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In this chapter, we apply the methodological analysis discussed in the predating chapter to find which theoretical account is suited for our clip series informations. The end of this chapter is to supply item analysis of our informations set in order to accomplish our aim, mentioned in chapter one. We begin by giving a brief overview of the information set used. This is followed by some initial informations analysis. Then, we proceed to the appraisal and theoretical account choice portion, where all the trials are conducted to happen the most appropriate theoretical account for our informations set.

Overview of Datas

The informations employed in this undertaking consists of 384 monthly Canadian unemployment rates runing from January 1980 to December 2011, for both sexes, where the topics are over the age of 15. The information has been extracted from OECD.Stat Extracts database. The first 300 observations ( January 1980 to December 2004 ) are used for parameter appraisal while the following 84 ( January 2005 to December 2011 ) are used for calculating rating[ 7 ]. All informations use and analysis are done in Eviews 7.0.

Datas Analysis

As discussed antecedently, clip series theoretical accounts are merely appropriate for stationary clip series. By looking at Figure 3.2. it appears that our unemployment rate clip series is non stationary. Furthermore, the clip series does non expose any tendency. However, a closer expression at the clip series shows that it displays some seasonality. From Figure 3.2. , we can see that the unemployment rate tends to increase in the month of March and lessening in September and October. Hence, seasonal accommodation is necessary. So, we transform our unemployment rate series into a seasonally adjusted one[ 8 ].

Figure 3.2. : Time Series Plot

Figure 3.2. : Time Series Plot by Seasons

Figure 3.2. : Seasonally Adjusted Time Series Plot

( Shaded countries denote recessions )

The descriptive statistics for the monthly seasonally adjusted unemployment rate is given in Table 3.2.. From the tabular array, it can be seen that our unemployment series has a mean of 8.595313, a standard divergence ( Std. Dev. ) of 1.737207, a positive lopsidedness of 0.557454 ( greater than nothing ) and kurtosis of 2.323963 ( less than 3 ) . The value of the lopsidedness implies that the series follows a right skewed distribution while the kurtosis value shows that this distribution is comparatively level ( platykurtic ) as compared to the normal. The series has a minimal value of 5.9 and a maximal value of 13. Furthermore, the Jarque-Bera besides rejects the void hypothesis for normalcy at 5 % degree. In add-on, the unit root trial outputs an Augmented Dickey Fuller ( ADF ) Test Statistic of -1.875236 with chance 0.3439, which rejects the void hypothesis for the presence of a unit root in the unemployment rate series at the 5 % significance degree. This consequence suggests that our clip series has to be differentiated.

Therefore, we convert the unemployment rates to first-differenced by utilizing the undermentioned expression:

( 34 )

where represents any observation at clip in the first-differenced clip series, and and are the unemployment rates at clip and in the original series severally. After taking first differences, the ADF Test Statistic is now -8.291770 and has a chance of 0.0000. Therefore, our new informations set is stationary and we can now continue to the appraisal phase. Ocular review of Figure 3.2. demonstrates that the first-differenced clip series is stationary. Random fluctuations around zero, which is a good mark of stationarity, can be observed. The first-differenced clip series show grounds of fat-tails, since the kurtosis exceeds 3, which is the normal value, and grounds of positive lopsidedness, which means that the series is skewed the right same as the original 1. Furthermore, the Jarque-Bera besides rejects the void hypothesis that the series follows a normal distribution at the 5 % significance degree.

Original

First-Differenced

A Observations

384

383

A Mean

8.595313

0.000000

A Median

8.000000

0.000000

A Maximum

13.00000

1.200000

A Minimum

5.900000

-0.600000

A Std. Dev.

1.737207

0.219948

A Lopsidedness

0.557454

0.935624

A Kurtosis

2.323963

6.181196

A Jarque-Bera

27.20077

217.3777

A Probability

0.000001

0.000000

ADF Test Statistic

-1.875236

-8.291770

Probability

( 0.3439 )

( 0.0000 )

Table 3.2. : Descriptive Statisticss of Canadian Unemployment Rate

Figure 3.2. : Differentiated Time Series Plot

Model Estimation and Evaluation

In this subdivision, we use the different clip series theoretical accounts, explained in the old chapter, to gauge our first-differenced clip series informations. After appraisal, we select the appropriate theoretical account based on the Schwarz Information Criterion ( SIC ) . The five possible campaigners from each theoretical account are selected for prediction exercisings since a theoretical account can suit our in-sample informations but is non suited for the out-of-sample 1.

Appraisal of ARMA Models

We apply 20 theoretical accounts for AR ( P ) and MA ( Q ) theoretical accounts ( p = Q = 1:20 ) and 25 theoretical accounts for ARMA ( P, Q ) theoretical accounts ( p = Q = 1, 2, 3, 4, 5 ) to our in-sample informations utilizing the Least Squares method.

Model Selection and Analysis

As mentioned earlier, theoretical account choice is based on SIC. However, for presentation intents, we examine the correlogram of the clip series to pull decisions about suited ARMA theoretical accounts. The designation of the ARMA theoretical accounts are based on the form of the autocorrelation map secret plan illustrated in Table 2.3.1. Figure 3.3. represents the correlogram of the unemployment rate after taking first differences.

The autocorrelations seem to disintegrate after a few slowdowns. Therefore, a mixture of autoregressive and moving mean theoretical account is suggested for our informations. It can be seen from the correlogram that both the autocorrelations and partial autocorrelations appear to cut off at slowdown 5. Henceforth, an ARMA ( 5, 5 ) theoretical account seems most appropriate. Table 3.3. below summarizes the SIC consequences of the five most plausible theoretical accounts from each clip series theoretical accounts estimated. The minimal SIC for the AR ( P ) theoretical accounts indicates an AR ( 1 ) theoretical account while that of the MA ( Q ) theoretical accounts shows a MA ( 1 ) theoretical account. Furthermore, the overall minimal SIC besides favours an ARMA ( 5, 5 ) theoretical account.

Figure 3.3. : Correlogram of DU

Model

SIC

Model

SIC

Model

SIC

AR ( 1 )

-0.066443

MA ( 1 )

-0.067312

ARMA ( 5,5 )

-0.124892

AR ( 3 )

-0.064332

MA ( 2 )

-0.051052

ARMA ( 3,3 )

-0.109100

AR ( 4 )

-0.052188

MA ( 3 )

-0.060788

ARMA ( 3,4 )

-0.106904

AR ( 2 )

-0.050607

MA ( 4 )

-0.049791

ARMA ( 4,3 )

-0.105106

AR ( 5 )

-0.048937

MA ( 5 )

-0.048254

ARMA ( 4,4 )

-0.096668

Table 3.3. : SIC of ARMA theoretical accounts

Model

Test Statistic

p-value

Critical Value

Consequence

AR ( 1 )

35.025

0.014

30.144

Cull

AR ( 2 )

29.923

0.038

28.869

Cull

AR ( 3 )

20.641

0.243

27.587

Do non reject

AR ( 4 )

17.346

0.364

26.296

Do non reject

AR ( 5 )

13.202

0.587

24.996

Do non reject

MA ( 1 )

37.471

0.007

30.144

Cull

MA ( 2 )

34.569

0.011

28.869

Cull

MA ( 3 )

25.204

0.090

27.587

Do non reject

MA ( 4 )

20.301

0.207

26.296

Do non reject

MA ( 5 )

16.654

0.340

24.996

Do non reject

ARMA ( 3,3 )

10.877

0.696

23.685

Do non reject

ARMA ( 3,4 )

16.481

0.224

22.362

Do non reject

ARMA ( 4,3 )

15.732

0.264

22.362

Do non reject

ARMA ( 4,4 )

11.142

0.517

21.026

Do non reject

ARMA ( 5,5 )

15.513

0.114

18.307

Do non reject

Table 3.3. : Ljung-Box-Pierce Q-test Consequences for Autocorrelation

As discussed antecedently, a clip series theoretical account is appropriate for our clip series merely if the remainders are random ( that is, white noise ) . In this survey, we use the Ljung-Box trial to find whether the theoretical accounts selected in the appraisal stage are appropriate for our informations set. From Table 3.3. , we can reason that the remainders, when tested for up to 20 slowdowns, are random in most instances except for AR ( 1 ) , AR ( 2 ) , MA ( 1 ) and MA ( 2 ) . Both the trial statistics and the p-values are important at the 5 % significance degree. Therefore, most of our theoretical accounts suit our in-sample informations. Nevertheless, a comparing of the consequences obtained from the appraisal phase and the Ljung-Box trial reveals that although a theoretical account has the smallest SIC, it may non ever be appropriate for our clip series ( for illustration, the AR ( 1 ) theoretical account ) .

Appraisal consequences

Table 3.3. studies consequences on the set of estimated ARMA theoretical accounts for the first-differenced and original unemployment rate informations over the period January 1980 to December 2004. The tabular array below lists merely theoretical accounts whose remainders are white noise.

Table 3.3. : ARMA Estimation Results

Model

Estimated Model

AR ( 3 )

AR ( 4 )

AR ( 5 )

MA ( 3 )

MA ( 4 )

MA ( 5 )

ARMA ( 3,3 )

ARMA ( 3,4 )

ARMA ( 4,3 )

ARMA ( 4,4 )

ARMA ( 5,5 )

The undermentioned section involves the application of non-linear univariate theoretical accounts to the remainders of the above selected conditional average theoretical accounts. In this undertaking, we allow the discrepancy of the remainders to follow both the symmetric and asymmetric GARCH theoretical accounts, viz. the GARCH ( P, Q ) , EGARCH ( P, Q ) and GJR-GARCH ( P, Q ) .

Remainders Nosologies

Before using any GARCH household theoretical accounts to the remainders of our conditional mean theoretical accounts, we examine the latter. In this survey, the undermentioned conditional mean theoretical accounts are used: AR ( P ) [ p = 1:5 ] , MA ( Q ) [ q = 1:5 ] , ARMA ( 3, 3 ) , ARMA ( 3, 4 ) , ARMA ( 4, 3 ) , ARMA ( 4, 4 ) and ARMA ( 5, 5 ) . It is of import to observe that GARCH household theoretical accounts can merely be applied to clip series that exhibits some signifier of heteroscedasticity ( as discussed in Chapter 2 ) . First, the remainders of the conditional mean theoretical accounts are checked for ARCH effects. Then, we are traveling to look into whether the remainders come from a normal distribution.

Mistake: Reference beginning non found gives the consequences of the ARCH trial applied up to 20 slowdowns. We can detect that both the F-statistics and the LM-statistics are important at the 5 % significance degree. Therefore, the remainders of our conditional mean theoretical accounts display ARCH effects. In other words, the discrepancy of the remainders ‘ series is non changeless throughout. Thus, GARCH household theoretical accounts can be employed. The descriptive statistics of the remainders for the different mean theoretical accounts considered are depicted in Table 3.3.. It is evident from the tabular array that the mean of the remainders are really near to nothing. In add-on, we can detect that the series show both positive lopsidedness and extra kurtosis ( kurtosis exceeds 3, which is the normal value ) in all instances. These values show grounds that the series follow a distribution, which is skewed to the right and peaked comparative to the normal ( leptokurtic ) . Furthermore, it is clear that the Jarque-Bera ( JB ) trial resolutely rejects the void hypothesis that the residuary series, { } , is Gaussian at the 5 % significance degree. For this ground, we suppose that the residuary series, { } , follows a Student-t Distribution or a Generalized Mistake Distribution ( GED ) during the appraisal of the GARCH household theoretical accounts[ 9 ].

Model

F-statistic

Prob. F

LM-statistic

Critical Value

Consequences

AR ( 1 )

5.403387

0.0000

82.29386

31.410

Cull

AR ( 2 )

5.119581

0.0000

79.13823

31.410

Cull

AR ( 3 )

3.989225

0.0000

65.77522

31.410

Cull

AR ( 4 )

3.847497

0.0000

63.94088

31.410

Cull

AR ( 5 )

4.971173

0.0000

77.29913

31.410

Cull

MA ( 1 )

4.581700

0.0000

73.12185

31.410

Cull

MA ( 2 )

4.333637

0.0000

70.15842

31.410

Cull

MA ( 3 )

3.299624

0.0000

56.82818

31.410

Cull

MA ( 4 )

3.18015

0.0000

55.18254

31.410

Cull

MA ( 5 )

3.668778

0.0000

61.77819

31.410

Cull

ARMA ( 3,3 )

4.128165

0.0000

67.50577

31.410

Cull

ARMA ( 3,4 )

3.131335

0.0000

54.41913

31.410

Cull

ARMA ( 4,3 )

3.030855

0.0000

52.98409

31.410

Cull

ARMA ( 4,4 )

3.027713

0.0000

52.86913

31.410

Cull

ARMA ( 5,5 )

3.776522

0.0000

62.99367

31.410

Cull

Table 3.3. : Engle ‘s ARCH trial for Heteroscedasticity

Model

Mean

Std. Dev

Lopsidedness

Kurtosis

JB

Prob.

AR ( 1 )

6.99e-10

0.230018

0.845509

5.948304

143.4378

0.0000

AR ( 2 )

4.67e-10

0.229624

0.808071

5.990846

143.0188

0.0000

AR ( 3 )

5.90e-10

0.225855

0.681100

5.865792

124.1763

0.0000

AR ( 4 )

-7.77e-13

0.225028

0.651610

5.836984

119.8052

0.0000

AR ( 5 )

1.01e-09

0.223197

0.691415

6.270273

154.4346

0.0000

MA ( 1 )

-3.40e-05

0.229929

0.857738

5.945441

144.7469

0.0000

MA ( 2 )

4.42e-05

0.229607

0.849049

5.996065

147.7552

0.0000

MA ( 3 )

-4.72e-05

0.226324

0.748202

5.907181

133.1911

0.0000

MA ( 4 )

-9.65e-05

0.225413

0.708232

5.796920

122.4546

0.0000

MA ( 5 )

-0.00020

0.223446

0.785030

6.435562

177.7577

0.0000

ARMA ( 3,3 )

0.00018

0.214578

0.770012

6.038092

143.0874

0.0000

ARMA ( 3,4 )

-0.00019

0.212759

0.731492

6.086970

143.9263

0.0000

ARMA ( 4,3 )

-6.04e-05

0.212905

0.710804

6.025342

137.3429

0.0000

ARMA ( 4,4 )

-0.00033

0.211755

0.607944

5.828064

116.4799

0.0000

ARMA ( 5,5 )

0.002475

0.204727

0.636479

5.586060

101.7746

0.0000

Table 3.3. : Descriptive Statisticss of Remainders

Appraisal of GARCH Models

In this subdivision, we estimate our stationary clip series utilizing four GARCH ( P, Q ) theoretical accounts [ p = 0, 1 and q = 1, 2 ] utilizing Marquadt algorithm. We use the aforesaid conditional mean theoretical accounts as average equations together with the two above mentioned mistake distributions, viz. the Student-t and the GED. The tabular array below shows the SIC consequences of the five theoretical accounts selected from each mean equations ( based on the minimal SIC attack ) . It is obvious from Mistake: Reference beginning non found that an ARMA ( 5, 5 ) -ARCH ( 2 ) theoretical account with the remainders following a Student-t distribution performs better. Yet, the adequateness of these theoretical accounts should be checked.

Table 3.3. : SIC consequences of GARCH theoretical accounts with different mean theoretical accounts

Model

Error Dist.

SIC

AR ( 1 ) -GARCH ( 1,1 )

Student-t

-0.1972

AR ( 1 ) -GARCH ( 1,2 )

Student-t

-0.180239

AR ( 2 ) -GARCH ( 1,1 )

Student-t

-0.174858

AR ( 1 ) -ARCH ( 1 )

Student-t

-0.174304

AR ( 1 ) -GARCH ( 1,1 )

GED

-0.167881

MA ( 1 ) -GARCH ( 1,1 )

Student-t

-0.19934

MA ( 2 ) -GARCH ( 1,1 )

Student-t

-0.18353

MA ( 1 ) -GARCH ( 1,2 )

Student-t

-0.18257

MA ( 1 ) -ARCH ( 1 )

Student-t

-0.17684

MA ( 1 ) -GARCH ( 1,1 )

GED

-0.17046

ARMA ( 5,5 ) -ARCH ( 2 )

Student-t

-0.23129

ARMA ( 3,3 ) -GARCH ( 1,1 )

Student-t

-0.19637

ARMA ( 3,3 ) -GARCH ( 1,1 )

GED

-0.18878

ARMA ( 5,5 ) -GARCH ( 1,1 )

GED

-0.1827

ARMA ( 3,4 ) -GARCH ( 1,1 )

Student-t

-0.17715

To look into the adequateness of these theoretical accounts, we apply the Engle ‘s ARCH trial up to 20 slowdowns to do certain that there are no ARCH effects left after appraisal. Both the F-statistic and the LM-statistic are undistinguished at the 5 % significance degree except for AR ( 1 ) -ARCH ( 1 ) , MA ( 1 ) -ARCH ( 1 ) and ARMA ( 5, 5 ) -ARCH ( 2 ) theoretical accounts. We can reason that using GARCH ( P, Q ) theoretical accounts to the average equations take the ARCH effects present in them except for these three theoretical accounts.

Model

Error Dist.

F-statistic

Prob. F

LM-

statistic

Critical

Value

AR ( 1 ) -GARCH ( 1,1 )

Student-t

0.675546

0.8490

13.88498

31.410

AR ( 1 ) -GARCH ( 1,2 )

Student-t

0.633418

0.8860

13.05977

31.410

AR ( 2 ) -GARCH ( 1,1 )

Student-t

0.651546

0.8708

13.41691

31.410

AR ( 1 ) -ARCH ( 1 )

Student-t

2.008828

0.0074

37.58399

31.410

AR ( 1 ) -GARCH ( 1,1 )

GED

0.719807

0.8047

14.74644

31.410

MA ( 1 ) -GARCH ( 1,1 )

Student-t

0.694896

0.8343

14.26093

31.410

MA ( 2 ) -GARCH ( 1,1 )

Student-t

0.669312

0.8549

13.76180

31.410

MA ( 1 ) -GARCH ( 1,2 )

Student-t

0.655572

0.8673

13.49295

31.410

MA ( 1 ) -ARCH ( 1 )

Student-t

2.018222

0.0071

37.74471

31.410

MA ( 1 ) -GARCH ( 1,1 )

GED

0.729213

0.7946

14.92752

31.410

ARMA ( 5,5 ) -ARCH ( 2 )

Student-t

2.365904

0.0011

43.17141

31.410

ARMA ( 3,3 ) -GARCH ( 1,1 )

Student-t

0.846606

0.5918

18.13368

31.410

ARMA ( 3,3 ) -GARCH ( 1,1 )

GED

0.849255

0.6519

17.23583

31.410

ARMA ( 5,5 ) -GARCH ( 1,1 )

GED

0.756760

0.7639

15.46625

31.410

ARMA ( 3,4 ) -GARCH ( 1,1 )

Student-t

0.897716

0.5903

18.15466

31.410

Table 3.3. : Engle ‘s ARCH trial for GARCH ( P, Q ) theoretical accounts

Appraisal consequences

Table 3.3. illustrates the consequences of the estimated GARCH theoretical accounts for the first-differenced and original unemployment rate informations over the period January 1980 to December 2004.

Table 3.3. : GARCH Estimation Results

Model

Error Dist.

Estimated Model

AR ( 1 ) -GARCH ( 1,1 )

Student-t

Average Equation:

Variance Equation:

AR ( 1 ) -GARCH ( 1,2 )

Student-t

Average Equation:

Variance Equation:

AR ( 2 ) -GARCH ( 1,1 )

Student-t

Average Equation:

Variance Equation:

AR ( 1 ) -GARCH ( 1,1 )

GED

Average Equation:

Variance Equation:

MA ( 1 ) -GARCH ( 1,1 )

Student-t

Average Equation:

Variance Equation:

MA ( 2 ) -GARCH ( 1,1 )

Student-t

Average Equation:

Variance Equation:

MA ( 1 ) -GARCH ( 1,2 )

Student-t

Average Equation:

Variance Equation:

MA ( 1 ) -GARCH ( 1,1 )

GED

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -GARCH ( 1,1 )

Student-t

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -GARCH ( 1,1 )

GED

Average Equation:

Variance Equation:

ARMA ( 5,5 ) -GARCH ( 1,1 )

GED

Average Equation:

Variance Equation:

ARMA ( 3,4 ) -GARCH ( 1,1 )

Student-t

Average Equation:

Variance Equation:

Appraisal of EGARCH Models

Following the same attack as the GARCH ( P, Q ) theoretical accounts, we apply four EGARCH ( P, Q ) theoretical accounts [ p = 0, 1 and q = 1, 2 ] to the antecedently mentioned conditional mean theoretical accounts utilizing the Marquadt algorithm. The same mistake distributions are used. EGARCH theoretical accounts with different asymmetric order ( up to 2 ) have been used[ 10 ]. Choice of the five most possible theoretical accounts from each mean equations are based on the SIC attack. On the other manus, we conduct an Engle ‘s ARCH trial up to 20 slowdowns to corroborate that no more Arch effects are left undermentioned appraisal.

Table 3.3. : SIC consequences of EGARCH theoretical accounts with different mean theoretical accounts

Model

Error Dist.

Order

SIC

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

1

-0.201592

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

0

-0.193925

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

2

-0.190718

AR ( 1 ) -EGARCH ( 1,2 )

Student-t

1

-0.185603

AR ( 2 ) -EGARCH ( 1,1 )

Student-t

1

-0.176634

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

1

-0.203512

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

0

-0.196295

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

2

-0.192631

MA ( 1 ) -EGARCH ( 1,2 )

Student-t

1

-0.188012

MA ( 2 ) -EGARCH ( 1,1 )

Student-t

1

-0.185917

ARMA ( 5,5 ) -EGARCH ( 1,1 )

Student-t

0

-0.211205

ARMA ( 3,3 ) -EGARCH ( 1,1 )

Student-t

1

-0.204488

ARMA ( 3,3 ) -EGARCH ( 1,1 )

Student-t

0

-0.199611

ARMA ( 5,5 ) -EGARCH ( 1,1 )

Student-t

1

-0.197941

ARMA ( 3,3 ) -EGARCH ( 1,1 )

GED

0

-0.197115

From the SIC consequences shown in Mistake: Reference beginning non found, we can see that an ARMA ( 5, 5 ) – EGARCH ( 1, 1 ) with mistake distribution Student-t best fits our stationary clip series while an AR ( 2 ) -EGARCH ( 1, 1 ) with the same mistake distribution, but with different asymmetric order, proves to be the worst.

Table 3.3. : Engle ‘s ARCH trial for EGARCH ( P, Q ) theoretical accounts

Model

Error Dist.

Order

F-statistic

Prob. F

LM-

statistic

Critical

Value

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

1

0.507321

0.9625

10.55864

31.410

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

0

0.802223

0.7103

16.33566

31.410

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

2

0.535564

0.9498

11.12294

31.410

AR ( 1 ) -EGARCH ( 1,2 )

Student-t

1

0.435689

0.9844

9.116690

31.410

AR ( 2 ) -EGARCH ( 1,1 )

Student-

1

0.466833

0.9795

9.747075

31.410

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

1

0.442449

0.9829

9.251913

31.410

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

0

0.793830

0.7204

16.17361

31.410

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

2

0.510236

0.9613

10.61547

31.410

MA ( 1 ) -EGARCH ( 1,2 )

Student-t

1

0.449520

0.9812

9.394805

31.410

MA ( 2 ) -EGARCH ( 1,1 )

Student-t

1

0.426076

0.9864

8.920499

31.410

ARMA ( 5,5 ) -EGARCH ( 1,1 )

Student-t

0

0.966395

0.5038

19.44658

31.410

ARMA ( 3,3 ) -EGARCH ( 1,1 )

Student-t

1

0.517041

0.9584

10.75622

31.410

ARMA ( 3,3 ) -EGARCH ( 1,1 )

Student-t

0

1.024084

0.4340

20.52051

31.410

ARMA ( 5,5 ) -EGARCH ( 1,1 )

Student-t

1

0.789841

0.7252

16.10260

31.410

ARMA ( 3,3 ) -EGARCH ( 1,1 )

GED

0

0.930336

0.5489

18.76947

31.410

It can be noticed from Table 3.3. that both the LM-statistic and F-statistic are undistinguished at the 5 % significance degree in most instances. This suggests that the estimated clip series theoretical accounts do non exhibit any ARCH effects. In contrast to the appraisal of the GARCH theoretical accounts, we find that the add-on of an EGARCH theoretical account to the average equations take about all ARCH effects from them. This may be because EGARCH theoretical accounts better capture the heteroscedasticity presents inn our informations.

Appraisal consequences

The consequences of the estimated EGARCH theoretical accounts for the first-differenced and original unemployment rate informations over the period January 1980 to December 2004 are depicted in Table 3.3..

Table 3.3. : EGARCH Appraisal Consequences

Model

Error Dist.

Order

Estimated Model

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

0

Average Equation:

Variance Equation:

AR ( 1 ) -EGARCH ( 1,1 )

Student-t

2

Average Equation:

Variance Equation:

AR ( 1 ) -EGARCH ( 1,2 )

Student-t

1

Average Equation:

Variance Equation:

AR ( 2 ) -EGARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

0

Average Equation:

Variance Equation:

MA ( 1 ) -EGARCH ( 1,1 )

Student-t

2

Average Equation:

Variance Equation:

MA ( 1 ) -EGARCH ( 1,2 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 2 ) -EGARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

ARMA ( 5,5 ) -EGARCH ( 1,1 )

Student-t

0

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -EGARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -EGARCH ( 1,1 )

Student-t

0

Average Equation:

Variance Equation:

ARMA ( 5,5 ) -EGARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -EGARCH ( 1,1 )

GED

0

Average Equation:

Variance Equation:

Appraisal of GJR-GARCH Models

In this subdivision, we study the behavior of the GJR-GARCH theoretical accounts when applied to the conditional mean theoretical accounts by changing the mistake distributions. In this survey, we employ the GJR-GARCH ( P, Q ) theoretical accounts [ p = 0, 1 and q = 1, 2 ] with different orders ( up to 2 ) utilizing the Marquadt algorithm. The same diagnostic trials are used, viz. the SIC and Engle ‘s ARCH trial.

Table 3.3. : SIC consequences of GJR-GARCH theoretical accounts with different mean theoretical accounts

Model

Error Dist.

Order

SIC

AR ( 1 ) -GJR-GARCH ( 1,1 )

Student-t

1

-0.19447

AR ( 1 ) -GJR-GARCH ( 1,2 )

Student-t

1

-0.17919

AR ( 1 ) – GJR-GARCH ( 1,1 )

Student-t

2

-0.178585

AR ( 2 ) – GJR-GARCH ( 1,1 )

Student-t

1

-0.17048

AR ( 1 ) -GJR-GARCH ( 0,1 )

Student-t

1

-0.164504

MA ( 1 ) – GJR-GARCH ( 1,1 )

Student-t

1

-0.196717

MA ( 1 ) – GJR-GARCH ( 1,2 )

Student-t

1

-0.181592

MA ( 1 ) – GJR-GARCH ( 1,1 )

Student-t

2

-0.180851

MA ( 2 ) – GJR-GARCH ( 1,1 )

Student-t

1

-0.179518

MA ( 1 ) – GJR-GARCH ( 0,1 )

Student-t

1

-0.167276

ARMA ( 5,5 ) – GJR-GARCH ( 0,1 )

GED

2

-0.214995

ARMA ( 5,5 ) – GJR-GARCH ( 1,1 )

Student-t

2

-0.187923

ARMA ( 3,3 ) – GJR-GARCH ( 1,1 )

Student-t

2

-0.178155

ARMA ( 3,3 ) – GJR-GARCH ( 1,2 )

Student-t

1

-0.177745

ARMA ( 3,4 ) – GJR-GARCH ( 1,1 )

Student-t

1

-0.175624

From the SIC consequences in Table 3.3. , we can happen that an ARMA ( 5, 5 ) – GJR- GARCH ( 0, 1 ) with GED mistake distribution fits best our clip series. In add-on, the adequateness of these theoretical accounts is checked by using the Engle ‘s ARCH trial up to 20 slowdowns. From the consequences given in the tabular array below, we can happen that the presence of ARCH effects is rejected at the 1 % significance degree except for an ARMA ( 5, 5 ) – GJR-GARCH ( 0, 1 ) with GED as mistake distribution. Again, the information from the tabular array is rather uncovering in the sense that even though a theoretical account is selected as the best by the SIC, it may non be appropriate ( for illustration, the ARMA ( 5, 5 ) – GJR-GARCH ( 0, 1 ) theoretical account ) .

Table 3.3. : Engle ‘s ARCH trial for GJR-GARCH ( P, Q ) theoretical accounts

Model

Error Dist.

Order

F-statistic

Prob. F

LM-

statistic

Critical

Value

AR ( 1 ) –

GJR-GARCH ( 1,1 )

Student-t

1

0.412627

0.9888

8.649146

31.410

AR ( 1 ) –

GJR-GARCH ( 1,2 )

Student-t

1

0.377797

0.9936

7.939917

31.410

AR ( 1 ) –

GJR-GARCH ( 1,1 )

Student-t

2

0.396152

0.9914

8.314127

31.410

AR ( 2 ) –

GJR-GARCH ( 1,1 )

Student-t

1

0.389273

0.9923

8.175488

31.410

AR ( 1 ) –

GJR-GARCH ( 0,1 )

Student-t

1

1.619829

0.0482

31.12079

31.410

MA ( 1 ) –

GJR-GARCH ( 1,1 )

Student-t

1

0.426327

0.9863

8.925585

31.410

MA ( 1 ) –

GJR-GARCH ( 1,2 )

Student-t

1

0.401919

0.9905

8.430014

31.410

MA ( 1 ) –

GJR-GARCH ( 1,1 )

Student-t

2

0.419640

0.9876

8.789995

31.410

MA ( 2 ) –

GJR-GARCH ( 1,1 )

Student-t

1

0.418999

0.9877

8.776992

31.410

MA ( 1 ) –

GJR-GARCH ( 0,1 )

Student-t

1

1.621309

0.0478

31.15044

31.410

ARMA ( 5,5 ) –

GJR-GARCH ( 0,1 )

GED

2

2.09918

0.0047

39.00887

31.410

ARMA ( 5,5 ) –

GJR-GARCH ( 1,1 )

Student-t

2

0.615771

0.8998

12.71854

31.410

ARMA ( 3,3 ) –

GJR-GARCH ( 1,1 )

Student-t

2

0.498700

0.9658

10.38903

31.410

ARMA ( 3,3 ) –

GJR-GARCH ( 1,2 )

Student-t

1

0.457992

0.9790

9.570406

31.410

ARMA ( 3,4 ) –

GJR-GARCH ( 1,1 )

Student-t

1

0.472570

0.9748

9.864150

31.410

Appraisal consequences

The tabular array below nowadayss the consequences of the estimated GJR-GARCH theoretical accounts for the first-differenced and original unemployment rate informations over the period January 1980 to December 2004.

Table 3.3. : GJR-GARCH Appraisal Consequences

Model

Error Dist.

Order

Estimated Model

AR ( 1 ) -GJR-GARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

AR ( 1 ) -GJR-GARCH ( 1,2 )

Student-t

1

Average Equation:

Variance Equation:

AR ( 1 ) -GJR-GARCH ( 1,1 )

Student-t

2

Average Equation:

Variance Equation:

AR ( 2 ) -GJR-GARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

AR ( 1 ) -GJR-GARCH ( 0,1 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 1 ) – GJR-GARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 1 ) -GJR-GARCH ( 1,2 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 1 ) -GJR-GARCH ( 1,1 )

Student-t

2

Average Equation:

Variance Equation:

MA ( 2 ) -GJR-GARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

MA ( 1 ) -GJR-GARCH ( 0,1 )

Student-t

1

Average Equation:

Variance Equation:

ARMA ( 5,5 ) -GJR-GARCH ( 1,1 )

Student-t

2

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -GJR-GARCH ( 1,1 )

Student-t

2

Average Equation:

Variance Equation:

ARMA ( 3,3 ) -GJR-GARCH ( 1,2 )

Student-t

1

Average Equation:

Variance Equation:

ARMA ( 3,4 ) -GJR-GARCH ( 1,1 )

Student-t

1

Average Equation:

Variance Equation:

Deductions

The consequences, obtained in the appraisal stage, are really interesting. First, we can detect that a theoretical account may non be ever equal ( based on Ljung-Box-Pierce Q-Test and Engle ‘s ARCH trial ) for our informations even though it is selected as the most appropriate by the SIC. Furthermore, it can be found that EGARCH ( P, Q ) theoretical accounts better capture the ARCH effects present in our selected conditional average equations. In add-on, it can be found that the mark consequence of the term depends on the asymmetric order chosen while the magnitude consequence depends on the order Q of the theoretical account. In contrast, the GJR-GARCH theoretical account merely considers the residuary if it is negative when we increase its order. Furthermore, it appears from the appraisal consequences that negative inventions have a smaller impact on the conditional discrepancy since in all instances[ 1 ].

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