regression arima garch var model
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Forecasting using Regression,ARIMA and VAR Models
Submitted to Prof. Sajal Ghosh for partial fulfillment of Part TimePGPM Oct 2011 batch
Modeling and Forecasting in Energy and Financial
Markets
Submitted by:
Manish Jindal (11PT2-36)
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1. RegressionInput Data Used (from E-Views Database):
Table 7.3
Real Gross Product, Labor Days and Real Capital Input in theAgricultural Sector, Taiwan, 1958-1972
YEAR = YearY = Real Gross Product, Millions of NT $X2 = Labor Days, Millions of DaysX3 = Real Capital Input, Millions of NT $
YEAR Y X2 X31958 16607.7 275.5 17803.71959 17511.3 274.4 18096.81960 20171.2 269.7 18271.81961 20932.9 267.0 19167.3
1962 20406.0 267.8 19647.61963 20831.6 275.0 20803.51964 24806.3 283.0 22076.61965 26465.8 300.7 23445.21966 27403.0 307.5 24939.01967 28628.7 303.7 26713.71968 29904.5 304.7 29957.81969 27508.2 298.6 31585.91970 29035.5 295.5 33474.51971 29281.5 299.0 34821.81972 31535.8 288.1 41794.3
E-views Output for multiple linear regression model:
ls y c x2 x3
Dependent Variable: Y
Method: Least Squares
Date: 04/24/13 Time: 17:39
Sample: 1958 1972
Included observations: 15
Variable Coefficient Std. Error t-Statistic Prob.
C -28067.17 9432.066 -2.975718 0.0116
X2 147.9362 36.44344 4.059338 0.0016
X3 0.403563 0.073561 5.486131 0.0001
R-squared 0.909559 Mean dependent var 24735.33
Adjusted R-squared 0.894485 S.D. dependent var 4874.173
S.E. of regression 1583.279 Akaike info criterion 17.74924
Sum squared resid 30081287 Schwarz criterion 17.89085
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Log likelihood -130.1193 F-statistic 60.34143
Durbin-Watson stat 1.039019 Prob(F-statistic) 0.000001
Analysis:
High R-squared and Adjusted R-squared indicate a good explanatory power of the
model.
t-test (Probability values < 0.05) indicates that both Labour Days (X2) and Real Capital
Input (X3) are significant in explaining the variability of the explained variable, Real
GDP.
Also, F-test (probability value < 0.05) indicates that the model as a whole is significant.
However, the Durbin-Watson stat value of 1.039 is in the doubtful range and therefore,
serial correlation LM test is performed to make sure no auto correlation exists in the
model. Lag order of 1 is used for LM test. The outcome is given below:
Serial Correlation LM Test:
Breusch-Godfrey Serial Correlation LM Test:
F-statistic 1.237281 Probability 0.289714
Obs*R-squared 1.516612 Probability 0.218133
Test Equation:
Dependent Variable: RESID
Method: Least SquaresDate: 04/24/13 Time: 18:00
Presample missing value lagged residuals set to zero.
Variable Coefficient Std. Error t-Statistic Prob.
C 3243.336 9784.721 0.331469 0.7465
X2 -12.76956 37.87033 -0.337192 0.7423
X3 0.016801 0.074393 0.225839 0.8255
RESID(-1) 0.333779 0.300071 1.112331 0.2897
R-squared 0.101107 Mean dependent var -1.36E-11
Adjusted R-squared -0.144045 S.D. dependent var 1465.832
S.E. of regression 1567.854 Akaike info criterion 17.77598
Sum squared resid 27039844 Schwarz criterion 17.96480
Log likelihood -129.3199 F-statistic 0.412427
Durbin-Watson stat 1.437784 Prob(F-statistic) 0.747397
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Here, the Null hypothesis is that no auto correlation exists in the model. A probability of
0.21 and 0.28 are greater than the threshold of 0.05, therefore, Null Hypothesis of No
Auto Correlation existing is accepted.
Check for Multicollinearity:
Correlation Matrix:
X2 X3
X2 1.000000 0.620523
X3 0.620523 1.000000
Correlation between X2 and X3 is found to be 0.620523 which is not high enough to
indicate a multicollinearity problem existing in the model. Therefore, the model can now
be used for forecasting.
The final regression model used for forecasting is as follows:Y = -28067.16664 + 147.9362123*X2 + 0.4035625699*X3
Forecasting using the model:
12000
16000
20000
24000
28000
32000
36000
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
YF
Forecast: YF
Actual: Y
Forecast sample: 1958 1972
Included observations: 15
Root Mean Squared Error 1416.128
Mean Absolute Error 1100.348
Mean Abs. Percent Error 5.169292
Theil Inequality Coefficient 0.028143
Bias Proportion 0.000000Variance Proportion 0.023694
Covariance Proportion 0.976306
Theil Inequality Coefficient of 0.023 is quite close to zero, which indicates a near perfect
fit of the model for forecasting purpose.
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2. ARIMAInput Data Used peak demand.xls:
peak demand.xls
E-views Output for correlogram:
Date: 06/23/13 Time: 14:59
Sample: 2000M04 2008M07
Included observations: 100
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
. |*******| . |*******| 1 0.970 0.970 96.941 0.000
. |*******| . | . | 2 0.940 -0.015 188.91 0.000
. |*******| . | . | 3 0.910 -0.015 275.99 0.000
. |*******| . | . | 4 0.880 -0.015 358.29 0.000
. |*******| . | . | 5 0.850 -0.015 435.90 0.000
. |****** | . | . | 6 0.820 -0.015 508.94 0.000
. |****** | . | . | 7 0.791 -0.015 577.50 0.000
. |****** | . | . | 8 0.761 -0.015 641.71 0.000
. |****** | . | . | 9 0.731 -0.015 701.67 0.000
. |***** | . | . | 10 0.702 -0.015 757.52 0.000
. |***** | . | . | 11 0.673 -0.015 809.38 0.000
. |***** | . | . | 12 0.643 -0.016 857.38 0.000
. |***** | . | . | 13 0.614 -0.015 901.64 0.000
. |**** | . | . | 14 0.586 -0.016 942.30 0.000
. |**** | . | . | 15 0.557 -0.015 979.50 0.000
. |**** | . | . | 16 0.528 -0.016 1013.4 0.000
. |**** | . | . | 17 0.500 -0.015 1044.1 0.000
. |**** | . | . | 18 0.472 -0.016 1071.8 0.000
. |*** | . | . | 19 0.444 -0.015 1096.6 0.000
. |*** | . | . | 20 0.416 -0.015 1118.6 0.000
. |*** | . | . | 21 0.389 -0.016 1138.1 0.000
. |*** | . | . | 22 0.361 -0.015 1155.2 0.000
. |*** | . | . | 23 0.334 -0.015 1170.0 0.000
. |** | . | . | 24 0.308 -0.016 1182.7 0.000
. |** | . | . | 25 0.281 -0.015 1193.5 0.000
. |** | . | . | 26 0.255 -0.016 1202.4 0.000
. |** | . | . | 27 0.229 -0.015 1209.8 0.000
. |** | . | . | 28 0.204 -0.016 1215.7 0.000
. |*. | . | . | 29 0.179 -0.015 1220.3 0.000
. |*. | . | . | 30 0.154 -0.016 1223.7 0.000
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. |*. | . | . | 31 0.130 -0.015 1226.2 0.000
. |*. | . | . | 32 0.106 -0.015 1227.9 0.000
. |*. | . | . | 33 0.082 -0.016 1228.9 0.000
. | . | . | . | 34 0.059 -0.015 1229.4 0.000
. | . | . | . | 35 0.036 -0.015 1229.6 0.000
. | . | . | . | 36 0.013 -0.016 1229.7 0.000
The correlogram shows signature of AR(1) process, however, it could be an illusion untilstationarity of the series is confirmed.
Performing the ADF Unit Root test for check of
stationarity:
ADF unit root test output at level:
Null Hypothesis: DEMAND has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=11)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -3.370152 0.0626
Test critical values: 1% level -4.073859
5% level -3.465548
10% level -3.159372
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(DEMAND)
Method: Least Squares
Date: 06/23/13 Time: 15:03
Sample (adjusted): 2000M05 2007M02
Included observations: 82 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
DEMAND(-1) -0.249099 0.073913 -3.370152 0.0012
C 4861.437 1432.497 3.393681 0.0011
@TREND(2000M04) 29.69826 9.602382 3.092802 0.0027
R-squared 0.125876 Mean dependent var 104.3415
Adjusted R-squared 0.103747 S.D. dependent var 946.6032
S.E. of regression 896.1555 Akaike info criterion 16.47000
Sum squared resid 63444484 Schwarz criterion 16.55806
Log likelihood -672.2702 F-statistic 5.688110
Durbin-Watson stat 1.724296 Prob(F-statistic) 0.004922
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The probability value of >5% indicates that H0 of unit root present is accepted at 5%
critical level and therefore, ADF Unit root test is done again on first difference with the
following outcome:
Null Hypothesis: D(DEMAND) has a unit root
Exogenous: Constant
Lag Length: 9 (Automatic based on SIC, MAXLAG=11)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -7.571975 0.0000
Test critical values: 1% level -3.524233
5% level -2.902358
10% level -2.588587
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(DEMAND,2)
Method: Least Squares
Date: 06/23/13 Time: 15:08
Sample (adjusted): 2001M03 2007M02
Included observations: 72 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
D(DEMAND(-1)) -4.811954 0.635495 -7.571975 0.0000
D(DEMAND(-1),2) 3.506694 0.574176 6.107346 0.0000
D(DEMAND(-2),2) 3.219195 0.511100 6.298564 0.0000
D(DEMAND(-3),2) 2.809151 0.456585 6.152531 0.0000
D(DEMAND(-4),2) 2.374129 0.408596 5.810456 0.0000
D(DEMAND(-5),2) 2.039335 0.353954 5.761583 0.0000
D(DEMAND(-6),2) 1.665176 0.295530 5.634549 0.0000
D(DEMAND(-7),2) 1.228536 0.244074 5.033460 0.0000
D(DEMAND(-8),2) 0.815078 0.190846 4.270875 0.0001
D(DEMAND(-9),2) 0.463414 0.121058 3.828023 0.0003
C 579.9569 122.9400 4.717396 0.0000
R-squared 0.692708 Mean dependent var -1.263889
Adjusted R-squared 0.642332 S.D. dependent var 1357.426
S.E. of regression 811.8135 Akaike info criterion 16.37618
Sum squared resid 40201512 Schwarz criterion 16.72401
Log likelihood -578.5425 F-statistic 13.75081
Durbin-Watson stat 1.980756 Prob(F-statistic) 0.000000
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The H0 of unit root is now rejected and therefore, the series is considered I(1)
Checking the correlogram with first difference series for
trend and seasonality:
Trend component is added in the first difference series series dd=d(demand, 1,0) and
correlogram checked with following outcome:
Date: 06/23/13 Time: 15:14
Sample: 2000M04 2008M07
Included observations: 82
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
. | . | . | . | 1 0.030 0.030 0.0747 0.785
. | . | . | . | 2 0.016 0.015 0.0967 0.953
.*| . | .*| . | 3 -0.111 -0.112 1.1721 0.760
**| . | **| . | 4 -0.199 -0.195 4.6597 0.324
.*| . | .*| . | 5 -0.112 -0.106 5.7834 0.328
.*| . | .*| . | 6 -0.134 -0.147 7.4030 0.285
.*| . | **| . | 7 -0.165 -0.229 9.9096 0.194
.*| . | **| . | 8 -0.131 -0.243 11.508 0.175
. | . | **| . | 9 -0.042 -0.200 11.675 0.232
.*| . | ***| . | 10 -0.094 -0.353 12.528 0.251
. |*** | . |*. | 11 0.352 0.085 24.557 0.011
. |** | . |*. | 12 0.252 0.079 30.822 0.002
. |** | . |*. | 13 0.260 0.167 37.577 0.000
. | . | . | . | 14 0.003 -0.016 37.578 0.001
.*| . | . | . | 15 -0.123 -0.046 39.137 0.001
.*| . | .*| . | 16 -0.151 -0.090 41.512 0.000
.*| . | . | . | 17 -0.099 -0.020 42.558 0.001
.*| . | . | . | 18 -0.090 -0.005 43.427 0.001
. | . | . |*. | 19 0.006 0.169 43.431 0.001
.*| . | . | . | 20 -0.129 -0.036 45.292 0.001
**| . | .*| . | 21 -0.195 -0.146 49.592 0.000
. |*. | . | . | 22 0.103 -0.018 50.814 0.000
. |*. | . | . | 23 0.098 -0.053 51.932 0.001
. |** | . | . | 24 0.308 0.057 63.234 0.000
. |*. | . | . | 25 0.156 0.035 66.175 0.000
. | . | . | . | 26 -0.022 -0.049 66.234 0.000
.*| . | . | . | 27 -0.084 -0.038 67.111 0.000
. | . | . | . | 28 -0.044 0.062 67.359 0.000
.*| . | . |*. | 29 -0.074 0.075 68.075 0.000
.*| . | .*| . | 30 -0.083 -0.096 68.979 0.000
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. | . | . | . | 31 -0.050 -0.055 69.315 0.000
.*| . | . | . | 32 -0.110 0.009 70.994 0.000
.*| . | . | . | 33 -0.077 -0.039 71.830 0.000
. |*. | . |*. | 34 0.079 0.173 72.732 0.000
. | . | .*| . | 35 0.061 -0.061 73.273 0.000
. |** | . | . | 36 0.221 0.034 80.603 0.000
Nothing can be derived from the correlogram regarding AR and MA components so
trying with the seasonal component (with 12 as seasonality): Series dd=d(demand,0,12)
Date: 06/23/13 Time: 15:15
Sample: 2000M04 2008M07
Included observations: 71
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
. |**** | . |**** | 1 0.486 0.486 17.479 0.000
. |**** | . |** | 2 0.475 0.313 34.414 0.000
. |*** | . |*. | 3 0.380 0.101 45.423 0.000
. |*** | . | . | 4 0.340 0.063 54.342 0.000
. |** | . | . | 5 0.303 0.050 61.572 0.000
. |** | . | . | 6 0.222 -0.041 65.492 0.000
. |*. | .*| . | 7 0.088 -0.166 66.118 0.000
. |*. | . | . | 8 0.132 0.054 67.557 0.000
. |** | . |** | 9 0.246 0.266 72.615 0.000
. |*. | .*| . | 10 0.072 -0.152 73.050 0.000
. |*. | . |*. | 11 0.174 0.085 75.652 0.000.*| . | ***| . | 12 -0.142 -0.398 77.429 0.000
. | . | . |** | 13 0.062 0.201 77.767 0.000
. | . | . | . | 14 0.043 0.058 77.939 0.000
. | . | . |*. | 15 0.051 0.093 78.178 0.000
. | . | . |*. | 16 0.031 0.072 78.271 0.000
. |*. | . |*. | 17 0.096 0.069 79.151 0.000
. |*. | . |*. | 18 0.141 0.072 81.100 0.000
. |*. | . | . | 19 0.195 -0.022 84.903 0.000
. |*. | . | . | 20 0.177 -0.053 88.092 0.000
. | . | . | . | 21 0.008 -0.047 88.098 0.000
. |*. | . | . | 22 0.145 0.001 90.326 0.000
. |*. | . |*. | 23 0.071 0.124 90.864 0.000
. |*. | .*| . | 24 0.125 -0.146 92.584 0.000
. | . | . |*. | 25 0.063 0.095 93.038 0.000
. |*. | . | . | 26 0.092 0.020 94.015 0.000
. | . | .*| . | 27 0.058 -0.062 94.406 0.000
. |*. | . | . | 28 0.106 -0.028 95.760 0.000
. | . | .*| . | 29 0.021 -0.084 95.812 0.000
.*| . | .*| . | 30 -0.123 -0.158 97.721 0.000
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.*| . | .*| . | 31 -0.151 -0.118 100.68 0.000
**| . | .*| . | 32 -0.237 -0.112 108.16 0.000
This correlogram shows an indication of AR(2), SMA(12) process. Confirming the
significance of ar(1), ar(2) and sma(12) using regression
Confirming AR and MA components from the
correlogram:
ls d(demand,0,12) c ar(1) ar(2) sma(12)
Dependent Variable: D(DEMAND,0,12)
Method: Least Squares
Date: 06/23/13 Time: 15:18
Sample (adjusted): 2001M06 2007M02
Included observations: 69 after adjustmentsConvergence achieved after 12 iterations
Backcast: 2000M06 2001M05
Variable Coefficient Std. Error t-Statistic Prob.
C 1701.522 214.0576 7.948896 0.0000
AR(1) 0.471909 0.118750 3.973966 0.0002
AR(2) 0.287270 0.117415 2.446625 0.0171
MA(12) -0.854022 0.041146 -20.75597 0.0000
R-squared 0.605499 Mean dependent var 1484.377
Adjusted R-squared 0.587291 S.D. dependent var 1103.399
S.E. of regression 708.8503 Akaike info criterion 16.02139
Sum squared resid 32660470 Schwarz criterion 16.15090
Log likelihood -548.7379 F-statistic 33.25499
Durbin-Watson stat 2.055818 Prob(F-statistic) 0.000000
Inverted AR Roots .82 -.35
Inverted MA Roots .99 .85+.49i .85-.49i .49+.85i
.49-.85i -.00-.99i -.00+.99i -.49-.85i
-.49+.85i -.85+.49i -.85-.49i -.99
All 3 processes are showing significance at 5% confidence level. Confirming whether the
Residual is white Noise of not.
Date: 06/23/13 Time: 15:20
Sample: 2001M06 2007M02
Included observations: 69
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Q-statisticprobabilities
adjusted for 3 ARMAterm(s)
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
. | . | . | . | 1 -0.043 -0.043 0.1322
.*| . | .*| . | 2 -0.075 -0.077 0.5457
. | . | . | . | 3 0.052 0.046 0.7499
. | . | . | . | 4 -0.022 -0.024 0.7863 0.375
. | . | . | . | 5 0.006 0.012 0.7892 0.674
. | . | . | . | 6 -0.029 -0.035 0.8565 0.836
**| . | **| . | 7 -0.203 -0.205 4.1276 0.389
. | . | . | . | 8 -0.011 -0.038 4.1382 0.530
. |*. | . |*. | 9 0.138 0.114 5.6902 0.459
.*| . | .*| . | 10 -0.097 -0.075 6.4719 0.486
. |*. | . |** | 11 0.182 0.199 9.2745 0.320
.*| . | .*| . | 12 -0.119 -0.145 10.488 0.312
. | . | . |*. | 13 0.043 0.070 10.650 0.385
. | . | . | . | 14 0.037 -0.051 10.772 0.463
. | . | . |*. | 15 0.035 0.073 10.885 0.539
. | . | . | . | 16 -0.047 -0.016 11.092 0.603
.*| . | .*| . | 17 -0.066 -0.078 11.508 0.646
. | . | . | . | 18 -0.029 0.004 11.587 0.710
. |*. | . |*. | 19 0.133 0.125 13.319 0.649
. | . | . | . | 20 0.000 -0.044 13.319 0.715
**| . | .*| . | 21 -0.227 -0.150 18.556 0.420
. | . | .*| . | 22 0.016 -0.075 18.583 0.484
.*| . | .*| . | 23 -0.109 -0.108 19.848 0.467
. | . | .*| . | 24 -0.007 -0.079 19.854 0.530
. | . | . |*. | 25 0.047 0.076 20.096 0.577
. | . | . |*. | 26 0.025 0.080 20.169 0.632
. | . | . | . | 27 -0.017 -0.006 20.202 0.685
. |*. | . | . | 28 0.136 0.050 22.401 0.612
Confirmed that the residuals components are white noise; so proceeding with forecasting:
Forecasting using the model:
Outcome of the static forecasting:
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18000
20000
22000
24000
26000
28000
30000
32000
34000
2001 2002 2003 2004 2005 2006
DEMANDF
Forecast: DEMANDF
Actual: DEMAND
Forecast sample: 2000M04 2008M07
Adjusted sample: 2001M06 2007M03
Included observations: 69
Root Mean Squared Error 687.9972
Mean Absolute Error 588.4692
Mean Abs. Percent Error 2.420978
Theil Inequality Coefficient 0.013758
Bias Proportion 0.013140
Variance Proportion 0.020256
Covariance Proportion 0.966604
Outcome of dynamic forecasting:
26000
28000
30000
32000
34000
36000
07M01 07M04 07M07 07M10 08M01 08M04 08M07
DEMANDF
Forecast: DEMANDF
Actual: DEMAND
Forecast sample: 2007M01 2008M07
Included observations: 2
Root Mean Squared Error 358.5803
Mean Absolute Error 358.5267
Mean Abs. Percent Error 1.239447
Theil Inequality Coefficient 0.006235
Bias Proportion 0.999701
Variance Proportion 0.000299
Covariance Proportion 0.000000
MAPE is between 1% and 3%, which is good enough so model can be used for forecasting.
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3. VARInput Data Used sensex_ex_oil_VAR.xls:
sensex_ex_oil_VAR.xls
Stationarity test for all the three series:
ADF Unit Root Test for Sensex:
Null Hypothesis: SEN has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=19)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -1.535520 0.8165
Test critical values: 1% level -3.972503
5% level -3.416877
10% level -3.130796
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(SEN)Method: Least Squares
Date: 06/23/13 Time: 15:54
Sample (adjusted): 2 639
Included observations: 638 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
SEN(-1) -0.008067 0.005254 -1.535520 0.1252
C 134.9426 66.52498 2.028450 0.0429
@TREND(1) 0.037049 0.087721 0.422351 0.6729
R-squared 0.007247 Mean dependent var 12.91376
Adjusted R-squared 0.004121 S.D. dependent var 229.7398
S.E. of regression 229.2660 Akaike info criterion 13.71233
Sum squared resid 33377444 Schwarz criterion 13.73330
Log likelihood -4371.235 F-statistic 2.317821
Durbin-Watson stat 1.896014 Prob(F-statistic) 0.099321
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Outcome with lag 2
Vector Autoregression Estimates
Date: 06/23/13 Time: 15:59
Sample (adjusted): 4 639
Included observations: 636 after adjustments
Standard errors in ( ) & t-statistics in [ ]
DLOG(SEN) DLOG(OIL) DLOG(EX)
DLOG(SEN(-1)) 0.062916 0.041426 -0.007926
(0.03938) (0.05878) (0.01270)
[ 1.59786] [ 0.70476] [-0.62406]
DLOG(SEN(-2)) -0.047235 0.069754 -0.032615
(0.03926) (0.05861) (0.01266)
[-1.20309] [ 1.19016] [-2.57540]
DLOG(OIL(-1)) 0.027833 -0.046923 0.000759
(0.02660) (0.03970) (0.00858)
[ 1.04652] [-1.18187] [ 0.08852]
DLOG(OIL(-2)) -0.046258 -0.019528 -0.012440
(0.02550) (0.03807) (0.00823)
[-1.81390] [-0.51295] [-1.51229]
DLOG(EX(-1)) -0.043623 -0.018486 -0.032198
(0.12249) (0.18286) (0.03951)
[-0.35613] [-0.10110] [-0.81489]
DLOG(EX(-2)) -0.461301 0.051077 -0.061351
(0.12216) (0.18236) (0.03940)
[-3.77614] [ 0.28008] [-1.55695]
C 0.000873 0.001302 -4.56E-05
(0.00063) (0.00094) (0.00020)
[ 1.39264] [ 1.39148] [-0.22567]
R-squared 0.033549 0.005954 0.018689
Adj. R-squared 0.024330 -0.003529 0.009328
Sum sq. resids 0.154639 0.344605 0.016089
S.E. equation 0.015680 0.023406 0.005058
F-statistic 3.639180 0.627866 1.996506
Log likelihood 1743.907 1489.092 2463.518
Akaike AIC -5.461973 -4.660667 -7.724899
Schwarz SC -5.412938 -4.611632 -7.675864
Mean dependent 0.000889 0.001300 -9.69E-05
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S.D. dependent 0.015874 0.023365 0.005081
Determinant resid covariance (dof adj.) 3.43E-12
Determinant resid covariance 3.32E-12
Log likelihood 5698.165
Akaike information criterion -17.85272Schwarz criterion -17.70561
The lag length criteria output is as follows:
VAR Lag Order Selection CriteriaEndogenous variables: DLOG(SEN) DLOG(OIL)DLOG(EX)
Exogenous variables: C
Date: 06/23/13 Time: 16:01
Sample: 1 639Included observations: 630
Lag LogL LR FPE AIC SC HQ
0 5655.428 NA 3.23e-12 -17.94421 -17.92304* -17.93599*
1 5658.995 7.088918 3.29e-12 -17.92697 -17.84229 -17.89408
2 5675.220 32.09070 3.21e-12 -17.94991 -17.80172 -17.89234
3 5697.734 44.31307* 3.08e-12* -17.99281* -17.78111 -17.91058
4 5703.470 11.23404 3.11e-12 -17.98244 -17.70723 -17.87554
5 5711.659 15.96324 3.12e-12 -17.97987 -17.64115 -17.84830
6 5715.384 7.225424 3.17e-12 -17.96312 -17.56089 -17.80689
7 5721.169 11.16626 3.20e-12 -17.95292 -17.48718 -17.77201
8 5727.525 12.20586 3.23e-12 -17.94452 -17.41527 -17.73895
* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level)
FPE: Final prediction error
AIC: Akaike information criterion
SC: Schwarz information criterion
HQ: Hannan-Quinn information criterion
The lag length 3 is selected by the criterion and therefore, it will be used for furtheranalysis.
Cointegration Test: The Johansen and Jeselius cointegration test is performed to validate the existence of
any cointegration between variables with the following outcome:
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Date: 06/23/13 Time: 16:06
Sample (adjusted): 6 639
Included observations: 634 after adjustments
Trend assumption: Linear deterministic trend
Series: DLOG(SEN) DLOG(OIL) DLOG(EX)
Lags interval (in first differences): 1 to 3
Unrestricted Cointegration Rank Test (Trace)
Hypothesized Trace 0.05
No. of CE(s) Eigenvalue Statistic Critical Value Prob.**
None * 0.272322 472.1336 29.79707 0.0001
At most 1 * 0.234106 270.5869 15.49471 0.0001
At most 2 * 0.147925 101.4914 3.841466 0.0000
Trace test indicates 3 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized Max-Eigen 0.05
No. of CE(s) Eigenvalue Statistic Critical Value Prob.**
None * 0.272322 201.5467 21.13162 0.0001
At most 1 * 0.234106 169.0955 14.26460 0.0001
At most 2 * 0.147925 101.4914 3.841466 0.0000
Max-eigenvalue test indicates 3 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):
DLOG(SEN) DLOG(OIL) DLOG(EX)
-106.7013 16.58454 -290.3165
12.35526 88.88217 94.37331
69.51591 11.45247 -287.3705
Unrestricted Adjustment Coefficients (alpha):
D(DLOG(SEN)) 0.007827 -0.001086 -0.003048
D(DLOG(OIL)) -0.003808 -0.012067 -0.001340
D(DLOG(EX)) 0.001309 -0.000759 0.001783
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1 Cointegrating Equation(s): Log likelihood 5590.689
Normalized cointegrating coefficients (standard error in parentheses)
DLOG(SEN) DLOG(OIL) DLOG(EX)
1.000000 -0.155430 2.720834
(0.05594) (0.25062)
Adjustment coefficients (standard error in parentheses)
D(DLOG(SEN)) -0.835138
(0.06505)
D(DLOG(OIL)) 0.406332
(0.11089)
D(DLOG(EX)) -0.139619
(0.02283)
2 Cointegrating Equation(s): Log likelihood 5675.236
Normalized cointegrating coefficients (standard error in parentheses)
DLOG(SEN) DLOG(OIL) DLOG(EX)
1.000000 0.000000 2.824833
(0.25049)
0.000000 1.000000 0.669108
(0.32764)
Adjustment coefficients (standard error in parentheses)
D(DLOG(SEN)) -0.848561 0.033247
(0.06532) (0.05498)
D(DLOG(OIL)) 0.257236 -1.135738(0.09882) (0.08318)
D(DLOG(EX)) -0.149002 -0.045799
(0.02275) (0.01915)
The H0 of Cointegration is rejected for all 3 possible pairs for existence of cointegration.
The unrestricted VAR can, therefore, be applied now.
Granger Causality Test:
VAR Granger Causality/Block Exogeneity Wald Tests
Date: 06/23/13 Time: 16:12
Sample: 1 639
Included observations: 635
Dependent variable: DLOG(SEN)
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Excluded Chi-sq df Prob.
DLOG(OIL) 5.287301 3 0.1519
DLOG(EX) 49.19361 3 0.0000
All 54.27170 6 0.0000
Dependent variable: DLOG(OIL)
Excluded Chi-sq df Prob.
DLOG(SEN) 1.856538 3 0.6027
DLOG(EX) 0.518338 3 0.9148
All 2.386966 6 0.8809
Dependent variable: DLOG(EX)
Excluded Chi-sq df Prob.
DLOG(SEN) 9.790470 3 0.0204
DLOG(OIL) 4.567472 3 0.2064
All 13.33178 6 0.0381
The granger causality test with H0 of non-causality running from the independent
variable DLOG(EX) to the dependent variable DLOG(SEN) has been rejected and
therefore, it is confirmed that Exchange Rate movement Granger causes Sensex
movement.
Similarly, the H0 of non-causality running from the independent variable DLOG(SEN) to
the dependent variable DLOG(EX) has been rejected and therefore, it is confirmed that
Sensex movement Granger causes Exchange rate movement.
In other words, there is a bidirectional causality that exists between the sensex and the
exchange rate movement.
Forecasting can be done now using the above VAR model with lag structure of 3 between
Sensex and Exchange Rate.
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