auto corelation
TRANSCRIPT
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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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