Term Paper Eco No Metric

8/3/2019 Term Paper Eco No Metric

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Submitted by:

Mandeep Kumar

Enrollment No.: 100165602

Master of Philosophy (M.Phil) in

Economics(Session 2010-11)

Term Paper

Basic Econometrics-REC 003GDP Forecast - Univariate Time Series

(BOX-JENKINS Methodology and

ARIMA Forecast Model)


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Univariate Time Series Analysis

FORECAST FOR US GDP (1992 1ST QUARTER) AND

INDIAN GDP (YEAR 2005-06) TIME SERIES: BOX-

JENKINS METHODOLOGY AND ARIMA

FORECAST MODEL

INTRODUCTION:

Time series data are always been challenge for econometricians and practitioners. purpose

of study of econometrics is not only to analysis the data but to forecast also. This term

paper is based on econometric analysis of time series. Generally, a time series is a sequence

of values a specific variable has taken on over some period of time. The observations have a

natural ordering in time. Usually when we refer to a series of observation as time series, we

assume some regularity of observation frequency. For example, one value is available for

each year e.g. annual GDP; the observation frequency could be more often than yearly. For

instance observation may be available for each quarter, each month or even each day of
http://espin086.files.wordpress.com/2011/01/bad-economy.jpeg


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particular period. Nowadays, time series of stock prices or other financial market variables

are even available at a much higher frequency such as every minute or seconds.

Many economic problems can be analysed using time series data. Forecasting the future

economic condition is one important objective of many analysis. For example, the

Government may wish to know the tax revenue for the next quarter or year, investors may

be interested in production or income of next year. In that case the forecast of the specific

variable is desired. GDP is a perfect example of Time Series data, to analysis and to forecast.

In this paper I have taken time series data of annual GDP, India for the period 1952 to 2005

and forecast for year 2006 is made. In first part I have taken US GDP quarterly data for the

period 1970 to 1991, from the text book "Basic Econometrics" by Damodar N. Gujrati and

Sangeeta, Fourth Edition for analysis and forecast of GDP for first quarter of 1992.

I used Box-Jenkins (BJ) methodology for forecasting the GDP, this methodology technically

known as ARIMA methodology, the emphasis of this method analyzing the probabilistic, or

stochastic nature of economic time series on their own under the philosophy let the data

speak for themselves. The objective of Box-Jenkins is to identify and estimate a statistical

model which can be interpreted as having generated the sample data. If this estimated

model is then to be used for forecasting, we must assume that the features of his model are

constant through time, and particularly over future time periods. Thus the simple reason for

requiring stationary data is that any model which is inferred from these data can itself be

interpreted as stationary or stable, therefore providing valid basis for forecasting.

BOX-JENKINS METHODOLOGY

1. MODEL IDENTIFICATION - TEST OF STATIONARITY:

I have used (A) Graphical analysis, (B) Auto correlation Function and Correlogram and (C)

Augmented Dickey Fuller Test to test stationarity of time series data of GDP, US.

A. Graphical Analysis:

Characteristics of the time series can seen from the plot of the series, US GDP. Such a plotgives an initial clues about the likely nature of the time series data.


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The plot of the time series, US GDP

Figure-1: GDP, United States, 1970-1991 (Quarterly)

Over the period of study US GDP has been increasing, that is, showing upward trend,

suggesting that the mean of US GDP has been changing. This suggests that the GDP seriesare not stationary.

B. AUTOCORRELATION FUNCTION (ACF), PARTIAL AUTOCORRELATIO

FUNCTION(PACF) AND CORRELOGRAM:

Date: 08/16/11 Time: 18:26

Sample: 1970Q1 1991Q4

Included observations: 88

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

. |******* . |******* 1 0.969 0.969 85.462 0.000

. |******* . | . | 2 0.935 -0.058 166.02 0.000

. |******| . | . | 3 0.901 -0.020 241.72 0.000

. |******| . | . | 4 0.866 -0.045 312.39 0.000

2,800

3,200

3,600

4,000

4,400

4,800

5,200

70 72 74 76 78 80 82 84 86 88 90

GDP


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. |******| . | . | 5 0.830 -0.024 378.10 0.000

. |******| . | . | 6 0.791 -0.062 438.57 0.000

. |***** | . | . | 7 0.752 -0.029 493.85 0.000

. |***** | . | . | 8 0.713 -0.024 544.11 0.000

. |***** | . | . | 9 0.675 0.009 589.77 0.000

. |***** | . | . | 10 0.638 -0.010 631.12 0.000

. |**** | . | . | 11 0.601 -0.020 668.33 0.000

. |**** | . | . | 12 0.565 -0.012 701.65 0.000

. |**** | . | . | 13 0.532 0.020 731.56 0.000

. |**** | . | . | 14 0.500 -0.012 758.29 0.000

. |*** | . | . | 15 0.468 -0.021 782.02 0.000

. |*** | . | . | 16 0.437 -0.001 803.03 0.000

. |*** | . | . | 17 0.405 -0.041 821.35 0.000

. |*** | . | . | 18 0.375 -0.005 837.24 0.000

. |** | . | . | 19 0.344 -0.038 850.79 0.000

. |** | . | . | 20 0.313 -0.017 862.17 0.000

. |** | .*| . | 21 0.279 -0.066 871.39 0.000

. |** | . | . | 22 0.246 -0.019 878.65 0.000

. |** | . | . | 23 0.214 -0.008 884.22 0.000

. |*. | . | . | 24 0.182 -0.018 888.31 0.000

. |*. | . | . | 25 0.153 0.017 891.25 0.000

Figure-2: Correlogram of U.S. GDP, 1970-I to 1991-IV. AC= autocorrelation, PAC= partial

autocorrelation, Q-stat= Q Statistics, Prob= Probability.

The Correlogram and partial Correlogram of the US GDP series, up to 25 lags is shown in

Figure-2. The autocorrelation coefficient (ACF) starts at a very high at lag 1 (0.969) and

decline slowly; ACF up to 23 lags are individually statistically different from zero, for they all

are outside the 95% confidence bounds. PACF drops dramatically after the first lag, and all

PACFs after lag 1 are statistically insignificant. This leading us to conclusion that time series

is non-stationary; It may be nonstationary in mean or variance or both.


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C. AUGMENTED DICKEY-FULLER TEST- UNIT ROOT TEST:-

1. GDP is a Random Walk without drift:

Null Hypothesis: US, GDP has a unit root

Exogenous: NoneLag Length: 1 (Automatic - based on SIC, maxlag=11)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic 3.449385 0.9998

Test critical values: 1% level -2.592129

5% level -1.944619

10% level -1.614288

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(GDP)

Method: Least Squares

Date: 08/16/11 Time: 18:28

Sample (adjusted): 1970Q3 1991Q4

Included observations: 86 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

GDP(-1) 0.003899 0.001130 3.449385 0.0009

D(GDP(-1)) 0.327024 0.103622 3.155918 0.0022

R-squared 0.088841 Mean dependent var 23.34535

Adjusted R-squared 0.077994 S.D. dependent var 35.93794

S.E. of regression 34.50803 Akaike info criterion 9.943242

Sum squared resid 100027.6 Schwarz criterion 10.00032

Log likelihood -425.5594 Hannan-Quinn criter. 9.966214

Durbin-Watson stat 2.034955

Random walk without drift, for US GDP. we rule out this model because the coefficient of

GDPt-1, which is equal to is positive. But since =(-1), a positive would imply that > 1.

We rule out this case because in this case the GDP time series would be explosive.


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2. GDP is a Random Walk with drift:

Null Hypothesis: GDP has a unit root

Exogenous: Constant

Lag Length: 1 (Automatic - based on SIC, maxlag=20)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -0.547205 0.8756


5% level -2.895512

10% level -2.584952





Date: 08/16/11 Time: 18:31




GDP(-1) -0.003304 0.006038 -0.547205 0.5857

D(GDP(-1)) 0.319711 0.103506 3.088807 0.0027

C 28.71900 23.65025 1.214321 0.2281






F-statistic 4.855544 Durbin-Watson stat 2.040544

Prob(F-statistic) 0.010134

Random walk with drift, for US GDP. In this case the estimated coefficient of GDPt-1, which is

equal to is negative, implying that the estimated value of is less than 1. For this model

the estimated value is -0.547205 which in absolute value is below even the 10 percent

critical value of -2.584952. Since in absolute terms, the estimated is smaller than critical

value, our conclusion is that US GDP series is not stationary.


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3. GDP is a Random Walk with drift and trend


Exogenous: Constant, Linear Trend


t-Statistic Prob.*



5% level -3.4629

10% level -3.1578





Date: 08/04/11 Time: 18:29




GDP(-1) -0.0786 0.0355 -2.2152 0.0295

D(GDP(-1)) 0.3557 0.1026 3.4647 0.0008

C 234.9729 98.5876 2.3833 0.0194

@TREND(1970Q1) 1.8921 0.8791 2.1522 0.0343





I have used augmented Dickey-Fuller (ADF) test with intercept and trend to test the

stationarity and results are given above. The t (= ) value of the GDPt-1 coefficient ( = ) is

-2.2152, but this value in absolute terms is much less than even the 10 percent critical

value of -3.1570, again suggesting that even after taking care of possible autocorrelation in

error term, the US GDP series is nonstationary.


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First Differences of US, GDP series

Figure-3: First Differences of US GDP, 1970 to 1991(quarterly)

To make US GDP series stationary, I have taken first differences of US GDP, using EVIEWs

and plotted it on graph in figure-3. Unlike figure-1, I do not observe any trend in this series,

perhaps suggesting that the first differenced US GDP time series is stationary. We can also

see this visually from the estimated ACF and PACF correlograms given in figure-4.

AUGMENTED DICKEY-FULLER TEST of First Difference of US GDP series.

Null Hypothesis: DGDP has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.895512

10% level -2.584952


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0

40

80

120

70 72 74 76 78 80 82 84 86 88 90 92

First Difference of US GDP


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Dependent Variable: D(DGDP)


Date: 08/19/11 Time: 23:38

Sample (adjusted): 1970Q3 1991Q4Included observations: 86 after adjustments


DGDP(-1) -0.682762 0.102975 -6.630339 0.0000

C 16.00498 4.396717 3.640211 0.0005








First difference US GDP series is tested by Augmented Dickey-Fuller test. The t(= )value of

the DGDPt-1coefficient (=) is -6.630339,the value in absolute terms is more than even the 1

percent critical value of-3.508326, the first difference US GDP series is stationary.

Correlogram of First differences of US GDP:

Date: 08/16/11 Time: 18:27

Sample: 1970Q1 1991Q4



. |** | . |** | 1 0.316 0.316 9.0136 0.003

. |*. | . |*. | 2 0.186 0.095 12.165 0.002

. | . | . | . | 3 0.049 -0.038 12.389 0.006

. | . | . | . | 4 0.051 0.033 12.631 0.013

. | . | . | . | 5 -0.007 -0.032 12.636 0.027

. | . | . | . | 6 -0.019 -0.020 12.672 0.049

.*| . | . | . | 7 -0.073 -0.062 13.188 0.068

**| . | **| . | 8 -0.289 -0.280 21.380 0.006

.*| . | . |*. | 9 -0.067 0.128 21.820 0.009

. | . | . |*. | 10 0.019 0.100 21.855 0.016


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. | . | . | . | 11 0.037 -0.008 21.991 0.024

**| . | **| . | 12 -0.239 -0.311 27.892 0.006

.*| . | . | . | 13 -0.117 0.011 29.314 0.006

.*| . | .*| . | 14 -0.204 -0.114 33.712 0.002

.*| . | . | . | 15 -0.128 -0.051 35.474 0.002

. | . | . | . | 16 -0.035 -0.021 35.610 0.003

. | . | . | . | 17 -0.056 -0.019 35.956 0.005

. | . | . |*. | 18 0.009 0.122 35.965 0.007

. | . | .*| . | 19 -0.045 -0.071 36.195 0.010

. | . | .*| . | 20 0.066 -0.126 36.694 0.013

. |*. | . |*. | 21 0.084 0.089 37.519 0.015

. | . | . | . | 22 0.039 -0.060 37.696 0.020

.*| . | .*| . | 23 -0.068 -0.121 38.259 0.024

. | . | . | . | 24 -0.032 -0.041 38.384 0.032

. | . | . |*. | 25 0.013 0.092 38.406 0.042

Figure-4: Correlogram of first differences of GDP, US, 1970-I to 1991-IV

The autocorrelations decline up to lag 4, the lags 1, 8 and 12 seem statistically different

from zero; but all other lags are not statistically different from zero (the solid lines shown in

his figure give the approximate 95% confidence limits). This is also true for partial

autocorrelations. Let us therefore assume that the process that generated the (first

differenced) GDP is at the most an AR(12) process. The Gujarati have included the only AR

terms at lag 1, 8 and 12 which are significant.

2. ESTIMATION OF THE ARIMA MODEL:

MODEL ACF BEHAVIOR PACF BEHAVIOR

AR(p) Decays Gradually Spike in lag p

MA(q) Spikes in lag q Decays Gradually

ARMA(p,q) Decays Gradually Decays Gradually

The above table is used to identify which kind of model we should be using AR, MA, or

mixture model ARIMA. Given the fact that there can be multiple models that are likely

candidates, I will use the (Schwartz Information Criterion) SIC statistics to choose the best

one. The model with the smallest SIC value will be the better fit.


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A simple regression on the first difference of GDP to its lag value will serve as this error

term. I have denoted the first differences of US GDP by DGDP. The tentative identified

ARIMA (1,8,12;1;0) model is DGDP = + 1

AR(1) + 8

AR(8 ) + 12

AR(12).

Using EViews, I obtained the following estimates:

Eviews code: dgdp c ar(1) ar(8) ar(12)

Dependent Variable: DGDP


Date: 08/20/11 Time: 23:11



Convergence achieved after 3 iterations


C 23.08936 2.980356 7.747181 0.0000

AR(1) 0.342768 0.098794 3.469531 0.0009

AR(8) -0.299466 0.101599 -2.947523 0.0043

AR(12) -0.264371 0.098582 -2.681742 0.0091








Inverted AR Roots .92-.28i .92+.28i .61-.59i .61+.59i

.31+.87i .31-.87i -.25+.88i -.25-.88i

-.57-.59i -.57+.59i -.85+.28i -.85-.28i

As per Gujarati, other models are also checked with dependent veriable as AR(1) only,

SIC=9.9863; with AR(1) and AR(8), SIC=9.9328; and with AR(8) and AR(12), SIC=10.0047, but

on Schwarz Information Criterion (SIC) statistics this AR model found the best (minimum

SIC=9.9056).


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I uesd the process of temporarily selecting several possible models with increasing order

and simply choosing the one that optimizes the model selection statistics (AIC and SIC).

Table 1 below summarizes the AIC (Akaike information criterion) and SIC (Schwarz criterion)

results. The overall minimum SIC indicates an ARIMA(1,8,12;1;8) model; the coefficients are

significant.

Model SIC AIC Model SIC AIC

ARMA(1,8,12;1) 9.92 9.76 MA(1) 10.0008 9.9513

ARMA(1,8,12;8) 9.6967 9.5422 MA(8) 9.9463 9.8896

ARMA(1,8,12;12) 9.7047 9.5502 MA(12) 9.9754 9.9187

ARMA(1,8,12;8,12) 9.7788 9.5934 MA(1,12) 9.8121 9.7271ARMA(1,8,12;1,12) 9.7098 9.5244 MA(1,8,12) 9.7665 9.6531

Table-1. Search for the Best ModelAkaike and Schwarz Criterion.

On the basis of above table -1, I have choosen the ARIMA (1,8,12;1;8). I have denoted the

first difference of US GDP by DGDP. The alternative to the model given in the book, is

identified as ARIMA (1,8,12;1;8) model because the SIC and AIC is much less in this model

i.e 9.6967 compared to 9.9056 choosen by Gujarati.

The identified ARIMA (1,8,12;1;8) model is

DGDP = + 1AR(1) + 8AR(8 ) + 12 AR(12) + 8 MA(8).


Eviews code: dgdp c ar(1) ar(8) ar(12) ma(8)

Dependent Variable: DGDPMethod: Least SquaresDate: 08/24/11 Time: 19:55Sample (adjusted): 1973Q2 1991Q4

Included observations: 75 after adjustmentsConvergence achieved after 15 iterationsMA Backcast: 1971Q2 1973Q1


C 25.75666 1.296402 19.86781 0.0000AR(1) 0.244920 0.086407 2.834503 0.0060AR(8) 0.266051 0.095613 2.782567 0.0069AR(12) -0.407692 0.096015 -4.246119 0.0001MA(8) -0.920604 0.023443 -39.26910 0.0000

R-squared 0.458503 Mean dependent var 21.52933Adjusted R-squared 0.427560 S.D. dependent var 36.55936


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S.E. of regression 27.66072 Akaike info criterion 9.542244Sum squared resid 53558.09 Schwarz criterion 9.696744Log likelihood -352.8342 Hannan-Quinn criter. 9.603934F-statistic 14.81781 Durbin-Watson stat 1.612204Prob(F-statistic) 0.000000

Inverted AR Roots .91-.20i .91+.20i .71-.68i .71+.68i.22+.89i .22-.89i -.18+.88i -.18-.88i-.66-.68i -.66+.68i -.87+.20i -.87-.20i

Inverted MA Roots .99 .70-.70i .70+.70i .00+.99i-.00-.99i -.70-.70i -.70-.70i -.99

3. DIGNOSTIC CHECKING:

To know that the model is fit to the data, Using EViews, I have obtained residuals from

model and ACF and PACF of these residuals, up to 25 lags.

Correlogram of the residuals from ARIMA Model

Date: 08/19/11 Time: 19:28

Sample: 1973Q2 1991Q4


Q-statistic probabilities adjusted for 3

ARMA term(s)


. |*. | . |*. | 1 0.102 0.102 0.8192

. |*. | . |*. | 2 0.087 0.077 1.4151

. | . | . | . | 3 0.051 0.035 1.6219

.*| . | .*| . | 4 -0.104 -0.120 2.4963 0.114

. | . | . | . | 5 -0.022 -0.008 2.5346 0.282

. | . | . | . | 6 0.026 0.047 2.5919 0.459

. | . | . | . | 7 0.009 0.016 2.5992 0.627

.*| . | .*| . | 8 -0.082 -0.105 3.1735 0.673

. |*. | . |*. | 9 0.132 0.146 4.6969 0.583

. |*. | . |*. | 10 0.132 0.137 6.2497 0.511

. |*. | . |*. | 11 0.118 0.087 7.5067 0.483

. | . | .*| . | 12 -0.062 -0.157 7.8561 0.549

. | . | . | . | 13 0.047 0.069 8.0595 0.623

.*| . | .*| . | 14 -0.160 -0.129 10.479 0.488

**| . | .*| . | 15 -0.211 -0.185 14.745 0.256

. | . | . | . | 16 -0.013 -0.012 14.761 0.322

.*| . | .*| . | 17 -0.205 -0.138 18.931 0.168

. | . | . | . | 18 0.026 0.072 19.001 0.214

. | . | . | . | 19 -0.002 -0.048 19.001 0.269


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.*| . | .*| . | 20 -0.107 -0.170 20.195 0.264

. | . | . | . | 21 0.036 0.073 20.331 0.314

. | . | . | . | 22 -0.002 0.001 20.332 0.375

.*| . | .*| . | 23 -0.073 -0.074 20.922 0.402

.*| . | . | . | 24 -0.076 -0.048 21.579 0.424

.*| . | . | . | 25 -0.084 0.003 22.393 0.437

Figure-5: The correlogram of ACF and PACF of the residuals estimated from ARIMA model

The estimated ACF and PACF in correlogram given above shows, none of the

autocorrelations and partial autocorrelations is individually statistically significant. The

correlogram of both autocorrelation and partial autocorrelation give the impression that the

residuals estimated from ARIMA model given above are purely random.

Graphical presentation of residuals:

Figure-6: The graph of the residuals estimated from ARIMA model

The errors of the model appear to be a white noise process, mean 0 and constant variance.

The Augmented Dickey Fuller test rejects the hypothesis that the models error term is non-

stationary. The diagnostic check of the models performance validates unbiased estimates

and the SIC criteria mentioned above used to select the simple model ensures maximum

efficiency is estimation.

4. FORECASTING GDP WITH THE MODEL SELECTED

-150

-100

-50

0

50

100

-120

-80

-40

0

40

80

120

70 72 74 76 78 80 82 84 86 88 90

Residual Actual Fitted


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The US GDP data are for the period 1970Q1 to 1991Q4. On the basis of above model I would

like to forecast the US GDP for the first quarter of 1992. In order to use the model that has

been selected a mathematical formulation of the ARIMA (1,8,12;1;0) model must be

outlines. After the model has been constructed using algebra the parameters estimated in

this model above will be useful in building our forecasting model. Breaking down the lags

and first differences yields the final forecasting model that will be used to forecast the GDP

numbers.

Thus, to obtain the forecast value of GDP (not DGDP) for 1992 Q1, Gujarati had rewriten the

GDP1992Q1 - GDP1994Q4 = + 1 (GDP1991Q4 - GDP1991Q3) + 8 (GDP1989Q4 - GDP1989Q3) + 12

(GDP1988Q4 - GDP1988Q3) + u1992Q1

That is,

GDP1992Q1= + (1+1) GDP1991Q4 - 1 GDP1991Q3+ 8 GDP1989Q4 - 8 GDP1989Q3+ 12 GDP1988Q4

- 12 GDP1988Q3 + u1992Q1

GDP1992Q1 = 23.0893 + ((1+0.3428) X 4868) - 0.3428 X (4862.7) - 0.2994 X (4859.7) + 0.2994 X

(4845.6) - 0.2644 X (4779.7) + 0.2644 X (4734.5)

GDP1992Q1 = 4876.7 (Approx.) By model given by Gujarati

Alternative Model

Now, to obtain the Forecast value of the GDP for 1992Q1, by alternative model

ARIMA(1,8,12;1;8) I have rewrite the model as

GDP1992Q1 - GDP1994Q4 = + 1 (GDP1991Q4 - GDP1991Q3) + 8 (GDP1989Q4 - GDP1989Q3) + 12

(GDP1988Q4 - GDP1988Q3) + 8 (Resid1989Q4 - Resid1989Q3) + u1992Q1

that is,

GDP1992Q1= + (1+1) GDP1991Q4 - 1 GDP1991Q3+ 8 GDP1989Q4 - 8 GDP1989Q3+ 12 GDP1988Q4

- 12 GDP1988Q3+ 8 (Resid1989Q4 + Resid1989Q3)/2 + u1992Q1

GDP1992Q1 = 25.75666 + ((1+0.24492) X 4868) - 0.24492 X (4862.7) + 0.266051 X (4859.7) -

0.266051 X (4845.6) - 0.407692 X (4779.7) + 0.407692 X (4734.5) -0.920604 X (11.0127-

13.1513)/2

GDP1992Q1 = 4873.865 (Approx.)

The actual value of GDP, US for 1992 Q1 was 4873.7 billion; the forecast error was an

overestimate of 3 billion by model given in Gujarati whereas the alternative model given in

this paper have an overestimate of 0.16 billion. Tha forecast by alternative model is more

close to actual value,


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Figure-7 : forecast series of DGDP for the period 1973Q2 to 1992Q1.

Figure-8 : Static forecast of US GDP for the period 1973Q2 to 1992Q1 .

To forecast a series of one step ahead, In Eviews I used the static forecast. The above figure-

8 shows the graph of US GDP static forecast and the plus and minus two standard error

bands.

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0

40

80

120

70 72 74 76 78 80 82 84 86 88 90 92

DGDP DGDPF

-100

-50

0

50

100

150

1974 1976 1978 1980 1982 1984 1986 1988 1990

DGDPF 2 S.E.

Forecast: DGDPF

Actual: DGDP

Forecast sample: 1970Q1 1992Q4

Adjusted sample: 1973Q2 1992Q1


Root Mean Squared Error 30.53202

Mean Absolute Error 23.43041

Mean Abs. Percent Error 151.7918

Theil Inequality Coefficient 0.427781

Bias Proportion 0.000000

Variance Proportion 0.297514

Covariance Proportion 0.702486


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Figure-9 : Dynamic forecast of US GDP for the period 1970Q1 to 1992Q4.

Using EViews, The above graph shows a dynamic forecast using the above mentioned

ARIMA model over the sample period 1970Q1 to 1992Q4. The forecast values is placed in

the series DGDPF, and EViews has produced a graph of the forecasts and the plus and minus

two standard errors bands, as well as a forecast evaluation given in box right to graph.

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-40

0

40

80

120

160

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992

DGDPF 2 S.E.

Forecast: DGDPF

Actual: DGDP

Forecast sample: 1970Q1 1992Q4

Adjusted sample: 1973Q2 1992Q4










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INDIAN GDP Forecast for year 2006: Box-Jenkins Methodology and

ARIMA Forecast Model

In this term paper, By using the Box-Jenkins methodology to fit an ARIMA forecast model to

the time series of Indian GDP for the period 1952 to 2005, forecast for year 2005-06 is made

and compared with actual value.

BOX-JENKINS METHODOLOGY

1) MODEL IDENTIFICATION - TEST OF STATIONARITY:

I will use (A) Graphical analysis, (B) Auto correlation Function and Correlogram and (C)

Augmented Dickey -Fuller test to test stationarity of time series data of GDP, INDIA.

(A) Graphical Analysis:

Characteristics of the time series can seen from the plot of the series, INDIAN GDP. Such a

plot gives an initial clues about the likely nature of the time series data.

Figure10: GDP, India, 1952-2005(annually)

Over the period of study GDP of India has been increasing, that is, showing upward trend,

suggesting that the mean of GDP has been changing. This suggests that the GDP series are

not stationary. India's GDP growth was very low till 1990's, but after economic reforms it

shows sufficient upward movement.

0

400,000

800,000

1,200,000

1,600,000

2,000,000

2,400,000

2,800,000

55 60 65 70 75 80 85 90 95 00 05

GDP at Factor Cost


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B. CORRELOGRAM AND PARTIAL CORRELOGRAM:

Date: 08/19/11 Time: 19:52

Sample: 1952 2005



. |******* . |******* 1 0.915 0.915 47.796 0.000

. |******| . | . | 2 0.837 -0.002 88.577 0.000

. |******| . | . | 3 0.768 0.013 123.58 0.000

. |***** | . | . | 4 0.699 -0.036 153.14 0.000

. |***** | . | . | 5 0.633 -0.017 177.89 0.000

. |**** | . | . | 6 0.568 -0.034 198.22 0.000

. |**** | . | . | 7 0.506 -0.018 214.72 0.000

. |*** | . | . | 8 0.449 -0.012 227.98 0.000

. |*** | . | . | 9 0.392 -0.034 238.29 0.000

. |** | . | . | 10 0.341 -0.002 246.26 0.000

. |** | . | . | 11 0.294 -0.009 252.33 0.000

. |** | . | . | 12 0.250 -0.013 256.81 0.000

. |*. | . | . | 13 0.208 -0.015 260.02 0.000

. |*. | . | . | 14 0.170 -0.015 262.20 0.000

. |*. | . | . | 15 0.129 -0.048 263.48 0.000

. |*. | . | . | 16 0.089 -0.028 264.10 0.000

. | . | . | . | 17 0.052 -0.017 264.32 0.000

. | . | . | . | 18 0.021 0.002 264.36 0.000

. | . | . | . | 19 -0.009 -0.023 264.37 0.000

. | . | . | . | 20 -0.037 -0.020 264.49 0.000

. | . | . | . | 21 -0.065 -0.027 264.88 0.000

.*| . | . | . | 22 -0.093 -0.031 265.69 0.000

.*| . | . | . | 23 -0.116 -0.010 267.02 0.000

.*| . | . | . | 24 -0.140 -0.031 269.00 0.000

.*| . | . | . | 25 -0.161 -0.012 271.70 0.000

Figure-11: Correlogram of GDP, India, 1951-52 to 2004-05.

The Correlogram of the India's GDP time series, up to 25 lags is shown in Figure-11. The

autocorrelation coefficient starts at a very high at lag 1 (0.915) and decline slowly, ACF up to

12 lags are individually statistically different from zero, for they all are outside the 95%

confidence bounds. PACF drops dramatically after the first lag, and all PACFs after lag 1 are

statistically insignificant. This leading us to conclusion that time series is non-stationary; It

may be nonstationary in mean or variance or both.


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C. AUGMENTED DICKEY -FULLER TEST: UNIT ROOT TEST

(i) RANDOM WALK WITH DRIFT


Exogenous: Constant


t-Statistic Prob.*



5% level -2.917650

10% level -2.596689





Date: 08/14/11 Time: 19:55

Sample (adjusted): 1953 2005



GDP(-1) 0.068553 0.005320 12.88656 0.0000

C -13117.18 5056.136 -2.594308 0.0123




Sum squared resid 2.19E+10 Schwarz criterion 22.82735

Log likelihood -600.9545 Hannan-Quinn criter. 22.78159F-statistic 166.0633 Durbin-Watson stat 2.308236


Random walk with drift, for India GDP. we rule out this model because the coefficient of

GDPt-1, which is equal to is positive. But since =(-1), a positive would imply that > 1.

We rule out this case because in this case the GDP time series would be explosive.


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(ii) RANDOM WALK WITH DRIFT AND TREND




t-Statistic Prob.*



5% level -3.496960

10% level -3.177579





Date: 08/14/11 Time: 19:56




GDP(-1) 0.079422 0.013847 5.735573 0.0000

C -10530.35 5912.168 -1.781132 0.0810

@TREND(1952) -412.0081 484.4021 -0.850550 0.3991






F-statistic 82.94312 Durbin-Watson stat 2.364045Prob(F-statistic) 0.000000

Random walk with drift and trend, for India GDP. we rule out this model because the

coefficient of GDPt-1, which is equal to is positive. But since =(-1), a positive would

imply that > 1. We rule out this case because in this case the GDP time series would be

explosive.


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FIRST DIFFERENCE OF INDIAN GDP

(i) Graphical Analysis of First differences of INDIAN GDP:

Figure-12 : First Difference of Indian GDP, for the period 1952 to 2005

The first defference of Indian GDP, over the period of study has been increasing, that is,

showing upward trend, suggesting that the mean of GDP has been changing. This suggests

that the First Difference of INDIAN GDP series is not stationary.

(ii) Correlogram and Partial Correlogram of First differences of INDIAN GDP:

Date: 08/20/11 Time: 22:24

Sample: 1952 2005



. |***** | . |***** | 1 0.650 0.650 23.680 0.000

. |**** | . |** | 2 0.555 0.230 41.310 0.000

. |**** | . |** | 3 0.558 0.238 59.449 0.000

. |*** | . | . | 4 0.462 -0.007 72.125 0.000

. |**** | . |** | 5 0.529 0.253 89.113 0.000

. |*** | . | . | 6 0.449 -0.058 101.64 0.000

. |*** | . | . | 7 0.403 0.032 111.92 0.000

. |*** | . | . | 8 0.420 0.036 123.33 0.000

. |** | .*| . | 9 0.281 -0.165 128.55 0.000

. |** | .*| . | 10 0.240 -0.086 132.45 0.000

. |** | . | . | 11 0.229 -0.014 136.09 0.000

. |*. | .*| . | 12 0.120 -0.142 137.11 0.000

-40,000

0

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80,000

120,000

160,000

200,000

55 60 65 70 75 80 85 90 95 00 05

DGDP


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. |*. | . | . | 13 0.142 0.022 138.57 0.000

. |*. | . | . | 14 0.127 0.046 139.78 0.000

. |*. | . |*. | 15 0.139 0.139 141.27 0.000

. | . | .*| . | 16 0.056 -0.176 141.51 0.000

. | . | . | . | 17 -0.022 -0.014 141.55 0.000

. | . | . | . | 18 -0.004 0.004 141.55 0.000. | . | . | . | 19 -0.030 -0.017 141.63 0.000

. | . | . | . | 20 -0.042 -0.018 141.79 0.000

. | . | . | . | 21 -0.045 0.010 141.97 0.000

.*| . | .*| . | 22 -0.129 -0.161 143.53 0.000

.*| . | . | . | 23 -0.112 0.024 144.75 0.000

.*| . | .*| . | 24 -0.182 -0.129 148.09 0.000

.*| . | . | . | 25 -0.203 0.024 152.39 0.000

Figure-13 : The Correlogram of First Difference of Indian GDP, for the period 1952 to 2005

The Correlogram of the First Difference of India's GDP time series, up to 11 lags is shown inFigure-13. The autocorrelation coefficient starts at a high at lag 1 (0.650) and decline slowly,

ACF up to 11 lags are individually statistically different from zero, for they all are outside the

95% confidence bounds. PACF drops dramatically after the third lag, and all PACFs after lag 5

are statistically insignificant. This leading us to conclusion that time series is non-stationary;

It may be nonstationary in mean or variance or both .

C. AUGMENTED DICKEY -FULLER TEST of First Difference of Indian GDP: UNIT

ROOT TEST


Null Hypothesis: DGDP has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.928142

10% level -2.602225



Dependent Variable: D(DGDP)


Date: 08/20/11 Time: 22:39

Sample (adjusted): 1961 2005Included observations: 45 after adjustments


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DGDP(-1) 0.544567 0.147329 3.696256 0.0007

D(DGDP(-1)) -1.628306 0.251825 -6.466034 0.0000

D(DGDP(-2)) -1.663902 0.318961 -5.216633 0.0000D(DGDP(-3)) -1.524341 0.339006 -4.496503 0.0001

D(DGDP(-4)) -1.532311 0.322380 -4.753121 0.0000

D(DGDP(-5)) -1.203777 0.320796 -3.752468 0.0006

D(DGDP(-6)) -0.901187 0.274534 -3.282603 0.0023

D(DGDP(-7)) -0.505443 0.193925 -2.606375 0.0132

C -587.9856 5469.960 -0.107494 0.9150





Random walk with drift and trend, for India GDP. we rule out this model because the

coefficient of GDPt-1, which is equal to is positive. But since =(-1), a positive would

imply that > 1. We rule out this case because in this case the GDP time series would be

explosive.

SECOND DEFFERENCE OF INDIAN GDP

Figure-14 : Second difference of India GDP for the period 1952 to 2005

-80,000

-40,000

0

40,000

80,000

120,000

55 60 65 70 75 80 85 90 95 00 05

DDGDP


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25

I have taken the second difference of Indian GDP, using EViews and plotted it on graph in

figure-14. The graph shows that the second difference of Indian GDP series is stationary. We

can also see this visually from the estimated ACF and PACF correlograms given below in

figure-15 and Augmented Dickey-Fuller test given below, which shows that Indian GDP time

series is stationary.

Augmented Dickey Fuller test of second difference of Indian GDP


Unit root with driftNull Hypothesis: DDGDP has a unit rootExogenous: ConstantLag Length: 1 (Automatic - based on SIC, maxlag=10)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -7.796844 0.0000Test critical values: 1% level -3.568308

5% level -2.92117510% level -2.598551


Augmented Dickey-Fuller Test EquationDependent Variable: D(DDGDP)Method: Least Squares

Date: 08/20/11 Time: 22:36Sample (adjusted): 1956 2005Included observations: 50 after adjustments


DDGDP(-1) -2.071391 0.265670 -7.796844 0.0000D(DDGDP(-1)) 0.363583 0.158469 2.294347 0.0263

C 5875.982 3692.866 1.591171 0.1183

R-squared 0.786065 Mean dependent var-

94.38000

Adjusted R-squared 0.776961 S.D. dependent var 54542.17S.E. of regression 25758.60 Akaike info criterion 23.20905Sum squared resid 3.12E+10 Schwarz criterion 23.32377Log likelihood -577.2262 Hannan-Quinn criter. 23.25274F-statistic 86.34650 Durbin-Watson stat 1.962118Prob(F-statistic) 0.000000

Second difference of INDIAN GDP series is tested by Augmented Dickey-Fuller test. The t (=

) value of the DGDPt-1coefficient (=) is -7.796844, the value in absolute terms is more than

even the 1 percent critical value of-3.568308, It shows that the second difference INDIAN

GDP series is stationary.


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(i) RANDOM WALK WITH DRIFT AND TREND

Null Hypothesis: DDGDP has a unit root



t-Statistic Prob.*



5% level -3.513075

10% level -3.186854



Dependent Variable: D(DDGDP)Method: Least Squares

Date: 08/20/11 Time: 22:37




DDGDP(-1) -8.301953 1.444256 -5.748257 0.0000

D(DDGDP(-1)) 6.163642 1.338779 4.603927 0.0001

D(DDGDP(-2)) 4.893284 1.142647 4.282409 0.0001

D(DDGDP(-3)) 3.669885 0.905690 4.052030 0.0003

D(DDGDP(-4)) 2.357670 0.670398 3.516823 0.0012

D(DDGDP(-5)) 1.304044 0.417466 3.123710 0.0035

D(DDGDP(-6)) 0.482988 0.188232 2.565919 0.0146

C -17869.72 8894.947 -2.008975 0.0521

@TREND(1952) 1183.450 306.2348 3.864517 0.0004




Sum squared resid 1.77E+10 Schwarz criterion 23.38737Log likelihood -509.0858 Hannan-Quinn criter. 23.16074



Second difference INDIAN GDP series is tested by Augmented Dickey-Fuller test. The t (= )

value of the DGDPt-1coefficient (=) is -5.748257, the value in absolute terms is more than

even the 1 percent critical value of-4.175640, It shows that the second difference INDIAN

GDP series is stationary.


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Correlogram of Second differences of INDIAN GDP:

Date: 08/20/11 Time: 22:35Sample: 1952 2005



****| . | ****| . | 1 -0.524 -0.524 15.122 0.000

. | . | **| . | 2 0.062 -0.293 15.341 0.000

. |*. | . | . | 3 0.097 -0.022 15.876 0.001

**| . | **| . | 4 -0.237 -0.246 19.160 0.001

. |*. | .*| . | 5 0.185 -0.094 21.202 0.001

. | . | . | . | 6 -0.046 -0.027 21.329 0.002

. | . | . | . | 7 -0.033 -0.034 21.397 0.003

. |*. | . |*. | 8 0.185 0.173 23.574 0.003

.*| . | . |*. | 9 -0.163 0.097 25.309 0.003

. | . | .*| . | 10 -0.024 -0.070 25.346 0.005. |** | . |** | 11 0.227 0.230 28.867 0.002

**| . | . |*. | 12 -0.236 0.099 32.774 0.001

. |*. | . |*. | 13 0.152 0.085 34.428 0.001

.*| . | .*| . | 14 -0.147 -0.144 36.019 0.001

. |*. | . |*. | 15 0.186 0.205 38.655 0.001

.*| . | . | . | 16 -0.112 -0.018 39.627 0.001

. | . | . | . | 17 -0.022 -0.059 39.665 0.001

. | . | .*| . | 18 0.063 -0.072 39.997 0.002

. | . | . | . | 19 -0.038 -0.035 40.120 0.003

. | . | .*| . | 20 -0.021 -0.106 40.157 0.005

. |*. | . |*. | 21 0.101 0.083 41.081 0.005

.*| . | .*| . | 22 -0.120 -0.106 42.426 0.006

. |*. | . |*. | 23 0.151 0.106 44.645 0.004

.*| . | . | . | 24 -0.149 -0.052 46.872 0.003

Figure-15: Correlogram of Second differences of GDP, India, 1952 to 2005

The autocorrelations at the lag 1 and 4 seem statistically different from zero; but all other

lags are not statistically different from zero (the solid lines shown in this figure gives the

approximate 95% confidence limits). The Partial Autocorrelations is also statistically

different from zero at lag 1, 2 and 4. Let us therefore assume that the process that

generated the (second differenced) GDP is at the most an ARIMA(4;2;4) process.

In practice, the identification of models can be difficult since the observed patterns of

autocorrelations only roughly correspond to the theoretical patterns, or the assignment is

ambiguous. However, analysts often circumvent this tricky process by temporarily selecting

several possible models with increasing order and simply choosing the one that optimizes

the model selection statistics (AIC and SIC). Table 1 below summarizes the AIC (Akaike

information criterion) and SC (Schwarz criterion) results. The overall minimum SIC indicatesan MA(1,4) model; the coefficients are significant.


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model SIC AIC Model SIC AIC

AR(1) 23.3301 23.2543 MA(1) 23.1521 23.0771

AR(1,2) 23.3237 23.209 MA(1,4) 23.0499 22.93

AR(4) 23.64 23.56 MA(4) 23.5698 23.4948

ARMA(1,1) 23.2411 23.1274 ARMA(1,4;1,4)23.1554 22.9605

ARMA(1;1,4) 23.1454 22.9930 ARMA(4,4) 23.6833 23.5663

Table-1. Search for the Best ModelAkaike and Schwarz Criterion.

On the basis of above table -1, I have choosen the ARIMA model (0;2;1,4). I have denoted

the second differences of INDIAN GDP by DDGDP. The tentative identified ARIMA (0;2;1,4)

model is DDGDP = + 1 MA(1) + 4 MA(4 ).


Eviews code: ddgdp c ma(1) ma(4)

Dependent Variable: DDGDP


Date: 08/20/11 Time: 23:28



Convergence achieved after 44 iterations

MA Backcast: 1950 1953


C 2643.353 792.4623 3.335620 0.0016

MA(1) -1.090621 0.035871 -30.40428 0.0000

MA(4) 0.341086 0.022874 14.91139 0.0000


Adjusted R-squared 0.460657 S.D. dependent var 30652.08S.E. of regression 22510.88 Akaike info criterion 22.93735





Inverted MA Roots .90-.33i .90+.33i -.36+.49i -.36-.49i


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3. DIGNOSTIC CHECKING:

To know that the model is fit to the data, Using EViews, I have obtained residuals from

model and ACF and PACF of these residuals, up to 25 lags.

Correlogram of the residuals from ARIMA Model

Date: 08/20/11 Time: 23:22

Sample: 1954 2005


Q-statistic probabilities adjusted for 2 ARMA

term(s)


. | . | . | . | 1 -0.001 -0.001 7.E-05

. | . | . | . | 2 0.055 0.055 0.1700

. |*. | . |*. | 3 0.191 0.192 2.2713 0.132

.*| . | .*| . | 4 -0.165 -0.173 3.8693 0.144

. |** | . |** | 5 0.237 0.233 7.2121 0.065

. |*. | . |*. | 6 0.139 0.118 8.3849 0.078

. |*. | . |*. | 7 0.084 0.135 8.8293 0.116

. |** | . |** | 8 0.332 0.240 15.848 0.015

. | . | . | . | 9 0.012 0.048 15.858 0.026

. | . | . | . | 10 0.045 0.001 15.997 0.042

. |*. | . |*. | 11 0.168 0.083 17.940 0.036

.*| . | .*| . | 12 -0.085 -0.071 18.449 0.048

. |*. | .*| . | 13 0.078 -0.077 18.887 0.063

. | . | .*| . | 14 0.019 -0.122 18.913 0.091

. |*. | . |*. | 15 0.164 0.175 20.963 0.074

. | . | **| . | 16 -0.012 -0.223 20.975 0.102

.*| . | .*| . | 17 -0.071 -0.100 21.385 0.125

. | . | . | . | 18 0.046 -0.047 21.561 0.158

. | . | . | . | 19 -0.033 0.010 21.656 0.198

. | . | . | . | 20 0.031 -0.017 21.741 0.244

. | . | . |*. | 21 0.072 0.085 22.214 0.274

.*| . | .*| . | 22 -0.129 -0.132 23.778 0.252

. | . | . | . | 23 0.030 0.008 23.866 0.300

.*| . | .*| . | 24 -0.177 -0.176 27.009 0.211

Figure-16: The correlogram of ACF and PACF of the residuals estimated from ARIMA model

The estimated ACF and PACF in correlogram given above shows, none of the

autocorrelations and partial autocorrelations is individually statistically significant. The

correlogram of both autocorrelation and partial autocorrelation give the impression that the

residuals estimated from ARIMA model given above are purely random.


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Graphical presentation of residuals:

Figure-17: The graph of the residuals estimated from ARIMA model

The errors of the model appear to be a white noise process, mean 0 and constant variance.

The Augmented Dickey Fuller test rejects the hypothesis that the models error term is non-

stationary. The diagnostic check of the models performance validates unbiased estimates

and the SIC criteria mentioned above used to select the simple model ensures maximum

efficiency is estimation.

4. FORECASTING GDP WITH THE MODEL SELECTED

The Indian GDP data are given for the period 1951-52 to 2004-05. On the basis of above

model I would like to forecast the Indian GDP for the Year 2005-06. In order to use the

model that has been selected a mathematical formulation of the ARIMA (0;2;1,4) model

must be outlines. After the model has been constructed using algebra the parameters

estimated in this model above will be useful in building our forecasting model. Breaking

down the Moving averages of errors and second differences yields the final forecasting

model that will be used to forecast the GDP numbers.

-60,000

-40,000

-20,000

0

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60,000

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-40,000

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55 60 65 70 75 80 85 90 95 00 05

Residual Actual Fitted


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Thus, to obtain the forecast value of GDP (not DDGDP) for year 2006, I rewrite the above

model as

(GDP2006 - GDP2005) - (GDP2005 -GDP2004) = + u2006 + 1 ma1+ 2 ma4

That is,

GDP2006= + GDP2005 + GDP2005 - GDP2004 + 1 (RESID2005+ RESID2004)/2 + 2 (RESID2002 +

RESID2001)/2 + u2006

GDP2006 = 2643.353 +2388768+2388768 - 2222758 + (- 1.090621 X (48417.7 - 50396.3) +

0.341086 X (13895.5 + 13368.5)

GDP2006 = 2568879 (Approx.)

The actual value of GDP, INDIA for YEAR 2005-06 was 2604532 crores; the forecast error is

an underestimate of 35653 crore. The growth of GDP in 2005-06 was 9.5, which was highest

ever.

Figure-18 : forecast series of DDGDP for the period 1952 to 2006.

-60,000

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55 60 65 70 75 80 85 90 95 00 05

DDGDPF


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Static Forecast for year 2005-06.

Figure-8 : Static forecast of Second difference of INDIAN GDP for the period 1951-52 to 2005-06.

Static Forecast, is to forecast a one step ahead in time series. By using Eviews, I plotted the

static forecast of second difference of GDP, India on graph. The above figure-8 shows the

graph of second difference of INDIAN GDP static forecast for year 2006 and the plus and

minus two standard error bands.

Dynamic Forecast of second difference of Indian GDP.

Figure-9 : Dynamic forecast of second difference INDIAN GDP for the period 1951-52 to 2005-06.

Using EViews, The above graph shows a dynamic forecast using the above mentioned

ARIMA model over the sample period 1952 to 2006. The forecast values is placed in the

series DGDPF, and EViews has produced a graph of the forecasts and the plus and minus

two standard errors bands, as well as a forecast evaluation given in box right to graph.

-120,000

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0

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120,000

55 60 65 70 75 80 85 90 95 00 05

DDGDPF 2 S.E.

Forecast: DDGDPF

Actual: DDGDP

Forecast sample: 1952 2006









-80,000

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DDGDPF 2 S.E.

Forecast: DDGDPF

Actual: DDGDP

Forecast sample: 1952 2006










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References:

1. Basic Econometrics, forth Edition, Damodar N. Gujarati and Sangeetha

2. Econometrics Methods, Forth Edition, J. Johnston and J. Dinardo

3. EViews user guides

4. Using EViews for Undergraduate Econometrics, Second edition, by R. Carter Hill, William

E. Griffiths and George G. Judge

5. RBI, Databank for Indian GDP series

6. Planning Commision official website for data.

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