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13. MODULE OBJECTIVE
This module attempts to explore the possibilities of correlation in cross sectional units of error
variables.
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
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 = β 0 + β 1 X t + 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
property.
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
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
Ö sluggishness
Ö Interpolation or extrapolation
Ö Misspecification of the random term
Ö An over determined model
Ö An under-determined model
Ö Lag explanatory variables
Ö Wrong data transformation
Ö Manipulation of data
Ö Non-stationarity
Ö Presence of lagged variable in the system
REMEDIAL MEASURES
• 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:
Variables used:
Dependent variable: Gold price
Independent variables: oil prices, dollar exchange rate, sensex
Model specification:
Gold price = β 1 + β 2 * USD exchange rate + β 3 * Sensex + β 4 * oil price per barrel
We expect the followings:
If β 2 is found to be statistically significant then, we can say USD exchange rate affects gold price;
If β 3 is found to be statistically significant then, we can say Sensex affects gold price; and
If β 4 is found to be statistically significant then, we can say oil price affects gold price.
Please note that the Durbin Watson test is valid only when we have an autocorrelation of the order 1, i.e.,
ui = ρ ui‐ 1 + ε t
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 u (^) i 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. 7 ‐Jun‐ 05 6110 43.53 6781.25 5317.07 4462.009 1647.9915 1538. 8 ‐Jun‐ 05 6090 43.53 6858.24 5338.66 4516.276 1573.7242 1647. : : 29 ‐Mar‐ 11 20610 44.67 19120.8 15967.23 16993.15 3616.8469 3470.
dcalc
X
30 ‐Mar‐ 11 20682 44.77 19290.18 16004.85 17211.27 3470.7318 3616. Gold* = β 1 + β 2 * USD exchange rate + β 3 * Sensex + β 4 * 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 order, i.e.,
ui
ui‐ 1
