# Autocorrelation Problem - Econometric Modeling - Lecture Notes, Study notes for Econometrics and Mathematical Economics. Agra University

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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

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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

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

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terms. If it is so, then there is no issue of serial correlation and estimated model can be used for

forecasting.

DETECTION CRITERIA

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:

RUNS TEST

DURBIN WATSON ‘D’ TEST

BREUSCH- GODFREY TEST

Q-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

sluggishness

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

Non-stationarity

Presence of lagged variable in the system

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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 = 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.

Gold price

‐46154.3

+

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

R2  0.857104

Durbin

Watson

0.039103  dL  = 1.645 (From Durbin‐

dU  = 1.692   Watson

Tables)

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,

i.e.,

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

=0.039

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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

order, i.e.,

ui

ui‐1

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