Unemployment Rate Time Series Biology Essay

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In this chapter, we apply the methodology discussed in the preceding chapter to determine which model is suitable for our time series data. The goal of this chapter is to provide detail analysis of our data set in order to achieve our objective, mentioned in chapter one. We begin by giving a brief overview of the data set used. This is followed by some initial data analysis. Then, we proceed to the estimation and model selection part, where all the tests are conducted to find the most appropriate model for our data set.

Overview of Data

The data employed in this project consists of 384 monthly Canadian unemployment rates ranging from January 1980 to December 2011, for both sexes, where the subjects are over the age of 15. The data has been extracted from OECD.Stat Extracts database. The first 300 observations (January 1980 to December 2004) are used for parameter estimation while the next 84 (January 2005 to December 2011) are used for forecasting evaluation [7] . All data manipulation and analysis are done in Eviews 7.0.

Data Analysis

As discussed previously, time series models are only appropriate for stationary time series. By looking at Figure 3.2. it appears that our unemployment rate time series is not stationary. Moreover, the time series does not display any trend. However, a closer look at the time 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 decrease in September and October. Hence, seasonal adjustment 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 areas denote recessions)

The descriptive statistics for the monthly seasonally adjusted unemployment rate is given in Table 3.2.. From the table, it can be seen that our unemployment series has a mean of 8.595313, a standard deviation (Std. Dev.) of 1.737207, a positive skewness of 0.557454 (greater than zero) and kurtosis of 2.323963 (less than 3). The value of the skewness implies that the series follows a right skewed distribution while the kurtosis value shows that this distribution is relatively flat (platykurtic) as compared to the normal. The series has a minimum value of 5.9 and a maximum value of 13. Moreover, the Jarque-Bera also rejects the null hypothesis for normality at 5% level. In addition, the unit root test yields an Augmented Dickey Fuller (ADF) Test Statistic of -1.875236 with probability 0.3439, which rejects the null hypothesis for the presence of a unit root in the unemployment rate series at the 5% significance level. This result suggests that our time series has to be differentiated.

Therefore, we convert the unemployment rates to first-differenced by using the following formula:

(34)

where represents any observation at time in the first-differenced time series, and and are the unemployment rates at time and in the original series respectively. After taking first differences, the ADF Test Statistic is now -8.291770 and has a probability of 0.0000. Thus, our new data set is stationary and we can now proceed to the estimation stage. Visual inspection of Figure 3.2. demonstrates that the first-differenced time series is stationary. Random fluctuations around zero, which is a good sign of stationarity, can be observed. The first-differenced time series show evidence of fat-tails, since the kurtosis exceeds 3, which is the normal value, and evidence of positive skewness, which means that the series is skewed the right same as the original one. Moreover, the Jarque-Bera also rejects the null hypothesis that the series follows a normal distribution at the 5% significance level.

Original

First-Differenced

 Observations

384

383

 Mean

8.595313

0.000000

 Median

8.000000

0.000000

 Maximum

13.00000

1.200000

 Minimum

5.900000

-0.600000

 Std. Dev.

1.737207

0.219948

 Skewness

0.557454

0.935624

 Kurtosis

2.323963

6.181196

 Jarque-Bera

27.20077

217.3777

 Probability

0.000001

0.000000

ADF Test Statistic

-1.875236

-8.291770

Probability

(0.3439)

(0.0000)

Table 3.2.: Descriptive Statistics of Canadian Unemployment Rate

Figure 3.2.: Differentiated Time Series Plot

Model Estimation and Evaluation

In this section, we use the different time series models, explained in the previous chapter, to estimate our first-differenced time series data. After estimation, we select the appropriate model based on the Schwarz Information Criterion (SIC). The five potential candidates from each model are selected for forecasting exercises since a model can fit our in-sample data but is not suitable for the out-of-sample one.

Estimation of ARMA Models

We apply twenty models for AR (p) and MA (q) models (p = q = 1:20) and twenty-five models for ARMA (p, q) models (p = q = 1, 2, 3, 4, 5) to our in-sample data using the Least Squares method.

Model Selection and Analysis

As mentioned earlier, model selection is based on SIC. However, for demonstration purposes, we examine the correlogram of the time series to draw conclusions about suitable ARMA models. The identification of the ARMA models are based on the shape of the autocorrelation function plot illustrated in Table 2.3.1. Figure 3.3. represents the correlogram of the unemployment rate after taking first differences.

The autocorrelations seem to decay after a few lags. Thus, a mixture of autoregressive and moving average model is suggested for our data. It can be seen from the correlogram that both the autocorrelations and partial autocorrelations appear to cut off at lag 5. Henceforth, an ARMA (5, 5) model seems most appropriate. Table 3.3. below summarizes the SIC results of the five most plausible models from each time series models estimated. The minimum SIC for the AR (p) models indicates an AR (1) model while that of the MA (q) models shows a MA (1) model. Moreover, the overall minimum SIC also favours an ARMA (5, 5) model.

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 models

Model

Test Statistic

p-value

Critical Value

Result

AR(1)

35.025

0.014

30.144

Reject

AR(2)

29.923

0.038

28.869

Reject

AR(3)

20.641

0.243

27.587

Do not reject

AR(4)

17.346

0.364

26.296

Do not reject

AR(5)

13.202

0.587

24.996

Do not reject

MA(1)

37.471

0.007

30.144

Reject

MA(2)

34.569

0.011

28.869

Reject

MA(3)

25.204

0.090

27.587

Do not reject

MA(4)

20.301

0.207

26.296

Do not reject

MA(5)

16.654

0.340

24.996

Do not reject

ARMA(3,3)

10.877

0.696

23.685

Do not reject

ARMA(3,4)

16.481

0.224

22.362

Do not reject

ARMA(4,3)

15.732

0.264

22.362

Do not reject

ARMA(4,4)

11.142

0.517

21.026

Do not reject

ARMA(5,5)

15.513

0.114

18.307

Do not reject

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

As discussed previously, a time series model is appropriate for our time series only if the residuals are random (that is, white noise). In this study, we use the Ljung-Box test to determine whether the models selected in the estimation phase are appropriate for our data set. From Table 3.3., we can conclude that the residuals, when tested for up to 20 lags, are random in most cases except for AR (1), AR (2), MA (1) and MA (2). Both the test statistics and the p-values are significant at the 5 % significance level. Thus, most of our models fit our in-sample data. Nevertheless, a comparison of the results obtained from the estimation stage and the Ljung-Box test reveals that although a model has the smallest SIC, it may not always be appropriate for our time series (for example, the AR (1) model).

Estimation results

Table 3.3. reports results on the set of estimated ARMA models for the first-differenced and original unemployment rate data over the period January 1980 to December 2004. The table below lists only models whose residuals 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 following segment involves the application of non-linear univariate models to the residuals of the above selected conditional mean models. In this project, we allow the variance of the residuals to follow both the symmetric and asymmetric GARCH models, namely the GARCH (p, q), EGARCH (p, q) and GJR-GARCH (p, q).

Residuals Diagnostics

Before applying any GARCH family models to the residuals of our conditional mean models, we examine the latter. In this study, the following conditional mean models 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 important to note that GARCH family models can only be applied to time series that exhibits some form of heteroscedasticity (as discussed in Chapter 2). First, the residuals of the conditional mean models are checked for ARCH effects. Then, we are going to check whether the residuals come from a normal distribution.

Error: Reference source not found gives the results of the ARCH test applied up to 20 lags. We can observe that both the F-statistics and the LM-statistics are significant at the 5 % significance level. Therefore, the residuals of our conditional mean models display ARCH effects. In other words, the variance of the residuals' series is not constant throughout. Thus, GARCH family models can be employed. The descriptive statistics of the residuals for the different mean models considered are depicted in Table 3.3.. It is apparent from the table that the mean of the residuals are very close to zero. In addition, we can notice that the series display both positive skewness and excess kurtosis (kurtosis exceeds 3, which is the normal value) in all cases. These values show evidence that the series follow a distribution, which is skewed to the right and peaked relative to the normal (leptokurtic). Furthermore, it is clear that the Jarque-Bera (JB) test decisively rejects the null hypothesis that the residual series, {}, is Gaussian at the 5% significance level. For this reason, we suppose that the residual series, {}, follows a Student-t Distribution or a Generalised Error Distribution (GED) during the estimation of the GARCH family models [9] .

Model

F-statistic

Prob. F

LM-statistic

Critical Value

Results

AR(1)

5.403387

0.0000

82.29386

31.410

Reject

AR(2)

5.119581

0.0000

79.13823

31.410

Reject

AR(3)

3.989225

0.0000

65.77522

31.410

Reject

AR(4)

3.847497

0.0000

63.94088

31.410

Reject

AR(5)

4.971173

0.0000

77.29913

31.410

Reject

MA(1)

4.581700

0.0000

73.12185

31.410

Reject

MA(2)

4.333637

0.0000

70.15842

31.410

Reject

MA(3)

3.299624

0.0000

56.82818

31.410

Reject

MA(4)

3.18015

0.0000

55.18254

31.410

Reject

MA(5)

3.668778

0.0000

61.77819

31.410

Reject

ARMA(3,3)

4.128165

0.0000

67.50577

31.410

Reject

ARMA(3,4)

3.131335

0.0000

54.41913

31.410

Reject

ARMA(4,3)

3.030855

0.0000

52.98409

31.410

Reject

ARMA(4,4)

3.027713

0.0000

52.86913

31.410

Reject

ARMA(5,5)

3.776522

0.0000

62.99367

31.410

Reject

Table 3.3.: Engle's ARCH test for Heteroscedasticity

Model

Mean

Std. Dev

Skewness

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 Statistics of Residuals

Estimation of GARCH Models

In this section, we estimate our stationary time series using four GARCH (p, q) models [p = 0, 1 and q = 1, 2] using Marquadt algorithm. We use the aforementioned conditional mean models as mean equations together with the two above mentioned error distributions, namely the Student-t and the GED. The table below shows the SIC results of the five models selected from each mean equations (based on the minimum SIC approach). It is obvious from Error: Reference source not found that an ARMA (5, 5)-ARCH (2) model with the residuals following a Student-t distribution performs better. Yet, the adequacy of these models should be checked.

Table 3.3.: SIC results of GARCH models with different mean models

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 check the adequacy of these models, we apply the Engle's ARCH test up to 20 lags to make sure that there are no ARCH effects left after estimation. Both the F-statistic and the LM-statistic are insignificant at the 5% significance level except for AR (1)-ARCH (1), MA (1)-ARCH (1) and ARMA (5, 5)-ARCH (2) models. We can conclude that applying GARCH (p, q) models to the mean equations remove the ARCH effects present in them except for these three models.

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 test for GARCH (p, q) models

Estimation results

Table 3.3. illustrates the results of the estimated GARCH models for the first-differenced and original unemployment rate data 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

Mean Equation:

Variance Equation:

AR(1)-GARCH(1,2)

Student-t

Mean Equation:

Variance Equation:

AR(2)-GARCH(1,1)

Student-t

Mean Equation:

Variance Equation:

AR(1)-GARCH(1,1)

GED

Mean Equation:

Variance Equation:

MA(1)-GARCH (1,1)

Student-t

Mean Equation:

Variance Equation:

MA(2)-GARCH(1,1)

Student-t

Mean Equation:

Variance Equation:

MA(1)-GARCH(1,2)

Student-t

Mean Equation:

Variance Equation:

MA(1)-GARCH(1,1)

GED

Mean Equation:

Variance Equation:

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

Student-t

Mean Equation:

Variance Equation:

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

GED

Mean Equation:

Variance Equation:

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

GED

Mean Equation:

Variance Equation:

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

Student-t

Mean Equation:

Variance Equation:

Estimation of EGARCH Models

Following the same approach as the GARCH (p, q) models, we apply four EGARCH (p, q) models [p = 0, 1 and q = 1, 2] to the previously mentioned conditional mean models using the Marquadt algorithm. The same error distributions are used. EGARCH models with different asymmetric order (up to 2) have been used [10] . Selection of the five most potential models from each mean equations are based on the SIC approach. On the other hand, we conduct an Engle's ARCH test up to 20 lags to confirm that no more ARCH effects are left following estimation.

Table 3.3.: SIC results of EGARCH models with different mean models

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 results shown in Error: Reference source not found, we can see that an ARMA (5, 5) - EGARCH (1, 1) with error distribution Student-t best fits our stationary time series while an AR (2)-EGARCH (1, 1) with the same error distribution, but with different asymmetric order, proves to be the worst.

Table 3.3.: Engle's ARCH test for EGARCH (p, q) models

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 insignificant at the 5% significance level in most cases. This suggests that the estimated time series models do not exhibit any ARCH effects. In contrast to the estimation of the GARCH models, we find that the addition of an EGARCH model to the mean equations remove almost all ARCH effects from them. This may be because EGARCH models better capture the heteroscedasticity presents inn our data.

Estimation results

The results of the estimated EGARCH models for the first-differenced and original unemployment rate data over the period January 1980 to December 2004 are depicted in Table 3.3..

Table 3.3.: EGARCH Estimation Results

Model

Error Dist.

Order

Estimated Model

AR(1)-EGARCH (1,1)

Student-t

1

Mean Equation:

Variance Equation:

AR(1)-EGARCH (1,1)

Student-t

0

Mean Equation:

Variance Equation:

AR(1)-EGARCH (1,1)

Student-t

2

Mean Equation:

Variance Equation:

AR(1)-EGARCH (1,2)

Student-t

1

Mean Equation:

Variance Equation:

AR(2)-EGARCH (1,1)

Student-t

1

Mean Equation:

Variance Equation:

MA(1)-EGARCH(1,1)

Student-t

1

Mean Equation:

Variance Equation:

MA(1)-EGARCH(1,1)

Student-t

0

Mean Equation:

Variance Equation:

MA(1)-EGARCH(1,1)

Student-t

2

Mean Equation:

Variance Equation:

MA(1)-EGARCH(1,2)

Student-t

1

Mean Equation:

Variance Equation:

MA(2)-EGARCH(1,1)

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

0

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

0

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

GED

0

Mean Equation:

Variance Equation:

Estimation of GJR-GARCH Models

In this section, we study the behaviour of the GJR-GARCH models when applied to the conditional mean models by varying the error distributions. In this study, we employ the GJR-GARCH (p, q) models [p = 0, 1 and q = 1, 2] with different orders (up to 2) using the Marquadt algorithm. The same diagnostic tests are used, namely the SIC and Engle's ARCH test.

Table 3.3.: SIC results of GJR-GARCH models with different mean models

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 results in Table 3.3., we can find that an ARMA (5, 5) - GJR- GARCH (0, 1) with GED error distribution fits best our time series. In addition, the adequacy of these models is checked by applying the Engle's ARCH test up to 20 lags. From the results given in the table below, we can find that the presence of ARCH effects is rejected at the 1% significance level except for an ARMA (5, 5) - GJR-GARCH (0, 1) with GED as error distribution. Again, the data from the table is quite revealing in the sense that even though a model is selected as the best by the SIC, it may not be appropriate (for example, the ARMA (5, 5) - GJR-GARCH (0, 1) model).

Table 3.3.: Engle's ARCH test for GJR-GARCH (p, q) models

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

Estimation results

The table below presents the results of the estimated GJR-GARCH models for the first-differenced and original unemployment rate data over the period January 1980 to December 2004.

Table 3.3.: GJR-GARCH Estimation Results

Model

Error Dist.

Order

Estimated Model

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

2

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

2

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

2

Mean Equation:

Variance Equation:

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

Student-t

2

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

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

Student-t

1

Mean Equation:

Variance Equation:

Implications

The results, obtained in the estimation phase, are very interesting. First, we can observe that a model may not be always adequate (based on Ljung-Box-Pierce Q-Test and Engle's ARCH test) for our data even though it is selected as the most appropriate by the SIC. Moreover, it can be found that EGARCH (p, q) models better capture the ARCH effects present in our selected conditional mean equations. In addition, it can be found that the sign effect of the term depends on the asymmetric order chosen while the magnitude effect depends on the order q of the model. In contrast, the GJR-GARCH model only considers the residual if it is negative when we increase its order. Furthermore, it appears from the estimation results that negative innovations have a smaller impact on the conditional variance since in all cases [1] .

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