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8/6/2019 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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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
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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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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
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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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ECONOMETRICSECONOMETRICSECONOMETRICSECONOMETRICS Time Series Analysis Final ExamTime Series Analysis Final ExamTime Series Analysis Final ExamTime Series Analysis Final Exam
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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
Sample (adjusted): 1971M02 1989M12
Included observations: 227 after adjustments
Convergence achieved after 2 iterations
Variable Coefficient Std. Error t-Statistic Prob.
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
Sum squared resid 0.402695 Schwarz criterion -3.472750
Log likelihood 396.8696 Durbin-Watson stat 2.677038
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)
Method: Least Squares
Date: 01/21/10 Time: 22:36
Sample (adjusted): 1971M02 1989M12
Included observations: 227 after adjustments
Convergence achieved after 7 iterations
Backcast: 1971M01
Variable Coefficient Std. Error t-Statistic Prob.
MA(1) -0.516094 0.056979 -9.057687 0.0000
R-squared 0.182529 Mean dependent var -0.000361
Adjusted R-squared 0.182529 S.D. dependent var 0.044773
S.E. of regression 0.040481 Akaike info criterion -3.571584
Sum squared resid 0.370345 Schwarz criterion -3.556496
Log likelihood 406.3747 Durbin-Watson stat 1.916350
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)
Method: Least Squares
Date: 01/21/10 Time: 22:51
Sample (adjusted): 1971M02 1989M12
Included observations: 227 after adjustments
Convergence achieved after 12 iterations
Backcast: 1970M01 1971M01
Variable Coefficient Std. Error t-Statistic Prob.
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
Adjusted R-squared 0.406720 S.D. dependent var 0.044773
S.E. of regression 0.034486 Akaike info criterion -3.887757
Sum squared resid 0.267588 Schwarz criterion -3.857581
Log likelihood 443.2604 Durbin-Watson stat 1.923616
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).
60
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100
110
120
130
140
90M01 90M07 91M01 91M07 92M01 92M07
_GREF
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ECONOMETRICSECONOMETRICSECONOMETRICSECONOMETRICS Time Series Analysis Final ExamTime Series Analysis Final ExamTime Series Analysis Final ExamTime Series Analysis Final Exam
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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.