What is autocorrelation?
The term autocorrelation may be defined as correlation between members of
series of observation ordered in time (as in time series data) or space (as in
cross sectional data).
Let us consider the general linear regression model isY=Xβ+ut
One of the basic assumption of this model is that the error termut’s are
mutually independent or uncorrelated, i.e.
cov(ut+ut+s)≠0, for all t and t+s.
but this assumption of uncorrelated error is not valid for certain cases such
as in time series data where the successive error tends to the highly
correlated i.e. cov(ut+ut+s)≠0, for all t≠s.
There is a correlation between successive values of ut’s. these types of
correlation is known as autocorrelation.
Serial correlation is defined as ‘lag correlation’ between two different series.
For example- the correlation between time series such as u1, u2,…..,u10 and
v2,v3,…v11, where u and v are two different time series is called serial
Distinguish between serial correlation and autocorrelation
When the correlation occurs in same When the correlation occurs in
series then the correlation is called
different series then the correlation
is called serial correlation
Auto correlation of a series with
In serial correlation we find
itself lagged by time unit
correlation lagged between two
Let us consider two time series
Let us consider two time series
What is auto regression and what is auto regression model? Write down
Auto regression: a regression is known as auto regression if one of the
explanatory variables is the lagged value of the dependent variable.
Auto regression model: Let Yt=β1+β2Xt+ut where t denotestheobservation at
time t. one can assume that disturbance term are generalised as usersut=ρut-1+vt
whereρ is known as coefficient of auto covariance and vt is the disturbance
term. This model is known as AR(1) model. Because it can be interpreted as
the regression of ut on itself lagged one period i.e. its immediate first value
If you depend on the value of two successive periods then the linear
relationship contain first and second order auto correlation coefficient.
Hence the second order auto regressive model is given byut=ρut-1+vt
Question: For an auto regressive model i.e.
where, │ρ│<1 and v[vt]=σv
We are given,
now taking expectation in the equation (i)
Var(ut)=1 and var(ut-1)=σu2
From equation (ii),
Question: find the first order auto regressive scheme or structure
Or, derive the consequence of autocorreltation.
Or, show that, first order autoregressive schemeu t=
⇒to find the consiqence of autocorrelation let us consider a sample
regression model with time t.
Where ut follows the first order auto regressive scheme,
whereρ is the coefficient of auto covariance.
i. e. -1≤ρ≤1
Vt is a random term; which fulfills all usual assumption of r. v
var[vt]=E[vt,vt-r]=σv2, when r=0 and var[vt]==0 when r≠0
Now we can write,
Now we perform continuous substitutions of lagged values of u in equation
(i) as follows substitute ut-1 and obtain-
Ut= ρ[ρut-2+vt-1] +vt
Again substitute ut-2,
ut= ρ[ρut-2+vt-1] +vt
What happens if the disturbance term are correlated?
CLRM⇒cov(ui,uj│xi,xj) = E(ui,uj)=0
The Classical Linear Regression Model assumes that the disturbance term in
any observation is not influenced by the disturbance term in any other
observation. For example- if we are dealing with quarterly time series data
involving the regression of output on labor and capital inputs. If say there is
a labor strike affecting output in one quarter there is no reason to believe that
this demonstration will be carried over to the next part i.e. if output is lower
to the first quarter there is no reason to believe that it will be continue to
lower next quarter.
If we are dealing with cross sectional data involving the regression of family
consumption expenditure on family income, the affect of an increase of one
family’s income on its consumption expenditure is not expected to affect the
consumption expenditure of another family.
Consequence of autocorrelation:
(1) When the disturbance terms (µ’s) are seriously correlated then the
least square (OLS) estimate are unbiased but optimality property
(M.V property) is not satisfy
(2) If the disturbance term µ’s are autocorrelated then the OLS variance is
greater than the variance of estimate calculated by other method then
the usual t and F test of significance are no longer misleading
conclusion about the estimate regression.
(3) If the disturbance term are autocorrelated then the OLS estimate are
(4) The variance of random term is may be seriously under estimated if
the µi’s are autocorrelated.
What remedial measures can be taken to alleviate autocorrelation
(1) Try to find out if the autocorrelation is pure correlation and not the
result of mis-specification of the model.
(2) If it is pure correlation one can use appropriate transformation of the
original model so that in the transformation model we don’t have the
problem of (Pure) autocorrelation as in the case of heteroscedasticity
we will have to use some type of generalised least square model or
(3) In large samples we can use Newey-West method to obtain standard
error of OLS estimators that are correlated from autocorrelation. This
method is actually an extension of whites’ heteroscedasticity