auto corelation

8/7/2019 auto corelation

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Prof. r c manocha

AUTOCORRELATION

WHAT HAPPENS IF THE

ERROR TERMS ARECORRELATED?


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TYPES OF DATA AVAILABLE

FOR EMPIRICAL ANALYSIS Generally, three types of data are

available for empirical analysis:

1. Time series data2. Cross-section data

3. Pooled-data : combination of time series

& cross-section data.


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TIME SERIES DATA

A time series data is a set of observation values that avariable take at different times.

State Y1 Y2 X 1 X2

Delhi 200 210 1200 1300Punjab 1500 1700 800 950

Haryana 1100 1050 930 1100

Y 1 =Potatoes produced in year 2007(tonnes)

Y2 =Potatoes produced in year 2008(tonnes)

X1= Price of potatoes per tonnee in year 2007

X 2 =Price of potatoes per tonnee in year 2008

Note: money supply is increasing each year.


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CROSS-SECTION DATA

Cross-section data are data on one or more

variables collected in the same point in time.

Example: census data collected by Census

Bureau every 10 years.

In above example, we have two cross sectional

tables for three states-one for the year 1990 &

the other for the year 1991 Cross sectional data create their own problems.

Specifically, heterogeneity.


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SPATIAL AUTO CORRELATION

EXAMPLE : cross section data:

In cross section studies, data are often collectedon the basis of a random sample of cross

sectional units, such as households(consumption function analysis) or firms( in

investment study analysis). If by chance, theerror pertaining to one household or firm is

correlated with the error term of another houseor firm, then such a correlation shall be calledspatial autocorrelation.


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SERIAL CORRELATION

If you observe stock prices (say in BSE) ,it is not unusual to find ups & downs inshares for several days in succession

( bulls or beers). Obviously, in such asituation, data follows a natural orderingover time so that successive observations

are likely to exhibit inter-correlations.This type of autocorrelation is called SerialCorrelation.


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POOLED DATA

In pooled data, data are of both types-

time series as well as cross section data.

Example: for each year, we have 3 crosssectional observations and for each state,

we have two time series data-one for the

year 1990 & the other for the year 1991.


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AUTOCORRELATION DEFINED

If there exists correlation between

members of the series ordered in time

(time series data) or space( cross section

data), it is called autocorrelation.

In first case, it is called serial

autocorrelation

In second case it is called space

autocorrelation.


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contd

Most economic data consists of time

series and there is very often a correlation

in the errors corresponding to successive

time-periods. This is the problem of

autocorrelation or serial correlation.

The error term t at time period t is

correlated with error terms t+1, t+2 and

t-1 , t-2 and so on.


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contd

Such correlation in the error term arises

due to omitted variables that the term

captures.

Correlation between t and t k is called

an auto correlation of the order k.

Correlation between tand

t 1 iscalled

an auto correlation of the first order and is

denoted by 1 and so on.


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Does autocorrelation exist in

classical regression model? No.

There is no autocorrelation in this case.

Hence there is no error in disturbances;Therefore E ( ui uj ) =0 i j

This means that disturbance term relating to

any one term is not influenced by thedisturbance of any other term.


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What happens when we have

autocorrelation? The disturbance of any one term can

effect the disturbance of another term.

E ( ui uj ) 0 i j


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PATTERNS OF

AUTOCORRELATIONThere are :

CYCLIC PATTERN

UPWARD PATTERN DOWNWARD PATTERN

BOTH-LINEAR & QUADRATIC

PATTERNSThese do not support the classical model as

the error terms are correlated.


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REASONS OF

AUTOCORRELATION1. Inertia or sluggishness of economic time

series.

2. Specification bias resulting fromexcluding important variables from the

model or using incorrect functional form.

3. The cobweb phenomenon

4. Data massaging; &

5. Data transformation


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EFFECT OF CORRELATION ON OLS

(ORDINARY LEAST SQUARES) OLS estimators remain unbiased.

OLS estimators remain consistent; &

Asymptotically normally distributed.

However, they are no longer efficient

(minimum variance is not there.)

2

Hence the usual t, F and X (chi square)

test cannot be legitimately applied.


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REMEDIAL MEASURES

The remedy depends upon the nature ofinterdependence among the disturbances ui. But as thedisturbances are unobservable, the common practice isto assume that they are generated by some mechanism.

The mechanism most commonly used is Markov firstorder autoregressive scheme, which assumes that thedisturbance in the current time period is linearly relatedto the disturbance term in the previous time period &(rho)-the coefficient of autocorrelation provides the

extent of interdependence. This mechanism is known as AR(I) scheme.


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DETECTION OF

AUTOCORRELATION The Durbin-Watson (DW) test is the most

often used to test for the presence of

autocorrelation.

This test is applicable only for small

samples and is appropriate only for the

first order autogressive scheme

( ut = (rho)ut-1 +vt) where rho is the

coefficient of first order serial correlation.


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DW TEST PROCEDURE

H0 : (rho)=0 i.t us are not autocorrelated with

first order scheme.

H1 : 0

To test the null hypothesis, we use the Durbin-

Watson statistic:

t=n 2 t=n 2

d=(t t-1) / tWHERE t is the estimated residual for period t

t=2 t=1


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Contd.

2 2

Since t and t-1 are approximately

equal , if the sample is large.We have d is apprx equal to 2(1-)

If = +1 d=0

If = -1 d=4From graph, we can see that

When d is close to 0 or 4, the residuals are


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contd

Highly correlated.

d=0 d=2 d=4

If d is less than dl we reject the null hyp ofno autocorrelation

If d is greater than du we donot reject the

null hyp of no autocorrelationIf d is greater than dl but less than du we say

that the test in inconclusive.


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