA.P
A
1.7
783
1.8
981
1.7
806
1.8
476
1.8
354
1.
6975
1.8
3121.
6986
1.7
532
1.7
552
1.7
101
1.85
29
1.7
106
1.7
417
1.7
732
1.7
072
1.7
074
CRG.I
R
2.2
364
2.4
119
2.2
406
2.3
228
2.3
118
2.
2284
2.4
3262.
2309
2.2
570
2.2
766
2.2
760
2.50
10
2.2
773
2.2
975
2.3
159
2.2
731
2.2
736
CS.P
A
5.2
962
5.6
475
5.3
146
6.4
790
5.3
263
5.
2718
5.6
6545.
2809
7.1
877
5.3
041
5.4
736
5.89
46
5.4
769
7.5
498
5.5
002
5.4
622
6.0
403
DAI.DE
2.3
475
2.4
922
2.3
479
2.4
751
2.4
163
2.
3008
2.4
7512.
3031
2.3
669
2.3
478
2.3
334
2.51
48
2.3
330
2.3
796
2.3
734
2.3
276
2.3
304
DB1.D
E
5.6
528
5.9
800
5.6
569
5.7
368
5.6
800
2.8
094
2.9
8372.
8054
2.9
097
2.8
400
2.8
157
3.03
42
2.8
136
2.8
679
2.8
356
2.
8038
2.9
608
DBK.D
E
2.7
007
2.8
958
2.7
037
2.8
189
2.8
080
2.6
744
2.8
868
2.
6741
2.7
476
2.7
655
2.7
293
2.95
74
2.7
284
2.7
789
2.8
702
2.7
208
2.7
224
DG.P
A
3.
1277
3.3
200
3.1
295
3.2
093
3.1
654
3.1
345
3.3
5483.
1365
3.1
804
3.1
603
3.2
405
3.47
40
3.2
409
3.2
686
3.2
737
3.2
344
3.4
804
DTE.D
E
2.5
324
2.7
233
2.5
364
2.6
314
2.5
900
2.4
383
2.6
5192.
4405
2.4
951
2.4
957
2.3
659
2.59
17
2.3
669
2.4
001
2.4
000
2.
3621
2.3
626
ENEL.M
I
1.5
414
1.6
477
1.5
432
1.6
769
1.6
430
1.5278
1.6
5301.
5292
1.6
190
1.6
178
1.5
665
1.70
29
1.5
671
1.6
348
1.6
958
1.5
654
1.5
689
ENI.MI
1.7
244
1.8
477
1.7
270
1.8
145
1.7
373
1.7024
1.8
4581.
7037
1.7
628
1.7
208
1.7
446
1.90
20
1.7
456
1.7
834
1.7
540
1.7
422
1.7
431
EOAN.D
E
2.5
767
2.7
669
2.5
837
2.8
690
2.7
113
2.5
640
2.7
8242.
5670
2.8
354
2.7
138
2.5
441
2.78
07
2.5
451
2.6
046
2.6
600
2.5387
2.6
286
FP.P
A
3.
8330
4.0
643
3.8
382
3.8
932
3.8
571
3.8
403
4.0
8433.
8419
3.8
830
3.8
402
3.9
848
4.24
85
3.9
858
4.0
010
3.9
960
3.9
772
4.3
575
FTE.P
A
1.7
132
1.8
539
1.7
168
1.8
053
1.7
724
1.5
668
1.7
0611.
5680
1.6
019
1.5
996
1.5
431
1.69
39
1.5
435
1.5
546
1.5
674
1.5394
1.5
396
GLE.P
A
2.5
801
2.7
429
2.5
819
2.7
151
2.6
845
2.5415
2.7
3722.
5437
2.5
932
2.5
820
2.6
106
2.81
83
2.6
107
2.6
529
2.6
363
2.6
057
2.6
089
G.M
I
1.6
760
1.7
734
1.6
761
1.7
494
1.7
429
1.
6579
1.7
7431.
6589
1.7
049
1.7
082
1.6
728
1.79
73
1.6
728
1.6
950
1.7
154
1.6
687
1.6
695
IBE.M
C
3.
8166
4.0
322
3.8
174
3.9
388
3.8
630
3.8
452
4.0
7723.
8460
3.8
767
3.8
784
3.9
917
4.23
75
3.9
914
3.9
937
4.0
220
3.9
799
4.3
596
16
-
7/29/2019 Are Short Term Stock Asset Returns Predictable? An Extended Empirical Analysis
18/38
LocalRMSEK
=40
LocalRM
SEK=100
Lo
calRMSEK=250
GlobalRMSE
Stock
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
GARCH
INGA.A
S
3.4
984
3.7
022
3.4
966
3.7
399
3.5
918
3.
3746
3.6
1803.
3775
3.5
001
3.4
772
3.4
808
3.73
37
3.4
798
3.5
674
3.5
585
3.4
694
3.9
115
ISP.M
I
2.5
033
2.6
791
2.5
045
2.6
310
2.5
651
2.4246
2.6
2432.
4257
2.4
786
2.4
660
2.4
571
2.67
14
2.4
570
2.4
983
2.4
958
2.4
504
2.4
539
MC.P
A
1.8
439
1.9
715
1.8
464
1.9
066
1.8
954
1.7925
1.9
4101.
7939
1.8
376
1.8
295
1.8
193
1.98
09
1.8
196
1.8
442
1.8
507
1.8
152
1.8
206
MUV2.D
E
2.0
257
2.1
634
2.0
279
2.1
330
2.0
633
1.8
117
1.9
6021.
8131
1.8
852
1.8
601
1.7
226
1.87
78
1.7
233
1.7
739
1.8
029
1.7
187
1.7186
OR.P
A
1.6
337
1.7
525
1.6
360
1.7
041
1.6
606
1.5579
1.6
9351.
5596
1.6
144
1.5
915
1.5
711
1.71
38
1.5
713
1.5
919
1.5
733
1.5
675
1.5
670
PHIA.A
S
2.7
297
2.9
183
2.7
331
2.8
332
2.7
925
2.6
511
2.8
6932.
6532
2.6
894
2.6
766
2.5
629
2.78
53
2.5
630
2.5
804
2.6
027
2.5565
2.5
570
REP.M
C
1.7
069
1.8
225
1.7
079
1.7
920
1.7
570
1.
6858
1.8
2011.
6870
1.7
362
1.7
442
1.7
419
1.88
70
1.7
420
1.7
681
1.7
555
1.7
375
1.7
374
RWE.D
E
1.8
070
1.9
346
1.8
092
2.0
040
1.9
080
1.6
653
1.7
9901.
6660
1.7
606
1.7
906
1.6
582
1.80
38
1.6
588
1.7
068
1.7
697
1.
6560
1.6
582
SAN.M
C
2.1
723
2.2
934
2.1
726
2.2
672
2.2
188
2.1
265
2.2
6402.
1266
2.1
770
2.1
766
2.0
873
2.23
30
2.0
864
2.1
092
2.1
135
2.
0800
2.2
472
SAN.P
A
1.7
148
1.8
422
1.7
181
1.7
757
1.7
605
1.
6507
1.7
9171.
6524
1.6
880
1.6
778
1.6
582
1.81
36
1.6
588
1.6
816
1.6
673
1.6
544
1.6
546
SAP.D
E
1.9
209
2.0
391
1.9
226
1.9
641
1.9
643
1.8
371
1.9
7321.
8378
1.8
788
1.8
753
1.7
193
1.85
99
1.7
199
1.7
559
1.7
475
1.7151
1.7
650
SGO.P
A
2.4
859
2.6
431
2.4
860
2.5
978
2.5
452
2.4172
2.6
1602.
4211
2.4
897
2.4
412
2.4
642
2.66
61
2.4
646
2.5
070
2.5
044
2.4
598
2.4
632
SIE.D
E
2.1
927
2.3
413
2.1
940
2.2
982
2.2
636
2.
1478
2.3
2582.
1492
2.2
123
2.2
068
2.1
757
2.36
46
2.1
756
2.2
095
2.2
424
2.1
686
2.2
493
SU.P
A
2.1
059
2.2
594
2.1
086
2.1
985
2.1
748
2.
0751
2.2
5872.
0773
2.1
369
2.1
031
2.1
289
2.32
91
2.1
294
2.1
686
2.1
477
2.1
244
2.1
254
TEF.M
C
1.9
318
2.0
666
1.9
338
2.0
168
1.9
836
1.8
886
2.0
4031.
8894
1.9
257
1.9
400
1.7
937
1.95
50
1.7
942
1.8
159
1.8
355
1.7905
1.7
910
TIT.M
I
5.8
648
6.3
562
5.8
833
6.2
027
5.8
856
5.3
006
5.7
7105.
3068
5.3
838
5.7
531
4.1
179
4.62
87
4.1
200
4.1
145
4.3
072
4.
1072
4.4
009
UCG.M
I
2.3
855
2.5
315
2.3
862
2.5
191
2.4
508
2.
3451
2.5
1572.
3468
2.4
162
2.3
812
2.3
920
2.57
18
2.3
915
2.4
487
2.4
353
2.3
857
2.4
655
UL.P
A
1.7791
1.9
116
1.7
822
1.8
470
1.8
325
1.7
825
1.9
3381.
7835
1.8
048
1.8
250
1.8
353
2.00
21
1.8
355
1.8
437
1.8
591
1.8
318
1.8
309
UNA.A
S
1.7665
1.9
037
1.7
698
1.8
734
1.7
991
1.7
891
1.9
4771.
7909
1.8
437
1.8
346
1.9
088
2.09
05
1.9
092
1.9
457
1.9
398
1.9
043
1.9
059
VIV.P
A
1.7
543
1.8
911
1.7
568
1.8
278
1.8
053
1.6
169
1.7
5501.
6181
1.6
533
1.6
508
1.5
668
1.70
69
1.5
672
1.5
887
1.5
894
1.5628
1.5
636
B.2.
RMSEofDowJone
sIndustrialAverageInde
x(DJI)Stocks
LocalRMSEK=
40
LocalRMS
EK=100
Loca
lRMSEK=250
GlobalR
MSE
Stock
Mean
HP
KF
E
WMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
MeanG
ARCH
AA
3.0
149
3.2
228
3.0
1783
.1141
3.1
188
2.9
891
3.2
319
2.99
12
3.0
558
3.0
886
2.9
598
3.2
077
2.9
587
3.0
249
3.0
676
2.
9501
2.9
506
AXP
2.7
836
2.9
831
2.7
8682
.9014
2.8
219
2.7
500
2.9
842
2.75
19
2.7
937
2.7
716
2.7
500
2.9
917
2.7
501
2.7
872
2.7
829
2.7433
2.8
644
17
-
7/29/2019 Are Short Term Stock Asset Returns Predictable? An Extended Empirical Analysis
19/38
LocalRMSEK=
40
LocalRMS
EK=100
Loca
lRMSEK=250
GlobalR
MSE
Stock
Mean
HP
KF
E
WMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
MeanG
ARCH
BA
2.1
733
2.3
042
2.1
7392
.2501
2.2
182
2.1
334
2.2
872
2.13
41
2.1
793
2.1
636
2.1
047
2.2
621
2.1
046
2.1
329
2.1
315
2.1
003
2
.0996
BAC
3.5
313
3.7
723
3.5
3553
.6699
3.6
907
3.5222
3.8
047
3.52
35
3.5
684
3.6
685
3.5
516
3.8
622
3.5
522
3.6
286
3.7
406
3.5
436
3.8
402
CAT
2.2
914
2.4
275
2.2
9022
.3671
2.3
610
2.2
436
2.4
132
2.24
54
2.2
741
2.2
791
2.2
250
2.3
969
2.2
245
2.2
434
2.2
384
2.
2182
2.2
190
CSCO
3.0
387
3.2
574
3.0
4363
.1379
3.1
051
2.9
708
3.2
188
2.97
27
3.0
088
3.0
031
2.8
531
3.1
062
2.8
535
2.8
700
2.8
963
2.
8478
2.8
496
CVX
1.8
139
1.9
436
1.8
1681
.9291
1.8
322
1.7
819
1.9
327
1.78
37
1.8
595
1.7719
1.7
863
1.9
486
1.7
873
1.8
495
1.7
933
1.7
837
1.7
842
DD
2.0
274
2.1
611
2.0
2912
.1037
2.0
745
1.9
597
2.1
171
1.96
15
2.0
001
1.9
884
1.9
243
2.0
846
1.9
245
1.9
508
1.9
635
1.
9188
1.9
193
DIS
2.2
613
2.4
174
2.2
6452
.3502
2.3
062
2.2
292
2.4
122
2.23
12
2.2
751
2.2
613
2.1
877
2.3
802
2.1
886
2.2
124
2.2
247
2.
1839
2.3
119
GE
2.2
720
2.4
190
2.2
7392
.3581
2.3
446
2.2
387
2.4
134
2.24
10
2.2
729
2.2
996
2.2
373
2.4
176
2.2
373
2.2
757
2.3
192
2.
2317
2.2
326
HD
2.4
353
2.5
995
2.4
3762
.5027
2.5
065
2.3
869
2.5
833
2.38
93
2.4
227
2.4
490
2.2
124
2.4
058
2.2
128
2.2
333
2.2
399
2.
2066
2.3
126
HPQ
2.6
861
2.8
699
2.6
8942
.7725
2.7
757
2.6
299
2.8
397
2.63
18
2.6
600
2.7
010
2.4
894
2.7
099
2.4
906
2.5
069
2.5
347
2.4859
2.5
672
IBM
1.9
367
2.0
531
1.9
3692
.0043
2.0
072
1.8
891
2.0
283
1.88
95
1.9
121
1.9
294
1.7
524
1.8
882
1.7
528
1.7
695
1.7
845
1.7474
1.8
124
INTC
2.9
163
3.1
320
2.9
1973
.0348
2.9
633
2.8
505
3.0
989
2.85
24
2.9
010
2.9
018
2.6
918
2.9
419
2.6
919
2.7
160
2.7
159
2.
6837
2.6
843
JNJ
1.4
008
1.5
048
1.4
0321
.4632
1.4
337
1.3
346
1.4
506
1.33
54
1.3
692
1.3
645
1.3
062
1.4
283
1.3
066
1.3
350
1.3
317
1.
3028
1.3
030
JPM
3.0
590
3.2
884
3.0
6553
.2097
3.1
703
3.0
355
3.3
006
3.03
78
3.0
801
3.1
091
3.0
350
3.3
260
3.0
362
3.1
382
3.1
485
3.
0279
3.1
598
KFT
1.4
872
1.5
843
1.4
8761
.5593
1.5
423
1.4
732
1.5
897
1.47
39
1.5
011
1.5
163
1.4
623
1.5
838
1.4
626
1.4
796
1.4
932
1.4576
1.4
580
KO
1.5
502
1.6
645
1.5
5321
.6386
1.6
059
1.4
652
1.5
826
1.46
60
1.5
176
1.5
094
1.3
810
1.4
941
1.3
809
1.4
303
1.4
316
1.
3766
1.3
772
MCD
1.7
664
1.8
910
1.7
6871
.8248
1.8
399
1.7
057
1.8
442
1.70
61
1.7
393
1.7
685
1.6
694
1.8
161
1.6
691
1.6
871
1.7
308
1.
6648
1
.6648
MMM
1.6
426
1.7
592
1.6
4441
.6926
1.7
037
1.5
940
1.7
272
1.59
53
1.6
249
1.6
302
1.5
683
1.7
102
1.5
688
1.5
817
1.5
888
1.5649
1.5
653
MRK
2.0
905
2.2
271
2.0
9292
.1708
2.1
791
2.0
483
2.2
007
2.04
90
2.0
704
2.1
038
2.0
445
2.2
123
2.0
450
2.0
545
2.0
843
2.
0410
2.0
477
MSFT
2.2
934
2.4
510
2.2
9442
.3819
2.3
524
2.1
932
2.3
730
2.19
43
2.2
368
2.2
226
2.0
773
2.2
666
2.0
779
2.0
952
2.0
848
2.0
708
2
.0704
PFE
1.8
746
2.0
157
1.8
7761
.9546
1.9
185
1.8
289
1.9
883
1.83
04
1.8
610
1.8
706
1.7
680
1.9
277
1.7
685
1.7
987
1.7
977
1.7636
1.7
644
PG
1.6
305
1.7
401
1.6
3351
.6743
1.6
662
1.3
681
1.4
820
1.36
90
1.3
969
1.4
010
1.3
017
1.4
216
1.3
023
1.3
242
1.3
232
1.
2994
1.3
306
T
2.0
094
2.1
557
2.0
1192
.0848
2.0
572
1.9
469
2.1
103
1.94
78
1.9
817
1.9
840
1.8
940
2.0
660
1.8
947
1.9
163
1.9
197
1.
8907
1.8
910
TRV
2.2
826
2.4
375
2.2
8532
.4106
2.3
564
2.2
216
2.4
127
2.22
38
2.3
234
2.2
614
2.2
056
2.4
092
2.2
063
2.2
841
2.2
647
2.
1997
2.2
008
UTX
2.0
056
2.1
431
2.0
0872
.0699
2.0
555
1.9
436
2.0
990
1.94
53
1.9
759
1.9
768
1.9
136
2.0
739
1.9
140
1.9
391
1.9
339
1.
9087
1.9
141
VZ
1.9
126
2.0
444
1.9
1401
.9851
1.9
923
1.8
743
2.0
299
1.87
51
1.9
121
1.9
491
1.7
844
1.9
469
1.7
849
1.8
079
1.8
432
1.7799
1.7
800
WMT
1.7
580
1.8
919
1.7
6071
.8168
1.7
998
1.6
716
1.8
224
1.67
33
1.6
954
1.6
899
1.5
452
1.6
910
1.5
459
1.5
621
1.5
624
1.5414
1
.5414
XOM
1.7
846
1.9
240
1.7
8881
.9122
1.8
262
1.7
516
1.9
087
1.75
36
1.8
361
1.7
780
1.7
485
1.9
168
1.7
496
1.8
190
1.7
770
1.7460
1
.7460
18
-
7/29/2019 Are Short Term Stock Asset Returns Predictable? An Extended Empirical Analysis
20/38
B.3.
RMSEofFinancial
TimesStockExchangeIn
dex(FTSE)Stocks
LocalRMSEK
=40
LocalRM
SEK=100
LocalRMSEK=250
Global
RMSE
Stock
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
GARCH
AAL.L
3.0
409
3.2
715
3.0
456
3.1
590
3.1
525
3.
0263
3.2
994
3.0300
3.0
968
3.0
970
3.1
289
3.424
5
3.1
294
3.1
684
3.1
945
3.1
209
3.2
795
ABF.L
1.2
963
1.3
815
1.2
978
1.3
579
1.3
321
1.
2764
1.3
786
1.2774
1.3
127
1.2
953
1.2
926
1.404
2
1.2
928
1.3
048
1.3
095
1.2
892
1.2
895
ADM.L
2.
2987
2.4
574
2.3
023
2.4
392
2.3
614
2.3
224
2.5
138
2.3238
2.4
660
2.3
679
2.4
256
2.649
2
2.4
269
2.4
891
2.4
640
2.4
224
2.4
208
AGK.L
2.4
149
2.5
769
2.4
171
2.5
278
2.4
745
2.3
637
2.5
546
2.3657
2.4
068
2.4
056
2.2
870
2.481
7
2.2
870
2.3
131
2.3
137
2.
2810
2.2
812
AMEC.L
2.2
203
2.3
746
2.2
224
2.3
315
2.3
206
2.
2010
2.3
800
2.2012
2.2
565
2.2
689
2.2
192
2.417
5
2.2
196
2.2
609
2.2
941
2.2
144
2.2
152
ANTO.L
4.
6118
4.8
808
4.6
159
4.6
858
4.6
876
4.6
247
4.9
348
4.6285
4.6
619
4.6
534
4.7
816
5.117
6
4.7
829
4.8
097
4.8
045
4.7
727
5.2
852
ARM.L
4.0
450
4.3
033
4.0
457
4.1
672
4.1
136
3.9
889
4.3
129
3.9929
4.0
167
4.0
220
3.8
289
4.140
3
3.8
284
3.9
087
3.8
512
3.
8183
4.1
647
AU.L
4.0
882
4.3
041
4.0
869
4.2
205
4.1
260
3.9
556
4.2
179
3.9556
4.0
329
3.9
806
3.4
568
3.721
8
3.4
551
3.4
904
3.5
026
3.4417
3.5
258
AV.L
2.9
569
3.1
675
2.9
605
3.0
738
3.0
272
2.
9008
3.1
409
2.9043
2.9
828
2.9
643
2.9
763
3.235
1
2.9
775
3.0
324
3.0
258
2.9
700
2.9
711
AZN.L
1.6
631
1.7
696
1.6
642
1.7
345
1.7
106
1.6
135
1.7
370
1.6148
1.6
441
1.6
471
1.6
089
1.741
8
1.6
092
1.6
254
1.6
348
1.6
054
1.
6049
BA.L
1.9
501
2.0
964
1.9
541
2.0
315
1.9
887
1.8
690
2.0
278
1.8703
1.9
027
1.8
743
1.8
328
2.001
9
1.8
336
1.8
556
1.8
418
1.
8303
1.8
307
BARC.L
3.5
367
3.7
523
3.5
395
3.7
050
3.6
489
3.5341
3.7
669
3.5347
3.5
978
3.6
030
3.6
622
3.909
1
3.6
617
3.7
234
3.7
048
3.6
505
3.9
254
BATS.L
2.4340
2.6
098
2.4
385
2.4
982
2.4
775
2.4
348
2.6
260
2.4373
2.4
772
2.4
787
2.5
006
2.708
4
2.5
015
2.5
370
2.5
451
2.4
953
2.8
425
BAY.L
2.8
878
3.0
805
2.8
894
2.9
722
2.9
611
2.
8016
3.0
246
2.8036
2.8
411
2.8
602
2.8
103
3.046
7
2.8
104
2.8
436
2.8
558
2.8
042
2.8
055
BG.L
2.1
106
2.2
650
2.1
155
2.1
888
2.1
652
2.
0856
2.2
635
2.0879
2.1
283
2.1
203
2.1
509
2.346
3
2.1
515
2.1
810
2.1
833
2.1
453
2.1
457
BLND.L
2.0
902
2.2
468
2.0
946
2.1
889
2.1
671
2.
0894
2.2
718
2.0917
2.1
261
2.1
045
2.0
982
2.290
1
2.0
987
2.1
175
2.1
226
2.0
947
2.0
946
BLT.L
2.7
814
2.9
788
2.7
841
2.8
889
2.8
318
2.7646
3.0
032
2.7678
2.8
328
2.8
118
2.8
448
3.107
0
2.8
458
2.8
912
2.8
972
2.8
382
2.8
394
BNZL.L
1.
6484
1.7
679
1.6
513
1.7
319
1.6
846
1.6
575
1.8
006
1.6587
1.6
954
1.6
850
1.7
195
1.885
4
1.7
201
1.7
487
1.7
553
1.7
150
1.7
154
BP.L
1.6
390
1.7
520
1.6
409
1.7
210
1.6
746
1.
6111
1.7
433
1.6126
1.6
695
1.6
469
1.6
448
1.793
9
1.6
455
1.6
889
1.6
868
1.6
415
1.6
417
BRBY.L
2.3
860
2.5
459
2.3
881
2.4
593
2.4
213
2.3
609
2.5
451
2.3614
2.3
884
2.3
920
2.3
468
2.547
1
2.3
468
2.3
825
2.3
546
2.
3433
2.3
450
BSY.L
3.5
161
3.7
513
3.5
297
4.2
781
3.5
050
3.5
212
3.7
846
3.5265
4.7
462
3.5
060
1.7
958
1.944
1
1.7
965
1.9
799
1.8
052
1.7
921
1.7920
BT-A.L
2.2
055
2.3
699
2.2
094
2.2
790
2.2
703
2.1
801
2.3
679
2.1813
2.2
179
2.2
334
2.0
519
2.239
7
2.0
520
2.0
767
2.0
827
2.
0482
2.0
487
CCL.L
2.
1098
2.2
583
2.1
117
2.2
173
2.1
846
2.1
140
2.2
907
2.1156
2.1
565
2.1
768
2.1
611
2.352
4
2.1
618
2.1
899
2.2
203
2.1
570
2.1
594
CNA.L
1.7414
1.8
456
1.7
447
1.8
434
1.7
797
1.7
489
1.8
743
1.7508
1.7
892
1.7
877
1.8
083
1.948
2
1.8
089
1.8
387
1.8
625
1.8
051
1.8
058
CNE.L
12.7
67
13.6
06
12.8
20
14.5
11
12.703
12.8
95
13.8
22
12
.915
15.5
65
12.8
60
13.4
17
14.40
4
13.4
25
16.2
41
13.4
45
13.3
91
13.1
08
COB.L
1.6
476
1.7
505
1.6
486
1.7
327
1.7
080
1.
6429
1.7
689
1.6445
1.6
885
1.6
775
1.6
808
1.820
2
1.6
815
1.6
977
1.7
045
1.6
769
1.6
778
CPG.L
2.2
083
2.3
665
2.2
113
2.3
076
2.2
590
2.1
969
2.3
786
2.1987
2.2
478
2.2
256
2.1
477
2.336
1
2.1
480
2.1
776
2.1
742
2.
1427
2.1
429
CPI.L
1.6
359
1.7
611
1.6
397
1.6
838
1.6
998
1.5
713
1.7
064
1.5729
1.5
866
1.6
241
1.4
713
1.609
8
1.4
719
1.4
836
1.5
018
1.4
683
1.4681
CW.L
2.0
723
2.2
121
2.0
744
2.1
451
2.1
241
1.9
465
2.0
946
1.9475
1.9
831
1.9
671
1.8
821
2.022
7
1.8
815
1.9
022
1.8
949
1.
8750
1.8
759
19
-
7/29/2019 Are Short Term Stock Asset Returns Predictable? An Extended Empirical Analysis
21/38
LocalRMSEK
=40
LocalRM
SEK=100
LocalRMSEK=250
Global
RMSE
Stock
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
HP
KF
EWMA
NW
Mean
GARCH
DGE.L
1.3
003
1.3
979
1.3
025
1.3
492
1.3
395
1.
2685
1.3
801
1.2698
1.2
976
1.2
985
1.2
758
1.397
8
1.2
765
1.2
970
1.3
061
1.2
734
1.2
736
EMG.L
3.
9686
4.2
633
3.9
784
4.2
214
4.1
830
4.0
697
4.4
097
4.0728
4.2
125
4.2
583
4.4
412
4.830
6
4.4
414
4.5
283
4.5
913
4.4
274
4.6
870
ENRC.L
5.7
062
6.1
952
5.7
191
5.9
252
5.8
463
5.8
936
6.4
642
5.8949
6.0
060
5.9
777
4.6
367
5.029
2
4.6
348
4.6
478
4.6
594
4.5
828
4.5808
EXPN.L
2.4486
2.6
116
2.4
503
2.5
571
2.4
873
2.4
935
2.7
064
2.4970
2.5
600
2.5
426
2.5
965
2.823
2
2.5
980
2.6
346
2.6
731
2.5
940
2.5
941
FRES.L
4.8
129
4.9
852
4.7
974
4.9
554
4.9
476
5.0
615
5.2
475
5.0475
5.1
670
5.2
004
3.7
035
3.949
5
3.6
986
3.6
830
3.7
660
3.6
719
3.
6543
GFS.L
1.7
506
1.8
716
1.7
524
1.8
275
1.7
938
1.7
327
1.8
778
1.7344
1.7
738
1.7
579
1.6
921
1.844
9
1.6
928
1.7
184
1.7
090
1.
6879
1.6
888
GSK.L
1.6
939
1.8
190
1.6
980
1.7
880
1.7
569
1.6
691
1.8
107
1.6708
1.7
932
1.7
099
1.6
563
1.807
7
1.6
569
1.6
711
1.6
643
1.6
526
1.
6502
HMSO.L
2.4
105
2.5
624
2.4
144
2.5
618
2.4
209
2.4
008
2.5
815
2.4026
2.6
236
2.
3976
2.4
557
2.648
6
2.4
558
2.7
011
2.4
375
2.4
508
2.4
618
HOME.L
3.9
261
4.1
652
3.9
306
3.9
879
3.9
702
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