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13. MODULE OBJECTIVE
This module attempts to explore the possibilities of correlation in cross sectional units of error
The estimated model by the application of OLS, discussed earlier, is based on the assumption
that there should not be any relationship among the error regressors. That is covariance between
two errors variables should equal to zero [i.e. Cov (Ui, Uj) = 0 for i ≠ j]. If this assumption is
violated, then there is chance of autocorrelation. It is otherwise called as serial correlation. So,
serial correlation occurs when the error in estimated econometric models are correlated.
In this module, we deal with the followings:
1. WHAT IS AUTOCORRELATION AND HOW IS ITS NATURE?
2. WHAT ARE ITS CONSEQUENCES?
3. DOES IT REALLY A PROBLEM?
4. DETECTION CRITERIA
5. CAUSES OF AUTOCORRELATION
6. REMEDIAL MEASURES
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WHAT IS AUTOCORRELATION?
In general, autocorrelation means the correlation among the error terms. If it is present in the
estimated model, it is the violation of OLS technique and hence, the estimated model cannot be
used for prediction and forecasting. The structure of auto correlation is as follows:
Yt = β0 + β1Xt + Ut and where Ut = ρUt‐1 + vt
If ρ = 0, then there is no serial correlation; otherwise, there is presence of serial correlation. The
range of ρ is between ‐1 and +1, indicating perfect negative and positive autocorrelation. So, if
ρ ≠ 0, it is autocorrelation and assumes that the error term follows the autoregressive scheme.
CONSEQUENCES OF AUTOCORRELATION
The estimation process requires that OLS applications of estimated parameters should follow the
BLUE theorem. If not, there is question on model reliability. In specific, the presence of
autocorrelation makes the estimated parameters highly volatile and their standard errors are
infinite. However, it will not affect the unbiasedness property; but affects minimum variance
DOES IT REALLY PROBLEM?
On the first instance, any estimated parameters whose value does not follow BLUE theorem
means it is really a problem. However, in the case of autocorrelation, it depends upon the
objective specification. If the objective is for prediction (or forecasting), then the existence of
autocorrelation (not in severe) is not a serious problem. But if the objective is model reliability,
then it is serious issue, even if it is at the minor level. So we assume that the disturbance term is
generated by a slightly different method and such error terms are also called the white noise error
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terms. If it is so, then there is no issue of serial correlation and estimated model can be used for
The detection of autocorrelation can be done only after the estimation process. So, first we
should have estimated model and then we can have the error term. Once we get the error term,
the process of detecting autocorrelation is feasible. The residuals in case of autocorrelation can
be calculated by plotting them in the time sequence plot or alternatively we can plot the
standardized residuals against time. Apart from these there are several quantitative tests that one
can apply in order to supplement the pure qualitative approach. These are as follows:
DURBIN WATSON ‘D’ TEST
BREUSCH- GODFREY TEST
VON-NEUMAN RATIO TEST
Among them, the most frequent used criteria to detect autocorrelation is
Here in case of detection of autocorrelation the most frequently used test is the Durbin
Watson‘d’ test. This can be analyzed as follows:
d = cov (ut, ut-1)/ var (ut)
With some simplification, we can have d = 2 (1-ρ)
If ρ = 0, d = 2 and the system has no autocorrelation;
If ρ = -1, d = 4 and the system has perfect negative autocorrelation;
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If ρ = 1, d = 0 and the system has perfect positive autocorrelation.
So, d varies from 0 - 4. But the value of d = 2 is the best for the estimated model.
The test, however, depends upon the following assumptions:
The errors follow the autoregressive model
There are no lagged dependent variables used as explanatory variables
There is an intercept in the original model
CAUSES OF AUTOCORRELATION
Interpolation or extrapolation
Misspecification of the random term
An over determined model
An under-determined model
Lag explanatory variables
Wrong data transformation
Manipulation of data
Presence of lagged variable in the system
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• First we try to find out whether the autocorrelation present is pure autocorrelation or not and
not the result of misspecification of the model
• We can then transform the original model just like in the case of heteroscedasticity we had to
use the generalized least square method
• In case of large samples we can use the Newey –West method
• In some situations we might continue to use the OLS method
THE SAMPLE PROBLEMS
Gold price determination:
Dependent variable: Gold price
Independent variables: oil prices, dollar exchange rate, sensex
Gold price = β1 + β2 * USD exchange rate + β3 * Sensex + β4 * oil price per barrel
We expect the followings:
If β2 is found to be statistically significant then, we can say USD exchange rate affects gold price;
If β3 is found to be statistically significant then, we can say Sensex affects gold price; and
If β4 is found to be statistically significant then, we can say oil price affects gold price.
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So, more than the values of β2, β3, and β14, we are more interested in finding whether each of these
coefficients is statistically significant.
When we ran this OLS, we got the following result for the βis.
1023.89 * USD Exch. Rate
0.60087 * Sensex
0.370835 * oil price
Std Error 786.1194 16.07572 0.036814 0.032304
t ‐58.7116 63.69173 16.32188 11.47951
0.039103 dL = 1.645 (From Durbin‐
dU = 1.692 Watson
Since t statistic is more than 2 for all the βis, the inference would be that all the βis are significant. That
is, we would conclude that each of these factors affect gold price significantly.
However, we notice that the Durbin Watson test statistic is very low, only 0.039, whereas, the Durbin
Watson Tables for 95% confidence level (or 5% significance level), N=1397 and 3 unknown variables plus
1 constant term was 1.645 – 1.692. If these are plotted it on the D‐W line, very strong indication of
positive auto‐correlation is found.
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Please note that the Durbin Watson test is valid only when we have an autocorrelation of the order 1,
ui = ρ ui‐1 + εt
and also, when there is an intercept in the original Regression Equation (here Eqn. 1), there are no
lagged terms of gold prices in Eqn. 1, and there are no missing observations.
Signs of autocorrelation is also found from the graphical representation of the error term ui vs. ui‐1 .
Date Gold USD Sensex Oil Gold* ui = Gold‐Gold* ui‐1
Column I II III IV V VI=I‐V VII
6‐Jun‐05 6080 43.6 6758.19 5375.35 4541.437 1538.563
7‐Jun‐05 6110 43.53 6781.25 5317.07 4462.009 1647.9915 1538.563
8‐Jun‐05 6090 43.53 6858.24 5338.66 4516.276 1573.7242 1647.9915
29‐Mar‐11 20610 44.67 19120.8 15967.23 16993.15 3616.8469 3470.732
=1.645 =1.692 =2.308 =2.355 dcalc
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30‐Mar‐11 20682 44.77 19290.18 16004.85 17211.27 3470.7318 3616.8469
Gold* = β1 + β2 * USD exchange rate + β3 * Sensex + β4 * oil price
Now, two more tests of autocorrelation can be done. One is the graphical representation and the other
is the Runs test
First, the graph, which is presented below:
We can see there is a strong positive relation between ui and ui‐1, suggesting autocorrelation of the first