modelling greek industrial production index

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ECONOMETRICSECONOMETRICSECONOMETRICSECONOMETRICS Time Series Analysis Final ExamTime Series Analysis Final ExamTime Series Analysis Final ExamTime Series Analysis Final Exam

Italo Raul A. ArbulItalo Raul A. ArbulItalo Raul A. ArbulItalo Raul A. Arbul VillanuevaVillanuevaVillanuevaVillanueva

1111

MASTER IN TOURISM AND ENVIRONMENTALECONOMICS

(MTEE)

Econometrics

Final Exam Time Series Analysis Part

MODELLINGGREEK INDUSTRIAL PRODUCTION INDEX

By

Italo Arbul Villanueva

January 2010


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INDEX

1. INTRODUCTION ............................................................................... 32. THE BOX-JENKINS METHODOLOGY ........................................ 42.1. Identification ..................................................................................... 52.2. Estimation........................................................................................ 132.3. Checking .......................................................................................... 162.4. Forecast............................................................................................ 183. MAIN CONLUSIONS....................................................................... 19


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1. INTRODUCTION

In this work are detailed the steps of the modeling analysis of the Industrial Greek

Production Index series by the Box-Jenkins methodology. Once the series is modeled,

the pattern is used to make projections for the following three years.

The Box-Jenkins methodology allows to make efficient forecasting, starting exclusively

from the information contained in a temporary series, but also this modeling univariante,is an indispensable tool to be able to build better and more complex models (that include

other variables) in the future.

The analysis has been made with 240 observations that correspond to those the values

of the monthly index of January from 1970 to December of 1989.


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2. THE BOX-JENKINS METHODOLOGY

The time series econometric models in general require of four stages for their

construction: Specification (in the language of Box-Jenkins, Identification), Estimation,

Checking and Forecast.

1. Identification: Choose the orders p, d, q, P, D and Q and of the ARIMA(p; d; q)

x SARIMA(P; D; Q) model.

a. Choose depicting the time series.

b. With the sample correlograms choose d and D.

c. With the sample correlograms of the transformed data choose p, q, P and

Q.

d. Check in the AR processes are stationary and if the MA ones are

invertible; otherwise modify the values of d and D.

2. Estimation: Once the ARIMA(p; d; q) x SARIMA(P; D; Q) is specified we have to

check it.

a. Check in the AR processes are stationary and if the MA ones are

invertible; otherwise modify the values of d and D.

b. Check if it possible to reject the null hypothesis in the individual

significance tests associated fitted parameters.

3. Checking: Check if the residuals of the fitted model behave like a white noise

process.

a. Visual inspection of the residuals correlograms.

b. Use of the Q Box-Pierce and Lunj-Box statistics.

c. Selection between alternative models using information criteria.

4. Forecast


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Although the stages are successive (they should be carried out first the identification of

the pattern and then its estimation), according to the result of each stage it can have

retro-feeding. After a first estimation it can be concluded about the necessity of change

the specification pattern.

2.1. Identification

The first stage of the identification of an ARIMA x SARIMA model is the graphical

analysis of the series.

30

40

50

60

70

80

90

100

70 72 74 76 78 80 82 84 86 88

GRE

As the series has an exponential behavior a transformation of the series is required in

order to give place to a stationary series. In this sense, the construction of a new series

wt is generally expressed as:

The function f(Yt) is in general a Box-Cox transformation which follows the expression:


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In this sense, the logarithmic transformation was established and the following graph

shows this estimation:

3.4

3.6

3.8

4.0

4.2

4.4

4.6

70 72 74 76 78 80 82 84 86 88

_LOG_GRE

As we can appreciate in the graph is that the new time series is not stationary because

the mean and the variance are no constant over the whole period analyzed. In this

sense we need to take a difference of the time series. In order to know how many

differences should be applied to the time series (d) we first check the correlogram.


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As we know in a stationary series it the correlogram tends exponentially to zero, but as

we can observe from this correlogram that this is not the case. This correlogram give us

the idea of the existence of a non stationary Autoregressive Process (AR) of order 1

because the Autocorrelation Function (AF) is not exponentially decaying to zero and the


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first coefficient of the Partial Autocorrelation Function (PAF) is statistically different from

zero1. In order to confirm this, we made the regression of the time series using as an

explanatory variable an AR(1).

Dependent Variable: _LOG_GRE

Method: Least Squares

Date: 12/30/09 Time: 10:57

Sample (adjusted): 1970M02 1989M12

Included observations: 239 after adjustments

Convergence achieved after 2 iterations

Variable Coefficient Std. Error t-Statistic Prob.

AR(1) 1.000731 0.000989 1012.052 0.0000

R-squared 0.918542 Mean dependent var 4.206885

Adjusted R-squared 0.918542 S.D. dependent var 0.225455

S.E. of regression 0.064347 Akaike info criterion -2.644885

Sum squared resid 0.985440 Schwarz criterion -2.630339

Log likelihood 317.0638 Durbin-Watson stat 2.648085

Inverted AR Roots 1.00

Estimated AR process is nonstationary

Now that it is confirmed the presence of a nonstationary process of order one, in order to

get a stationary series we apply the first difference of the series (d=1). The following

graph shows the new series (_D_log_GRE)

1We can affirm that because the null hypothesis of no autocorrelation is rejected because the p-

value of the Q-stat is lower than the confidence level established (5%).


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-.2

-.1

.0

.1

.2

.3

70 72 74 76 78 80 82 84 86 88

_D_LOG_GRE

The correlogram of the new series is shown in the following graph. As we can

appreciate, the PAF and the AF show the possible presence of a season autoregressive

process in the twelve period (SAR-12) since the PAF shows a big value on this period

and the AF shows coefficients not decaying exponentially towards zero, this means, thepresence of peaks over twelve periods (12, 24, 36).


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In order to confirm the presence of a seasonal process we also estimated a regression

of the series over a SAR(12) process.


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Dependent Variable: _D_LOG_GRE

Method: Least SquaresDate: 12/30/09 Time: 10:57





AR(12) 0.770461 0.043171 17.84656 0.0000

R-squared 0.583739 Mean dependent var 0.003512

Adjusted R-squared 0.583739 S.D. dependent var 0.065426S.E. of regression 0.042212 Akaike info criterion -3.487838



Inverted AR Roots .98 .85+.49i .85-.49i .49-.85i

.49+.85i .00+.98i -.00-.98i -.49-.85i

-.49+.85i -.85-.49i -.85+.49i -.98

As we can see in the results, two of the inverted AR roots are almost equal to one, so in

this case we apply the 12th difference of the series, this means that D=1. In this sense,

the following table shows the correlogram of the new series which is D(log(GRE),1,12)


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The AF shows a value statically different from zero and the PAF is exponentially

decaying to zero, this could mean the presence of a Moving Average process of first

order (MA(1)). In this sense, the following chart shows the estimation output of the time


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series. As we can see, the inverted MA root is inferior to 1, in this sense, there is no

need to make a modification (differentiate) of the time series.

Dependent Variable: D(LOG(GRE),1,12)


Date: 01/21/10 Time: 22:36




Backcast: 1971M01


MA(1) -0.516094 0.056979 -9.057687 0.0000

R-squared 0.182529 Mean dependent var -0.000361





Inverted MA Roots .52

2.2. Estimation

Once we have specified the time series to estimate, we see the correlogram of the last

regression in order to allow the identification of any order process.


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As we can see, the AF shows a coefficient statistically different from zero and the PAF

shows that the twelve period coefficients (12, 24, and 36) are exponentially decaying to

zero. In this sense, the correlogram show the presence of a seasonal moving average of

order 12) and in order to confirm this, we estimate the previous equation including the

SMA(12) as a regressor, the following chart shows the results for this estimation.


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Dependent Variable: D(LOG(GRE),1,12)


Date: 01/21/10 Time: 22:51




Backcast: 1970M01 1971M01


MA(1) -0.490891 0.058300 -8.420103 0.0000

SMA(12) -0.725088 0.041443 -17.49616 0.0000

R-squared 0.409345 Mean dependent var -0.000361





Inverted MA Roots .97 .84-.49i .84+.49i .49

.49+.84i .49-.84i .00-.97i -.00+.97i

-.49-.84i -.49+.84i -.84-.49i -.84+.49i

-.97

In order to culminate this stage of the Box-Jenkins method, we need to check if the AR

processes are stationary and if the MA ones are invertible, otherwise we would need to

modify the values of d and D. However, in this case, we can see that there is no AR

process and that the MA process is invertible.

Finally, we check that it possible to reject the null hypothesis in the individual

significance tests associated with the fitted parameters.


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2.3. Checking

Once we have estimated the final equation, the next step is to check if the residuals of

the fitted model behave like a white noise process. In order to do this we apply a visual

inspection of the residuals correlogram.


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The last two columns reported in the correlogram are the Ljung-Box Q-statistics and their

p-values. The Q-statistic at lag k is a test statistic for the null hypothesis that there is no

autocorrelation up to order k. .

-.15

-.10

-.05

.00

.05

.10

.15

-.2

-.1

.0

.1

.2

72 74 76 78 80 82 84 86 88

Residual Actual Fitted

0

5

10

15

20

25

30

-0.15 -0.10 -0.05 0.00 0.05 0.10

Series: Residuals

Sample 1971M02 1989M12

Observations 227

Mean -0.002511

Median 0.001124

Maximum 0.105467

Minimum -0.141326

Std. Dev. 0.034317Skewness -0.179095

Kurtosis 4.125722

Jarque-Bera 13.19958

Probability 0.001361


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In this sense, the correlogram shows that we accept the null hypothesis and we can

describe the residuals as white noise and the main statistics confirm that the expected

value is almost zero and the Jarque-Bera test accepts the null hypothesis of normality.

Now that we have checked the results we can confirm that the Greek Industrial

Production Index follows the following process:

ARIMA(0; 1; 1) x SARIMA(0;1;1)

2.4. Forecast

Once we have identified the process we can use it in order to forecast the following three

years (1990, 1991 and 1992). The following graph show the results (blue line) and the

confidence interval related with the forecast estimation (red lines).

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70

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100

110

120

130

140

90M01 90M07 91M01 91M07 92M01 92M07

_GREF


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3. MAIN CONLUSIONS

The main objective of this work was to understand the behavior of the Greek Industrial

Production Index. This objective was successfully achieved through the use of the Box-

Jenkins methodology.

In this work we used 240 observations of the monthly series (from January of 1970 to

December of 1989). The Box-Jenkins method allowed us the identification and

estimation of the time series as an ARIMA(0; 1; 1) x SARIMA(0;1;1). These results were

checked by the use of the residuals test which showed all the characteristics of a white

noise (no autocorrelation, mean equal to zero and normality).

Finally, we use the model in order to forecast the value of the following three years

(1990, 1991 and 1992) of the series.

modelling greek industrial production index

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