# Time Series Analysis Exercises

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```					    Universität Potsdam

Time Series Analysis
Exercises

Hans Gerhard Strohe

Potsdam 2005

1
I   Typical exercises and solutions

1   For theoretically modelling the economic development of national economy scenarios
the following 2 models for GDP increment are analysed:

a. Yt = Yt-1 + at

b. Yt = 1.097 Yt-1 - 0,97 Yt-2 + at ,

where Y       -      GDP increment

a       -      White noise with zero mean and constant variance σ2=100

t       -      Time (quarters starting with Q1, 1993)

Check by an algebraic criterion which one is stationary.

2

a)   The process can be written
φ(L)Yt = at with the lag polynom
φ(L) = 1-L and L being the lag or backshift operator LYt.= Yt-1
From this follows the characteristic equation
φ(z) = 1-z = 0
The root, i.e. the solution of it, is z=1. It is not higher than 1, i.e. it is not placed
outside the unit circle. The process is not stationary. It is a “unit root” process or
more specific a random walk.

b)   The process can be written
φ(L)Yt = at with the lag polynom
φ(L) = 1 – 1,079L + 0,97 L2.
From this follows the characteristic equation
φ(z) = 1- 1,097z + 0,97 z2 = 0
The roots (complex numbers) of it
are     z1 = 0,57 + 0,84 i
z2 = 0,57 - 0,84 i
with    | z1/ 2 |= 0,57 2 + 0,84 2 = 1,015 > 1.
That means that both of the roots are outside the unit circle what is a sufficient
condition for the stationarity of the process.

3
2 The variable l (=labour) in the file employees.dat denotes the quantity of labour
force, i. e. the number of employees, in a big German company from January 1995
till December 2004.

i.        Display the graph of the time series lt.

ii.       What are the characteristics of a stationary time series? Is the time series lt likely
to be stationary? Check it first by the naked eye.

iii.      Test the stationarity of lt by a suitable procedure. Determine the degree of
integration.

iv.       Estimate the correlation and the partial correlation function of the process. Give
a description and an interpretation.

v.        What type of basic model could fit the time series lt. Why?

vi.       Estimate the parameters concerning the model assumed in question v .

vii.      Estimate alternative models and compare them by suitable indicators.

viii.     Forecast the time series for 2005.

4
ix.         A particular analysis method of time series lt results in the following graph.

1.6

Bartlett
1.4

1.2

Tukey

1.0

0.8

Parzen

0.6
0               1                   2             3               4

Circular frequency

Fig. 1: A special diagnostic function

What is the name of this particular analysis method? What can you derive from the curve.
What should be the typical shape of the function estimated for a process type assumed in
question v?

5

i.      The graph of the time series lt.

6000

5600

5200

4800

4400

4000
95     96    97     98     99    00   01    02    03      04

L

Fig. 2: Graph of the employees time series

ii.     The main characteristics of a stationary time series are that the mean µ t and the variance
of the stochastic process generating this special time series are independent from time t,
µ t = µ = const
σ t2 = σ 2 = const,

and that the autocovariances γ t , t depend only on the time difference τ :
1   2

γ t , t = γ t - t = γ τ with τ = t 1 – t 2   (Lag)
1   2     1   2

A check of the graph by the naked eye gives no reason to assume the time series not to be
stationary. At the first glance there does not appear any trend or relevant development of the
variance.

6
iii.    Test of stationarity by DF Test.

The following Dickey Fuller regression of ∆l on lt-1 produces a t-value – 3,63. Application of
an augmented Dickey-Fuller regression is not necessary because the augmented model would
have higher values of the Schwarz criterion (SIC, version on the base of the error variance).

Table 1:
Dickey-Fuller Test Equation
Dependent Variable: ∆lt
Method: Least Squares
Sample (adjusted): 1995M02 2004M12; observations: 119

Variable                Coefficient Std. Error    t-Statistic    Prob.

lt-1                    -0.200468 0.055277        -3.626566      0.0004
C                       1096.460 301.8879         3.632011       0.0004

R-squared               0.101051      Mean dependent var         2.038202
Adjusted R-squared      0.093368      S.D. dependent var         92.83931
S.E. of regression      88.39903      Akaike info criterion(σ)   11.81826
Sum squared resid       914283.5      Schwarz criterion(σ)       11.86497

The critical value of the t-statistic for a model with intercept c is -2,89 on the 5 % level:

Table 2:
Null Hypothesis: l has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=12)

t-Statistic

Test critical values:   1% level                  -3.486064
5% level                  -2.885863

The t-value measured exceeds the critical value downwards. That means that the null hypothesis
of nonstationarity or a unit root can be rejected on the 5 % level. The time series is stationary at
least with a probability of 95 %. As lt is stationary its degree of integration is 0 ( I(0)).

7
iv.    Sample correlation and partial correlation functions of the process.

The sample autocorrelation function (AC) for a short series can be calculated using the formulae
for the sample autocovariance (9.7)

T −τ
∑ ( xt − x )( xt +τ − x )
cτ =   t =1
T   − τ

and the sample autocorrelation (9.8)
cτ
rτ =      2
sx
with   x   and sx being the average and the standard deviation of the time series, respectively.

The partial sample autocorrelation function (PAC) can be obtained by linear OLS regression of xt
on xt-1, xt-2,…, xt-τ corresponding to formula (9.9):

xt = φ 0 + φ1τ xt −1 + ... + φττ xt −τ

The partial correlation coefficient ρpart(τ) is the coefficient φττ of the highest order lagged variable
xt-τ.

By calculating the first 5 regressions this way we obtain the following highest order coefficients
for each of them, respectively:

ф11= 0,799532; ф22= -0,037588;                       ф33= 0,075722; ф44= 0,008910; ф55= 0,115970

We can compare these „manually“ calculated coefficient with the partial autocorrelation
functions displayed in the following table and figure together with the sample autocorrelation
function delivered as a whole in one step by eViews. The differences between the regression
results and the eViews output probably are caused by different estimation methods.

8
Table 3:

AC        PAC                     τ      AC     PAC                τ      AC       PAC
τ
1     0.794        0.794                    9    0.039   -0.008             17   -0.157     -0.066
2     0.616        -0.042                  10    0.014   0.049              18   -0.154     0.038
3     0.495        0.052                   11   -0.004   -0.027             19   -0.125     0.042
4     0.390        -0.028                  12   -0.050   -0.092             20   -0.096     0.038
5     0.348        0.118                   13   -0.061   0.069              21   -0.098     -0.083
6     0.298        -0.038                  14   -0.100   -0.082             22   -0.142     -0.127
7     0.205        -0.115                  15   -0.139   -0.042             23   -0.150     0.046
8     0.109        -0.083                  16   -0.137   0.013              24   -0.183     -0.134

Correlogram

1,2
1
0,8
0,6
AC
0,4
PAC
0,2
0
-0,2 0   1   2    3    4   5   6   7   8   9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

-0,4

Fig. 3: Correlogram

The bold curve shows the sample autocorrelation function. The thin one displays the partial
autocorrelation, both depending on the lag τ. While the smooth autocorrelation continuously
decreases from 1 towards zero and below to a limit of about 0,1 the partial autocorrelation starts
as well at the value 1 and has the same value as the autocorrelation for τ=1 but then drops down
to zero and remains there oscillating between -0,1 and 0,1.

9
v.     Possible type of basic model fitting the time series l.

The basic model could be a first order autoregressive process , because AC is exponentially
decreasing and PAC is dropping down after τ=1. The shape of the AC is typical for the
autocorrelation of an AR process with positive coefficients. The partial autocorrelation drops
down to and remains close to zero after the lag τ=1 what indicates an AR(1) process.

vi.    Estimated parameters concerning the model assumed in answer v.

Table 4:
Dependent Variable: lt
Method: Least Squares
Sample 1995M02 2004M12, observations: 119

Variable               Coefficient Std. Error   t-Statistic      Prob.

CONST                  1096.460    301.8879      3.632011        0.0004
lt-1                   0.799532    0.055277      14.46398        0.0000

R-squared              0.641332      Mean dependent var          5461.386
Adjusted R-squared     0.638266      S.D. dependent var          146.9782
S.E. of regression     88.39903      Akaike info criterion       11.81826
Sum squared resid      914283.5      Schwarz criterion           11.86497

Thus we get the empirical model:
lt = 1096.5+ 0.7995 lt-1 + at

Because the t-statistic of both exceeds the 5-% critical value 1,96, both coefficients are
significant on this level, at least. The prob. values behind indicate that the even are on the 1 %
level.

10
Another way of estimation is to take advantage of the special ARMA estimation procedure of
eViews. Here first the mean of the variable is estimated and then the AR(1) coefficient of an AR
model of the deviations from the mean:
Table 5:
Dependent Variable: l
Method: Least Squares
Sample: 1995M02 2004M12; observations: 119
Convergence achieved after 3 iterations

Variable                Coefficient Std. Error    t-Statistic    Prob.

CONST                   5469.515    40.52025      134.9823       0.0000
AR(1)                   0.799532    0.055277      14.46398       0.0000

R-squared               0.641332      Mean dependent var         5461.386
Adjusted R-squared      0.638266      S.D. dependent var         146.9782
S.E. of regression      88.39906      Akaike info criterion(σ)   11.81826
Sum squared resid       914284.2      Schwarz criterion(σ)       11.86497

The equivalent model obtained this way can be written:

(lt - 5469,5) = 0,7995 ( lt-1 - 5469.5) + at

11
vii.   As an alternative, an ARMA(2,1) model would fit the data. It can be found by trial and
error via several different models and the aim of obtaining significant coefficients and
compared by Schwarz criterion.

As the model contains an MA term ordinary least squares is not practicable for estimating the
coefficients. Therefore again the iterative nonlinear least squares procedure by eViews is used:

Table 6:
Dependent Variable: l
Method: Least Squares
Sample 1995M03 2004M12; observations: 118
Convergence achieved after 7 iterations
Backcast: 1994M12

Variable                Coefficient Std. Error      t-Statistic   Prob.

CONST                   5469.200    37.73310        144.9444      0.0000
AR(2)                   0.600156    0.093632        6.409735      0.0000
MA(1)                   0.856271    0.060133        14.23960      0.0000

R-squared               0.646928      Mean dependent var          5462.352
Adjusted R-squared      0.640787      S.D. dependent var          147.2253
S.E. of regression      88.23855      Akaike info criterion(σ)    11.82306
Sum squared resid       895394.8      Schwarz criterion(σ)        11.89350

Again here is to be considered that the “constant” is the mean and the coefficients belong to a
model of derivations from it.
Thus the ARMA(2,1) model is to be written:
(lt - 5469,2) = 0,6002 ( lt-2 - 5469.5) + 0,8563 at + at

In order to have the model in the explicit form, the mean is subtracted:
lt = 5469,2 +     0,6002 ( lt-2 - 5469.2) + 0,8563 at + at
= 5469,2 - 5469.2 × 0,6002 .+         0,6002 lt-2 + 0,8563 at + at
and at last
lt =   2186,6 .+ 0,6002 lt-2 + 0,8563 at + at

This model does not show an improvement compared by Akaike and Schwarz criteria with the
AR(1) estimated earlier. It has slightly higher values. EViews provides these criteria on the base
of the error variance which is to be minimised (in contrast to those on likelihood base such as in
Microfit that are to be maximised).

12
viii.   Forecast of the number of employees for 2005:

The somewhat uneasy shape of both models:

(lt - 5469,5) = 0,7995 ( lt-1 - 5469.5) + at
and
(lt - 5469,2) = 0,6002 ( lt-2 - 5469.5) + 0,8563 at + at

give us the advantage of easily forecasting the process in the long run because the stochastic
limes for time tending to infinity is given by what we called the means 5469,5 and 5469,2,
respectively:

plim (lt - 5469,5)      = plim (0,7995 ( lt-1 - 5469.5) + at)
= plim (0,7995 ( lt-1 - 5469.5) + plim at
= 0                           + 0.
That means      plim lt = 5469,5 and we can take this constant as suitable forecast for the farther
future.
In the same way we obtain from the second model the forecast 5469,2 that does not differ very
much.
The following graph displays the forecast for the years 2004 and 2005. In 2004 we obtain an
irregular curve, because the one-month ahead forecast for every month can be calculated on the
base of the varying values of the previous month. But in 2005, there is a smooth exponential
curve because the forecasts can be calculated only on the base of the previous forecast instead of
the real data.

13
5800

5700

5600

5500

5400

5300

5200

5100

5000
2004M01        2004M07         2005M01          2005M07

Forecast of the number of employees

Fig. 4: Forecast

ix.    Spectral analysis is the name of this method.

Basically, the main peaks of spectral density occur at the circular frequencies
ω1= 0,71       ω2 = 1,86.
These correspond to frequencies
f1 = 0,114      f2 = 0,295
and to periods of average length
p1 = 8,8        p2 = 3,4 month, respectively.
But these periodicities are not of any practical importance. They superimpose to the point of
being unrecognizable. They are not visible in the original time series and the correlogram. Taking
standard errors of the spectral estimates into consideration, peaks and troughs of the spectral
density curve does not significantly differ.

The typical shape of the spectral density function for an        AR(1) process is similar to the
following

14
Estimates of spectral density of an AR(1) process
8

Bartlett

6

4
Tukey

2

Parzen

0
0         1                  2              3              4

Circular frequency

Fig. 5: Estimates of spectral density of an AR(1) process

In case of an additional seasonal component the spectral density would show a peak over the
frequency f = 1/12 or the circular frequency ω = 2π/12 = 0,52.

15
3   The file dax_j95_a04.txt contains the daily closing data of the main German stock
price index DAX from January 1995 by August 2004, i.e. 2498 values.

i.     Display the graph of the time series dax.

ii.    Determine the order of integration of dax.

iii.   Generate the series of the growth rate or rate of return r in the direct way and in
the logarithmic way. Display both graphs of r.

iv.    Compare the r data. Characterize the general patterns of both graphs of r

v.     Check whether or not r is stationary.

vi.    Estimate the correlation function and the partial correlation function till a lag of
20. Characterize generally the extent of correlation in this series.

vii.   Estimate an AR(8) model to r that contains only coefficients significant at least on
the five percent level

viii. Estimate an ARIMA(8,d,8) model to r that contains only coefficients significant at
least on the one percent level.

ix.    Test the residuals of this model for autoregressive conditional heteroscedasticity
by a rough elementary procedure

x.     Make visible conditional heteroscedasticity by estimating a 5-day moving variance
of the residuals

xi.    Estimate an ARCH(1) model on the base of the ARIMA(8,d,8) model estimated in
viii. You can change the ARMA part in order to keep only coefficients significant
on the one percent level.

xii.   Estimate a GARCH(1,1) model on the base of the simple model r=const.

16

i.       The graph of dax.

9000

8000

7000

6000

5000

4000

3000

2000

1000

0
500       1000        1500       2000

DAX

Fig. 6: The graph of the German share price index

This graph indicates a non-stationary process, perhaps a random walk. It is characterized by
changing stochastic trends and increasing variance.

17
ii.      The order of integration of dax.

For testing whether or not dax is stationary first the original date are tested by the Dickey-Fuller
test of the levels. The following table shows the estimation of the coefficient of the Dickey-
Fuller Test regression of ∆daxt on daxt-1 :

Table 7:
Dickey-Fuller Test Equation
Dependent Variable: ∆daxt
Method: Least Squares
Sample (adjusted): 2 2498; observations: 2497

Variable                Coefficient Std. Error       t-Statistic    Prob.

daxt-1                  -0.001453 0.000919           -1.580936      0.1140
C                       6.970114 4.213823            1.654107       0.0982

R-squared               0.001001         Mean dependent var         0.701418
Adjusted R-squared      0.000600         S.D. dependent var         71.28183
S.E. of regression      71.26043         Akaike info criterion(σ)   11.37136
Sum squared resid       12669731         Schwarz criterion(σ)       11.37602

The t-value of the slope coefficient is -1.580936. As the following table demonstrates it is not
less than the 5%-critical t-value for a model with a constant, i.e. -2.862497.

Table 8:
Null Hypothesis: DAX has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=26)

t-Statistic     Prob.*

Augmented Dickey-Fuller test statistic               -1.580936      0.4922
Test critical values: 1% level                       -3.432775
5% level                       -2.862497

That means that dax is on the 5% level not stationary. The next step is to check the first
differences of dax in the same way.

18
The Dickey-Fuller test equation of the first differences is a regression of the second differences
∆2daxt on the lagged first differences ∆daxt-1 .

Table 9:
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(DAX,2) = ∆2daxt
Method: Least Squares
Sample: 3 2498; observations: 2496

Variable                     Coefficient Std. Error           t-Statistic    Prob.

D(DAX(-1))= ∆daxt-1          -0.996234 0.020026               -49.74669      0.0000
C                            0.702902 1.427408                0.492432       0.6225

R-squared                    0.498061             Mean dependent var         -0.017540
Adjusted R-squared           0.497860             S.D. dependent var         100.6319
S.E. of regression           71.30959             Akaike info criterion(σ)   11.37274
Sum squared resid            12682133             Schwarz criterion(σ)       11.37741

The t-value -49.74669 obviously exceeds all thinkable critical values.
Thus the first differences are stationary and dax itself is first order integrated I(1).

iii.    The growth rate or rate of return r can be generated as the ratio

( dax t − dax t − 1 )
rt =
dax t − 1

That is displayed in the following graph:

19
.08

.04

.00

-.04

-.08

-.12
500            1000            1500             2000

R _ R A T IO

Fig. 7: Graph of the growth rate of DAX

Another way of calculating the rate of return mostly used in financial market analysis is the
logarithmic way:
rt = ln dax t − ln dax t −1
The next figure shows the graph of this logarithmically generated rate of return r.
.08

.04

.00

-.04

-.08

-.12
500            1000            1500             2000

R _ LO G

Fig. 8: Graph of the logarithmic return rate of DAX

20
iv.    Both curves are very similar to each other and can hardly be distinguished by the naked
eye. An enlarged display of the differences as shown in the next figure demonstrates that
the ratio always slightly exceeds the logarithmic approximation. In the following analysis
the latter will be used.

Differences between alternative series of dax return rate r

.005

.004

.003

.002

.001

.000
500          1000            1500           2000

D IF F _ R

Fig. 9: Graph of the differences between both ways of calculating return rates

Both graphs of r show certain common general patterns. These are significant clusters of high
variability or volatility separated by quieter periods. This changing behaviour of variance is
typical for ARCH or GARCH processes.

v.     Stationarity test for r.

Particularly for later fitting an ARCH or GARCH model, the time series should be stationary.
The following Dickey Fuller Regression of ∆rt on rt-1 produces a negative t-value of -50.377
that lies below of all possible critical values: The return rate r proves stationary.

21
Table 10:
Dickey-Fuller Test Equation
Dependent Variable: ∆rt
Method: Least Squares
Sample (adjusted): 3 2498; observations: 2496

Variable                         Coefficient Std. Error      t-Statistic    Prob.

rt-1                             -1.008850 0.020026          -50.37696      0.0000
C                                0.000250 0.000320           0.780708       0.4350

R-squared                        0.504356        Mean dependent var         -3.68E-06
Adjusted R-squared               0.504157        S.D. dependent var         0.022675
S.E. of regression               0.015967        Akaike info criterion(σ)   -5.435790
Sum squared resid                0.635829        Schwarz criterion(σ)       -5.431125

vi.          Sample correlation function and partial correlation functions of r

Correlogram

1,2

1

0,8

0,6                                                                                   AC
0,4                                                                                   PAC

0,2

0
0   1   2   3   4   5   6   7   8    9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
-0,2

Fig. 10: Graph of sample correlation and partial correlation function

22
Table 11:

τ     AC        PAC                     τ      AC        PAC                  τ    AC       PAC
1     -0.009   -0.009                   13     -0.017    -0.015               25   0.005    0.004
2     -0.005   -0.005                   14     0.071     0.076                26   -0.048   -0.048
3     -0.032   -0.032                   15     0.027     0.028                27   -0.046   -0.036
4     0.027    0.026                    16     -0.024    -0.026               28   0.021    0.014
5     -0.041   -0.040                   17     -0.038    -0.030               29   0.056    0.048
6     -0.042   -0.043                   18     0.001     -0.001               30   0.025    0.033
7     -0.006   -0.006                   19     -0.048    -0.048               31   0.020    0.030
8     0.042    0.038                    20     0.023     0.029                32   0.010    0.001
9     -0.005   -0.005                   21     0.025     0.031                33   -0.001   -0.001
10    -0.007   -0.007                   22     0.005     -0.008               34   -0.014   -0.006
11    0.012    0.011                    23     -0.002    -0.004               35   -0.021   -0.022
12    0.014    0.010                    24     0.033     0.033                36   0.019    0.017

Let us try a general characterisation of the extent of correlation in this series:
The sample correlation and partial correlation function do not differ very much. The graphs of
both of them coincide. There is not any significant correlation value. This could be the functions
of a white noise. Anyway, because of the little peaks at 5, 6 and 8 it could be sensible try to fit
an AR(8) model.

vii.        Example: AR(8) model coefficients significant on the five percent level.

Several trials with AR coefficient up to the order 8 finally resulted in three significant
coefficients and a missing constant. The criterion for rejecting other variants was non-
significance of coefficients.

Table 12:
Dependent Variable: r
Method: Least Squares
Sample (adjusted): 10 2498; observations: 2489

Variable                     Coefficient Std. Error     t-Statistic      Prob.

rt-5                         -0.039842 0.020023         -1.989799        0.0467
rt-6                         -0.041799 0.020013         -2.088571        0.0368
rt-8                         0.039960 0.020028          1.995155         0.0461

Mean dependent var 0.000244              S.D. dependent var            0.015982
S.E. of regression   0.015951              Akaike info criterion         -5.437444
Sum squared resid    0.632487              Schwarz criterion             -5.430429

23
Thus we obtain by OLS regression the AR(8) model

rt = -0.039842 rt-5 - 0.041799 rt-6 + 0.039960 rt-8 + et ,

where et is the error term of the process estimated.

We get exactly the same results, if we consider this AR model as a special case of an ARMA
model estimated by iterative nonlinear least squares

Table 13:
Dependent Variable: r
Method: Least Squares
Sample (adjusted): 10 2498; observations: 2489
Convergence achieved after 3 iterations

Variable               Coefficient Std. Error    t-Statistic    Prob.

AR(5)                  -0.039842 0.020023        -1.989799      0.0467
AR(6)                  -0.041799 0.020013        -2.088571      0.0368
AR(8)                  0.039960 0.020028         1.995155       0.0461

Mean dependent var     0.000244      S.D. dependent var         0.015982
S.E. of regression     0.015951      Akaike info criterion(σ)   -5.437444
Sum squared resid      0.632487      Schwarz criterion(σ)       -5.430429

viii.   Example: ARIMA(8,d,8) model for r that contains only coefficients significant at least on
the one percent level.

Now we try to improve the model by including a moving average term. In this case we must use
the iterative nonlinear least squares method. Now the aim is to obtain coefficients, significant on
the 1% level. Again after a series of trials with varying ARMA(8,8) models we got finally the
following results by omitting non significant coefficients:

24
Table 14:
Dependent Variable: r
Method: Least Squares
Sample (adjusted): 10 2498; observations: 2489
Convergence achieved after 15 iterations

Variable                 Coefficient Std. Error     t-Statistic       Prob.

AR(3)                    -0.339258   0.025338       -13.38956         0.0000
AR(8)                    -0.613633   0.025849       -23.73905         0.0000
MA(3)                    0.296335    0.027198       10.89544          0.0000
MA(6)                    -0.075689   0.013945       -5.427549         0.0000
MA(8)                    0.654681    0.020088       32.59034          0.0000

Mean dependent var       0.000244       S.D. dependent var            0.015982
S.E. of regression       0.015874       Akaike info criterion(σ)      -5.446226
Sum squared resid        0.625950       Schwarz criterion(σ)          -5.434536

The        model       resulting       from        this          estimation      is   the   following:

rt = - 0.339 rt-3 - 0.614 rt-8 + et + 0.296 et-3 - 0.076 et-6 + 0.655 et-8

where et is the error term of the estimated ARMA process.

Now the goodness of fit of both models should be compared. Besides considering the
significance level of the coefficients in both models a powerful mean of comparison are the
Akaike and Schwarz criteria, which are to be minimised.
Here both criteria are very close to each other. But the slightly smaller (negative!) values of
both criteria in the second case give a certain preference to the ARMA model.
Because we know from v. that r is stationary, that means integrated of order 0, this model is an
ARIMA(1,0,1) for r at the same time.

25
ix.     Rough and preliminary test of the residuals for autoregressive conditional
heteroscedasticity by OLS regression

A rough check for first order autoregressive conditional heteroscedasticity (ARCH(1)) is the
estimation of a regression of the squared residuals et2 on et-12 :

Table 15:
Dependent Variable: et2
Method: Least Squares
Sample (adjusted): 11 2498; observations: 2488

Variable               Coefficient Std. Error    t-Statistic     Prob.

C                      0.000208    1.18E-05      17.59238        0.0000
et-12                  0.175073    0.019746      8.866325        0.0000

R-squared              0.030652      Mean dependent var          0.000252
Adjusted R-squared     0.030263      S.D. dependent var          0.000543
S.E. of regression     0.000535      Akaike info criterion (σ)   -12.22961
Sum squared resid      0.000710      Schwarz criterion (σ)       -12.22493

For testing this dependence we cannot simply use the usual t-test because the t-values estimated
do not meet an exact student distribution. Anyway, because here the t-values immensely exceed
the 5-percent and 1-percent critical values, we can with great practical confidence assume, that
there exist a highly significant relationship between      et2 and et-12 i.e. high conditional
heteroscedasticity.

x.      Visualisation of conditional heteroscedasticity by moving 5-day residual variance

The existence of conditional heteroscedasticity can be visualised by smoothing the series of
squared residuals that means by moving averages of the et2 or moving variances.

26
.004

.003

.002

.001

.000
500            1000              1500          2000

5 -d a y m o v in g re s id u a l v a ria n c e

Fig. 11:

At the figure, you can see typical clusters of higher variance, i.e. volatility, changing with
intervals of lower variance. The similarity of neighbouring variances is one more indicator for
conditional heteroscedasticity: On the base of knowing the variance at time t (i. e. conditionally)
you can forecast the variance at time t+1.

xi.     ARCH(1) model on the base of the AR(8) model

Because of the latter analytical results it would be worthwhile to estimate an ARCH(1) model.
Then the conditional variance of the error et is

ht2 = var(et | et −1 ) = E(et2 | et −1 ) = λ0 + λ1et2−1 ,
where practically ht2 is estimated by et2

In the software used this is a special case of the more general GARCH model. An ARCH(1)
there corresponds to a GARCH(0,1) model. We assume here the time series to follow an AR(8)
process as estimated earlier without ARCH:

27
Table 16:
Dependent Variable: r
Method: ML – ARCH
Sample (adjusted): 10 2498; observations: 2489
GARCH = C(4) + C(5)*RESID(-1)^2

Coefficient Std. Error    z-Statistic    Prob.

AR(5)                  -0.045702 0.013304        -3.435162      0.0006
AR(6)                  -0.046538 0.014090        -3.303005      0.0010
AR(8)                  0.044716 0.013812         3.237367       0.0012

Variance Equation

C                      0.000194    4.67E-06      41.55146       0.0000
RESID(-1)^2            0.243443    0.026156      9.307385       0.0000

R-squared              0.004705      Mean dependent var         0.000244
Adjusted R-squared     0.003102      S.D. dependent var         0.015982
S.E. of regression     0.015958      Akaike info criterion(σ)   -5.493524
Sum squared resid      0.632539      Schwarz criterion(σ)       -5.481833

The coefficients differ from the earlier estimated ones because here they are estimated
simultaneously with the ARCH term. The newly estimated model is:

rt = -0.0457 rt-5 - 0.0465 rt-5 + 0.0447 rt-8 + et ,
with the error process et2 = 0.000194 + 0.2434 et-12 + at

where at should be a pure random series. The AR representation of the error process gives
the opportunity to forecast the volatility.

28
xii.      GARCH(1,1) model on the base of the simple model r=const.

Here you should model the return rate as a real GARCH process.

The generalized autoregressive conditional heteroscedasticity model (GARCH (p,q)) describes a
process where the conditional error variance on all information Ωt-1 available at time t
ht2 = var (u t Ωt −1 )
is assumed to obey an ARMA(p,q) model:
ht2 = α 0 + α 1ht2−1 + ... + α p ht2− p + β 1ut2−1 + β 2 ut2−2 + ... + β q ut2−q

with u t being the error process .

Table 17:
Dependent Variable: r
Method: ML - ARCH (Marquardt) - Normal distribution
Sample (adjusted): 10 2498; observations: 2489
Convergence achieved after 12 iterations
GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*GARCH(-1)

Coefficient Std. Error            z-Statistic      Prob.

C                                0.000710       0.000246           2.882234         0.0039
rt-5                             -0.029571      0.020063           -1.473904        0.1405
rt-5                             -0.036495      0.020360           -1.792478        0.0731
rt-8                             0.012575       0.020570           0.611309         0.5410

Variance Equation

C                                2.01E-06       4.03E-07           4.980196         0.0000
RESID(-1)^2                      0.082160       0.009593           8.564982         0.0000
GARCH(-1)                        0.911339       0.009476           96.17172         0.0000

R-squared                        0.003340          Mean dependent var               0.000244
Adjusted R-squared               0.000930          S.D. dependent var               0.015982
S.E. of regression               0.015975          Akaike info criterion(σ)         -5.752055
Sum squared resid                0.633406          Schwarz criterion (σ)            -5.735688

While in the original AR(8) model the constant could be omitted here the constant is the only
significant term in the regression part of the model. Therefore all lagged r terms can be omitted
now:

29
Table 18:
Dependent Variable: r
Method: ML – ARCH
Sample (adjusted): 2 2498; observations: 2497
Convergence achieved after 12 iterations
GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1)

Coefficient Std. Error    z-Statistic    Prob.

C                      0.000662    0.000240      2.753134       0.0059

Variance Equation

C                      2.08E-06    4.06E-07      5.116231       0.0000
RESID(-1)^2            0.083192    0.009602      8.664398       0.0000
GARCH(-1)              0.910103    0.009445      96.35969       0.0000

R-squared              -0.000681     Mean dependent var         0.000245
Adjusted R-squared     -0.001886     S.D. dependent var         0.015961
S.E. of regression     0.015977      Akaike info criterion(σ)   -5.756824
Sum squared resid      0.636337      Schwarz criterion(σ)       -5.747496

The model proves very simple: DAX return equals the constant 0,00066 (i.e. 0,06 % per day on
an average) with sort of an ARMA(1,1) conditional variance that describes the development of
volatility or risk:

ht2 = 2,08 · !0-06 + 0,9101 ht-12 + 0.08319 et-12 + at

2
with ht being the conditional variance of rt on base of the information by time t, and et being
the deviation of r from its mean in this model.
In the meaning of Akaike and Schwarz criteria, the model fits better than all considered before:
The values of these criteria are the lower ones despite the model is extremely simple. The model
shows that conditional variance as a measure of volatility and investment risk is highly
determined by the variance of the last day, i.e. rather by the more theoretical conditional variance
ht-12 than by the directly measurable deviation et-1.

30

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