Gas furnace data, cont

Gas furnace data, cont.

How should the sample cross correlations between the prewhitened series be interpreted?

pwgas<-prewhiten(gasrate,CO2,ylab="CCF")print(pwgas)$ccf

Autocorrelations of series ‘X’, by lag

-21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -0.069 -0.044 -0.060 -0.092 -0.073 -0.069 -0.053 -0.015 -0.054 -0.026 -0.059 -0.060 -0.081 -0.141 -0.208 -0.299 -0.318 -0.292 -0.226 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -0.116 -0.077 -0.083 -0.108 -0.093 -0.073 -0.058 -0.018 -0.013 -0.030 -0.077 -0.065 -0.045 0.039 0.047 0.067 0.041 -0.004 -0.026 17 18 19 20 21 -0.058 -0.060 -0.038 -0.091 -0.075

$model

Call:ar.ols(x = x)

Coefficients: 1 2 1.0467 -0.2131

Intercept: -0.2545 (2.253)

Order selected 2 sigma^2 estimated as 1492

k = –3

What does this say about

t

br

r

ss

tb XB

BB

BBCXB

B

BC

1

1

1

1

Since the first significant spike occurs at k = –3 it is natural to set b = 3

tr

r

ss XBBB

BBC 3

1

1

1

1

Rules-of-thumb:

•The order of the numerator polynomial, i.e. s is decided upon how many significant spikes there are after the first signifikant spike and before the spikes start to die down. s =2•The order of the denominator polynomial, i.e. r is decided upon how the spikes are dying down. exponential : choose r = 1 damped sine wave: choose r = 2 cuts of after lag b + s : choose r = 0 r = 1

Using the arimax command we can now try to fit this model without specifying any ARMA-structure in the full model

and then fit an ARMA to the residuals.

tXB

B

BBC 3

1

221

1

1

tt

bt e

B

BXB

B

BCY

0

Alternatively we can expand

to get an infinite terms expression

tXB

B

BBC 3

1

221

1

1

tt XBBBBCXBBBBC 63

52

41

3333

2211

in which we cut the terms corresponding with lags after the last lag with a significant spike in the sample CCF.

The last significant spike is for k = –8

867564534231

85

74

63

52

41

3

tttttt

t

XXXXXX

XBBBBBBC

Hence, we can use OLS (lm) to fit the temporary model

and then model the residuals with a proper ARMA-model

tttttttt XXXXXXY 8675645342310

model1_temp2 <- lm(CO2[1:288]~gasrate[4:291]+gasrate[5:292]+gasrate[6:293]+gasrate[7:294]+gasrate[8:295]+gasrate[9:296])summary(model1_temp2)

Call:lm(formula = CO2[1:288] ~ gasrate[4:291] + gasrate[5:292] + gasrate[6:293] + gasrate[7:294] + gasrate[8:295] + gasrate[9:296])

Residuals: Min 1Q Median 3Q Max -7.1044 -2.1646 -0.0916 2.2088 6.9429

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 53.3279 0.1780 299.544 < 2e-16 ***gasrate[4:291] -2.7983 0.9428 -2.968 0.00325 ** gasrate[5:292] 3.2472 2.0462 1.587 0.11365 gasrate[6:293] -0.8936 2.3434 -0.381 0.70325 gasrate[7:294] -0.3074 2.3424 -0.131 0.89570 gasrate[8:295] -0.6443 2.0453 -0.315 0.75300 gasrate[9:296] 0.3574 0.9427 0.379 0.70491 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.014 on 281 degrees of freedomMultiple R-squared: 0.1105, Adjusted R-squared: 0.09155 F-statistic: 5.82 on 6 and 281 DF, p-value: 9.824e-06

acf(rstudent(model1_temp2))pacf(rstudent(model1_temp2))

AR(2) or AR(3), probably AR(2)

model1_gasfurnace <- arima(CO2[1:288],order=c(2,0,0),xreg=data.frame(gasrate_3=gasrate[4:291],gasrate_4=gasrate[5:292],gasrate_5=gasrate[6:293],gasrate_6=gasrate[7:294],gasrate_7=gasrate[8:295],gasrate_8=gasrate[9:296]))print(model1_gasfurnace)

Call:arimax(x = CO2[1:288], order = c(2, 0, 0), xreg = data.frame(gasrate_3 = gasrate[4:291], gasrate_4 = gasrate[5:292], gasrate_5 = gasrate[6:293], gasrate_6 = gasrate[7:294], gasrate_7 = gasrate[8:295], gasrate_8 = gasrate[9:296]))

Coefficients: ar1 ar2 intercept gasrate_3 gasrate_4 gasrate_5 gasrate_6 gasrate_7 gasrate_8 1.8314 -0.8977 53.3845 -0.3382 -0.2347 -0.1517 -0.3673 -0.2474 -0.3395s.e. 0.0259 0.0263 0.3257 0.1151 0.1145 0.1111 0.1109 0.1139 0.1119

sigma^2 estimated as 0.1341: log likelihood = -122.35, aic = 262.71

Note! With use only of xreg we can stick to the command arima

tsdiag(model1_gasfurnace)

Some autocorrelation left in residuals

Forecasting a lead of two time-points(Works since the CO2 is modelled to depend on gasrate lagged 3 time-points)

predict(model1_gasfurnace,n.ahead=2,newxreg=data.frame(gasrate[2:3],gasrate[3:4],gasrate[4:5],gasrate[5:6],gasrate[6:7],gasrate[7:8]))

$predTime Series:Start = 289 End = 290 Frequency = 1 [1] 56.76652 57.52682

$seTime Series:Start = 289 End = 290 Frequency = 1 [1] 0.3662500 0.7642258

Modelling Heteroscedasticity

Exponential increase, but it can also be seen that the variance is not constant with time.

A log transformation gives a clearer picture

Normally we would take first-order differences of this series to remove the trend (first-order non-stationarity)

The variance seems to increase with time, but we can rather see clusters of values with constant variance.

For financial data, this feature is known as volatility clustering.

Correlation pattern?

Not indicating non-stationarity. A clearly significant correlation at lag 19, though.

Autocorrelation of the series of absolute changes

Clearly non-stationary!

If two variables X and Y are normally distributed

Corr(X,Y) = 0 X and Y are independent Corr(|X|,|Y|) = 0

If the distributions are not normal, this equivalence does not hold

In a time series where data are non-normally distributed

0,0, kttkttk YYCorrYYCorr

The same type of reasoning can be applied to squared values of the time series.

Hence, a time series where the autocorrelations are zero may still possess higher-order dependencies.

Autocorrelation of the series of squared changes

Another example :

Stationary?

Just noise spikes?

Ljung-Box tests for spurious data

No evidence for any significant autocorrelations!

LB_plot <- function(x,K1,K2,fitdf=0) {output <- matrix(0,K2-K1+1,2)for (i in 1:(K2-K1+1)) { output[i,1]<-K1-1+i output[i,2]<-Box.test(x,K1-1+i,type="Ljung-Box",fitdf)$p.value}win.graph(width=4.875,height=3,pointsize=8)plot(y=output[,2],x=output[,1],pch=1,ylim=c(0,1),xlab="Lag",ylab="P-value")abline(h=0.05,lty="dashed",col="red")

Autocorrelations for squared values

A clearly significant spike at lag 1

McLeod-LI’s test

Ljung-Box’ test applied to the squared values of a series

McLeod.Li.test(y=spurdata)

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

Lag

P-v

alue

All P-values are below 0.05!

qqnorm(spurdata)qqline(spurdata)

Slight deviation from the normal distribution Explains why there are linear dependencies between squared values, although not between original values

Jarque-Bera test

A normal distribution is symmetric

03 YE

Moreover, the distribution tails are such that

43 3 YE

The sample functions

sample skewness sample kurtosis

can therefore be used as indicators of whether the sample comes from a normal distribution or not (they should both be close to zero for normally distributed data).

3 and

41

4

231

3

1

sn

YYg

sn

YYg

n

i i

n

i i

The Jarque-Bera test statistic is defined as

246

22

21 gngn

JB

and follows approximately a 2-distribution with two degrees of freedom if the data comes from a normal distribution.

Moreover, each term is approximately 2-distributed with one degree of freedom under normality.

Hence, if data either is too skewed or too heavily or lightly tailed compared with normally distributed data, the test statistic will exceed the expected range of a 2-distribution .

With R skewness can be calculated with the command skewness(x, na.rm = FALSE)

and kurtosis with the commandkurtosis(x, na.rm = FALSE)

(Both come with the package TSA.)

Hence, for the spurious time series we compute the Jarque-Bera test statistic and its P-value as

JB1<-NROW(spurdata)*((skewness(spurdata))^2/6+ (kurtosis(spurdata))^2/24)print(JB1)[1] 31.057341-pchisq(JB1,2)[1] 1.802956e-07

Hence, there is significant evidence that data is not normally distributed

For the changes in log(General Index) we have the QQ plot

In excess of the normal distribution

JB2< NROW(difflogIndex)*((skewness(difflogIndex))^2/6+ (kurtosis(difflogIndex))^2/24)print(JB2)[1] 313.48691-pchisq(JB2,2)[1] 0

P-value so low that it is represented by 0!

ARCH-models

ARCH stands for Auto-Regressive Conditional Heteroscedasticity

112

1 ,,Let tttt YYYVar

The conditional variance or conditional volatility given the previous values of the time series.

The ARCH model of order q [ARCH(q)] is defined as

2222

211

21

1

qtqtttt

tttt

YYY

Y

Engle R.F, Winner of Nobel Memorial Prize in Economic Sciences 2003 (together with C. Granger)

where {t } is white noise with zero mean and unit variance (a special case of et )

Modelling ARCH in practice

Consider the ARCH(1).model

21

21

1

ttt

tttt

Y

Y

dataon directly applied becannot model theobserved becannot Since 21tt

21

21

2221

221

,, of

tindependen is

1122

1nexpectatio in theconstant a like behave""

makeson ngConditioni

1122

11122

1112

01

,,

,,,,,,

thatNote

11

2

1

1

tttttt

YYttt

YY

ttttttttt

YYEY

YYEY

YYYEYYEYYYE

t

ttt

Now, let

21

2 tttt Y

{t } will be white noise [not straightforward to prove, but Yt is constructed so it will have zero mean. Hence the difference between Yt

2 and E(Yt2 | Y1, … , Yt – 1 )

(which is t ) can intuitively be realized must be something with zero mean and no correlation with values at other time-points]

model1AR and satisifies

gives modelin ngSubstituti

2

21

221

2

221

t

tttttt

tttt

Y

YYYY

Y

This extends by similar argumentation to ARCH(q)-models, i.e. the squared values Yt

2 should satisfy an AR(q)

Application to “spurious data” example

If an AR(q) model should be satisfied, the order can be 1 or two (depending on whether the interpretation of the SPAC renders one or two significant spikes.

Try AR(1)

spur_model<-arima(spurdata^2,order=c(1,0,0))print(spur_model)

Call:arima(x = spurdata^2, order = c(1, 0, 0))

Coefficients: ar1 intercept 0.4410 0.3430s.e. 0.0448 0.0501

sigma^2 estimated as 0.3153: log likelihood = -336.82, aic = 677.63

tsdiag(spur_model)

Satisfactory?

Thus, the estimated ARCH-model is

21

21

1

441.0343.0

ttt

tttt

Y

Y

Prediction of future conditional volatility:

etc.

ˆˆˆˆ

ˆˆˆˆˆˆˆˆ

,,,,

,,,,

,,,,ˆ

,,,,

,,,,,,ˆ

22

23

221

22

21

12

1

1

21

21

of and ,, of

tindependen is

122

112

21

21

21

21

12

1

11122

2

11

tttt

ttttt

tht

ththtththt

YYthththttht

tttttttt

thttht

tthttthttht

Y

YYYEEYYE

YYEYYYE

YYYYEYYE

YYYEYYYVar

YYYYYVarEYYE

hthtt

ht

GARCH-models

For purposes of better predictions of future volatility more of the historical data should be allowed to be involved in the prediction formula.

The Generalized Auto-Regressive Conditional Heteroscedastic model of orders p and q [ GARCH(p,q) ] is defined as

2211

21

2211

21

1

qtqtptptptttt

tttt

YY

Y

i.e. the current conditional volatility is explained by previous squared values of the time series as well as previous conditional volatilities.

For estimation purposes, we again substitute

qjpk

YYY

YY

YYY

Y

jk

ptptt

ptqpqptt

qtqt

ptptptttt

tttt

for 0;for 0

for

11

2,max,max

2111

2

2211

21

211

2

21

2

Hence, Yt2 follows an ARMA(max(p,q), p )-model

Note! An ARCH(q)-model is a GARCH(0,q)-model

A GARCH(p,q)-model can be shown to be weakly stationary if

1),max(

1

qp

iii

A prediction formula can be derived as a recursion formula, now including conditional volatilities several steps backwards.

See Cryer & Chan, page 297 for details.

Example GBP/EUR weekly average exchange rates 2007-2012 (Source: www.oanda.com)

Non-stationary in mean (to slow decay of spikes)

First-order differences

Apparent volatility clustering.

Seems to be stationary in mean, though

Correlation pattern of squared values?

Difficult to judge upon

EACF table

AR/MA 0 1 2 3 4 5 6 7 8 9 10 11 12 130 x x x x x x x x x o o x o o 1 x o o x o o o o o o o o o o 2 x x o x o o o o o o o o o o 3 x x x x o o o o o o o o o o 4 x x o o x o x o o o o o o o 5 x x x o o o o o o o o o o o 6 x x o o o o x o o o o o o o 7 o o x o o o x o o o o o o o

ARMA(1,1)

ARMA(2,2)

ARMA(1,1) 1 = max(p,q) GARCH(0,1) [=ARCH(1)] or GARCH(1,1)ARMA(2,2) 2 = max(p,q) GARCH(0,2, GARCH(1,2) or GARCH(2,2)

Candidate GARCH models:

Try GARCH(1,2) !

model1<-garch(diffgbpeur,order=c(1,2))

***** ESTIMATION WITH ANALYTICAL GRADIENT *****

I INITIAL X(I) D(I)

1 1.229711e-04 1.000e+00 2 5.000000e-02 1.000e+00 3 5.000000e-02 1.000e+00 4 5.000000e-02 1.000e+00

IT NF F RELDF PRELDF RELDX STPPAR D*STEP NPRELDF 0 1 -1.018e+03 1 7 -1.019e+03 8.87e-04 1.15e-03 1.0e-04 1.2e+10 1.0e-05 6.73e+06 2 8 -1.019e+03 2.13e-04 2.95e-04 1.0e-04 2.0e+00 1.0e-05 5.90e+00 3 9 -1.019e+03 2.34e-05 3.00e-05 7.0e-05 2.0e+00 1.0e-05 5.69e+00 4 17 -1.025e+03 5.32e-03 9.66e-03 4.0e-01 2.0e+00 9.2e-02 5.68e+00 5 18 -1.025e+03 6.78e-04 1.61e-03 2.2e-01 2.0e+00 9.2e-02 7.59e-02 6 19 -1.026e+03 8.91e-04 1.80e-03 5.0e-01 1.9e+00 1.8e-01 5.02e-02 7 21 -1.030e+03 3.30e-03 2.14e-03 2.5e-01 1.8e+00 1.8e-01 7.51e-02 8 23 -1.030e+03 5.68e-04 5.78e-04 3.9e-02 2.0e+00 3.7e-02 1.87e+01 9 25 -1.030e+03 6.15e-05 6.64e-05 3.7e-03 2.0e+00 3.7e-03 1.00e+00 10 28 -1.031e+03 2.15e-04 2.00e-04 1.5e-02 2.0e+00 1.5e-02 1.11e+00 11 30 -1.031e+03 4.32e-05 4.02e-05 2.9e-03 2.0e+00 2.9e-03 8.17e+01 12 32 -1.031e+03 8.65e-06 8.04e-06 5.7e-04 2.0e+00 5.9e-04 4.13e+03 13 34 -1.031e+03 1.73e-05 1.61e-05 1.1e-03 2.0e+00 1.2e-03 3.89e+04 14 36 -1.031e+03 3.46e-05 3.22e-05 2.3e-03 2.0e+00 2.3e-03 8.97e+05 15 38 -1.031e+03 6.96e-06 6.43e-06 4.5e-04 2.0e+00 4.7e-04 1.29e+04 16 41 -1.031e+03 2.23e-07 1.30e-07 9.0e-06 2.0e+00 9.4e-06 8.06e+03 17 43 -1.031e+03 2.04e-08 2.57e-08 1.8e-06 2.0e+00 1.9e-06 8.06e+03 18 45 -1.031e+03 6.24e-08 5.15e-08 3.6e-06 2.0e+00 3.8e-06 8.06e+03 19 47 -1.031e+03 1.03e-07 1.03e-07 7.2e-06 2.0e+00 7.5e-06 8.06e+03 20 49 -1.031e+03 2.93e-08 2.06e-08 1.4e-06 2.0e+00 1.5e-06 8.06e+03 21 59 -1.031e+03 -4.30e-14 1.70e-16 6.5e-15 2.0e+00 8.9e-15 -1.37e-02

***** FALSE CONVERGENCE *****

FUNCTION -1.030780e+03 RELDX 6.502e-15 FUNC. EVALS 59 GRAD. EVALS 21 PRELDF 1.700e-16 NPRELDF -1.369e-02

I FINAL X(I) D(I) G(I)

1 3.457568e-05 1.000e+00 -1.052e+01 2 2.214434e-01 1.000e+00 -6.039e+00 3 3.443267e-07 1.000e+00 -5.349e+00 4 5.037035e-01 1.000e+00 -1.461e+01

summary(model1)

Call:garch(x = diffgbpeur, order = c(1, 2))

Model:GARCH(1,2)

Residuals: Min 1Q Median 3Q Max -3.0405 -0.6766 -0.0592 0.6610 2.5436

Coefficient(s): Estimate Std. Error t value Pr(>|t|) a0 3.458e-05 1.469e-05 2.353 0.01861 * a1 2.214e-01 8.992e-02 2.463 0.01379 * a2 3.443e-07 1.072e-01 0.000 1.00000 b1 5.037e-01 1.607e-01 3.135 0.00172 **---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Diagnostic Tests: Jarque Bera Test

data: Residuals X-squared = 1.1885, df = 2, p-value = 0.552

Box-Ljung test

data: Squared.Residuals X-squared = 0.1431, df = 1, p-value = 0.7052

Not every parameter significant, but diagnostic tests are OK.

However, the problem with convergence should make us suspicious!

Try GARCH(0.2), i.e. ARCH(2)

model11<-garch(diffgbpeur,order=c(0,2)) ***** ESTIMATION WITH ANALYTICAL GRADIENT *****

I INITIAL X(I) D(I)

1 1.302047e-04 1.000e+00 2 5.000000e-02 1.000e+00 3 5.000000e-02 1.000e+00

IT NF F RELDF PRELDF RELDX STPPAR D*STEP NPRELDF 0 1 -1.018e+03 1 7 -1.019e+03 8.27e-04 1.07e-03 1.0e-04 1.1e+10 1.0e-05 5.83e+06 2 8 -1.019e+03 2.25e-04 3.01e-04 1.0e-04 2.1e+00 1.0e-05 5.80e+00 3 9 -1.019e+03 1.37e-05 1.58e-05 7.0e-05 2.0e+00 1.0e-05 5.61e+00 4 17 -1.024e+03 5.37e-03 9.58e-03 4.0e-01 2.0e+00 9.2e-02 5.59e+00 5 18 -1.025e+03 2.04e-04 1.47e-03 2.4e-01 2.0e+00 9.2e-02 6.28e-02 6 28 -1.025e+03 4.45e-06 8.10e-06 2.6e-06 5.7e+00 9.9e-07 1.36e-02 7 37 -1.025e+03 4.75e-04 8.77e-04 1.1e-01 1.8e+00 4.8e-02 9.43e-03 8 38 -1.026e+03 1.03e-03 7.23e-04 1.3e-01 6.1e-02 4.8e-02 7.24e-04 9 39 -1.027e+03 5.04e-04 4.04e-04 9.8e-02 3.0e-02 4.8e-02 4.04e-04 10 40 -1.027e+03 8.14e-05 6.74e-05 4.0e-02 0.0e+00 2.3e-02 6.74e-05 11 41 -1.027e+03 5.52e-06 5.00e-06 1.1e-02 0.0e+00 7.0e-03 5.00e-06 12 42 -1.027e+03 1.30e-07 1.12e-07 1.4e-03 0.0e+00 7.8e-04 1.12e-07 13 43 -1.027e+03 2.19e-08 1.56e-08 5.2e-04 0.0e+00 3.5e-04 1.56e-08 14 44 -1.027e+03 7.58e-09 6.98e-09 6.2e-04 3.0e-02 3.5e-04 6.98e-09 15 45 -1.027e+03 2.30e-10 2.11e-10 1.1e-04 0.0e+00 5.8e-05 2.11e-10 16 46 -1.027e+03 3.88e-12 3.72e-12 8.2e-06 0.0e+00 5.8e-06 3.72e-12

***** RELATIVE FUNCTION CONVERGENCE *****

FUNCTION -1.026796e+03 RELDX 8.165e-06 FUNC. EVALS 46 GRAD. EVALS 17 PRELDF 3.722e-12 NPRELDF 3.722e-12

I FINAL X(I) D(I) G(I)

1 8.171538e-05 1.000e+00 1.048e-01 2 2.631538e-01 1.000e+00 -3.790e-06 3 1.466673e-01 1.000e+00 8.242e-05

summary(model11)

Call:garch(x = diffgbpeur, order = c(0, 2))

Model:GARCH(0,2)

Residuals: Min 1Q Median 3Q Max -3.25564 -0.66044 -0.05575 0.63832 2.81831

Coefficient(s): Estimate Std. Error t value Pr(>|t|) a0 8.172e-05 1.029e-05 7.938 2e-15 ***a1 2.632e-01 9.499e-02 2.770 0.00560 ** a2 1.467e-01 5.041e-02 2.910 0.00362 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

All parameters are significant!

Diagnostic Tests: Jarque Bera Test

data: Residuals X-squared = 3.484, df = 2, p-value = 0.1752

Box-Ljung test

data: Squared.Residuals X-squared = 0.0966, df = 1, p-value = 0.7559

Diagnostic tests OK!

In summary, we have the model

15.0ˆ;26.0ˆ;000082.0ˆ

estimates parameter with

1

21

222

211

21

1

tttt

ttttt

YY

YYBCoefficient(s): Estimate Std. Error a0 8.172e-05 1.029e-05 a1 2.632e-01 9.499e-02 a2 1.467e-01 5.041e-02

Extension: Firstly, model the conditional mean, i.e. apply an ARMA-model to the first order differences ( an ARIMA(p,1,q) to original data)

An MA(1) to the first-order differences seems appropriate

cm_model<-arima(gbpeur,order=c(0,1,1))

Coefficients: ma1 0.4974s.e. 0.0535

sigma^2 estimated as 0.0001174: log likelihood = 804.36, aic = -1606.72

tsdiag(cm_model)

Seems to be in order

Now, investigate the correlation pattern of the squared residuals from the model

AR/MA 0 1 2 3 4 5 6 7 8 9 10 11 12 130 o x x x x o o x x o o o o o 1 x o o o x o o o x o o o o o 2 x o o o x o o o o o o o o o 3 x o x o o o o o o o x o o o 4 x x o o o o o o o o o o o o 5 x x o o x o o o o o o o o o 6 x x x x x o o o o o o o o o 7 o x o x o o o o o o o o o o

ARMA(1,1) ?

GARCH(0,1) or GARCH(1,1) Try GARCH(1,1)!

model2<-garch(residuals(cm_model),order=c(1,1))

Again, we get a message about false convergence Caution!However, we proceed here as we are looking for a model with better prediction capability.

summary(model2)Call:garch(x = residuals(cm_model), order = c(1, 1))

Model:GARCH(1,1)

Residuals: Min 1Q Median 3Q Max -3.31234 -0.68936 -0.01541 0.56837 3.55745

Coefficient(s): Estimate Std. Error t value Pr(>|t|) a0 8.306e-06 6.203e-06 1.339 0.18058 a1 1.069e-01 3.959e-02 2.701 0.00691 ** b1 8.163e-01 8.799e-02 9.278 < 2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Significant parameters

Diagnostic Tests: Jarque Bera Test

data: Residuals X-squared = 3.173, df = 2, p-value = 0.2046

Box-Ljung test

data: Squared.Residuals X-squared = 1.7533, df = 1, p-value = 0.1855

Diagnostic tests OK

In summary, the model used is

221

21

21

1

11

ttttt

tttt

ttt

e

e

eeYB

The parameter estimates are MA(1)Coefficients: ma1 0.4974s.e. 0.0535

GARCH(1,1)Coefficient(s): Estimate Std. Error a0 8.306e-06 6.203e-06 a1 1.069e-01 3.959e-02 b1 8.163e-01 8.799e-02

82.0ˆ

11.0ˆ

0000008.0ˆ

50.0ˆ

stationaryWeakly

193.082.011.0ˆˆ