are short term stock asset returns predictable? an extended empirical analysis

7/29/2019 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: [email protected]

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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

11


13/38

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,

12


14/38

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

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

13


15/38

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.

14


16/38

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)

15


17/38

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


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


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


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


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

are short term stock asset returns predictable? an extended empirical analysis

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