Unemployment Rate Time Series Biology Essay

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May 5, 2019
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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.

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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 PlotFigure 3.2. : Time Series Plot by SeasonsFigure 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

384383

A Mean

8.5953130.000000

A Median

8.0000000.000000

A Maximum

13.

000001.200000

A Minimum

5.900000-0.600000

A Std. Dev.

7372070.219948

A Lopsidedness

0.5574540.

935624

A Kurtosis

2.3239636.181196

A Jarque-Bera

27.20077217.3777

A Probability

0000010.000000

ADF Test Statistic

-1.875236-8.291770

Probability

( 0.3439 )( 0.0000 )Table 3.2. : Descriptive Statisticss of Canadian Unemployment RateFigure 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 AnalysisAs 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 )

-0.096668Table 3.3.

: SIC of ARMA theoretical accounts

Model

Test Statistic

p-value

Critical Value

Consequence

AR ( 1 )

35.0250.01430.

144Cull

AR ( 2 )

29.9230.03828.869Cull

AR ( 3 )

20.

6410.24327.587Do non reject

AR ( 4 )

17.3460.36426.296Do non reject

AR ( 5 )

13.2020.

58724.996Do non reject

MA ( 1 )

37.4710.

00730.144Cull

MA ( 2 )

34.5690.01128.869Cull

MA ( 3 )

25.2040.09027.

587Do non reject

MA ( 4 )

20.3010.20726.296Do non reject

MA ( 5 )

16.6540.34024.996Do non reject

ARMA ( 3,3 )

10.8770.

69623.685Do non reject

ARMA ( 3,4 )

16.4810.22422.362Do non reject

ARMA ( 4,3 )

15.

7320.26422.362Do non reject

ARMA ( 4,4 )

11.

1420.51721.026Do non reject

ARMA ( 5,5 )

15.5130.11418.307Do non rejectTable 3.3. : Ljung-Box-Pierce Q-test Consequences for AutocorrelationAs 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 consequencesTable 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 )

5.4033870.000082.2938631.410Cull

AR ( 2 )

5.1195810.000079.1382331.

410Cull

AR ( 3 )

3.9892250.000065.7752231.

410Cull

AR ( 4 )

3.8474970.000063.9408831.

410Cull

AR ( 5 )

4.9711730.000077.2991331.410Cull

MA ( 1 )

4.5817000.

000073.1218531.410Cull

MA ( 2 )

4.3336370.000070.1584231.

410Cull

MA ( 3 )

3.2996240.000056.

8281831.410Cull

MA ( 4 )

3.180150.

000055.1825431.410Cull

MA ( 5 )

6687780.000061.7781931.

410Cull

ARMA ( 3,3 )

4.1281650.000067.5057731.410Cull

ARMA ( 3,4 )

1313350.000054.4191331.410Cull

ARMA ( 4,3 )

3.0308550.000052.9840931.410Cull

ARMA ( 4,4 )

0277130.000052.8691331.410Cull

ARMA ( 5,5 )

3.7765220.

000062.9936731.410CullTable 3.3. : Engle ‘s ARCH trial for Heteroscedasticity

Model

Mean

Std. Dev

Lopsidedness

Kurtosis

JB

Prob.

AR ( 1 )

99e-100.2300180.8455095.

948304143.43780.0000

AR ( 2 )

4.67e-100.2296240.8080715.990846143.01880.

0000

AR ( 3 )

5.90e-100.2258550.6811005.

865792124.17630.0000

AR ( 4 )

-7.77e-130.2250280.6516105.

836984119.80520.0000

AR ( 5 )

1.01e-090.2231970.6914156.270273154.43460.

0000

MA ( 1 )

-3.40e-050.2299290.8577385.

945441144.74690.0000

MA ( 2 )

4.42e-050.2296070.

8490495.996065147.75520.0000

MA ( 3 )

-4.72e-050.2263240.

7482025.907181133.19110.0000

MA ( 4 )

-9.65e-050.2254130.7082325.

796920122.45460.0000

MA ( 5 )

-0.

000200.2234460.7850306.435562177.75770.

0000

ARMA ( 3,3 )

0.000180.2145780.7700126.038092143.08740.

0000

ARMA ( 3,4 )

-0.000190.2127590.7314926.086970143.92630.0000

ARMA ( 4,3 )

-6.04e-050.

2129050.7108046.025342137.

34290.0000

ARMA ( 4,4 )

-0.000330.2117550.

6079445.828064116.47990.0000

ARMA ( 5,5 )

0.0024750.2047270.6364795.

586060101.77460.0000Table 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-0.17715To 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-t0.6755460.849013.8849831.410

AR ( 1 ) -GARCH ( 1,2 )

Student-t0.6334180.

886013.0597731.410

AR ( 2 ) -GARCH ( 1,1 )

Student-t0.6515460.870813.

4169131.410

AR ( 1 ) -ARCH ( 1 )

Student-t2.0088280.007437.5839931.

410

AR ( 1 ) -GARCH ( 1,1 )

GED0.7198070.804714.7464431.

410

MA ( 1 ) -GARCH ( 1,1 )

Student-t0.6948960.834314.

2609331.410

MA ( 2 ) -GARCH ( 1,1 )

Student-t0.6693120.854913.7618031.410

MA ( 1 ) -GARCH ( 1,2 )

Student-t0.6555720.867313.

4929531.410

MA ( 1 ) -ARCH ( 1 )

Student-t2.0182220.007137.7447131.410

MA ( 1 ) -GARCH ( 1,1 )

GED0.7292130.

794614.9275231.410

ARMA ( 5,5 ) -ARCH ( 2 )

Student-t2.3659040.001143.

1714131.410

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

Student-t0.8466060.591818.1336831.410

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

GED0.8492550.

651917.2358331.410

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

GED0.

7567600.763915.4662531.410

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

Student-t0.8977160.

590318.1546631.410Table 3.

3. : Engle ‘s ARCH trial for GARCH ( P, Q ) theoretical accountsAppraisal consequencesTable 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-tAverage Equation:Variance Equation:AR ( 1 ) -GARCH ( 1,2 )Student-tAverage Equation:Variance Equation:AR ( 2 ) -GARCH ( 1,1 )Student-tAverage Equation:Variance Equation:AR ( 1 ) -GARCH ( 1,1 )GEDAverage Equation:Variance Equation:MA ( 1 ) -GARCH ( 1,1 )Student-tAverage Equation:Variance Equation:MA ( 2 ) -GARCH ( 1,1 )Student-tAverage Equation:Variance Equation:MA ( 1 ) -GARCH ( 1,2 )Student-tAverage Equation:Variance Equation:MA ( 1 ) -GARCH ( 1,1 )GEDAverage Equation:Variance Equation:ARMA ( 3,3 ) -GARCH ( 1,1 )Student-tAverage Equation:Variance Equation:ARMA ( 3,3 ) -GARCH ( 1,1 )GEDAverage Equation:Variance Equation:ARMA ( 5,5 ) -GARCH ( 1,1 )GEDAverage Equation:Variance Equation:ARMA ( 3,4 ) -GARCH ( 1,1 )Student-tAverage 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.

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

GED

0

-0.197115From 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

0

0.9303360.548918.7694731.410It 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 consequencesThe 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

Order

F-statistic

Prob. F

LM-

statistic

1

0.4579920.97909.57040631.410

ARMA ( 3,4 ) –

GJR-GARCH ( 1,1 )

Student-t

1

0.4725700.97489.86415031.410Appraisal consequencesThe 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

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 ].