EXERCISE ON COINTEGRATION

Hands-on-Session
Market Integration, Price Discovery, and Volatility in Agricultural Commodity Futures
P.Ramasundaram* and Sendhil R**
*National Coordinator (M&E), National Agricultural Innovation Project (NAIP), Krishi Anusandhan Bhavan II,
Pusa, New Delhi - 110 012, India.
**Scientist, Directorate of Wheat Research, Karnal – 132001, Haryana, India.
Cointegration: If a linear combination of two non-stationary time series data results in a
stationary error term, then the two series are cointegrated
Price Discovery: It is a continuous process of arriving at a price at which a person buys
and another sells a futures contract (commodity) in a commodity exchange
Volatility: Volatility is an uncertain movement of a random variable over time
Steps in Cointegration Test
1. Collect the time series data
2. Convert to natural logarithm
3. Check the original time series for unit root test
(Augmented Dickey Fuller or Phillips Perron)
4. Check the first differenced series for unit root
5. Run the cointegration test
6. For cointegrated series, run the error correction model
7. Interpret the coefficients
Steps in GARCH Model
1. Collect the time series data
2. Convert to natural logarithm
3. Take first difference (2 and 3 can be performed as a single EVIEWS command)
4. Run ARIMA filtration analysis and find the AR value
5. Run GARCH (Trial and Error). Choose the best fit model
6. Interpret the coefficients
A. Cointegration: Illustrated Example in EVIEWS
When a time series model is estimated, the first thing to make sure is that either all
time series variables in the model are stationary or they are cointegrated, which means that
they are integrated of the same order and errors are stationary, in which case the model
defines a long run equilibrium relationship among the cointegrated variables. Therefore, a
cointegration test generally takes two steps. The first step is to conduct a unit root test on
each variable to find the order of integration. If all variables are integrated of the same
order, the second step is to estimate the model, also called a “cointegrating equation,” and
test whether the residual of the model is stationary. The purpose of this exercise is to
implement the cointegration test in EVIEWS and also estimate the error correction model.
Hands-on-Session
Spot and Futures Prices
For this exercise time series data on wheat futures price (FP) of contract ending
March 2010 and spot price (SP_Karnal) of Karnal market are used for illustrative purpose
(www.ncdex.com). Since the fundamentals evolve over time, the prices changes over time.
It may or may not be stationary (checked by unit root test).
Date
10-Sep-09
11-Sep-09
12-Sep-09
14-Sep-09
15-Sep-09
16-Sep-09
17-Sep-09
18-Sep-09
19-Sep-09
21-Sep-09
22-Sep-09
23-Sep-09
24-Sep-09
25-Sep-09
26-Sep-09
28-Sep-09
29-Sep-09
30-Sep-09
1-Oct-09
3-Oct-09
5-Oct-09
6-Oct-09
7-Oct-09
8-Oct-09
9-Oct-09
10-Oct-09
12-Oct-09
13-Oct-09
14-Oct-09
15-Oct-09
16-Oct-09
17-Oct-09
19-Oct-09
20-Oct-09
21-Oct-09
22-Oct-09
23-Oct-09
24-Oct-09
26-Oct-09
27-Oct-09
28-Oct-09
29-Oct-09
30-Oct-09
31-Oct-09
2-Nov-09
3-Nov-09
4-Nov-09
5-Nov-09
FP
1234.80
1234.80
1198.00
1198.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1190.00
1220.20
1223.60
1192.40
1209.00
1236.00
1236.00
1249.20
1249.20
1249.20
1249.20
1249.20
1242.80
1242.80
1256.40
1276.60
1276.60
1276.60
1277.40
1305.20
1336.80
1349.00
1349.00
1349.00
1378.40
1352.20
1358.20
SP_Karnal
1132.50
1140.00
1137.50
1130.00
1132.50
1157.50
1130.00
1132.50
1130.00
1130.00
1135.00
1140.00
1135.00
1142.50
1125.00
1125.00
1130.00
1130.00
1135.00
1135.00
1140.00
1132.50
1132.50
1140.00
1142.50
1136.00
1167.50
1167.50
1135.00
1125.00
1200.00
1200.00
1200.00
1180.00
1165.00
1145.00
1260.00
1275.00
1250.00
1337.50
1300.00
1250.00
1305.00
1325.00
1325.00
1300.00
1300.00
1350.00
Log_FP
7.12
7.12
7.09
7.09
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.08
7.11
7.11
7.08
7.10
7.12
7.12
7.13
7.13
7.13
7.13
7.13
7.13
7.13
7.14
7.15
7.15
7.15
7.15
7.17
7.20
7.21
7.21
7.21
7.23
7.21
7.21
Log_SP_Karnal
7.03
7.04
7.04
7.03
7.03
7.05
7.03
7.03
7.03
7.03
7.03
7.04
7.03
7.04
7.03
7.03
7.03
7.03
7.03
7.03
7.04
7.03
7.03
7.04
7.04
7.04
7.06
7.06
7.03
7.03
7.09
7.09
7.09
7.07
7.06
7.04
7.14
7.15
7.13
7.20
7.17
7.13
7.17
7.19
7.19
7.17
7.17
7.21
Hands-on-Session
6-Nov-09
7-Nov-09
9-Nov-09
10-Nov-09
11-Nov-09
12-Nov-09
13-Nov-09
14-Nov-09
16-Nov-09
17-Nov-09
18-Nov-09
19-Nov-09
20-Nov-09
21-Nov-09
23-Nov-09
24-Nov-09
25-Nov-09
26-Nov-09
27-Nov-09
30-Nov-09
1-Dec-09
2-Dec-09
3-Dec-09
4-Dec-09
5-Dec-09
7-Dec-09
8-Dec-09
9-Dec-09
10-Dec-09
11-Dec-09
12-Dec-09
14-Dec-09
15-Dec-09
16-Dec-09
17-Dec-09
18-Dec-09
19-Dec-09
21-Dec-09
22-Dec-09
23-Dec-09
24-Dec-09
26-Dec-09
28-Dec-09
29-Dec-09
30-Dec-09
31-Dec-09
1-Jan-10
2-Jan-10
4-Jan-10
5-Jan-10
6-Jan-10
7-Jan-10
8-Jan-10
9-Jan-10
11-Jan-10
12-Jan-10
1353.60
1354.60
1335.20
1307.20
1336.80
1337.20
1317.60
1304.80
1324.20
1343.20
1361.80
1372.00
1357.60
1365.00
1365.00
1370.20
1351.20
1360.00
1358.80
1369.00
1355.20
1355.20
1361.40
1366.00
1367.60
1375.20
1358.00
1363.80
1339.20
1344.00
1342.40
1334.00
1297.40
1276.40
1279.00
1278.40
1297.60
1275.40
1270.80
1266.20
1280.00
1290.60
1284.80
1280.00
1286.00
1285.20
1278.40
1296.80
1291.60
1292.00
1294.00
1299.00
1313.40
1322.80
1327.80
1341.00
1400.00
1400.00
1400.00
1400.00
1412.50
1425.00
1400.00
1405.50
1405.50
1400.00
1400.00
1407.50
1400.00
1400.00
1400.00
1400.00
1390.00
1392.50
1400.00
1400.00
1395.00
1400.00
1390.00
1390.00
1390.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1400.00
1390.00
1390.00
1400.00
1370.00
1385.00
1390.00
1390.00
1385.00
1392.50
1395.00
1395.00
1375.00
1390.00
1387.50
1412.50
1400.00
1400.00
1370.00
1370.00
7.21
7.21
7.20
7.18
7.20
7.20
7.18
7.17
7.19
7.20
7.22
7.22
7.21
7.22
7.22
7.22
7.21
7.22
7.21
7.22
7.21
7.21
7.22
7.22
7.22
7.23
7.21
7.22
7.20
7.20
7.20
7.20
7.17
7.15
7.15
7.15
7.17
7.15
7.15
7.14
7.15
7.16
7.16
7.15
7.16
7.16
7.15
7.17
7.16
7.16
7.17
7.17
7.18
7.19
7.19
7.20
7.24
7.24
7.24
7.24
7.25
7.26
7.24
7.25
7.25
7.24
7.24
7.25
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.24
7.22
7.23
7.24
7.24
7.23
7.24
7.24
7.24
7.23
7.24
7.24
7.25
7.24
7.24
7.22
7.22
Hands-on-Session
13-Jan-10
14-Jan-10
15-Jan-10
16-Jan-10
18-Jan-10
19-Jan-10
20-Jan-10
21-Jan-10
22-Jan-10
23-Jan-10
25-Jan-10
27-Jan-10
28-Jan-10
29-Jan-10
30-Jan-10
1-Feb-10
2-Feb-10
3-Feb-10
4-Feb-10
5-Feb-10
6-Feb-10
8-Feb-10
9-Feb-10
10-Feb-10
11-Feb-10
12-Feb-10
13-Feb-10
15-Feb-10
16-Feb-10
17-Feb-10
18-Feb-10
19-Feb-10
20-Feb-10
22-Feb-10
23-Feb-10
24-Feb-10
25-Feb-10
26-Feb-10
27-Feb-10
1-Mar-10
2-Mar-10
3-Mar-10
4-Mar-10
5-Mar-10
6-Mar-10
8-Mar-10
9-Mar-10
10-Mar-10
11-Mar-10
12-Mar-10
13-Mar-10
15-Mar-10
16-Mar-10
17-Mar-10
18-Mar-10
19-Mar-10
1322.80
1325.40
1317.00
1310.60
1296.00
1291.80
1309.00
1295.80
1290.00
1295.20
1285.60
1278.40
1271.40
1279.20
1281.00
1285.00
1292.00
1285.80
1281.00
1286.00
1290.60
1284.00
1285.40
1282.80
1289.60
1289.60
1306.60
1304.20
1309.20
1320.40
1314.80
1312.80
1319.60
1286.60
1287.80
1275.00
1293.20
1290.80
1284.20
1284.20
1266.20
1263.80
1253.00
1260.40
1262.60
1241.60
1220.20
1224.80
1229.80
1234.00
1241.60
1260.00
1266.60
1271.80
1272.80
1279.40
1390.00
1400.00
1400.00
1400.00
1415.00
1407.50
1400.00
1407.50
1400.00
1425.00
1380.00
1400.00
1375.00
1385.00
1410.00
1385.00
1370.00
1385.00
1387.50
1385.00
1400.00
1370.00
1400.00
1390.00
1400.00
1400.00
1392.50
1385.00
1385.00
1387.50
1375.00
1400.00
1400.00
1385.00
1385.00
1370.00
1385.00
1362.50
1387.50
1387.50
1385.00
1385.00
1375.00
1375.00
1350.00
1345.00
1347.50
1320.00
1312.50
1305.00
1300.00
1315.00
1337.50
1300.00
1305.00
1300.00
7.19
7.19
7.18
7.18
7.17
7.16
7.18
7.17
7.16
7.17
7.16
7.15
7.15
7.15
7.16
7.16
7.16
7.16
7.16
7.16
7.16
7.16
7.16
7.16
7.16
7.16
7.18
7.17
7.18
7.19
7.18
7.18
7.19
7.16
7.16
7.15
7.16
7.16
7.16
7.16
7.14
7.14
7.13
7.14
7.14
7.12
7.11
7.11
7.11
7.12
7.12
7.14
7.14
7.15
7.15
7.15
7.24
7.24
7.24
7.24
7.25
7.25
7.24
7.25
7.24
7.26
7.23
7.24
7.23
7.23
7.25
7.23
7.22
7.23
7.24
7.23
7.24
7.22
7.24
7.24
7.24
7.24
7.24
7.23
7.23
7.24
7.23
7.24
7.24
7.23
7.23
7.22
7.23
7.22
7.24
7.24
7.23
7.23
7.23
7.23
7.21
7.20
7.21
7.19
7.18
7.17
7.17
7.18
7.20
7.17
7.17
7.17
Hands-on-Session
In this exercise, unit root test and the integration between spot and futures prices are
examined for the above dataset.
1. Collect the time series data. Save as a single sheet in Excel 2003 format for
compatibility purpose. Recent versions of EVIEWS like EVIEWS 7 (Enterprise
Edition) will take any excel extension.
2. Import the original data to EVIEWS just by dragging it to the software window or
copy and paste or menu driven
3.
Convert the original data to log values in excel and then import to EVIEWS or
import the original data to EVIEWS and then convert to log values with the following
command. Transform the SP into its natural log by Genr LSP = log(SP), and
similarly transform FP also into its natural log and save the newly generated log
variables.
4. Do the unit root test. The first step of testing cointegration is to test all the time
series variables for stationarity. Therefore, conduct the augmented Dickey Fuller
unit root test or Phillips Perron test on each of the series: LFP and LSP_Karnal,
and verify that each of these series is integrated of order one. Check the graphs too.
Hands-on-Session
In the unit root test of levels, always include intercept and also time trend if the data
has a trend. In the unit root test of first differences, include only the intercept
Note: View – Unit Root Test – Choose the Option – OK
5. Now carry out the cointegration test. If two time series variables are nonstationary,
but cointegrated, at any point in time the two variables may drift apart, but there will
always be a tendency for them to retain a reasonable proximity to each other. There
may be more than one cointegrating relationship among cointegrated variables.
Johansen test provides estimates of all such cointegrating equations and provides a
test statistic for the number of cointegrating equations.1
1
It is a likelihood ratio test statistic that Johansen test presents along with the critical values.
Hands-on-Session
Note: Open the Log Transformed Data – View – Cointegration Test - OK.
A Johansen cointegration test window appears. Choose linear deterministic trend in
data, select Intercept (no trend) in CE and test VAR2 (Option No. 3). Specify the
appropriate number of lag intervals (1 1 in our case, i.e., 1 lag and 1 4 if the series is
quarterly)3. Finally, if there is any truly exogenous variable it has to be specified, other than
the intercept and the time trend, included in the model. In the present illustration there is no
such exogenous variable; and therefore, do not enter any name for exogenous series. The
Johansen test uses the VAR method, in which all cointegrated series are considered
endogenous. Click OK to get the cointegration test result.
In the very first table of this result, start from the first row and compare the likelihood
ratio (LR) value or trace statistic with the 5 percent critical value. If the value exceeds the
critical value, go down to the next row and compare the value with the critical value in that
row. Repeat this process until you reach the row in which the trace statistic is lower than
the critical value. Stop at that row; do not move down any further. The last column in this
row gives you the number of cointegrating equations for the integrated variables, and at the
bottom of this table, the conclusion of the test, as to how many cointegrating equations are
indicated, is stated.
Below the likelihood ratio test table, there would be a number of other tables. Only
look for the table(s) that has normalized cointegrating coefficients, in which the coefficient of
one of the two variables is normalized to one. There may be more than one table with
normalized coefficients (in case of more than two variables). If the above mentioned LR test
indicates one cointegrating equation, look at the first normalized coefficient table only. If
the test indicates two cointegrating equations, look at the second normalized coefficient
table, and so on. A normalized coefficient table presents the estimate of the model
(cointegrating equation) with all variables taken to the left hand side. Below each coefficient
estimate, the standard error is given within parentheses. The ratio of the coefficient to its
standard error is the t-statistic.
2
The test allows a choice among three options regarding the deterministic time trend of data: no trend, linear
trend and quadratic trend. Select the appropriate nature of trend.
3
The appropriate lag length may be decided through the AIC or SIC criterion.
Hands-on-Session
Estimation of Error Correction Model
According to the Granger representation theorem, when variables are cointegrated,
there must also be an error correction model (ECM) that describes the short-run dynamics
or adjustments of the cointegrated variables towards their equilibrium values. ECM consists
of one-period lagged cointegrating equation and the lagged first differences of the
endogenous variables. Using the Vector Autoregression (VAR) method, ECM can be
estimated.
The model involves two nonstationary variables; therefore, ECM would be a
simultaneous equation system of two equations, one for each variable describing the short
run adjustment of that variable towards the long run equilibrium. The adjustment process
may take a number of periods and thus each equation in the ECM will have lagged
variables. It is important to include the appropriate number of lags.
Note: Select the Variables - Open - as VAR. A new window on Vector Autoregression
appears. Under VAR specification, click on Vector Error Correction, type in lag intervals
1 1 to allow for one period lag length, check that sample period is correct (if necessary,
correct it), type in endogenous variables (which would be all the series in this illustration),
type in any exogenous variable (none in this case, leave it blank), choose the trend in the
cointegrating equation, as was done above for the Johansen test (VAR assumes linear
trend in data: intercept (no trend in CE), type in the number of CE’s (1 in our case), and
click OK.4 The ECM estimates will appear immediately.
The first table presents the estimates of the cointegrating equation, and the second
table presents the rest of the ECM. The first row in the second table presents the estimates
of the speed of adjustment coefficient for each variable, their standard errors and the
t-statistics. Present the results and interpret the coefficients.
4
See footnote three for the appropriate number of lags.
Hands-on-Session
B. GARCH: Illustrated Example in EVIEWS
For this exercise, time series data on wheat spot price (SP_Karnal) of Karnal market is
used for illustration. Since the economic fundamentals evolve over a period of time, prices
tends to be volatile (random movement) over time which can be captured by the GARCH
coefficients.
Note: Import the variable (SP_Karnal) for which volatility has to be measured. Now select
Quick – Estimate Equation from the menu. A new window will appear. Under Methods
select ARCH, a new window will appear again and in that type the order of ARCH and
GARCH coefficients. Type the dependent variable dlog(sp_karnal) in the space and then
click OK to get the GARCH estimates. Run different models and choose the best model
based on the AIC or LR criterion. Present the results and interpret the coefficients.
Hands-on-Session
Hands-on-Session
Exercise for Trainees
Table 1. Estimated ADF and PP statistic for unit root test in wheat
Futures market price
Test statistic
Contract period
st
Level
1 difference
ADF
10.09.09 to 19.03.10
PP
Spot market price
st
Level
1 difference
Order
Note: * indicates significance at one per cent of MacKinnon (1996) one-sided p-values
Table 2. Estimated AIC and SIC value for optimum lag length
Criteria
Contract period
Value
AIC
SIC
Order of lag length
10.09.09 to 19.03.10
Table 3. Estimates of Johansen’s cointegration test
Eigen
Contract period
Correlation
value
Trace
statistic
Null
hypothesis
Log
likelihood
10.09.09 to 19.03.10
Note: ***, ** and * denote the rejection of null hypothesis at 1, 5 and 10 per cent level of significance
^ indicates the significance of correlation coefficient at 1 per cent level of probability (2 tailed)
Table 4. Estimates of vector error correction model
Cointegration equation
Contract period
Constant
Coefficient
10.09.09 to 19.03.10
(0.1704)
Error correction estimates
Futures price
Spot price
(0.0340)
(0.0276)
Note: Figures in parentheses indicate the standard error
Table 5. Estimates of fitted GARCH model for wheat spot price
Particulars
Observations (days)
Standard deviation
C.V (%)
GARCH estimates
Estimates
GARCH fit order
Constant
Estimates of ARCH term (αi)
 2 t 1
 2 t 2
 2t 3
Estimates of GARCH term (βi)
 t21
 t22
 t23
Log likelihood
αi + βi
Volatility level
Note: ** Significant at 1 per cent level of probability (z statistic) and * Significant at 5 per cent level of probability (z statistic)
Hands-on-Session
Suggested Readings
Bollerslev, T. (1986). “Generalized autoregressive conditional heteroscedasticity”, Journal
of Econometrics, 31: 307-327.
Dickey, D and Fuller, W.A. (1979). “Distribution of the estimators for autoregressive time
series regressions with unit roots”, Journal of American Statistical Association, 74:
427-431.
Easwaran, S.R., and Ramasundaram, P., Whether the Commodity Futures in Agriculture
are Efficient in Price Discovery? - An Econometric Analysis. Agricultural Economics
Research Review, 2008, 21, 337-344.
Engle, R.F and Granger, C.W.J. (1987). “Cointegration and error-correction:
Representation, estimation and testing”, Econometrica, 55: 251-276.
Engle, R.F. (1982). “Autoregressive conditional heteroscedasticity with estimates of the
variance of United Kingdom inflation”, Econometrica, 50(4), 987-1007.
Fackler, P. (1996). “Spatial Price Analysis: A Methodological Review”, Mimeo.North
Carolina State University.
Garbade, K.D and Silber, W.L. (1982). “Price movements and price discovery in future and
cash markets”, Review of Economics and Statistics, 65: 289-297.
Garbade, K.D and Silber, W.L. (1983). “Dominant satellite relationship between live cattle
cash and futures markets”, The Journal of Futures Markets, 10(2): 123-136.
Goodwin, B.K and Schroeder, T.C. (1991). “Cointegration tests and spatial price linkages in
regional cattle markets”, American Journal of Agricultural Economics, 73: 452-64.
Granger, C. (1981). “Some properties of time series data and their use in econometric
model specification”, Journal of Econometrics, 16: 121-130.
Guida, T and Matringe, O. (2004). Application of GARCH models in forecasting the volatility
of Agricultural commodities, UNCTAD Publications.
Johansen, S. (1988). “Statistical analysis of cointegration vectors”, Journal of Economic
Dynamics and Control, 12: 231-254.
Johansen, S. (1991). “Estimation and hypothesis testing of cointegration vectors in
Gaussian vectors auto regression models”, Econometrica, 59: 1551-80.
Johansen, S. (1994). “The role of the constant and linear terms in cointegration analysis of
nonstationary variables”, Econometric Reviews, 13: 205-229.
Johansen, S. (1995). Likelihood-based interference in cointegrated vector autoregressive
models. Oxford: Oxford University Press.
Sendhil, R., Amit Kar, Mathur, V.C. and Jha, G.K., Price discovery, transmission and
volatility: Evidence from agricultural commodity futures. Agricultural Economics
Research Review, 2013, 26 (1): 41-54.
Singh, N.P., Kumar, R., Singh, R.P and Jain, P.K. (2005). “Is futures market mitigating price
risk: An exploration of wheat and maize market”, Agricultural Economics Research
Review, 18: 35-46.